~, ~~ ~-~,~-~~ ~ -L~e ~L ~~IL~~-- ~-5~5~5~5~5~5~5~5~5~5~5~5~5~5~5~5~5~5~5~ 1 -~~ ~(~__ll)~C I ~- -~ 84Pm - -1 I Il I-- ~ I' I i 1_ 1111.-~-"-~-91 /WA r e r: /v 2, 7 2 - THE SCIENCE OF MECHANICS A CRITICAL AND HISTORICAL ACCOUNT OF ITS DEVELOPMENT BY DR. ERNST MACH PROFESSOR OF THE HISTORY AND I HEORY OF INDUCTIVE SCIENCE IN THE UNIVERSITY OF VIENNA TRANSLATED FROM THE GERMAN BY THOMAS J. McCORMACK \VTr 1') 1 UN1DRED ANDI FIFTY CUTS AND ILLUSTRATIONS FOURTHI E)ITION C I C A G O L O N D O N T E11 OPEN COURT PUBLISHING CO. 1919 A ti TRANSLATOR'S PREFACE TO THE SECOND ENGLISH EDITION. SINCE the appearance of the first edition of the present translation of Mach's Mechanics,* the views which Professor Mach has advanced on the philosophy of science have found wide and steadily increasing acceptance. Many fruitful and elucidative controversies have sprung from his discussions of the historical, logical, and psychological foundations of physical science, and in consideration of the great ideal success which his works have latterly met with in Continental Europe, the time seems ripe for a still wider dissemination of his views in English-speaking couitries. The study of the history and theory of science is finding fuller and fuller recognition in our universities, and it is to be hoped that the present exemplary treatment of the simplest and most typical branch of physics will stimulate further progress in this direction. The text of the present edition, which contains the extensive additions ' e by the author to the * Die Mechanik in ihrer Entwickelung,istorisch-kritisch dargestellt. Von Dr. Ernst Mach, Professor an der Universitit zu Wien. Mit 257 Abbildungen. First German edition, 1883. Fourth German edition, 1901O. First edition of the English translation, Chicago, The Open Court Publishing Co., 1893. vi TRANSLA TOR'S PREFA CE. latest German editions, has been thoroughly revised by the translator. All errors, either of substance or typography, so far as they have come to the translator's notice, have been removed, and in many cases the phraseology has been altered. The sub-title of the work has, in compliance with certain criticisms, also been changed, to accord more with the wording of the original title and to bring out the idea that the work treats of the principles of mechanics predominantly under the aspect of their development (Entwickelung). To avoid confusion in the matter of references, the main title stands as in the first edition. The author's additions, which are considerable, have been relegated to the Appendix. This course has been deemed preferable to that of incorporating them in the text, first, because the numerous references in other works to the pages of the first edition thus hold good for the present edition also, and secondly, because with few exceptions the additions are either supplementary in character, or in answer to criticisms. A list of the subjects treated in these additions is given in the Table of Contents, under the heading "Appendix" on page xix. Special reference, however, must be made to the additions referring to Hertz's Mechanics (pp. 548-555), and to the history of the development of Professor Mach's own philosophical and scientific views, notably to his criticisms of the concepts of mass, inertia, absolute motion, etc., on pp. 542-547, 555-574, and 579 TRANSLA TOR'S PREFA CE. vii -583. The remarks here made will be found highly elucidative, while the references given to the rich literature dealing with the history and philosophy of science will also be found helpful. As for the rest, the text of the present edition of the translation is the same as that of the first. It has had the sanction of the author and the advantage of revision by Mr. C. S. Peirce, well known for his studies both of analytical mechanics and of the history and logic of physics. Mr. Peirce read the proofs of the first edition and rewrote Sec. 8 in the chapter on Units and Measures, where the original was inapplicable to the system commonly taught in this country. THOMAS J. MCCORMACK. LA SALLE, ILL., February, 1902. AUTHOR'S PREFACE TO THE TRANSLATION. Having read the proofs of the present translation of my work, Die fec/iain zik izlrer E nlwicke/ung, I can testify that the publishers have supplied an excellent, accurate, and faithful rendering of it, as their previous translations of essays of mine gave me every reason to expect. My thanks are due to all concerned, and especially to Mr. McCormack, whose intelligent care in the conduct of the translation has led to the discovery of many errors, heretofore overlooked. I may, thus, confidently hope, that the rise and growth of the ieeas of the great inquirers, which it was my task to portray, will appear to my new public in distinct and sharp outlines. E. MACH. PRAGUE, April 8th, 1893. PREFACE TO THE THIRD EDITION. THAT the interest in the foundations of mechanics is still unimpaired, is shown by the works published since 1889 by Budde, P. and J. Friedlander, H. Hertz, P. Johannesson, K. Lasswitz, MacGregor, K. Pearson, J. Petzoldt, Rosenberger, E. Strauss, Vicaire, P. Volkmann, E. Wohlwill, and others, many of which are deserving of consideration, even though briefly. In Prof. Karl Pearson (Grammar of Science, London, 1892), I have become acquainted with an inquirer with whose epistemological views I am in accord at nearly all essential points, and who has always taken a frank and courageous stand against all pseudoscientific tendencies in science. Mechanics appears at present to be entering on a new relationship to physics, as is noticeable particularly in the publication of H. Hertz. The nascent transformation in our conception of forces acting at a distance will perhaps be influenced also by the interesting investigations of H. Seeliger (" Ueber das Newton'sche Gravitationsgesetz," Sitzungsbericht der Miinchener Akademie, 1896), who has shown the incompatibility of a rigorous interpretation of Newton's law with the assumption of an unlimited mass of the universe. VIENNA, January, 1897. E. Mach. x PREFACE TO THE FIRST EDITION. ing these cases must ever remain the method at once the most effective and the most natural for laying this gist and kernel bare. Indeed, it is not too much to say that it is the only way in which a real comprehension of the general upshot of mechanics is to be attained. I have framed my exposition of the subject agreeably to these views. It is perhaps a little long, but, on the other hand, I trust that it is clear. I have not in every case been able to avoid the use of the abbreviated and precise terminology of mathematics. To do so would have been to sacrifice matter to form; for the language of everyday life has not yet grown to be sufficiently accurate for the purposes of so exact a science as mechanics. The elucidations which I here offer are, in part, substantially contained in my treatise, Die Geschiczte und die Wurzel des Satzes von der Erhaltung der A-rbei (Prague, Calve, 1872). At a later date nearly the same views were expressed by KIRCHHOFF ( Vorlesungen zber nmathematische Physik: Mechanik, Leipsic, 1874) and by HELMHOLTZ (Die T/hasachen n i der Wa/zrnehimung, Berlin, 1879), and have since become commonplace enough. Still the matter, as I conceive it, does not seem to have been exhausted, and I cannot deem my exposition to be at all superfluous. In my fundamental conception of the nature of science as Economy of Thought,-a view which I indicated both in the treatise above cited and in my PREFA CE TO THE FIRST EDITION. xi pamphlet, Die Gestalten der Fliissigkeit (Prague, Calve, 1872), and which I somewhat more extensively developed in my academical memorial address, Die 6konomische Natur der physikalischen Forschung (Vienna, Gerold, 1882,-I no longer stand alone. I have been much gratified to find closely allied ideas developed, in an original manner, by Dr. R. AVENARIUS (Philosophie als Denken der Welt, genmiss dem Princij des kleinsten Kraftmaasses, Leipsic, Fues, 1876). Regard for the true endeavor of philosophy, that of guiding into one common stream the many rills of knowledge, will not be found wanting in my work, although it takes a determined stand against the encroachments of metaphysical methods. The questions here dealt with have occupied me since my earliest youth, when my interest for them was powerfully stimulated by the beautiful introductions of LAGRANGE to the chapters of his Analytic Mechanics, as well as by the lucid and lively tract of JOLLY, Principien der AMechanik (Stuttgart, 1852). If DUEHRING'S estimable work, Kritische Geschichte der Principien der Mechanik (Berlin, 1873), did not particularly influence me, it was that at the time of its appearance, my ideas had been not only substantially worked out, but actually published. Nevertheless, the reader will, at least on the destructive side, find many points of agreement between Diihring's criticisms and those here expressed. The new apparatus for the illustration of the subject, here figured and described, were designed entirely xii PREFA'CE TO T77A FIRST EDITION. by me and constructed by Mr. F. Hajek, the mechanician of the physical institute under my control. In less immediate connection with the text stand the fac-simile reproductions of old originals in my possession. The quaint and naive traits of the great inquirers, which find in them their expression, have always exerted upon me a refreshing influence in my studies, and I have desired that my readers should share this pleasure with me. E. MACH. PRAGUE, May, 1883. PREFACE TO THE SECOND EDITION. IN consequence of the kind reception which this book has met with, a very large edition has been exhausted in less than five years. This circumstance and the treatises that have since then appeared of E. Wohlwill, H. Streintz, L. Lange, J. Epstein, F. A. Milller, J. Popper, G. Helm, M. Planck, F. Poske, and others are evidence of the gratifying fact that at the present day questions relating to the theory of cognition are pursued with interest, which twenty years ago scarcely anybody noticed. As a thoroughgoing revision of my work did not yet seem to me to be expedient, I have restricted myself, so far as the text is concerned, to the correction of typographical errors, and have referred to the works that have appeared since its original publication, as far as possible, in a few appendices. E. MACH. PRAGUE, June, I888. PREFACE TO THE FIRST EDITION. THE present volume is not a treatise upon the application of the principles of mechanics. Its aim is to clear up ideas, expose the real significance of the matter, and get rid of metaphysical obscurities. The little mathematics it contains is merely secondary to this purpose. Mechanics will here be treated, not as a branch of mathematics, but as one of the physical sciences. If the reader's interest is in that side of the subject, if he is curious to know how the principles of mechanics have been ascertained, from what sources they take their origin, and how far they can be regarded as permanent acquisitions, he will find, I hope, in these pages some enlightenment. All this, the positive and physical essence of mechanics, which makes its chief and highest interest for a student of nature, is in existing treatises completely buried and concealed beneath a mass of technical considerations. The gist and kernel of mechanical ideas has in almost every case grown up in the investigation of very simple and special cases of mechanical processes; and the analysis of the history of the discussions concern PREFACE TO THE FOURTH EDITION. THE number of the friends of this work appears to have increased in the course of seventeen years, and the partial consideration which my expositions have received in the writings of Boltzmann, F6ppl, Hertz, Love, Maggi, Pearson, and Slate, have awakened in me the hope that my work shall not have been in vain. Especial gratification has been afforded me by finding in J. B. Stallo (The Concepts of Modern Physics) another staunch ally in my attitude toward mechanics, and in W. K. Clifford (Lectures and Essays and The Common Sense of the Exact Sciences), a thinker of kindred aims and points of view. New books and criticisms touching on my discussions have received attention in special additions, which in some instances have assumed considerable proportions. Of these strictures, 0. Holder's note on my criticism of the Archimedean deduction (Denken und Anschauung in der Geometric, p. 63, note 62) has been of special value, inasmuch as it afforded me the opportunity of establishing my view on still firmer foundations (see pages 512-517). I do not at all dispute that rigorous demonstrations are as possible in mechanics as in mathematics. But with respect to xvi PREFACE TO THE FOURTHTI EDITION. the Archimedean and certain other deductions, I am still of the opinion that my position is the correct one. Other slight corrections in my work may have been made necessary by detailed historical research, but upon the whole I am of the opinion that I have correctly portrayed the picture of the transformations through which mechanics has passed, and presumably will pass. The original text, from which the later insertions are quite distinct, could therefore remain as it first stood in the first edition. I also desire that no changes shall be made in it even if after my death a new edition should become necessary. E. MACH. VIENNA, January, go1901. TABLE OF CONTENTS. Translator's Preface to the Second Edition Author's Preface to the Translation... Preface to the First Edition Preface to the Second Edition..... Preface to the Third Edition Preface to the Fourth Edition.. Table of Contents.......... Introduction.......... PAGE V.......... viii ix........ xiii....... xiv xv.... XV....... xvii...... I" CHAPTER I. THE DEVELOPMENT OF THE PRINCIPLES OF STATICS. I. The Principle of the Lever......... 8 II. The Principle of the Inclined Plane...... 24 Ill. The Principle of the Compositicn of Forces.... 33 IV. The Principle of Virtual Velocities....... 49 V. Retrospect of the Development of Statics..... 77 VI. The Principles of Statics in Their Application to Fluids 86 VII. The Principles of Statics in Their Application to Gaseous Bodies............... Io CHAPTER II. THE DEVELOPMENT OF THE PRINCIPLES OF DYNAMICS. I. Galileo's Achievements............ 128 II. The Achievements of Huygens......... 155 III. The Achievements of Newton.......... 187 IV. Discussion and Illustration of the Principle of Reaction 201 V. Criticism of the Principle of Reaction and of the Concept of Mass...............216 VI. Newton's Views of Time, Space, and Motion.... 222 xviii THE SCIENCE OF MECHAANICS. PAGE VII. Synoptical Critique of the Newtonian Enunciations. 238 VIII. Retrospect of the Development of Dynamics... 245 CHAPTER III. THE EXTENDED APPLICATION OF THE PRINCIPLES OF MECHANICS AND THE DEDUCTIVE DEVELOPMENT OF THE SCIENCE. I. Scope of the Newtonian Principles........ 256 II. The Formulae and Units of Mechanics.....269 III. The Laws of the Conservation of Momentum, of the Conservation of the Centre of Gravity, and of the Conservation of Areas.......... 287 IV. The Laws of Impact........... 305 V. D'Alembert's Principle..........331 VI. The Principle of Vis Viva.......... 343 VII. The Principle of Least Constraint....... 350 VIII. The Principle of Least Action........ 364 IX. Hamilton's Principle............. 380 X. Some Applications of the Principles of Mechanics to Hydrostatic and Hydrodynamic Questions.. 384 CHAPTER IV. THE FORMAL DEVELOPMENT OF MECHANICS. I. The Isoperimetrical Problems......... 421 II. Theological, Animistic, and Mystical Points of View in Mechanics.............. 446 III. Analytical Mechanics.......... 465 IV. The Economy of Science......... 481 CHAPTER V. THE RELATION OF MECHANICS TO OTHER DEPARTMENTS OF KNOWLEDGE. I. The Relations of Mechanics to Physics.... 495 II. The Relations of Mechanics to Physiology..... 504 TABLE OF CONTENTS. xix PAGE Appendix................. 509 I. The Science of Antiquity, 509.-II. Mechanical Researches of the Greeks, 510o.-III.. and IV. The Archimedean Deduction of the Law of the Lever, 512, 514.-V. Mode of Procedure of Stevinus, 515. -VI. Ancient Notions of the Nature of the Air, 517.-VII. Galileo's Predecessors, 520.-VIII. Galileo on Falling Bodies, 522.-IX. Galileo on the Law of Inertia, 523.-X. Galileo on the Motion of Projectiles, 525.-XI. Deduction of the Expression for Centrifugal Force (Hamilton's Hodograph), 527.-XII. Descartes and Huygens on Gravitation, 528.-XIII. Physical Achievements of Huygens, 530.XIV. Newton's Predecessors, 531.-XV. The Explanations of Gravitation, 533 -XVI. Mass and Quantity of Matter, 536.-XVII. Galileo on Tides, 537.-XVIII. Mach's Definition of Mass, 539.-XIX. Mach on Physiological Time, 541.-XX. Recent Discussions of the Law of Inertia and Absolute Motion, 542.-XXI. Hertz's System of Mechanics, 548.-XXII. History of Mach's Views of Physical Science (Mass, Inertia, etc.), 555.-XXIII. Descartes's Achievements in Physics, 574.-XXIV. Minimum Principles, 575.-XXV. Grassmann's Mechanics, 577.-XXVI. Concept of Cause, 579.-XXVII. Mach's Theory of the Economy of Thought, 579.--XXVIII. Description of Phenomena by Differential Equations, 583.-XXIX. Mayer and the Mechanical Theory of Heat, 584.-XXX. Principle of Energy, 585. Chronological Table of a Few Eminent Inquirers and of Their More Important Mechanical Works...... 589 Index................. 593 THE SCIENCE OF MECHANICS INTRODUCTION. I. THAT branch of physics which is at once the old- The science of mechanest and the simplest and which is therefore treated ics. as introductory to other departments of this science, is concerned with the motions and equilibrium of masses. It bears the name of mechanics. 2. The history of the development of mechanics, is quite indispensable to a full comprehension of the science in its present condition. It also affords a simple and instructive example of the processes by which natural science generally is developed. An instinctive, irreflective knowledge of the processes Instinctive knowledge. of nature will doubtless always precede the scientific, conscious apprehension, or investigalion, of phenomena. The former is the outcome of the relation in which the processes of nature stand to the satisfaction of our wants. The acquisition of the most elementary truth does not devolve upon the individual alone: it is pre-effected in the development of the race. In point of fact, it is necessary to make a dis-Mechanical tinction between mechanical experience and mechan- experiences ical science, in the sense in which the latter term is at present employed. Mechanical experiences are, unquestionably, very old. If we carefully examine the ancient Egyptian and Assyrian monuments, we shall find there pictorial representations of many kinds of THE SCIENCE OF MECHANICS. The me- implements and mechanical contrivances; but acchanical knowledge counts of the scientific knowledge of these peoples of antiquity of antiquitare either totally lacking, or point conclusively to a very inferior grade of attainment. By the side of highly ingenious appliances, we behold the crudest and roughest expedients employed-as the use of sleds, for instance, for the transportation of enormous blocks of stone. All bears an instinctive, unperfected, accidental character. So, too, prehistoric -graves contain impleSmentswhoseconstrucStion and employment ( imply no little skill S I and much mechanical experience. Thus,long before theory was dreamed of, implements, machines, mechanical experiences, and mechanical knowledge were abundant. Have we 3. The idea often underrated itd? suggests itself that perhaps the incomplete accounts we pos INTROD UCTION. sess have led us to underrate the science of the ancient world. Passages occur in ancient authors which seem to indicate a profounder knowledge than we are wont to ascribe to those nations. Take, for instance, the following passage from Vitruvius, De Architectura, Lib. V, Cap. III, 6: "The voice is a flowing breath, made sensible to A passage from Vitru'the organ of hearing by the movements it produces vius. "in the air. It is propagated in infinite numbers of "circular zones exactly as when a stone is thrown "into a pool of standing water countless circular un"dulations are generated therein, which, increasing "as they recede from the centre, spread out over a "great distance, unless the narrowness of the locality "or some obstacle prevent their reaching their terSmination; for the first line of waves, when impeded "by obstructions, throw by their backward swell the "succeeding circular lines of waves into confusion. "Conformably to the very same law, the voice also ' generates circular motions; but with this distinction, " that in water the circles, remaining upon the surface, "are propagated horizontally only, while the voice is "propagated both horizontally and vertically." Does not this sound like the imperfect exposition controverted by other of a popular author, drawn from more accurate disqui- evidence. sitions now lost? In what a strange light should we ourselves appear, centuries hence, if our popular literature, which by reason of its quantity is less easily destructible, should alone outlive the productions of science? This too favorable view, however, is very rudely shaken by the multitude of other passages containing such crude and patent errors as cannot be conceived to exist in any high stage of scientific culture. (See Appendix, I., p. 509.) THE SCIENCE OF MECHANICS. The origin 4. When, where, and in what manner the developof science.ment of science actually began, is at this day difficult historically to determine. It appears reasonable to assume, however, that the instinctive gathering of experiential facts preceded the scientific classification of them. Traces of this process may still be detected in the science of to-day; indeed, they are to be met with, now and then, in ourselves. The experiments that man heedlessly and instinctively makes in his struggles to satisfy his wants, are just as thoughtlessly and unconsciously applied. Here, for instance, belong the primitive experiments concerning the application of the lever in all its manifold forms. But the things that are thus unthinkingly and instinctively discovered, can never appear as peculiar, can never strike us as surprising, and as a rule therefore will never supply an impetus to further thought. The func- The transition from this stage to the classified, tions of spe- s and of firs becial classes scientific knowledge and apprehension of facts, first bein the development comes possible on the rise of special classes and proof science, fessions who make the satisfaction of definite social wants their lifelong vocation. A class of this sort occupies itself with particular kinds of natural processes. The individuals of the class change; old members drop out, and new ones come in. Thus arises a need of imparting to those who are newly come in, the stock of experience and knowledge already possessed; a need of acquainting them with the conditions of the The com- attainment of a definite end so that the result may be munication of knowl- determined beforehand. The communication of knowledge. edge is thus the first occasion that compels distinct reflection, as everybody can still observe in himself. Further, that which the old members of a guild mechanically pursue, strikes a new member as unusual INTROD UCTONT. and strange, and thus an impulse is given to fresh reflection and investigation. When we wish to bring to the knowledge of a per- Involves description. son any phenomena or processes of nature, we have the choice of two methods: we may allow the person to observe matters for himself, when instruction comes to an end; or, we may describe to him the phenomena in some way, so as to save him the trouble of personally making anew each experiment. Description, however, is only possible of events that constantly recur, or of events that are made up of component parts that constantly recur. That only can be described, and conceptually represented which is uniform and conformable to law; for description presupposes the employment of names by which to designate its elements; and names can acquire meanings only when applied to elements that constantly reappear. 5. In the infinite variety of nature many ordinaryA unitary conception events occur; while others appear uncommon, per-of nature. plexing, astonishing, or even contradictory to the ordinary run of things. As long as this is the case we do not possess a well-settled and unitary conception of nature. Thence is imposed the task of everywhere seeking out in the natural phenomena those elements that are the same, and that amid all multiplicity are ever present. By this means, on the one hand, the most economical and briefest description and communication are rendered possible; and on the other, The nature of knowlwhen once a person has acquired the skill of recog-edge. nising these permanent elements throughout the greatest range and variety of phenomena, of seeing them in the same, this ability leads to a comprehensive, compact, consistent, and facile conception of thefacts. When once we have reached the point where we are everywhere TIE SCIENCE OF MECHANICS. The adap- able to detect the same few simple elements, combintation of thoughts to ing in the ordinary manner, then they appear to us as facts. things that are familiar; we are no longer surprised, there is nothing new or strange to us in the phenomena, we feel at home with them, they no longer perplex us, they are explained. It is a process of adaptation of thoughts to facts with which we are here concerned. The econ- 6. Economy of communication and of apprehenomy of thought sion is of the very essence of science. Herein lies its pacificatory, its enlightening, its refining element. Herein, too, we possess an unerring guide to the historical origin of science. In the beginning, all economy had in immediate view the satisfaction simply of bodily wants. With the artisan, and still more so with the investigator, the concisest and simplest possible knowledge of a given province of natural phenomena-a knowledge that is attained with the least intellectual expenditure-naturally becomes in itself an economical aim; but though it was at first a means to an end, when the mental motives connected therewith are once developed and demand their satisfaction, all thought of its original purpose, the personal need, disappears. Further de- To find, then, what remains unaltered in the phevelopment of these nomena of nature, to discover the elements thereof ideas and the mode of their interconnection and interdependence--this is the business of physical science. It endeavors, by comprehensive and thorough description, 4o make the waiting for new experiences unnecessary; it seeks to save us the trouble of experimentation, by making use, for example, of the known interdependence of phenomena, according to which, if one kind of event occurs, we may be sure beforehand that a certain other event will occur. Even in the description itself labor may be saved, by discovering methods of de LVTROD UCTIOA. scribing the greatest possible number of different ob-Their present discusjects at once and in the concisest manner. All this will sion merely be made clearer by the examination of points of detailpreparatory than can be done by a general discussion. It is fitting, however, to prepare the way, at this stage, for the most important points of outlook which in the course of our work we shall have occasion to occupy. 7. We now propose to enter more minutely into the Proposed plan of subject of our inquiries, and, at the same time, without treatment. making the history of mechanics the chief topic of discussion, to consider its historical development so far as this is requisite to an understanding of the present state of mechanical science, and so far as it does not conflict with the unity of treatment of our main subject. Apart from the consideration that we cannot afford to neglect the great incentives that it is in our power to derive from the foremost intellects of all Theincentives deepochs, incentives which taken as a whole are more rived from contact fruitful than the greatest men of the present day are with the great intelable to offer, there is no grander, no more intellectually lets of the elevating spectacle than that of the utterances of theworld. fundamental investigators in their gigantic power. Possessed as yet of no methods, for these were first created by their labors, and are only rendered comprehensible to us by their performances, they grapple with and subjugate the object of their inquiry, and imprint upon it the forms of conceptual thought. They that know the entire course of the development of science, will, as a matter of course, judge more freely and And the increase of more correctly of the significance of any present scien-power which such tific movement than they, who limited in their views acontact to the age in which their own lives have been spent,lends. contemplate merely the momentary trend that the course of intellectual events takes at the present moment. CHAPTER I. THE DEVELOPMENT OF THE PRINCIPLES OF STATICS. I. THE PRINCIPLE OF THE LEVER. Theearliest I. The earliest investigations concerning mechanresearches ics of which we have any account, the investigations related to statics. of the ancient Greeks, related to statics, or to the doctrine of equilibrium. Likewise, when after the taking of Constantinople by the Turks in 1453 a fresh impulse was imparted to the thought of the Occident by the ancient writings that the fugitive Greeks brought with them, it was investigations in statics, principally evoked by the works of Archimedes, that occupied the foremost investigators of the period. (See p. 510.) Archimedes 2. ARCHIMEDES of Syracuse (287-212 B. C.) left or Syracuse (287-212e. behind him a number of writings, of which several c.). have come down to us in complete form. We will first employ ourselves a moment with his treatise De AEquiponderantibus, which contains propositions respecting the lever and the centre of gravity. In this treatise Archimedes starts from the following assumptions, which he regards as self-evident: Axiomatic a. Magnitudes of equal weight acting at equal assumptions of Ar- distances (from their point of support) are in equichimedes. l m. librium. TIHE PRIVCIPLES OF STATICS. b. Magnitudes of equal weight acting at une- Axiomatic assumpqual distances (from their point of support) are tions of Archimedes. not in equilibrium, but the one acting at the greater distance sinks. From these assumptions he deduces the following proposition: c. Commensurable magnitudes are in equilibrium when they are inversely proportional to their distances (from the point of support). It would seem as if analysis could hardly go behind these assumptions. This is, however, when we carefully look into the matter, not the case. Imagine (Fig. 2) a bar, the weight of which is neglected. The bar rests on a fulcrum. At equal distances from the fulcrum we append two equal weights. That the two weights, thus circumstanced, are in equilibrium, is Fig. 2. the assumption from which Archimedes starts. We might suppose that this was self- Analysis of the Archievident entirely apart from any experience, agreeably to medean asthe so-called principle of sufficient reason; that in view sumptions. of the symmetry of the entire arrangement there is no reason why rotation should occur in the one direction rather than in the other. But we forget, in this, that a great multitude of negative and positive experiences is implicitly contained in our assumption; the negative, for instance, that dissimilar colors of the lever-arms, the position of the spectator, an occurrence in the vicinity, and the like, exercise no influence; the positive, on the other hand, (as it appears in the second assumption,) that not only the weights but also their distances from the supporting point are decisive factors in the disturbance of equilibrium, that they also are cir 10 THEll' SCIENCE OF MECHANICS. cumstances determinative of motion. By the aid of these experiences we do indeed perceive that rest (no motion) is the only motion which can be uniquely* determined, or defined, by the determinative conditions of the case.f character Now we are entitled to regard our knowledge of and value of the Archi- the decisive conditions of any phenomenon as sufficient medean results. only in the event that such conditions determine the phenomenon precisely and uniquely. Assuming the fact of experience referred to, that the weights and their distances alone are decisive, the first proposition of Archimedes really possesses a high degree of evidence and is eminently qualified to be made the foundation of further investigations. If the spectator place himself in the plane of symmetry of the arrangement in question, the first proposition manifests itself, moreover, as a highly imperative stinsictive perception,-a result determined by the symmetry of our own body. The pursuit of propositions of this character is, furthermore, an excellent means of accustoming ourselves in thought to the precision that nature reveals in her processes. Thegeneral 3. We will now reproduce in general outlines the proposition of the lever train of thought by which Archimedes endeavors to rereduced to the simple duce the general proposition of the lever to the parand partIcular caste ticular and apparently self-evident case. The two equal weights i suspended at a and b (Fig. 3) are, if the bar ab be free to rotate about its middle point c, in equilibrium. If the whole be suspended by a cord at c, the cord, leaving out of account the weight of the * So as to leave only a single possibility open. t If, for example, we were to assume that the weight at the right descended, then rotation in the opposite direction also would be determined by the spectator, whose person exerts no influence on the phenomenon, taking up his position on the opposite side. TIHE PRINCIPLES OF STATICS. II bar, will have to support the weight 2. The equal The general proposition weights at the extremities of the bar supply accor- of the lever reduced to dingly the place of the double weight at the centre. the simple and particSular case. o o a b... | ---- c ---- |r-- j------ 1 Q Lii 2 Lii Fig. 3. Fig. 4. On a lever (Fig. 4), the arms of which are in the proportion of I to 2, weights are suspended in the proportion of 2 to i. The weight 2 we imagine replaced by two weights i, attached on either side at a distance I from the point of suspension. Now again we have complete symmetry about the point of suspension, and consequently equilibrium. On the lever-arms 3 and 4 (Fig. 5) are suspended the weights 4 and 3. The lever-arm 3 is prolonged the distance 4, the arm 4 is prolonged the distance 3, and the weights 4 and 3 are replaced respectively by *1.-~- ---, H\' I I--T r I - 1 ý, ri, r ril r.1 -,' rn r -, r LJ LJ. L.- L- L l J LJ [L Fig. 5 4 and 3 pairs of symmetrically attached weights 1, in the manner indicated in the figure. Now again we have perfect symmetry. The preceding reasoning, The generwhich we have here developed with specific figures, is alisation. easily generalised. 4. It will be of interest to look at the manner in which Archimedes's mode of view, after the precedent of Stevinus, was modified by GALILEO. 12 THE SCIENCE OF MECHANICS. Galileo's Galileo imagines (Fig. 6) a heavy horizontal prism, mode of treatment, homogeneous in material composition, suspended by its extremities from a homogeneous bar of the same length. The bar is provided at its middle point with a suspensory attachm n m n ment. In this case equilibrium will obtain; this we perceive at once. But in this case is contained every other case,-which 2m 2nf ______ Galileo shows in the Fig. 6. following manner. Let us suppose the whole length of the bar or the prism to be 2(m- +n). Cut the prism in two, in such a manner that one portion shall have the length 2m and the other the length 2n. We can effect this without disturbing the equilibrium by previously fastening to the bar by threads, close to the point of proposed section, the inside extremities of the two portions. We may then remove all the threads, if the two portions of the prism be antecedently attached to the bar by their centres. Since the whole length of the bar is 2(m -+ n), the length of each half is m + n. The distance of the point of suspension of the right-hand portion of the prism from the point of suspension of the bar is therefore m, and that of the left-hand portion n. The experience that we have here to deal with the weight, and not with the form, of the bodies, is easily made. It is thus manifest, that equilibrium will still subsist if any weight of the magnitude 2m be suspended at the distance n on the one side and any weight of the magnitude 2n be suspended at the distance m on the other. The instinctive elements of our perception of this phenomenon are even more THE PRINCIPLES OF STA TICS. 13 prominently displayed in this form of the deduction than in that of Archimedes. We may discover, moreover, in this beautiful presentation, a remnant of the ponderousness which was particularly characteristic of the investigators of antiquity. How a modern physicist conceived the same prob- Lagrange's presentalem, may be learned from the following presentation of tion. LAGRANGE. Lagrange says: Imagine a horizontal homogeneous prism suspended at its centre. Let this prism (Fig. 7) be conceived divided into two prisms of the lengths 2m and 2n. If now we consider the centres of gravity of these two parts, at which we may imagine weights to act proportional to 2m and 2n, the 2m 2n x x S' S Fig. 7. two centres thus considered will have the distances n and m from the point of support. This concise disposal of the problem is only possible to the practised mathematical perception. 5. The object that Archimedes and his successors Object of S. Archimedes sought to accomplish in the considerations we have here and his sucpresented, consists in the endeavor to reduce the morecessrs complicated case of the lever to the simpler and apparently self-evident case, to discern the simpler in the more complicated, or vice versa. In fact, we regard a phenomenon as explained, when we discover in it known simpler phenomena. But surprising as the achievement of Archimedes and his successors may at the first glance appear to us, doubts as to the correctness of it, on further reflec 14 TIHE SCIENCE' OF IMECHANIIICS. Critique of tion, nevertheless spring up. From the mere assumptheir meth-. ods. tion of the equilibrium of equal weights at equal distances is derived the inverse proportionality of weight and lever-arm! How is that possible? If we were unable philosophically and a priori to excogitate the simple fact of the dependence of equilibrium on weight and distance, but were obliged to go for that result to experience, in how much less a degree shall we be able, by speculative methods, to discover the form of this dependence, the proportionality! Thestatical As a matter of fact, the assumption that the equimoment involved in librium-disturbing effect of a weight P at the distance all their deductions. L from the axis of rotation is measured by the product P.L (the so-called statical moment), is more or less covertly or tacitly introduced by Archimedes and all his successors. For when Archimedes substitutes for a large weight a series of symmetrically arranged pairs of small weights, which weights extend beyond the point of support, he employs in this very act the doctrine of the centre of gravity in its more general form, which iitself nothing else than the doctrine of the lever in its more general form. (See Appendix, III., p. 512.) Without it Without the assumption above mentioned of the imdemonstration is i- port of the product P.L, no one can prove (Fig. 8) possible, that a bar, placed in any way on the fulcrum S, is supported, 8 string attached to its "5s centre of gravity and Fig 8. carried over a pulley, by a weight equal to its own weight. But this is contained in the deductions of Archimedes, Stevinus, Galileo, and also in that of Lagrange. THE PRINCIPLES OF ST4 77CS. 15 6. HUYGENS, indeed, reprehends this method, and Huygens's criticism. gives a different deduction, in which he believes he has avoided the error. If in C the presentation of Lagrange we imagine the two portions into which G E the prism is divided turned ninety degrees about two vertical axes A 4 passing through the centres of gravity s,s' of the prism-portions (see Fig. 9a), and it be shown i that under these circum- I stances equilibrium still D H continues to subsist, we Fig. 9. shall obtain the Huygenian deduction. Abridged and simplified, it is as follows. In a rigid weightless rig. 9a. rig.9 a. plane (Fig. 9) through the point S we draw a straight line, on which we cut off on the one side the length i 16 THE SCIENCE OF MEC HANICS. His own de- and on the other the length 2, at A and B respectively. auction. On the extremities, at right angles to this straight line, we place, with the centres as points of contact, the heavy, thin, homogeneous prisms CD and EF, of the lengths and weights 4 and 2. Drawing the straight line HSG (where A G= _ A C) and, parallel to it, the line CF, and translating the prism-portion CG by parallel displacement to FH, everything about the axis GH is symmetrical and equilibrium obtains. But equilibrium also obtains for the axis AB; obtains consequently for every axis through S, and therefore also for that at right angles to AB: wherewith the new case of the lever is given. Apparently Apparently, nothing else is assumed here than that unimpeachable. equal weights p,p (Fig. io) in the same plane and at equal distances /,/ from an axis AA' (in this plane) equilibrate one another. If we place ourselves in the plane passing through AA' perpendicularly to /,/, say Y Y, / P M 0 X A' A; Fig. o1. Fig. Ii. at the point M, and look now towards A and now towards A', we shall accord to this proposition the same evidentness as to the first Archimedean proposition. The relation of things is, moreover, not altered if we institute with the weights parallel displacements with respect to the axis, as Huygens in fact does. THE PRINCIPLES OF STA TICS. I7 The error first arises in the inference: if equilib- Yet involving in the rium obtains for two axes of the plane, it also obtains final inference an erfor every cther axis passing through the point of inter- ror. section ot the first two. This inference (if it is not to be regarded as a purely instinctive one) can be drawn only upon the condition that disturbant effects are ascribed to the weights p /rorlional to their distances from the axis. But in this is contained the very kernel of the doctrine of the lever and the centre of gravity. Let the heavy points of a plane be referred to a system of rectangular cofrdinates (Fig. II). The coordinates of the centre of gravity of a system of masses mI m' Mi... having the co6rdinates x x' x"... y y' y". are, as we know, Mathemat~mx 1x _711y icAl discusS71. sion of I- -l Huygens's inference. If we turn the system through the angle a, the new coordinates of the masses will be X1 = x cosa --_ 1' sina, y 1 =-y cosa + x sina and consequently the co6rdinates of the centre of gravity ~ m (x cosa -,y sina) /mx 2nmy 1 --- Cos - sina-- $ cosa - 1/ sina and, similarly, 7/I = I/ cosa + sina. \Ve accordingly obtain the co6rdinates of the new centre of gravity, by simply transforming the co6rdinates of the first centre to the new axes. The centre of gravity remains therefore the se/f-samie point. If we select the centre of gravity itself as origin, then 2' im x 2'my- 0. On turning the system of axes, this relation continues to subsist. If, accordingly, equi 18 THE SCIENCE OF MECHANICS. librium obtains for two axes of a plane that are perpendicular to each other, it also obtains, and obtains then only, for every other axis through their point of intersection. Hence, if equilibrium obtains for any two axes of a plane, it will also obtain for every other axis of the plane that passes through the point of intersection of the two. The infer- These conclusions, however, are not deducible if ence admissible only the co6rdinates of the centre of gravity are determined on one condition. by some other, more general equation, say - mf(x) m+ If(x') m" 'f(x") +.. mn + i' 42 i" +. The Huygenian mode of inference, therefore, is inadmissible, and contains the very same error that we remarked in the case of Archimedes. Self-decep- Archimedes's self-deception in this his endeavor to tion of Archimedes. reduce the complicated case of the lever to the case instinctively grasped, probably consisted in his unconscious employment of studies previously made on the centre of gravity by the lelp of the very proposition he sought to prove. It is characteristic, that he will not trust on his own authority, perhaps even on that of others, the easily presented observation of the import of the product P.L, but searches after a further verification of it. Now as a matter of fact we shall not, at least at this stage of our progress, attain to any comprehension whatever of the lever unless we directly discern in the phenomena the product P.L as the factor decisive of the disturbance of equilibrium. In so far as Archimedes, in his Grecian mania for demonstration, strives to get around this, his deduction is defective. But regarding the import of P.L as given, the Archimedean THE PRNLCIPLES OF S TA TICS. 19 deductions still retain considerable value, in so far as Function of the Archithe modes of conception of different cases are supported medean deduction. the one on the other, in so far as it is shown that one simple case contains all others, in so far as the same mode of conception is established for all cases. Imagine (Fig. 12) a homogeneous prism, whose axis is AB, supported at its centre C. To give a graphical representation of the sum of the products of the weights and distances, the sum decisive of the disturbance of equilibrium, let us erect upon the elements of the axis, which are proportional to the elements of the weight, the distances as ordinates; the ordinates to the right U P'/ A S M Fig. 12. of C (as positive) being drawn upwards, and to the left Illustration of C (as negative) downwards. The sum of the areas of the two triangles, A CD + CBE - 0, illustrates here the subsistence of equilibrium. If we divide the prism into two parts at M, we may substitute the rectangle MUWB for MTEB, and the rectangle MVXA for TMCAD, where TP - TE and TR =- TD, and the prism-sections MB, MA are to be regarded as placed at right angles to AB by rotation about Q and S. THE SCIENCE OF MECHANICS. 20 In the direction here indicated the Archimedean view certainly remained a serviceable one even after no one longer entertained any doubt of the significance of the product P.L, and after opinion on this point had been established historically and by abundant verification. (See Appendix, IV., p. 514.) Treatment 7. The manner in which the laws of the lever, as of the lever by modern handed down to us from Archimedes in their original physicists. simple form, were further generalised and treated by modern physicists, is very interesting and instructive. LEONARDO DA VINCI (1452-1519), the famous painter and investigator, appears to have been the first to recognise the importance of the general notion of the soC 0 P 9 Fig. 13. Leonardo called statical moments. In the manuscripts he has Da Vinci 1452-519). left us, several passages are found from which this clearly appears. He says, for example: We have a bar AD (Fig. 13) free to rotate about A, and suspended from the bar a weight P, and suspended from a string which passes over a pulley a second weight Q. What must be the ratio of the forces that equilibrium may obtain? The lever-arm for the weight P is not AD, but the "potential" lever AB. The lever-arm for the weight Q is not AD, but the "potential" lever AC. The method by which Leonardo arrived at this view is difficult to discover. But it is clear that he recog THE PRINCIPLES OF STA 7TCS. 21 nised the essential circumstances by which the effect of the weight is determined. Considerations similar to those of Leonardo da Guido Ubald; Vinci are also found in the writings of GUIDO UBALDI. 8. We will now endeavor to obtain some idea of the way in which the notion of statical moment, by which as we know is understood the product of a force into the perpendicular let fall from the axis of rotation upon the line of direction of the force, could have been arrived at,-although the way that really led to this idea is not now fully ascertainable. That equilibrium exists (Fig. 14) if we lay a cord, subjected at both sides E to equal tensions, over a pulley, is perceived without difficulty. We shall always find a plane of symmetry for the apparatus-the plane which stands at right angles Fig. 14. to the plane of the cord and bisects (EE) the angle made by its two parts. The motion that might be supposed A method by which possible cannot in this case be precisely determined or the notion of the statdefined by any rule whatsoever: no motion will there- icalmoment might fore take place. If we note, now, further, that the mate- have been arrived at. rial of which the pulley is made is essential only to the extent of determining the form of motion of the points of application of the strings, we shall likewise readily perceive that almost any portion of the pulley may be removed without disturbing the equilibrium of the machine. The rigid radii that lead out to the tangential points of the string, are alone essential. We see, thus, that the rigid radii (or the perpendiculars on the linear directions of the strings) play here a part similar to the lever-arms in the lever of Archimedes. 22 THE SCIENCE OF MECHANICS. This notion Let us examine a so-called wheel and axle (Fig. derived from the 15) of wheel-radius 2 and axle-radius i, provided reconsideration of a spectively with the cord-hung loads I and 2; an appawheel and axle. ratus which corresponds in every respect to the lever of Archimedes. If now we place about the axle, in any manner we may choose, a second cord, which we subject at each side to the tension of a weight 2, the second cord will not disturb the equilibrium. It is plain, however, that we are also permitted to regard 2 1 2 22 2 1 Fig. 15. Fig. 16. the two pulls marked in Fig. 16 as being in equilibrium, by leaving the two others, as mutually destructive, out of account. But we arrive in so doing, dismissing from consideration all unessential features, at the perception that not only the pulls exerted by the weights but also the perpendiculars let fall from the axis on the lines of the pulls, are conditions determinative of motion. The decisive factors are, then, the products of the weights into the respective perpendiculars let fall from the axis on the directions of the pulls; in other words, the so-called statical moments. The princi- 9. What we have so far considered, is the develple of the lever all- opment of our knowledge of the principle of the lever. sufficient to explain he Quite independently of this was developed the knowlother machines, edge of the principle of the inclined plane. It is not necessary, however, for the comprehension of the ma THE PRIANCIPLES OF STA TICS. 23 chines, to search after a new principle beyond that of the lever; for the latter is sufficient by itself. Galileo, for example, explains the inclined plane from the lever in the following manner. We have before us (Fig. o 17) an inclined plane, on A which rests the weight P Q, held in equilibrium by the weight P. Gali- a" leo, now, points out the Fig. 17. fact, that it is not requisite that Q should lie directly upon the inclined plane, but that the essential point is rather the form, or character, of the motion of Q. We may, consequently, conceive the weight attached to the bar AC, perpendicular to the inclined plane, and rotatable about C. If then we institute a Galileo's explanation very slight rotation about the point C, the weight will of the inclined move in the element of an arc coincident with the in-plane by the lever. dined plane. That the path assumes a curve on the motion being continued is of no consequence here, since this further movement does not in the case of equilibrium take place, and the movement of the instant alone is decisive. Reverting, however, to the observation before mentioned of Leonardo da Vinci, we readily perceive the validity of the theorem Q. CB SP. CA or Q/P = CA/CB = ca/cb, and thus reach the law of equilibrium on the inclined plane. Once we have reached the principle of the lever, we may, then, easily apply that principle to the comprehension of the other machines. THE SCIENCE OF 'ME CH4ANICS. II. THE PRINCIPLE OF THE INCLINED PLANE. Stevinus I. STEVINUS, or STEVIN, (1548-1620) was the first (1548-1620) first investi- who investigated the mechanical properties of the ingates the mechanics lined plane; and he did so in an eminently original ot the inclined manner. If a weight lie (Fig. plane. 18) on a horizontal table, we perceive at once, since the pressure is directly perpendicular to the plane of the table, by the principle of symmetry, Fig. I8. that equilibrium subsists. On a vertical wall, on the other hand, a weight is not at all obstructed in its motion of descent. The inclined plane accordingly will present an intermediate case between these two limiting suppositions. Equilibrium will not exist of itself, as it does on the horizontal support, but it will be maintained by a less weight than that necessary to preserve it on the vertical wall. The ascertainment of the statical law that obtains in this case, caused the earlier inquirers considerable difficulty. Hismodeof Stevinus's manner of procedure is in substance as law.hin its follows. He imagines a triangular prism with horizontally placed edges, a cross-section of which ABC is represented in Fig. 19. For the sake of illustration we will say that AB 2BC; also that AC is horizontal. Over this prism Stevinus lays an endless string on which 14 balls of equal weight are strung and tied at equal distances apart. We can advantageously replace this string by an endless uniform chain or cord. The chain will either be in equilibrium or it will not. If we assume the latter to be the case, the chain, since THE PRINCIPLES OF STA TICS. 25 the conditions of the event are not altered by its motion, must, when once actually in motion, continue to move for ever, that is, it must present a perpetual motion, which Stevinus deems absurd. Consequently only Stevinus's deduction the first case is conceivable. The chain remains in equi- o the law of the inlibrium. The symmetrical portion ADC may, there- lined fore, without disturbing the equilibrium, be removed. plane. The portion AB of the chain consequently balances the portion BC. Hence: on inclined planes of equal heights equal weights act in the inverse proportion of the lengths of the planes. B / VC. Fig. 19. Fig. 20. In the cross-section of the prism in Fig. 20 let us imagine AC horizontal, BC vertical, and AB = 2BC; furthermore, the chain-weights Q and P on AB and BC proportional to the lengths; it will follow then that 26 TIIL SCIENCE OF AMEICIANICS. Q/P= AB/BC= 2. The generalisation is self-evident. The as- 2. Unquestionably in the assumption from which sumptions of Stevi- Stevinus starts, that the endless chain does not move, nus's deduction. there is contained primarily only a purely instinctive cognition. He feels at once, and we with him, that we have never observed anything like a motion of the kind referred to, that a thing of such a character does not exist. This conviction has so much logical cogency that we accept the conclusion drawn from it respecting the law of equilibrium on the inclined plane without the thought of an objection, although the law if presented as the simple result of experiment, or otherwise put, Their in- would appear dubious. We cannot be surprised at this stinctive character, when we reflect that all results of experiment are obscured by adventitious circumstances (as friction, etc.), and that every conjecture as to the conditions which are determinative in a given case is liable to error. That Stevinus ascribes to instinctive knowledge of this sort a higher authority than to simple, manifest, direct observation might excite in us astonishment if we did not ourselves possess the same inclination. The question accordingly forces itself upon us: Whence does this higher authority come? If we remember that scientific demonstration, and scientific criticism generally can only have sprung from the consciousness of the individual fallibility of investigators, the explanation is not far Their cog- to seek. We feel clearly, that we ourselves have concy. tributed nothing to the creation of instinctive knowledge, that we have added to it nothing arbitrarily, but that it exists in absolute independence of our participation. Our mistrust of our own subjective interpretation of the facts observed, is thus dissipated. Stevinus's deduction is one of the rarest fossil in THE PRIIVCIPLES OF S TA TICS. 27 dications that we possess in the primitive history of Hihistorical value of mechanics, and throws a wonderful light on the pro- tevins's cess of the formation of science generally, on its rise from instinctive knowledge. We will recall to mind that Archimedes pursued exactly the same tendency as Stevinus, only with much less good fortune. In later times, also, instinctive knowledge is very frequently taken as the starting-point of investigations. Every experimenter can daily observe in his own person the guidance that instinctive knowledge furnishes him. If he succeeds in abstractly formulating what is contained in it, he will as a rule have made an important advance in science. Stevinus's procedure is no error. If an error were The trustworthiness contained in it, we should all share it. Indeed, it is of instinctive knowl perfectly certain, that the union of the strongest in-edge. stinct with the greatest power of abstract formulation alone constitutes the great natural inquirer. This by no means compels us, however, to create a new mysticism out of the instinctive in science and to regard this factor as infallible. That it is not infallible, we very easily discover. Even instinctive knowledge of so great logical force as the principle of symmetry employed by Archimedes, may lead us astray. Many of my readers will recall to mind,perhaps, the intellectual shock they experienced when they heard for the first time that a magnetic needle lying in the magnetic meridian is deflected in a definite direction away from the meridian by a wire conducting a current being carried along in a parallel direction above it. The instinctive is just as fallible as the distinctly conscious. Its only value is in provinces with which we are very familiar. Let us rather put to ourselves, in preference to pursuing mystical speculations on this subject, the 28 THE SCIENCE OF MECHANICS. The origin question: How does instinctive knowledge originate of instinctive knowl- and what are its contents? Everything which we obedge. serve in nature imprints itself uncomprehended and unanalysed in our percepts and ideas, which, then, in their turn, mimic the processes of nature in their most general and most striking features. In these accumulated experiences we possess a treasure-store which is ever close at hand and of which only the smallest portion is embodied in clear articulate thought. The circumstance that it is far easier to resort to these experiences than it is to nature herself, and that they are, notwithstanding this, free, in the sense indicated, from all subjectivity, invests them with a high value. It is a peculiar property of instinctive knowledge that it is predominantly of a negative nature. We cannot so well say what must happen as we can what cannot happen, since the latter alone stands in glaring contrast to the obscure mass of experience in us in which single characters are not distinguished. Instinctive Still, great as the importance of instinctive knowlknowledge and extern-edge may be, for discovery, we must not, from our al realities mutually point of view, rest content with the recognition of its condition each other. authority. We must inquire, on the contrary: Under what conditions could the instinctive knowledge in question have originated? We then ordinarily find that the very principle to establish which we had recourse to instinctive knowledge, constitutes in its turn the fundamental condition of the origin of that knowledge. And this is quite obvious and natural. Our instinctive knowledge leads us to the principle which explains that knowledge itself, and which is in its turn also corroborated by the existence of that knowledge, which is a separate fact by itself. This we will find on close examination is the state of things in Stevinus's case. THE PRINCIPLES OF STATICS. 29 3. The reasoning of Stevinus impresses us as so The ingenuity of Stehighly ingenious because the result at which he arrives vinus's reaapparently contains more than the assumption fromsoning. which he starts. While on the one hand, to avoid contradictions, we are constrained to let the result pass, on the other an incentive remains which impels us to seek further insight. If Stevinus had distinctly set forth the entire fact in all its aspects, as Galileo subsequently did, his reasoning would no longer strike us as ingenious; but we should have obtained a much more satisfactory and clear insight into the matter. In the endless chain which does not glide upon the prism, is contained, in fact, everything. We might say, the chain does not glide because no sinking of heavy bodies takes place here. This would not be accurate, however, for when the chain moves many of its links really do descend, while others rise in their place. We must say, therefore, more accurately, the chain does not glide because for every body that could possibly de- ritiquesf scend an equally heavy body would have to ascend deduction. equally high, or a body of double the weight half the height, and so on. This fact was familiar to Stevinus, who presented it, indeed, in his theory of pulleys; but he was plainly too distrustful of himself to lay down the law, without additional support, as also valid for the inclined plane. But if such a law did not exist universally, our instinctive knowledge respecting the endless chain could never have originated. With this our minds are completely enlightened.-The fact that Stevinus did not go as far as this in his reasoning and rested content with bringing his (indirectly discovered) ideas into agreement with his instinctive thought, need not further disturb us. (See p. 515.) The service which Stevinus renders himself and his THE SCIENCE OF MECHANICS. 30 The meritg readers, consists, therefore, in the contrast and comof Stevinus'sproce- parison of knowledge that is instinctive with knowledge dure. that is clear, in the bringing the two into connection and accord with one another, and in the supporting Fig. 21. the one upon the other. The strengthening of mental view which Stevinus acquired by this procedure, we learn from the fact that a picture of the endless chain and the prism graces as vignette, with the inscription "Wonder en is gheen wonder," the title-page of his THEI P/RLVCIPLES OF S TA TICS. 31 work Hypomnemata Mathiemalica (Leyden, 1605).* As Enlightenment in a fact, every enlightening progress made in science is science always acaccompanied with a certain feeling of disillusionment. companied with disillu We discover that that which appeared wonderful to sionment. us is no more wonderful than other things which we know instinctively and regard as self-evident; nay, that the contrary would be much more wonderful; that everywhere the same fact expresses itself. Our puzzle turns out then to be a puzzle no more; it vanishes into nothingness, and takes its place among the shadows of history. 4. After he had arrived at the principle of the in- Explanation of the dined plane, it was easy for Stevinus to apply that other rachines by principle to the other machines and to explain by it Stevinus's principle. their action. He makes, for example, the following application. We have, let us suppose, an inclined plane (Fig. 22) and on it a load Q. We pass a string over the pulley A at the summit and imagine the load Q held in equilibrium by the load P. D Stevinus, now, proceeds by 0 a method similar to that later taken by Galileo. He R \ c remarks that it is not necessary that the load Q should lie directly on the -IF,\E c inclined plane. Provided Fig. 22 only the form of the machine's motion be preserved, the proportion between force and load will in all cases remain the same. We may therefore equally well conceive the load Q to be attached to a properly weighted string passing over a pulley D: which string is normal to the *The title given is that of Willebrord Snell's Latin translation (1608) of Simon Stevin's Wisconstige Gedachtenissen, Leyden, 1605.-Trans. 32 THE SCIENVCE OF MECIHANVICS. The funicu-inclined plane. If we carry out this alteration, we lr machine shall have a so-called funicular machine. We now perceive that we can ascertain very easily the portion of weight with which the body on the inclined plane tends downwards. We have only to draw a vertical line and to cut off on it a portion ab corresponding to the load Q. Then drawing on aA the perpendicular bc, we have P/Q =-A CIAB = ac/ab. Therefore ac represents the tension of the string aA. Nothing prevents us, now, from making the two strings change And the functions and from imagining the load Q to lie on the special case oftheparal-dotted inclined plane EDF. Similarly, here, we oblelogram of forces. tain ad for the tension of the second string. In this manner, accordingly, Stevinus indirectly arrives at a knowledge of the statical relations of the funicular machine and of the so-called parallelogram of forces; at first, of course, only for the particular case of strings (or forces) ac, ad at right angles to one another. Thegeneral Subsequently, indeed, Stevinus employs the prinform of the last-men- ciple of the composition and resolution of forces in tioned principle also a more general form; yet the method by which he employed. I' IE/ I Fig. 23, Fig. 24. reached the principle, is not very clear, or at least is not obvious. He remarks, for example, that if we have three strings AB, AC, AD, stretched at any THE PRINCIPLES OF STATICS. 33 given angles, and the weight P is suspended from the first, the tensions may be determined in the following manner. We produce (Fig. 23) AB to X and cut off on it a portion AE. Drawing from the point E, EF parallel to AD and EG parallel to A C, the tensions of AB, AC, AD are respectively proportional to AE, AF, AG. With the assistance of this principle of construction Stevinus solves highly compli- Fig. 25. cated problems. He determines, for instance, the solution of other corntensions of a system of ramifying strings like that plicated illustrated in Fig. 24; in doing which of course he problems. starts from the given tension of the vertical string. The relations of the tensions of a funicular polygon are likewise ascertained by construction, in the manner indicated in Fig. 25. We may therefore, by means of the principle of the General reinclined plane, seek to elucidate the conditions of op-sul eration of the other simple machines, in a manner similar to that which we employed in the case of the principle of the lever. III. THE PRINCIPLE OF THE COMPOSITION OF FORCES. i. The principle of the parallelogram of forces, at The principle o the which STEVINUS arrived and employed, (yet without ex- parallelogram of pressly formulating it,) consists, as we know, of theforces. following truth. If a body A (Fig. 26) is acted upon by two forces whose directions coincide with the lines AB and A C, and whose magnitudes are proportional to the lengths AB and A C, these two forces produce the 34 TIHE SCIENCE OF MECHIANICS. same effect as a single force, which acts in the direction of the diagonal AD of the parallelogram ABCD and is proportional to that diagonal. For instance, if on the strings AB, AC weights A exactly proportional to the lengths AB, AC be supSposed to act, a single weight acting on the string Fig. 26. AD exactly proportional to the length AD will produce the same effect as the first two. The forces AB and AC are called the components, the force AD the resultant. It is furthermore obvious, that conversely, a single force is replaceable by two or several other forces. Method by 2. We shall now endeavor, in connection with the which the general no- investigations of Stevinus, to give ourselves some idea tion of the parallelo- of the manner in which the gram of Z U forces general proposition of the might have been ar- parallelogram of forces rived at. S - rived at. might have been arrived at. The relation,-disY covered by Stevinus,q r that exists between two mutually perpendicular w n 0 m v forces and a third force that equilibrates them, we shall assume as (indirectly) given. We suppose now (Fig. 27) that there act on three strings Fig. 27. OX, OY, OZ, pulls which balance each other. Let us endeavor to determine the nature of these pulls. Each pull holds the two remaining ones in equilibrium. The pull OYwe will replace TIHE PRIVCIPLES OF STATICS. 35 (following Stevinus's principle) by two new rectangular The deduction of the pulls, one in the direction Ou (the prolongation of general principle OX), and one at right angles thereto in the direction from the special case Ov. And let us similarly resolve the pull OZ in the of tevinus. directions Ou and Ow. The sum of the pulls in the direction Ou, then, must balance the pull OX, and the two pulls in the directions Ov and Ow must mutually destroy each other. Taking the two latter as equal and opposite, and representing them by Om and On, we determine coincidently with the operation the components Op and Oq parallel to Ou, as well also as the pulls Or, Os. Now the sum Op + Og is equal and opposite to the pull in the direction of OX; and if we draw st parallel to O Y, or rt parallel to OZ, either line will cut off the portion Ot Op+ Og: with which result the general principle of the parallelogram of forces is reached. The general case of composition may be deduced A different mode of the in still another way from the special composition of same derectangular forces. Let OA and OB be the two forces tion. acting at O. For OB substitute a force OC acting parallel to O C A E OA and a force OD acting at right angles to OA. There then act for OA and OB the B F two forces OE OA + OC Fig. 28. and OD, the resultant of which forces OF is at the same time the diagonal of the parallelogram OAFB constructed on OA and OB as sides. 3. The principle of the parallelogram of forces, The principle here when reached by the method of Stevinus, presents it- presents itself as an self as an indirect discovery. It is exhibited as a con- indirect discovery. sequence and as the condition of known facts. Wediscovery. perceive, however, merely that it does exist, not, as yet 36 THE SCIENCE OF MECHANICS. And is first why it exists; that is, we cannot reduce it (as in dyclearly enunciated namics) to still simpler propositions. In statics, inby Newton and Varig- deed, the principle was not fully admitted until the non. time of Varignon, when dynamics, which leads directly to the principle, was already so far advanced that its adoption therefrom presented no difficulties. The principle of the parallelogram of forces was first clearly enunciated by NEWTON in his Princizles ofNatural Philosophy. In the same year, VARIGNON, independently of Newton, also enunciated the principle, in a work submitted to the Paris Academy (but not published until after its author's death), and made, by the aid of a geometrical theorem, extended practical application of it.* The geo- The geometrical theorem referred to is this. If we metrical theorem consider (Fig. 29) a parallelogram the sides of which employed by Varig- arep and q, and the diagonal is r, and from any point m non. in the plane of the parm allelogram we draw perv pendiculars on these VUm three straight lines, Swhich perpendiculars we will designate as u, v, w, then p. u - q. v = r. w. This is easily proved by drawFig. 29. Fig. 30. ing straight lines from m to the extremities of the diagonal and of the sides of the parallelogram, and considering the areas of the triangles thus formed, which are equal to the halves of the products specified. If the point m be taken within the parallelogram and perpendiculars then be * In the same year, 1687, Father Bernard Lami published a little appendix to his Traiti de mechanique, developing the same principle.- Trans. THE PRINCIPLES OF STA TICS. 37 drawn, the theorem passes into the form. u - q. v = r. w. Finally, if mn be taken on the diagonal and perpendiculars again be drawn, we shall get, since the perpendicular. let fall on the diagonal is now zero, p. u-- q. v = 0 or p. u= - q. v. With the assistance of the observation that forces The deduction. are proportional to the motions produced by them in equal intervals of time, Varignon easily advances from the composition of motions to the composition of forces. Forces, which acting at a point are represented in magnitude and direction by the sides of a parallelogram, are replaceable by a single force, similarly represented by the diagonal of that parallelogram. If now, in the parallelogram considered, p and q Moments of represent the concurrent forces (the components) and r forces. the force competent to take their place (the resultant), then the products pu, qv, rw are called the moments of these forces with respect to the point m. If the point im lie in the direction of the resultant, the two moments pu and qv are with respect to it equal to each other. 4. With the assistance of this principle Varignon is varignon's. *treatment now in a position to treat y o thesnitto t X * \ pie mathe machines in a much chines. simpler manner than were his predecessors. Let us consider, for example, (Fig. 31) a rigid body / capable of rotation about an axis passing through B O. Perpendicular to the axis we conceive a plane, P O and select therein two Fig. 31. points A, B, on which two forces P and Q in the plane are supposed to act. We recognise with Varignon 38 THE SCIENCE OF MECHANICS. The deduc- that the effect of the forces is not altered if their points tion of the law of the of application be displaced along their line of action, lever from the paral- since all points in the same direction are rigidly conlelogramprinciple, nected with one another and each one presses and pulls the other. We may, accordingly, suppose P applied at any point in the direction AX, and Q at any point in the direction BY, consequently also at their point of intersection M. With the forces as displaced to M, then, we construct a parallelogram, and replace the forces by their resultant. We have now to do only with the effect of the latter. If it act only on movable points, equilibrium will not obtain. If, however, the direction of its action pass through the axis, through the point O, which is not movable, no motion can take place and equilibrium will obtain. In the latter case 0 is a point on the resultant, and if we drop the perpendiculars u and v from 0 on the directions of the forces p, q, we shall have, in conformity with the theorem before mentioned, p. u =-. v. With this we have deduced the law of the lever from the principle of the parallelogram of forces. The statics Varignon explains in like manner a number of other of Varignon aydarnmioal cases of equilibrium by the equilibration of the resultstatics. ant force by some obstacle or restraint. On the inclined plane, for example, equilibrium exists if the resultant is found to be at right angles to the plane. In fact, Varignon rests statics in its entirety on a dynamic foundation; to his mind, it is but a special case of dynamics. The more general dynamical case constantly hovers before him and he restricts himself in his investigation voluntarily to the case of equilibrium. We are confronted here with a dynamical statics, such as was possible only after the researches of Galileo. Incidentally, it may be remarked, that from Varignon 6 THE PRINCIPLES OF STA TICS. 39 is derived the majority of the theorems and methods of presentation which make up the statics of modern elementary text-books. 5. As we have already seen, purely statical consid- Special statical conerations also lead to the proposition of the parallel- siderations also lead to ogram of forces. In special cases, in fact, the principle the principle. admits of being very easily verified. We recognise at once, for instance, that any number whatsoever of equal forces acting (by pull or pressure) in the same plane at a point, around which their successive lines make equal angles, B are in equilibrium. If, for exampie, (Fig. 32) the three equal forces OA, OB, OC act on the 0 point 0 at angles of 1200, each two of the forces holds the third in equilibrium. We see immediately that the resultant of OA Fig. 32. and OB is equal and opposite to OC. It is represented by OD and is at the same time the diagonal of the parallelogram OADB, which readily follows from the fact that the radius of a circle is also the side of the hexagon included by it. 6. If the concurrent forces act in the same or in The case of coincident opposite directions, the resultant is equal to the sum forces merely a or the difference of the particular comp A caste of the components. e rec- general ognise both cases with- B' B c/ C principle. out any difficulty as O particular cases of the principle of the paral- Fig. 33 lelogram of forces. If in the two drawings of Fig. 33 we imagine the angle A OB to be gradually reduced to the value o~, and the angle A' O' B' increased to the 40 THE SCIENCE OF MECHANICS. value I800, we shall perceive that OC passes into OA + AC- OA + OB and 0' C' into O'A'- A' C' O' 0A' - ' B'. The principle of the parallelogram of forces includes, accordingly, propositions which are generally made to precede it as independent theorems. The princi- 7. The principle of the parallelogram of forces, in p e a proposition de- the form in which it was set forth by Newton and rived from experience. Varignon, clearly discloses itself as a proposition derived from experience. A point acted on by two forces describes with accelerations proportional to the forces two mutually independent motions. On this fact the parallelogram construction is based. DANIEL BERNOULLI, however, was of opinion that the proposition of the parallelogram of forces was a geometrical truth, independent of physical experience. And he attempted to furnish for it a geometrical demonstration, the chief features of which we shall here take into consideration, as the Bernoullian view has not, even at the present day, entirely disappeared. Daniel Ber- If two equal forces, at right angles to each other noulli's attempted (Fig. 34), act on a point, there can be no doubt, acderonstra- cording to Bernoulli, that the line torfthe. of bisection of the angle (con7 formably to the principle of symp metry) is the direction of the resultant r. To determine geometrically also the magnitude of the r resultant, each of the forces p is Fig 34. decomposed into two equal forces q, parallel and perpendicular to r. The relation in respect of magnitude thus produced between p and q is consequently the same as that between r and p. We have, accordingly: p =. q and r =. p; whence r = pq. THE PRINCIPLES OF STA TICS. 41 Since, however, the forces q acting at right angles to r destroy each other, while those parallel to r constitute the resultant, it further follows that r -- 2; hence u = 1/2 and r = V/2. p. The resultant, therefore, is represented also in respect of magnitude by the diagonal of the square constructed on p as side. Similarly, the magnitude may be determined of the The case of unequal resultant of unequal rectangular components. Here, rectangular components however, nothing is known before- components hand concerning the direction of S the resultant r. If we decompose u the components p, q (Fig. 35), A parallel and perpendicular to the yet undetermined direction r, into" ~ the forces u, s and v, t, the new forces will form with the components p, q the same angles that p, Fig. 35. q form with r. From which fact the following relations in respect of magnitude are determined: r p n r q r r q - and - and, p it V s pv t' from which two latter equations follows s = = q/r. On the other hand, however, P2 q2 r -u - V. - + - or r2 =-2 + q2. r r The diagonal of the rectangle constructed onp and q represents accordingly the magnitude of the resultant. Therefore, for all rhombs, the direction of the re- General resultant is determined; for all rectangles, the magni-sults. tude; and for squares both magnitude and direction. Bernoulli then solves the problem of substituting for 42 TIHE SCIENCE OF MiECHANICS. two equal forces acting at one given angle, other equal, equivalent forces acting at a different angle; and finally arrives by circumstantial considerations, not wholly exempt from mathematical objections, but amended later by Poisson, at the general principle. Critique of 8. Let us now examine the physical aspect of this Bernoulli's method, question. As a proposition derived from experience, the principle of the parallelogram of forces was already known to Bernoulli. What Bernoulli really does, therefore, is to simulate towards himself a comple/e ignorance of the proposition and then attempt to philosophise it abstractly out of the fewest possible assumptions. Such work is by no means devoid of meaning and purpose. On the contrary, we discover by such procedures, how few and how imperceptible the experiences are that suffice to supply a principle. Only we must not deceive ourselves, as Bernoulli did; we must keep before our minds all the assumptions, and should overlook no experience which we involuntarily employ. What are the assumptions, then, contained in Bernoulli's deduction? The as- 9. Statics, primarily, is acquainted with force only sumptions of his de- as a pull or a pressure, that from whatever source it duction derived from may come always admits of being replaced by the pull experience. or the pressure of a weight. All forces thus may be regarded as quantities of the same kind and be measured by weights. Experience further instructs us, that the particular factor of a force which is determinative of equilibrium or determinative of motion, is contained not only in the magnitude of the force but also in its direction, which is made known by the direction of the resulting motion, by the direction of a stretched cord, or in some like manner. We may ascribe magnitude indeed to other things given in physical experience, THE PRINCIPLES OF STA TICS. 43 such as temperature, potential function, but not direction. The fact that both magnitude and direction are determinative in the efficiency of a force impressed on a point is an important though it may be an unobtrusive experience. Granting, then, that the magnitude and direction Magnitude and direcof forces impressed on a point alone are decisive, it will tion the sole decisive be perceived that two equal and opposite forces, as they factors. cannot uniquely and precisely determine any motion, are in equilibrium. So, also, at right angles to its direction, a force/ is unable uniquely to determine a motional effect. But if a force p is inclined at an angle to another direction s s' (Fig. 36), it is able to determine a motion in that direction. Yet ex- S perience alone can inform us, Fig. 36. that the motion is determined in the direction of s's and not in that of ss'; that is to say, in the direction of the side of the acute angle or in the direction of the projection of s on s's. Now this latter experience is made use of by Ber- Thee~ectof direction noulli at the very start. The sense, namely, of the re- derivable only from sultant of two equal forces acting at right angles to one experience. another is obtainable only on the ground of this experience. From the principle of symmetry follows only, that the resultant falls in the plane of the forces and coincides with the line of bisection of the angle, not however that it falls in the acute angle. But if we surrender this latter determination, our whole proof is exploded before it is begun. 10. If, now, we have reached the conviction that our knowledge of the effect of the direction of a force is 44 THE SCIENCE OF MECHIANVICS. So also solely obtainable from experience, still less then shall must the form of the we believe it in our power to ascertain by any other way effect be thus de- theform of this effect. It is utterly out of our power, rived. to divine, that a force p acts in a direction s that makes with its own direction the angle a, exactly as a force p cos a in the direction s; a statement equivalent to the proposition of the parallelogram of forces. Nor was it in Bernoulli's power to do this. Nevertheless, he makes use, scarcely perceptible it is true, of experiences that involve by implication this very mathematical fact. The man- A person already familiar with the composition ner in which the and resolution of forces is well aware that several forces assumptions men- acting at a point are, as regards their effect, replaceable, tioned enter into Ber- in every respect and in every direction, by a single force. noulli's deduction. This knowledge, in Bernoulli's mode of proof, is expressed in the fact that the forces p, q are regarded as absolutely qualified to replace in all respects the forces s, u and t, v, as well in the direction of r as in every other direction. Similarly r is regarded as the equivalent of p and q. It is further assumed as wholly indifferent, whether we estimate s, u, i, v first in the directions of p, q, and thenp, q in the direction of r, or s, u, t, v be estimated directly and from the outset in the direction of r. But this is something that a person only can know who has antecedently acquired a very extensive experience concerning the composition and resolution of forces. We reach most simply the knowledge of the fact referred to, by starting from the knowledge of another fact, namely that a force p acts in a direction making with its own an angle a, with an effect equivalent to p. cos a. As a fact, this is the way the perception of the truth was reached. Let the coplanar forces P, P', P"... be applied to THE7 PRINCIPLES OF STA TICS. 45 one and the same point at the angles a, a', a"... with Mathematical analya given direction X. These forces, let us suppose, are sis of the results of replaceable by a single force 11, which makes with Xthe true and necessary an angle ju. By the familiar principle we have then assumption. 2Pcosa= - 1 cosp. If H is still to remain the substitute of this system of forces, whatever direction X may take on the system being turned through any angle 6, we shall further have 2P cos (a + 6) cos (u + ), or (2P cosa- Hcosji) cos6 - (:P sina - H sinp) sind = 0. If we put 21P cosa -- 7 cos/ = A, - (2P sina - T sinu-) = B, B tanr it follows that A cos6 + B sind =- /A2 2 B2sin (6 + r) = 0, which equation can subsist for every 6 only on the condition that A = 2- P cosa - - cos/ = 0 and B = (2P sina - H sin)u) = 0; whence results H cosu = 2P cosa H sinu = 'P sina. From these equations follow for H and Ju the determinate values H7 = /[(2ýPsina)2 + (2ýPcosa)2] and YP' sina tany = - co 2P cosa* 46 THE SCIENCE OF ME CHANICS. The actual Granting, therefore, that the effect of a force in every results not deducible direction can be measured by its projection on that dion any other sup- rection, then truly every system of forces acting at a position. point is replaceable by a single force, determinate in magnitude and direction. This reasoning does not hold, however, if we put in the place of cos a any general function of an angle,

Fig. 47. ( - r/R) Q, the cylinder, on the application of the force, will roll itself up on the string. Roberval's Roberval's Balance (Fig. 48) consists of a paralbalance. lelogram with variable angles, two opposite sides of which, the upper and lower, are capable of rotation about their middle points A, B. To the two remaining sides, which are always vertical, horizontal rods are THE PRINCIPLES OF STA TICS. 61 fastened. If from these rods we suspend two equal weights P, equilibrium will subsist independently of the position of the points A of suspension, because on displacement the descent of the one weight is always equal to the ascent of the other. At three fixed points A, I I Discussion of the case B, C (Fig. 49) let pulleys Fig. 48. of equilibrium of be placed, over which three strings are passed loaded three knotwith equal weights and knotted at O. In what posi- ted strings tion of the strings will equilibrium exist? We will call the lengths of the three strings AO =-s, BO =-s2, A P 0 B s B A s, 0 cc 0p S i Fig. 49. Fig. 50. CO s s. To obtain the equation of equilibrium, let us displace the point 0 in the directions s, and s3 the infinitely small distances 6ds and ds., and note that by so doing every direction of displacement in the plane ABC (Fig. 50) can be produced. The sum of the virtual moments is P&s2 - PTs cos a P,ds cos (a ) 0 20 + P6s P6s3 cos3 P+ P&s cos (a, + ) or [1 - cos cos (a+ )] 6S + - cos/ + cos (a+ f)] 6s, - 0. But since each of the displacements 6s2, 6s3 is ar 62 THE SCIENCE OF ME CHANICS. bitrary, and each independent of the other, and may by themselves be taken = 0, it follows that 1 - cos a cos (a+ /3) = 0 1 - cos3 + cos (a - /3) = 0. Therefore cos a = cos /3, and each of the two equations may be replaced by 1 - cos a -- cos 2a = 0; or cos a 2= ', wherefore a + /--= 1200. Remarks on Accordingly, in the case of equilibrium, each of the the preceding case. strings makes with the others angles of 120; which is, moreover, directly obvious, since three equal forces can be in equilibrium only when such an arrangement exists. This once known, we may find the position of the point O with respect to ABC in a number of different ways. We may proceed for instance as follows. We construct on AB, BC, CA, severally, as sides, equilateral triangles. If we describe circles about these triangles, their common point of intersection will be the point O sought; a result which easily follows from the well-known relation of the angles at the centre and circumference of circles. The case of A bar OA (Fig. 51) is revolvable about O in the a bar revolvable plane of the paper and makes with a fixed straight line about one of its ex- OX the variable angle tremities. A p a. At A there is applied a force P which B makes with OX the o a X angle y, and at B, on Fig. 51. a ring displaceable along the length of the bar, a force Q, making with OX the angle 3. We impart to the bar an infinitely THE PRINCIPLES OF STATICS. 63 small rotation, in consequence of which B and A move The case of a bar reforward the distances 6s and 6s, at right angles to OA, volvable about one and we also displace the ring the distance 6r along the of its extremities, bar. The variable distance OB we will call r, and we will let OA a. For the case of equilibrium we have then Q6r cos (/ - a) + Q6s sin (3 - a) + P6s1 sin (a - y) = 0. As the displacement 6r has no effect whatever on the other displacements, the virtual moment therein involved must, by itself, - 0, and since 6r may be of any magnitude we please, the coefficient of this virtual moment must also - 0. We have, therefore, Q cos (/ - a) = 0, or when Q is different from zero, 3 - a =- 90~. Further, in view of the fact that 6s, = (a/r) 6s, we also have rQ sin (/3 - a) + a P sin (a - y) = 0, or since sin (/3- a) = I, rQ + aP sin (a - y) = 0; wherewith the relation of the two forces is obtained. ii. An advantage, not to be overlooked, which Every general prinevery general principle, and therefore also the prin- ciple involves an ciple of virtual displacements, fur- -, economy of nishes, consists in the fact that it thought. saves us to a great extent the necessity of considering every new particular case presented. In the possession of this principle we need not, for Fig. 52. example, trouble ourselves about the details of a machine. If a new machine say were so enclosed in a 64 TIHE SCI'NCE OF IMICHANICS. box (Fig. 52), that only two levers projected as points of application for the force P and the weight P', and we should find the simultaneous displacements of these levers to be,h and /h, we should know immediately that in the case of equilibrium P/i= P' h', whatever the construction of the machine might be. Every principle of this character possesses therefore a distinct economical value. Further re- 12. We return to the general expression of the prinmarks on the general ciple of virtual displacements, in order to add a few expression of the prin- further remarks. If ciple. A " at the points A, B, SC.... the forces P' P, 1P', P".... act, ( B and p, f"f I. S v I are the projections Fig. 53. of infinitely small mutually compatible displacements, we shall have for the case of equilibrium Pp+ P'f' + p"' +.. = 0. If we replace the forces by strings which pass over pulleys in the directions of the forces and attach thereto the appropriate weights, this expression simply asserts that the centre of gravity of the system of weights as a whole cannot descend. If, however, in certain displacements it were possible for the centre of gravity to rise, the system would still be in equilibrium, as the heavy bodies would not, of themselves, enter on any Modifica- such motion. In this case the sum above given would tion of the previous be negative, or less than zero. The general expression equation of condition. of the condition of equilibrium is, therefore, PWh + i i' + splace +. t t e e s. When for every virtual displacement there exists TIIE PRINICIPLE S OF STA TICS. 65 another equal and opposite to it, as is the case for example in the simple machines, we may restrict ourselves to the upper sign, to the equation. For if it were possible for the centre of gravity to ascend in certain displacements, it would also have to be possible, in consequence of the assumed reversibility of all the virtual displacements, for it to descend. Consequently, in the present case, a possible rise of the centre of gravity is incompatible with equilibrium. The question assumes a different aspect, however, The condition is, that when the displacements are not all reversible. Two the sum of the virtual bodies connected together by strings can approach moments shall be each other but cannot recede from each other beyond equal to or less than the length of the strings. A body is able to slide or zero. roll on the surface of another body; it can move away from the surface of the second body, but it cannot penetrate it. In these cases, therefore, there are displacements that cannot be reversed. Consequently, for certain displacements a rise of the centre of gravity may take place, while the contrary displacements, to which the descent of the centre of gravity corresponds, are impossible. We must therefore hold fast to the more general condition of equilibrium, and say, the sum of the virtual moments is equal to or less than zero. 13. LAGRANGE in his Analytical Mechanics attempted The Lagrangian a deduction of the principle of virtual displacements, deduction of the prinwhich we will now consider. At the points A, B, ciple. C.... (Fig. 54) the forces P, P', P".... act. We imagine rings placed at the points in question, and other rings A', B', C'.... fastened to points lying in the directions of the forces. We seek some common measure Q/2 of the forces P, P', P".. that enables us to put: 66 THE SCIENCE OF MECHANICS. Effected by Q means of a 2n, - = P, set of pul- 2 leys and a single Pt weight. 2n. = P, 2n". - ", where n, n', n".... are whole numbers. Further, we make fast to the ring A' a string, carry this string back and forth n times between A' and A, then through B',, c P' Q B' L-2 Fig. 54. n' times back and forth between B' and B, then through C', n" times back and forth between C' and C, and, finally, let it drop at C', attaching to it there the weight Q/2. As the string has, now, in all its parts the tension Q/2, we replace by these ideal pulleys all the forces present in the system by the single force Q/2. If then the virtual (possible) displacements in any given configuration of the system are such that, these displacements occurring, a descent of the weight Q/2 can take place, the weight will actually descend and produce those displacements, and equilibrium therefore will not obtain. But on the other hand, no motion will ensue, if the displacements leave the weight Q/2 in its original position, or raise it. The expression of this condition, reckoning the projections of the virtual displacements in the directions of the forces positive, THE PRINCIPLES OF STATICS. 67 and having regard for the number of the turns of the string in each single pulley, is 2np + 2n'p' + 2n"p" +... 0. Equivalent to this condition, however, is the expression 2n p2 + 2 p ' + 2n".. +... 0, or pP + p'p+-4- p +. 7 0. 14. The deduction of Lagrange, if stripped of the The convincing fearather odd fiction of the pulleys, really possesses con- tures of Lagrange's vincing features, due to the fact that the action of a deduction. single weight is much more immediate to our experience and is more easily followed than the action of several weights. Yet it is not proved by the Lagrangian deduction that work is the factor determinative of the disturbance of equilibrium, but is, by the employment of the pulleys, rather assumed by it. As a matter of fact every pulley involves the fact enunciated and recognised by the principle of virtual displacements. The replacement of all the forces by a single weight that does the same work, presupposes a knowledge of the import of work, and can be proceeded with on this assumption alone. The fact that some certain cases are It is not, however, a more familiar to us and more immediate to our expe- proof. rience has as a necessary result that we accept them without analysis and make them the foundation of our deductions without clearly instructing ourselves as to their real character. It often happens in the course of the development of science that a new principle perceived by some inquirer in connection with a fact, is not immediately recognised and rendered familiar in its entire generality. 68 THE SCIENCE OF AMECHIANICS. The expe- Then, every expedient calculated to promote these dients employed to ends, is, as is proper and natural, called into service. support all new prin- All manner of facts, in which the principle, although ciples. contained in them, has not yet been recognised by inquirers, but which from other points of view are more familiar, are called in to furnish a support for the new conception. It does not, however, beseem mature science to allow itself to be deceived by procedures of this sort. If, throughout all facts, we clearly see and discern a principle which, though not admitting of proof, can yet be known to prevail, we have advanced much farther in the consistent conception of nature than if we suffered ourselves to be overawed by a specious Value of the demonstration. If we have reached this point of view, Lagrangian proof. we shall, it is true, regard the Lagrangian deduction with quite different eyes; yet it will engage nevertheless our attention and interest, and excite our satisfaction from the fact that it makes palpable the similarity of the simple and complicated cases. 15. MAUPERTUIS discovered an interesting proposition relating to equilibrium, which he communicated to the Paris Academy in 1740 under the name of the " Loi de repos." This principle was more fully discussed by EULER in 1751 in the Proceedings of the Berlin Academy. If we cause infinitely small displaceThe Loide ments in any system, we produce a sum of virtual morepos. ments P P'p' P"'" -..., which only reduces to zero in the case of equilibrium. This sum is the work corresponding to the displacements, or since for infinitely small displacements it is itself infinitely small, the corresponding element of work. If the displacements are continuously increased till a finite displacement is produced, the elements of the work will, by summation, produce a finite amount of work. So, if we THEr PRIVCIPLES OF STATICS. 69 start from any given initial configuration of the system Statement of the prinand pass to any given final configuration, a certainciple. amount of work will have to be done. Now Maupertuis observed that the work done when a final configuration is reached which is a configuration of equilibrium, is generally a maximum or a minimum; that is, if we carry the system through the configuration of equilibrium the work done is previously and subsequently less or previously and subsequently greater than at the configuration of equilibrium itself. For the configuration of equilibrium Ppt + P'p' + P"p" +...= 0, that is, the element of the work or the differential (more correctly the variation) of the work is equal to zero. If the differential of a function can be put equal to zero, the function has generally a maximum or minimum value. 16. We can produce a very clear representation to Graphical illustration the eye of the import of Maupertuis's principle. of the import of the We imagine the forces of a system replaced by principle. Lagrange's pulleys with the weight Q/2. We suppose that each point of the system is restricted to movement on a certain curve and that the motion is such that when one point occupies a definite position on its curve all the other points assume uniquely determined positions on their respective curves. The simple machines are as a rule systems of this kind. Now, while imparting displacements to the system, we may carry a vertical sheet of white paper horizontally over the weight Q/2, while this is ascending and descending on a vertical line, so that a pencil which it carries shall describe a curve upon the paper (Fig. 55). When the pencil stands at the points a, c, d of the curve, there are, 70 THE SCIENCE OF MECHANICS. Interpreta- we see, adjacent positions in the system of points at tion of the diagram, which the weight Q/2 will stand higher or lower than in the configuration given. The weight will then, if the system be left to itself, pass into this lower position and Qa c e Fig. 55. displace the system with it. Accordingly, under conditions of this kind, equilibrium does not subsist. If the pencil stands at e, then there exist only adjacent configurations for which the weight Q/2 stands higher. But of itself the system will not pass into the lastnamed configurations. On the contrary, every displacement in such a direction, will, by virtue of the tendency of the weight to move downwards, be reversed. Stable equilibrium, therefore, is the condition Stableequi- that corresponds to the lowest position of the weziht or to librium. a maximnum of work done in the system. If the pencil stands at b, we see that every appreciable displacement brings the weight Q/2 lower, and that the weight therefore will continue the displacement begun. But, assuming infinitely small displacements, the pencil moves in the horizontal tangent at b, in which event the weight cannot descend. Therefore, unstable equiUnstable librium is the state that corresponds to the highest position equilibrium of the weight Q/2, or to a minimum of work done in the system. It will be noted, however, that conversely TIE PRINCIPLES OF STA TICS. 7I every case of equilibrium is not the correspondent of a maximum or a minimum of work performed. If the pencil is atf, at a point of horizontal contrary flexure, the weight in the case of infinitely small displacements neither rises nor falls. Equilibrium exists, although the work done is neither a maximum nor a minimum. The equilibrium of this case is the socalled mixed equilibrium *: for some disturbances it is Mixed equilibrium. stable, for others unstable. Nothing prevents us from regarding mixed equilibrium as belonging to the unstable class. When the pencil stands at g, where the curve runs along horizontally a finite distance, equilibrium likewise exists. Any small displacement, in the configuration in question, is neither continued nor reversed. This kind of equilibrium, to which likewise neither a maximum nor a minimum corresponds, is termed [neutral or] indifferent. If the curve described Neutral by Q/2 has a cusp pointing upwards, this indicates a equilibrium minimum of work done but no equilibrium (not even unstable equilibrium). To a cusp pointing downwards a maximum and stable equilibrium correspond. In the last named case of equilibrium the sum of the virtual moments is not equal to zero, but is negative. 17. In the reasoning just presented, we have as- Theprecedsumed that the motion of a point of a system on one nnillustiread curve determines the motion of all the other points oft b anodifthe system on their respective curves. The movabilityficult cases. of the system becomes multiplex, however, when each point is displaceable on a surface, in a manner such that the position of one point on its surface determines *This term is not used in English, because our writers hold that no equilibrium is conceivable which is not stable or neutral for some possible displacements. Hence what is called mixed equilibrium in the text is called unstable equilibrium by English writers, who deny the existence of equilibrium unstable in every respect.-Trans. THE SCIENCE OF AE CI.4NICS. 72 uniquely the position of all the other points on their surfaces. In this case, we are not permitted to consider the curve described by Q/2, but are obliged to picture to ourselves a surface described by Q/2. If, to go a step further, each point is movable throughout a space, we can no longer represent to ourselves in a purely geometrical manner the circumstances of the motion, by means of the locus of Q/2. In a correspondingly higher degree is this the case when the position of one of the points of the system does not determine conjointly all the other positions, but the character of the system's motion is more multiplex still. In all these cases, however, the curve described by Q/2 (Fig. 55) can serve us as a symbol of the phenomena to be considered. In these cases also we rediscover the Maupertuisian propositions. Further ex- We have also supposed, in our considerations up to tension of the same this point, that constant forces, forces independent of idea. the position of the points of the system, are the forces that act in the system. If we assume that the forces do depend on the position of the points of the system (but not on the time), we are no longer able to conduct our operations with simple pulleys, but must devise apparatus the force active in which, still exerted by Q/2, varies with the displacement: the ideas we have reached, however, still obtain. The depth of the descent of the weight Q/2 is in every case the measure of the work performed, which is always the same in the same configuraFig. 56. tion of the system and is independent of the path of transference. A contrivance which would develop by means of a constant weight a force varying with the displacement, would be, for example, a wheel THE PRINCIPLES OF STATICS. 73 and axle (Fig. 56) with a non-circular wheel. It would not repay the trouble, however, to enter into the details of the reasoning indicated in this case, since we perceive at a glance its feasibility. 18. If we know the relation that subsists between The principle of the work done and the so-called vis viva of a sys-courtivron. tem, a relation established in dynamics, we arrive easily at the principle communicated by COURTIVRON in 1749 to the Paris Academy, which is this: For the stable configuration of unstable equilibrium, at which the maximum work done is a nimum, the vis viva of the system, maximum in motion, is also a minim in its transit through these configurations. 19. A heavy, homogeneous triaxial ellipsoid resting Illustration of the varion a horizontal plane is admirably adapted to illustrate ous kindsof equilibrium the various classes of equilibrium. When the ellipsoid rests on the extremity of its smallest axis, it is in stable equilibrium, for any displacement it may suffer elevates its centre of gravity. If it rest on its longest axis, it is in unstable equilibrium. If the ellipsoid stand on its mean axis, its equilibrium is mixed. A homogeneous sphere a or a homogeneous right cylinder on a horizontal plane illus- Fig. 57. trates the case of indifferent equilibrium. In Fig. 57 we have represented the paths of the centre of gravity of a cube rolling on a horizontal plane about one of its edges. The position a of the centre of gravity is the position of stable equilibrium, the position 6, the position of unstable equilibrium. THE SCIENCE OF MECHANICS. 74 The caten- 20. We will now consider an example which at ary. first sight appears very complicated but is elucidated at once by the principle of virtual displacements. John and James Bernoulli, on the occasion of a conversation on mathematical topics during a walk in Basel, lighted on the question of what form a chain would take that was freely suspended and fastened at both ends. They soon and easily agreed in the view that the chain would assume that form of equilibrium at which its centre of gravity lay in the lowest possible position. As a matter of fact we really do perceive that equilibrium subsists when all the links of the chain have sunk as low as possible, when none can sink lower without raising in consequence of the connections of the system an equivalent mass equally high or higher. When the centre of gravity has sunk as low as it possibly can sink, when all has happened that can happen, stable equilibrium exists. The physical part of the problem is disposed of by this consideration. The determination of the curve that has the lowest centre of gravity for a given length between the two points A, B, is simply a mathematical problem. (See Fig. 58.) Theprinci- 21. Collecting all that has been presented, we see, pie is simply the rc- that there is contained in the principle of virtual disognition of afn ctf placements simply the recognition of a fact that was instinctively familiar to us long previously, only that we had not apprehended it so precisely and clearly. This fact consists in the circumstance that heavy bodies, of themselves, move only downwards. If several such bodies be joined together so that they can suffer no displacement independently of each other, they will then move only in the event that some heavy mass is on the whole able to descend, or as the principle, with a more perfect adaptation of our ideas to THE PRINCIPLES OF STATICS. 75 76 THLE SCIENCE OF JIIECtl.IVli/CS. What this the facts, more exactly expresses it, only in the event fact is. that work can be performed. If, extending the notion of force, we transfer the principle to forces other than those due to gravity, the recognition is again contained therein of the fact that the natural occurrences in question take place, of themselves, only in a definite sense and not in the opposite sense. Just as heavy bodies descend downwards, so differences of temperature and electrical potential cannot increase of their own accord but only diminish, and so on. If occurrences of this kind be so connected that they can take place only in the contrary sense, the principle then establishes, more precisely than our instinctive apprehension could do this, the factor work as determinative and decisive of the direction of the occurrences. The equilibrium equation of the principle may be reduced in every case to the trivial statement, that when nothing can lhaipen nothing does happen. The prin- 22. It is important to obtain clearly the perception, ciple in the light of that we have to deal, in the case of all principles, Gauss's view. merely with the ascertainment and establishment of a fact. If we neglect this, we shall always be sensible of some deficiency and will seek a verification of the principle, that is not to be found. Jacobi states in his Leclures on Dynamics that Gauss once remarked that Lagrange's equations of motion had not been proved, but only historically enunciated. And this view really seems to us to be the correct one in regard to the principle of virtual displacements. The differ- The task of the early inquirers, who lay the founent tasks of early and of dations of any department of investigation, is entirely subsequent inquirers in different from that of those who follow. It is the busiany departmnent. e ness of the former to seek out and to establish the facts of most cardinal importance only; and, as history THE PRINCIPLES OF STATICS. 77 teaches, more brains are required for this than is generally supposed. When the most important facts are once furnished, we are then placed in a position to work them out deductively and logically by the methods of mathematical physics; we can then organise the department of inquiry in question, and show that in the acceptance of some one fact a whole series of others is included which were not to be immediately discerned in the first. The one task is as important as the other. We should not however confound the one with the other. We cannot prove by mathematics that nature must be exactly what it is. But we can prove, that one set of observed properties determines conjointly another set which often are not directly manifest. Let it be remarked in conclusion, that the princi- Every general principle of virtual displacements, like every general prin- pie brings with it disciple, brings with it, by the insight which it furnishes, illusionment as disillusionment as well as elucidation. It brings with well as eluScidation. it disillusionment to the extent that we recognise in it facts which were long before known and even instinctively perceived, our present recognition being simply more distinct and more definite; and elucidation, in that it enables us to see everywhere throughout the most complicated relations the same simple facts. V. RETROSPECT OF THE DEVELOPMENT OF STATICS. i. Having passed successively in review the prin- Review of statics as a ciples of statics, we are now in a position to take a whole. brief supplementary survey of the development of the principles of the science as a whole. This development, falling as it does in the earliest period of mechanics, -the period which begins in Grecian antiquity and 78 THE SCIENCE OF MECHANICS. reaches its close at the time when Galileo and his younger contemporaries were inaugurating modern mechanics,-illustrates in an excellent manner the process of the formation of science generally. All conceptions, all methods are here found in their simplest form, and as it were in their infancy. These beginnings The origin point unmistakably to their origin in the experiences of of science. the manual arts. To the necessity of putting these experiences into commnunicable form and of disseminating them beyond the confines of class and craft, science owes its origin. The collector of experiences of this kind, who seeks to preserve them in written form, finds before him many different, or at least supposably different, experiences. His position is one that enables him to review these experiences more frequently, more variously, and more impartially than the individual workingman, who is always limited to a narrow province. The facts and their dependent rules are brought into closer temporal and spatial proximity in his mind and writings, and thus acquire the opportunity of revealing The econo- their relationship, their connection, and their gradual my of communication. transition the one into the other. The desire to simplify and abridge the labor of communication supplies a further impulse in the same direction. Thus, from economical reasons, in such circumstances, great numbers of facts and the rules that spring from them are condensed into a system and comprehended in a single expression. The gene- 2. A collector of this character has, moreover, opral character of prin- portunity to take note of some new aspect of the facts ciples. before him-of some aspect which former observers had not considered. A rule, reached by the observation of facts, cannot possibly embrace the entire fact, in all its infinite wealth, in all its inexhaustible manifoldness; THIL PRIVCIPL ES OF STA4TICS. 79 on the contrary, it can furnish only a rough outline of the fact, one-sidedly emphasising the feature that is of importance for the given technical (or scientific) aim in view. lWat aspects of a fact are taken notice of, will consequently depend upon circumstances, or even on Their form in many asthe caprice of the observer. Hence there is always op- pects, accidental. portunity for the discovery of new aspects of the fact, which will lead to the establishment of new rules of equal validity with, or superior to, the old. So, for instance, the weights and the lengths of the lever-arms were regarded at first, by Archimedes, as the conditions that determined equilibrium. Afterwards, by Da Vinci and Ubaldi the weights and the perpendicular distances from the axis of the lines of force were recognised as the determinative conditions. Still later, by Galileo, the weights and the amounts of their displacements, and finally by Varignon the weights and the directions of the pulls with respect to the axis were taken as the elements of equilibrium, and the enunciation of the rules modified accordingly. 3. Whoever makes a new observation of this kind, our liability to error and establishes such a new rule, knows, of course, our in the mental reconliability to error in attempting mentally to represent struction of the fact, whether by concrete images or in abstract conceptions, which we must do in order to have the mental model we have constructed always at hand as a substitute for the fact when the latter is partly or wholly inaccessible. The circumstances, indeed, to which we have to attend, are accompanied by so many other, collateral circumstances, that it is frequently difficult to single out and consider those that are essential to the purpose in view. Just think how the facts of friction, the rigidity of ropes and cords, and like conditions in machines, obscure and obliterate the pure outlines of 8o THE SCIENCE OF ME CHANICS. Thisliabil- the main facts. No wonder, therefore, that the discovity impels us to seek erer or verifier of a new rule, urged by mistrust of himafter proofs of all new self, seeks after a proof of the rule whose validity he rules. believes he has discerned. The discoverer or verifier does not at the outset fully trust in the rule; or, it may be, he is confident only of a part of it. So, Archimedes, for example, doubted whether the effect of the action of weights on a lever was proportional to the lengths of the lever-arms, but he accepted without hesitation the fact of their influence in some way. Daniel Bernoulli does not question the influence of the direction of a force generally, but only the form of its influence. As a matter of fact, it is far easier to observe that a circumstance has influence in a given case, than to determine /what influence it has. In the latter inquiry we are in much greater degree liable to error. The attitude of the investigators is therefore perfectly natural and defensible. The natural The proof of the correctness of a new rule can be methods of proof. attained by the repeated application of it, the frequent comparison of it with experience, the putting of it to the test under the most diverse circumstances. This process would, in the natural course of events, get carried out in time. The discoverer, however, hastens to reach his goal more quickly. He compares the results that flow from his rule with all the experiences with which he is familiar, with all older rules, repeatedly tested in times gone by, and watches to see if he do not light on contradictions. In this'procedure, the greatest credit is, as it should be, conceded to the oldest and most familiar experiences, the most thoroughly tested rules. Our instinctive experiences, those generalisations that are made involuntarily, by the irresistible force of the innumerable facts that press in upon THE PRINCIPLES OF STA TICS. us, enjoy a peculiar authority; and this is perfectly warranted by the consideration that it is precisely the elimination of subjective caprice and of individual error that is the object aimed at. In this manner Archimedes proves his law of the illustration of the prelever, Stevinus his law of inclined pressure, Daniel ceding remarks. Bernoulli the parallelogram of forces, Lagrange the principle of virtual displacements. Galileo alone is perfectly aware, with respect to the last-mentioned principle, that his new observation and perception are of equal rank with every former one-that it is derived from the same source of experience. He attempts no demonstration. Archimedes, in his proof of the principle of the lever, uses facts concerning the centre of gravity, which he had probably proved by means of the very principle now in question; yet we may suppose that these facts were otherwise so familiar, as to be unquestioned,-so familiar indeed, that it may be doubted whether he remarked that he had employed them in demonstrating the principle of the lever. The instinctive elements embraced in the views of Archimedes and Stevinus have been discussed at length in the proper place. 4. It is quite in order, on the making of a new dis- The position that adcovery, to resort to all proper means to bring the new vanced science should rule to the test. When, however, after the lapse of a occupy. reasonable period of time, it has been sufficiently often subjected to direct testing, it becomes science to recognise that any other proof than that has become quite needless; that there is no sense in considering a rule as the better established for being founded on others that have been reached by the very same method of observation, only earlier; that one well-considered and tested observation is as good as another. To-day, we 82 T7HE SCIENCE OF AIE CHANICS. should regard the principles of the lever, of statical moments, of the inclined plane, of virtual displacements, and of the parallelogram of forces as discovered by equivalent observations. It is of no importance now, that some of these discoveries were made directly, while others were reached by roundabout ways and as dependent upon other observations. It is more in keeping, furthermore, with the economy of thought and with Insight bet- the aesthetics of science, directly to recognise a principle ter than artificialdem- (say that of the statical moments) as the key to the unonstration. derstanding of all the facts of a department, and really see how it pervades all those facts, rather than to hold ourselves obliged first to make a clumsy and lame deduction of it from unobvious propositions that involve the same principle but that happen to have become earlier familiar to us. This process science and the individual (in historical study) may go through once for all. But having done so both are free to adopt a more convenient point of view. The mis- 5. In fact, this mania for demonstration in science take of the mania for results in a rigor that is false and mistaken. Some prodemonstration.s- positions are held to be possessed of more certainty than others and even regarded as their necessary and incontestable foundation; whereas actually no higher, or perhaps not even so high, a degree of certainty attaches to them. Even the rendering clear of the degree of certainty which exact science aims at, is not attained here. Examples of such mistaken rigor are to be found in almost every text-book. The deductions of Archimedes, not considering their historical value, are infected with this erroneous rigor. But the most conspicuous example of all is furnished by Daniel Bernoulli's deduction of the parallelogram of forces (Comment. Acad. Petrop. T. I.). THE PRINCIPLES OF STATICS. 83 6. As already seen, instinctive knowledge enjoys The characterof inour exceptional confidence. No longer knowing how stinctive knowledge. we have acquired it, we cannot criticise the logic by which it was inferred. We have personally contributed nothing to its production. It confronts us with a force and irresistibleness foreign to the products of voluntary reflective experience. It appears to us as something free from subjectivity, and extraneous to us, although we have it constantly at hand so that it is more ours than are the individual facts of nature. All this has often led men to attribute knowledge of Its authority not absothis kind to an entirely different source, namely, to view lutely supreme. it as existing a priori in us (previous to all experience). pre That this opinion is untenable was fully explained in our discussion of the achievements of Stevinus. Yet even the authority of instinctive knowledge, however important it may be for actual processes of development, must ultimately give place to that of a clearly and deliberately observed principle. Instinctive knowledge is, after all, only experimental knowledge, and as such is liable, we have seen, to prove itself utterly insufficient and powerless, when some new region of experience is suddenly opened up. 7. The true relation and connection of the different The truerelation of the principles is the /istorical one. The one extends farther principles an historiin this domain, the other farther in that. Notwith- cal one. standing that some one principle, say the principle of virtual displacements, may control with facility a greater number of cases than other principles, still no assurance can be given that it will always maintain its supremacy and will not be outstripped by some new principle. All principles single out, more or less arbitrarily, now this aspect now that aspect of the same facts, and contain an abstract summarised rule for the 84 TIE SCIENCE OF MECHANICS. refigurement of the facts in thought. We can never assert that this process has been definitively completed. Whosoever holds to this opinion, will not stand in the way of the advancement of science. Conception 8. Let us, in conclusion, direct our attention for a of force in statics, moment to the conception of force in statics. Force is any circumstance of which the consequence is motion. Several circumstances of this kind, however, each single one of which determines motion, may be so conjoined that in the result there shall be no motion. Now statics investigates what this mode of conjunction, in general terms, is. Statics does not further concern itself about the particular character of the motion conditioned by the forces. The circumstances determinative of motion that are best known to us, are our own voThe origin litional acts-our innervations. In the motions which of the notion of we ourselves determine, as well as in those to which pressure e are forced by external circumstances, we are always sensible of a pressure. Thence arises our habit of representing all circumstances determinative of motion as something akin to volitional acts-as pressures. The attempts we make to set aside this conception, as subjective, animistic, and unscientific, fail invariably. It cannot profit us, surely, to do violence to our own natural-born thoughts and to doom ourselves, in that regard, to voluntary mental penury. We shall subsequently have occasion to observe, that the conception referred to also plays a part in the foundation of dynamics. We are able, in a great many cases, to replace the circumstances determinative of motion, which occur in nature, by our innervations, and thus to reach the idea of a gradation of the intensity of forces. But in the estimation of this intensity we are thrown entirely on the THE PRINCIPLES OF STATICS. 85 resources of our memory, and are also unable to com- The common charmunicate our sensations. Since it is possible, how- acterof all forces. ever, to represent every condition that determines motion by a weight, we arrive at the perception that all circumstances determinative of motion (all forces) are alike in character and may be replaced and measured by quantities that stand for weight. The measurable weight serves us, as a certain, convenient, and communicable index, in mechanical researches, just as the thermometer in thermal researches is an exacter substitute for our perceptions of heat. As has pre- The idea of motion an viously been remarked, statics cannot wholly rid itself auxiliary conceptin of all knowledge of phenomena of motion. This par- statics. ticularly appears in the determination of the direction of a force by the direction of the motion which it would produce if it acted alone. By the point of application of a force we mean that point of a body whose motion is still determined by the force when the point is freed from its connections with the other parts of the body. Force accordingly is any circumstance that de- The general attritermines motion; and its attributes may be stated as butesof follows. The direction of the force is the direction of motion which is determined by that force, alone. The point of application is that point whose motion is determined independently of its connecti9ns with the system. The magnitude of the force is that weight which, acting (say, on a string) in the direction determined, and applied at the point in question, determines the same motion or maintains the same equilibrium. The other circumstances that modify the determination of a motion, but by themselves alone are unable to produce it, such as virtual displacements, the arms of levers, and so forth, may be termed collateral conditions determinative of motion and equilibrium. 85 THE SCIENCE OF AIfELCHANICS. VI. THE PRINCIPLES OF STATICS IN THEIR APPLICATION TO FLUIDS. No essen- i. The consideration of fluids has not supplied stattially new points of iCS with many essentially new points of view, yet nuview involved in merous applications and confirmations of the principles this suject already known have resulted therefrom, and physical experience has been greatly enriched by the investigations of this domain. We shall devote, therefore, a few pages to this subject. 2. To ARCHIMEDES also belongs the honor of founding the domain of the statics of liquids. To him we owe the well-known proposition concerning the buoyancy, or loss of weight, of bodies immersed in liquids, of the discovery of which Vitruvius, De Arc/hiectura, Lib. IX, gives the following account: Vitruvius's " Though Archimedes discovered many curious account of Archime- "matters that evince great intelligence, that which I am des's discovery. "about to mention is the most extraordinary. Hiero, "when he obtained the regal power in Syracuse, hav"ing, on the fortunate turn of his affairs, decreed a "votive crown of gold to be placed in a certain temple "to the immortal gods, commanded it to be made of "great value, and assigned for this purpose an appro"priate weight of the metal to the manufacturer. The "latter, in due time, presented the work to the king, "beautifully wrought; and the weight appeared to cor"respond with that of the gold which had been as"signed for it. "But a report having been circulated, that some of ':the gold had been abstracted, and that the deficiency THE PRINrCIPLES OF STA TICS. 87 "thus caused had been supplied by silver, Hiero was The account of Vi"indignant at the fraud, and, unacquainted with thetruvius. "method by which the theft might be detected, re"quested Archimedes would undertake to give it his "attention. Charged with this commission, he by "chance went to a bath, and on jumping into the tub, "perceived that, just in the proportion that his body "became immersed, in the same proportion the water "ran out of the vessel. Whence, catching at the "method to be adopted for the solution of the proposi"tion, he immediately followed it up, leapt out of the "vessel in joy, and returning home naked, cried out "with a loud voice that he had found that of which he "was in search, for he continued exclaiming, in Greek, "'fp7l(a Evp7lpja, (I have found it, I have found it!)" 3. The observation which led Archimedes to his Statement of the Arproposition, was accordingly this, that a body im- chimedean proposition mersed in water must raise an equivalent quantity ofprsiti water; exactly as if the body lay on one pan of a balance and the water on the other. This conception, which at the present day is still the most natural and the most direct, also appears in Archimedes's treatises On Floating Bodies, which unfortunately have not been completely preserved but have in part been restored by F. Commandinus. The assumption from which Archimedes starts reads thus: "It is assumed as the essential property of a liquid The Archi-. medean asthat in all uniform and continuous positions of its parts sumption. the portion that suffers the lesser pressure is forced upwards by that which suffers the greater pressure. But each part of the liquid suffers pressure from the portion perpendicularly above it if the latter be sinking or suffer pressure from another portion." 88 TIHE SCIENCE OF MECANAICS. Analysis of Archimedes now, to present the matter briefly, the principle. conceives the entire spherical earth as fluid in constitution, and cuts out of it pyramids the vertices of which lie at the centre (Fig. 59). All these pyramids a must, in the case of equilibrium, have the same weight, and the similarly situated b parts of the same must all suffer the same pressure. If we plunge a body a of the same specific gravity as water into one of the pyramids, the body will comFig. 59. pletely submerge, and, in the case of equilibrium, will supply by its weight the pressure of the displaced water. The body b, of less specific gravity, can sink, without disturbance of equilibrium, only to the point at which the water beneath it suffers the same pressure from the weight of the body as it would if the body were taken out and the submerged portion replaced by water. The body c, of a greater specific gravity, sinks as deep as it possibly can. That its weight is lessened in the water by an amount equal to the weight of the water displaced, will be manifest if we imagine the body joined to another of less specific gravity so that a third body is formed having the same specific gravity as water, which just completely submerges. rhe state of 4. When in the sixteenth century the study of the the science In the six- works of Archimedes was again taken up, scarcely the teenth century. principles of his researches were understood. The complete comprehension of his deductions was at that time impossible. STEVINUS rediscovered by a method of his own the THE PRLVCIPLES OF STATICS. 89 most important principles of hydrostatics and the de- The discoveries of Steductions therefrom. It was principally two ideas from vinus. which Stevinus derived his fruitful conclusions. The one is quite similar to that relating to the endless chain. The other consists in the assumption that the solidification of a fluid in equilibrium does not disturb its equilibrium. Stevinus first lays down this principle. Any given The first fundamenmass of water A (Fig. 60), immersed in water, is in talprinciple. equilibrium in all its parts. If A p were not supported by the surrounding water but should, let us 74.. say, descend, then the portion of water taking the place of A and placed thus in the same circumstances, would, on the same assumption, also have to descend. Fig. 6o. This assumption leads, therefore, to the establishment of a perpetual motion, which is contrary to our experience and to our instinctive knowledge of things. Water immersed in water loses accordingly itS The second fundamenwhole weight. If, now, we imagine the surface of the talprincisubmerged water solidified, the vessel formed by thisple. surface, the vzas szuprc;iciarium as Stevinus calls it, will still be subjected to the same circumstances of pressure. If emlty, the vessel so formed will suffer an upward pressure in the liquid equal to the weight of the water displaced. If we fill the solidified surface with some other substance of any specific gravity we may choose, it will be plain that the diminution of the weight of the body will be equal to the weight of the fluid displaced on immersion. In a rectangular, vertically placed parallelepipedal vessel filled with a liquid, the pressure on the horizontal 90o THE SCIENCE OF MECIL4NICS. Stevinus's base is equal to the weight of the liquid. The pressure deductions. ctions.s equal, also, for all parts of the bottom of the same area. When now Stevinus imagines portions of the liquid to be cut out and replaced by rigid immersed bodies of the same specific gravity, or, what is the same thing, imagines parts of the liquid to become solidified, the relations of pressure in the vessel will not be altered by the procedure. But we easily obtain in this way a clear view of the law that the pressure on the base of a vessel is independent of its form, as well as of the laws of pressure in communicating vessels, and so forth. Galileo, in 5. GALILEO treats the equilibrium of liquids in comthe treatment of this municating vessels and the problems connected theresubject, employs the with by the help of the principle of virtual displaceprinciple of virtual dis- ments. ANN (Fig. 61) being the placements A A common level of a liquid in equilibrium in two communicating vessels, SGalileo explains the equilibrium S here presented by observing that in the case of any disturbance the displacements of the columns are to each other in the inverse proportion Fig. 61. of the areas of the transverse sections and of the weights of the columns-that is, as with machines in equilibrium. But this is not quite correct. The case does not exactly correspond to the cases of equilibrium investigated by Galileo in machines, which present indifferent equilibrium. With liquids in communicating tubes every disturbance of the common level of the liquids produces an elevation of the centre of gravity. In the case represented in Fig. 61, the centre of gravity S of the liquid displaced from the shaded space in A is elevated to S', and we may THE PRINCIPLES OF STATICS. 9I regard the rest of the liquid as not having been moved. Accordingly, in the case of equilibrium, the centre of gravity of the liquid lies at its lowest possible point. 6. PASCAL likewise employs the principle of virtual The same principle displacements, but in a more correct manner, leaving made useof Sby Pascal. the weight of the liquid out of account and considering only the pressure at the surface. If we imagine two communicating vessels to be closed by pistons (Fig. 62), and these pistons loaded with weights proportional to their surfaceareas, equilibrium will obtain, because in consequence of the invariability of the volume of the liquid the displacements in every disturbance are in- - versely proportional to the weights. Fig. 62. For Pascal, accordingly, it follows, as a necessary consequence, from the principle of virtual displacements, that in the case of equilibrium every pressure on a superficial portion of a liquid is propagated with undiminished effect to every other superficial portion, however and in whatever position it be placed. No objection is to be made to discovering the principle in this way. Yet we shall see later on that the more natural and satisfactory conception is to regard the principle as immediately given. 7. We shall now, after this historical sketch, again Detailed consideraexamine the most important cases of liquid equilibrium, tion of the subject. and from such different points of view as may be convenient. The fundamental property of liquids given us by experience consists in the flexure of their parts on the slightest application of pressure. Let us picture to ourselves an element of volume of a liquid, the gravity of which we disregard-say a tiny cube. If the slightest TIHE SCIENCE OF MECHANICS. 92 The funda- excess of pressure be exerted on one of the surfaces of mental property of this cube, (which we now conceive, for the moment, liquids thec mobility of as a fixed geometrical locus, containing the fluid but their parts. not of its substance) the liquid (supposed to have previously been in equilibrium and at rest) will yield and pass out in all directions through the other five surfaces of the cube. A solid cube can stand a pressure on its upper and lower surfaces different in magnitude from that on its lateral surfaces; or vice versa. A fluid cube, on the other hand, can retain its shape only if the same perpendicular pressure be exerted on all its sides. A similar train of reasoning is applicable to all polyhedrons. In this conception, as thus geometrically elucidated, is contained nothing but the crude experience that the particles of a liquid yield to the slightest pressure, and that they retain this property also in the interior of the liquid when under a high pressure; it being observable, for example, that under the conditions cited minute heavy bodies sink in fluids, and so on. A second With the mobility of their parts liquids combine he com- still another property, which we will now consider. Lipressibility of theirvol- quids suffer through pressure a diminution of volume ume. which is proportional to the pressure exerted on unit of surface. Every alteration of pressure carries along with it a proportional alteration of volume and density. If the pressure diminish, the volume becomes greater, the density less. The volume of a liquid continues to diminish therefore on the pressure being increased, till the point is reached at which the elasticity generated within it equilibrates the increase of the pressure. 8. The earlier inquirers, as for instance those of the Florentine Academy, were of the opinion that liquids were incompressible. In 1761, however, JOHN CANTON performed an experiment by which the compressibility THE PRINCIPLES OF STA TICS. 93 of water was demonstrated. A thermometer glass is The first demonstrafilled with water, boiled, and then sealed. (Fig. 63.) tionofthe compressiThe liquid reaches to a. But since the space above a is bility of liquids. airless, the liquid supports no atmospheric pressure. If the sealed end be broken off, the liquid will sink to b. Only a portion, however, of this displacement is to be placed to the credit of the Cb compression of the liquid by atmospheric pressure. For if we place the glass before breaking off the top under an air-pump and exhaust the chamber, the liquid will sink to c. This last phe- Fig. 63. nomenon is due to the fact that the pressure that bears down on the exterior of the glass and diminishes its capacity, is now removed. On breaking off the top, this exterior pressure of the atmosphere is compensated for by the interior pressure then introduced, and an enlargement of the capacity of the glass again sets in. The portion cb, therefore, answers to the actual compression of the liquid by the pressure of the atmosphere. The first to institute exact experiments on the con- The experiments of pressibility of water, was OERSTED, who employed to Oersted on this subject. this end a very ingenious method. A thermometer glass A (Fig. 64) is filled with boiled water and is inverted, with B open mouth, into a vessel of mercury. Near it stands a manometer tube B filled with air and likewise inverted with open mouth in the mercury. The whole apparatus is then placed in a vessel filled with water, which is compressed by the aid of a pump. By this means the water Fig. 64. in A is also compressed, and the filament of quicksilver which rises in the capillary tube of the thermometer 94 THEL SCIENCE OF MEOCHANICS. glass indicates this compression. The alteration of capacity which the glass A suffers in the present instance, is merely that arising from the pressing together of its walls by forces which are equal on all sides. The experi- The most delicate experiments on this subject have ments of Grassi. been conducted by GRASSI with an apparatus constructed by Regnault, and computed with the assistance of Lame's correction-formulae. To give a tangible idea of the compressibility of water, we will remark that Grassi observed for boiled water at 0~ under an increase of one atmospheric pressure a diminution of the original volume amounting to 5 in 1oo,ooo parts. If we imagine, accordingly, the vessel A to have the capacity of one litre (Iooo ccm.), and affix to it a capillary tube of i sq. mm. cross-section, the quicksilver filament will ascend in it 5 cm. under a pressure of one atmosphere. Surface- 9. Surface-pressure, accordingly, induces a physical pressure induces in alteration in a liquid (an alteration in density), which liquids an alteration can be detected by sufficiently delicate means-even of density. optical. We are always at liberty to think that portions of a liquid under a higher pressure are more dense, though it may be very slightly so, than parts under a less pressure. The impli- Let us imagine now, we have in a liquid (in the incations of this fact. terior of which no forces act and the gravity of which we accordingly neglect) two portions subjected to unequal pressures and contiguous to one another. The portion under the greater pressure, being denser, will expand, and press against the portion under the less pressure, until the forces of elasticity as lessened on the one side and increased on the other establish equilibrium at the bounding surface and both portions are equally compressed. THE PRINCIPLES OF STA TICS. 95 If we endeavor, now, quantitatively to elucidate our The statement of mental conception of these two facts, the easy mobility these implications. and the compressibility of the parts of a liquid, so that they will fit the most diverse classes of experience, we shall arrive at the following proposition: When equilibrium subsists in a liquid, in the interior of which no forces act and the gravity of which we neglect, the same equal pressure is exerted on each and every equal surface-element of that liquid however and wherever situated. The pressure, therefore, is the same at all points and is independent of direction. Special experiments in demonstration of this principle have, perhaps, never been instituted with the requisite degree of exactitude. But the proposition has by our experience of liquids been made very familiar, and readily explains it. o1. If a liquid be enclosed in a vessel (Fig. 65)Preliminary rewhich is supplied with a piston A, the cross-section marks to the discussof which is unit in area, and with a piston B which ion of Pascal's deducfor the time being is made station- tion. ary, and on the piston A a load p be placed, then the same pressure p, gravity neglected, will prevail throughout all the parts of the vessel. The piston will penetrate inward and the walls of the vessel will continue to be deformed till the point is reached Fig. 65. at which the elastic forces of the rigid and fluid bodies perfectly equilibrate one another. If then we imagine the piston B, which has the cross-section f, to be movable, a force f.p alone will keep it in equilibrium. Concerning Pascal's deduction of the proposition before discussed from the principle of virtual displacements, it is to be remarked that the conditions of dis 96 THE SCIENCE OF MECHANCICS. criticism of placement which he perceived hinge wholly upon the Pascal's deduction. fact of the ready mobility of the parts and on the equality of the pressure throughout every portion of the liquid. If it were possible for a greater compression to take place in one part of a liquid than in another, the ratio of the displacements would be disturbed and Pascal's deduction would no longer be admissible. That the property of the equality of the pressure is a property given in experience, is a fact that cannot be escaped; as we shall readily admit if we recall to mind that the same law that Pascal deduced for liquids also holds good for gases, where even approximately there can be no question of a constant volume. This latter fact does not afford any difficulty to our view; but to that of Pascal it does. In the case of the lever also, be it incidentally remarked, the ratios of the virtual displacements are assured by the elastic forces of the lever-body, which do not permit of any great deviation from these relations. The behav- i. We shall now consider the action of liquids uniour of liquidsunder der the influence of gravity. The upper surface of a the action of gravity. liquid in equilibrium is horizontal, S' NS^AN(Fig. 66). This fact is at once rendered intelligible when we reflect that every alteration of the sur- face in question elevates the centre of gravity of the liquid, and pushes Fig. 66. the liquid mass resting in the shaded space beneath NN and having the centre of gravity S into the shaded space above NN having the centre of gravity S'. Which alteration, of course, is at once reversed by gravity. Let there be in equilibrium in a vessel a heavy liquid with a horizontal upper surface. We consider THE PRINCIPLES OF STA4 TICS. 97 (Fig. 67) a small rectangular parallelepipedon in the The con-. ditions of interior. The area of its horizontal base, we will say, is equilibrium in liquids a, and the length of its vertical edges dh. The weight subjected to the a'of this parallelepipedon is therefore a dh s, where s is tion of gravity. its specific gravity. If the paral-ty. lelepipedon do not sink, this is possible only on the condition that a greater pressure is exerted on the pdp lower surface by the fluid than on the upper. The pressures on the upper and lower surfaces we will Fig. 67. respectively designate as ap and a (p + dp). Equilibrium obtains when a dh.s-- adp or df/dh i s, where h in the downward direction is reckoned as positive. We see from this that for equal increments of h vertically downwards the pressure p must, correspondingly, also receive equal increments. So that p hs - q; and if q, the pressure at the upper surface, which is usually the pressure of the atmosphere, becomes - 0, we have, more simply, p -- s, that is, the pressure is proportional to the depth beneath the surface. If we imagine the liquid to be pouring into a vessel, and this condition of affairs i;- vet attained, every liquid particle will then sink until the compressed particle beneath balances by the elasticity developed in it the weight of the particle above. From the view we have here presented it will be fur- Different force-relather apparent, that the increase of pressure in a liquid tions exist only in the takes place solely in the direction in which gravity line of the action of acts. Only at the lower surface, at the base, of the gravity. parallelepipedon, is an excess of elastic pressure on the part of the liquid beneath required to balance the weight of the parallelepipedon. Along the two sides of the vertical containing surfaces of the parallelepipedon, 98 THE SCIENCE OFi MECHANICS. the liquid is in a state of equal compression, since no force acts in the vertical containing surfaces that would determine a greater compression on the one side than on the other. Level sur- If we picture to ourselves the totality of all the faces. S points of the liquid at which the same pressure p acts, we shall obtain a surface-a so-called level surface. If we displace a particle in the direction of the action of gravity, it undergoes a change of pressure. If we displace it at right angles to the direction of the action of gravity, no alteration of pressure takes place. In the latter case it remains on the same level surface, and the element of the level surface, accordingly, stands at right angles to the direction of the force of gravity. Imagining the earth to be fluid and spherical, the level surfaces are concentric spheres, and the directions of the forces of gravity (the radii) stand at right angles to the elements of the spherical surfaces. Similar observations are admissible if the liquid particles be acted on by other forces than gravity, magnetic forces, for example. Their func- The level surfaces afford, in a certain sense, a diation in thought. gram of the force-relations to which a fluid is subjected; a view further elaborated by analytical hydrostatics. 12. The increase of the pressure with the depth below the surface of a heavy liquid may be illustrated by a series of experiments which we chiefly owe to Pascal. These experiments also well illustrate the fact, that the pressure is independent of the direction. In Fig. 68, i, is an empty glass tube g ground off at the bottom and closed by a metal disc pp, to which a string is attached, and the whole plunged into a vessel of water. When immersed to a sufficient depth we may let the string go, without the metal disc, which is THE PRINCIPLES OF STA TICS. 99 supported by the pressure of the liquid, falling. In 2, Pascal's experinments the metal disc is replaced by a tiny column of mer- on the pressure of cury. If (3) we dip an open siphon tube filled with liquids. mercury into the water, we shall see the mercury, in consequence;I ': i of the pressure at a, rise into so the longer arm. In 4, we see a tube, at the lower extremity of which a leather bag filled with i -l mercury is tied: continued im- i i mersion forces the mercury I i.. higher and higher into the tube. In 5, a piece of wood h is driven by the pressure of the water into I I the small arm of an empty siphon i l tube. In 6, a piece of woodH I immersed in mercury adheres to the bottom of the vessel, and is; pressed firmly against it for as 'I long a time as the mercury is I kept from working its way underneath it. 13. Once we have made quite ' ' The presOc ' qiesure at the clear to ourselves that the pres- I I ase of a vessel indesure in the interior of a heavy' pendent of. its form. liquid increases proportionally to its the depth below the surface, the law that the pressure at the base of a vessel is independent of its i-" form will be readily perceived. The pressure increases as we descend at an equal rate, whether the vessel (Fig. 69) has the form abcad or ebcf. In both cases the walls of the vessel where they meet the liquid, go on deforming Ioo THE SCIENCE OF MECHANICS. till the point is reached at which they equilibrate by the elasticity developed in them the pressure exerted by the fluid, that is, take the place as regards pressure of the fluid adjoining. This fact is S/a direct justification of Stevinus's fiction of the solidified fluid supplying the place of the walls of the vessel. \-c The pressure on the base Fig. 69. always remains P-- A s,.where A denotes the area of the base, h the depth of the horizontal plane base below the level, and s the specific gravity of the liquid. Elucida- The fact that, the walls of the vessel being negtion of this fact. lected, the vessels I, 2, 3 of Fig. 70 of equal basearea and equal pressure-height weigh differently in the / 2 3 balance, of course _in no wise contradicts the laws of pressure men1 1 tioned. If we take Fig. 70. into account the lateral pressure, we shall see that in the case of I we have left an extra component downwards, and in the case of 3 an extra component upwards, so that on the whole the resultant superficial pressure is always equal to the weight. The princi- 14. The principle of virtual displacements is adple of virtual disr- mirably adapted to the acquisition of clearness and placements applied to comprehensiveness in cases of this character, and we the consideration of shall accordingly make use of it. To begin with, howproblems of this class, ever, let the following be noted. If the weight q (Fig. 71) descend from position I to position 2, and a weight of exactly the same size move at the same time from THE PRINCIPLES OF STA TICS. 101 2 to 3, the work performed in this operation is q h1 Preliminary reqh2 q= (hI + /12), the same, that is, as if the weightmarks. q passed directly from I to 3 and the weight at 2 remained in its original position. The observation is easily generalised. I 4 3 Fig. 71. Fig. 72. Let us consider a heavy homogeneous rectangular parallelepipedon, with vertical edges of the length i, base A, and the specific gravity s (Fig. 72). Let this parallelepipedon (or, what is the same thing, its centre of gravity) descend a distance dh. The work done is then A hs.dh, or, also, A dhs./h. In the first expression we conceive the whole weight A h s displaced the vertical distance dh; in the second we conceive the weight Adhs as having descended from the upper shaded space to the lower shaded space the distance h, and leave out of account the rest of the body. Both methods of conception are admissible and equivalent. 15. With the aid of Pascal's. -paradox. this observation we shall paradox. obtain a clear insight into Fig. 73. the paradox of Pascal, which consists of the following. The vessel g (Fig. 73), fixed to a separate support and consisting of a narrow upper and a very broad lower cylinder, is closed at the bottom by a movable piston, 102 THE SCIENCE OF AMECHIIANVICS. which, by means of a string passing through the axis of the cylinders, is independently suspended from the extremity of one arm of a balance. If g be filled with water, then, despite the smallness of the quantity of water used, there will have to be placed on the other scale-pan, to balance it, several weights of considerable size, the sum of which will be A/is, where A is the piston-area, h the height of the liquid, and s its specific gravity. But if the liquid be frozen and the mass loosened from the walls of the vessel, a very small weight will be sufficient to preserve equilibrium. The expla- Let us look to the virtual displacements of the two nation of theparadox cases (Fig. 74). In the first case, supposing the piston to be lifted a distance dh, the virtual moment is Adhs./ or Ahs.d/. It thus comes to the same thing, dk whether we consider the mass / 2 that the motion of the piston h displaces to be lifted to the dk......upper surface of the fluid Fig. 74. through the entire pressureheight, or consider the entire weight A/is lifted the distance of the piston-displacement d/. In the second case, the mass that the piston displaces is not lifted to the. upper surface of the fluid, but suffers a displacement which is much smaller-the displacement, namely, of the piston. If A, a are the sectional areas respectively of the greater and the less cylinder, and k and I their respective heights, then the virtual moment of the present case is Adhs.k + ad/s. / (Ak + a!) s.d/; which is equivalent to the lifting of a much smaller weight (Ak + a ) s, the distance dh. 16. The laws relating to the lateral pressure of liquids are but slight modifications of the laws of basal THE PRIACIPLES OF STA TICS. 103 pressure. If we have, for example, a cubical vessel The laws of lateral of i decimetre on the side, which is a vessel of litre pressure. capacity, the pressure on any one of the vertical lateral walls ABCD, when the vessel is filled with water, is easily determinable. The deeper the migratory element considered descends beneath the surface, the greater the pressure will be to which it is subjected. We easily perceive, thus, that the pressure on a lateral wall is represented by a wedge of water ABCDHI resting upon the wall horizontally A E placed, where ID is at B F right angles to BD and ID HC- A C. The lateral pressure accor- H L dingly is equal to half D G a kilogramme. To determine the i 7 point of application of the resultant pressure, conceive ABCD again horizontal with the water-wedge resting upon it. We cut off AK= BL -= 2AC, draw the straight line KL and bisect it at M; AM is the point of application sought, for through this point the vertical line cutting the centre of gravity of the wedge passes. A plane inclined figure forming the base of a vessel The presPsure on a filled with a liquid, is divided into the elements a, a, plane inclined base. "... with the depths,h, h', h"... below the level ofinedbase the liquid. The pressure on the base is (a hL + a' h' + a" h" +...) s. If we call the total base-area A, and the depth of its centre of gravity below the surface H, then ah e+ tahe 4+ arr'h" +..._ o + t' + s... H. a + a' + ' +... A a whence the pressure on the base is AHis. 104 THE SCIENCE OF MECHANICS. The deduc- 17. The principle of Archimedes can be deduced in tion of the principle of various ways. After the manner of Stevinus, let us Archime- 1 des maybe conceive in the interior of the liquid a portion of it effected in various solidified. This portion now, as before, will be supways. ported by the circumnatant liquid. The resultant of the forces of pressure acting on the surfaces is accordingly applied at the centre of gravity of the liquid displaced by the solidified body, and is equal and opposite to its weight. If now we put in the place of the solidified liquid another different body of the same form, but of a different specific gravity, the forces of pressure at the surfaces will remain the same. Accordingly, there now act on the body two forces, the weight of the body, applied at the centre of gravity of the body, and the upward buoyancy, the resultant of the surface-pressures, applied at the centre of gravity of the displaced liquid. The two centres of gravity in question coincide only in the case of homogeneous solid bodies. One meth- If we immerse a rectangular parallelepipedon of alod. titude / and base (, with edges vertically placed, in a liquid of specific gravity s, then the pressure on the upper basal surface, when at a depth k below the level of the liquid is aks, while the pressure on the lower surface is a (k -- ) s. As the lateral pressures destroy each other, an excess of pressure a1/s upwards remains; or, where v denotes the volume of the parallelepipedon, an excess v. s. Another We shall approach nearest the fundamental conmethod involving the ception from which Archimedes started, by recourse to princiole of virtial dis- the principle of virtual displacements. Let a paralplacements.1 lelepipedon (Fig. 76) of the specific gravity o, base a, and height sink the distance dh. The virtual moment of the transference from the upper into the lower shaded space of the figure will be a d h. h/. But while THfLE PRIILCI'LES OF ST74 TICS. I05 this is done, the liquid rises from the lower into the upper space, and its moment is adhsi. The total virtual moment is therefore ah ( - s) dh - (p - q) dh, where p denotes the weight of the body and q the weight of the displaced liquid. B htFig. 76. Fig. 77. 18. The question might occur to us, whether the Isthe buoyancy of a upward pressure of a body in a liquid is affected by the body in a liquid afimmersion of the latter in another liquid. As a fact, fected by the if nerthis very question has been proposed. Let therefore sion ot that liquid in a (Fig. 77) a body K be submerged in a liquid A and the second liquid with the containing vessel in turn submerged in another liquid B. If in the determination of the loss of weight in A it were proper to take account of the loss of weight of A in B, then K's loss of weight would necessarily vanish when the fluid B became identical with A. Therefore, K immersed in A would suffer a loss of weight and it would suffer none. Such a rule would be nonsensical. With the aid of the principle of virtual displace- The elucidation of ments, we easily comprehend the more complicated morecomplicated cases of this character. If a body be first gradually casesofthis class. immersed in B, then partly in B and partly in A, finally in A wholly; then, in the second case, considering the virtual moments, both liquids are to be taken into account in the proportion of the volume of the body immersed in them. But as soon as the body is wholly immersed in A, the level of A on further dis io6 THE SCIENCE OF MECHANICS. placement no longer rises, and therefore B is no longer of consequence. The Archi- 19. Archimedes's principle may be illustrated by a medean principle il- pretty experiment. From the one extremity of a scalelustrated by an experi- beam (Fig. 78) we hang a hollow cube H, and beneath ment.it a solid cube AM, which exactly fits into the first cube. We put weights into the opposite pan, until the scales are in equilibrium. If now M be submerged in water by lifting a vessel which stands H beneath it, the equilibrium will be disturbed; but it will be immediately restored if H, the hollow cube, be filled with water. The coun- / A counter-experiment is the followm1peri. ing. H is left suspended alone at the S one extremity of the balance, and into the opposite pan is placed a vessel of Fig. 78. water, above which on an independent support Mhangs by a thin wire. The scales are brought to equilibrium. If now Al be lowered until it is immersed in the water, the equilibrium of the scales will be disturbed; but on filling H with water, it will be restored. Remarks on At first glance this experiment appears a little parathe experiment.pe doxical. We feel, however, instinctively, that AM cannot be immersed in the water without exerting a pressure that affects the scales. When we reflect, that the level of the water in the vessel rises, and that the solid body M equilibrates the surface-pressure of the water surrounding it, that is to say represents and takes the place of an equal volume of water, it will be found that the paradoxical character of the experiment vanishes. THE PRINCIPLES OF STA TICS. o07 20. The most important statical principles have The general princibeen reached in the investigation of solid bodies. This ples of statics might course is accidentally the historical one, but it is by no have been reached in means the only possible and necessary one. The dif- the investigation of ferent methods that Archimedes, Stevinus, Galileo, and fluid bodies the rest, pursued, place this idea clearly enough before the mind. As a matter of fact, general statical principles, might, with the assistance of some very simple propositions from the statics of rigid bodies, have been reached in the investigation of liquids. Stevinus certainly came very near such a discovery. We shall stop a moment to discuss the question. Let us imagine a liquid, the weight of which we neg- The discussion and lect. Let this liquid be enclosed in a vessel and sub- illustration of this jected to a definite pressure. A portion of the liquid, statement. let us suppose, solidifies. On the closed surface normal forces act proportional to the elements of the area, and we see without difficulty that their resultant will always be = 0. If we mark off by a closed curve a portion of the closed surface, we obtain, on either side of it, a nonclosed surface. All surfaces which are bounded by the same curve (of double curvature) and on which forces act normally (in the same sense) pro- \ portional to the elements of the area, have lines coincident in position for the resultants of these forces. Let us suppose, now, that a fluid / cylinder, determined by any closed /t plane curve as the perimeter of its Fig. 79. base, solidifies. We may neglect the two basal surfaces, perpendicular to the axis. And instead of the cylindrical surface the closed curve simply may be considered. From this method follow quite analogous Io8 THE SCIENCE OF AfECHAICS. The dis- propositions for normal forces proportional to the elecussion and illustration ments of a plane curve. of this statement. If the closed curve pass into a triangle, the consideration will shape itself thus. The resultant normal forces applied at the middle points of the sides of the triangle, we represent in direction, sense, and magnitude by straight lines (Fig. 8o). The +-.- lines mentioned intersect at a point the centre of the circle described about the triangle. It will further be noted, Fig. 80. that by the simple parallel displacement of the lines representing the forces a triangle is constructible which is similar and congruent to the original triangle. Thededuc- Thence follows this proposition: tion of the triangle of Any three forces, which, acting at a point, are proforces by this method portional and parallel in direction to the sides of a triangle, and which on meeting by parallel displacement form a congruent triangle, are in equilibrium. We see at once that this proposition is simply a different form of the principle of the parallelogram of forces. If instead of a triangle we imagine a polygon, we shall arrive at the familiar proposition of the polygon of forces. We conceive now in a heavy liquid of specific gravity m a portion solidified. On the element a of the closed encompassing surface there acts a normal force a 7 z, where z is the distance of the element from the level of the liquid. We know from the outset the result. Similar de- If normal forces which are determined by au z, duction of another im- where a denotes an element of area and z its perpenportant proposition. dicular distance from a given plane E, act on a closed surface inwards, the resultant will be V. x, in which expression V represents the enclosed volume. The THE PRTVICIPILES OF STA TICS. o109 resultant acts at the centre of gravity of the volume, is perpendicular to the plane mentioned, and is directed towards this plane. Under the same conditions let a rigid curved surface The proposition here be bounded by a plane curve, which encloses on the deduced, a special case plane the area A. The resultant of the forces acting of Green's Theorem. on the curved surface is R, where 2= (AZ j)2+ ( V) - AZVj2 cos y, in which expression Z denotes the distance of the centre of gravity of the surface A from E, and v the normal angle of E and A. In the proposition of the last paragraph mathematically practised readers will have recognised a particular case of Green's Theorem, which consists in the reduction of surface-integrations to volume-integrations or vice versa. We may, accordingly, see into the force-system of a The implications of fluid in equilibrium, or, if you please, see out of it, sys-the view tems of forces of greater or less complexity, and thusdiscussed. reach by a short path propositions a posteriori. It is a mere accident that Stevinus did not light on these propositions. The method here pursued corresponds exactly to his. In this manner new discoveries can still be made. 21. The paradoxical results that were reached in Fruitful results of the the investigation of liquids, supplied a stimulus to fur- investigations of this ther reflection and research. It should also not be left domain. unnoticed, that the conception of a physico-mechanicat continuum was first formed on the occasion of the investigation of liquids. A much freer and much more fruitful mathematical mode of view was developed thereby, than was possible through the study even of io THE SCIENCE OF MECHANICS. systems of several solid bodies. The origin, in fact, of important modern mechanical ideas, as for instance that of the potential, is traceable to this source. VII. THE PRINCIPLES OF STATICS IN THEIR APPLICATION TO GASEOUS BODIES. Character i. The same views that subserve the ends of science of this departmentofin the investigation of liquids are applicable with but inquiry. slight modifications to the investigation of gaseous bodies. To this extent, therefore, the investigation of gases does not afford mechanics any very rich returns. Nevertheless, the first steps that were taken in this province possess considerable significance from the point of view of the progress of civilisation and so have a high import for science generally. The elus- Although the ordinary man has abundant opporiveness of its subject- tunity, by his experience of the resistance of the air, by matter. the action of the wind, and the confinement of air in bladders, to perceive that air is of the nature of a body, yet this fact manifests itself infrequently, and never in the obvious and unmistakable way that it does in the case of solid bodies and fluids. It is known, to be sure, but is not sufficiently familiar to be prominent in popular thought. In ordinary life the presence of the air is scarcely ever thought of. (See p. 517.) The effect Although the ancients, as we may learn from the of the first disclosures 'accounts of Vitruvius, possessed instruments which, in this province. like the so-called hydraulic organs, were based on the condensation of air, although the invention of the airgun is traced back to Ctesibius, and this instrument was also known to Guericke, the notions which people held with regard to the nature of the air as late even THIE PRINLVCIPLES OF STA TICS. I OTTO De GUERICKE Serenifs.. Potentifs: Elector: Brandebl Confiliarius et Civitat: Magdeb.Confiu: II2 THE SCIENCE OF MECHANICS. as the seventeenth century were exceedingly curious and loose. We must not be surprised, therefore, at the intellectual commotion which the first more important experiments in this direction evoked. The enthusiastic description which Pascal gives of Boyle's air-pump experiments is readily comprehended, if we transport ourselves back into the epoch of these discoveries. What indeed could be more wonderful than the sudden discovery that a thing which we do not see, hardly feel, and take scarcely any notice of, constantly envelopes us on all sides, penetrates all things; that it is the most important condition of life, of combustion, and of gigantic mechanical phenomena. It was on this occasion, perhaps, first made manifest by a great and striking disclosure, that physical science is not restricted to the investigation of palpable and grossly sensible processes. The views 2. In Galileo's time philosophers explained the entertained on this sub-phenomenon of suction, the action of syringes and ject in Galileo's timie pumps by the so-called horror vacui-nature's abhorrence of a vacuum. Nature was thought to possess the power of preventing the formation of a vacuum by laying hold of the first adjacent thing, whatsoever it was, and immediately filling up with it any empty space that arose. Apart from the ungrounded speculative element which this view contains, it must be conceded, that to a certain extent it really represents the phenomenon. The person competent to enunciate it must actually have discerned some principle in the phenomenon. This principle, however, does not fit all cases. Galileo is said to have been greatly surprised at hearing of a newly constructed pump accidentally supplied with a very long suction-pipe which was not able to raise water to a height of more than eighteen Italian THE PRINCIPLES OF STATICS. 113 ells. His first thought was that the horror vacui (or the resistenza del vacuo) possessed a measurable power. The greatest height to which water could be raised by suction he called al/ezza limitatissima. He sought, moreover, to determine directly the weight able to draw out of a closed pump-barrel a tightly fitting piston resting on the bottom. 3. TORRICELLI hit upon the idea of measuring the Torricelli's experiment resistance to a vacuum by a column of mercury instead of a column of water, and he expected to obtain a column of about -1 of the length of the water column. His expectation was confirmed by the experiment performed in 1643 by Viviani in the well-known manner, and which bears to-day the name of the Torricellian experiment. A glass tube somewhat over a metre in length, sealed at one end and filled with mercury, is stopped at the open end with the finger, inverted in a dish of mercury, and placed in a vertical position. Removing the finger, the column of mercury falls and remains stationary at a height of about 76 cm. By this experiment it was rendered quite probable, that some very definite pressure forced the fluids into the vacuum. What pressure this was, Torricelli very soon divined. Galileo had endeavored, some time before this, to Galileo's determine the weight of the air, by first weighing a weigair glass bottle containing nothing but air and then again weighing the bottle after the air had been partly expelled by heat. It was known, accordingly, that the air was heavy. But to the majority of men the horror vacui and the weight of the air were very distantly connected notions. It is possible that in Torricelli's case the two ideas came into sufficient proximity to lead him to the conviction that all phenomena ascribed to the horror vacui were explicable in a simple and 114 THE SCIENCE OF MECHANICS. Atmospher- logical manner by the pressure exerted by the weight ic pressure discovered of a fluid column-a column of air. Torricelli discovby Torricelli. ered, therefore, the pressure of the atmosphere; he also first observed by means of his column of mercury the variations of the pressure of the atmosphere. 4. The news of Torricelli's experiment was circulated in France by Mersenne, and came to the knowledge of Pascal in the year 1644. The accounts of the theory of the experiment were presumably so imperfect that PASCAL founrd it necessary to reflect independently thereon. (Pesanteur de l'air. Paris, 1663.) Pascal's ex- He repeated the experiment with mercury and with periments. a tube of water, or rather of red wine, 40 feet in length. He soon convinced himself by inclining the tube that the space above the column of fluid was really empty; and he found himself obliged to defend this view against the violent attacks of his countrymen. Pascal pointed out an easy way of producing the vacuum which they regarded as impossible, by the use of a glass syringe, the nozzle of which was closed with the finger under water and the piston then drawn back without much difficulty. Pascal showed, in addition, that a curved siphon 40 feet high filled with water does not flow, but can be made to do so by a sufficient inclination to the perpendicular. The same experiment'was made on a smaller scale with mercury. The same siphon flows or does not flow according as it is placed in an inclined or a vertical position. In a later performance, Pascal refers expressly to the fact of the weight of the atmosphere and to the pressure due to this weight. He shows, that minute animals, like flies, are able, without injury to themselves, to stand a high pressure in fluids, provided only the pressure is equal on all sides; and he applies this THE PRINCIPLES OF STATICS. 115 at once to the case of fishes and of animals that live in The analogy between the air. Pascal's chief merit, indeed, is to have estab- liquid and atmospherlished a complete analogy between the phenomena con- ic pressure. ditioned by liquid pressure (water-pressure) and those conditioned by atmospheric pressure. 5. By a series of experiments Pascal shows that mercury in consequence of atmospheric pressure rises into a space containing no air in the same way that, in consequence of water-pressure, it rises into a space containing no water. If into a deep vessel filled with water (Fig. 81) a tube be sunk at the lower end of which a bag of mercury is tied, but so inserted that the upper end of the tube projects out of the water and thus contains only air, then the deeper the tube is sunk into the water the higher will the mercury, subjected Fig. 8I. to the constantly increasing pressure of the water, ascend into the tube. The experiment can also be made, with a siphon-tube, or with a tube open at its lower end. Undoubtedly it was the attentive consideration of The height of mounthis very phenomenon that led Pascal to the idea that tains deter-mined by the barometer-column must necessarily stand lower at the barorneter. the summit of a mountain than at its base, and that it could accordingly be employed to determine the height of mountains. He communicated this idea to his brother-in-law, Perier, who forthwith successfully performed the experiment on the summit of the Puy de Dome. (Sept. 19, 1648.) Pascal referred the phenomena connected with ad- Adhesion plates. hesion-plates to the pressure of the atmosphere, and gave as an illustration of the principle involved the resistance experienced when a large hat lying flat on a table is suddenly lifted. The cleaving of wood to the n6 THE SCIENCE OF MECILANICS. bottom of a vessel of quicksilver is a phenomenon of the same kind. A siphon Pascal imitated the flow produced in a siphon by which acts by water- atmospheric pressure, by the use of water-pressure. pressure. The two open unequal arms a and b of a three-armed tube a b c (Fig. - 82) are dipped into the vessels of mercury e and d. If the whole arrangement then be immersed in a deep vessel of water, yet so that the long open branch shall always __ -_ project above the upper surface, the mercury will gradually rise in Fig. 82. the branches a and b, the columns finally unite, and a stream begin to flow from the vessel d to the vessel e through the siphon-tube open above to the air. Pascals d The Torricellian experiment was modimodification of the fled by Pascal in a very ingenious manner. Torricellian experi- A tube of the form abed (Fig. 83), of ment. double the length of an ordinary barometer-tube, is filled with mercury. The openings a and b are closed with the fingers and the tube placed in a dish of b mercury with the end a downwards. If now a be opened, the mercury in cd will all fall into the expanded portion at c, and the mercury in ab will sink to the height of the ordinary barometer-column. A vacuum is produced at b which presses the finger closing the hole painfully inwards. Fig. 83. If b also be opened the column in a b will sink completely, while the mercury in the expanded portion c, being now exposed to the pressure of the THE PRINCIPLES OF STATICS. II7 atmosphere, will rise in cd to the height of the barometer-column. Without an air-pump it was hardly possible to combine the experiment and the counterexperiment in a simpler and more ingenious manner than Pascal thus did. 6. With regard to Pascal's mountain-experiment, Supplementary rewe shall add the following brief supplementary remarks. marks on Pascal's Let b be the height of the barometer at the level of mountainO experiment the sea, and let it fall, say, at an elevation of m metres, to kb0, where k is a proper fraction. At a further elevation of m metres, we must expect to obtain the barometer-height k. k bo, since we here pass through a stratum of air the density of which bears to that of the first the proportion of k: 1. If we pass upwards to the altitude h- n. m metres, the barometer-height corresponding thereto will be log b, - log bo b, = k". b, or n = or S ~log /k i ~ (log b, - log bo). log k The principle of the method is, we see, a very simple one; its difficulty arises solely from the multifarious collateral conditions and corrections that have to be looked to. 7. The most original and fruitful achievements in The experiments of the domain of aerostatics we owe to OTTO VON GUE-Otto von RICKE. His experiments appear to have been suggestedGuericke. in the main by philosophical speculations. He proceeded entirely in his own way; for he first heard of the Torricellian experiment from Valerianus Magnus at the Imperial Diet of Ratisbon in 1654, where he demonstrated the experimental discoveries made by him about 165o. This statement is confirmed by his method i8 THE SCIENCE OF MECHANICS. of constructing a water-barometer which was entirely different from that of Torricelli. The histori- Guericke's book (Experinenila 7ova, Ul1 vocantur, cal value of Guericke's Afagdebui-r'ica. Amsterdam. 1672) makes us realise book. the narrow views men took in his time. The fact that he was able gradually to abandon these views and to acquire broader ones by his individual endeavor speaks favorably for his intellectual powers. We perceive with astonishment how short a space of time separates us from the era of scientific barbarism, and can no longer marvel that the barbarism of the social order still so oppresses us. Its specula- In the introduction to this book and in various other tive character. places, Guericke, in the midst of his experimental investigations, speaks of the various objections to the Copernican system which had been drawn from the Bible, (objections which he seeks to invalidate,) and discusses such subjects as the locality of heaven, the locality of hell, and the day of judgment. Disquisitions on empty space occupy a considerable portion of the work. Guericke's Guericke regards the air as the exhalation or odor notion of the air. of bodies, which we do not perceive because we have been accustomed to it from childhood. Air, to him, is not an element. He knows that through the effects of heat and cold it changes its volume, and that it is compressible in Hero's Ball, or Pita Ifcronis; on the basis of his own experiments he gives its pressure at 20 ells of water, and expressly speaks of its weight, by which flames are forced upwards. 8. To produce a vacuum, Guericke first employed a wooden cask filled with water. The pump of a fireengine was fastened to its lower end. The water, it was thought, in following the piston and the action of THE PRIVCIPLES OF STA TICS. Sig bt - Guericke's First Experiments. (Exterinm. Magdeb.) 120 THE SCIENCE OF MECHANICS. His at- gravity, would fall and be pumped out. Guericke extempts to produce a pected that empty space would remain. The fastenings vacuum. of the pump repeatedly proved to be too weak, since in consequence of the atmospheric pressure that weighed on the piston considerable force had to be applied to move it. On strengthening the fastenings three powerful men finally accomplished the exhaustion. But, meantime the air poured in through the joints of the cask with a loud blast, and no vacuum was obtained. In a subsequent experiment the small cask from which the water was to be exhausted was immersed in a larger one, likewise filled with water. But in this case, too, the water gradually forced its way into the smaller cask. His final Wood having proved in this way to be an unsuitsuccess. able material for the purpose, and Guericke having remarked in the last experiment indications of success, the philosopher now took a large hollow sphere of copper and ventured to exhaust the air directly. At the start the exhaustion was successfully and easily conducted. But after a few strokes of the piston, the pumping became so difficult that four stalwart men (viri quadrati), putting forth their utmost efforts, could hardly budge the piston. And when the exhaustion had gone still further, the sphere suddenly collapsed, with a violent report. Finally by the aid of a copper vessel of perfect spherical form, the production of the vacuum was successfully accomplished. Guericke describes the great force with which the air rushed in on the opening of the cock. 9. After these experiments Guericke constructed an independent air-pump. A great glass globular receiver was mounted and closed by a large detachable tap in which was a stop-cock. Through this opening the objects to be subjected to experiment were placed THIE PRINCIPLES OF STATICS. I2I in the receiver. To secure more perfect closure the Guericke's w. air-pump. receiver was made to stand, with its stop-cock under water, on a tripod, beneath which the pump proper was Guericke's Air-pump. (Experim. Magdeb.) placed. Subsequently, separate receivers, connected with the exhausted sphere, were also employed in the experiments. 122 7THE1 SCIENCE OF MECIANICS. The curious The phenomena which Guericke observed with this phenomena observed by apparatus are manifold and various. The noise which means of the air- water in a vacuum makes on striking the sides of the tump. glass receiver, the violent rush of air and water into exhausted vessels suddenly opened, the escape on exhaustion of gases absorbed in liquids, the liberation of their fragrance, as Guericke expresses it, were immediately remarked. A lighted candle is extinguished on exhaustion, because, as Guericke conjectures, it derives its nourishment from the air. Combustion, as his striking remark is, is not an annihilation, but a transformation of the air. A bell does not ring in a vacuum. Birds die in it. Many fishes swell up, and finally burst. A grape is kept fresh in vacuo for over half a year. By connecting with an exhausted cylinder a long tube dipped in water, a water-barometer is constructed. The column raised is 19-20 ells high; and Von Guericke explained all the effects that had been ascribed to the horror 7vacui by the principle of atmospheric pressure. An important experiment consisted in the weighing of a receiver, first when filled with air and then when exhausted. The weight of the air was found to vary with the circumstances; namely, with the temperature and the height of the barometer. According to Guericke a definite ratio of weight between air and water does not exist. The experi- But the deepest impression on the contemporary ments relating to at- world was made by the experiments relating to atmosmospheric pressure, pheric pressure. An exhausted sphere formed of two hemispheres tightly adjusted to one another was rent asunder with a violent report only by the traction of sixteen horses. The same sphere was suspended from THE PRIVCIPLLES OF S TA TICS. 123 a beam, and a heavily laden scale-pan was attached to the lower half. The cylinder of a large pump is closed by a piston. To the piston a rope is tied which leads over a pulley and is divided into numerous branches on which a great number of men pull. The moment the cylinder is connected with an exhausted receiver, the men at the ropes are thrown to the ground. In a similar manner a hhge weight is lifted. Guericke mentions the compressed-air gun as some- Guericke's air-gun. thing already known, and constructs independently an instrument that might appropriately be called a rarified-air gun. A bullet is driven by the external atmospheric pressure through a suddenly exhausted tube, forces aside at the end of the tube a leather valve which closes it, and then continues its flight with a considerable velocity. Closed vessels carried to the summit of a mountain and opened, blow out air; carried down again in the same manner, they suck in air. From these and other experiments Guericke discovers that the air is elastic. 10. The investigations of Guericke were continued Theinvestigations of by an Englishman, ROBERT BOYLE.* The new experi- Robert Boyle. ments which Boyle had to supply were few. He observes the propagation of light in a vacuum and the action of a magnet through it; lights tinder by means of a burning glass; brings the barometer under the receiver of the air-pump, and was the first to construct a balance-manometer ["the statical manometer"]. The ebullition of heated fluids and the freezing of water on exhaustion were first observed by him. Of the air pump experiments common at the present day may also be mentioned that with falling bodies, *And published by him in 1660, before the work of Von Guericke.-Trans. 124 THE SCIENCE OF MECHANICS. The fall of which confirms in a simple manner the view of Galileo bodies in a vacuum, that when the resistance of the air has been eliminated light and heavy bodies both fall with the same velocity. In an exhausted glass tube a leaden bullet and a piece of paper are placed. Putting the tube in a vertical position and quickly turning it about a horizontal axis through an angle of I800, both bodies will be seen to arrive simultaneously at the bottom of the tube. Quantita- Of the quantitative data we will mention the foltive data. lowing. The atmospheric pressure that supports a column of mercury of 76 cm. is easily calculated from the specific gravity 13-60 of mercury to be 1-0336 kg. to I sq.cm. The weight of 1ooo cu.cm. of pure, dry air at 00 C. and 760 mm. of pressure at Paris at an elevation of 6 metres will be found to be i 293 grams, and the corresponding specific gravity, referred to water, to be 0'001293. The discov- II. Guericke knew of only one kind of air. We ery of other gaseous may imagine therefore the excitement it created when u in 1755 BLACK discovered carbonic acid gas (fixed air) and CAVENDISH in 1766 hydrogen (inflammable air), discoveries which were soon followed by other similar ones. The dissimilar 1physical properties of Sgases are very striking. Faraday has ilA / lustrated their great inequality of weight Sby a beautiful lectureexperiment. If from Fig. 84. a balance in equilibrium, we suspend (Fig. 84) two beakers A, B, the one in an upright position and the other with its opening downwards, we may pour heavy carbonic acid gas from THE PRINCIPLES OF STATICS. 125 above into the one and light hydrogen from beneath into the other. In both instances the balance turns in the direction of the arrow. To-day, as we know, the decanting of gases can be made directly visible by the optical method of Foucault and Toeppler. 12. Soon after Torricelli's discovery, attempts were The mercurial airmade to employ practically the vacuum thus produced. pump. The so-called mercurial air-pumps were tried. But no such instrument was successful until the present century. The mercurial air-pumps now in common use are really barometers of which the extremities are supplied with large expansions and so connected that their difference of level may be easily varied. The mercury takes the place of the piston of the ordinary air-pump. 13. The expansive force of the air, a property ob- Boyle'slaw. served by Guericke, was more accurately investigated by BOYLE, and, later, by MARIOTTE. The law which both found is as follows. If V be called the volume of a given quantity of air and P its pressure on unit area of the containing vessel, then the product V. P is always = a constant quantity. If the volume of the enclosed air be reduced one-half, the air will exert double the pressure on unit of area; if the volume of the enclosed quantity be doubled, the pressure will sink to one-half; and so on. It is quite correct-as a number of English writers have maintained in recent times-that Boyle and not Mariotte is to be regarded as the discoverer of the law that usually goes by Mariotte's name. Not only is this true, but it must also be added that Boyle knew that the law did not hold exactly, whereas this fact appears to have escaped Mariotte. The method pursued by Mariotte in the ascertainment of the law was very simple. He partially filled 126 ITHEML SCIENCE OF MECHANICS. Mariotte's Torricellian tubes with mercury, measured the volume experiments. of the air remaining, and then performed the Torricellian experiment. The new volume of air was thus obtained, and by subtract~ ing the height of the column of mercury from the barometer-height, also the new pressure to which the same quantity of air was now subjected. - - - To condense the air Mariotte employed a siphon-tube with vertical Fig. 85. arms. The smaller arm in which the air was contained was sealed at the upper end; the longer, into which the mercury was poured, was open at the r, upper end. The volume of the air His appa- was read off on the graduated tube, ratus. and to the difference of level of the mercury in the two arms the barometerheight was added. At the present day r both sets of experiments are performed in the simplest manner by fastening a cylindrical glass tube (Fig. 86) rr, closed at the top, to a vertical scale r -r, and connecting it by a caoutchouc tube kk with a second open glass tube r' r', which is movable up and down k k the scale. If the tubes be partly filled with mercury, any difference of level whatsoever of the two surfaces of merFig. 86. cury may be produced by displacing r' r', and the corresponding variations of volume of the air enclosed in r r observed. It struck Mariotte on the occasion of his investigations that any small quantity of air cut off completely THE PRINCIPLES OF STATICS. 127 from the rest of the atmosphere and therefore not The expan sive force of directly affected by the latter's weight, also supported isolated portions of the barometer-column; as where, to give an instance, the atmosthe open arm of a barometer-tube is closed. The simple explanation of this phenomenon, which, of course, Mariotte immediately found, is this, that the air before enclosure must have been compressed to a point at which its tension balanced the gravitational pressure of the atmosphere; that is to say, to a point at which it exerted an equivalent elastic pressure. We shall not enter here into the details of the arrangement and use of air-pumps, which are readily understood from the law of Boyle and Mariotte. 14. It simply remains for us to remark, that the discoveries of a~rostatics furnished so much that was new and wonderful that a valuable inteliectual stimulus proceeded from the science. CHAPTER II. THE DEVELOPMENT OF THE PRINCIPLES OF DYNAMICS. I. GALILEO'S ACHIEVEMENTS. Dynamics I. We now pass to the discussion of the fundamodern mental principles of dynamics. This is entirely a modscience. ern science. The mechanical speculations of the ancients, particularly of the Greeks, related wholly to statics. Dynamics was founded by GALILEO. We shall readily recognise the correctness of this assertion if we but consider a moment a few propositions held by the Aristotelians of Galileo's time. To explain the descent of heavy bodies and the rising of light bodies, (in liquids for instance,) it was assumed that every thing and object sought its place: the place of heavy bodies was below, the place of light bodies was above. Motions were divided into natural motions, as that of descent, and violent motions, as, for example, that of a projectile. From some few superficial experiments and observations, philosophers had concluded that heavy bodies fall more quickly and lighter bodies more slowly, or, more precisely, that bodies of greater weight fall more quickly and those of less weight more slowly. It is sufficiently obvious from this that the dynamical knowledge of the ancients, particularly of the Greeks, was very insignificant, and that it was left to modern TLE PRINCIPLES OF D) YiNTAGS.l1CS. I29 times to lay the true foundations of this department of inquiry. (See Appendix, VII., p. 520.) LEANNVM AG ENS ANNVML AG ENS 2. 'ihe treatise Discorsi e daimost/razzoni i mailmeatic/e, in which Galileo communicated to the world the first 130 THE SCIENCE OF MECCHANICS. Galileo's dynamical investigation of the laws of falling bodies, ton o te appeared in 1638. The modern spirit that Galileo dislaws of falling bodies. covers is evidenced here, at the very outset, by the fact that he does not ask why heavy bodies fall, but propounds the question, How do heavy bodies fall? in agreement with what law do freely falling bodies move? The method he employs to ascertain this law is this. He makes certain assumptions. He does not, however, like Aristotle, rest there, but endeavors to ascertain by trial whether they are correct or not. His first, The first theory on which he lights is the following. erroneous theory. It seems in his eyes plausible that a freely falling body, inasmuch as it is plain that its velocity is constantly on the increase, so moves that its velocity is double after traversing double the distance, and triple after traversing triple the distance; in short, that the velocities acquired in the descent increase proportionally to the distances descended through. Before he proceeds to test experimentally this hypothesis, he reasons on it logically, implicates himself, however, in so doing, in a fallacy. He says, if a body has acquired a certain velocity in the first distance descended through, double the velocity in double such distance descended through, and so on; that is to say, if the velocity in the second instance is double what it is in the first, then the double distance will be traversed in the same time as the original simple distance. If, accordingly, in the case of the double distance we conceive the first half traversed, no time will, it would seem, fall to the account of the second half. The motion of a falling body appears, therefore, to take place instantaneously; which not only contradicts the hypothesis but also ocular evidence. We shall revert to this peculiar fallacy of Galileo's later on. THE PRINCIPLES OF D YNAAMICS. I3P 3. After Galileo fancied he had discovered this as- His second, correct, assumption to be untenable, he made a second one, ac- sumption. cording to which the velocity acquired is proportional to the time of the descent. That is, if a body fall once, and then fall again during twice as long an interval of time as it first fell, it will attain in the second instance double the velocity it acquired in the first. He found no self-contradiction in this theory, and he accordingly proceeded to investigate by experiment whether the assumption accorded with observed facts. It was difficult to prove by any direct means that the velocity acquired was proportional to the time of descent. It was easier, however, to investigate by what law the distance increased with the time; and he consequently deduced from his assumption the relation that obtained between the distance and the time, and tested this by experiment. The deduction Discussion B and eluciis simple, distinct, and per- dation of the true fectly correct. He draws theory. (Fig. 87) a straight line, and on it cuts off successive por- OCA tions that represent to him Fig. 87. the times elapsed. At the extremities of these portions he erects perpendiculars (ordinates), and these represent the velocities acquired. Any portion OG of the line OA denotes, therefore, the time of descent elapsed, and the corresponding perpendicular GH the velocity acquired in such time. If, now, we fix our attention on the progress of the velocities, we shall observe with Galileo the following fact: namely, that at the instant C, at which one-half OC of the time of descent OA has elapsed, the velocity CD is also one-half of the final velocity AB. If now we examine two instants of time, E and G, 132 THE SCIENCE OF MECILHNICS. Uniformly equally distant in opposite directions from the instant accelerated motion. C, we shall observe that the velocity HG exceeds the mean velocity CD by the same amount that EF falls short of it. For every instant antecedent to C there exists a corresponding one equally distant from it subsequent to C. Whatever loss, therefore, as compared with uniform motion with half the final velocity, is suffered in the first half of the motion, such loss is made up in the second half. The distance fallen through we may consequently regard as having been uniformily described with half the final velocity. If, accordingly, we make the final velocity v proportional to the time of descent /, we shall obtain 7 gt, where g denotes the final velocity acquired in unit of time--the so-called acceleration. The space s descended through is therefore given by the equation s = (g /2) / or s = 2 /2. Motion of this sort, in which, agreeably to the assumption, equal velocities constantly accrue in equal intervals of time, we call uniformly accelera/ed motlion. Table of the If we collect the times of descent, the final velocitimes, velocitiesnd ties, and the distances traversed, we shall obtain the distances of descent, following table: I. V. S. 1. 1^. 1 X 1 " -o 2. 2g. 2 X 2. - 3. 30'. 3X3 ' 4. 49. 4 X 4. ~2, tg. tX THE PRINCIPLES OF D YNAIAMICS. I33 4. The relation obtaining between / and s admits Experimental verificaof experimental proof; and this Galileo accomplished tion of the law. in the manner which we shall now describe. We must first remark that no part of the knowledge and ideas on this subject with which we are now so familiar, existed in Galileo's time, but that Galileo had to create these ideas and means for us. Accordingly, it was impossible for him to proceed as we should do to-day, and he was obliged, therefore, to pursue a different method. He first sought to retard the motion of descent, that it might be more accurately observed. He made observations on balls, which he caused to roll down inclined planes (grooves); assuming that only the velocity of the motion would be lessened here, but that the form of the law of descent would remain unmodified. If, beginning from the upper extremity, the The artifices emdistances I, 4, 9, 16... be notched off on the groove, ployed. the respective times of descent will be representable, it was assumed, by the numbers I, 2, 3, 4...; a result which was, be it added, confirmed. The observation of the times involved, Galileo accomplished in a very ingenious manner. There were no clocks of the modern kind in his day: such were first rendered possible by the dynamical knowledge of which Galileo laid the foundations. The mechanical clocks which were used were very inaccurate, and were available only for the measurement of great spaces of time. Moreover, it was chiefly water-clocks and sand-glasses that were in use-in the form in which they had been handed down from the ancients. Galileo, now, constructed a very simple clock of this kind, which he especially adjusted to the measurement of small spaces of time; a thing not customary in those days. It consisted of a vessel of water of very large transverse dimensions, having in 134 THE SCIENCE OF MECHANICS. Galileo's the bottom a minute orifice which was closed with the clc. finger. As soon as the ball began to roll down the inclined plane Galileo removed his finger and allowed the water to flow out on a balance; when the ball had arrived at the terminus of its path he closed the orifice. As the pressure-height of the fluid did not, owing to the great transverse dimensions of the vessel, perceptibly change, the weights of the water discharged from the orifice were proportional to the times. It was in this way actually shown that the times increased simply, while the spaces fallen through increased quadratically. The inference from Galileo's assumption was thus confirmed by experiment, and with it the assumption itself. The rela- 5. To form some notion of the relation which subtion of motion on an sists between motion on an inclined plane and that of inclined plane to free descent, Galileo made the assumption, that a body that of free descent, which falls through the height of an inclined plane attains the same final velocity as a body which falls through its length. This is an assumption that will strike us as rather a bold one; but in the manner in which it was enunciated and employed by Galileo, it is quite natural. We shall endeavor to explain the way by which he was led to it. He says: If a body fall freely downwards, its velocity increases proportionally to the time. When, then, the body has arrived at a point below, let us imagine its velocity reversed and directed upwards; the body then, it is clear, will rise. We make the observation that its motion in this case is a reflection, so to speak, of its motion in the first case. As then its velocity increased proportionally to the time of descent, it will now, conversely, diminish in that proportion. When the body has continued to rise for as long a time as it descended, and has reached the height from which it originally fell, its velocity will be reduced to THE PRINCIPLES OF D YNAMICS. i35 zero. We perceive, therefore, that a body will rise, Justification of the in virtue of the velocity acquired in its descent, just as assumption toat the high as it has fallen. If, accordingly, a body fallingfinalvelocities of such down an inclined plane could acquire a velocity which notions are the same. would enable it, when placed on a differently inclined plane, to rise higher than the point from which it had fallen, we should be able to effect the elevation of bodies by gravity alone. There is contained, accordingly, in this assumption, that the velocity acquired by a body in descent depends solely on the vertical height fallen through and is independent of the inclination of the path, nothing more than the uncontradictory apprehension and recognition of thefact that heavy bodies do not possess the tendency to rise, but only the tendency to fall. If we should assume that a body falling down the length of an inclined plane in some way or other attained a greater velocity than a body that fell through its height, we should only have to let the body pass with the acquired velocity to another inclined or vertical plane to make it rise to a greater vertical height than it had fallen from. And if the velocity attained on the inclined plane were less, we should only have to reverse the process to obtain the same result. In both instances a heavy body could, by an appropriate arrangement of inclined planes, be forced continually upwards solely by its own weight-a state of things which wholly contradicts our instinctive knowledge of the nature of heavy bodies. (See p. 522.) 6. Galileo, in this case, again, did not stop with the mere philosophical and logical discussion of his assumption, but tested it by comparison with experience. He took a simple filar pendulum (Fig. 88) with a heavy ball attached. Lifting the pendulum, while 136 THE SCIENCE OF 0MEICtANICS. Galileo's elongated its full length, to the level of a given altitude, experimnental verifica- and then letting it fall, it ascended to the same level tion of this assumption on the opposite side. If it does not do so exac/ly, Galileo said, the resistance of the air must be the cause of the deficit. This is inferrible from the fact that the deficiency is greater in the case of a cork ball than it is C n;n ~6E Fig. 88. Effected by in the case of a heavy metal one. However, this negpartially impeding lected, the body ascends to the same altitude on the the motion ofapendo- opposite side. Now it is permissible to regard the molm string. tion of a pendulum in the arc of a circle as a motion of descent along a series of inclined planes of different inclinations. This seen, we can, with Galileo, easily cause the body to rise on a different arc-on a different series of inclined planes. This we accomplish by driving in at one side of the thread, as it vertically hangs, a nail for g, which will prevent any given portion of the thread from taking part in the second half of the motion. The moment the thread arrives at the line of equilibrium and strikes the nail, the ball, which has fallen through ba, will begin to ascend by a different series of inclined planes, and describe the arc a li or a n. Now if the inclination of the planes had any influence THEI PJRINCIPLES OF D YNAfMICS. I37 on the velocity of descent, the body could not rise to the same horizontal level from which it had fallen. But it does. By driving the nail sufficiently low down, we may shorten the pendulum for half of an oscillation as much as we please; the phenomenon, however, always remains the same. If the nail h be driven so low down that the remainder of the string cannot reach to the plane E, the ball will turn completely over and wind the thread round the nail; because when it has attained the greatest height it can reach it still has a residual velocity left. 7. If we assume thus, that the same final velocity is The assumption attained on an inclined plane whether the body fall leads to the law of relathrough the height or the length of the plane,-in which tive accelerations assumption nothing more is contained than that a body sought. rises by virtue of the velocity it has acquired in falling just as high as it has fallen,-we shall easily arrive, with Galileo, at the perception that the times of the descent along the height and the length of an inclined plane are in the simple proportion of the height and the length; or, what is the same, that the accelerations are inversely proportional to the times of descent. The acceleration along the height will consequently bear to the acceleration along A the length the proportion of the length to the height. Let AB (Fig. 89) be the height and A C B --- C the length of the inclined plane. Fig. 89. Both will be descended through in uniformly accelerated motion in the times / and t1 with the final velocity v. Therefore, v v AB t AB -t and AC- - t 2 2 1"AC - 1 138 THE SCIENCE OF ME CIANICS. If the accelerations along the height and the length be called respectively g and g,, we also have g t/ AC v-(gI and v- g_ /1, whence sm-- - sna. In this way we are able to deduce from the acceleration on an inclined plane the acceleration of free descent. A corollary From this proposition Galileo deduces several corof the preceding law. ollaries, some of which have passed into our elementary text-books. The accelerations along the height and length are in the inverse proportion of the height and length. If now we cause one body to fall along the length of an inclined plane and simultaneously another to fall freely along its height, and ask what the distances are that are traversed by the two in equal intervals of time, the solution of the problem will be readily found (Fig. go) by simply letting fall from B a perpendicular on the length. The part AD, thus cut off, will be the distance traversed by the one body on the inclined plane, while the second body is freely falling through the height of the plane. A A D B Cc Fig. 90. Fig. 91. Relative If we describe (Fig. 91) a circle on AB as diametimes of description of ter, the circle will pass through D, because D is a nddiae- right angle. It will be seen thus, that we can imagine ters of cirScles, any number of inclined planes, AE, AF, of any degree of inclination, passing through A, and that in every THE PRINCIPLES OF D YNAAMICS. I39 case the chords A G, AH drawn in this circle from the upper extremity of the diameter will be traversed in the same time by a falling body as the vertical diameter itself. Since, obviously, only the lengths and inclinations are essential here, we may also draw the chords in question from the lower extremity of the diameter, and say generally: The vertical diameter of a circle is described by a falling particle in the same time that any chord through either extremity is so described. We shall present another corollary, which, in the The figures formed by pretty form in which Galileo gave it, is usually no bodies falling in the longer incorporated in elementary expositions. We chords of circles. imagine gutters radiating in a vertical plane from a common point A at a number of different degrees of inclination to the horizon (Fig. 92). We place at their common extremity A a like number of heavy bodies and cause them to begin simultaneously their motion of descent. The bodies will always form at any one Fig. 92. instant of time a circle. After the lapse of a longer time they will be found in a circle of larger radius, and the radii increase proportionally to the squares of the times. If we imagine the gutters to radiate in a space instead of a plane, the falling bodies will always form a sphere, and the radii of the spheres will increase proportionally to the squares of the times. This will be 140 THE SCIENCE OF AfE CilANIC S. perceived by imagining the figure revolved about the vertical A /7 Character 8. We see thus,-as deserves again to be briefly of Galileo's inquiries. noticed,-that Galileo did not supply us with a tl/eory of the falling of bodies, but investigated and established, wholly without preformed opinions, the actual facts of falling. Gradually adapting, on this occasion, his thoughts to the facts, and everywhere logically abiding by the ideas he had reached, he hit on a conception, which to himself, perhaps less than to his successors, appeared in tne light of a new law. In all his reasonings, Galileo followed, to the greatest advantage of science, a principle which might appropriately be called the principle The prin- of continuity. Once we have reached a theory that apciple of continuity, plies to a particular case, we proceed gradually to modify in thought the conditions of that case, as far as it is at all possible, and endeavor in so doing to adhere throughout as closely as we can to the conception originally reached. There is no method of procedure more surely calculated to lead to that comprehension of all natural phenomena which is the simplejst and also attainable with the least expenditure of mentality and feeling. (Compare Appendix, IX., p. 523.) A particular instance will show more clearly than any general remarks what we mean. Galileo conA C D F B H Fig. 93. siders (Fig. 93) a body which is falling down the inclined plane AB, and which, being placed with the THE PRNLCIPLES OF D YNAMICS. 141 velocity thus acquired on a second plane BC, for ex- Galileo's discovery ample, ascends this second plane. On all planes BC, of thesocalled law BD, and so forth, it ascends to the horizontal plane of inertia. that passes through A. But, just as it falls on BD with less acceleration than it does on BC, so similarly it will ascend on BD with less retardation than it will on BC. The nearer the planes BC, BD, BE, BF approach to the horizontal plane BH, the less will the retardation of the body on those planes be, and the longer and further will it move on them. On the horizontal plane BH the retardation vanishes entirely (that is, of course, neglecting friction and the resistance of the air), and the body will continue to move infinitely long and infinitely far with constant velocity. Thus advancing to the limiting case of the problem presented, Galileo discovers the so-called law of inertia, according to which a body not under the influence of forces, i. e. of special circumstances that change motion, will retain forever its velocity (and direction). We shall presently revert to this subject. 9. The motion of falling that Galileo found actually The deduction of the to exist, is, accordingly, a motion of which the velocity idea of uniformly acincreases proportionally to the time-a so-called uni- celerated motion. formly accelerated motion. motion. It would be an anachronism and utterly unhistorical to attempt, as is sometimes done, to derive the uniformly accelerated motion of falling bodies from the constant action of the force of gravity. " Gravity is a constant force; consequently it generates in equal elements of time equal increments of velocity; thus, the motion produced is uniformly accelerated." Any exposition such as this would be unhistorical, and would put the whole discovery in a false light, for the reason that the notion of force as we hold it to-day was first created 142 THE SCIENCE OF MECHANICS. Forces and by Galileo. Before Galileo force was known solely as accelerations. r ressure. Now, no one can know, who has not learned it from experience, that generally pressure produces motion, much less in what manner pressure passes into motion; that not position, nor velocity, but acceleration, is determined by it. This cannot be philosophically deduced from the conception, itself. Conjectures may be set up concerning it. But experience alone can definitively inform us with regard to it. 10. It is not by any means self-evident, therefore, that the circumstances which determine motion, that is, forces, immediately produce accelerations. A glance at other departments of physics will at once make this clear. The differences of temperature of bodies also determine alterations. However, by differences of temperature not compensatory accelerations are determined, but compensatory velocities. The fact That it is accelerations which are the immediate efthat forces determine fects of the circumstances that determine motion, that iones sran is, of the forces, is a fact which Galileo perceived in the experimental fact natural phenomena. Others before him had also perceived many things. The assertion that everything seeks its place also involves a correct observation. The observation, however, does not hold good in all cases, and it is not exhaustive. If we cast a stone into the air, for example, it no longer seeks its place; since its place is below. But the acceleration towards the earth, the retardation of the upward motion, the fact that Galileo perceived, is still present. His observation always remains correct; it holds true more generally; it embraces in one mental effort much more. I1. We have already remarked that Galileo discovered the so-called law of inertia quite incidentally. A body on which, as we are wont to say, no force acts, THE PRINCIPLES OF DYNAMICS. I43 preserves its direction and velocity unaltered. The History of the sofortunes of this law of inertia have been strange. It called law. of inertia. appears never to have played a prominent part in Galileo's thought. But Galileo's successors, particularly Huygens and Newton, formulated it as an independent law. Nay, some have even made of inertia a general property of matter. We shall readily perceive, however, that the law of inertia is not at all an independent law, but is contained implicitly in Galileo's perception that all circumstances determinative of motion, or forces, produce accelerations. In fact, if a force determine, not position, not velo- The law a simple incity, but acceleration, change of velocity, it stands to ference from Galireason that where there is no force there will be no leo's funda. mental obchange of velocity. It is not necessary to enunciate servation. this in independent form. The embarrassment of the neophyte, which also overcame the great investigators in the face of the great mass of new material presented, alone could have led them to conceive the same fact as two different facts and to formulate it twice. In any event, to represent inertia as self-evident, or Erroneous methods of to derive it from the general proposition that "the ef- deducing it. fect of a cause persists," is totally wrong. Only a mistaken straining after rigid logic can lead us so out of the way. Nothing is to be accomplished in the present domain with scholastic propositions like the one just cited. We may easily convince ourselves that the contrary proposition, "cessante causa cessat effectus," is as well supported by reason. If we call the acquired velocity "the effect," then the first proposition is correct; if we call the acceleration "effect," then the second proposition holds. 12. We shall now examine Galileo's researches from another side. He began his investigations with the 144 THE SCIENCE OF AMECHANICS. Notion of notions familiar to his time-notions developed mainly velocity as p it existed in in the practical arts. One notion of this kind was that Galileo's time. of velocity, which is very readily obtained from the consideration of a uniform motion. If a body traverse in every second of time the same distance c, the distance traversed at the end of / seconds will be s -- ct. The distance c traversed in a second of time we call the velocity, and obtain it from the examination of any portion of the distance and the corresponding time by the help of the equation c st, that is, by dividing the number which is the measure of the distance traversed by the number which is the measure of the time elapsed. Now, Galileo could not complete his investigations without tacitly modifying and extending the traditional idea of velocity. Let us represent for distinctness sake B A A At A 0At A Fig. 94. in I (Fig. 94) a uniform motion, in 2 a variable motion, by laying off as abscissae in the direction OA the elapsed times, and erecting as ordinates in the direction AB the distances traversed. Now, in i, whatever increment of the distance we may divide by the corresponding increment of the time, in all cases we obtain for the velocity c the same value. But if we were thus to proceed in 2, we should obtain widely differing values, and therefore the word "velocity " as ordinarily understood, ceases in this case to be unequivocal. If, however, we consider the increase of the distance in a sufficiently THE PRINCIPLES OF D YNAMICS. 145 small element of time, where the element of the curve Galileo's modificain 2 approaches to a straight line, we may regard the tion of this notion. increase as uniform. The velocity in this element of the motion we may then define as the quotient, d s/A t, of the element of the time into the corresponding element of the distance. Still more precisely, the velocity at any instant is defined as the limiting value which the ratio A s/A t assumes as the elements become infinitely small-a value designated by ds dt. This new notion includes the old one as a particular case, and is, moreover, immediately applicable to uniform motion. Although the express formulation of this idea, as thus extended, did not take place till long after Galileo, we see none the less that he made use of it in his reasonings. 13. An entirely new notion to which Galileo was The notion of accelera. led is the idea of acceleration. In uniformly acceler- tion. ated motion the velocities increase with the time agreeably to the same law as in uniform motion the spaces increase with the times. If we call v the velocity acquired in time t, then v = t. Here g denotes the increment of the velocity in unit of time or the acceleration, which we also obtain from the equation g v--/t. When the investigation of variably accelerated motions was begun, this notion of acceleration had to experience an extension similar to that of the notion of velocity. If in i and 2 the times be again drawn as abscissae, but now the velocities as ordinates, we may go through anew the whole train of the preceding reasoning and define the acceleration as dv/dt, where dv denotes an infinitely small increment of the velocity and dt the corresponding increment of the time. In the notation of the differential calculus we 146 THE SCIENCE OF AMECHANICS. have for the acceleration of a rectilinear motion, p = dv/dt = d2 s/d2. Graphic The ideas here developed are susceptible, moreover, representation of of graphic representation. If we lay off the times as these ideas. abscissae and the distances as ordinates, we shall perceive, that the velocity at each instant is measured by the slope of the curve of the distance. If in a similar manner we put times and velocities together, we shall see that the acceleration of the instant is measured by the slope of the curve of the velocity. The course of the latter slope is, indeed, also capable of being traced in the curve of distances, as will be perceived from the following considerations. Let us imagine, in the E F 0 CA o A- B a c d e Fig. 95. Fig. 96. The curve usual manner (Fig. 95), a uniform motion represented of distance. by a straight line OCD. Let us compare with this a motion OCE the velocity of which in the second half of the time is greater, and another motion OCF of which the velocity is in the same proportion smaller. In the first case, accordingly, we shall have to erect for the time OB - 2 OA, an ordinate greater than BD 2 AC; in the second case, an ordinate less than BD. We see thus, without difficulty, that a curve of distance convex to the axis of the time-abscissao corresponds to accelerated motion, and a curve concave thereto to retarded motion. If we imagine a lead-pencil to perform a vertical motion of any kind and in THE PRINCIPLES OF D YNAMICS. I47 front of it during its motion a piece of paper to be uniformly drawn along from right to left and the pencil to thus execute the drawing in Fig. 96, we shall be able to read off from the drawing the peculiarities of the motion. At a the velocity of the pencil was directed upwards, at b it was greater, at c it was - 0, at d it was directed downwards, at e it was again - 0. At a, 6, d, e, the acceleration was directed upwards, at c downwards; at c and e it was greatest. 14. The summary representation of what Galileo Tabular presentdiscovered is best made by a table of times, acquired inent of Ga. lileo's dis-. v7. S. covery. 1 g 2 2 2g 4 3 3g 9 - t t< tj velocities, and traversed distances. But the numbers The table may be refollow so simple a law,-one immediately recognisable, placed by rules for its -that there is nothing to prevent our replacing the construction. table by a rule for its construction. If we examine thet relation that connects the first and second columns, we shall find that it is expressed by the equation --= gt, which, in its last analysis, is nothing but an abbreviated direction for constructing the first two columns of the table. The relation connecting the first and third columns is given by the equation s = g 2/2. The connection of the second and third columns is represented by s = v2/2g. 148 THE SCIENCE OF MECHANICS. The rules. Of the three relations 2 V2 S2g' strictly, the first two only were employed by Galileo. Huygens was the first who evinced a higher appreciation of the third, and laid, in thus doing, the foundations of important advances. A remark 15. We may add a remark in connection with on the relation of the this table that is very valuable. It has been stated spaces and the times. previously that a body, by virtue of the velocity it has acquired in its fall, is able to rise again to its original height, in doing which its velocity diminishes in the same way (with respect to time and space) as it increased in falling. Now a freely falling body acquires in double time of descent double velocity, but falls in this double time through four times the simple distance. A body, therefore, to which we impart a vertically upward double velocity will ascend twice as long a time, but four times as high as a body to which the simple velocity has been imparted. Thedispute It was remarked, very soon after Galileo, that there of the Cartesians and is inherent in the velocity of a body a something that Leibnitzians on the corresponds to a force-a something, that is, by which measure of force, a force can be overcome, a certain "efficacy," as it has been aptly termed. The only point that was debated was, whether this efficacy was to be reckoned proportional to the velocity or to the square of the zvelocity. The Cartesians held the former, the Leibnitzians the latter. But it will be perceived that the question involves no dispute whatever. The body with the double velocity overcomes a given force through double the THE PRINCIPLES OF DYNAMICS. I49 time, but through four times the distance. With respect to time, therefore, its efficacy is proportional to the velocity; with respect to distance, to the square of the velocity. D'Alembert drew attention to this misunderstanding, although in not very distinct terms. It is to be especially remarked, however, that Huygens's thoughts on this question were perfectly clear. r6. The experimental procedure by which, at the Thepresent experimenpresent day, the laws of falling bodies are verified, is talmeansof verifying somewhat different from that of Galileo. Two methods the laws of falling bodmay be employed. Either the motion of falling, which ies. from its rapidity is difficult to observe directly, is so retarded, without altering the law, as to be easily observed; or the motion of falling is not altered at all, but our means of observation are improved in delicacy. On the first principle Galileo's inclined gutter and Atwood's machine rest. Atwood's machine consists (Fig. 97) of an easily running pulley, over which is thrown a thread, to whose extremities two equal weights P are attached. If upon one of the weights P we lay a third small weight p, a uniformly accel- P erated motion will be set up by the added Fig. 97. weight, having the acceleration (/2 P + _p) g-a result that will be readily obtained when we shall have discussed the notion of "mass." Now by means of a graduated vertical standard connected with the pulley it may easily be shown that in the times I, 2, 3, 4.... the distances, 4, 9, 16.... are traversed. The final velocity corresponding to any given time of descent is investigated by catching the small additional weight, p, which is shaped so as to project beyond the outline of P, in a ring through which the falling body passes, after which the motion continues without acceleration. 150 TIHE SCIENCE OF MLECIiHANVICS. The appa- The apparatus of Morin is based on a different prinratus of Morin, La- ciple. A body to which a writing pencil is attached borde, Lippich, and describes on a vertical sheet of paper, which is drawn Von Babo. uniformly across it by a clock-work, a horizontal straight line. If the body fall while the paper is not in motion, it will describe a vertical straight line. If the two motions are combined, a parabola will be produced, of which the horizontal abscissae correspond to- the elapsed times and the vertical ordinates to the distances of descent described. For the abscissae i, 2, 3, 4.... we obtain the ordinates i, 4, 9, 6.... By an unessential modification, Morin employed instead of a plane sheet of paper, a rapidly rotating cylindrical drum with vertical axis, by the side of which the body fell down a guiding wire. A different apparatus, based on the same principle, was invented, independently, by Laborde, Lippich, and Von Babo. A lampblacked sheet of glass (Fig. 98a) falls freely, while a horizontally vibrating vertical rod, which in its first transit through the position of equilibrium starts the motion of descent, traces, by means of a quill, a curve on the lampblacked surface. Owing to the constancy of the period of vibration of the rod combined with the increasing velocity of the descent, the undulations traced by the rod become longer and longer. Thus (Fig. 98) bc 3aby, cd- 5ab, de- 7ab, and so forth. The law of falling bodies is clearly exhibited by this, since ab cb = 4ab, ab +bc+ cd = gab, and so forth. The law of the velocity is confirmed by the inclinations of the tangents at the points a, b, c, d, and so forth. If the time of oscillation of the rod be known, the value of g is determinable from an experiment of this kind with considerable exactness. Wheatstone employed for the measurement of mi THE PRINCIPLES OF D YNA,4MICS. 151 nute portions of time a rapidly operating clock-work The devices of called a chronoscope, which is set in motion at the be- Wheatstone and ginning of the time to be measured and stopped at the Hipp. termination of it. Hipp has advantageously modified e I d Fig. 98. Fig. 98. Fig. 98a. this method by simply causing a light index-hand to be thrown by means of a clutch in and out of gear with a rapidly moving wheel-work regulated by a vibrating reed of steel tuned to a high note, and acting as an es 152 7HE SCIENCE OF MECHANICS. capement. The throwing in and out of gear is effected by an electric current. Now if, as soon as the body begins to fall, the current be interrupted, that is the hand thrown into gear, and as soon as the body strikes the platform below the current is closed, that is the hand thrown out of gear, we can read by the distance the index-hand has travelled the time of descent. Galileo's 17. Among the further achievements of Galileo we minor investiga- have yet to mention his ideas concerning the motion tions. of the pendulum, and his refutation of the view that bodies of greater weight fall faster than bodies of less weight. We shall revert to both of these points on another occasion. It may be stated here, however, that Galileo, on discovering the constancy of the period of pendulum-oscillations, at once applied the pendulum to pulse-measurements at the sick-bed, as well as proposed its use in astronomical observations and to a certain extent employed it therein himself. The motion 18. Of still greater importance are his investigaof projectiles. tions concerning the motion of projectiles. A free body, according to Galileo's view, constantly experiences a vertical acceleration g towards the earth. If at the beginning of its motion it is affected with a vertical A - velocity c, its velocity at the end of the time / will be v c g t. An initial velocity upwards would have to be reckoned negative here. The disx tance described at the end of Fig. 99. time t is represented by the equation s a - ct - t2, where c and 1g12 are the portions of the traversed distance that correspond respectively to the uniform and the uniformly accelerated motion. The constant a is to be put 0 when we reckon THEL PRINCIPLES OF D YNAAMICS. 153 the distance from the point that the body passes at time / = 0. When Galileo had once reached his fundamental conception of dynamics, he easily recognised the case of horizontal projection as a combination of two independent motions, a horizontal uniform motion, and a vertical uniformly accelerated motion. He thus introduced into use the principle of the parallelogran of motions. Even oblique projection no longer presented the slightest difficulty. If a body receives a horizontal velocity c, it de-The curve of projecscribes in the horizontal direction in time t the distance tion a parabola. y c t, while simultaneously it falls in a vertical direction the distance x =g2/2. Different motion-determinative circumstances exercise no mutual effect on one another, and the motions determined by them take place independently of each other. Galileo was led to this assumption by the attentive observation of the phenomena; and the assumption proved itself true. For the curve which a body describes when the two motions in question are compounded, we find, by employing the two equations above given, the expression y =/(2 c2/g) x. It is the parabola of Apollonius having its parameter equal to c 2/g and its axis vertical, as Galileo knew. We readily perceive with Galileo, that oblique pro- Oblique projection. jection involves nothing new. The velocity c impartedprojec to a body at the angle a with the horizon is resolvable into the horizontal component c. cos a and the vertical component c. sin a. With the latter velocity the body ascends during the same interval of time t which it would take to acquire this velocity in falling vertically downwards. Therefore, c. sin a =-gt. When it has reached its greatest height the vertical component of its initial velocity has vanished, and from the point S TIH SCIENCE OF MECHANICS. I54 onward (Fig. Ioo) it continues its motion as a horizontal projection. If we examine any two epochs equally distant in time, before and after the transit through S, s we shall see that the body at these two epochs is equally distant from the perpendicular through S and situated the same distance below the horiSzontal line through S. The Fig. ioo. curve is therefore symmetrical with respect to the vertical line through S. It is a parabola with vertical axis and the parameter (c COS a) 2 /g. The range To find the so-called range of projection, we have tirojec- simply to consider the horizontal motion during the time of the rising and falling of the body. For the ascent this time is, according to the equations above given, / = c sin a/g, and the same for the descent. With the horizontal velocity c. cos a, therefore, the distance is traversed c sin a c c2 w =- c cos a. 2 - 2 sin a cos a sin 2 a. g g g The range of projection is greatest accordingly when a t 450, and equally great for any two angles a = 450 ~. The mutual 19. The recognition of the mutual independence of independence of the forces, or motion-determinative circumstances ocforcescurring in nature, which was A aB reached and found expression in the investigations relating to SD projection, is important. A body Fig. roi. may move (Fig. 101) in the direction AB, while the space in which this motion occurs is displaced in the direction A C. The body then 77IE PRINCIPLES OF D YN.4AIICS. 155 goes from A to D. Now, this also happens if the two circumstances that simultaneously determine the motions AB and AC, have no influence on one another. It is easy to see that we may compound by the parallelogram not only displacements that have taken place but also velocities and accelerations that simultaneously take place. (See Appendix, X., p. 525.) II. THE ACHIEVEMENTS OF HUYGENS. i. The next in succession of the great mechanical in- Huygens's high rank quirers is HUYGENS, who in every respect must be as aninranked as Galileo's peer. If, perhaps, his philosophical qurer. endowments were less splendid than those of Galileo, this deficiency was compensated for by the superiority of his geometrical powers. Huygens not only continued the researches which Galileo had begun, but he also solved the first problems in the dynamics of several masses, whereas Galileo had throughout restricted himself to the dynamics of a single body. The plenitude of Huygens's achievements is best Enumeration of Huyseen in his HorologiumOscillatorium, which appeared in gens's achieve1673. The most important subjects there treated of forments. the first time, are: the theory of the centre of oscillation, the invention and construction of the pendulumclock, the invention of the escapement, the determination of the acceleration of gravity, g, by pendulumobservations, a proposition regarding the employment of the length of the seconds pendulum as the unit of length, the theorems respecting centrifugal force, the mechanical and geometrical properties of cycloids, the doctrine of evolutes, and the theory of the circle of curvature. 156 TH7E SCIENCE OF MVECZHANICS. 2. With respect to the form of presentation of his work, it is to be remarked that Huygens shares with CHRISTIANUS HUGENIUS nattus 14 Aprilis 16*29. denatLLs 8 Junii 1695. Galileo, in all its perfection, the latter's exalted and inimitable candor. He is frank without reserve in the presentment of the methods that led him to his dis THE PRINCIPLES OF D YNAMICS. I57 coveries, and thus always conducts his reader into the full comprehension of his performances. Nor had he cause to conceal these methods. If, a thousand years from now, it shall be found that he was a man, it will likewise be seen what manner of man he was. In our discussion of the achievements of Huygens, however, we shall have to proceed in a somewhat different manner from that which we pursued in the case of Galileo. Galileo's views, in their classical simplicity, could be given in an almost unmodified form. With Huygens this is not possible. The latter deals with more complicated problems; his mathematical methods and notations become inadequate and cumbrous. For reasons of brevity, therefore, we shall reproduce all the conceptions of which we treat, in modern form, retaining, however, Huygens's essential and characteristic ideas. Characterisation of Huygens's performances. Huygens's Pendulum Clock. 158 TIE SCIENCE OF MECHANICS. Centrifugal 3. We begin with the investigations concerning and centripetal force. centrifugal force. When once we have recognised with Galileo that force determines acceleration, we are impelled, unavoidably, to ascribe every change of velocity and consequently also every change in the direction of a motion (since the direction is determined by three velocity-components perpendicular to one another) to a force. If, therefore, any body attached to a string, say a stone, is swung uniformly round in a circle, the curvilinear motion which it performs is intelligible only on the supposition of a constant force that deflects the body from the rectilinear path. The tension of the string is this force; by it the body is constantly deflected from the rectilinear path and made to move towards the centre of the circle. This tension, accordingly, represents a centripetal force. On the other hand, the axis also, or the fixed centre, is acted on by the tension of the string, and in this aspect the tension of the string appears as a centrifugal force. I1. I II Fig. 102. Fig. 103. Let us suppose that we have a body to which, a velocity has been imparted and which is maintained in uniform motion in a circle by an acceleration constantly directed towards the centre. The conditions on which this acceleration depends, it is our purpose to investigate. We imagine (Fig. 102) two equal circles uni THE, PRINCIPLES OF D YNAAMICS. 159 formly travelled round by two bodies; the velocities in Uniform motion in the circles I and II bear to each other the proportion equal circles. S:2. If in the two circles we consider any same arcelement corresponding to some very small angle a, then the corresponding element s of the distance that the bodies in consequence of the centripetal acceleration have departed from the rectilinear path (the tangent), will also be the same. If we call ~P, and g2 the respective accelerations, and r and r/2 the time-elements for the angle a, we find by Galileo's law 2s 2s P 2, P2 = 4 r,2 that is to say ~P = 42 p. Therefore, by generalisation, in equal circles the centripetal acceleration is proportional to the square of the velocity of the motion. Let us now consider the motion in the circles I and Uniform motion in II (Fig. 103), the radii of which are to each other as unequal circles. I: 2, and let us take for the ratio of the velocities of the motions also 1:2, so that like arc-elements are travelled through in equal times. pl, p2, s, 2s denote the accelerations and the elements of the distance traversed; r is the element of the time, equal for both cases. Then 2s 4s P 1= 2 2 2, that is to say 92 = 29. If now we reduce the velocity of the motion in II one-half, so that the velocities in I and II become equal, 2, will thereby be reduced one-fourth, that is to say to 9l/2. Generalising, we get this rule: when the velocity of the circular motion is the same, the centripetal acceleration is inversely proportional to the radius of the circle described. 4. The early investigators, owing to their following 160 THE SCIENCE OF MECHANICS. Deduction the conceptions of the ancients, generally obtained their of the general law of propositions in the cumbersome form of proportions. circular motion. We shall pursue a different method. On a movable object having the velocity v let a force act during the element of time r which imparts to the object perpendicularly to the direction of its motion the acceleration CS. 213 "And what was the nature of this element, which now "opposed and now obeyed the dominion of the hours?' " As the king concluded from what had happened that "the fixed time for the return of the tide was after "sunrise, he set out, in order to anticipate it, at mid" night, and proceeding down the river with a few "ships he passed the mouth and, finding himself at "last at the goal of his wishes, sailed out 400 stadia "into the ocean. He then offered a sacrifice to the "divinities of the sea, and returned to his fleet." 8. The essential point to be noted in the explana- The explaration of tion of the tides is, that the earth as a rigid body can the phenomena of receive but one determinate acceleration towards the the tides. moon, while the mobile particles of water on the sides nearest to and remotest from the moon can acquire various accelerations. E A.8 Fig. 137. Let us consider (Fig. 137) on the earth T, opposite which stands the moon M, three points A, B, C. The accelerations of the three points in the direction of the moon, if we regard them as free points, are respectively 9 + J- 9, q), () - dA (. The earth as a whole, however, has, as a rigid body, the acceleration p. The acceleration towards the centre of the earth we will call g. Designating now all accelerations to the left as negative, and all to the right as positive, we get the following table: 214 THE SCIENCE OF AMECHANICS. A B C -(O9+9JOY), -99, -(9-J9P) + "r_ (r -9, -9, -9 50 g-~Jp0, 0, - (g- P), where the symbols of the first and second lines represent the accelerations which the free points that head the columns receive, those of the third line the acceleration of corresponding rigid points of the earth, and those of the fourth line, the difference, or the resultant accelerations of the free points towards the earth. It will be seen from this result that the weight of the water at A and C is diminished by exactly the same amount. The water will rise at A and C (Fig. 137). A tidal wave will be produced at these points twice every day. A variation It is a fact not always sufficiently emphasised, that of the phenomenon. the phenomenon would be an essentially different one if the moon and the earth were not affected with accelerated motion towards each other but were relatively fixed and at rest. If we modify the considerations presented to comprehend this case, we must put for the rigid earth in the foregoing computation, q) = 0 simply. We then obtain for the free points.... A C the accelerations.. - (9 + dp ), - (p - dp), or........... (-.( )- q)-, -(- ) - (p or...............g' - cp, where g'= - A p. In such case, therefore, the weight of the water at A would be diminished, and the weight at C increased; the height of the water at A THE PRINCIPLES OF D YNAMICS. 215 would be increased, and the height at C diminished. The water would be elevated only on the side facing the moon. (Fig. 138.) E A.B C Fig. 138. 9. It would hardly be worth while to illustrate An illustrative experipropositions best reached deductively, by experiments ment. that can only be performed with difficulty. But such experiments are not beyond the limits of possibility. If we imagine a small iron sphere K to swing as a conical pendulum about the pole of a magnet N (Fig. 139), and cover the sphere with a solution of magnetic sulphate of iron, the fluid drop should, if the magnet is sufficiently powerful, rep- K resent the phenomenon of the tides. But if we imagine the sphere to be fixed and at rest with respect to the pole of the magnet, the fluid drop will certainly not be found tapering to a point both on the side facing and the side opposite to Fig. 139. the pole of the magnet, but will remain suspended only on the side of the sphere towards the pole of the magnet. 10. We must not, of course, imagine, that the some further conentire tidal wave is produced at once by the action siderations of the moon. We have rather to conceive the tide as an oscillatory movement maintained by the moon. If, for example, we should sweep a fan uniformly and 216 THE SCIENCE OF IMECGII4INICS. continuously along over the surface of the water of a circular canal, a wave of considerable magnitude following in the wake of the fan would by this gentle and constantly continued impulsion soon be produced. In like manner the tide is produced. But in the latter case the occurrence is greatly complicated by the irregular formation of the continents, by the periodical variation of the disturbance, and so forth. (See Appendix, XVII., p. 537.) v. CRITICISM OF THE PRINCIPLE OF REACTION AND OF THE CONCEPT OF MASS. The con- I. Now that the preceding discussions have made cept of mass. us familiar with Newton's ideas, we are sufficiently prepared to enter on a critical examination of them. We shall restrict ourselves primarily in this, to the consideration of the concept of mass and the principle of reaction. The two cannot, in such an examination, be separated; in them is contained the gist of Newton's achievement. The expres- 2. In the first place we do not find the expression sio "ll quantity of mat-' quantity of matter " adapted to explain and elucidate ter." the concept of mass, since that expression itself is not possessed of the requisite clearness. And this is so, though we go back, as many authors have done, to an enumeration of the hypothetical atoms. We only complicate, in so doing, indefensible conceptions. If we place together a number of equal, chemically homogeneous bodies, we can, it may be granted, connect some clear idea with "quantity of matter," and we perceive, also, tlat the resistance the bodies offer to motion increases with this quantity. But the moment we suppose chemical heterogeneity, the assumption that THE PRINCIPLES OF D YNAVIlCS. 217 there is still something that is measurable by the same Newton's formulation standard, which something we call quantity of matter, of the concept. may be suggested by mechanical experiences, but is an assumption nevertheless that needs to be justified. When therefore, with Newton, we make the assumptions, respecting pressure due to weight, that p m g, p' 'g, and put in conformity with such assumptions pip' ~i m/i', we have made actual use in the operation thus performed of the supposition, yet to be justified, that different bodies are measurable by the same standard. We might, indeed, arbitrarily posit, that m/r/11' p/p'; that is, might define the ratio of mass to be the ratio of pressure due to weight when g was the same. But we should then have to substantiate the use that is made of this notion of mass in the principle of reaction and in other relations. <- --_> <-_A B> Fig. 14o a. Fig. 140 b. 3. When two bodies (Fig. 140), perfectly equal Anewformulation of in all respects, are placed opposite each other, we ex- the conpect, agreeably to the principle of symmetry, that they will produce in each other in the direction of their line of junction equal and opposite accelerations. But if these bodies exhibit any difference, however slight, of form, of chemical constitution, or are in any other respects different, the principle of symmetry forsakes us, unless we assiume or know before/and that sameness of form or sameness of chemical constitution, or whatever else the thing in question may be, is not determinative. If, however, mechanical experiences clearly and indubitably point to the existence in bodies of a special and distinct property determinative of accelerations, 218 THE SCIENCE OF MECHANICS. nothing stands in the way of our arbitrarily establishing the following definition: Definition All those bodies are bodies of equal mass, owhich, muof equal inassesf. tually acting on each other, produce in each other equal and opposite accelerations. We have, in this, simply designated, or named, an actual relation of things. In the general case we proceed similarly. The bodies A and B receive respectively as the result of their mutual action (Fig. 140 b) the accelerations - cp and +- q', where the senses of the accelerations are indicated by the signs. We say then, B has cp/(p' times the mass of A. If we take A as our unit, we assigz to /that body th/e miass 1i whichz imparts to A mi t imes the accelerationl that A in t/e ireaction imparts to it. The ratio of the masses is the negative inverse ratio of the counter-accelerations. That these accelerations always have opposite signs, that there are therefore, by our definition, only positive masses, is a point that experience teaches, and experience alone Character can teach. In our concept of mass no theory is inof the definition. volved; " quantity of matter" is wholly unnecessary in it; all it contains is the exact establishment, designation, and denomination of a fact. (Compare Appendix, XVIII., p. 539.) 4. One difficulty should not remain unmentioned in this connection, inasmuch as its removal is absolutely necessary to the formation of a perfectly clear concept of mass. We consider a set of bodies, A,, C, D..., and compare them all with A as unit. A, B, C, D, E, F. 1, m, m', lm", ml"', m"" We find thus the respective mass-values, 1, m, in', m"...., and so forth. The question now arises, If we TIIE PRINCIPLES OF D YNAMIICS. 219 select B as our standard of comparison (as our unit), Discussion of a diffishall we obtain for C the mass-value m'/mi, and for D culty involved in the value n"/Im, or will perhaps wholly different values the preceding formuresult? More simply, the question may be put thus: lation. Will two bodies B, C, which in mutual action with A have acted as equal masses, also act as equal masses in mutual action with each other? No logical necessity exists whatsoever, that two masses that are equal to a third mass should also be equal to each other. For we are concerned here, not with a mathematical, but with a physical question. This will be rendered quite clear by recourse to an analogous relation. We place by the side of each other the bodies A, B, C in the proportions of weight a, b, c in which they enter into the chemical combinations AB and A C. There exists, now, no logical necessity at all for assuming that the same proportions of weight b, c of the bodies B, C will also enter into the chemical combination BC. Experience, however, informs us that they do. If we place by the side of each other any set of bodies in the proportions of weight in which they combine with the body A, they will also unite with each other in the same proportions of weight. But no one can know this who has not tried it. And this is precisely the case with the mass-values of bodies. If we were to assume that the order of combination The order of combiof the bodies, by which their mass-values are deter- nation not influential mined, exerted any influence on the mass-values, the consequences of such an assumption would, we should find, lead to conflict with experience. Let us suppose, for instance (Fig. 141), that we have three elastic bodies, A, B, C, movable on an absolutely smooth and rigid ring. We presuppose that A and B in their mutual relations comport themselves like equal masses 220 THE SCIENCE OF ME ZCHANICS. and that B and C do the same. We are then also obliged to assume, if we wish to avoid conflicts with experience, that C and A in their mutual relations act like equal masses. If we impart to A a velocity, A will transmit this velocity by impact to B, and B to C. But if C were to act towards A, say, as a greater mass, A on impact would acquire a greater velocity than it originally had while C would still retain a residue of what it had. With every revolution in the direction of the hands of a watch the vis viva of the system would be increased. If C were the Fig. 141. smaller mass as compared with A, reversing the motion would produce the same result. But a constant increase of vis viva of this kind is at decided variance with our ex.perience. The new 5. The concept of mass when reached in the manconcept of mass in- ner just developed renders unnecessary the special volves implicitly the enunciation of the principle of reaction. In the conprinciple of reaction cept of mass and the principle of reaction, as we have stated in a preceding page, the same fact is twice formulated; which is redundant. If two masses I and 2 act on each other, our very definition of mass asserts that they impart to each other contrary accelerations which are to each other respectively as 2:1. 6. The fact that mass can be measured by weight, where the acceleration of gravity is invariable, can also be deduced from our definition of mass. We are sensible at once of any increase or diminution of a pressure, but this feeling affords us only a very inexact and indefinite measure of magnitudes of pressure. An exact, serviceable measure of pressure springs from the observation that every pressure is replaceable by T7111 J'A IVCIPLES OF1)D YVAAIICS 22 221 the pressure of a number of like and commensurable It also involves the weights. Every pressure can be counterbalanced by fact that,ass can be the pressure of weights of this kind. Let two bodies measured in and in' be respectively affected in opposite directions by weight. with the accelerations p and q/, determined by external circumstances. And let the bodies be joined by a string. If equilibrium prevails, the acceleration q- in in and the acceleration q' in in' are exactly balanced by inlcl-aclion. For this case, accordingly, in 9 -- m'iq'. When, <_ therefore, pqi-p', as is the case P g9 when the bodies are abandoned to the acceleration of gravity, we have, in the case of equilibrium, also in --i'. It is obviously immaterial whether we make the bodies act on each other directly by means of a string, or by means of a string passed over a pulley, or by placing them on the two pans of a balance. The fact that mass can be measured by weight is evident from our definition without recourse or reference to "quantity of matter." 7. As soon therefore as we, our attention being The general results of drawn to the fact by experience, have &crccizvd in bod- this view. ies the existence of a special-property determinative of accelerations, our task with egard to it ends with the recognition and unequivocal designation of this fact. Beyond the recognition of this fact we shall not get, and every venture beyond it will only be productive of obscurity. All uneasiness will vanish when once we have made clear to ourselves that in the concept of mass no theory of any kind whatever is contained, but simply a fact of experience. The concept has hitherto held good. It is very improbable, but not impossible, that it will be shaken in the future, just as the concep 222 TIIHE SCIENCE OF MECHANICS. tion of a constant quantity of heat, which also rested on experience, was modified by new experiences. VI. NEWTON'S VIEWS OF TIME, SPACE, AND MOTION. i. In a scholium which he appends immediately to his definitions, Newton presents his views regarding time and space-views which we shall now proceed to examine more in detail. We shall literally cite, to this end, only the passages that are absolutely necessary to the characterisation of Newton's views. Newton's "So far, my object has been to explain the senses views of time, space, "in which certain words little known are to be used in and motion. moion.the sequel. Time, space, place, and motion, being "words well known to everybody, I do not define. Yet "it is to be remarked, that the vulgar conceive these "quantities only in their relation to sensible objects. "And hence certain prejudices with respect to them (have arisen, to remove which it will be convenient to "distinguish them into absolute and relative, true and "apparent, mathematical and common, respectively. Absolute ''I. Absolute, true, and mathematical time, of itand relative time, "self, and by its own nature, flows uniformly on, with"out regard to anything external. It is also called "duranion. "Relative, apparent, and common time, is some "sensible and external measure of absolute time (dura"tion), estimated by the motions of bodies, whether "accurate or inequable, and is commonly employed 1" in place of true time; as an hour, a day, a month, "a year... c"The natural days, which, commonly, for the pur" pose of the measurement of time, are held as equal. "are in reality unequal. Astronomers correct this in THE PRVINCIPLES OF D YNAMICS. 223 "equality, in order that they may measure by a truer "time the celestial motions. It may be that there is " no equable motion, by which time can accurately be "measured. All motions can be accelerated and re"tarded. But the flow of absolute time cannot be "changed. Duration, or the persistent existence of " things, is always the same, whether motions be swift "or slow or null." 2. It would appear as though Newton in the re-Discussion of Newton's marks here cited still stood under the influence of the view of mediaeval philosophy, as though he had grown unfaith-me. ful to his resolve to investigate only actual facts. When we say a thing A changes with the time, we mean simply that the conditions that determine a thing A depend on the conditions that determine another thing B. The vibrations of a pendulum take place in time when its excursion depenads on the position of the earth. Since, however, in the observation of the pendulum, we are not under the necessity of taking into account its dependence on the position of the earth, but may compare it with any other thing (the conditions of which of course also depend on the position of the earth), the illusory notion easily arises that all the things with which we compare it are unessential. Nay, we may, in attending to the motion of a pendulum, neglect entirely other external things, and find that for every position of it our thoughts and sensations are different. Time, accordingly, appears to be some particular and independent thing, on the progress of which the position of the pendulum depends, while the things that we resort to for comparison and choose at random appear to play a wholly collateral part. But we must not forget that all things in the world are connected with one another and depend on one another, and that 224 TIHE SCIENCE OF MECHANICS. General we ourselves and all our thoughts are also a part of discussion of the con- nature. It is utterly beyond our power to measure the cept of time. changes of things by time. Quite the contrary, time is an abstraction, at which we arrive by means of the changes of things; made because we are not restricted to any one definite measure, all being interconnected. A motion is termed uniform in which equal increments of space described correspond to equal increments of space described by some motion with which we form a comparison, as the rotation of the earth. A motion may, with respect to another motion, be uniform. But the question whether a motion is in itself uniform, is senseless. With just as little justice, also, may we speak of an " absolute time "-of a time iindependent of change. This absolute time can be measured by comparison with no motion; it has therefore neither a practical nor a scientific value; and no one is justified in saying that he knows aught about it. It is an idle metaphysical conception. Further elu- It would not be difficult to show from the points of cidation of the idea. view of psychology, history, and the science of language (by the names of the chronological divisions), that we reach our ideas of time in and through the interdependence of things on one another. In these ideas the profoundest and most universal connection of things is expressed. When a motion takes place in time, it depends on the motion of the earth. This is not refuted by the fact that mechanical motions can be reversed. A number of variable quantities may be so related that one set can suffer a change without the others being affected by it. Nature behaves like a machine. The individual parts reciprocally determine one another. But while in a machine the position of one part determines the position of all the other parts, in nature THE PRIArCIPLES OF D YAAMICS. 225 more complicated relations obtain. These relations are best represented under the conception of a number, 1n, of quantities that satisfy a lesser number, n', of equations. Were n -- n', nature would be invariable. Were n'= n - 1, then with one quantity all the rest would be controlled. If this latter relation obtained in nature, time could be reversed the moment this had been accomplished with any one single motion. But the true state of things is represented by a different relation between n and n'. The quantities in question are partially determined by one another; but they retain a greater indeterminateness, or freedom, than in the case last cited. We ourselves feel that we are such a partially determined, partially undetermined element of nature. In so far as a portion only of the changes of nature depends on us and can be reversed by us, does time appear to us irreversible, and the time that is past as irrevocably gone. We arrive at the idea of time,-to express it briefly Some psy-. chological and popularly,-by the connection of that which is considera-. tions. contained in the province of our memory with that which is contained in the province of our sense-perception. When we say that time flows on in a definite direction or sense, we mean that physical events generally (and therefore also physiological events) take place only in a definite sense.* Differences of temperature, electrical differences, differences of level generally, if left to themselves, all grow less and not greater. If we contemplate two bodies of different temperatures, put in contact and left wholly to themselves, we shall find that it is possible only for greater differences of temperature in the field of memory to * Investigations concerning the physiological nature of the sensations of time and space are here excluded from consideration. 226 THE SCIENCE OF MEOCHANICS. exist with lesser ones in the field of sense-perception, and not the reverse. In all this there is simply expressed a peculiar and profound connection of things. To demand at the present time a full elucidation of this matter, is to anticipate, in the manner of speculative philosophy, the results of all future special investigation, that is a perfect physical science. (Compare Appendix, XIX., p. 541.) Newton's 3. Views similar to those concerning time, are deviews of space and veloped by Newton with respect to space and motion. motion. We extract here a few passages which characterise his position. "II. Absolute space, in its own nature and with"out regard to anything external, always remains sim"ilar and immovable. "Relative space is some movable dimension or "measure of absolute space, which our senses deter"mine by its position with respect to other bodies, " and which is commonly taken for immovable [abso"lute] space.... " IV. Absolute motion is the translation of a body "from one absolute place* to another absolute place; " and relative motion, the translation from one relative " place to another relative place.... Passages ".... And thus we use, in common affairs, instead from his works. of absolute places and motions, relative ones; and "that without any inconvenience. But in physical "disquisitions, we should abstract from the senses. " For it may be that there is no body really at rest, to "which the places and motions of others can be re"ferred.... " The effects by which absolute and relative motions * The place, or locus of a body, according to Newton, is not its position, but the/art of space which it occupies. It is either absolute or relative.-Trans THE PRINCIPLES OF D YNAMI4ICS. 227 "are distinguished from one another, are centrifugal "forces, or those forces in circular motion which pro"duce a tendency of recession from the axis. For in 'a circular motion which is purely relative no such "forces exist; but in a true and absolute circular mo"tion they do exist, and are greater or less according "to the quantity of the [absolute] motion. " For instance. If a bucket, suspended by a long The rotating bucket. Scord, is so often turned about that finally the cord istingbucket 'strongly twisted, then is filled with water, and held "at rest together with the water; and afterwards by Sthe action of a second force, it is suddenly set whirlSing about the contrary way, and continues, while the 'cord is untwisting itself, for some time in this mo"tion; the surface of the water will at first be level, "just as it was before the vessel began to move; but, "subsequently, the vessel, by gradually communicatSing its motion to the water, will make it begin sens"ibly to rotate, and the water will recede little by little Sfrom the middle and rise up at the sides of the ves'sel, its surface assuming a concave form. (This ex( periment I have made myself.)..... At first, when the -relative motion of the wa- Relative and real 'ter in the vessel was greatest, that motion produced motion. ( no tendency whatever of recession from the axis; the Swater made no endeavor to move towards the cir"cumference, by rising at the sides of the vessel, but "remained level, and for that reason its true circular "motion had not yet begun. But afterwards, when Sthe relative motion of the water had decreased, the "rising of the water at the sides of the vessel indicated can endeavor to recede from the axis; and this en" deavor revealed the real circular motion of the water, "continually increasing, till it had reached its greatest 228 THE SCIENCEO OFF MECII4ANCS. "point, when relatively the water was at rest in the Svessel.... " It is indeed a matter of great difficulty to discover "and effectually to distinguish the true from the ap"parent motions of particular bodies; for the parts of " that immovable space in which bodies actually move, " do not come under the observation of our senses. Newton's "Yet the case is not altogether desperate; for there criteria for distinguish- "exist to guide us certain marks, abstracted partly ing absolute from rela- "from the apparent motions, which are the differences tive motion. "of the true motions, and partly from the forces that "are the causes and effects of the true motions. If, "for instance, two globes, kept at a fixed distance "from one another by means of a cord that connects "them, be revolved about their common centre of "gravity, one might, from the simple tension of the "cord, discover the tendency of the globes to recede "from the axis of their motion, and on this basis the "quantity of their circular motion might be computed. "And if any equal forces should be simultaneously "impressed on alternate faces of the globes to augment " or diminish their circular motion, we might, from "the increase or decrease of the tension of the cord, " deduce the increment or decrement of their motion; "and it might also be found thence on what faces "forces would have to be impressed, in order that the "motion of the globes should be most augmented; "that is, their rear faces, or those which, in the cirScular motion, follow. But as soon as we knew which "faces followed, and consequently which preceded, we "should likewise know the direction of the motion. " In this way we might find both the quantity and the "direction of the circular motion, considered even in "an immense vacuum, where there was nothing ex THE PRINCIPLES OF D YNA4MICS. 229 "ternal or sensible with which the globes could be "compared..... 4. It is scarcely necessary to remark that in the re- The predications of flections here presented Newton has again acted con- Newton are not the trary to his expressed intention only to investigate aclual expression of actual facts. No one is competent to predicate things about facts. absolute space and absolute motion; they are pure things of thought, pure mental constructs, that cannot be produced in experience. All our principles of mechanics are, as we have shown in detail, experimental knowledge concerning the relative positions and motions of bodies. Even in the provinces in which they are now recognised as valid, they could not, and were not, admitted without previously being subjected to experimental tests. No one is warranted in extending these principles beyond the boundaries of experience. In fact, such an extension is meaningless, as no one possesses the requisite knowledge to make use of it. Let us look at the matter in detail. When we say that Detailed view of the a body K alters its direction and velocity solely through matter. the influence of another body K', we have asserted a conception that it is impossible to come at unless other bodies A, B, C... are present with reference to which the motion of the body K has been estimated. In reality, therefore, we are simply cognisant of a relation of the body K to A, B, C.... If now we sud-A denly neglect A, B, C.... and attempt to speak of the deportment of the body K in absolute space, we implicate ourselves in a twofold error. In the first place, we cannot know how K would act in the absence of A, B, C...; and in the second place, every means would be wanting of forming a judgment of the behaviour of K and of putting to the test what we had 230 THE SCIENCE OF MEClHANICS. predicated,-which latter therefore would be bereft of all scientific significance. The part Two bodies K and K", which gravitate toward each which the bodies of other, impart to each other in the direction of their space play in the de- line of junction accelerations inversely proportional to terminationP of motion. their masses im, m'. In this proposition is contained, not only a relation of the bodies K and K' to one another, but also a relation of them to other bodies. For the proposition asserts, not only that K and K' suffer with respect to one another the acceleration designated by n (min- m'/r2), but also that K experiences the acceleration - 71 m'/r2 and K' the acceleration +- um/r2 in the direction of the line of junction; facts which can be ascertained only by the presence of other bodies. The motion of a body K can only be estimated by reference to other bodies A,, C.... But since we always have at our disposal a sufficient number of bodies, that are as respects each other relatively fixed, or only slowly change their positions, we are, in such reference, restricted to no one definite body and can alternately leave out of account now this one and now that one. In this way the conviction arose that these bodies are indifferent generally. The hy- It might be, indeed, that the isolated bodies A, B, pothesis of a medium C.... play merely a collateral r61e in the determinain space determinative tion of the motion of the body K", and that this motion of motion. is determined by a medium in which K exists. In such a case we should have to substitute this medium for Newton's absolute space. Newton certainly did not entertain this idea. Moreover, it is easily demonstrable that the atmosphere is not this motion-determinative medium. We should, therefore, have to picture to ourselves some other medium, filling, say, all space, with respect to the constitution of which and its kinetic TIHE PRICIPLES OF D YNA'AIICS. 231 relations to the bodies placed in it we have at present no adequate knowledge. In itself such a state of things would not belong to the impossibilities. It is known, from recent hydrodynamical investigations, that a rigid body experiences resistance in a frictionless fluid only when its velocity chainges. True, this result is derived theoretically from the notion of inertia; but it might, conversely, also be regarded as the primitive fact from which we have to start. Although, practically, and at present, nothing is to be accomplished with this conception, we might still hope to learn more in the future concerning this hypothetical medium; and from the point of view of science it would be in every respect a more valuable acquisition than the forlorn idea of absolute space. When we reflect that we cannot abolish the isolated bodies A, B, C...., that is, cannot determine by experiment whether the part they play is fundamental or collateral, that hitherto they have been the sole and only competent means of the orientation of motions and of the description of mechanical facts, it will be found expedient provisionally to regard all motions as determined by these bodies. 5. Let us now examine the point on which New- critical S.. examinaton, apparently with sound reasons, rests his distinc- tion of. Newton's tion of absolute and relative motion. If the earth is distinction of absolute affected with an atsolute rotation about its axis, cen- from relative motion. trifugal forces are set up in the earth: it assumes an oblate form, the acceleration of gravity is diminished at the equator, the plane of Foucault's pendulum rotates, and so on. All these phenomena disappear if the earth is at rest and the other heavenly bodies are affected with absolute motion round it, such that the same relative rotation is produced. This is, indeed, the case, if we start ab initio from the idea of absolute space. 232 THE SCIENCE OF IMECHIIANICS. But if we take our stand on the basis of facts, we shall find we have knowledge only of relative spaces and motions. Relatively, not considering the unknown and neglected medium of space, the motions of the universe are the same whether we adopt the Ptolemaic or the Copernican mode of view. Both views are, indeed, equally correct; only the latter is more simple and more practical. The universe is not twice given, with an earth at rest and an earth in motion; but only once, with its relative motions, alone determinable. It is, accordingly, not permitted us to say how things would be if the earth did not rotate. We may interpret the one case that is given us, in different ways. If, however, we so interpret it that we come into conflict with experience, our interpretation is simply wrong. The principles of mechanics can, indeed, be so conceived, that even for relative rotations centrifugal forces arise. Interpreta- Newton's experiment with the rotating vessel of tion of the experiment water simply informs us, that the relative rotation of with the rotating the water with respect to the sides of the vessel probucket of water. duces no noticeable centrifugal forces, but that such forces are produced by its relative rotation with respect to the mass of the earth and the other celestial bodies. No one is competent to say how the experiment would turn out if the sides of the vessel increased in thickness and mass till they were ultimately several leagues thick. The one experiment only lies before us, and our business is, to bring it into accord with the other facts known to us, and not with the arbitrary fictions of our imagination. 6. We can have no doubts concerning the significance of the law of inertia if we bear in mind the manner in which it was reached. To begin with, Galileo discovered the constancy of the velocity and direction TIE PFRINCIPLES OF D YN4A4IICS. 233 of a body referred to terrestrial objects. Most terres- The law of inertia in trial motions are of such brief duration and extent, that the light of this view. it is wholly unnecessary to take into account the earth's rotation and the changes of its progressive velocity with respect to the celestial bodies. This consideration is found necessary only in the case of projectiles cast great distances, in the case of the vibrations of Foucault's pendulum, and in similar instances. When now Newton sought to apply the mechanical principles discovered since Galileo's time to the planetary system, he found that, so far as it is possible to form any estimate at all thereof, the planets, irrespectively of dynamic effects, appear to preserve their direction and velocity with respect to bodies of the universe that are very remote and as regards each other apparently fixed, the same as bodies moving on the earth do with respect to the fixed objects of the earth. The comportment of terrestrial bodies with respect to the earth is reducible to the comportment of the earth with respect to the remote heavenly bodies. If we were to assert that we knew more of moving objects than this their last- mentioned, experimentally- given comportment with respect to the celestial bodies, we should render ourselves culpable of afalsity. When, accordingly, we say, that a body preserves unchanged its direction and velocity zi space, our assertion is nothing more or less than an abbreviated reference to the entic-c unlivcrse. The use of such an abbreviated expression is permitted the original author of the principle, because he knows, that as things are no difficulties stand in the way of carrying out its implied directions. But no remedy lies in his power, if difficulties of the kind mentioned present themselves; if, for example, the requisite, relatively fixed bodies are wanting. 234 TIlE SCIENCE OF MlECHIANiICS. The rela- 7. Instead, now, of referring a moving body K to tion of the bodies of space, that is to say to a system of co6rdinates, let us the universe to view directly its relation to the bodies of the universe, each other. by which alone such a system of co6rdinates can be determined. Bodies very remote from each other, moving with constant direction and velocity with respect to other distant fixed bodies, change their mutual distances proportionately to the time. We may also say, All very remote bodies-all mutual or other forces neglected-alter their mutual distances proportionately to those distances. Two bodies, which, situated at a short distance from one another, move with constant direction and velocity with respect to other fixed bodies, exhibit more complicated relations. If we should regard the two bodies as dependent on one another, and call r-the distance, / the time, and a a constant dependent on the directions and velocities, the formula would be obtained: dtr/d2 =/2 (1/r) [a2 -- (r/dI)2]. It is manifestly much simplcr and clearer to regard the two bodies as independent of each other and to consider the constancy of their direction and velocity with respect to other bodies. Instead of saying, the direction and velocity of a mass p in space remain constant, we may also employ the expression, the mean acceleration of the mass with respect to the masses in, iii', i".. at the distances r, r', r"... is -=0, or d2(,m,/N m)/a'2 -0. The latter expression is equivalent to the former, as soon as we take into consideration a sufficient number of sufficiently distant and sufficiently large masses. The mutual influence of more proximate small masses, which are apparently not concerned about each other, is eliminated of itself. That the constancy of direction and velocity is given by the condition adduced, will be THE PRINCIPLES OF D YNAflICS. 235 seen at once if we construct through iy as vertex cones The expression of the that cut out different portions of space, and set up the law of inertia in terms condition with respect to the masses of these separate of this relation. portions. We may put, indeed, for the entire space encompassing p, d2 (2;m' -/Ym) /d12 - 0. But the equation in this case asserts nothing with respect to the motion of p, since it holds good for all species of motion where,u is uniformly surrounded by an infinite number of masses. If two masses p,, u2 exert on each other a force which is dependent on their distance r, then d2r/dt2 _= (/I + /U2)f(r). But, at the same time, the acceleration of the centre of gravity of the two masses or the mean acceleration of the mass-system with respect to the masses of the universe (by the principle of reaction) remains = 0; that is to say, d1t2 r Y P7 2 -- 0. d 2 2mr 2mr, 1 When we reflect that the time-factor that enters The necessity in sciinto the acceleration is nothing more than a quantity ence of a considerathat is the measure of the distances (or angles of rota- tion of the All. tion) of the bodies of the universe, we see that even in the simplest case, in which apparently we deal with the mutual action of only two masses, the neglecting of the rest of the world is impzossible. Nature does not begin with elements, as we are obliged to begin with them. It is certainly fortunate for us, that we can, from time to time, turn aside our eyes from the overpowering unity of the All, and allow them to rest on individual details. But we should not omit, ultimately to complete and correct our views by a thorough consideration of the things which for the time being we left out of account. 8. The considerations just presented show, that it 236 THE SCIENCE OF IMECHANICS. The law of is not necessary to refer the law of inertia to a special inertia does not involve absolute space. On the contrary, it is perceived that absolute space. the masses that in the common phraseology exert forces on each other as well as those that exert none, stand with respect to acceleration in quite similar relations. We may, indeed, regard all masses as related to each other. That accelerations play a prominent part in the relations of the masses, must be accepted as a fact of experience; which does not, however, exclude attempts to elucidate this fact by a comparison of it with other facts, involving the discovery of new points of view. In all the processes of nature the differences of certain quantities 1t play a determinative role. DifferS ences of temperature, of potential function, and so Str forth, induce the natural - - --p processes, which consist Fig. 143. in the equalisation of Natural these differences. The familiar expressions d(12udx2, processes consist in d'21 i/ 2, d2u/dz2, which are determinative of the the equalisation of character of the equalisation, may be regarded as the the differences of measure of the departure of the condition of any point quantities. from the mean of the conditions of its environmentto which mean the point tends. The accelerations of masses may be analogously conceived. The great distances between masses that stand in no especial forcerelation to one another, change proportionately to each other. If we lay off, therefore, a certain distance p as abscissa, and another r as ordinate, we obtain a straight line. (Fig. 143.) Every i-ordinate corresponding to a definite p-value represents, accordingly, the mean of the adjacent ordinates. If a force-relation exists between the bodies, some value dr2r/dl2 is determined THE PR'IVCIPLES OF DYN.4MICS. 237 by it which conformably to the remarks above we may replace by an expression of the form d2r/dp2. By the force-relation, therefore, a derarture of the r-ordinate from the mean; f fte adjacent ordinates is produced, which would not exist if the supposed force-relation did not obtain. This intimation will suffice here. 9. We have attempted in the foregoing to give the character bof the new law of inertia a different expression from that in ordi- expression for the law nary use. This expression will, so long as a suffi-of inertia. cient number of bodies are apparently fixed in space, accomplish the same as the ordinary one. It is as easily applied, and it encounters the same difficulties. In the one case we are unable to come at an absolute space, in the other a limited number of masses only is within the reach of our knowledge, and the summation indicated can consequently not be fully carried out. It is impossible to say whether the new expression would still represent the true condition of things if the stars were to perform rapid movements among one another. The general experience cannot be constructed from the particular case given us. We must, on the contrary, wait until such an experience presents itself. Perhaps when our physico-astronomical knowledge has been extended, it will be offered somewhere in celestial space, where more violent and complicated motions take place than in our environment. The most impor- The simplest printant result of our reflexions is, however, tiat precisely ciples of mechanics tie apparcntly simplest mccaical principles are f a vrare of a highly comcompip'cated c/aracler, t/at tf/se pri'nciples are founded on plicated nature and are uncomplfleted experiences, 1nay on experiences ia(t llnever canl all derived from expele fully conmpleted, tliaft practical'l, llindeed, tcy al'e suf- rience. ficiently secured, iin view of the tolerable stalbility of our environment, to serve as t/e fouindalion of imatuematical deduclion', but tialt thery can byi no means /themselves be re 238 7THE SCIENCA OF AfE'CILAANICS. gardcd as mniatihenmatically established truths but only as principles that not only admit of cons/ant control by, experience but actually require it. This perception is valuable in that it is propitious to the advancement of science. (Compare Appendix, XX., p. 542.) VII. SYNOPTICAL CRITIQUE OF THE NEWTONIAN ENUNCIATIONS. Newton's i. Now that we have discussed the details with Definitions...., Definitionssufficient particularity, we may pass again under review the form and the disposition of the Newtonian enunciations. Newton premises to his work several definitions, following which he gives the laws of motion. We shall take up the former first. Mass. "Definition I. The quantity of any matter is the "measure of it by its density and volume conjointly.... This quantity is what I shall understand by the "term mass or body in the discussions to follow. It is "ascertainable from the weight of the body in ques"tion. For I have found, by pendulum-experiments "of high precision, that the mass of a body is propor"tional to its weight; as will hereafter be shown. Quantity of "Definition II. Quantity of motion is the measure motion, inertia, "of it by the velocity and quantity of matter conforce, and accelera- 'jointly. ion. "Definition III. The resident force [zvis insita, i. e. "the inertia] of matter is a power of resisting, by "which every body, so far as in it lies, perseveres in "its state of rest or of uniform motion in a straight "line. "Definition IV. An impressed force is any action "upon a body which changes, or tends to change, its "state of rest, or of uniform motion in a straight line. THE PRINCIPLES OF D YLNAMJICS. 239 "Definition V. A centripetal force is any force by " which bodies are drawn or impelled towards, or tend "in any way to reach, some point as centre. "Definition VI. The absolute quantity of a centri- Forcesciassiftied as abSpetal force is a measure of it increasing and dimin- solute, accelerative, "ishing with the efficacy of the cause that propagates and nov"it from the centre through the space round about. n. "Definition VII. The accelerative quantity of a "centripetal force is the measure of it proportional to "the velocity which it generates in a given time. "Definition VIII. The moving quantity of a cen"tripetal force is the measure of it proportional to the "motion [See Def. II.] which it generates in a given "time. "The three quantities or measures of force thus dis- The relations of the "tinguished, may, for brevity's sake, be called abso- forces thus distin"lute, accelerative, and moving forces, being, for dis- guished. "tinction's sake, respectively referred to the centre of "force, to the places of the bodies, and to the bodies "that tend to the centre: that is to say, I refer moving "force to the body, as being an endeavor of the whole "towards the centre, arising from the collective en"deavors of the several parts; accelerative force to the "place of the body, as being a sort of efficacy originat"ing in the centre and diffused throughout all the sev"eral places round about, in moving the bodies that "are at these places; and absolute force to the centre, "as invested with some cause, without which moving "forces would not be propagated through the space "round about; whether this latter cause be some cen"tral body, (such as is a loadstone in a centre of mag"netic force, or the earth in the centre of the force of "gravity,) or anything else not visible. This, at least, "is the mathematical conception of forces; for their 240 TIHE SCIANCE OF AMfECHANICS. "physical causes and seats I do not in this place conSsider. The dis- "Accelerating force, therefore, is to moving force, tinction mathemat- i as velocity is to quantity of motion. For quantity ical and not physical. "of motion arises from the velocity and the quantity " of matter; and moving force arises from the accel" erating force and the same quantity of matter; the " sum of the effects of the accelerative force on the sev"eral particles of the body being the motive force of "the whole. Hence, near the surface of the earth, "where the accelerative gravity or gravitating force is "in all bodies the same, the motive force of gravity or " the weight is as the body [mass]. But if we ascend "to higher regions, where the accelerative force of " gravity is less, the weight will be equally diminished, " always remaining proportional conjointly to the mass " and the accelerative force of gravity. Thus, in those "regions where the accelerative force of gravity is half " as great, the weight of a body will be diminished by "one-half. Further, I apply the terms accelerative and "motive in one and the same sense to attractions and "to impulses. I employ the expressions attraction, im"pulse, or propensity of any kind towards a centre, "promiscuously and indifferently, the one for the other; Sconsidering those forces not in a physical sense, but "mathematically. The reader, therefore, must not "infer from any expressions of this kind that I may "use, that I take upon me to explain the kind or the "mode of an action, or the causes or the physical rea"son thereof, or that I attribute forces in a true or "physical sense, to centres (which are only mathemat"ical points), when at any time I happen to say that "centres attract or that central forces are in action." THE PRINCIPLES OF D YNA4MICS. 241 2. Definition i is, as has already been set forth, a Criticism of N ewton's pseudo-definition. The concept of mass is not made Definitions. clearer by describing mass as the product of the volume into the density, as density itself denotes simply the mass of unit of volume. The true definition of mass can be deduced only from the dynamical relations of bodies. To Definition IT, which simply enunciates a mode of computation, no objection is to be made. Definition ini (inertia), however, is rendered superfluous by Definitions v-viii of force, inertia being included and given in the fact that forces are accelerative. Definition iv defines force as the cause of the acceleration, or tendency to acceleration, of a body. The latter part of this is justified by the fact that in the cases also in which accelerations cannot take place, other attractions that answer thereto, as the compression and distension etc. of bodies occur. The cause of an acceleration towards a definite centre is defined in Definition v as centripetal force, and is distinguished in vi, vii, and viii as absolute, accelerative, and motive. It is, we may say, a matter of taste and of form whether we shall embody the explication of the idea of force in one or in several definitions. In point of principle the Newtonian definitions are open to no objections. 3. The Axioms or Laws of Motion then follow, of Newton's Laws of which Newton enunciates three: Motion. -' Law I. Every body perseveres in its state of rest "or of uniform motion in a straight line, except in so "far as it is compelled to change that state by im"pressed forces." " Law II. Change of motion [i. e. of momentum] is "proportional to the moving force impressed, and takes 242 THE SCIENCE OF MECHANICS. "place in the direction of the straight line in which "such force is impressed." "Law III. Reaction is always equal and opposite "to action; that is to say, the actions of two bodies "upon each other are always equal and directly opSposite." Newton appends to these three laws a number of Corollaries. The first and second relate to the principle of the parallelogram of forces; the third to the quantity of motion generated in the mutual action of bodies; the fourth to the fact that the motion of the centre of gravity is not changed by the mutual action of bodies; the fifth and sixth to relative motion. criticism of 4. We readily perceive that Laws I and II are conNewton's laws of tained in the definitions of force that precede. Acmotion. cording to the latter, without force there is no acceleration, consequently only rest or uniform motion in a straight line. Furthermore, it is wholly unnecessary tautology, after having established acceleration as the measure of force, to say again that change of motion is proportional to the force. It would have been enough to say that the definitions premised were not arbitrary mathematical ones, but correspond to properties of bodies experimentally given. The third law apparently contains something new. But we have seen that it is unintelligible without the correct idea of mass, which idea, being itself obtained only from dynamical experience, renders the law unnecessary. The corol- The first corollary really does contain something laries to theselaws. new. But it regards the accelerations determined in a body K by different bodies M, N, P as self-evidently independent of each other, whereas this is precisely what should have been explicitly recognised as a fact of experience. Corollary Second is a simple applica TIE PRINCIPLES OF DYNAMIIlCS. 243 tion of the law enunciated in corollary First. The remaining corollaries, likewise, are simple deductions, that is, mathematical consequences, from the conceptions and laws that precede. 5. Even if we adhere absolutely to the Newtonian points of view, and disregard the complications and indefinite features mentioned, which are not removed but merely concealed by the abbreviated designations "Time" and " Space," it is possible to replace Newton's enunciations by much more simple, methodically better arranged, and more satisfactory propositions. Such, in our estimation, would be the following: a. Experimental Proposition. Bodies set opposite Proposed substitueach other induce in each other, under certain circum- tions for the Newstances to be specified by experimental physics, con- tonian laws and definitrary accelerations in the direction of their line of junc- tions. tion. (The principle of inertia is included in this.) b. Definition. The mass-ratio of any two bodies is the negative inverse ratio of the mutually induced accelerations of those bodies. c. Experimental Proposition. The mass-ratios of bodies are independent of the character of the physical states (of the bodies) that condition the mutual accelerations produced, be those states electrical, magnetic, or what not; and they remain, moreover, the same, whether they are mediately or immediately arrived at. d. Experimental Proposition. The accelerations which any number of bodies A, B, C.... induce in a body K, are independent of each other. (The principle of the parallelogram of forces follows immediately from this.) e. Definition. Moving force is the product of the mass-value of a body into the acceleration induced in that body. 244 THE SCIENCE OF MECHANICS. Extent and Then the remaining arbitrary definitions of the alcharacter of the pro- gebraical expressions "momentum," "vis viva," and posed substitutions. the like, might follow. But these are by no means indispensable. The propositions above set forth satisfy the requirements of simplicity and parsimony which, on economico-scientific grounds, must be exacted of them. They are, moreover, obvious and clear; for no doubt can exist with respect to any one of them either concerning its meaning or its source; and we always know whether it asserts an experience or an arbitrary convention. The 6. Upon the whole, we may say, that Newton disachievements of cerned in an admirable manner the concepts and princiNewton from the ples that were suficiently assured to allow of being furpoint of view of his ther built upon. It is possible that to some extent he time. was forced by the difficulty and novelty of his subject, in the minds of the contemporary world, to great amplitude, and, therefore, to a certain disconnectedness of presentation, in consequence of which one and the same property of mechanical processes appears several times formulated. To some extent, however, he was, as it is possible to prove, not perfectly clear himself concerning the import and especially concerning the source of his principles, This cannot, however, obscure in the slightest his intellectual greatness. He that has to acq~ a new point of view naturally cannot possess it so securely from the beginning as they that receive it unlaboriously from him. He has done enough if he has discovered truths on which future generations can further build. For every new inference therefrom affords at once a new insight, a new control, an extension of our prospect, and a clarification of our field of view. Like the commander of an army, a great discoverer cannot stop to institute petty THE PRINCIPLES OF D YNA42MI2CS. 245 inquiries regarding the right by which he holds each The achievepost of vantage he has won. The magnitude of the ments of Newton in problem to be solved leaves no time for this. But at the light of subsequent a later period, the case is different. Newton might research. well have expected of the two centuries to follow that they should further examine and confirm the foundations of his work, and that, when times of greater scientific tranquillity should come, the principles of the subject might acquire an even higher philosophical interest than all that is deducible from them. Then problems arise like those just treated of, to the solution of which, perhaps, a small contribution has here been made. We join with the eminent physicists Thomson and Tait, in our reverence and admiration of Newton. But we can only comprehend with difficulty their opinion that the Newtonian doctrines still remain the best and most philosophical foundation of the science that can be given. VIII. RETROSPECT OF THE DEVELOPMENT OF DYNAMICS. i. If we pass in review the period in which the de- The chief result, the velopment of dynamics fell,-a period inaugurated by discovery of one great Galileo, continued by Huygens, and brought to a close fact. by Newton,-its main result will be found to be the perception, that bodies mutually determine in each other acce/lralions dependent on definite spatial and material circumstances, and that there are masses. The reason the perception of these facts was embodied in so great a number of principles is wholly an historical one; the perception was not reached at once, but slowly and by degrees. In reality only one great fact was established. Different pairs of bodies determine, independently of each other, and mutually, in themselves, 246 THEA SCIENCE.OF Ifl1,'CIIAANICS. pairs of accelerations, whose terms exhibit a constant ratio, the criterion and characteristic of each pair. This fact Not even men of the calibre of Galileo, Huygens, even the greatest in- and Newton were able to perceive this fact at once. could per- Even they could only discover it piece by piece,asi onlyag is expressed in the law of falling bodies, in the special net. law of inertia, in the principle of the parallelogram of forces, in the concept of mass, and so forth. To-day, no difficulty any longer exists in apprehending the unity of the whole fact. The practical demands of communication alone can justify jts piecemeal presentation in several distinct principles, the number of which is really only determined by scientific taste. What is more, a reference to the reflections above set forth respecting the ideas of time, inertia, and the like, will surely conVince us that, accurately viewed, the entire fact has, in all its aspects, not yet been perfectly apprehended. The results The point of view reached has, as Newton expressly re ached have noth- states, nothing to do with the "unknown causes" of ing to do with the so- natural phenomena. That which in the mechanics of 4cassthe present day is called Jor-cc is not a something that of phenomenla. lies latent in the natural processes, but a measurable, actual circumstance of motion, the product of the mass into the acceleration. Also when we speak of the attractions or repulsions of bodies, it is not necessary to think of any hidden causes of the motions produced. We signalise by the term attraction merely an actually existing;-cscmblance between events determined by conditions of motion and the results of our volitional impulses. In both cases either actual motion occurs or, when the motion is counteracted by some other circumstance of motion, distortion, compression of bodies, and so forth, are produced. 2. The work which devolved on genius here, was TILLH PRINCIPLES OF D YVNAMICS. 247 the noting of the connection of certain determinative Theformof the meelements of the mechanical processes. The precise es- chanical principles, tablishment of the form of this connection was rather a in the main of histortask for plodding research, which created the different ical origin. concepts and principles of mechanics. We can determine the true value and significance of these principles and concepts only by the investigation of their historical origin. In this it appears unmistakable at times, that accidental circumstances have given to the course of their development a peculiar direction, which under other conditions might have been very different. Of this an example shall be given. Before Galileo assumed the familiar fact of the de- For example, Galipendence of the final velocity on the time, and put it to leo's laws of falling the test of experiment, he essayed, as we have already bodies might have seen, a different hypothesis, and made the final velocity taken a difSferent form. proportional to the space described. He imagined, by a course of fallacious reasoning, likewise already referred to, that this assumption involved a self-contradiction. His reasoning was, that twice any given distance of descent must, by virtue of the double final velocity acquired, necessarily be traversed in the same time as the simple distance of descent. But since the first half is necessarily traversed first, the remaining half will have to be traversed instantaneously, that is in an interval of time not measurable. Whence, it readily follows, that the descent of bodies generally is instantaneous. The fallacies involved in this reasoning are manifest. Galileo's reasoning Galileo was, of course, not versed in mental integra- and its errors. tions, and having at his command no adequate methods for the solution of problems whose facts were in any degree complicated, he could not but fall into mistakes whenever such cases were presented. If we call s the distance and / the time, the Galilean assumption reads 248 THEI SCIEANCE OF MELCILA-VICS. in the language of to-day ds/d t as, from which follows s = A E, where a is a constant of experience and A a constant of integration. This is an entirely different conclusion from that drawn by Galileo. It does not conform, it is true, to experience, and Galileo would probably have taken exception to a result that, as a condition of motion generally, made s different from 0 when / equalled 0. But in itself the assumption is by no means sc/f-contradictory. The suppo- Let us suppose that Kepler had put to himself the sition that Kepler ha same question. Whereas Galileo always sought after made GaliSeo's re- the very simplest solutions of things, and at once researches. jected hypotheses that did not fit, Kepler's mode of procedure was entirely different. He did not quail before the most complicated assumptions, but worked his way, by the constant gradual modification of his original hypothesis, successfully to his goal, as the history of his discovery of the laws of planetary motion fully shows. Most likely, Kepler, on finding the assumption ds/ldt = as would not work, would have tried a nunmber of others, and among them probably the correct one ds/dit a1 /s. But from this would have resulted an essentially different course of development for the science of dynamics. In such a It was only gradually and with great difficulty that case the consept the concept of "work" attained its present position might have of importance; and in our judgment it is to the aboveoriginal mentioned trifling historical circumstance that the difficoncept of mechanics. culties and obstacles it had to encounter are to be ascribed. As the interdependence of the velocity and the time was, as it chanced, first ascertained, it could not be otherwise than that the relation v g- R should appear as the original one, the equation s =-t 2/2 as the next immediate, and gs - v 2/2 as a remoter inference. In Till PRINCIPLESS OF D YNVAAlICS. 249 troducing the concepts mass (im) and force (j), where p = m g, we obtain, by multiplying the three equations by mi, the expressions mv p t, ms ~ 2/2, ps mv2 /2-the fundamental equations of mechanics. Of necessity, therefore, the concepts force and momentum (m v) appear more primitive than the concepts work ( ps) and vis viva (my2). It is not to be wondered at, accordingly, that, wherever the idea of work made its appearance, it was always sought to replace it by the historically older concepts. The entire dispute of the Leibnitzians and Cartesians, which was first composed in a manner by D'Alembert, finds its complete explanation in this fact. From an unbiassed point of view, we have exactly Justification of this the same right to inquire after the interdependence of view. the final velocity and the time as after the interdependence of the final velocity and the distance, and to answer the question by experiment. The first inquiry leads us to the experiential truth, that given bodies in contraposition impart to each other in given times definite increments of velocity. The second informs us, that given bodies in contraposition impart to each other for given mutual displacciemts definite increments of velocities. Both propositions are equally justified, and both may be regarded as equally original. The correctness of this view has been substantiated Exemplification of it in our own day by the example of J. R. Mayer. Mayer, in modern a modern mind of the Galilean stamp, a mind whollyes. free from the influences of the schools, of his own independent accord actually pursued the last-named method, and produced by it an extension of science which the schools did not accomplish until later in a much less complete and less simple form. For Mayer, work was the original concept. That which is called 250 THE SCIENCE OF MECHANICS. work in the mechanics of the schools, he calls force. Mayer's error was, that he regarded his method as the only correct one. The results 3. We may, therefore, as it suits us, regard the time which flow from it. of descent or the distance of descent as the factor determinative of velocity. If we fix our attention on the first circumstance, the concept of force appears as the original notion, the concept of work as the derived one. If we investigate the influence of the second fact first, the concept of work is the original notion. In the transference of the ideas reached in the observation of the motion of descent to more complicated relations, force is recognised as dependent on the distance between the bodies-that is, as a function of the distance, f(r). The work done through the element of distance dr is thenf(r) dr. By the second method of investigation work is also obtained as a function of the distance, F (r); but in this case we know force only in the form d. F (r)/dr-that is to say, as the limiting value of the ratio: (increment of work)/(increment of distance.) The prefer- Galileo cultivated by preference the first of these ences of the differentin- two methods. Newton likewise preferred it. Huygens quirers. pursued the second method, without at all restricting himself to it. Descartes elaborated Galileo's ideas after a fashion of his own. But his performances are insignificant compared with those of Newton and Huygens, and their influence was soon totally effaced. After Huygens and Newton, the mingling of the two spheres of thought, the independence and equivalence of which are not always noticed, led to various blunders and confusions, especially in the dispute between the Cartesians and Leibnitzians, already referred to, concerning the measure of force. In recent times, however, inquirers turn by preference now to the one and now to THE PRINCIPLES OF D YNAIMICS. 251 the other. Thus the Galileo-Newtonian ideas are cultivated with preference by the school of Poinsot, the Galileo-Huygenian by the school of Poncelet. 4. Newton operates almost exclusively with the no- The importance and tions of force, mass, and momentum. His sense of the history of the Newvalue of the concept of mass places him above his prede- tonian concept of cessors and contemporaries. It did not occur to Galileo mass. that mass and weight were different things. Huygens, too, in all his considerations, puts weights for masses; as for example in his investigations concerning the centre of oscillation. Even in the treatise De Percussione (On Impact), Huygens always says "corpus majus," the larger body, and " corpus minus," the smaller body, when he means the larger or the smaller mass. Physicists were not led to form the concept mass till they made the discovery that the same body can by the action of gravity receive different accelerations. The first occasion of this discovery was the pendulum-observations of Richer (1671-1673),-from which Huygens at once drew the proper inferences,-and the second was the extension of the dynamical laws to the heavenly bodies. The importance of the first point may be inferred from the fact that Newton, to prove the proportionality of mass and weight on the same spot of the earth, personally instituted accurate observations on pendulums of different materials (Princz ia. Lib. II, Sect. VI, De Motu et Resistentia Corporum Funependulorum). In the case of John Bernoulli, also, the first distinction between mass and weight (in the Meditatio de Natura Centri Oscillationis. Opera Omnia, Lausanne and Geneva, Vol. II, p. 168) was made on the ground of the fact that the same body can receive different gravitational accelerations. Newton, accordingly, disposes of all dynamical questions involving the relations 252 THE SCIENCE OF AlECHA4NICS. of several bodies to each other, by the help of the ideas of force, mass, and momentum. The meth- 5. Huygens pursued a different method for the soods of Huygens. lution of these problems. Galileo had previously discovered that a body rises by virtue of the velocity acquired in its descent to exactly the same height as that from which it fell. Huygens, generalising the principle (in his fHorologium Oscillatorium) to the effect that the centre of gravity of any system of bodies will rise by virtue of the velocities acquired in its descent to exactly the same height as that from which it fell, reached the principle of the equivalence of work and vis vivza. The names of the formulae which he obtained, were, of course, not supplied until long afterwards. The Huygenian principle of work was received by the contemporary world with almost universal distrust. People contented themselves with making use of its brilliant consequences. It was always their endeavor to replace its deductions by others. Even after John and Daniel Bernoulli had extended the principle, it was its fruitfulness rather than its evidency that was valued. The meth- We observe, that the Galileo-Newtonian principles ods of Newton and were, on account of their greater simplicity and apHuygens conpared. parently greater evidency, invariably preferred to the Galileo-Huygenian. The employment of the latter is exacted only by necessity in cases in which the employment of the former, owing to the laborious attention to details demanded, is impossible; as in the case of John and Daniel Bernoulli's investigations of the motion of fluids. If we look at the matter closely, however, the same simplicity and evidency will be found to belong to the Huygenian principles as to the Newtonian proposi THE PRINCIPLES OF D YNAMICS. 253 tions. That the velocity of a body is determined by the time of descent or determined by the distance of descent, are assumptions equally natural and equally simple. The fo-rm of the law must in both cases be supplied by experience. As a starting-point, therefore, pt = - v and p s m= v 2 /2 are equally well fitted. 6. When we pass to the investigation of the motion The necessity and of several bodies, we are again compelled, in both cases, universality of the to take a second step of an equal degree of certainty, two methods. The Newtonian idea of mass is justified by the fact, that, if relinquished, all rules of action for events would have an end; that we should forthwith have to expect contradictions of our commonest and crudest experiences; and that the physiognomy of our mechanical environment would become unintelligible. The same thing must be said of the Huygenian principle of work. If we surrender the theorem 2ps = 2mvZ2/2, heavy bodies will, by virtue of their own weights, be able to ascend higher; all known rules of mechanical occurrences will have an end. The insti/ctive factors which entered alike into the discovery of the one view and of the other have been already discussed. The two spheres of ideas could, of course, have The points of contact grown up much more independently of each other. But of the two methods. in view of the fact that the two were constantly in contact, it is no wonder that they have become partially merged in each other, and that the Huygenian appears the less complete. Newton is all-sufficient with his forces, masses, and momenta. Huygens would likewise suffice with work, mass, and vis viva. But since he did not in his time completely possess the idea of mass, that idea had in subsequent applications to be borrowed from the other sphere. Yet this also could have been avoided. If with Newton the mass-ratio of 254 THE SCIENCE OF MIECHANICS. two bodies can be defined as the inverse ratio of the velocities generated by the same force, with Huygens it would be logically and consistently definable as the inverse ratio of the squares of the velocities generated by the same work. The respec- The two spheres of ideas consider the mutual detive merits of each. pendence on each other of entirely different factors of the same phenomenon. The Newtonian view is in so far more complete as it gives us information regarding the motion of each mass. But to do this it is obliged to descend greatly into details. The Huygenian view furnishes a rule for the whole system. It is only a convenience, but it is then a mighty convenience, when the relative velocities of the masses are previously and independently known. The gen- 7. Thus we are led to see, that in the developeral development of ment of dynamics, just as in the development of statics, dynamics in the light the connection of widely different features of mechanical of the preceding re- phenomena engrossed at different times the attention. marks. of inquirers. We may regard the momentum of a system as determined by the forces; or, on the other hand, we may regard its vis vivza as determined by the work. In the selection of the criteria in question the individuality of the inquirers has great scope. It will be conceived possible, from the arguments above presented, that our system of mechanical ideas might, perhaps, have been different, had Kepler instituted the first investigations concerning the motions of falling bodies, or had Galileo not committed an error in his first speculations. We shall recognise also that not only a knowledge of the ideas that have been accepted and cultivated by subsequent teachers is necessary for the historical understanding of a science, but also that the rejected and transient thoughts of the inquirers, THIE PRINCIPLES OF D YNAMIICS. 255 nay even apparently erroneous notions, may be very important and very instructive. The historical investigation of the development of a science is most needful, lest the principles treasured up in it become a system of half-understood prescripts, or worse, a system of prejudices. Historical investigation not only promotes the understanding of that which now is, but also brings new possibilities before us, by showing that which exists to be in great measure convcntional and accidental. From the higher point of view at which different paths of thought converge we may look about us with freer powers of vision and discover routes before unknown. In all the dynamical propositions that we have dis- The substitution of cussed, velocity plays a prominent role. The reason "integral" for "differ. of this, in our view, is, that, accurately considered, ential" laws may every single body of the universe stands in some defi- some day make the nite relation with every other body in the universe; concept of force super, that any one body, and consequently also any several fluous. bodies, cannot be regarded as wholly isolated. Our inability to take in all things at a glance alone compels us to consider a few bodies and for the time being to neglect in certain aspects the others; a step accomplished by the introduction of velocity, and therefore of time. We cannot regard it as impossible that integral laws, to use an expression of C. Neumann, will some day take the place of the laws of mathematical elements, or differential laws, that now make up the science of mechanics, and that we shall have direct knowledge of the dependence on one another of the positions of bodies. In such an event, the concept of force will have become superfluous. (See Appendix, XXI., p. 548, on Hertz's Mechanics; also Appendix XXII., p. 555, in answer to criticisms of the views expressed by the author in Chapters I. and II.) CHAPTER III. THE EXTENDED APPLICATION OF THE PRINCIPLES OF MECHANICS AND THE DEDUCTIVE DEVELOPMENT OF THE SCIENCE. SCOPE OF THE NEWTONIAN PRINCIPLES. Newton's i. The principles of Newton suffice by themselves, principles areuni- without the introduction of any new laws, to explore versal in scope and thoroughly every mechanical phenomenon practically power. occurring, whether it belongs to statics or to dynamics. If difficulties arise in any such consideration, they are invariably of a mathematical, or 2 V2 formal, character, and in no rem3 spect concerned with questions V of principle. We have given, Ms ".3 let us suppose, a number of masses in1, i, m21M....in space, with definite initial velocities vl, rV2, 7,.... We imagine, further, lines M# V. of junction drawn between every Fig. 144. two masses. In the directions of these lines of junction are set up the accelerations and counter-accelerations, the dependence of which on the distance it is the business of physics to determine. In a small element of time r the mass m., for example, will traverse in the direction of its initial velocity the distance -, r, and in the directions of the lines joining TIE EXTENSION OF TIHE PRAIVCIPLES. 257 it with the masses ml, m2, n,...., being affected in Schematic illustration such directions with the accelerations p5, q~, Qp...., of the preceding the distances (p9/2)r2, (p5/2)T2, (// 2)r2.... If staterent. we imagine all these motions to be performed independently of each other, we shall obtain the new position of the mass m5 after lapse of time r. The composition of the velocities v5 and T, pr, qp7r, p.... gives the new initial velocity at the end of time r. We then allow a second small interval of time r to elapse, and, making allowance for the new spatial relations of the masses, continue in the same way the investigation of the motion. In like manner we may proceed with every other mass. It will be seen, therefore, that, in point of principle, no embarrassment can arise; the difficulties which occur are solely of a mathematical character, where an exact solution in concise symbols, and not a clear insight into the momentary workings of the phenomenon, is demanded. If the accelerations of the mass m7, or of several masses, collectively neutralise each other, the mass m5 or the other masses mentioned are in equilibrium and will move uniformly onwards with their initial velocities. If, in addition, the initial velocities in question are = 0, both equilibriumn and rest subsist for these masses. Nor, where a number of the masses mi, M2... The same idea aphave considerable extension, so that it is impossible to plied to aggregates of speak of a single line joining every two masses, is the dif- material ficulty, in point of principle, any greater. We divideparticles. the masses into portions sufficiently small for our purpose, and draw the lines of junction mentioned between every two such portions. We, furthermore, take into account the reciprocal relation of the parts of the same large mass; which relation, in the case of rigid masses for instance, consists in the parts resisting 258 THE SCIENCE OF MJECHANICS. every alteration of their distances from one another. On the alteration of the distance between any two parts of such a mass an acceleration is observed proportional to that alteration. Increased distances diminish, and diminished distances increase in consequence of this acceleration. iBy the, displacement of the parts with respect to one another, the familiar forces of elasticity are aroused. When masses meet in impact, their forces of elasticity do not come into play until contact and an incipient alteration of form take place. A practical 2. If we imagine a heavy perpendicular column illustration of th~e scope resting on the earth, any particle in in the interior of of Newton's principles. the column which we may choose to isolate in thought, is in equilibrium and at rest. A vertical downward acceleration g is produced by the earth in the particle, which acceleration the particle obeys. But in so doing it approaches nearer to the particles lying beneath it, and the elastic forces thus awakened generate in vina vertical acceleration upwards, which ultimately, when the particle has approached near enough, becomes equal to gk The particles lying above in likewise approach vi with the acceleration g. Here, again, acceleration and counter- acceleration are produced, whereby the particles situated above are brought to rest, but whereby vin continues to be forced nearer and nearer to the particles beneath it until the acceleration downwards, which it receives from the particles above it, increased by g, is equal, to the acceleration it receives in the upward direction from the particles beneath it. We may apply the same reasoning to every portion of the column and the earth beneath it, readily perceiving that the lower portions lie nearer each other and are more violently pressed together than the parts above. Every portion lies between a less closely pressed THE EXTEZNSION OF THE PRINCIPLES. 259 upper portion and a more closely pressed lower por- Rest in the light of tion; its downward acceleration g is neutralised by a these principles apsurplus of acceleration upwards, which it experiences pears as a special case from the parts beneath. We comprehend the equilib- of motion. rium and rest of the parts of the column by imagining all the accelerated motions which the reciprocal relation of the earth and the parts of the column determine, as in fact simultaneously performed. The apparent mathematical sterility of this conception vanishes, and it assumes at once an animate form, when we reflect that in reality no body is completely at rest, but that in all, slight tremors and disturbances are constantly taking place which now give to the accelerations of descent and now to the accelerations of elasticity a slight preponderance. Rest, therefore, is a case of motion, very infrequent, and, indeed, never completely realised. The tremors mentioned are by no means an unfamiliar phenomenon. When, however, we occupy ourselves with cases of equilibrium, we are concerned simply with a schematic reproduction in thought of the mechanical facts. We then purposely neglect these disturbances, displacements, bendings, and tremors, as here they have no interest for us. All cases of this class, which have a scientific or practical importance, fall within the province of the so-called theory of elasticity. The whole The unity and homooutcome of Newton's achievements is that we every- geneity which these where reach our goal with one and the same idea, and principles introduce by means of it are able to reproduce and construct be- into the forehand all cases of equilibrium and motion. All phenomena of a mechanical kind now appear to us as uniform throughout and as made up of the same elements. 3. Let us consider another example. Two masses m, m are situated at a distance a from each 260 tIlLH SCIENCE OF MECGHANICS. A general other. (Fig. 145.) When displaced with respect to exemplification of each other, elastic forces proportional to the change the power of the prin- - of distance are supposed to be ciples. 0CIPle. D awakened. Let the masses be x, movable in the X-direction parFig. 145. allel to a, and their co6rdinates be x1, x2. If a forcef is applied at the point x., the following equations obtain: jd2 X1 -.1 m --2 p[(xx2+-f.I).-- a]........(1) d2 X 2 (2) where p stands for the force that one mass exerts on the other when their mutual distance is altered by the value i. All the quantitative properties of the mechanical process are determined by these equations. But we obtain these properties in a more comprehensible form by the integration of the equations. The ordinary procedure is, to find by the repeated differentiation of the equations before us new equations in sufficient number to obtain by elimination equations in x, alone or x2 alone, which are afterwards integrated. We shall here pursue a different method. By subtracting the first equation from the second, we get j/2(x- x1) 2 -The devel- ---- -.) - 2p 2 X - X1) - 1 + or opminent of 1/ L2 the equationsob- putting X 2, tained in - - - this exampie. t[2 j pe _ - -2-2.p[u-- 4] - f......... (3) d2 and by the addition of the first and the second equations (12 (X 2 + X1) d2 -2 - _f1,--f or, putting x.2 + X I NEIL LX EX8NIONV OF TH11 PRLV CIPLES. 261 d 2 /if P........... (4) The integrals of (3) and (4) are respectively The integrals of these 2f dlevelopa A 4sin\. /4- B cos4. / a and ents. 1/2p f t/ SCI + D; whence A.2p B+ 1f2t x sin t C os2 2 4-C + C 4p 2 2X2 / B 4 /1 in 2 x2As/n.+ 2 cos. /2 /71 1/ 1 H afD + 4- C- f- D 4p To take a particular case, we will assume that theA particular case of action of the force f begins at t= -, and that at this thle xantpie. time dx dx2 that is, the initial positions are given and the initial velocities are- 0. The constants A, B, C, D being eliminated by these conditions, we get 9; f t/2 f (5) x C- cos./ 2+.0 1, f f_1 2 4 f (6)) x. f cos4. 2 ta (4, and x) x x1 f Cos 2p. a. 2 --./ + a 262 THIE SCIENCE OF AE L, CANiICS. Theinform-We see from (5) and (6) that the two masses, in addiation which theresult- tion to a uniformly accelerated motion with half the ant equations give acceleration that the force f would impart to one of concerning this exam- these masses alone, execute an oscillatory motion symple. metrical with respect to their centre of gravity. The duration of this oscillatory motion, T 2 7tr /m/2f, is smaller in proportion as the force that is awakened in the same mass-displacement is greater (if our attention is directed to two particles of the same body, in proportion as the body is harder). The amplitude of oscillation of the oscillatory motion f/2p likewise decreases with the magnitudep of the force of displacement generated. Equation (7) exhibits the periodic change of distance of the two masses during their progressive motion. The motion of an elastic body might in such case be characterised as vermicular. With hard bodies, however, the number of the oscillations is so great and their excursions so small that they remain unnoticed, and may be left out of account. The oscillatory motion, furthermore, vanishes, either gradually through the effect of some resistance, or when the two masses, at the moment the force fbegins to act, are a distance a -f/21 apart and have equal initial velocities. The distance a - f/2p that the masses are apart after the vanishing of their vibratory motion, isf/2P greater than the distance of equilibrium a. A tension y, namely, is set up by the action off, by which the acceleration of the foremost mass is reduced to onehalf whilst that of the mass following is increased by the same amount. In this, then, agreeably to our asThis in- sumption, pjy/m =f/2 m or y =f/2p. As we see, it is formation isexhaus- in our power to determine the minutest details of a tive. phenomenon of this character by the Newtonian principles. The investigation becomes (mathematically, TIHE EXTENSION OF THE PRINICIPLES. 263 yet not in point of principle) more complicated when we conceive a body divided up into a great number of small parts that cohere by elasticity. Here also in the case of sufficient hardness the vibrations may be neglected. Bodies in which we purposely regard the mutual displacement of the parts as evanescent, are called -iriid bodies. 4. We will now consider a case that exhibits the The deduction of the schema of a lever. We imagine the masses M, mn, m2 laws of the lever by arranged in a triangle and joined by elastic connec- Newton's principles. tions. Every alteration of the sides, and consequently also every alteration of the angles, gives rise to accelerations, as the result of which the triangle endeavors to assume its previous form and size. By the aid of the Newtonian principles we can deduce from such a schema the laws of the lever, and at the same time feel that the form of the deduction, although it may be more complicated, still remains admissible when we pass from a schematic b ' lever composed of three s d e masses to the case of a i real lever. The mass A1 Fig. 146. we assume either to be in itself very large or conceive it joined by powerful elastic forces to other very large masses (the earth for instance). AM then represents an immovable fulcrum. I Let ml, now, receive from the action of some ex- The method of the ternal force an acceleration f perpendicular to the line deduction of junction Mm, = c -- d. Immediately a stretching of the lines m, m12 = and m1 M - a is produced, and in the directions in question there are respectively set up the accelerations, as yet undetermined, s and or, of which the components s (e/b) and o (e/a) are directed 264 THI/ SCIEVNCE OF J1 ECI/ IVICS. oppositely to the accelerationf. Here e is the altitude of the triangle mim,2M. The mass m1 receives the acceleration s', which resolves itself into the two components s'(d/b) in the direction of M and s'(e/b) parallel to f. The former of these determines a slight approach of Im2 to M. The accelerations produced in AM by the reactions of m, and m2, owing to its great mass, are imperceptible. We purposely neglect, therefore, the motion of M. The deduc- The mass mn, accordingly, receives the accelerad1on obtailcodby tion f- s(e/b) - (e/a), whilst the mass m2 suffers hlic considcrationof the parallel acceleration s'(e/b). Between s and a a acceleratiea. simple relation obtains. If, by supposition, we have a very rigid connection, the triangle is only imperceptibly distorted. The components of s and a perpendicular to f destroy each other. For if this were at any one moment not the case, the greater component would produce a further distortion, which would immediately counteract its excess. The resultant of s and a is therefore directly contrary to f, and consequently, as is readily obvious, a (c/a) - s (d bl). Between s and s', further, subsists the familiar relation;ti s.-i 1//2 s' or s =s'(m2/mi). Altogether m, and a1, receive respectively the accelerations s'(e/b) and f- s'(e/b) (mn2/1) (c + d/c), or, introducing in the place of the variable value s'(e/b) the designation p, the accelerations ( and f- 9 (m2 /m ) (c -- d/c). On t1ie pre- At the commencement of the distortion, the accelceding suppositions eration of m1, owing to the increase of (p, diminishes, the laws for he rotation whilst that of mn increases. If we make the altitude e of the lever are easily of the triangle very small, our reasoning still remains deduced.applicable. In this case, however, a becomes - c = r, and a - b c +d= r2. We see, moreover, that the distortion must continue, (p increase, and the accelera THIE EXTENSION OF THE PRINCIPLES. 265 tion of Im1 diminish until the stage is reached at which the accelerations of m, and m12 bear to each other the proportion of r, to r2. This is equivalent to a rotation of the whole triangle (without further distortion) about M, which mass by reason of the vanishing accelerations is at rest. As soon as rotation sets in, the reason for further alterations of q ceases. In such a case, consequently, 2' f 2- 2 r^\ i S 2 (/- (P 2 or2 ) - /- 12 1 f/ 2 -orr--- m.., m --r! -2 - m 2 For the angular acceleration /i of the lever we get '2 ll211 12 + 1122 '2 Nothing prevents us from entering still more into Discussion of the charthe details of this case and determining the distortions acter of the preceding and vibrations of the parts with respect to each other. result. With sufficiently rigid connections, however, these details may be neglected. It will be perceived that we have arrived, by the employment of the Newtonian principles, at the same result to which the Huygenian view also would have led us. This will not appear strange to us if we bear in mind that the two views are in every respect equivaient, and merely start from different aspects of the same subject-matter. If we had pursued the Huygenian method, we should have arrived more speedily at our goal but with less insight into the details of the phenomenon. We should have employed the work done in some displacement of mil to determine the vires vizvc of m1 and mn,, wherein we should have assumed that the velocities in question v1, v, maintained the ratio v1/v72 r-- /r.. The example here treated is very well adapted to illustrate what such an equation of condition means. The equation 266 THiE SCIENCE OF MECItA NICS. simply asserts, that on the slightest deviations of v /v2 from rl/r2 powerful forces are set in action which in Point offacl prevent all further deviation. The bodies obey of course, not the equalions, but the forces. A simple 5. We obtain a very obvious case if we put in the case of the same exam-example just treated m 11 -- = i and a - b (Fig. ple. 147). The dynamical state of the system ceases to change when ( - 2 (/f- 2 q), that is, when the accelerations of the masses - - at the base and the ver-..b-- el'".M tex are given by 2f15 F and f/5. At the comFig. ' mencement of the distortion p increases, and simultaneously the acceleration of the mass at the vertex is decreased by double that amount, until the proportion subsists between the two of 2: i. The equi- Ve have yet to consider the case of equilibriumi of librium of the lever a schematic lever, consisting (Fig. 148) of three masses deduced from the d /u, i,, and J/, of which the last is again supposed same considerations. //S Fig. 148. to be very large or to be elastically connected with very large masses. We imagine two equal and opposite forces s, - s applied to in1 and in in the direction z1m,V or, what is the same thing, accelerations impressed inversely proportional to the masses in,.Y2* The stretching of the connection vin 1n2 also generates TILE EXTENSION OF TILE IPRI.VCILES. 267 accelerations inversely proportional to the masses Mr, W 2, which neutralise the first ones and produce equilibrium. Similarly, along m1 M imagine the equal and contrary forces1/, - / operative; and along in2 Ai the forces it, - M. In this case also equilibrium obtains. If MA be elastically connected with masses sufficiently large, ---m and - I need not be applied, inasmuch as the last-named forces are spontaneously evoked the moment the distortion begins, and always balance the forces opposed to them. Equilibrium subsists, accordingly, for the two equal and opposite forces s, - s as well as for the wholly arbitrary forces /, u. As a matter of fact s, - s destroy each other and 1, u pass through the fixed mass MA, that is, are destroyed on distortion setting in. The condition of equilibrium readily reduces itself The reduction of the to the common form when we reflect that the mo- preceding case to the ments of / and i, forces passing through ML, are with common form. respect to AM zero, while the moments of s and - s are equal and opposite. If we compound / and s to p, and u and - s to q, then, by Varignon's gcomnc/rica/ principle of the parallelogram, the moment of p is equal to the sum of the moments of s and /, and the moment of q is equal to the sum of the moments of u and - s. The moments of p and q are therefore equal and opposite. Consequently, any two forcesp and q will be in equi/ibriuni if they produce in the direction vi11 1112 equal and opposite components, by which condition the equality of the moments with respect to AL is posited. That then the resultant of _p and q also passes through AM, is likewise obvious, for s and - s destroy each other and t and u pass through AL. 6. The Newtonian point of view, as the example just developed shows us, includes that of Varignon. 268 THE SCIENCE OF lAE CIIANICS. Newton's We were right, therefore, when we characterised the point of view in- statics of Varignon as a dynamical statics, which, startcludes Varignon's. ing from the fundamental ideas of modern dynamics, voluntarily restricts itself to the investigation of cases of equilibrium. Only in the statics of Varignon, owing to its abstract form, the significance of many operations, as for example that of the translation of the forces in their own directions, is not so distinctly exhibited as in the instance just treated. The econ- The considerations here developed will convince ony and wealth of us that we can dispose by the Newtonian principles theNewtonian ideas, of every phenomenon of a mechanical kind which may arise, provided we only take the pains to enter far enough into details. We literally see through the cases of equilibrium and motion which here occur, and behold the masses actually impressed with the accelerations they determine in one another. It is the same grand fact, which we recognise in the most various phenomena, or at least can recognise there if we make a point of so doing. Thus a unity, homogeneity, and economy of thought were produced, and a new and wide domain of physical conception opened which before Newton's time was unattainable. The New- Mechanics, however, is not altogether an end in ittonian and themodern, self; it has also problems to solve that touch the needs routine methods. of practical life and affect the furtherance of other sciences. Those problems are now for the most part advantageously solved by other methods than the Newtonian,-methods whose equivalence to that has already been demonstrated. It would, therefore, be mere impractical pedantry to contemn all other advantages and insist upon always going back to the elementary Newtonian ideas. It is sufficient to have once convinced ourselves that this is always possible. Yet the New TIE i'XTA'EVSION OF 7THA I'LVCZII)LES. 269 tonian conceptions are certainly the most satifactory and the most lucid; and Poinsot shows a noble sense of scientific clearness and simplicity in making these conceptions the sole foundation of the science. II. THE FORMULAE AND UNITS OF MECHANICS. I. All the important formulae of modern mechanics History of the formuwere discovered and employed in the period of Galileo Ie and units of and Newton. The particular designations, which, mechanics. owing to the frequency of their use, it was found convenient to give them, were for the most part not fixed upon until long afterwards. The systematical mechanical units were not introduced until later still. Indeed, the last named improvement,.cannot be regarded as having yet reached its completion. 2. Let s denote the distance, / the time, v' the in- The origstantaneous velocity, and p the acceleration of a uni- tionsofaGalileo and formly accelerated motion. From the researches of Huygens. Galileo and Huygens, we derive the following equations:,-- pt 1 2................. (1) _2 (S U 2J Multiplying throughout by the mass mi, these equa- The introduction tions give the following: c ''mas and " miov1ll 7 --= - / m y ing force." M s -- -/ 2 11 q)S =. 270 TYlE SCIEN(_CE OF ALE CIf4NA "ICS Final form and, denoting the moving force in p by the letter p, we of the fundamental obtain equations. SviZ --/ Ins 2 /2) 2....................2) M V2 2 Equations (i) all contain the quantity (p; and each contains in addition two of the quantities s, 1, v, as exhibited in the following table: V, I Equations (2) contain the quantities m, p, s, l, v; each containing in, _p and in addition to m, 5 two of the three quantities s, /, v, according to the following table v, /;n,. s, / sn, SY, The scope Questions concerning motions due to constant forces and application of are answered by equations (2) in great variety. If, for these equations, example, we want to know the velocity v that a mass vi acquires in the time / through the action of a force.p, the first equation gives v =zzJl/m. If, on the other hand, the time be sought during which a mass in with the velocity v can move in opposition to a force p, the same equation gives us t - mv/f. Again, if we inquire after the distance through which in will move with velocity v in opposition to the force 1, the third equation gives s- 1v2/2p. The two last questions illustrate, also, the futility of the Descartes-Leibnitzian dispute concerning the measure of force of a body in motion. The use of these equations greatly contributes TIIE EXIEN.SION 0OF T/iE PARINCIPLES. 271 to confidence in dealing with mechanical ideas. Suppose, for instance, we put to ourselves, the question, what force p will impart to a given mass in the velocity v; we readily see that between m, p, and v alone, no equation exists, so that either s or I must be supplied, and consequently the question is an indelterminale one. We soon learn to recognise and avoid indeterminate cases of this kind. The distance that a mass m acted on by the force p describes in the time t, if moving with the initial velocity 0, is found by the second equation s = t2/2 M. 3. Several of the formulae in the above-discussed The names which the equations have received particular names. The force formul; of of a moving body was spoken of by Galileo, who al-t inshave received. ternately calls it "momentum, " impulse, " and " en-re ergy." He regards this momentum as proportional to the product of the mass (or rather the weight, for Galileo had no clear idea of mass, and for that matter no more had Descartes, nor even Leibnitz) into the velocity of the body. Descartes accepted this view. He put the force of a moving body m 1171, called it quantity of motion, and maintained that the sum-total of the quantity of motion in the universe remained constant, so that when one body lost momentum the loss was compensated for by an increase of momentum in other bodies. Newton also employed the designation "quantity of motion'" for m v', and this name has been retained to the Momentum and present day. [But momentum is the more usual term.] impulse. For the second member of the first equation, viz. t, Belanger, proposed, as late as 1847, the name impulse.* The expressions of the second equation have received * See, also, Maxwell, Matter and Motion, American edition, page 72. But this word is commonly used in a different sense, namely, as "the limit of a force which is infinitely great but acts only during an infinitely short time." See Routh, Rigid Dynamics, Part I, pages 65 66.-Trans. 272 THE SCIENVCE OF MlE CICANICS. fis viva no particular designations. Leibnitz (1695) called the and work. expression mvv2 of the third equation vis viva or living, force, and he regarded it, in opposition to Descartes, as the true measure of the force of a body in motion, calling the pressure of a body at rest vis moritu, or dead force. Coriolis found it more appropriate to give the term -imv2 the name vis viva. To avoid confusion, Belanger proposed to call mi v2 livinZg force and i/l Zv2 living power [now commonly called in English kinctic energy]. For ps Coriolis employed the name work. Poncelet confirmed this usage, and adopted the kilograimme-metre (that is, a force equal to the weight of a kilogramme acting through the distance of a metre) as the unil of work. The history 4. Concerning the historical details of the origin of of the ideas quantity of these notions "quantity of motion " and " vis viva," motion and vis viva. a glance may now be cast at the ideas which led Descartes and Leibnitz to their opinions. In his Princi/ia Phiilosophiae, published in I644, II, 36, DESCARTES expressed himself as follows: "Now that the nature of motion has been examined, "we must consider its cause, which may be conceived "in two senses: first, as a universal, original cause( the general cause of all the motion in the world; and Ssecond, as a special cause, from which the individual "parts of matter receive motion which before they did "not have. As to the universal cause, it can maniSfestly be none other than God, who in the beginning "created matter with its motion and rest, and who now "preserves, by his simple ordinary concurrence, on the "whole, the same amount of motion and rest as he "originally created. For though motion is only a con"dition of. moving matter, there yet exists in matter "a definite quantity of it, which in the world at large THE EXTENSION OF THE PRINCIPILES. 273 Snever increases or diminishes, although in single por- Passage from Des"tions it changes; namely, in this way, that we must carte.'s Princi'ia "assume, in the case of the motion of a piece of matter "which is moving twice as fast as another piece, but in " quantity is only one half of it, that there is the same "amount of motion in both, and that in the proportion "as the motion of one part grows less, in the same pro"portion must the motion of another, equally large "part grow greater. We recognise it, moreover, as "a perfection of God, that He is not only in Himself "unchangeable, but that also his modes of operation ' are most rigorous and constant; so that, with the ex"ception of the changes which indubitable experience Sor divine revelation offer, and which happen, as our cfaith or judgment show, without any change in the "Creator, we are not permitted to assume any others " in his works-lest inconstancy be in any way pre"dicated of Him. Therefore, it is wholly rational to "assume that God, since in the creation of matter he " imparted different motions to its parts, and preserves "all matter in the same way and conditions in which "he created it, so he similarly preserves in it the same "quantity of motion." (See Appendix, XXIII., p. 574-) The merit of having first sought after a more uni- The merits and defects versal and more fruitful point of view in mechanics, of Descartes's physcannot be denied Descartes. This is the peculiar task icalinquirof the philosopher, and it is an activity which constantly exerts a fruitful and stimulating influence on physical science. Descartes, however, was infected with all the usual errors of the philosopher. He places absolute confidence in his own ideas. He never troubles himself to put them to experiential test. On the contrary, a minimum of experience always suffices him for a maximum 274 7Til'E SCIENCE O MIE CHAINICS. of inference. Added to this, is the indistinctness of his conceptions. Descartes did not possess a clear idea of mass. It is hardly allowable to say that Descartes defined mi v as momentum, although Descartes's scientific successors, feeling the need of more definite notions, adopted this conception. Descartes's greatest error, however,-and the one that vitiates all his physical inquiries,-is this, that many propositions appear to him self-evident i pr'iori concerning the truth of which experience alone can decide. Thus, in the two paragraphs following that cited above (~~37-39) it is asserted as a self-evident proposition that a body preserves unchanged its velocity and direction. The experiences cited in ~38 should have been employed, not as a confirmation of an 2t priori law of inertia, but as a foundation on which this law in an empirical sense should be based. Leibnitz Descartes's view was attacked by LEIBNITZ (1686) on quantity of motion. in the Acta Eruditorum, in a little treatise bearing the title: " A short Demonstration of a Remarkable Error of Descartes and Others, Concerning the Natural Law by which they think that the Creator always preserves the same Quantity of Motion; by which, however, the Science of Mechanics is totally perverted." In machines in equilibrium, Leibnitz remarks, the loads are inversely proportional to the velocities of displacement; and in this way the idea arose that the product of a body ("i corpus, " "moles ") into its velocity is the measure of force. This product Descartes regarded as a constant quantity. Leibnitz's opinion, however, is, that this measure of force is only accidentally the correct measure, in the case of the machines. The true measure of force is different, and must be determined by the method which Galileo and THE EX TEASION OF 11,THE PRLVCIP'LES. 275 Huygens pursued. Every body rises by virtue of the Leibnitz on the measvelocity acquired in its descent to a height exactly nre orf orce. equal to that from which it fell. If, therefore, we assume, that the same "force" is requisite to raise a body m a height 4/ as to raise a body 4m a height /, we must, since we know that in the first case the velocity acquired in descent is but twice as great as in the second, regard the product of a "body" into the square of its velocity as tie measure of force. In a subsequent treatise (1695), Leibnitz reverts to this subject. He here makes a distinction between simple pressure (vis mortua) and the force of a moving body (vis viva), which latter is made up of the sum of the pressure-impulses. These impulses produce, indeed, an "impetus" (m 7v), but the impetus produced is not the true measure of force; this, since the cause must be equivalent to the effect, is (in conformity witl the preceding considerations) determined by miv2. Leibnitz remarks further that the possibility of perpetual motion is excluded only by the acceptance of his measure of force. Leibnitz, no more than Descartes, possessed a gen- The idea or mIass ill uine concept of mass. Where the necessity of such Leibnitz's vi ewx. an idea occurs, he speaks of a body (corius), of a load (moles), of different-sized bodies of the same specific gravity, and so forth. Only in the second treatise, and there only once, does the expression "massa " occur, in all probability borrowed from Newton. Still, to derive any definite results from Leibnitz's theory, we must associate with his expressions the notion of mass, as his successors actually did. As to the rest, Leibnitz's procedure is much more in accordance with the methods of science than Descartes's. Two things, however, are confounded: the question of the measure of for;ce 276 THE SCIENCE OF AIfECItANICS. In a sense, and the question of the constancy of the sums 2 n v and Descartes and Leib- i vL2. The two have in reality nothing to do with nitz were each right. each other. With regard to the first question, we now know that both the Cartesian and the Leibnitzian measure of force, or, rather, the measure of the effectiveness of a body in motion, have, each in a different sense, their justification. Neither measure, however, as Leibnitz himself correctly remarked, is to be confounded with the common, Newtonian, measure of force. The dis- With regard to the second question, the later inpute, the result ofenis- vestigations of Newton really proved that for free maunderstandings. terial systems not acted on by external forces the Cartesian sum 2inv is a constant; and the investigations of Huygens showed that also the sum 2m1,2 is a constant, provided work performed by forces does not alter it. The dispute raised by Leibnitz rested, therefore, on various mzisiunderstandings. It lasted fifty-seven years, till the appearance of D'Alembert's Tr-a/i de dyj'namiqu(e, in 1743. To the theological ideas of Descartes and Leibnitz, we shall revert in another place. The appli- 5. The three equations above discussed, though cation of the fuinda- they are only applicable to rectilinear motions produced mental equations by constant forces, may yet be considered the fundato variable forces. mental equations of mechanics. If the motion be rectilinear but the force variable, these equations pass by a slight, almost self-evident, modification into others, which we shall here only briefly indicate, since mathematical developments in the present treatise are wholly subsidiary. From the first equation we get for variable forces m v = dt -]- C, where p is the variable force, dt the time-element of the action, fidt the sum of all the THE EXTENSION OF T1H PRIVCIPLES. 277 products. d1t from the beginning to the end of the action, and C a constant quantity denoting the value of mt v before the force begins to act. ilTe second equation passes in like manner into the forms s= j d/ --dt - CIt - D, with two so-called constants of integration. The third equation must be replaced by /f71 2 _ S2 =fP/ds + C. Curvilinear motion may always be conceived as the product of the simultaneous combination of three rectilinear motions, best taken in three mutually perpendicular directions. Also for the components of the motion of this very general case, the above-given equations retain their significance. 6. The mathematical processes of addition, sub- Theunitso mechanics. traction, and equating possess intelligible meaning only when applied to quantities of the same kind. We cannot add or equate masses and times, or masses and velocities, but only masses and masses, and so on. When, therefore, we have a mechanical equation, the question immediately presents itself whether the members of the equation are quantities of tic samei kind, that is, whether they can be measured by t//i sameic unit, or whether, as we usually say, the equation is /lomogr(cncoZts. The units of the quantities of mechanics will form, therefore, the next subject of our investigations. The choice of units, which are, as we know, quantities of the same kind as those they serve to measure, is in many cases arbitrary. Thus, an arbitrary mass is employed as the unit of mass, an arbitrary length is employed as the unit of length, an arbitrary time as the unit of time. The mass and the length employed as units can be preserved; the time can be reproduced 278 TIlL SCIANCA 0/'" 1Z Lt'ILPiCS. Arbitrary by pendulum-experiments and astronomical observaimits, and derived or tions. But units like a unit of velocity, or a unit of absolute utu1its. acceleration, cannot be preserved, and are much more difficult to reproduce. These quantities are consequently so connected with the arbitrary fundamental units, mass, length, and time, that they can be easily and at once derived from them. Units of this class are called derived or absolute units. This latter designation is due to GAUSS, who first derived the magnetic units from the mechanical, and thus created the possibility of a universal comparison of magnetic measurements. The name, therefore, is of historical origin. The de- As unit of velocity we might choose the velocity rived units ofvelocity, with which, say, q units of length are travelled over in acceleration, and unit of time. But if we did this, we could not express force. the relation between the time /, the distance s, and the velocity v? by the usual simple formula s---v,/, but should have to substitute for it s--q. v7 7. If, however, we define the unit of velocity as the velocity with which the unit of length is travelled over in unit of time, we may retain the form s -v 7i. Among the derived units the simplest possible relations are made to obtain. Thus, as the unit of area and the unit of volume, the square and cube of the unit of length are always employed. According to this, we assume then, that by unit velocity unit length is described in unit time, that by unit acceleration unit velocity is gained in unit time, that by unit force unit acceleration is imparted to unit mass, and so on. The derived units depend on the arbitrary fundamental units; they are functions of them. The function which corresponds to a given derived unit is called its dimensiois. The theory of dimensions was laid down THE EXTENSION OF THE PRLICIPLES. 2 279 by FOURIER, in 1822, in his Theory of Heat. Thus, if /The theory of dimrendenote a length, / a time, and mi a mass, the dimen- sions. sions of a velocity, for instance, are 1// or //-1. After this explanation, the following table will be readily understood: NAMES SYMBOLS Velocity............ Acceleration........... Force.............. Momentum......... m 7 Impulse............ pt W ork............. ps Vis viva........... Moment of inertia...... 9. Statical moment...... D DIMENSIONS //-1 /1-2 m t1-2 Im /-1 l t//-1 ml21-2 //1/21-2 m112/-2 This table shows at once that the above-discussed equations are homogeneous, that is, contain only members of /ie same kind. Every new expression in mechanics might be investigated in the same manner. 7. The knowledge of the dimensions of a quantity The usefulness of the is also important for another reason. Namely, if tthe osry of ditoenvalue of a quantity is known for one set of fundamental Si"on units and we wish to pass to another set, the value of the quantity in the new units can be easily found from the dimensions. The dimensions of an acceleration, which has, say, the numerical value q), are //-2. If we pass to a unit of length A times greater and to a unit of time r times greater, then a number A times smaller must take the place of / in the expression lt-2, and a number r times smaller the place of /. The numerical value of the same acceleration referred to the new units will consequently be (r2 /A) p. If we 280 THE SCIENCE OF ME CHANICS. take the metre as our unit of length, and the second as our unit of time, the acceleration of a falling body for example is 9-81, or as it is customary to write it, indicating at once the dimensions and the fundamental measures: 9-81 (metre/second2). If we pass now to the kilometre as our unit of length (A 10Ioo0), and to the minute as our unit of time (r -- 6o), the value of the same acceleration of descent is (60 X 60/1000) 981, or 35'316 (kilometre/minute2). The Inter- [8. The following statement of the mechanical units national Bureau of at present in use in the United States and Great Britain Weights and Meas- is substituted for the statement by Professor Mach of ures. the units formerly in use on the continent of Europe. All the civilised governments have united in establishing an International Bureau of Weights and Measures in the Pavillon de Breteuil, in the Parc of St. Cloud, at Sevres, near Paris. In some countries, the standards emanating from this office are exclusively legal; in others, as the United States and Great Britain, they are optional in contracts, and are usual with physicists. These standards are a standard of length and a standard of mass (not wegl/t.) The inter- The unit of length is the International Metre, which national unit of is defined as the distance at the melting point of ice length, between the centres of two lines engraved upon the polished surface of a platiniridium bar, of a nearly X-shaped section, called the International Prototype Metre. Copies of this, called National Prototype Metres, are distributed to the different governments. The international metre is authoritatively declared to be identical with the former French metre, used until the adoption of the international standard; and it is impossible to ascertain any error in this statement, be 7/E YEXTESIONV O1F 'lE PRINACIPLES. 281 cause of doubt as to the length of the old metre, owing partly to the imperfections of the standard, and partly to obstacles now intentionally put in the way of such ascertainment. The French metre was defined as the distance, at the melting-point of ice, between the ends of a platinum bar, called the mitre des archivcs. It was against the law to touch the ends, which made it difficult to ascertain the distance between them. Nevertheless, there was a strong suspicion they had been dented. The mtire des archiveCs was intended to be one ten-millionth of a quadrant of a terrestrial meridian. In point of fact such a quadrant is, according to Clarke, 32814820 feet, which is 10002015 metres. The international unit of mass is the kilogramme, The international which is the mass of a certain cylinder of platiniridium unit of Smass. called the International Prototype Kilogramme. Each government has copies of it called National Prototype Kilogrammes. This mass was intended to be identical with the former French kilogramme, which was defined as the mass of a certain platinum cylinder called the kilogramme des archivcs. The platinum being somewhat spongy contained a variable amount of occluded gases, and had perhaps suffered some abrasion. The kilogramme is iooo grammes; and a gramme was intended to be the mass of a cubic centimetre of water at its temperature of maximum density, about 3-930 C. It is not known with a high degree of precision how nearly this is so, owing to the difficulty of the determination. The regular British unit of length is the Imperial The British unit of Yard which is the distance at 620 F. between the cen- length. tres of two lines engraved on gold plugs inserted in a bronze bar usually kept walled up in the Houses of Parliament in Westminster. These lines are cut rela 282 IlA SCIENC ( )LL'J 0/L"CtIA7NCS. Condition,; tively deep, and the burr is rubbed off and the surface of comiparison of the rendered mat, by rubbing with charcoal. The centre Imperial Yard whith of such a line can easily be displaced by rubbing; which other itteasu,-., is probably not true of the lines on the Prototype metres. The temperature is, by law, ascertained by a mercurial thermometer; but it was not known, at the time of the construction of the standard, that such thermometers may give quite different readings, according to the mode of their manufacture. The quality of glass makes considerable difference, and the mode of determining the fixed points makes still more. The best way of marking these points is first to expose the thermometer for several hours to wet aqueous vapor at a known pressure, and mark on its stem the height of the column of mercury. The thermometer is then brought down to the temperature of melting ice, as rapidly as possible, and is immersed in pounded ice which is melting and from which the water is not allowed to drain off. The mercury being watched with a magnifying glass is seen to fall, to come to rest, and to commence to rise, owing to the lagging contraction cf the glass. Its lowest point is marked on the stem. The interval between the two marks is then divided into equal degrees. When such a thermometer is used, it is kept at the temperature to be determined for as long a time as possible, and immediately after is cooled as rapidly as it is safe to cool it, and its zero is redetermined. Thermometers, so made and treated, will give very constant indications. But the thermometers made at the Kew observatory, which are used for determining the temperature of the yard, are otherwise constructed. Namely the melting-point is determined first and the boiling-point afterwards; and the thermometers are exposed to both tempera THE ELXTENSION OFV T7HE PRINLVCIPLES. 283 tures for many hours. The point which upon such a Relative lengths of thermometer will appear as 62' will really be consider- the metre and vard. ably hotter (perhaps a third of a centigrade degree) than if its melting point were marked in the other way. If this circumstance is not attended to in making comparisons, there is danger of getting the yard too short by perhaps one two-hundred-thousandth part. General Comstock finds the metre equal to 39 36985 inches. Several less trustworthy determinations give nearly the same value. This makes the inch 2 540014 centimetres. At the time the United States separated from Eng- The American unit of land, no precise standard of length was legal*; and length. none has ever been established. We are, therefore, without any precise legal yard; but the United States office of weights and measures, in the absence of any legal authorisation, refers standards to the British Imperial Yard. The regular British unit of mass is the Pound, de- The British unit of fined as the mass of a certain platinum weight, called mass the Imperial Pound. This was intended to be so constructed as to be equal to 7000 grains, each the 5760th part of a former Imperial Troy pound. This would be within 3 grains, perhaps closer, of the old avoirdupois pound. The British pound has been determined by Miller to be o 4535926525 kilogramme; that is the kilogramme is 2-204621249 pounds. At the time the United States separated from Great Britain, there were two incommensurable units of weight, the avoirdupois poind and the Troy pound. Congress has since established a standard Troy pound, which is kept in the Mint in Philadelphia. It was a copy of the old Imperial Troy pound which had been adopted in England after American independence. It * The so-called standard of 1758 had not been legalised. 284 THE SCIENCE OF AMECIIHANICS. TheAmeri- is a hollow brass weight of unknown volume; and no can unit of mass. accurate comparisons of it with modern standards have ever been published. Its mass is, therefore, unknown. The mint ought by law to use this as the standard of gold and silver. In fact, they use weights furnished by the office of weights and measures, and no doubt derived from the British unit; though the mint officers profess to compare these with the Troy pound of the United States, as well as they are able to do. The old avoirdupois pound, which is legal for most purposes, differed without much doubt quite appreciably from the British Imperial pound; but as the Office of Weights and Measures has long been, without warrant of law, standardising pounds according to this latter, the legal avoirdupois pound has nearly disappeared from use of late years. The makers of weights could easily detect the change of practice of the Washington Office. Measures of capacity are not spoken of here, because they are not used in mechanics. It may, however, be well to mention that they are defined by the weight of water at a given temperature which they measure. The unit of The universal unit of time is the mean solar day or time. its one 864ooth part, which is called a second. Sidereal time is only employed by astronomers for special purposes. Whether the International or the British units are employed, there are two methods of measurement of mechanical quantities, the absolute and the gravitational. The absolute is so called because it is not relative to the acceleration of gravity at any station. This method was introduced by Gauss. The special absolute system, widely used by physicists in the United States and Great Britain, is called THE EXX7TENSION OF THIIE PRlINCIIPLESI. 285 the Centimetre-Gramme-Second system. In this sys- The absolute system tem, writing C for centimetre, G for gramme mass, of the 0United and S for second, States and Great Britain. the unit of length is........... C; the unit of mass is............ G; the unit of time is............... S; the unit of velocity is............ C/S the unit of acceleration (which might be called a "galileo," because Galileo Galilei first measured an acceleration) is............... C/S; the unit of density is......... G/C; the unit of momentum is....... G C/S; the unit of force (called a dine) is... G C/S 2; the unit of pressure (called one millionth of an absolute atmosphere) is.. G/CS2; the unit of energy (vis viva, or work, called an erg) is.......... GC2/S2; etc. The gravitational system of measurement of me- TheGravitational chanical quantities, takes the kilogramme or pound, or System. rather the attraction of these towards the earth, compounded with the centrifugal force,-which is the acceleration called gravity, and denoted by g, and is different at different places,-as the unit of force, and the foot-pound or kilogramme-metre, being the amount of gravitational energy transformed in the descent of a pound through a foot or of a kilogramme through a metre, as the unit of energy. Two ways of reconciling these convenient units with the adherence to the usual standard of length naturally suggest themselves, namely, first, to use the pound weight or the kilogramme weight divided by g as the unit of mass, and, second, to adopt 286 THE SCIENCL OF IE CHAIIACS. such a unit of time as will make the acceleration of g, at an initial station, unity. Thus, at Washington, the acceleration of gravity is 980 - 05 galileos. If, then, we take the centimetre as the unit of length, and the 0-031943 second as the unit of time, the acceleration of gravity will be I centimetre for such unit of time squared. The latter system would be for most purposes the more convenient; but the former is the more familiar. Compari- In either system, the formula / = mg is retained; son of the absolute but in the former g retains its absolute value, while in and gravitational the latter it becomes unity for the initial station. In Paris, g is 980-96 galileos; in Washington it is 980-05 galileos. Adopting the more familiar system, and taking Paris for the initial station, if the unit of force is a kilogramme's weight, the unit of length a centimetre, and the unit of time a second, then the unit of mass will be 1/981 o kilogramme, and the unit of energy will be a kilogramme-centimetre, or (1/2)(1ooo/981-0) G C 2/S 2. Then, at Washington the gravity of a kilogramme will be, not I, as at Paris, but 980 -1/98 I o = o'99907 units or Paris kilogrammeweights. Consequently, to produce a force of one Paris kilogramme-weight we must allow Washington gravity to act upon 981 -o/980-1 =i -00092 kilogrammes.]* In mechanics, as in some other branches of physics closely allied to it, our calculations involve but three fundamental quantities, quantities of space, quantities of time, and quantities of mass. This circumstance is a source of simplification and power in the science which should not be underestimated. * For some critical remarks on the preceding method of expcs:tion, see Nature, in the issue for November 15, 1894. TIE EXTEANSION OF THE PRLVCYZIPLES. 287 III. THE LAWS OF THE CONSERVATION OF MOMENTUM, OF THE CONSERVATION OF THE CENTRE OF GRAVITY, AND OF THE CONSERVATION OF AREAS. i. Although Newton's principles are fully adequate Specialisaion of the to deal with any mechanical problem that may arise, mechanical laws. it is yet convenient to contrive for cases more frequently occurring, particular rules, which will enable us to treat problems of this kind by routine forms and to dispense with the minute discussion of them. Newton and his successors developed several such principles. Our first subject will be NEWTON'S doctrines concerning freely mov.able material systems. 2. If two free masses mi and mi' are subjected in Mutual action of free the direction of their line of junction to the action of masses. forces that proceed from other masses, then, in the interval of time /, the velocities v, v' will be generated, and the equation (p - -p') t = /11 v 1- m'v' will subsist. This follows from the equations pt nv and p't' Iiv'. The sum iv - m'v' is called the momiicenZtum of the system, and in its computation oppositely directed forces and velocities are regarded as having opposite signs. If, now, the masses mi, ii' in addition to being subjected to the action of the external forces p, p' are also acted upon by interna/l forces, that is by such as are mutually exerted by the masses on one anot/her, these forces will, by Newton's third law, be equal and opposite, q, - q. The sum of the impressed impulses is, then, (p + -f q - q)t ( + p')/, the same as before; and, consequently, also, the total momentum of the system will be the same. The momentum of a 288 THE SCIENCE OF MECHANICS. system is thus determined exclusively by external forces, that is, by forces which masses outside of the system exert on its parts. Law of the Imagine a number of free masses m, mi', m".. ConservationofMo- distributed in any manner in space and acted on by mentum. external forces p, ', ".... whose lines have any directions. These forces produce in the masses in the interval of time t the velocities v, v', v".....Resolve all the forces in three directions x, y, z at right angles to each other, and do the same with the velocities. The sum of the impulses in the x-direction will be equal to the momentum generated in the x-direction; and so with the rest. If we imagine additionally in action between the masses m, m', m"...., pairs of equal and opposite internal forces q, - q, r, - i, s, - s, etc., these forces, resolved, will also give in every direction pairs of equal and opposite components, and will consequently have on the sum-total of the impulses no influence. Once more the momentum is exclusively determined by external forces. The law which states this fact is called the law of the conservation of nmomentunm. Law of the 3. Another form of the same principle, which NewConservation of the ton likewise discovered, is called the law of the conserCentre of Gravity. vation of /he centre of grar2 mn m D A S B C ity. Imagine inA and B (Fig. 149) two masses, 2 m Fig. 149. and m, in mutual action, say that of electrical repulsion; their centre of gravity is situated at S, where BS= 2AS. The accelerations they impart to each other are oppositely directed and in the inverse proportion of the masses. If, then, in consequence of the mutual action, 2m describes a distance AD, m will necessarily describe a distance BC THE EXT'ENSIONA OF 7THL PRIACIPLES. 289 2AD. The point S will still remain the position of the centre of gravity, as CS-- 2DS. Therefore, two masses cannot, by mutual action, displace their common centre of gravity. If our considerations involve several masses, dis- This law applied to tributed in any way in space, the same result will also systems of be found to hold good for this case, For as no two of the masses can displace their centre of gravity by mutual action, the centre of gravity of the system as a whole cannot be displaced by the mutual action of its parts. Imagine freely placed in space a system of masses /i, i1I', 7/.... acted on by external forces of any kind. We refer the forces to a system of rectangular co6rdinates and call the co6rdinates respectively x, y, z, x', ', z', and so forth. The co6rdinates of the centre of gravity are then S 2 m x i my ~ 2n z in which expressions x, y, z may change either by uniform motion or by uniform acceleration or by any other law, according as the mass in question is acted on by no external force, by a constant external force, or by a variable external force. The centre of gravity will have in all these cases a different motion, and in the first may even be at rest. If now internal forces, acting between every two masses, im' and im", come into play in the system, opposite displacements w', w" will thereby be produced in the direction of the lines of junction of the masses, such that, allowing for signs, im'w' + mn'"w" = 0. Also with respect to the components xI and x2 of these displacements the equation mn'x + /m"x2 = 0 will hold. The internal forces consequently 290 TIIHE SCIETNCE OF MECHIIANICS. produce in the expressions $, 7/, only such additions as mutually destroy each other. Consequently, the motion of the centre of gravity of a system is determined by external forces only. Accelera- If we wish to know the acceleration of the centre of tion of the centre of gravity of the system, the accelerations of the system's gravity of a r. system. piarts must be similarly treated. If 9p, qp',.p".. denote the accelerations of 7, m', nm"... in any direction, and p the acceleration of the centre of gravity in the same direction, p -= 2p/nm, or putting the total mass 2im = M, 9) = 2m np/M. Accordingly, we obtain the acceleration of the centre of gravity of a system in any direction by taking the sum of all the forces in that direction and dividing the result by the total mass. The centre of gravity of a system moves exactly as if all the masses and all the forces of the system were concentrated at that centre. Just as a single mass can acquire no acceleration without the action of some external force, so the centre of gravity of a system can acquire no acceleration without the action of external forces. 4. A few examples may now be given in illustration of the principle of the conservation of the centre of gravity. Movement Imagine an animal free in space. If the animal ial free in move in one direction a portion in of its mass, the respace. mainder of it AMwill be moved in the opposite direction, always so that its centre of gravity retains its original position. If the animal draw back the mass in, the motion of M also will be reversed. The animal is unable, without external supports or forces, to move itself from the spot which it occupies, or to alter motions impressed upon it from without. A lightly running vehicle A is placed on rails and 77IEf I XTENSION OF THE PRINCIPLES.2'9S 29I loaded with stones. A man stationed in the vehicle of a vehicle, from casts out the stones one after another, in the same di- which stones are rection. The vehicle, supposing the friction to be suf- cast. ficiently slight, will at once be set in motion in the opposite direction. The centre of gravity of the system as a whole (of the vehicle -+- the stones) will, so far as its motion is not destroyed by external obstacles, continue to remain in its original spot. If the same man were to pick up the stones from without and place them in the vehicle, the vehicle in this case would also be set in motion; but not to the same extent as before, as the following example will render evident. A projectile of mass nz is thrown with a velocity v, Motion of a cannon and from a cannon of mass AM. In the reaction, M also re-its projectile. ceives a velocity, V, such that, making allowance for the signs, AV -+- mv7, - 0. This explains the so-called recoil. The relation here is VK -- (In A/) 7,; or, for equai velocities of flight, the recoil is less according as the mass of the cannon is greater than the mass of the projectile. If the work done by the powder be expressed by A, the vir-cs vivce will be determined by the equation MV2 /2 +- mv2 /2 A; and, the sum of the momenta being by the first-cited equation - 0, we readily obtain V= V2A 7 /iAM [(M -- in). Consequently, neglecting the mass of the exploded powder, the recoil vanishes when the mass of the projectile vanishes. If the mass in were not expelled from the cannon but sucked into it, the recoil would take place in the opposite direction. But it would have no time to make itself visible since before any perceptible distance had been traversed, mi would have reached the bottom of the bore. As soon, however, as M and in are in rigid connection with each other, as soon, that is, as they are relatively at rest to each other, they must be absolutely at rest, 292 7TIE SCIENCE OF MIECCHANICS. for the centre of gravity of the system as a whole has no motion. For the same reason no considerable motion can take place when the stones in the preceding example are taken into the vehicle, because on the establishment of rigid connections between the vehicle and the stones the opposite momenta generated are destroyed. A cannon sucking in a projectile would experience a perceptible recoil only if the sucked in projectile could fly through it. oscilla- Imagine a locomotive freely suspended in the air, tions of the body of a or, what will subserve the same purpose, at rest with locomotive. insufficient friction on the rails. By the law of the conservation of the centre of gravity, as soon as the heavy masses of iron in connection with the pistonrods begin to oscillate, the body of the locomotive will be set in oscillation in a contrary direction-a motion which may greatly disturb its uniform progress. To eliminate this oscillation, the motion of the masses of iron worked by the piston-rods must be so compensated for by the contrary motion of other masses that the centre of gravity of the system as a whole will remain in one position. In this way no motion of the body of the locomotive will take place. This is done by affixing masses of iron to the driving-wheels. Illustration The facts of this case may be very prettily shown of the last case. by Page's electromotor (Fig. 150). When the iron core in the bobbin AB is projected by the internal forces acting between bobbin and core to the right, the body of the motor, supposing it to rest on lightly movable wheels rr, will move to the left. But if to a spoke of the fly-wheel R we affix an appropriate balance-weight a, which always moves in the contrary direction to the iron core, the sideward movement of the body of the motor may be made totally to vanish. THE EXTENSION OF THE PRINACIPLES. 293 Of the motion of the fragments of a bursting bomb A bursting bomb. we know nothing. But it is plain, by the law of the conservation of the centre of gravity, that, making allowance for the resistance of the air and the obstacles the individual parts may meet, the centre of gravity of the system will continue after the bursting to describe the parabolic path of its original projection. 5. A law closely allied to the law of the centre of Law of the Conservagravity, and similarly applicable to free systems, is the tion of Areas. principle of the conservation of areas. Although Newton A B Fig. 150. had, so to say, this principle within his very grasp, it was nevertheless not enunciated until a long time afterwards by EULER, D'ARCY, and DANIEL BERNOULLI. Euler and Daniel Bernoulli discovered the law almost simultaneously (1746), on the occasion of treating a problem proposed by Euler concerning the motion of balls in rotatable tubes, being led to it by the consideration of the action and reaction of the balls and the tubes. D'Arcy (1747) started from Newton's investigations, and generalised the law of sectors which the latter had employed to explain Kepler's laws. 294 THE SCIENCE OF A/ME CHAtNICS. Deduction Two masses mi, m' (Fig. 151) are in mutual action. of the law. By virtue of this action the masses describe the distances AB, CD in the direction of their line of junction. Allowing for the signs, then, mi. AB + mi'. CD 0. Drawing radii vectores to the moving masses from any point 0, and regarding the areas described in opposite senses by the \7i' radii as having opposite \, signs, we further obtain \. OAB + ji'. OCD- 0. \ Which is to say, if two \ / / masses mutually act on \ /each other, and radii 7ec/ // tores be drawn to these S\// masses from any point, the sum of the areas 0 described by the radii Fig. 151. multiplied by the respective masses is --= 0. If the masses are also acted on by external forces and as the effect of these the areas OAE and OCF are described, the joint action of the internal and external forces, during any very small period of time, will produce the areas OA G and OCH. But it follows from Varignon's theorem that m OA G m' O CH m OAE ' OCF - imOAB i m'OCD = mOAE + m'OC'; in other words, the sum of the products of the areas so described into the respective masses whi/ic compose a system is unaltered by the action of internal forces. If we have several masses, the same thing may be asserted, for every two masses, of the projection on any given plane of the motion. If we draw radii from THE EXTENSION OF THE P RIrNCIPLES. 2 295 any point to the several masses, and project on any plane the areas the radii describe, the sum of the products of these areas into the respective masses will be independent of the action of internal forces. This is the law of the conserCvation of areas. If a single mass not acted on by forces is moving Interpretation of the uniformly forward in a straight line and we draw a law. radius vector to the mass from any point 0, the area described by the radius increases proportionally to the time. The same law holds for 2I f, in cases in which several masses not acted on by forces are moving, where we signify by the summation the algebraic sum of all the products of the areas (f) into the moving masses-a sum which we shall hereafter briefly refer to as the sum of the mass-areas. If inte-nal forces come into play between the masses of the system, this relation will remain unaltered. It will still subsist, also, if external forces be applied whose lines of action pass through the fixed point O, as we know from the researches of Newton. If the mass be acted on by an external force, the area f described by its radius vector will increase in time by the law f= a /2 2- b + c, where a depends on the accelerative force, b on the initial velocity, and c on the initial position. The sum 2n1if increases by the same law, where several masses are acted upon by external accelerative forces, provided these may be regarded as constant, which for sufficiently small intervals of time is always the case. The law of areas in this case states that the internal forces of the system have no influtencie on the increase of the sum of the massareas. A free rigid body may be regarded as a system whose parts are maintained in their relative positions 296 TIlE SCIENCE" OF 0 IE GIL IVJCS. Uniform ro- by internal forces. The law of areas is applicable theretation of a free rigid fore to this case also. A simple instance is afforded body. by the uniform rotation of a rigid body about an axis passing through its centre of gravity. If we call in a portion of its mass, r the distance of the portion from the axis, and at its angular velocity, the sum of the mass-areas produced in unit of time will be 2w (/-/2) /- T-- (0V/2) 2/in r2, or, the product of the moment of inertia of the system into half its angular velocity. This product can be altered only by external forces. Illustrative 6. A few examples maynow be cited in illustration examples, of the law. If two rigid bodies K and K' are connected, and K is brought by the action of internal forces into rotation relatively to K', immediately K' also will be set in rotation, in the opposite direction. The rotation of K generates a sum of mass-areas which, by the law, must be compensated for by the production of an equal, but opposite, sum by K'. Opposite This is very prettily exhibited by the electromotor rotation of the wheel of Fig. 152. The fly-wheel of the motor is placed in and body of a free elec- a horizontal plane, and the motor thus attached to a tro-motor. vertical axis, on which it can freely turn. The wires conducting the current dip, in order to prevent their interference with the rotation, into two conaxial gutters of mercury fixed on the axis. The body of the motor (K') is tied by a thread to the stand supporting the axis and the current is turned on. As soon as the flywheel (K), viewed from above, begins to rotate in the direction of the hands of a watch, the string is drawn taut and the body of the motor exhibits the tendency to rotate hi (ie oposi/e direction-a rotation which immediately takes place when the thread is burnt away. The motor is, with respect to rotation about its THE1 EXTNSION OF THEI PRNCILS. 297 axis, a free system. The sum of the mass-areas gen- Its explanation by the erated, for the case of rest is = 0. But the w/icel of law. the motor being set in rotation by the action of the internal electro-magnetic forces, a sum of mass-areas is produced which, as the total sum must remain - 0, is compensated for by the rotation in the opposite direction of the body of the motor. If an index be attached to the body of the motor and kept in a fixed position 298 THE111 SCIENCE OF MTECIZANICS. by an elastic spring, the rotation of the body of the. motor cannot take place. Yet every acceleration of the wheel in the direction of the hands of a watch (produced by a deeper immersion of the battery) causes the index to swerve in the opposite direction, and every retardation produces the contrary eff ect, A variation A beautiful but curious phenomenon presents itself of the samne pihenonme- when the current to tbe motor is interrupted. Wheel non, and motor continue at first their movements in opposite directions. But the effect of the friction of the axes soon becomes apparent and the parts gradually assume with respect to each other relative rest. The motion of the body of the motor is seen to diminish; for a moment it ceases; and, finally, when the state of relative rest is reached, it is reversed and assumes the direction of the original motion of the wheel. The -whole motor now rotates in the direction the wheel did at the start. The explanation of the phenomenon is obvious. The motor is not a pe; belly free system. It is impeded by the friction of the axes. In a perfectly free system the sum of the mass-areas, the moment the parts re-entered the state of relative rest, would again necessarily be 0. But in the present instance, an external force is introduced-the friction of the axes. The friction on the axis of the wheel diminishes the mass-areas generated by the wheel and body of the motor alike. But tbe friction on the axis of the body of the motor only diminishes the sum of the massareas generated by the body. The wheel retains, thus, an excess of mass-area, which when the parts are relatively at rest is rendered apparent in the motion of the entire motor. The phenomenon subsequent to the interruption of the current supplies us with a model of what according to the hypothesis of astronomers has TIHE EXTENSION OF THE PRINCIPLES. 299 taken place on the moon. The tidal wave created by Itsillustration in the the earth has reduced to such an extent by friction the case of the velocity of rotation of the moon that the lunar day hasmoon grown to a month. The fly-wheel represents the fluid mass moved by the tide. Another example of this law is furnished by rcac- Reactionwheels. tion-whcels. If air or gas be emitted from the wheel (Fig. 153a) in the direction of the short arrows, the whole wheel will be set in rotation in the direction of the large arrow. In Fig. 1536, another simple reaction-wheel is represented. A brass tube rr plugged at both ends and appropriately perforated, is placed on a second brass tube R, supplied with a thin steel pivot through which air can be blown; the air escapes at the apertures 0, 0'. It might be supposed that sucking on the reaction- variation of the phliewheels would produce the opposite motion.to that re- noenaitof reactionsulting from blowing. Yet this does not usually take wheels. place, and the reason is obvious. The air that is sucked into the spokes of the wheel must take part immediately in the motion of the wheel, must enter the condition of relative rest with respect to the wheel; and when the system is completely at rest, the sum of its mass-areas must be 0. Generally, no perceptible rotation takes place on the sucking in of the air. The circumstances are similar to those of the recoil of a cannon which sucks in a projectile. If, therefore, an elastic ball, which has but one escape-tube, be attached to the reaction-wheel, in the manner represented in Fig. 153a, and be alternately squeezed so that the same quantity of air is by turns blown out and sucked in, the wheel will continue rapidly to revolve in the same direction as it did in the case in which we blew into it. This is partly due to the fact that the air 300 TlfF.'SOJENGIX OF MECLIANICS. Fig. 153 a. Fig. 153 b. THIE EXTENSION OF 11THE PRINCIPLES. 301 sucked into the spokes must participate in the motion Explanation of the of the latter and therefore can produce no reactional variations. rotation, but it also partly results from the difference of the motion which the air outside the tube assumes in the two cases. In blowing, the air flows out in jets, and performs rotations. In sucking, the air comes in from all sides, and has no distinct rotation. The correctness of this view is easily demonstrated. If we perforate the bottom of a hollow cylinder, a closed band-box for instance, and place the cylinder on the steel pivot of the tube R, after the side has been slit and bent in the manner indicated in Fig. 154, the box will turn in the direction of the long arrow when blown into and in the Fig. 154. direction of the short arrow when sucked on. The air, here, on entering the cylinder, can continue its rotation unimpeded, and this motion is accordingly compensated for by a rotation in the opposite direction. 7. The following case also exhibits similar condi- Reactiontubes. tions. Imagine a tube (Fig. 155a) which, running straight from a to b, turns at right angles a to itself at the latter point, passes to c, describes the circle cdef, whose plane d is at right angles to ab, and whose cen- b e tre is at b, then proceeds from f to g, and, finally, continuing the straight line ab, runs from g to. The entire tube is free to turn on an axis a /. If we pour into this tube, in the manner in- Fi. 55 dicated in Fig. I55b, a liquid, which flows in the direction cdef, the tube will immediately begin to turn 302 THE SCIENCE OF i.E1' 'CIHANI CS. in the direction fe dc. This impulse, however, ceases, the moment the liquid reaches the pointf, and flowing out into the radius fg is obliged to join in the motion of the latter. By the use of a constant stream of liquid, therefore, the rotation of the tube may soon be stopped. But if the stream be interrupted, the fluid, in flowing off through the radius fg, will impart to the tube a motional impulse in the direction of its own motion, cdef, and the tube will turn in this direction. All these phenomena are easily explained by the law of areas. The trade-winds, the deviation of the oceanic currents and of rivers, Foucault's pendulum experiment, and the like, may also be treated Fig. 155b. as examples of the law Additional of areas. Another pretty illustration is afforded by illustrationss bodies with variable moments of inertia. Let a body with the moment of inertia @ rotate with the angular velocity a and, during the motion, let its moment of inertia be transformed by internal forces, say by springs, into 9', a will then pass into a', where a 9 a'O', that is a' = a (9/@'). On any considerable diminution of the moment of inertia, a great increase of HE A EXTENSION OF T01,HE PRINCIPLIS. 303 angular velocity ensues. The principle might conceivably be employed, instead of Foucault's method, to demonstrate the rotation of the earth, [in fact, some attempts at this have been made, with no very marked success]. A phenomenon which substantially embodies the Rotating liquid in a conditions last suggested is the following. A glass funnel. funnel, with its axis placed in a vertical position, is rapidly filled with a liquid in such a manner that the stream does not enter in the direction of the axis but strikes the sides. A slow rotatory motion is thereby set up in the liquid which as long as the funnel is full, is not noticed. But when the fluid retreats into the neck of the funnel, its moment of inertia is so diminished and its angular velocity so increased that a violent eddy with considerable axial depression is created. Frequently the entire effluent jet is penetrated by an axial thread of air. 8. If we carefully examine the principles of the Both principles are centre of gravity and of the areas, we shall discover in simplyspecial casesof both simply convenient the law of Sr Z- action and modes of expression, for 2m, 2W reaction. practical purposes, of s a well-known property of mechanical phenomena. To the accelera- O tion qp of one mass m Fig. 156. there always corresponds a contrary acceleration tc/y determined velocity after the collision is the velocity 0. If, further, we make the observation that only the difference of the velocities, that is only relative velocity, determines the phenomenon of impact, we shall, by imagin-;ig the environment to move, (which experience 7"ILE EXTENSION OF TILL' PRINCIPLES. 319 tells us has no influence on the occurrence,) also readily perceive additional cases. For equal inelastic masses with velocities v and 0 or v and v' the velocity after impact is v/2 or (v +- 1')/2. It stands to reason that we can pursue such a line of reflection only after experience has informed us whial the essential and decisive features of the phenomena are. If we pass to unequal masses, we must not only The experiential know from experience that mass generally is of conse- conditions of this quence, but also in what manner its influence is effec- method. tive. If, for example, two bodies of masses i and 3 with the velocities v and V collide, we might reason 03?^ m m, Fig. 162. Fig. 163. thus. We cut out of the mass 3 the mass i (Fig. 162), and first make the masses i and i collide: the resultant velocity is (v + V)/2. There are now left, to equalise the velocities (v -+- V)/2 and V, the masses i +-- i -= 2 and 2, which applying the same principle gives V+ + V 2 v+ V 3 _ v+3V 2 4 1~+3 Let us now consider, more generally, the masses min and in', which we represent in Fig. 163 as suitably proportioned horizontal lines. These masses are affected with the velocities v and v', which we represent by ordinates erected on the mass-lines. Assun-Kig that 320 TlHE SCIENCE OF MLECHIANICS. Its points ofM -c m', we cut off from im' a portion m. The offsetting contact with the of m and mi gives the mass 2 v with the velocity (v -- Newtonian. v')/2. The dotted line indicates this relation. We proceed similarly with the remainder /i'- vm. We cut off from 2 z a portion im'- m, and obtain the mass 2m-v - (/m'- m) with the velocity (v -- v')/2 and the mass 2 (vi'- mi) with the velocity [(v V- ')/2 + v']/2. In this manner we may proceed till we have obtained for the whole mass m +- mz' the samze velocity i. The constructive method indicated in the figure shows very plainly that here the surface equation (in +- m') u = v - mz'v' subsists. We readily perceive, however, that we cannot pursue this line of reasoning except the sum mi v -- m'v', that is the form of the influence of mi and v, has through some experience or other been previously suggested to us as the determinative and decisive factor. If we renounce the use of the Newtonian principles, then some other specific experiences concerning the import of mi v which are equivalent to those principles, are indispensable. Second, the 7. The impact of elastic masses may also be treated impact of elastic by the Newtonian principles. The sole observation masses in Newton's here required is, that a deformation of elastic bodies view. calls into play forces of restitution, which directly depend on the deformation. Furthermore, bodies possess impenetrability; that is to say, when bodies affected with unequal velocities meet in impact, forces which equalise these velocities are produced. If two elastic masses M,, mi with the velocities C, c collide, a deformation will be effected, and this deformation will not cease until the velocities of the two bodies are equalised. At this instant, inasmuch as only internal forces are involved and therefore the momentum and THE EXTENSION OF THE PRINCIPLES. 321 the motion of the centre of gravity of the system remain unchanged, the common equalised velocity will be MC+ mc Au f M+ Consequently, up to this time, M's velocity has suffered a diminution C- u; and m's an increase u - c. But elastic bodies being bodies that recover their forms, in perfectly elastic bodies the very same forces that produced the deformation, will, only in the inverse order, again be brought into play, through the very same elements of time and space. Consequently, on the supposition that m is overtaken by M, A will a second time sustain a diminution of velocity C- u, and mi will a second time receive an increase of velocity 1i - c. Hence, we obtain for the velocities V, v after impact the expressions V= 2 zu- C and v = 2 z - c, or MC + m (2 c- C) mc mc+ M(2C- c) M + /m ' +- m If in these formulae we put Jf= mz, it will followThe deduction by this that V= c and v = C; or, if the impinging masses are view of all the laws. equal, the velocities which they have will be interchanged. Again, since in the particular case Mi/m- c/C or MC - mc 0= also u =0, it follows that V= 2u- C= - C and v = 2 u- c - c; that is, the masses recede from each other in this case with the same velocities (only oppositely directed) with which they approached. The approach of any two masses M, in affected with the velocities C, c, estimated as positive when in the same direction, takes place with the velocty C- c; their separation with the velocity V-v. But it follows at once from V- 2 u- C, v =2 u - c, that V-v - (C- c); that is, the relative velocity of approach and recession is the same. 322 THE SCIFE NC[P OF MECHANICS. By the use of the expressions V=-2 ui- C and v2 uM- c, we also very readily find the two theorems Af V+ in v MC-+ inic and MAV2 -+-- I7/72 --1C02 +IIC 2, which assert that the quantity of motion before and after impact, estimated in the same direction, is the same, and that also the vis viva of the system before and after impact is the same. We have reached, thus, by the use of the Newtonian principles, all of Huygens's results. The impli- 8. If we consider the laws of impact from Huygens's cations of HIuygens's point of view, the following reflections immediately view. claim our attention. The height of ascent which the centre of gravity of any system of masses can reach is given by its vis viva, ~;;iv2. In every case in which work is done by forces, and in such cases the masses follow the forces, this sum is increased by an amount equal to the work done. On the other hand, in every case in which the system moves in opposition to forces, that is, when work, as we may say, is donze u /on the system, this sum is diminished by the amount of work done. As long, therefore, as the algebraical sum of the work done on the system and the work done by the system is not changed, whatever other alterations may take place, the sum ~2 17v2 also remains unchanged. Huygens now, observing that this first property of material systems, discovered by him in his investigations on the pendulum, also obtained in the case of impact, could not help remarking that also the sum of the vbl-es vivae must be the same before and after impact. For in the mutually effected alteration of the forms of the colliding bodies the material system considered has the same amount of work done on. it as, on THE EXTENSION OF THE PRINLCIPLIS. 3 323 the reversal of the alteration, is done 3b it, provided always the bodies develop forces wholly determined by the shapes they assume, and that they regain their original form by means of the same forces employed to effect its alteration. That the latter process takes place, definite experience alone can inform us. This law obtains, furthermore, only in the case of so-called ferfectly elastic bodies. Contemplated from this point of view, the majority The deduction of the of the Huygenian laws of impact follow at once. Equal laws of impact by the masses, which strike each other with equal but oppo- notion of z 7s viva and site velocities, rebound with the same velocities. The work. velocities are uniquely determined only when they are equal, and they conform to the principle of vis viva only by being the saome before and after impact. Further it is evident, that if one of the unequal masses in impact change only the sign and not the magnitude of its velocity, this must also be the case with the other. On this supposition, however, the relative velocity of separation after impact is the same as the velocity of approach before impact. Every imaginable case can be reduced to this one. Let c and c' be the velocities of the mass m before and after impact, and let them be of any value and have any sign. We imagine the whole/ system to receive a velocity n of such magnitude that ii + c - - (u + c') or -=- (c - c') i/2. It will be seen thus that it is always possible to discover a velocity of transportation for the system such that the velocity of one of the masses will only change its sign. And so the proposition concerning the velocities of approach and recession holds generally good. As Huygens's peculiar group of ideas was not fully perfected, he was compelled, in cases in which the velocity-ratios of the impinging masses were not origin 324 THE SCIENCE OF MECHANICS. Huygens's ally known, to draw on the Galileo-Newtonian system tacit appropriation of for certain conceptions, as was pointed out above. the idea of mass. Such an appropriation of the concepts mass and momentum, is contained, although not explicitly expressed, in the proposition according to which the velocity of each impinging mass simply changes its sign when before impact M/m i - c/C. If Huygens had wholly restricted himself to his own point of view, he would scarcely have discovered this proposition, although, once discovered, he was able, after his own fashion, to supply its deduction. Here, owing to the fact that the momenta produced are equal and opposite, the equalised velocity of the masses on the completion of the change of form will be u 0. When the alteration of form is reversed, and the same amount of work is performed that the system originally suffered, the same velocities with opposite signs will be restored. Construe- If we imagine the entire system affected with a vetive comparison of locity of translation, this particular case will simultathe special and general neously present the grneralcase. pact.ofim- D --H Let the impinging masses be /" "-- K represented in the figure by L -- M -BC and mi==AC (Fig. 164), and their respective velocities by C- AD and c -BE. A C B On AB erect the perpendicular Fig. 164. CF, and through F draw IK parallel to AB. Then ID - (mi. C--c)/(M+- mi) and KE - (M. C- c)/(M-+- vi). On the supposition now that we make the masses M and m collide with the velocities ID and KE, while we simultaneously impart to the system as a whole the velocity n --= AI= KB = C- (m. C-- )/(MA+ ni) -- c+ (MA. C- c)/(M+,) =- (AMC+ mc)/(M+ - ), THE EXTENSION OF THE PRINCIPLES. 325 the spectator who is moving forwards with the velocity ii will see the particular case presented, and the spectator who is at rest will see the general case, be the velocities what they may. The general formulae of impact, above deduced, follow at once from this conception. We obtain: V AG C- (C-c) MC+ m(2c-C) V AG= C-2 -- BH -- 2 - - _ + M cMA + m M+ m Huygen's successful employment of the fictitious significance of the motions is the outcome of the simple perception that fictitious motions. bodies not affected with differences of velocities do not act on one another in impact. All forces of impact are determined by differences of velocity (as all thermal effects are determined by differences of temperature). And since forces generally determine, not velocities, but only changes of velocities, or, again, differences of velocities, consequently, in every aspect of impact the sole decisive factor is diffrences of velocity. With respect to which bodies the velocities are estimated, is indifferent. In fact, many cases of impact which from lack of practice appear to us as different cases, turn out on close examination to be one and the same. Similarly, the capacity of a moving body for work, velocity, a physical whether we measure it with respect to the time of its level. action by its momentum or with respect to the distance through which it acts by its vis viva, has no significance referred to a single body. It is invested with such, only when a second body is introduced, and, in the first case, then, it is the difference of the velocities, and in the second the square of the difference that is decisive. Velocity is a physical level, like temperature, potential function, and the like. 326 THE SCIENCE OF MEI CI4NICS. Possible It remains to be remarked, that Huygens could different origin of have reached, originally, in the investigation of the Huygens's ideas, phenomena of impact, the same results that he previously reached by his investigations of the pendulum. In every case there is one thing and one thing only to be done, and that is, to discover in all the facts the same elements, or, if we will, to rediscover in one fact the elements of another which we already know. From which facts the investigation starts, is, however, a matter of historical accident. Conserva- 9. Let us close our examination of this part of the tion of momentn in- subject with a few general remarks. The sum of the terpreed.momenta of a system of moving bodies is preserved in impact, both in the case of inelastic and elastic bodies. But this preservation does not take place precisely in the sense of Descartes. The momentum of a body is not diminished in proportion as that of another is increased; a fact which Huygens was the first to note. If, for example, two equal inelastic masses, possessed of equal and opposite velocities, meet in impact, the two bodies lose in the Cartesian sense their entire momentum. If, however, we reckon all velocities in a.given direction as positive, and all in the opposite as negative, the sum of the momenta is preserved. Quantity of motion, conceived in this sense, is always preserved. Thevs is vva of a system of inelastic masses is altered in impact; that of a system of perfectly elastic masses is preserved. The diminution of vis viva produced in the impact of inelastic masses, or produced generally when the impinging bodies move with a common velocity, after impact, is easily determined. Let M, m be the masses, C, c their respective velocities be THE EXTEYSION OF THE PRINCIPLES. 327 fore impact, and i their common velocity after impact; onservation of vis then the loss of vis viva is viva in inpact inter1AMC2 + 1_ 2 -12 - (J+1 m) I/i/,........ (1) preted. which in view of the fact that z = (AMC + /i c) /(M-+ n') may be expressed in the form ~(AMn/A-- v) (C-c)2. Carnot has put this loss in the form (C- )2 + (mu-c)2.......... (2) If we select the latter form, the expressions M(C- )2 and -m1_ (/ - c)2 will be recognised as the vis viva generated by the vwork of the internal forces. The loss of vis viva in impact is equivalent, therefore, to the work done by the internal or so-called molecular forces. If we equate the two expressions (i) and (2), remembering that (MA+ in) u =i l MC n+ ic, we shall obtain an identical equation. Carnot's expression is important for the estimation of losses due to the impact of parts of machines. In all the preceding expositions we have treated Oblique impact. the impinging masses as points which moved only in the direction of the lines joining them. This simplification is admissible when the centres of gravity and the point of contact of the impinging masses lie in one straight line, that is, in the case of so-called direct impact. The investigation of what is called oblique impact is somewhat more complicated, but presents no especial interest in point of principle. A question of a different character was treated by The centre of percusWALLIS. If a body rotate about an axis and its motion sion. be suddenly checked by the retention of one of its points, the force of the percussion will vary with the position (the distance from the axis) of the point arrested. The point at which the intensity of the impact is greatest is called by Wallis the centre of percussion. 328 THE SCIENCE OF MECHANICS. If this point be checked, the axis will sustain no pressure. We have no occasion here to enter in detail into these investigations; they were extended and developed by Wallis's contemporaries and successors in many ways. Theballis- io. We will now briefly examine, before concluding tic pendulum. this section, an interesting application of the laws of impact; namely, the determination of the velocities of projectiles by the ballistic pendulum. A mass M is suspended by a weightless and massless string (Fig. 165), so as to oscillate as a pendulum. While in the position of S equilibrium it suddenly receives the horizontal velocity V. It ascends by virtue of this velocity to an altitude h/ (/) (1 cos a) V2/2 g, where denotes the i, V length of the pendulum, a the angle of Fig. 65. elongation, and g the acceleration of gravity. As the relation T=- 7l///g subsists between the time of oscillation T and the quantities /, g, we easily obtain V (T/rR) 1/2 (1 - cos a), and by the use of a familiar trigonometrical formula, also V=7t 2 Vy -g Tsm -" Itsformula. If now the velocity V is produced by a projectile of the mass m which being hurled with a velocity v and sinking in M.is arrested in its progress, so that whether the impact is elastic or inelastic, in any case the two masses acquire after impact the common velocity V, it follows that my = (M + m) V; or, if m be sufficiently small compared with M, also v (M/m) V; whence finally 2 M a v... Tsin 7. //.) 2 TIHE EXTENSION OF THEII PRINCIPLES. 329 If it is not permissible to regard the ballistic pen- A different deduction. dulum as a simple pendulum, our reasoning, in conformity with principles before employed, will take the following shape. The projectile m with the velocity v has the momentum mv, which is diminished by the pressurej due to impact in a very short interval of time r to mi V. Here, then, m (v - V) = pr, or, if V compared with v is very small, my =- p. With Poncelet, we reject the assumption of anything like instatanaeous forces, which generate instantler velocities. There are no instantaneous forces. What has been called such are very great forces that produce perceptible velocities in very short intervals of time, but which in other respects do not differ from forces that act continuously. If the force active in impact cannot be regarded as constant during its entire period of action, we have only to put in the place of the expression tr the expressionfp dta. In other respects the reasoning is the same. A force equal to that which destroys the momentum The vis viva and of the projectile, acts in reaction on the pendulum. If work of the pendulum. we take the line of projection of the shot, and consequently also the line of the force, perpendicular to the axis of the pendulum and at the distance b from it, the moment of this force will be bp, the angular acceleration generated bp/ m r2, and the angular velocity produced in time r l. b r m v 2'Imr2 IrT2* The vis viva which the pendulum has at the end of time r is therefore 7)2 m 2 v 2 1 (p22mr2 /- 1 _1 '- 2 m'r2 330 THE SCIENCE OF ME CHANICS. The result, By virtue of this vis viva the pendulum performs the same. the excursion a, and its weight Mg, (a being the distance of the centre of gravity from the axis,) is lifted the distance a (1 - cos a). The work performed here is Mga (1-cos a), which is equal to the above-mentioned vis viva. Equating the two expressions we readily obtain 1/ 2 Mga m r2 (1 - cos a) vib zmb and remembering that the time of oscillation is T / 2mri T 7r Mga ' and employing the trigonometrical reduction which was resorted to immediately above, also 2 Ma a v.. - -rT. sin 7t m b 2 Interpreta- This formula is in every respect similar to that obtion of the result, tained for the simple case. The observations requisite for the determination of v, are the mass of the pendulum and the mass of the projectile, the distances of the centre of gravity and point of percussion from the axis, and the time and extent of oscillation. The formula also clearly exhibits the dimensions of a velocity. The expressions 2/7t and sin (a/2) are simple numbers, as are also Al/m and a/b, where both numerators and denominators are expressed in units of the same kind. But the factor g T has the dimensions //-1, and is consequently a velocity. The ballistic pendulum was invented by ROBINS and described by him at length in a treatise entitled New Prinzcipls of Gunnery, published in 1742. TlE EXTEZNSI"ON OF THE PRINCIPLE S. 331 V. D'ALEMBERT'S PRINCIPLE. i. One of the most important principles for the History of the prin-,rapid and convenient solution of the problems of me- ciple. chanics is the princziple of D'Alembert. The researches concerning the centre of oscillation on which almost all prominent contemporaries and successors of Huygens had employed themselves, led directly to a series of simple observations which D'ALEMBERT ultimately generalised and embodied in the principle which goes by his name. We will first cast a glance at these preliminary performances. They were almost without exception evoked by the desire to replace the deduction of Huygens, which did not appear sufficiently obvious, by one that was more convinciig. Although this desire was founded, as we have already seen, on a miscomprehension due to historical circumstances, we have, of course, no occasion to regret the new points of view which were thus reached. 2. The first in importance of the founders of the James Berno0111's theory of the centre of oscillation, after Huygens, iscontributions to the JAMES BERNOULLI, who sought as early as 1686 to ex- theory of the centre plain the compound pendulum by the lever. He ar- of oscillation. rived, however, at results which not only were obscure but also were at variance with the conceptions of Huygens. The errors of Bernoulli were animadverted on by the Marquis de L'HOPITAL in the Journal de Rotterdam, in 1690. The consideration of velocities acquired in infinitely small intervals of time in place of velocities acquired in finite times-a consideration which the lastnamed mathematician suggested-led to the removal 332 THE SCIENCE OF MEC/IIANACS. of the main difficulties that beset this problem; and in 1691, in the Acta Eruditorum, and, later, in 1703, in the Proceedings of Mhe Paris Academy James Bernoulli corrected his error and presented his results in a final and complete form. We shall here reproduce the essential points of his final deduction. James Ber- A horizontal, massless bar AB (Fig. 166) is free to noulli's deduction of rotate about A; and at the distances r, r' from A the the law of the corn- masses in, i' are attached. The accelerations with which pound pendulumn from these masses as thus connected the princi- m'y m, r pie of the - A will fall must be different from lever. le ^^the accelerations which they Fig. 166. would assume if their connecFig. 166. tions were severed and they fell freely. There will be one point and one only, at the distance x, as yet unknown, from A which will fall with the same acceleration as it would have if it were free, that is, with the acceleration g. This point is termed the centre of oscillation. If ni and m' were to be attracted to the earth, not proportionally to their masses, but m so as to fall when free with the acceleration p gr/x and in' with the acceleration p' ==gr'/x, that is to say, if the natural accelerations of the masses were proportional to their distances from A, these masses would not interfere with one another when connected. In reality, however, mi sustains, in consequence of the connection, an upward component acceleration g - (p, and i' receives in virtue of the same fact a downward component acceleration p' -g; that is to say, the former suffers an upward force of mi (g- -p) g(x- r/x) mi and the latter a downward force of mi' (p - g) - g (r' - x/x) i'. Since, however, the masses exert what influence they have on each other solely through the medium of THE EXTELNSION OF THE PRINCIPLES. 333 the lever by which they are joined, the upward force The law of the distriupon the one and the downward force upon the other bution of the effects must satisfy the law of the lever. If m in conse- of theimpressed quence of its being connected with the lever is held forces, in James Berback by a force f from the motion which it would take, noiili's example. if free, it will also exert the same forcefon the leverarm r by reaction. It is this reaction pull alone that can be transferred to im' and be balanced there by a pressure f'= (r/r')f, and is therefore equivalent to the latter pressure. There subsists, therefore, agreeably to what has been above said, the relation g (r' - x/x) v' = r/r'. g(x - r/x) i or, (x - r) i r = (r' - x) m'r', from which we obtain x = (mr2 - + 'r'2)/(mr + m'r'), exactly as Huygens found it. The generalisation of this reasoning, for any number of masses, which need not lie in a single straight line, is obvious. 3. JOHN BERNOULLI (in 1712) attacked in a different Theprinciple of manner the problem of the centre of oscillation. His John Bernoulli's soperformances are easiest consulted in his Collected lution of theproblemn Works (Opera, Lausanne and Geneva, 1762, Vols. II ofthe centre of osciland IV). We shall examine in detail here the main lation. ideas of this physicist. Bernoulli reaches his goal by conceiving the masses and forces separated. First, let us consider two simple pendulums of dif- The first step in John ferent lengths 1, 1' whose bobs are affected with gravi- Bernoulli's deduction. tational accelerations proportional to the lengths of the pendulums, that is, let us put ///' =g/g'. As the time of oscillation of a pendulum is T= rr1/ /g, it follows that the times of oscillation of these pendulums will be the same. Doubling the length of a pendulum, accordingly, while at the same time doubling the acceleration of gravity does not alter the period of oscillation. Second, though we cannot directly alter the accel 334 THE SCIENCE OF AME CHANICS. The second eration of gravity at any one spot on the earth, we step in John Bernoulli's can do what amounts virtually to this. Thus, imagine deduction. a straight massless bar of length 2a, free to rotate about its middle point; and attach to the one exS,' tremity of it the mass m and to the other the a mass im'. Then the total mass is m + - m' at the distance a from the axis. But the force a which acts on it is (in - m') -, and the ac-.1' celeration, consequently, (m - m'/m -+ n') g. Fig. 167. Hence, to find the length of the simple pendulum, having the ordinary acceleration of gravity g, which is isochronous with the present pendulum of the length a, we put, employing the preceding theorem, (/ ' nI + -n a g" or /a - a in - PI W - l The third Th/ird, we imagine a simple pendulum of length I step, or the determina- with the mass m at its extremity. The weight of m tion of the centre of produces, by the principle of the lever, the same acgyration.celeration as half this force at a distance 2 from the point of suspension. Half the mass m placed at the distance 2, therefore, would suffer by the action of the force impressed at I the same acceleration, and a fourth of the mass m would suffer double the acceleration; so that a simple pendulum of the length 2 having the original force at distance I from the point of suspension and one-fourth the original mass at its extremity would be isochronous with the original one. Generalising this reasoning, it is evident that we may transfer any force f acting on a compound pendulum at any distance r, to the distance I by making its value rf, and any and every mass placed at the distance r to the distance i by making its value r2m, without changing THE EXTENSION OF THE PRINCIPLES. 335 the time of oscillation of the pendulum. If a force f act on a lever-arm a (Fig. 168) while at the distance r from the axis a mass m is attached, f will be equivalent to a force af/r impressed on m and will impart to it the linear A acceleration afl/m rand the angu- a lar acceleration afl/nr2. Hence, to find the angular acceleration Fig. 168. of a compound pendulum, we divide the sum of the statical momnents by the sum of the moments of inertia. BROOK TAYLOR, an Englishman,* also developed Theresearches of this idea, on substantially the same principles, but Brook Tayquite independently of John Bernoulli. His solution,r. however, was not published until some time later, in 1715, in his work, Methodus Incrementorunm. The above are the most important attempts to solve the problem of the centre of oscillation. We shall see that they contain the very same ideas that D'Alembert enunciated in a generalised form. 4. On a system of points AM, M', AM"... connected Motion of a system of with one another in any way,- the forces P, P', P".. pointssubJect to conare impressed. (Fig. 169.) These forces would im-straints. part to the free points of the system certain determinate motions. To the connected points, however, different motions are usually imparted-motions which could be produced by the forces W, W', W".... These last are the motions which we shall study. Conceive the force P resolved into W and V, the force P' into W' and V', and the force P" into W" * Author of Taylor's theorem, and also of a remarkable work on perspective.- 7rans. t In precise technical language, they are subject to constraints, that is, forces regarded as infinite, which compel a certain relation between their motions.- Trans. 336 THE SCIENCE OF AME'CHANICS. Statement and V", and so on. Since, owing to the connections, of D'Alembert'sprin- only the components V, TV', TV.... are effective, ciple. therefore, the forces V, V', V"... must be equilibrated by the connections. We will call the forces P, P', P" the impressed forces, V/ the forces W, W', TV"...., which produce the ac" tual motions, the effective " M'-- / forces, and the forces V, V'.... the forces gained (and lost, or the Fig. 16. equilibrated forces. We Fig. 169. perceive, thus, that if we resolve the impressed forces into the effective forces and the equilibrated forces, the latter form a system balanced by the connections. This is the principle of D'Alembert. We have allowed ourselves, in its exposition, only the unessential modification of putting forces for the momenta generated by the.forces. In this form the principle was stated by D'ALEMBERT in his Traidt de dynamique, published in 1743. Various As the system V, V', V"...is in equilibrium, the forms in which the principle of virtual displacements is applicable thereto. principle may beex- This gives a second form of D'Alembert's principle. presse. A third form is obtained as follows: The forces P, P'.... are the resultants of the components V, W'.... and, V'.... If, therefore, we combine with the forces W, WT'.... and V, V'.... the forces -P, -P'...., equilibrium will obtain. The force-system --P, W, V is in equilibrium. But the system Vis independently in equilibrium. Therefore, also the system -P, W is in equilibrium, or, what is the same thing, the system P, - W is in equilibrium. Accordingly, if the effective forces with opposite signs be joined to the impressed THE EXTENSION OF THE PRINCIPLES. 337 forces, the two, owing to the connections, will balance. The principle of virtual displacements may also be applied to the system P, - W. This LAGRANGE did in his Mdca'nque analytique, 1788. The fact that equilibrium subsists between the sys- An equivalent princitem P and the system - IV, may be expressed in still pie employed by another way. We may say that Hermann the system W/is equivalent to the _p ad system P. In this form HERMANN (Phloronoi00 c'a, 1716) and EULER (Comm0ent. Acad. Petrop., Fig. 170 Old Series, Vol. VII, 1740) employed the principle. It is substantially not different from that of D'Alembert. 5. We will now illustrate D'Alembert's principle by one or two examples. On a massless wheel and axle with the radii R, r the Illustration of D'Alemloads P and Q are hung, which are not in equilibrium, bert's principle by the We resolve the force P into (I) W motion of a wheel and (the force which would produce the axle. actual motion of the mass if this were R r free) and (2) V, that is, we put P- WV+ V and also Q TV'+ V'; it being evident that we may here disregard all motions that are not p Q in the vertical. We have, accordFig. 171. ingly, V= P - V and V'- Q - W', and, since the forces V, V' are in equilibrium, also V. R = V'. r. Substituting for V, V' in the last equation their values in the former, we get (P- W)_R=-(Q- W')r......... (1) which may also be directly obtained by the employment of the second form of D'Alembert's principle. From the conditions of the problem we readily perceive 338 THE SCIENCE OF IIECHANICS. that we have here to deal with a uniformly accelerated motion, and that all that is therefore necessary is to ascertain the acceleration. Adopting gravitation measure, we have the forces W and W', which produce in the masses P/g and Q/g the accelerations y and y'; wherefore, W=(P/g) y and W'- (Q/g) y'. But we also know that y'= - y(r/R). Accordingly, equation (i) passes into the form (P -.. Y)...........(2) whence the values of the two accelerations are obtained PR - Q r, PR - Q r Y R PR and y = Q g. y __R2+ Qr-2 c"'aPR2 + Q I" These last determine the motion. Employ- It will be seen at a glance that the same result can ment of the ideas stat- be obtained by the employment of the ideas of statical ical moment and moment and moment of inertia. We get by this method moment of inertia. to for the angular acceleration obtain this result. PR - Qr PR - Qr P= PP Q R 2 ~ + Qr29 _R2 + -- and as y = R p and y'= - r p we re-obtain the preceding expressions. When the masses and forces are given, the problem of finding the motion of a system is determinate. Suppose, however, only the acceleration y is given with which P moves, and that the problem is to find the loads P and Q that produce this acceleration. We obtain easily from equation (2) the result P- Q (R g ry) r/(g- y) R2, that is, a relation between P and Q. One of the two loads therefore is arbitrary. The prob THE EXTENSION OF THE PRINCIPLES. 339 lem in this form is an indeterminate one, and may be solved in an infinite number of different ways. The following may serve as a second example. A weight P (Fig. 172) free to move on a vertical A second il lustration straight line AB, is attached to a cord of the prinC A ciple. passing over a pulley and carrying a weight Q at the other end. The cord makes with the line AB the variable a p angle a. The motion of the present W case cannot be uniformly accelerated. /LV But if we consider only vertical mo- y' B tions we can easily give for every Fig. 172. value of a the momentary acceleration (y and y') of P and Q. Proceeding exactly as we did in the last case, we obtain P = W+ V, Q = '+ V' also V' cosa = V, or, since y' = - cosa,. ( Q cos a cos a = - y; whence P - Q cosa S J Q2 cos 2 a P P- Qcosa Y = cos + cos a g. S -cos- a co Again the same result may be easily reached by the solution of this case employment of the ideas of statical moment and mo- also by the ideas of ment of inertia in a more generalised form. The fol- statical moInent and lowing reflexion will render this clear. The force, or moment of. inertia genstatical moment, that acts on P is P - Q cos a. But eralised. the weight Q moves cos a times as fast as P; consequently its mass is to be taken cos 2a times. The acceleration which P receives, accordingly is, 340 77/; SCIENCE OF MECHANICS. P- Qcos P- Qcosa - Q 2 Q(cos 2a +P cos2 a coS 2 4 _ In like manner the corresponding expression for y' may be found. The foregoing procedure rests on the simple remark, that not the circular path of the motion of the masses is of consequence, but only the relative velocities or relalive displacements. This extension of the concept moment of inertia may often be employed to advantage. Import and 6. Now that the application of D'Alembert's princharacter of D'Aler- ciple has been sufficiently illustrated, it will not be diffibert's principle. cult to obtain a clear idea of its significance. Problems relating to the motion of connected points are here disposed of by recourse to experiences concerning the mutual actions of connected bodies reached in the investigation of problems of equilibrium. Where the last mentioned experiences do not suffice, D'Alembert's principle also can accomplish nothing, as the examples adduced will amply indicate. We should, therefore, carefully avoid the notion that D'Alembert's principle is a general one which renders special experiences superfluous. Its conciseness and apparent simplicity are wholly due to the fact that it refers us to experiences already in our possession. Detailed knowledge of the subject under consideration founded on exact and minute experience, cannot be dispensed with. This knowledge we must obtain either from the case presented, by a direct investigation, or we must previously have obtained it, in the investigatioi of some other subject, and carry it with us to the problem in hand. We learn, in fact, from D'Alembert's principle, as our examples show, nothing that we could not also have learned by TIHE EXTENSION OF TIE PRINCIPLES. 341 other methods. The principle fulfils in the solution of problems, the office of a routine-form which, to a certain extent, spares us the trouble of thinking out each new case, by supplying directions for the employment of experiences before known and familiar to us. The principle does not so much promote our insigý/t into the processes as it secures us a practical mastery of them. The value of the principle is of an economical character. When we have solved a problem by D'Alembert's The relation of principle, we may rest satisfied with the experiences D'Alenbert's prinpreviously made concerning equilibrium, the applica- ciple to the other printion of which the principle implies. But if we wish ciples of mechanics. clearly and thoroiughly to apprehend the phenomenon, that is, to rediscover in it the simplest mechanical elements with which we are familiar, we are obliged to push our researches further, and to replace our experiences concerning equilibrium either by the Newtonian or by the Huygenian conceptions, in some way similar to that pursued on page 266. If we adopt the former alternative, we shall mentally see the accelerated motions enacted which the mutual action of bodies on one another produces; if we adopt the second, we shall directly contemplate the wo-rk done, on which, in the Huygenian conception, the vis viva depends. The latter point of view is particularly convenient if we employ the principle of virtual displacements to express the conditions of equilibrium of the system V or P - WV. D'Alembert's principle then asserts, that the sum of the virtual moments of the system V, or of the system P - W, is equal to zero. The elementary work o' ihe equilibrated forces, if we leave out of account the straining of the connections, is equal to zero. The total work done, then, is performed solely by the system P, 342 THE SCIENCE OF iMECIL4NICS. and the work performed by the system Wmust, accordingly, be equal to the work done by the system P. All the work that can possibly be done is due, neglecting the strains of the connections, to the impressed forces. As will be seen, D'Alembert's principle in this form is not essentially different from the principle of vis viva. Form of ap- 7. In practical applications of the principle of plication of D'Alem- D'Alembert it is convenient to resolve every force P bert's prin-. ciple, and impressed on a mass m of the system into the mutually the resultingequa- perpendicular components X, Y, Z parallel to the axes tions of motion. of a system of rectangular co6rdinates; every effective force W into corresponding components m$, mi, mv, where $, r/, 4 denote accelerations in the directions of the co6rdinates; and every displacement, in a similar manner, into three displacements dx, 6y, 6z. As the work done by each component force is effective only in displacements parallel to the directions in which the components act, the equilibrium of the system (P,- W) is given by the equation ) (X-m, /) 6 + (Y-m ( Y- y) 6y + (Z-m-,,) 6z-=(0 (1) or 2' (X6x Yy I -- Zzz) = -2m ($6x + y 6 + 6d'z).. (2) These two equations are the direct expression of the proposition above enunciated respecting the possible work of the impressed forces. If this work be - 0, the particular case of equilibrium results. The principle of virtual displacements flows as a special case from this expression of D'Alembert's principle; and this is quite in conformity with reason, since in the general as well as in the particular case the experimental perception of the import of work is the sole thing of consequence. Equation (i) gives the requisite equations of mo TIIE EXTENYSION OF THE PRINCIPLES. 343 tion; we have simply to express as many as possible of the displacements 6x, ),, 6z by the others in terms of their relations to the latter, and put the coefficients of the remaining arbitrary displacements= 0, as was illustrated in our applications of the principle of virtual displacements. The solution of a very few problems by D'Alem- convenience and bert's principle will suffice to impress us with a full utility of D'Alemsense of its convenience. It will also give us the con- bert's prinviction that it is possible, in every case in which it mayciple. be found necessary, to solve directly and with perfect insight the very same problem by a consideration of elementary mechanical processes, and to arrive thereby at exactly the same results. Our conviction of the feasibility of this operation renders the performance of it, in cases in which purely practical ends are in view, unnecessary. VI. THE PRINCIPLE OF VIS VIVA. T. The principle of vis viva, as we know, was first The original. historemployed by HUYGENS. JOHN and DANIEL BERNOULLI ical fori of had simply to provide for a greater generality of ex- ciple.i pression; they added little. If p, p', p"... are weights, mI, '11, i'.... their respective masses, /, /i', /"... the distances of descent of the free or connected masses, and v, v', v".... the velocities acquired, the relation obtains If the initial velocities are not - 0, but are vo, tv, 70"...., the theorem will refer to the increment of the vis viva by the work and read 2.i j = -- - in (v2 - o72). 344 THlE SCIENCE OF AMECHANICS. Theprinci- The principle still remains applicable when p... pie applied w - c a tc forces of are, not weights, but any constant forces, and h... any kind. not the vertical spaces fallen through, but any paths in the lines of the forces. If the forces considered are variable, the expressions /h, p'/i'.. must be replaced by the expressions p ds, p'ds'...., in which _p denotes the variable forces and ds the elements of distance described in the lines of the forces. Then ds 4- '' +ds' +. - or pds=2m(v-( 712)............(1) The princi- 2. In illustration of the principle of vis viva we ple illustrated by shall first consider the simple problem which we treated the motion of a wheel by the principle of D'Alembert. On and axle. rnil ~-smeL n ax a wheel and axle with the radii R, I R hang the weights P, Q. When this machine is set in motion, work is performed by which the acquired vis viva is fully determined. For a rotation of Sthe machine through the angle a, the P[ w70'ok is Fig. 173. P.la Q. ra' 'a(PR-Qr). Calling the angular velocity which corresponds to this angle of rotation, p, the vis viva generated will be PJ (Rq,)2 Q (,p)2 pq2 + rW (PR2 +Qr2). 6 2 Consequently, the equation obtains a(PR- Q-r) - (P Qr....... (1) Now the motion of this case is a uniformly accelerated motion; consequently, the same relation obtains here between the angle a, the angular velocity p, and the THILE EXTENSION OF THE PRILVCIPLES. 345 angular acceleration /., as obtains in free descent between s, v, g. If in free descent s= v2 /2g, then here a= )p /2. Introducing this value of a in equation (i), we get for the angular acceleration of P, I- =(PR- Qr/ PR2 + Qr2)g, and, consequently, for its absolute acceleration y = (PR - Q/PR2 -+ Qr2) Rg, exactly as in the previous treatment of the problem. As a second example let us consider the case of a A rolling cylinder on massless cylinder of radius r, in the surface of which, an inclined plane. diametrically opposite each other, are fixed two equal masses m, and which in consequence of the weight of U Fig. 174. Fig. 175. these masses rolls without sliding down an inclined plane of the elevation a. First, we must convince ourselves, that in order to represent the total vis viva of the system we have simply to sum up the vis viva of the motions of rotation and progression. The axis of the cylinder has acquired, we will say, the velocity u in the direction of the length of the inclined plane, and we will denote by v the absolute velocity of rotation of the surface of the cylinder. The velocities of rotation v of the two masses im make with the velocity of progression zi the angles 0 and 0' (Fig. 175), where 0 0' i= 8o~. The compound velocities w and z satisfy therefore the equations w2 =1u2 + - 2 uv cos z2 - u2 + 2 -2uvcos0'. --. I7 o ' 346 THE SCIENCE OF MECIHANICS. The law of But since cos 0 - cos 0', it follows that motion of such a ') 2 Z 2 2+ 2O cylinder. 2.- ' 2, or, 'mn'2,,z -},1z2 -- /1,/i 2 + 4.1 / / 2 7 -- _ _ 2 -4- 1 1,2 If the cylinder moves through the angle,<, m describes in consequence of the rotation the space r p, and the axis of the cylinder is likewise displaced a distance rp. As the spaces traversed are to each other, so also are the velocities v and it, which therefore are equal. The total vis viva may accordingly be expressed by 2m1,u2. If I is the distance the cylinder travels along the length of the inclined plane, the work done is 2mg. I/sin a 2m u2; whence u -= 1'. sin a. If we compare with this result the velocity acquired by a body in sliditng down an inclined plane, namely, the velocity 1/ 2gl sin a, it will be observed that the contrivance we are here considering moves with only one-half the acceleration of descent that (friction neglected) a sliding body would under the same circumstances. The reasoning of this case is not altered if the mass be uniformly distributed over the entire surface of the cylinder. Similar considerations are applicable to the case of a sp/here rolling down an inclined plane. It will be seen, therefore, that Galileo's experiment on falling bodies is in need of a quantitative correction. A modifica- Next, let us distribute the mass in uniformly over tion of the preceding the surface of a cylinder of radius R, which is coaxal case. with and rigidly joined to a massless cylinder of radius r, and let the latter roll down the inclined plane. Since here v/u =-- /r, the principle of vis viva gives mgl/ sina= ~-mu2(1 R- R2/r2), whence 2 g sin a ---- _R-- it R2 THE EXTENSION OF THE PRINCIPLES. 347 For R/r = i the acceleration of descent assumes its previous value g/2. For very large values of R/r the acceleration of descent is very small. When R/r =c it will be impossible for the machine to roll down the inclined plane at all. As a third example, we will consider the case of a The motion of a chain chain, whose total length is 1, and which lies partly on on an inclined a horizontal plane and partly on a plane having the plane. angle of elevation a. If we imagine the surface on which the chain - rests to be very smooth, any very small portion of the chain left hang-.~ Fig. 176. ing over on the inclined plane will draw the remainder after it. If pu is the mass of unit of length of the chain and a portion x is hanging over, the principle of vis viva will give for the velocity v acquired the equation pl,2 x x2 2 -- xg sina,g - sina, or v -- x /g sin a/. In the present case, therefore, the velocity acquired is proportional to the space described. The very law holds that Galileo first conjectured was the law of freely falling bodies. The same reflexions, accordingly, are admissible here as at page 248. 3. Equation (i), the equation of vis viva, can always Extension b.e employed, to solve problems of moving bodies, ciple ofi when the total distance traversed and the force that 0. acts in each element of the distance are known. It was disclosed, however, by the labors of Euler, Daniel Bernoulli, and Lagrange, that cases occur in which the 348 THE SCIENCE OF MECH.4NICS. principle of vis viva can be employed without a knowledge of the actual path of the motion. We shall see later on that Clairaut also rendered important services in this field. The re- Galileo, even, knew that the velocity of a heavy searches of Euler. falling body depended solely on the vertical hezight descended through, and not on the length or form of the path traversed. Similarly, Huygens finds that the vis viva of a heavy material system is dependent on the vertical Zcigzhts of the masses of S" the system. Euler was able to make a further step in advance. S \ If a body K (Fig. 177) is attracted towards a fixed centre C C in obedience to some given law, the increase of the vis viva 8 / in the case of rectilinear approach is calculable from the initial and terminal distances Fig. 77. (ro, r,). But the increase is the same, if K passes at all from the position ro to the position r,, independently of the form of its path, KB. For the elements of the work done must be calculated from the projections on the radius of the actual displacements, and are thus ultimately the same as before. The re- If K is attracted towards several fixed centres C, searches of niel Ber- C', C"...., the increase of its vis viva depends on the noulli and Lagrange. initial distances ro, r, ro'.... and on the terminal distances r,,,', r,"...., that is on the initial and terminal positions of K. Daniel Bernoulli extended this idea, and showed further that where movable bodies are in a state of mutual attraction the change of vis viva is determined solely by their initial and terminal dis THE EXTENSION OF THE PRINCIPLES. 349 tances from one another. The analytical treatment of these problems was perfected by Lagrange. If we join a point having the co6rdinates a, 1, c with a point having the co6rdinates x, y, z, and denote by r the length of the line of junction and by a, /3, y the angles that line makes with the axes of x, y, z, then, according to Lagrange, because r (2 = ( - a)2 + (y - ))2 ( - c)2, x- a dr -b dr cos a, - -cos - - r dx r dy z -c dr Cos y = --- - r -rdz" Accordingly, if f(r) - is the repulsive force, or The force Sdr components, parthe negative of the attractive force acting between the tial differential coeftwo points, the components will be ficients of the same dPf(r) dr dF(r) function of Sf(r) cos a dr dx dx ' cates. os dF(r) dr _ dF(r) Y r c dr dy dy dF(r) )dr dF(r) Z =f(lr) cosy = ----dr d- -dz The force-components, therefore, are the partial differential coefficients of one and the same function of r, or of the co6rdinates of the repelling or attracting points. Similarly, if several points are in mutual action, the result will be dU X dx - dU dy dz 350 THE SCIENCE OF AIE CIIANICS. The force- where U is a function of the co6rdinates of the points. unction.This function was subsequently called by Hamilton* the force-function. Transforming, by means of the conceptions here reached, and under the suppositions given, equation (i) into a form applicable to rectangular coordinates, we obtain 2ýf(Xdx + Ydy + Zdz) = 2.m (v2 -,o2) or, since the expression to the left is a complete differential, dU dU dU 2 -- dx o+ - Y- + z - (f dx dy dz) dU= 2 (UU - Uo) 2 m(v2 - vo2), where U, is a function of the terminal values and Uo the same function of the initial values of the co6rdinates. This equation has received extensive applications, but it simply expresses the knowledge that under the conditions designated the work done and therefore also the vis viva of a system is dependent on the positions, or the coordinates, of the bodies constituting it. If we imagine all masses fixed and only a single one in motion, the work changes only as U changes. The equation U- constant defines a so-called level surface, or surface of equal work. Movement upon such a surface produces no work. U increases in the direction in which the forces tend to move the bodies. VII. THE PRINCIPLE OF LEAST CONSTRAINT. I. GAUSS enunciated (in Crelle'sJournalfiir Afat/iematik, Vol. IV, 1829, p. 233) a new law of mechanics, the principle of least constraint. He observes, that, in * On a General Method in Dynamics, Phil. Trans. for 1834. See also C. G. ]. Jacobi, Vorlesungen iiber Dynzmik, edited by Clebsch, 1866. T, E EXTENSION OF THE PRIN"CIPLES. 351 the form which mechanics has historically assumed, dy- History of. the princinamics is founded upon statics, (for example, D'Alem-ple of least. constraint. bert's principle on the principle of virtual displace-constraint. ments,) whereas one naturally would expect that in the highest stage of the science statics would appear as a particular case of dynamics. Now, the principle which Gauss supplied, and which we shall discuss in this section, includes both dynamical and statical cases. It meets, therefore, the requirements of scientific and logical aesthetics. We have already pointed out that this is also true of D'Alembert's principle in its Lagrangian form and the mode of expression above adopted. No essentially nzew rinciple, Gauss remarks, can now be established in mechanics; but this does not exclude the discovery of new points of view, from which mechanical phenomena may be fruitfully contemplated. Such a new point of view is afforded by the principle of Gauss. 2. Let m, m,.... be masses, connected in any man- Statement of the prinner with one another. These masses, i;free, would, under ciple. the action of the forces impressed on them, describe in a very short element of time the spaces a b, a, b,....; but in a b consequence of their connec- m m lions they describe in the same c, Fig. 178. element of time the spaces a c,, c,.... Now, Gauss's principle asserts, that the motion of the connected points is such that, for the motion actually taken, the sum of the products of the mass of each material particle into the square of the distance of its deviation from the position it would have reached if free, namely m(b c)2 + m, (b,c,)2...... 2 m(b c)2, is a minimum, that is, is smaller for the actual motion 352 TIE SCIEN CE OF 1ME CHANICS. than for any other conceivable motion in t/e stc ri connections. If this sum,.2 m(bc)2, is less for;-rst than for any motion, equilibrium will obtain. The principle includes, thus, both statical and dynamical cases. Definition The sum 2 m(b c)2 is called the "constraint."* In of "constraint. forming this sum it is plain that the velocities present in the system may be neglected, as the relative positions of a, b, c are not altered by them. 3. The new principle is equivalent to that of D'Alembert; it may be used in place of the latter; and, as Gauss has shown, can also be deduced from it. The impressed forces carry the free mass in in an element of time through the space a b, the effective forces carry the same mass in the same time in consequence of the connections through the space a c. We resolve ab into a c and cb; and do the same for all the a6- masses. It is thus evident that forces corresponding to the disc y tances cb, c, b,... and proportional to mcb, m, c, b,..., do not, Fig. 179. owing to the connections, become effective, but form with the connections an equilibrating system. If, therefore, we erect at the terminal positions c, c,, c,,.... the virtual displacements cy, c, y,.... forming with cb, c, b,.... the angles 0, 8,.... we may apply, since by D'Alembert's principle forces proportional to mcb, m, c, b,.... are here in equilibrium, the principle of virtual velocities. Doing so, we shall have * Professor Mach's term is Abweickungssumme. The Abweichung is the declination or departure from free motion, called by Gauss the Ablenkung. (See Diihring, Princihien der Mechanik, ~~ 168, 169; Routh, Rigid Dynamics, Part I, ~~ 390-394.) The quantity in m (bc)2 is called by Gauss the Zwang; and German mathematicians usually follow this practice. In English, the term constraint is established in this sense, although it is also used with another, hardly quantitative meaning, for the force which restricts a body absolutely to moving in a certain way.-Trans. 7'TL EXTENSION OF T/ HE PRINVCIPLES. 353 mchb..Cy COS 0 0............... (1)The deduction of the But principle of least (b/) 2 = () c)2 + ( y) 2 - 2 b. C y cOS 0, constraint. (b y)2 (bC)2 (C y)2 - 2 C. c y cos 0, and m (by)2-2m(bc) 2=2m(cy)2-22m l bc.cycosO (2) Accordingly, since by (i) the second member of the right-hand side of (2) can only be = 0 or negative, that is to say, as the sum 2m(c y)2 can never be diminished by the subtraction, but only increased, therefore the left-hand side of (2) must also always be positive and consequently 2m(by)2 always greater than 2mi (bc)2, which is to say, every conceivable constraint from unhindered motion is greater than the constraint for the actual motion. 4. The declination, bc, for the very small element various fornis in of time r, may, for purposes of practical treatment, be which the principle designated by s, and following Scheffler (Schl6milch's may be expressed. Zeitschrift fiir Mathematik und Physik, 1858, Vol. II I, p. 197), we may remark that s yr 2/2, where y denotes acceleration. Consequently, 2ms2 may also be expressed in the forms T2 r2 r4 2Nm. ss 2 m y. s - 2fp. s -. 2 my2 wherep denotes the force that produces the declination from free motion. As the constant factor in no wise affects the minimum condition, we may say, the actual motion is always such that 2 ms2...................... (1) or 2ps..................... (2) or S2my2.................... (3) is a minimum. 354 THE SCIENCE OF MEI CHANICS. The motion 5. We will first employ, in our illustrations, the of a wheel and axle. third form. Here again, as our first example, we select the motion of a wheel and axle by the overweight of one of its parts R r and shall use the designations above frequently employed. Our problem is, to so determine the actual accelerations y of P and y, of Q, that p Q (P/g) (g - y) + (Q/g) ( - y,)2 shall be a minimum, or, since y, Fig. 180. -y(r/R), so that P (g- y)2 + Q(g + y.rlR)2 N shall assume its smallest value. Putting, to this end, dN r\ r dy R=- (-r) (? +r =0, we get y - (PR - Qr/PR2 + Qr2) R, exactly as in the previous treatments of the problem. Descent on As our second example, the motion of descent on an inclined plane. an inclined plane may be taken. In this case we shall employ the first form, i zs 2. rf Since we have here only to g deal with one mass, our inquiry will be directed to findS ing that acceleration of descent y for the plane by Fig. r8I. which the square of the declination (s2) is made a minimum. By Fig. 181 we have s2 =g- +( -2r~ 2 -.- y )sin a, and putting d(s 2)/dy = 0, we obtain, omitting all constant factors, 2y - 2g sin a = 0 or y =g. sin a, the familiar result of Galileo's researches. THE EXTENSION OF THE PRIiVCIPLES. 355 The following example will show that Gauss's prin- A case of equilibciple also embraces cases of equilibrium. On the arms rium. a, a' of a lever (Fig. 182) are hung the heavy masses /, m'. The principle requires that m (g-- y)2 mi'(g- y')2 shall be a minimum. But y'= - y(a'/a). Further, if the masses are ina a" versely proportional to the Z a lengths of the lever-arms, that is to say, if m/m' = a'/a, then y = - y (im/ '). Conse- Fig. 182. quently, m (g - y) 2 +_ m'(g + y. m/mr')2 = N must be made a minimum. Putting dN/dy = 0, we get i (I -- m/ m')y = 0 or y= 0. Accordingly, in this case equilibrium presents the least constraint from free motion. Every new cause of constraint, or restriction upon New causes of conthe freedom of motion, increases the quantity of con- straint increase the straint, but the increase is always the least possible. departure from free If two or more systems be connected, the motion of motion. least constraint from the motions of the unconnected systems is the actual motion. If, for example, we join together several simple pendulums so as to form a compound linear pendulum, the latter will oscillate with the motion of least constraint from the motion of the single pendulums. The simple pendulum, for any excursion a, receives, in the direction of its path, the acceleration g sin a. Denoting, therefore by y sin a the acceleration corresponding to this excur- Fig. 183. sion at the axial distance I on the compound pendulum, im (g sin a - r y sin a)2 or Yim (g - ry)2 will be the quantity to be made a minimum. Consequently, 2 n(g - r y)r = 0, and y g( m i; r/ im r2). 356 THE SCIE.NCL OF ME CHANICS. The problem is thus disposed of in the simplest manner. But this simple solution is possible only because the expericlnces that Huygens, the Bernoullis, and others long before collected, are implicitly contained in Gauss's principle. Illustra- 6. The increase of the quantity of constraint, or tions of the preceding declination, from free motion by new causes of constatement. straint may be exhibited by the following examples. Over two stationary pulleys A, B, and beneath a movable pulley C (Fig. 184), a cord is passed, each B P P 2P+ p Fig. 184. Fig. 185. extremity of which is weighted with a load P; and on C a load 2P +- is placed. The movable pulley will now descend with the acceleration (/4 P + /) g. But if we make the pulley A fast, we impose upon the system a new cause of constraint, and the quantity of constraint, or declination, from free motion will be increased. The load suspended from B, since it now moves with double the velocity, must be reckoned as possessing four times its original mass. The movable pulley accordingly sinks with the acceleration (p/6P + p) g. A simple calculation will show that the constraint in the latter case is greater than in the former. THE EXTENSION OF THE PRINCIPLE S. 357 A number, n, of equal weights, p, lying on a smooth horizontal surface, are attached to n small movable pulleys through which a cord is drawn in the manner indicated in the figure and loaded at its free extremity with p. According as all the pulleys are movable or all except one arefixed, we obtain for the motive weightp, allowing for the relative velocities of the masses as referred to p, respectively, the accelerations (4 n/i +4 4 n) g and (4/5) g. If all the n - i masses are movable, the deviation assumes the valuepg/4 n +, which increases as n, the number of the movable masses, is decreased. Fig. 186. 7. Imagine a body of weight Q, movable on rollers Treatment of a imeon a horizontal surface, and having an inclined plane chanical problem by face. On this inclined face a body of weight P is different mechanical placed. We now perceive instinctively that P will de- principles. scend with quicker acceleration when Q is movable and can give way, than it will when Q is fixed and P's descent more hindered. To any distance of descent h of P a horizontal velocity v and a vertical velocity u of P and a horizontal velocity w of Q correspond. Owing to the conservation of the quantity of horizontal motion, (for here only internal forces act,) we have Pv Qw, and for obvious geometrical reasons (Fig. 186) also u = (v + ) tan a The velocities, consequently, are Zit u 358 THE SCIENCE OF MECHANICS. First, by the = COt a. U, principles P -- ( a of the conservation of P momentum _____ cot a / and of vis + -- -p - cot a u. viva. -* Q For the work Ph performed, the principle of vis viva gives P u2 P Q 2 u2 Ph2= +-Qcota 2+ Q( P 2 u2,P+_-Qcot a) 2 Multiplying by -, we obtain ( Q cos2a U2 g +P Q sin2a 2 To find the vertical acceleration y with which the space h is described, be it noted that = u2/2 y. Introducing this value in the last equation, we get (P+ Q)sin 2 a Y Psin2a Q For Q =, y = g sin2 a, the same as on a stationary inclined plane. For Q = 0, y =-g, as in free descent. For finite values of Q =- mP, we get, 1 + m since -.n2a > 1, (1 + in) sin 2 a -y.. > g > g g sin 2 a. Y w +4i stn2 The making of Q stationary, being a newly imposed cause of constraint, accordingly increases the quantity of constraint, or declination, from free motion. To obtain y, in this case, we have employed the principle of the conservation of momentum and the TIHLL EXTENSION OF THE PRINCIPLES. 359 principle of vis ziva. Employing Gauss's principle, Second, by the prinwe should proceed as follows. To the velocities de- ciple of noted as iu, v, w the accelerations y, 6, E correspond.Gauss, Remarking that in the free state the only acceleration is the vertical acceleration of P, the others vanishing, the procedure required is, to make P --y)2 + 62 + Q a minimum. As the problem possesses significance only when the bodies P and Q touch, that is only when y- (6 - E) tan a, therefore, also N__ g --(6 + r )6tan a]2 62 QE Forming the differential coefficients of this expression with respect to the two remaining independent variables 6 and e, and putting each equal to zero, we obtain - [ - (6 +- e) tan a] P tan (a Px + - 0 and -- [g - (6 - +) tan x] P tan ( Qx + e - 0. From these two equations follows immediately Pd-- Q = 0, and, ultimately, the same value for y that we obtained before. We will now look at this problem from another point of view. The body P describes at an angle/ f with the horizon the space s, of which the horizontal and vertical components are 7' and i, while simultaneously Q describes the horizontal distance w. The force-component that acts in the direction of s is Psin /3, consequently the acceleration in this direction, allowing for the relative velocities of P and Q, is P. sin / P Q/zwi\ S 360 THE SCIENCE OF ALME CANICS. Third, by Employing the following equations which are dithe extended con- rectly deducible, cept of noinent of in- Q W P= P ertia. V = S COS / u = v tan p. the acceleration in the direction of s becomes Q sin [3 Q + P cos2/J and the vertical acceleration corresponding thereto is Q sin2/ Y Q +P'cos2/' g' an expression, which as soon as we introduce by means of the equation u = (v +- v) tan a, the angle-functions of a for those of f/, again assumes the form above given. By means of our extended conception of moment of inertia we reach, accordingly, the same result as before. Fourth, by Finally we will deal with this problem in a direct direct principles. manner. The body P does not descend on the movable inclined plane with the vertical acceleration g, with which it would fall if free, but with a different vertical acceleration, y. It sustains, therefore, a vertical counterforce (P/g) (r- y). But as P and Q, friction neglected, can only act on each other by means of a pressure S, normal to the inclined plane, therefore P S(g - y) - Scosa and QSsinP Q. g g From this is obtained P Q (g -- )= cota, 9 - THE EXTENSION OF THE PRINCIPLES. 36I and by means of the equation y = ( + E) tan a, ultimately, as before, (P+Q)sin ( P sinm2 + Q Q sin a cos a Psin2 X............(2) P sin a cos a P sin2,Qg............. (3) P sin2a Q+ Q If we put P = Q and a = 450, we obtain for this Discussion of the reparticular case y -=, 6 =., E =. For Pr -suits. Q/g - I we find the "constraint," or declination from free motion, to be g 2/3. If we make the inclined plane stationary, the constraint will be g2/2. If P moved on a stationary inclined plane of elevation /3, where tan /3 y/6, that is to say, in the same path in which it moves on the movable inclined plane, the constraint would only be g2 /5. And, in that case it would, in reality, be less impeded than if it attained the same acceleration by the displacement of Q. 8. The examples treated will have convinced us that Gauss's principle no substantially new insight or perception is afforded by affords no -" newinsiglit Gauss's principle. Employing form (3) of the principle and resolving all the forces and accelerations in the mutually perpendicular co6rdinate-directions, giving here the letters the same significations as in equation (i) on page 342, we get in place of the declination, or constraint, ~-I y 2, the expression N 2 (YL 2X 2 Y 2 and by virtue of the minimum condition dN= 2 2m [ - i dR[+ ( -- d1 rl Il1H 362 THE SCIENCE OF MECHANICS. (z o0. or 2[(X-- m 47) d; + (Y-- i /) di7 + (Z- im,)d] = 0. Gauss's and If no connections exist, the coefficients of the (in D'Alembert's prin- that case arbitrary) d4, dq7, dt, severally made = 0, ciples conmiutable. give the equations of motion. But if connections do exist, we have the same relations between d$, dr, do as above i n equation (), at page 342, between 8 x, 6y, 6z. The equations of motion come out the same; as the treatment of the same example by D'Alembert's principle and by Gauss's principle fully demonstrates. The first principle, however, gives the equations of motion directly, the second only after differentiation. If we seek an expression that shall give by differentiation D'Alembert's equations, we are led perforce to the principle of Gauss. The principle, therefore, is new only in form and not in miatter. Nor does it, further, possess any advantage over the Lagrangian form of D'Alembert's principle in respect of competency to comprehend both statical and dynamical problems, as has been before pointed out (page 342). The phys- There is no need of seeking a mystical or ume/ap/i/ysical basis of the prin- ica/ reason for Gauss's principle. The expression ' least ciple. constraint" may seem to promise something of the sort; but the name proves nothing. The answer to the question, "'In what does this constraint consist? " cannot be derived from metaphysics, but must be sought in the facts. The expression (2) of page 353, or (4) of page 361, which is made a minimum, represents the work done in an element of time by the deviation of the constrained motion from the free motion. This work, the work due to the constraint, is less for the motion actually performed than for any other possible motion. THIE EXTEN'XSIONV OF TILE I'IVCIFLES. 363 Once we have recognised work as the factor deter- RC,1e of the f actor work. minative of motion, once we have grasped the meaning of the principle of virtual displacements to be, that motion can never take place except where work can be performed, the following converse truth also will involve no difficulty, namely, that all the work that can be performed in an element of time actually is performed. Consequently, the total diminution of work due in an element of time to the connections of the system's parts is restricted to the portion annulled by the counler-worlk of those parts. It is again merely a new aspect of a familiar fact with which we have here to deal. This relation is displayed in the very simplest cases. The foundations of Let there be two masses in and in at A, the one im- the principressed with a force p, the other with nisablerin t sie - the force q. If we connect the two, we q B Bplest cases. shall have the mass 21n acted on by a resultant force r. Supposing the spaces N. described in an element of time by the free masses to be represented by AC, " D A B, the space described by the conjoint, or double, mass will be AO- 0 - c - A D. The deviation, or constraint, Fig. 187. is i(OB2 + 0C2). It is less than it would be if the mass arrived at the end of the element of time in Ml or indeed in any point lying outside of B C, say Xi, as the simplest geometrical considerations will show. The deviation is proportional to the expression p2- q2 2 +_2pq cos 0/2, which in the case of equal and opposite forces becomes 2p2, and in the case of equal and like-directed forces zero. Two forces p and q act on the same mass. The force q we resolve parallel and at right angles to the 364 THE SCIENCE OF AMECHANICS. Even in the direction of p in r and s. The work done in an element principle of the compo- of time is proportional to the squares of the forces, and sition of forces its if there be no connections is expressible by p2 q2 qproperties2 are found. p2 + r 2 - s2. If now r act directly counter to the force p, a diminution of work will be effected and the sum mentioned becomes (p -r)2 -- S2. Even in the principle of the composition of forces, or of the mutual independence of forces, the properties are contained which Gauss's principle makes use of. This will best be perceived by imagining all the accelerations simultaneously performed. If we discard the obscure verbal form in which the principle is clothed, the metaphysical impression which it gives also vanishes. We see the simple fact; we are disillusioned, but also enlightened. The elucidations of Gauss's principle here presented are in great part derived from the paper of Scheffler cited above. Some of his opinions which I have been unable to share I have modified. We cannot, for example, accept as new the principle which he himself propounds, for both in form and in import it is identical with the D'Alembert-Lagrangian. VIII. THE PRINCIPLE OF LEAST ACTION. The orig- I. MAUPERTUIS enunciated, in 1747, a principle inal, obscureform which he called " le princije de la moindre quantitd d'acipteof" tion, the principle of least action. He declared this leastaction.principle to be one which eminently accorded withthe wisdom of the Creator. He took as the measure of the "action" the product of the mass, the velocity, and the space described, or mvs. Why, it must be confessed, is not clear. By mass and velocity definite quantities may be understood; not so, however, by THE EXTENSION OF TIlHE PRINCIPLES. 365 space, when the time is not stated in which the space is described. If, however, unit of time be meant, the distinction of space and velocity in the examples treated by Maupertuis is, to say the least, peculiar. It appears that Maupertuis reached this obscure expression by an unclear mingling of his ideas of vis viva and the principle of virtual velocities. Its indistinctness will be more saliently displayed by the details. 2. Let us see how Maupertuis applies his principle. Determination of the If A-, mi be two inelastic masses, C and c their velocities laws of impact by this before impact, and u their common velocity after im- principle. pact, Maupertuis requires, (putting here velocities for spaces,) that the "action" expended in the change of the velocities in impact shall be a minimum. Hence, M(C - u) 2 1 m (c - _) 2 is a minimum; that is, A/ (C - u) + (c- u) - 0; or AMC+ m c For the impact of elastic masses, retaining the same designations, only substituting V and v for the two velocities after impact, the expression A'(C- V)2 + -m(c -v)2 is a minimum; that is to say, AI (C- V)dV- nm (c- v,)dv- 0.... (1) In consideration of the fact that the velocity of approach before impact is equal to the velocity of recession after impact, we have C-c=-- (V- v) or c+ V-(c v+)-.............. (2) and dV-dv=........................(3) The combination of equations (i), (2), and (3) readily gives the familiar expressions for V and v. These two cases may, as we see, be viewed as pro 366 THE SCIE.NCE OF MECHANICS. cesses in which the least change of vis viva by reaction takes place, that is, in which the least counter-work is done. They fall, therefore, under the principle of Gauss. Mauper- 3. Peculiar is Maupertuis's deduction of the law of tuis's deduction of thie lever. Two masses M and mi (Fig. 188) rest on a the law of the lever by bar a, which the fulcrum divides into the portions this principle. x and a - x. If lie bar be set in rotation, the velocities and the spaces described will be proportional to the lengths of the lever-arms, and Mx2 +_v m (a-x)2 is the quantity to be made a minimum, that is Mx - m (a - x) 0; whence x = via/M+ -- m, a condition that in the case of eqiilib______ rium is actually fulfilled. In M-., criticism of this, it is to be -x _remarked, first, that masses Fig. i88. not subject to gravity or other forces, as Maupertuis here tacitly assumes, are always in equilibrium, and, secondly, that the inference from Maupertuis's deduction is that the principle of least action is fulfilled only in the case of equilibrium, a conclusion which it was certainly not the author's intention to demonstrate. The correc- If it were sought to bring this treatment into aption ofMaupertuis's proximate accord with the preceding, we should have deduction. to assume that the heavy masses M and mi constantly produced in each other during the process the least possible change of vis viva. On that supposition, we should get, designating the arms of the lever briefly by a, b, the velocities acquired in unit of time by u, v, and the acceleration of gravity by g, as our minimum expression, M(g- U)2 -+- v(i- v)2; whence M(g- u) du 4- v(g- v)dv 0. But in view of the connection of the masses as lever, THE EXTENSION0 OF 71THE PRINCIPLES. 367 ' and a b' du =. dv; whence these equations correctly follow Ma - m b Ma -m b u= a v= j ~-b r a a+ mb2 ' Ma2+,z mb2 g' and for the case of equilibrium, where u = v - 0, Ma - mb= 0. Thus, this deduction also, when we come to rectify it, leads to Gauss's principle. 4. Following the precedent of Fermat and Leib- Treatment of the monitz, Maupertuis also treats by his method the nmoion tion of light by the prinof light. Here again, however, ciple of least ache employs the notion "least ac- A I tion. tion" in a totally different sense. m R D The expression which for the C case of refraction shall be a min- 11 imum, is m. AR + n. RB, where AR and RB denote the paths described by the light in Fig. 189. the first and second media respectively, and m and n the corresponding velocities. True, we really do obtain here, if R be determined in conformity with the minimum condition, the result sin a/sin / = n/mn const. But before, the " action" consisted in the change of the expressions mass X velocity X distance; now, however, it is constituted of the sum of these expressions. Before, the spaces described in unit of time were considered; in the present case the total spaces traversed are taken. Should not m. A R- n. RB or m- n)(AR- RB) be taken as a minimum, and if not, why not? But 368 THE SCIENCE OF AECLCIL4NICS. even if we accept Maupertuis's conception, the reciprocal values of the velocities of the light are obtained, and not the actual values. characteri- It will thus be seen that Maupertuis really had no sation of Mauper- principle, properly speaking, but only a vague formtuis's principle. ula, which was forced to do duty as the expression of different familiar phenomena not really brought under one conception. I have found it necessary to enter into some detail in this matter, since Maupertuis's performance, though it has been unfavorably criticised by all mathematicians, is, nevertheless, still invested with a sort of historical halo. It would seem almost as if something of the pious faith of the church had crept into mechanics. However, the mere eiueavor to gain a more extensive view, although beyond the powers of the author, was not altogether without results. Euler, at least, if not also Gauss, was stimulated by the attempt of Maupertuis. Euler'scon- 5. Euler's view is, that the purposes of the phetributions to this sub- nomena of nature afford as good a basis of explanaject. tion as their causes. If this position be taken, it will be presumed a priori that all natural phenomena present a maximum or minimum. Of what character this maximum or minimum is, can hardly be ascertained by metaphysical speculations. But in the solution of mechanical problems by the ordinary methods, it is possible, if the requisite attention be bestowed on the matter, to find the expression which in all cases is made a maximum or a minimum. Euler is thus not led astray by any metaphysical propensities, and proceeds much more scientifically than Maupertuis. He seeks an expression whose variation put = 0 gives the ordinary equations of mechanics. For a single body moving under the action of forces THE EXTEiNSION OF THE PRINCIPLES. 369 Euler finds the requisite expression in the formula Theform which the fv ds, where ds denotes the element of the path and principle f assumed in v the corresponding velocity. This expression is smaller Euler's hands for the path actually taken than for any other infinitelyh adjacent neighboring path between the same initial and terminal points, which the body may be constrained to take. Conversely, therefore, by seeking the path that makesf v ds a minimum, we can also determine the path. The problem of minimising fv ds is, of course, as Euler assumed, a permissible one, only when v depends on the position of the elements ds, that is to say, when the principle of vis viva holds for the forces, or a force-function exists, or what is the same thing, when v is a simple function of co6rdinates. For a motion in a plane the expression would accordingly assume the form f< P(xfY) I+(<2.dx In the simplest cases Euler's principle is easily verified. If no forces act, v is constant, and the curve of motion becomes a straight line, for which fv ds = vf ds is unquestionably shorter than for any other curve between the same terminal points. Also, a body moving on a curved surface A without the action of forces or friction, preserves its velocity, and describes on J the surface a shortest line. The consideration of the motion of a \ Euler's projectile in a parabola ABC (Fig. 190) ppled to will also show that the quantity fv ds OC of a roecis smaller for the parabola than for any Fig. 9 le other neighboring curve; smaller, even, than for the stra*igt line ABC between the same terminal points. The velocity, here, depends solely on the 370 THE SCIENCIE 01" iIIECHANVICS. Mathemat- vertical space described by the body, and is therefore ical development of the same for all curves whose altitude above OC is the this case. same. If we divide the curves by a system of horizontal straight lines into elements which severally correspond, the elements to be multiplied by the same v's, though in the upper portions smaller for the straight line AD than for A B, are in the lower portions just the reverse; and as it is here that the larger zr's come into play, the sum upon the whole is smaller for A B C than for the straight line. Putting the origin of the co6rdinates at A, reckoning the abscissas x vertically downwards as positive, and calling the ordinates perpendicular thereto jy, we obtain for the expression to be minimised fp 2g(- +X)\/+(dy)22dx, 0 where g denotes the acceleration of gravity and a the distance of descent corresponding to the initial velocity. As the condition of minimum the calculus of variations gives 2(a + x) - C or dy C + d 1/^.) - W Cd f.. 2c~(a-+-x)- C2 and, ultimately, C + C. -. /2(ra+x)C2+ C, g /,(a+X T77/E EXTENSION OF THE PRINCIPLES. 371 where C and C' denote constants of integration that pass into C- 1 2 ga and C'- 0, if for x = 0, dx/dy = 0 and y 0 be taken. Therefore, y = 21/ x. By this method, accordingly, the path of a projectile is shown to be of parabolic form. 6. Subsequently, Lagrange drew express attention The additions of Lato the fact that Euler's principle is applicable only in grange and Jacobi. cases in which the principle of vis viva holds. Jacobi pointed out that we cannot assert that fv ds for the actual motion is a miniuzmum, but simply that the variation of this expression, in its passage to an infinitely adjacent neighboring path, is - 0. Generally, indeed, this condition coincides with a maximum or minimum, but it is possible that it should occur without such; and the minimum property in particular is subject to certain limitations. For example, if a body, constrained to move on a spherical surface, is set in motion by some impulse, it will describe a great circle, generally a shortest line. But if the length of the arc described exceeds I8o0, it is easily demonstrated that there exist shorter infinitely adjacent neighboring paths between the terminal points. 7. So far, then, this fact only has been pointed out, Euler's expression that the ordinary equations of motion are obtained by but one of many which equating the variation of fv ds to zero. But since the give the f equations properties of the motion of bodies or of their paths may of motion. always be defined by differential expressions equated to zero, and since furthermore the condition that the variation of an integral expression shall be equal to zero is likewise given by differential expressions equated to zero, unquestionably various other integral expressions may be devised that give by variation the ordinary equations of motion, without its following that the 372 TILE SCIENCE OF AIECHANICS. integral expressions in question must possess on that account any particular physical significance. Yet the ex- 8. The striking fact remains, however, that so simple pression rsst pos- an expression as v ds does possess the property mensess a phvs- 1 ical import, tioned, and we will now endeavor to ascertain its physical import. To this end the analogies that exist between the motion of masses and the motion of light, as well as between the motion of masses and the equilibrium of strings-analogies noted by John Bernoulli and by M6bius-will stand us in stead. A body on which no forces act, and which therefore preserves its velocity and direction constant, describes a straight line. A ray of light passing through a homogeneous medium (one having everywhere the same index of refraction) describes a straight line. A string, acted on by forces at its extremities only, assumes the shape of a straight line. Elucidation A body that moves in a curved path from a point of this import by the A to a point B and whose velocity v q ^(x, y, Z) is a miotion of a miass, the function of co6rdinates, describes between A and B a motion of a ray of light, curve for which generally fv ds is a minimum. A ray an~d the equilibrium of light passing from A to B describes the same curve, of a string, if the refractive index of its medium,;i q q(x, y, z), is the same function of co6rdinates; and in this case fids is a minimum. Finally, a string passing from A to B will assume this curve, if its tension S q) (x, y, z) is the same above-mentioned function of co6rdinates; and for this case, also, fSds is a minimum. The mo'ion of a mass may be readily deduced from the equilibrium of a s/ring, as follows. On an element ds of a string, at its two extremities, the tensions S, S' act, and supposing the force on unit of length to be P, in addition a force P. ds. These three forces, which we shall represent in magnitude and direction by BA, THE EXTENSION OF THE PRLVCIPLES. 373 BC, BD (Fig. 191), are in equilibrium. If now, a body, The motion of a mass with a velocity v represented in magnitude and direc- deduced from the tion by AB, enter the element of the path ds, and re- equilibrium of a string. ceive within the same the velocity component BF= s - BD, the body will proceed onward with the velocity v' - BC. Let Q be an accelerating force whose action is directly opposite B to that of P; then for unit of time the acceleration of this force will DE be Q, for unit of length of the Fig. 191. string Q/v1, and for the element of the string (Q/i) ds. The body will move, therefore, in the curve of the string, if we establish between the forces P and the tensions S, in the case of the string, and the accelerating forces Q and the velocity v in the case of the mass, the relation P: - _ -S:. V The minus sign indicates that the directions of P and Q are opposite. A closed circular string is in equilibrium when be- The equilibrium of tween the tension S of the string, everywhere constant, closed strings. and the force P falling radially outwards on unit of length, the relation PJ) S/r obtains, where r is the radius of the circle. A body will move with the constant velocity 7v in a circle, when between the velocity and the accelerating force Q acting radially inwards the relation 7Q 7v2 - or Q =- obtains. 7 r r A body will move with constant velocity vz in any curve when an accelerating force Q- = 2/ r constantly acts 374 THE SCIENCE OF MECHANICS. on it in the direction of the centre of curvature of each element. A string will lie under a constant tension S in any curve if a force P S/r acting outwardly from the centre of curvature of the element is impressed on unit of length of the string. The deduc- No concept analogous to that of force is applicable tion of the motion of to the motion of ligrzt. Consequently, the deduction of light from the motions the motion of light from the equilibrium of a string or of masses and the the motion of a mass must be differently effected. A equilibriumn of strings, mass, let us say, is moving with the velocity AB =-v. (Fig. 192.) A force in the direction A H BD is impressed on the mass which produces an increase of velocity BE, I B AR so that by the composition of the velocities BC 4AB and BE the new S\ velocity BF-- v' is produced. If we resolve the velocities v, v' into comE ponents parallel and perpendicular to the force in question, we shall perD ceive that the parallel components alone are changed by the action of the force. Fig. 192. This being the case, we get, denoting by k the perpendicular component, and by a and a' the angles zv and v' make with the direction of the force, k = v sin a k =- v sin a' or sin a 7' sin a' v If, now, we picture to ourselves a ray of light that penetrates in the direction of v a refracting plane at right angles to the direction of action of the force, and thus passes from a medium having the index of refrac /111 EXTENSION OF TIlHE PRINCIPLES. 375 tion 1I into a medium having the index of refraction n', Development of this where n/n' - v/v', this ray of light will describe the illustration. Samre path as the body in the case above. If, therefore, we wish to imitate the motion of a mass by the motion of a ray of licht (in the same curve), we must everywhere put the indices of refraction, n, proportional to the velocities. To deduce the indices of refraction from the forces, we obtain for the velocity v2 d (--= Pdq, and \2)_ for the index of refraction, by analogy, where P denotes the force and dq a distance-element in the direction of the force. If ds is the element of the path and a the angle made by it with the direction of the force, we have then S -i ()= P cos a. ds d 2)-- P cos a. ds. For the path of a projectile, under the conditions above assumed, we obtained the expression y = i2 1/a x. This same parabolic path will be described by a ray of light, if the law n = /2 g(a x) be taken as the index of refraction of the medium in which it travels. 9. We will now more accurately investigate the Relation of the minimanner in which this minimum property is related to mum property to the the form of the curve. Let us take, first, (Fig. 193) a form of curves. broken straight line ABC, which intersects the straightc line MN, put AB = s, BC= s', and seek the condition that makes vs +-v's' a minimum for the line that passes 376 THE SCIENCE OF ME ChANICS. First, de- through the fixed points A and B, where v and v' are duction of the mini- supposed to have different, though constant, values mum condition. above and below MN. If we displace the point B an infinitely small distance to D, the new line through A and C will remain parallel to the original one, as the drawing symbolically shows. The expression s v's' is increased hereby by an amount - vm sina +v' m sin a', where min=DB. The alteration is accordingly proportional to -vsin a -v'sin a', and the condition of minimum is that S. sin ' - v sina -- v'sin a -- 0, or - sin a V A4 A\ A1 41 \ B N Fig. 193. Fig. 19. If the expression s/v1 + s'/v' is to be made a minimum, we have, in a similar way, sin a( 7 sinl ( ' v " Second, the If, next, we consider the case of a string stretched ofapiiscon- in the direction ABC, the tensions of which S and S' equilibrium are different above and below MN, in this case it is of a string. the minimum of Ss + S's' that is to be dealt with. To obtain a distinct idea of this case, we may imagine the THE EXTENSION OF THE PRINCIPLES. 377 string stretched once between A and B and thrice between B and C, and finally a weight P attached. Then S = P and S' - 3 P. If we displace the point B a distance m, any diminution of the expression Ss - S's' thus effected, will express the increase of work which the attached weight P performs. If - Sm sin a + S'm sin a'- 0, no work is performed. Hence, the minimumz of Ss + S's' corresponds to a maximum of work. In the present case the principle of least action is simply a differcnt form of the principle of virtual displacements. Now suppose that ABC is a ray of light, whose ve- Third, the application locities v and v' above and below AIN are to each other of this condition tothe as 3 to i. The motion of light be- motion of a.. ray of light. tween two points A and B is such that the light reaches B in a minimum of time. The physical reason of this is simple. The light travels from A to B, in the form of elementary waves, by different routes. Owing to the periodicity of the light, the waves generally destroy each other, and only those that reach the M SA D k N B E Fig. 195. designated point in equal times, that is, in equal phases, produce a result. But this is true only of the waves that arrive by the mnimiiium prath/ and its adjacent neighboring paths. Hence, for the path actually taken by the light s/v +- s'/v' is a minimum. And since the indices of refraction n are inversely proportional to the velocities 7v of the light, therefore also ns -n- 's' is a minimum. In the consideration of the molion of a mass the conlition that vs m- v's' shall be a minimum, strikes us as something novel. (Fig. 195.) If a mass, in its passage 378 38 TIE SCIENCE OF ML' IZCHANICS. Fourth. its through a plane MN, receive, as the result of the action toete ion of a force impressed in the direction DB, an increase of tion of a mass. velocity, by which v, its original velocity, is made v', we have for the path actually taken by the mass the equation v sin Y = v' sin ca' k. This equation, /whic/ is also /the condition of minii1mum, sifply s tasIes that only the velocit p-component paracllel to the dircCLion of ie force is altered, but thait the componcth k at rzhit ang/ces thereto remains unchiangced. Thus, here also, Euler's principle simply states a familiar fact i a new form. (See p.575.) Form of the io. The minimum condition - 7r sin a + v'sin aY' 0 minimum condition may also be written, if we pass from a finite broken applicable to curves, straight line to the elements of curves, in the form - v sin a +[ ( v dv ) sin(a + da) 0 or d(v sin a) = 0 or, finally, v sin a - const. In agreement with this, we obtain for the motion of light d (n sin a) = 0, n sin c - conist, ~(sin, ) sin const and for the equilibrium of a string d(Ssina) --- 0, Ssina - const. To illustrate the preceding remarks by an example, let us take the parabolic path of a projectile, where a always denotes the angle that the element of the path makes with the perpendicular. Let the velocity be v 1/2g(a a x), and let the axis of they-ordinates be horizontal. The condition v. sin a = const, or 1 /2g(a -(+x). dy/ds -- const, is identical with that which the calculus of variation gives, and we now know THE EXTENSION OF THEl PRINCIPLES. 379 its simple p/ysicalsignificance. If we picture to ourselves Illustration L..... 7of the three a string whose tension is S = V' 2g(a + ), an arrange- typical cases by ment which might be effected by fixing frictionless curvilinear motions. pulleys on horizontal parallel rods placed in a vertical plane, then passing the string through these a sufficient number of times, and finally attaching a weight to the extremity of the string, we shall obtain again, for equilibrium, the preceding condition, the physical significance of which is now obvious. When the distances between the rods are made infinitely small the string assumes the parabolic form. In a medium, the refractive index of Fig.96. which varies in the vertical direction by the law n /2 g(a + x), or the velocity of light in which similarly varies by the law v 1/1/ 2.g(a -- x), a ray of light will describe a path which is a parabola. If we should make the velocity in such a medium S- / 2g(a-+x), the ray would describe a cycloidal path, for which, not f1/2g(a - x). ds, but the expression fds/v/2g(a + x) would be a minimum. I1. In comparing the equilibrium of a string with the motion of a mass, we may employ in place of a string wound round pulleys, a simple homogeneous cord,, provided we subject the cord to an appropriate system of forces. We readily observe A that the systems of forces that make the tension, or, as the case may be, the ve- Fig. 197. locity, the same function of coardinates, are diZferent. If we consider, for example, the force of gravity, 38o T1fE SCIENCE OF MECIHA NICS. The condi- v = 7 /2g(a - x). A string, however, subjected to the tions and con action of gravity, forms a catenary, the tension of quences of the preccd- which is given by the formula S -- - -nx, where mi ing analogies. and in are constants. The analogy subsisting between the equilibrium of a string and the motion of a mass is substantially conditioned by the fact that for a string subjected to the action of forces possessing a forcefunction U, there obtains in the case of equilibrium the easily demonstrable equation U + S= const. This pJiysical interpretation of the principle of least action is here illustrated only for simple cases; but it may also be applied to cases of greater complexity, by imagining groups of surfaces of equal tension, of equal velocity, or equally refractive indices constructed which divide the string, the path of the motion, or the path of the'light into elements, and by making a in such a case represent the angle which these elements make with the respective surface-normals. The principle of least action was extended to systems of masses by Lagrange, who presented it in the form ( 2m vf ds =-- 0. If we reflect that the principle of vis viva, which is the real foundation of the principle of least action, is not annulled by the connection of the masses, we shall comprehend that the latter principle is in this case also valid and physically intelligible. IX. HAMILTON'S PRINCIPLE. I. It was above remarked that various expressions can be devised whose variations equated to zero give the ordinary equations of motion. An expression of this kind is contained in Hamilton's principle THIE EXTENSION OF THE PRLVCIPLES. 381 6 r(U T) d 0, or The points U of identity to of Hamiltt ton's and (6U 6T)ert's printo ciples. where 6 U and ( Tdenote the variations of the work and the vis viva, vanishing for the initial and terminal epochs. Hamilton's principle is easily deduced from D'Alembert's, and, conversely, D'Alembert's from Hamilton's; the two are in fact identical, their difference being merely that of form.* 2. We shall not enter here into any extended in- Hamilton's principle vestigation of this subject, but simply exhibit the iden- applied to the motion tity of the two principles by an exampie- of a wheel and axle. the same that served to illustrate the principle of D'Alembert: the motion of a wheel and axle by the over-weight of one of its parts. In place of the actual motion, we may imagine, performed in the same interval of time, a different motion, varying infinitely little from the actual motion, but Fig. 198. coinciding exactly with it at the beginning and end. There are thus produced in every element of time dt, variations of the work (6 U) and of the vis viva (6T); variations, that is, of the values U and T realised in the actual motion. But for the actual motion, the integral expression, above stated, is = 0, and may be employed, therefore, to determine the actual motion. If the angle of rotation performed varies in the element of time dl an amount a from the angle of the actual motion, the variation of the work corresponding to such an alteration will be 6 U= (PR - Q r) a = M a. * Compare, for example, Kirchhoff, Vorlesungen iiber mathematische Physik, Mechanik, p. 25 et seqq., and Jacobi, Vorl sungen iiber Dynamik, p. 58. 382 TZIE SCIENCE OF ME CHANICS. Mathemat- The vis viva, for any given angular velocity O, is ical development of co2 this case. T- = (PR 2 Q;2 0 2 and for a variation 6w of this velocity the variation of the vis viva is 6T- = (PR2 2+ -Q-2) Cow. But if the angle of rotation varies in the element dt an amount a, 6 CO and dt 1 da da 67T >=-j (PR2+ Q 2) a Ng dt di The form of the integral expression, accordingly, is 1d M[ N d IV t =l 0. to But as d dN da (Neex) __ __a + _Z dt dt dt' therefore, IV (A, d _-). dl + (-Zva) -t 0. to The second term of the left-hand member, though, drops out, because, by hypothesis, at the beginning and end of the motion a = 0. Accordingly, we have ft fMf- d) adl - 0 to an expression which, since a in every element of time is arbitrary, cannot subsist unless generally dN Mf- =0. dt' THE EXTENSION OF THE PRINCIPLES. 383 Substituting for the symbols the values they represent, we obtain the familiar equation d(c PR- Qr t JR2 Q/-r2 D'Alembert's principle gives the equation The same results ob( I } b tained by (M 1 =y 0 the use of D'Alembert's prinwhich holds for everyjossiblc displacement. We might, ciple. in the converse order, have started from this equation, have thence passed to the expression (M_ d/a)dt - 0 ( dN to and, finally, from the latter proceeded to the same result fMa + < ) t(t - (NVa) d.t to to t5 J "(Ma + N d t - 0. to 3. As a second and more simple example let us Illustration of this point consider the motion of vertical descent. For every bythe point tion of verinfinitely small displacement s the equation subsists icalodee[m, r-/- il (dv/dl)]s __ 0, in which the letters retain scent. their conventional significance. Consequently, this equation obtains mg' ~ m s. d~t = 0, dt.f(mr, $)sdto which, as the result of the relations (my z' s) dv dH7 S d ddS + MyV and didi dti 384 THE SCIEANCE OF IME ClHANI CS. t(n 71 S) __ (n St1S)t t 0, to to provided s vanishes at both limits, passes into the form t ds ( (mgs- myv -7di--= 0, to that is, into the form of Hamilton's principle. Thus, through all the apparent differences of the mechanical principles a common fundamental sameness is seen. These principles are not the expression of different facts, but, in a measure, are simply views of different aspects of the same fact. x. SOME APPLICATIONS OF THE PRINCIPLES OF MECHANICS TO HYDROSTATIC AND HYDRODYNAMIC QUESTIONS. Method of I. We will now supplement the examples which eliminating the action we have given of the application of the principles of gravity on liquid of mechanics, as they applied to rigid bodies, by a maiss~es. few hydrostatic and hydrodynamic illustrations. We shall first discuss the laws of equilibrium of a weightless liquid subjected exclusively to the action of so-called molecular forces. The forces of gravity we neglect in our considerations. A liquid may, in fact, be placed in circumstances in which it will behave as if no forces of gravity acted. The method of this is due to PLATEAU.* It is effected by immersing olive oil in a mixture of water and alcohol of the same density as the oil. By the principle of Archimedes the gravity of the masses of oil in such a mixture is exactly counterbalanced, and the liquid really acts as if it were devoid of weight. * Statique experimentale et thkorique des liquides, 1873. THE EXTENSION OF TIIE PRAINCIPLES. 385 2. First, let us imagine a weightless liquid mass Theworkot molecular free in space. Its molecular forces, we know, act only forces dependent on at very small distances. Taking as our radius the dis- a change in the liquid's tance at which the molecular forces cease to exert a superficial area. measurable influence, let us describe about a particlearea. a, b, c in the interior of the mass a sphere-the socalled sphere of action. This sphere of action is regularly and uniformly filled with other particles. The resultant force on the central particles a, b, c is therefore zero. Those parts only that lie at a distance from the bounding surface less than the radius of the sphere of action are in different dynamic conditions from the particles in the interior. If the radii of curvature of 0 Fig. 199. Fig. 200. the surface-elements of the liquid mass be all regarded as very great compared with the radius of the sphere of action, we may cut off from the mass a superficial stratum of the thickness of the radius of the sphere of action in which the particles are in different physical conditions from those in the interior. If we convey a particle a in the interior of the liquid from the position a to the position b or c, the physical condition of this particle, as well as that of the particles which take its place, will remain unchanged. No work can be done in this way. Work can be done only when a particle is conveyed from the superficial stratum into the interior, or, from the interior into the superficial stratum. That is to say, work can be done only by a 386 TLHE SCIENCE OF A L" GCIIAEHNICS. change of size of the surface. The consideration whether the density of the superficial stratum is the same as that of the interior, or whether it is constant throughout the entire thickness of the stratum, is not primarily essential. As will readily be seen, the variation of the surface-area is equally the condition of the performance of work when the liquid mass is immersed in a second liquid, as in Plateau's experiments. Diminution We now inquire whether the work which by the of superficial area transportation of particles into the interior effects a due to positive work. diminution of the surface-area is positive or negative, that is, whether work is performed or work is expended. If we put two fluid drops in contact, they will coalesce of their own accord; O/ and as by this action the area of the surface is diminished, it follows that the work that produces a diminution of superfiFig. 20. cial area in a liquid mass is Josilive. Van der Mensbrugghe has demonstrated this by a very pretty experiment. A square wire frame is dipped into a solution of soap and water, and on the soap-film formed a loop of moistened thread is placed. If the film within the loop be punctured, the film outside the loop will contract till the thread bounds a circle in the middle of the liquid surface. But the circle, of all plane figures of the same circumference, has the greatest area; consequently, the liquid film has contracted to a minimum. Consequent The following will now be clear. A weightless conditionb of liquid liquid, the forces acting on which are molecular forces, equilibriumwill be in equilibrium in all forms in which a system of virtual displacements produces no alteration of the liquid's superficial area. But all infinitely small changes THE EXTENSION OF THE PRIVCIPLELS 387 of form may be regarded as virtual which.the liquid admits without alteration of its volume. Consequently, equilibrium subsists for all liquid forms for which an infinitely small deformation produces a superficial variation 0. For a given volume a minimum of superficial area gives stable equilibrium; a maximum unstable equilibrium. Among all solids of the same volume, the sphere has the least superficial area. Hence, the form which a free liquid mass will assume, the form of stable equilibrium, is the sphere. For this form a maximum of work is done; for it, no more can be done If the liquid adheres to rigid bodies, the form assumed is dependent on various collateral conditions, which render the problem more complicated. 3. The connection between the size and the form of Mode of determining the liquid surface may be investigated as follows. We the connection of the imagine the closed outer sur- size and form of a face of the liquid to receive dJ liquid surface. without alteration of the liquid's volume an infinitely small variation. By two sets of mutually perpendicular lines Fig. 202. of curvature, we cut up the original surface into infinitely small rectangular elements. At the angles of these elements, on the original surface, we erect normals to the surface, and determine thus the angles of the corresponding elements of the varied surface. To every element dO of the original surface there now corresponds an element dO' of the varied surface; by an infinitely small displacement, dn, along the normal, outwards or inwards, dO passes into dO' and into a corresponding variation of magnitude. Let dp, dq be the sides of the element dO. For the 388 THE SCIENCE OF MIE CHANICS. The mathe- sides dp',.d' of the element d0', then, these relations matical development obtain of this mnethod. d' / ~ + r where r and r' are the radii of curvature of the principal sections touching the elements of the lines of curvature,, q, or the so-called principal radii of curvature.* The radius of curvature of an outwardly convex element is reckoned as positive, that of an outwardly concave element as negative, in the usual manner. For the variation of the element we obtain, accordingly, 6. dO - dO'- dO dqI + 1 + 7 - d. Neglecting the higher powers of 6 n we dp'/ get 6n 11 dp 6. dO = + . dO. r r r The variation of the whole surface, then, is expressed by 60=.l + 1,) .dO,.... (1) Furthermore, the normal displacements must be so chosen that Fig. 203. f n. dO-0.......... (2) that is, they must be such that the sum of the spaces produced by the outward and inward displacements of * The normal at any point of a surface is cut by normals at infinitely neighboring points that lie in two directions on the surface from the original point, these two directions being at right angles to each other; and the distances from the surface at which these normals cut are the two principal, or extreme, radii of curvature of the surface.-Trans. THIE EXTENSION OF THE PRINCIPLES. 389 the superficial elements (in the latter case reckoned as negative) shall be equal to zero, or the volume remain constant. Accordingly, expressions (I) and (2) can be put A condition on which simultaneously = 0 only if I/r + i/r' has the same value the generality of the exfor all points of the surface. This will be readily seen pressions obtained, from the following consideration. Let the elements depends. dO of the original surface be symbolically represented by the elements of the line AX (Fig. 204) and let the normal displacements 6n be erected as ordinates thereon in the plane E, the outward displacements upwards as positive and the inward displacements downwards as negative. Join the extremities E of these ordinates so as to form a curve, / and take the quadra- z ture of the curve,Fi. 2 Fig. 204. reckoning the surface above AX as positive and that below it as negative. For all systems of 6n for which this quadrature = 0, the expression (2) also 0, and all such systems of displacements are admissible, that is, are virtual displacements. Now let us erect as ordinates, in the plane E', the values of I/r i- i/r' that belong to the elements dO. A case may be easily imagined in which the expressions (i) and (2) assume coincidently the value zero. Should, however, I/r +- I/r' have different values for different elements, it will always be possible without altering the zero-value of the expression (2), so to distribute the displacements 6n that the expression (i) shall be different from zero. Only on the condition that I/r 7 -i/r' has the same value for all the elements, is expres 390 TIlE SCIENCE OF ME CIIAAICS. sion (i) necessarily and universally equated to zero with expression (2). The sum Accordingly, from the two conditions (i) and (2) it which for equilibrium follows that 1/r +- 1/r'- coist; that is to say, the sum mullst be constant for of the reciprocal values of the principal radii of curvathe whole surface, ture, or of the radii of curvature of the principal normal sections, is, in the case of equilibrium, constant for the whole surface. By this theorem the dependence of the area of a liquid surface on its superficialform is defined. The train of reasoning here pursued was first developed by GAuss,* in a much fuller and more special form. It is not difficult, however, to present its essential points in the foregoing simple manner. Application 4. A liquid mass, left wholly to itself, assumes, as of this general condi- we have seen, the spherical form, and presents an abtion to interrupted solute minimum of superficial area. The equation liquid lnas-I SSu. 1 /-r + I/r' = cons/ is here visibly fulfilled in the form 2/R = Const, R being the radius of the sphere. If the free surface of the liquid mass be bounded by two solid circular rings, the planes of which are parallel to each other and perpendicular to the line joining their middle points, the surface of the liquid mass will assume the form of a surface of revolution. The nature of the meridian curve and the volume of the enclosed mass are determined by the radius of the rings _R, by the distance between the circular planes, and by the value of the expression 1/r -+- Ir' for the surface of revolution. When 1 1 1 1 1 r r.'', I or n' the surface of revolution becomes a cylindrical surface. For 1/r -+- 1/r' 0, where one normal section is con* Princifia Generalia Theoriae Figure Fluidorum in Stalu /Equilibril G6ttingen, 1830; Werke, Vol. V, 29, Gottingen, 1867. It E EXTLENSION OF TIE PRVICIPLE'S. 391 vex and the other concave, the meridian curve assumes the form of the catenary. Plateau visibly demonstrated these cases by pouring oil on two circular rings of wire fixed in the mixture of alcohol and water above mentioned. Now let us picture to ourselves a liquid mass Liquid masses whose bounded by surface-parts for which the expression surfacesare partly conl/r+ l1/r' has a positive value, and by other parts cave and partly confor which the same expression has a negative value, vex or, more briefly expressed, by convex and concave surfaces. It will be readily seen that any displacement of the superficial elements outwards along the normal will produce in the concave parts a diminution of the superficial area and in the convex parts an increase. Consequently, work is performcd when concave surfaces move outwards and convex surfaces inwards. Work also is performed when a superficial portion moves outwards for which 1/r 1/ir'= a, while simultaneously an equal superficial portion for which 1 'r1/,' > a moves inwards. Hence, when dijfe'retly curved surfaces bound a liquid mass, the convex parts are forced inwards and the concave outwards till the condition 1/r +- 1/r'1 const is fulfilled for the entire surface. Similarly, when a connected liquid mass has several isolated surfaceparts, bounded by rigid bodies, the value of the expression 1/r 1 /'r' must, for the state of equilibrium be the same for all free portions of the surface. For example, if the space between the two circular ExperiSmental rings in the mixture of alcohol and water above re- illustration of these ferred to, be filled with oil, it is possible, by the use conditions. of a sufficient quantity of oil, to obtain a cylindrical surface whose two bases are spherical segments. The curvatures of the lateral and basal surfaces will accord 392 THE SCIENCE OF AMECII/4NICS. ingly fulfil the condition 1/R - 1/oo =- 1/p +- 1/, or p 2R, where p is the radius of the sphere and R that of the circular rings. Plateau verified this conclusion by experiment. Liquidmas- 5. Let us now study a weightless liquid mass which ses enclosing a hoi- encloses a hollow space. The condition that 1/r 1- 1/r' low space. w shall have the same value for the interior and exterior surfaces, is here not realisable. On the contrary, as this sum has always a greater positive value for the closed exterior surface than for the closed interior surface, the liquid will perform work, and, flowing from the outer to the inner surface, cause the hollow space to disappear. If, however, the hollow space be occupied by a fluid or gaseous substance subjected to a determinate pressure, the work done in the last-mentioned process can be counteracted by the work expended to produce the compression, and thus equilibrium may be produced. The me- Let us picture to ourselves a liquid mass confined chanical properties between two similar and similarly situated surfaces of bubbles. very near each other. A bubble is such a system. Its primary condition of equilibrium is the exertion of an excess of pressure by the inclosed gaseous contents. If the sum 1/rI -+ Ir' has the Svalue -- a for the exterior surface, it will Fig. 205. have for the interior surface very nearly the value - a. A bubble, left wholly to itself, will always assume the spherical form. If we conceive such a spherical bubble, the thickness of which we neglect, the total diminution of its superficial area, on the shortening of the radius r by dr, will be 16 r 7dr. If, therefore, in the diminution of the surface by tnit of area the work A is performed, then A. 6r-nrdr will THII EXTENSION OF1 THILE PRINCIPLES. 393 be the total amount of work to be compensated for by the work of compression p.4r27 rdr expended by the pressure p on the inclosed contents. From this follows 4A/r- =p; from which A may be easily calculated if the measure of r is obtained and p is found by means of a manometer introduced in the bubble. An open spierical bubble cannot subsist. If an Open bubbles. open bubble is to become a figure of equilibrium, the sum 1/r +- 1lr' must not only be constant for each of the two bounding surfaces, but must also be equal for both. Owing to the opposite curvatures of the surfaces, then, 1/r I- 1/r' -= 0. Consequently, r = - r' for all points. Such a surface is called a minimal surface; that is, it has the smallest area consistent with its containing certain closed contours. It is also a surface of zero-sum of principal curvatures; and its elements, as we readily see, are saddle-shaped. Surfaces of this kind are obtained by constructing closed spacecurves of wire and dipping the wire into a solution of soap and water.* The soap-film assumes of its own accord the form of the curve mentioned. 6. Liquid figures of equilibrium, made up of thin Plateau's liquid figfilms, possess a peculiar property. The work of the resofequilibrium. forces of gravity affects the entire mass of a liquid; that of the molecular forces is restricted to its superficial film. Generally, the work of the forces of gravity preponderates. But in thin films the molecular forces come into very favorable conditions, and it is possible to produce the figures in question without difficulty in the open air. Plateau obtained them by dipping wire polyhedrons into solutions of soap and water. Plane liquid films are thus formed, which meet * The mathematical problem of determining such a surface, when the forms of the wires are given, is called Plateau's Problem.-Trans. 394 THE SCIENCE OF ME CHIANICS. one another at the edges of the framework. When thin plane films are so joined that they meet at a hollow edge, the law 1/r +- 1/r' + const no longer holds for the liquid surface, as this sum has the value zero for plane surfaces and for the hollow edge a very large negative value. Conformably, therefore, to the views above reached, the liquid should run out of the films, the thickness of which would constantly decrease, and escape at the edges. This is, in fact, what happens. But when the thickness of the films has decreased to a certain point, then, for p/phsical reasons, which are, as it appears, not yet perfectly known, a state of equ'libriium is effected. Yet, notwithstanding the fact that the fundamental equation 1/r+- 1/r' = const is not fulfilled in these figures, because very thin liquid films, especially films of viscous liquids, present physical conditions somewhat different from those on which our original suppositions were based, these figures present, nevertheless, in all cases a,minimizumz of superficial area. The liquid films, connected with the wire edges and with one another, always meet at the edges by threes at approximately equal angles of 120~, and by fours in corners at approximately equal angles. And it is geometrically demonstrable that these relations correspond to a minimum of superficial area. In the great diversity of phenomena here discussed but one fact is expressed, namely that the molecular forces do work,.positive work, when the superficial area is diminished. The reason 7. The figures of equilibrium which Plateau obthe forms of equlibriumm tained by dipping wire polyhedrons in solutions of are symetricaI. soap, form systems of liquid films presenting a remarkable syimmetry. The question accordingly forces itself upon us, What has equilibrium to do with sym THE EXTENSION OF THE PRINCIPLES. 395 metry and regularity? The explanation is obvious. In every symmetrical system every deformation that tends to destroy the symmetry is complemented by an equal and opposite deformation that tends to restore it. In each deformation positive or negative work is done. One condition, therefore, though not an absolutely sufficient one, that a maximum or minimum of work corresponds to the form of equilibrium, is thus supplied by symmetry. Regularity is successive symmetry. There is no reason, therefore, to be astonished that the forms of equilibrium are often symmetrical and regular. 8. The science of mathematical hydrostatics arose The figure oftheearth in connection with a special problem-that of the figure B C \\lB A A 1 2 3 Fig. 206. of the earth. Physical and astronomical data had led Newton and Huygens to the view that the earth is an oblate ellipsoid of revolution. NEWTON attempted to calculate this oblateness by conceiving the rotating earth as a fluid mass, and assuming that all fluid filaments drawn from the surface to the centre exert the same pressure on the centre. HUYGENS'S assumption was that the directions of the forces are perpendicular to the superficial elements. BOUGUER combined both assumptions. CLAIRAUT, finally (Tic'orie de la figure de la terre, Paris, 1743), pointed out that the fulfilment of both conditions does not assure the subsistence of equilibrium. 396 THE SCIENCE OF IME CHANICS. Clairaut's Clairaut's starting-point is this. If the fluid earth vitof is in equilibrium, we may, without disturbing its equilibrium, imagine any portion of it solidified. Accordingly, let all of it be solidified but a canal AB, of any form. The liquid in this canal must also be in equilibrium. But now the conditions which control equilibrium are more easily investigated. If equilibrium exists in ezvery imaginable canal of this kind, then the entire mass will be in equilibrium. Incidentally Clairaut remarks, that the Newtonian assumption is realised when the canal passes through the centre (illustrated in Fig. 206, cut 2), and the Huygenian when the canal passes along the surface (Fig. 206, cut 3). Conditions But the kernel of the problem, according to Claiof equilibrinum of raut, lies in a different view. In all imaginable canals, Clairaut's canals. z Q X p M Fig. 207. Fig. 208. even in one which returns into itself, the fluid must be in equilibrium. Hence, if cross-sections be made at any two points M and N of the canal of Fig. 207, the two fluid columns MPN and MQN must exert on the surfaces of section at M and N equal pressures. The terminal pressure of a fluid column of any such canal cannot, therefore, depend on the length and the form of the fluid column, but must depend solely on the position of its terminal points. Imagine in the fluid in question a canal MN of any form (Fig. 208) referred to a system of rectangular co THE EXTEI'NSION OF THE PRINCIPLES. 397 ordinates. Let the fluid have the constant density p Mathematical expresand let the force-components X, l, Z acting on unit of sion of these conmass of the fluid in the coordinate directions, be func- ditions, and the consetions of the co6rdinates x, y', z of this mass. Let the quent general condielement of length of the canal be called ds, and let its tion of liquid equiprojections on the axes be dx, dy, dz. The force-corn-librium. ponents acting on unit of mass in the direction of the canal are then X(ix/ds), Y(dy/ds), Z(dz/ds). Let q be the cross-section; then, the total force impelling the element of mass pqds in the direction ds, is pqds X +Y"I +Z I~ ds ds ds This force must be balanced by the increment of pressure through the element of length, and consequently must be put equal to q. dp. We obtain, accordingly, dp = p (Xdx + Ydy + Zdz). The difference of pressure (p) between the two extremities M and N is found by integrating this expression from MAto N. But as this difference is not dependent on the form of the canal but solely on the position of the extremities MA and N, it follows that p (Xdx - Ydy + Zdz), or, the density being constant, Xdx + Ydy - Zdz, must be a complete differential. For this it is necessary that dU dU dU ix' i'' dz ' where Uis a function of coordinates. Hence, according to Clairaut, the general condition of liquid equilibriunl is, that the liquid be controlled by forces which can be expressed as the partial differential coefficients of one and the same function of co'rdinates. 9. The Newtonian forces of gravity, and in fact all central forces,-forces that masses exert in the directions of their lines of junction and which are functions 398 THE SCIENCE OF AMECIANICS. Character Of the distances between these masses,-possess this of the forces property. Under the action of forces of this character requisite to produce the equilibrium of fluids is possible. If we know U, equilibrium we may replace the first equation by d +U dU y dU z dx 7 dy d z or f = p(dU and p = pU+ const. The totality of all the points for which U= const is a surface, a so-called level surface. For this surface alsop = const. As all the force-relations, and, as we now see, all the pressure-relations, are determined by the nature of the function U, the pressure-relations, accordingly, supply a diagram of the force-relations, as was before remarked in page 98. Clairaut's In the theory of Clairaut, here presented, is contheory the germ of the tained, beyond all doubt, the idea that underlies the doctrine of potential. doctrine of force-function or potential, which was afterwards developed with such splendid results by Laplace, Poisson, Green, Gauss, and others. As soon as our attention has been directed to this property of certain forces, namely, that they can be expressed as derivatives of the same function U, it is at once recognised as a highly convenient and economical course to investigate in the place of the forces themselves the function U. If the equation dp = p (XdX + Ydy + Zdz) = pdU be examined, it will be seen that Xdx+ Ydy-- Zdz is the element of the work performed by tie forces on unit of mass of the fluid in the displacement ds, whose projections are dx, ay, dz. Consequently, if we transport unit mass from a point for which U-= C1 to an TIE EXTENSION OF THE PRILCIPLES. 399 other point, indifferently chosen, for which U- C, Characteristics of the or, more generally, from the surface U- C1 to the force-function. surface U-=- C,, we perform, no matter by what path the conveyance has been effected, the same amount of work. All the points of the first surface present, with respect to those of the second, the same difference of pressure; the relation always being such, that S--2, = p(C2 - C1), where the quantities designated by the same indices belong to the same surface. o1. Let us picture to ourselves a group of such Characteristics of very closely adjacent surfaces, of which every two suc- level, or equipotencessive ones differ from each other by the same, very tial, surfaces. small, amount of work required to transfer a mass from face one to the other; in other words, imagine the surfaces U= C, U= C+ d C, U= C+ 2 dC, and so forth. A mass moving on a level surface evidently performs no work. Hence, every component force in a direction tangential to the surface is = 0; and the direction of the resultant forceis everywhere normal to the surface. If we call dn the element of the normal intercepted between two C+ 3dC consecutive surfaces,andf \+C 2adC the force requisite to con- C+ C/C vey unit mass from the c one surface to the other Fig. 209. through this element, the work done isf. dn - d C. As dCis by hypothesis everywhere constant, the force f= dC/dn is inversely proportional to the distance between the surfaces consid THE SCIENCE OF AIECIHANICS. 400 ered. If, therefore, the surfaces U are known, the directions of t/ie forces are given by the elements of a system of curves everywhere at right angles to these surfaces, and the inverse distances between the surfaces measure the ma gnitude of the forces.* These surfaces and curves also confront us in the other departments of physics. We meet them as equipotential surfaces and lines of force in electrostatics and magnetism, as isothermal surfaces and lines of flow in the theory of the conduction of heat, and as equipotential surfaces and lines of flow in the treatment of electrical and liquid currents. Illustration Ii. We will now illustrate the fundamental idea of of Clairautsdoc- Clairaut's doctrine by another, very simple example. simple Imagine two mutually perpendicular planes to cut the exampe. paper at right angles in the straight lines OX and 0 Y (Fig. 210). We assume that a force-function exists U - xy, where x and y are the distances from the two planes. The force-components parallel to OX and 0 Y are then respectively dU X --- -- Y dx and dU Y --x. dy * The same conclusion may be reached as follows. Imagine a water pipe laid from New York to Key West, with its ends turning up vertically, and of glass. Let a quantity of water be poured into it, and when equilibrium is attained, let its height be marked on the glass at both ends. These two marks will be on one level surface. Now pour in a little more water and again mark the heights at both ends. The additional water in New York balances the additional water in Key West. The gravities of the two are equal. EBut their quantities are proportional to the vertical distances between the marks. Hence, the force of gravity on a fixed quantity of water is inversely as those vertical distances, that is, inversely as the distances between con;ecutive level surfaces.--Trans. THE EXTENSION OF THE PRINCIPLES. 401 The level surfaces are cylindrical surfaces, whose generating lines are at right angles to the plane of the paper, and whose directrices, xy -= const, are equilateral hyperbolas. The lines of force are obtained by turning the first mentioned system of curves through an angle of 450 in the plane of the paper about O. If a unit of mass pass from the point r to O 0 by the route rfYO, or rqO, or by any other route, the work done is always Op X O. q If we imagine a ___ X closed canal OprqO filled with a liquid, the liquid in the canal will be in equilibrium. If transverse sections be made at any two points, each Fig. 21o. section will sustain at both its surfaces the same pressure. We will now modify the example slightly. Let the A modification of this forces be X= -y, Y= - a, where a has a constant example. value. There exists now no function Uso constituted that X= dU/dx and Y== dU/dy; for in such a case it would be necessary that dX/dy = d Y/dx, which is obviously not true. There is therefore no force-function, and consequently no level surfaces. If unit of mass be transported from r to 0 by the way of p, the work done is a X Oq. If the transportation be effected by the route r qO, the work done is a X 0 q + Op X 0 q. If the canal OprqO were filled with a liquid, the liquid could not be in equilibrium, but would be forced to 402 THE SCIENCE OF ME CHANICS. rotate constantly in the direction OprqO. Currents of this character, which revert into themselves but continue their motion indefinitely, strike us as something quite foreign to our experience. Our attention, however, is directed by this to an important property of the forces of nature, to the property, namely, that the work of such forces may be expressed as a function of co6rdinates. Whenever exceptions to this principle are observed, we are disposed to regard them as apparent, and seek to clear up the difficulties involved. Torricelli's 12. We shall now examine a few problems of liquid researches on the velo- mnotion. The founder of the theory of hydrodynamics is city of liquid efflux. TORRICELLI. Torricelli,* by observations on liquids discharged through orifices in the bottom of vessels, discovered the following law. If the time occupied in the complete discharge of a vessel be divided into n equal intervals, and the quantity discharged in the last, the 1th, interval be taken as the unit, there will be discharged in the (n -1)th, the (n- 2)th, the ( -3)th.. interval, respectively, the quantities 3, 5, 7.... and so forth. An analogy between the motion of falling bodies and the motion of liquids is thus clearly suggested. Further, the perception is an immediate one, that the most curious consequences would ensue if the liquid, by its reversed velocity of efflux, could rise higher than its original level. Torricelli remarked, in fact, that it can rise at the utmost to this height, and assumed that it would rise exactly as high if all resistances could be removed. Hence, neglecting all resistances, the velocity of efflux, v, of a liquid discharged through an orifice in the bottom of a vessel is connected with the height / of the surface of the liquid by the equation v =- 1/ 2/1; that is to say, the velocity * De Motut Gravium Projectorum, 1643. THE EXTENSION OF THE PRINCIPLES. 403 of efflux is the final velocity of a body freely falling through the height h, or liquid-head; for only with this velocity can the liquid just rise again to the surface. * Torricelli's theorem consorts excellently with the arignon's deduction rest of our knowledge of natural processes; but we of the velocity of feel, nevertheless, the need of a more exact insight. efflux. VARIGNON attempted to deduce the principle from the relation between force and the momentum generated by force. The familiar equation pt i v gives, if by a we designate the area of the basal orifice, by the pressure-head of the liquid, by s its specific gravity, by g the acceleration of a freely falling body, by v the velocity of efflux, and by r a small interval of time, this result avrs ahs. T--. v or v2 = gh. 6 Here ahs represents the pressure acting during the time r on the liquid mass avrs/g. Remembering that v is a final velocity, we get, more exactly, a T. S ah/s. r -.v, and thence the correct formula v2 - 2g/h. 13. DANIEL BERNOULLI investigated the motions of fluids by the principle of vis viva. We will now treat the preceding case from this point of view, only rendering the idea more modern. The equation which we employ is ps mv2 /2. In a vessel of transverse section q (Fig. 211), into which a liquid of the specific * The early inquirers deduct cheir propositions in the incomplete form of proportions, and therefore usually put v proportional to V gh or V. 404 THE SCIENCE OF MECHA/NICS. Daniel Ber- gravity s is poured till the head h is reached, the surface noulli's treatment sinks, say, the small distance d/i, and the liquid mass of the same problem. q. da/. s g is discharged with the velocity v. The work done is the same as though the weight q. dh. s had descended the distance h. The path of the motion in the vessel is not of consequence here. It makes no difference whether the stratum q. dh A - is discharged directly through the a b basal orifice, or passes, say, to a position a, while the liquid at a is displaced to b, that at b displaced to Fig. 211. c, and that at c discharged. The work done is in each case q. dh. s. h. Equating this work to the vis viva of the discharged liquid, we get q. dh. s v2 q. d. s. h-. s or g 2 v = /2 gh. The sole assumption of this argument is that all the work done in the vessel appears as vis viva in the liquid discharged, that is to say, that the velocities within the vessel and the work spent in overcoming friction therein may be neglected. This assumption is not very far from the truth if vessels of sufficient width are employed, and no violent rotatory motion is set up. The law of Let us neglect the gravity of the liquid in the veswhen pro- sel, and imagine it loaded by a movable piston, on duced by thepres- whose surface-unit the pressure p falls. If the piston sure of pistons. be displaced a distance dh, the liquid volume q. dh will be discharged. Denoting the density of the liquid by p and its velocity by v, we then shall have v2 I2p q.p. dh = q. dh. p 2, or= - ^' \P THE EXTENSION OF THE PRINCIPLES. 405 Wherefore, under the same pressure, different liquids are discharged with velocities inversely proportional to the square root of their density. It is generally supposed that this theorem is directly applicable to gases. Its form, indeed, is correct; but the deduction frequently employed involves an error, which we shall now expose. 14. Two vessels (Fig. 212) of equal cross-sections The application of are placed side by side and connected with each other this last re-,suIt to the by a small aperture in the base of their dividing walls. flow of For the velocity of flow through this aperture we ob-gases. tain, under the same suppositions as before, di/ s v2 q. di. s (9' - 2) q-^- - or v 1 /2(/ i-/i2). If we neglect the gravity of the liquid and imagine the pressures p, and P2 produced by pistons, we shall similarly have v -/ V 2(P --P2)/p- For example, if the pistons employed be loaded with the weights P and P/2, the weight P will sink the distance h and P/2 will rise the distance h. The work (P1/2)/h is thus left, to generate the vis viva of the effluent fluid. A gas under such circumstances would behave dif-The behaviour of a ferently. Supposing the gas to flow from the vessel gas under the ascontaining the load Pinto that contain- sumed conditions. ing the load P/2, the first weight will _ dA o fall a distance /, the second, however, since under half the pressure a gas dou- - _ h bles its volume, will rise a distance 2/i, so that the work P1i - (P/2) 2 h /z 0 would be performed. In the case of Fig. 22. *r Fig. 212. gases, accordingly, some additional work, competent to produce the flow between the vessels must be performed. This work the gas itself performs, by expanding, and by overcoming by its force of expan 406 THE SCIENCE: OF MECHANICS. The result sion a pressure. The expansive force p and the volume the same in form but w of a gas stand to each other in the familiar relation different in mnagnitude.Wp - w k, where k, so long as the temperature of the gas remains unchanged, is a constant. Supposing the volume of the gas to expand under the pressure p by an amount dw, the work done is For an expansion from zwo to w,, or for an increase of pressure from p0 to p, we get for the work k log (---- -- klog. Conceiving by this work a volume of gas wo of density p, moved with the velocity v, we obtain,/2po log (-o) SV The velocity of efflux is, accordingly, in this case also inversely proportional to the square root of the density; Its magnitude, however, is not the same as in the case of a liquid. Incom- But even this last view is very defective. Rapid pleteness of this view. changes of the volumes of gases are always accompanied with changes of temperature, and, consequently also with changes of expansive force. For this reason, questions concerning the motion of gases cannot be dealt with as questions of pure mechanics, but always involve questions of heat. [Nor can even a thermodynamical treatment always suffice: it is sometimes necessary to go back to the consideration of molecular motions.] 15. The knowledge that a compressed gas contains stored-up work, naturally suggests the inquiry, whether TIE EXTENSION OF THE PRINCIPLES. 407 this is not also true of compressed liquids. As a mat- Relative volumes of ter of fact, every liquid under pressure is compressed, compressed gases and To effect compression work is requisite, which reap- liquids. pears the moment the liquid expands. But this work, in the case of the mobile liquids, is very small. Imagine, in Fig. 213, a gas and a mobile liquid of the same volume, measured by OA, subjected to the same pressure, a pressure of one atmosphere, designated by AB. If the pressure be reduced to one-half an atmosphere, the volume of the gas will be doubled, while that of the liquid will be increased by only about 25 millionths. The expansive work of the gas is represented by the surface ABDC, that of the liquid by ABLK, where D F FH AKI C Fig. 213. AK- o'oo0 25 OA. If the pressure decrease till it become zero, the total work of the liquid is represented by the surface ABI, where AI=0 o'0oo5 OA, and the total work of the gas by the surface contained between AB, the infinite straight line AICEG....., and the infinite hyperbola branch BDFH.... Ordinarily, therefore, the work of expansion of liquids may be neglected. There are however phenomena, for example, the soniferous vibrations of liquids, in which work of this very order plays a principal part. In such cases, the changes of temperature the liquids undergo must also be considered. We thus see that it is only by a fortunate concatenation of circumstances that we are at liberty to consider a phenomenon with any close 408 THE SCIENCE OF ALE CIIANICS. approximation to the truth as a mere matter of molar mechanics. The hydro- 16. We now come to the idea which DANIEL BERdynamic principle NOULLI sought to apply in his work Hydrodynamiica, sive of Daniel Bernoulli. de Viribus et Motibus Fluidorum Commentarii (1738). When a liquid sinks, the space through which its centre of gravity actually descends (descensus actualis) is equal to the space through which the centre of gravity of the separated parts affected with the velocities acquired in the fall can ascend (ascensus yotentialis). This idea, we see at once, is identical with that employed by Huygens. Imagine a vessel filled with a liquid (Fig. 214); and let its horizontal cross(/ section at the distance x from the plane S of the basal orifice, be calledf(x). Let the liquid move and its surface descend a distance dx. The centre of gravity, then, descends the distance xf(x). cdx IM, where M ff(x) dx. If k is the space of potential ascent of the liquid in a crossseciion equal to unity, the space of potential ascent in the cross-section f(x) will be k/f(x)2, and the space of potential ascent of the centre of gravity will be kCdx Jj(x)- N M M' where r dx N f(x) For the displacement of the liquid's surface through a distance dx, we get, by the principle assumed, both N and k changing, the equation - xf(x) dx = Ndk +- kdN. THE lEXTENSION OF TLHE PRLVCIPLES. 409 This equation was employed by Bernoulli in the solu- The parallelism of tion of various problems. It will be easily seen, that strata. Bernoulli's principle can be employed with success only when the relative velocities of the single parts of the liquid are known. Bernoulli assumes,-an assumption apparent in the formulae,-that all particles once situated in a horizontal plane, continue their motion in a horizontal plane, and that the velocities in the different horizontal planes are to each other in the inverse ratio of the sections of the planes. This is the assumption of the parallelism of strata. It does not, in many cases, agree with the facts, and in others its agreement is incidental. When the vessel as compared with the orifice of efflux is very wide, no assumption concerning the motions within the vessel is necessary, as we saw in the development of Torricelli's theorem. 17. A few isolated cases of liquid motion were Thewaterpendulum treated by NEWTON and JOHN BERNOULLI. We shallof Newton. consider here one to which a familiar law is directly applicable. A cylindrical U-tube with vertical branches is filled with x_ a liquid (Fig. 215). The length x of the entire liquid column is /. If in one of the branches the column be forced a distance x below the level, the column in Fig. 215. the other branch will rise the distance x, and the difference of level corresponding to the excursion x will be 2 x. If a is the transverse section of the tube and s the liquid's specific gravity, the force brought into play when the excursion x is made, will be 2 asx, which, since it must move a mass als/g will determine the acceleration (2 asx)/(als/g) = (2 g/) x, or, for unit 4o TIHE SCIENCE OF MECHANICS. excursion, the acceleration 2g//. We perceive that pendulum vibrations of the duration T--- will take place. The liquid column, accordingly, vibrates the same as a simple pendulum of half the length of the column. The liquid A similar, but somewhat more general, problem was pendulum of John treated by John Bernoulli. The two branches of a Bernoulli. cylindrical tube (Fig. 216), curved in any manner, make with the horizon, at the points at which the x surfaces of the liquid x ] move, the angles a and /p. Displacing one of the surfaces the disFig. 216. tance x, the other surFig. 216. face suffers an equal displacement. A difference of level is thus produced x (sin a + sin/ ), and we obtain, by a course of reasoning similar to that of the preceding case, employing the same symbols, the formula g (sin sin ) ) The laws of the pendulum hold true exactly for the liquid pendulum of Fig. 215 (viscosity neglected), even for vibrations of great amplitude; while for the filar pendulum the law holds only approximately true for small excursions. 18. The centre of gravity of a liquid as a whole can rise only as high as it would have to fall to produce its velocities. In every case in which this principle appears to present an exception, it can be shown that the excep THE EXTENSION OF THE PRINCIPLES. 411 tion is only apparent. One example is Hero's fountain. Hero's fountain. This apparatus, as we know, consists of three vessels, which may be designated in the descending order as A, B, C. The water in the open vessel A falls through a tube into the closed vessel C; the air displaced in C exerts a pressure on the water in the closed vessel B, and this pressure forces the water in B in a jet above A whence it falls back to its original level. The water in B rises, it is true, considerably above the level of B, but in actuality it merely flows by the circuitous route of the fountain and the vessel A to the much lower level of C. Another ap- rnt hparent exception draulic ram. to the principle A in question is that of Montgolfier's hv draulic i m R I ram, in which the liquid by its own B gravitational work appears to rise considerably above its original level. The liquid flows (Fig. 217) from a cistern A Fig. 217. through a long pipe RR and a valve V, which opens inwards, into a vessel B. When the current becomes rapid enough, the valve V is forced shut, and a liquid mass iz affected with the velocity v is suddenly arrested in RR, which must 412 THE SCIENCE OF MECHANICS. be deprived of its momentum. If this be done in the time t, the liquid can exert during this time a pressure q- mi v/t, to which must be added its hydrostatical pressure p. The liquid, therefore, will be able, during this interval of time, to penetrate with a pressure/p + q through a second valve into a pila Heronis, H, and in consequence of the circumstances there existing will rise to a higher level in the ascension-tube SS than that corresponding to its simple pressure p. It is to be observed here, that a considerable portion of the liquid must first flow off into B, before a velocity requisite to close V is produced by the liquid's work in RR. A small portion only rises above the original level; the greater portion flows from A into B. If the liquid discharged from SS were collected, it could be easily proved that the centre of gravity of the quantity thus discharged and of that received in B lay, as the result of various losses, actually below the level of A. An illustra- The principle of the hydraulic ram, that of the tion, which elucidates transference of work done by a large liquid mass to a the action of the hy- smaller one, which draulic ram 0 thus acquires a great vis viva, may be illustrated in the following very simple manner. Close the narrow T/ \ T opening 0 of a funnel S and plunge it, with its Fig. 2s8. wide opening downwards, deep into a large vessel of water. If the finger closing the upper opening be quickly removed, the space inside the funnel will rapidly fill with water, and the surface of the water outside the funnel will sink. The work performed THE EXTENSION OF THE PRINCIPLES. 413 is equivalent to the descent of the contents of the funnel from the centre of gravity S of the superficial stratum to the centre of gravity S' of the contents of the funnel. If the vessel is sufficiently wide the velocities in it are all very small, and almost the entire vis viva is concentrated in the contents of the funnel. If all the parts of the contents had the same velocities, they could all rise to the original level, or the mass as a whole could rise to the height at which its centre of gravity was coincident with S. But in the narrower sections of the funnel the velocity of the parts is greater than in the wider sections, and the former therefore contain by far the greater part of the vis viva. Consequently, the liquid parts above are violently separated from the parts below and thrown out through the neck of the funnel high above the original surface. The remainder, however, are left considerably below that point, and the centre of gravity of the whole never as much as reaches the original level of S. 19. One of the most important achievements of Hydrostatic and hydroDaniel Bernoulli is his distinction of /hydrostatic and dynamic pressure. /hydrodynamic pressure. The pressure pr which liquids exert is altered by motion; and the pressure of a liquid in motion may, according to the circumstances, be A O greater or less than that of the liquid at rest with the same arrangement of parts. We will illustrate this by a simple example. The vessel A, which has the form of a body k of revolution with vertical axis, is kept Fig. 219. constantly filled with a frictionless liquid, so that its surface at mnn does not change during the discharge at kl. We will reckon the vertical distance of a particle 414 THE SCIENCE OF IMECHIANICS. Determina- from the surface m n downwards as positive and call tion of the pressures it z. Let us follow the course of a prismatic element of generally acting in li- volume, whose horizontal base-area is a and height /3, quids in motion. in its downward motion, neglecting, on the assumption of the parallelism of strata, all velocities at right angles to z. Let the density of the liquid be p, the velocity of the element v, and the pressure, which is dependent on z, p. If the particle descend the distance dz, we have by the principle of vis viva a/pd a/3 pgdz - /a~dz.... (1) that is, the increase of the vis viva of the element is equal to the work of gravity for the displacement in question, less the work of the forces of pressure of the liquid. The pressure on the upper surface of the element is ap, that on the lower surface is ac [p - (dp/dz)/3]. The element sustains, therefore, if the pressure increase downwards, an upward pressure a (dp/dz)/3; and for any displacement dz of the element, the work a (dp/dz)/3dz must be deducted. Reduced, equation (i) assumes the form p pgd gdz- dz p 2. d (- 2 dz and, integrated, gives v 2 p. - -pgz-- + const.......... (2) If we express the velocities in two different horizontal cross-sections al and a2 at the depths z1 and z, below the surface, by v7, '2, and the corresponding pressures by p1, p2, we may write equation (2) in the form S.(v - - ) pz1- z2) + (P, -_. (2) THE EXTENSION OF THE PRINCIPLES. 415 Taking for our cross-section al the surface, z, = 0, The hydrodynamic p, 0; and as the same quantity of liquid flows through pressure varies with all cross-sections in the same interval of time, a1 v1 = the circumstances of a2 v2. Whence, finally, the motion. P a2(- a2 p2 gz2 + a (, 2 -a 2) 2 The pressure p2 of the liquid in motion (the hydrodynamic pressure) consists of the pressure pgz2 of the liquid at rest (the hydrostatic pressure) and of a pressure (p/2)v, [(a2 - a2)/aj] dependent on the density, the velocity of flow, and the cross-sectional areas. In cross-sections larger than the surface of the liquid, the hydrodynamic pressure is greater than the hydrostatic, and vice versa. A clearer idea of the significance of Bernoulli's llustration of these reprinciple may be obtained by imagining the liquid in suits by the flow of lithe vessel A unacted on by gravity, and its outflow quidsunder pressures produced by a constant pressure p, on the surface. produced by pistons. Equation (3) then takes the form by pistons. p2=T + )-(v2-v2). If we follow the course of a particle thus moving, it will be found that to every increase of the velocity of flow (in the narrower cross-sections) a decrease of pressure corresponds, and to every decrease of the velocity of flow (in the wider cross-sections) an increase of pressure. This, indeed, is evident, wholly aside from mathematical considerations. In the present case every change of the velocity of a liquid element must be exclusively produced by the work of the liquid's forces ofpressure. When, therefore, an element enters into a narrower cross-section, in which a greater velocity of flow prevails, it can acquire this higher velocity only 416 TIHE SCIENCE OF AE CIL4ANICS. on the condition that a greater pressure acts on its rear surface than on its front surface, that is to say, only when it moves from points of higher to points of lower pressure, or when the pressure decreases in the direction of the motion. If we imagine the pressures in a wide section and in a succeeding narrower section to be for a moment equal, the acceleration of the elements in the narrower section will not take place; the elements will not escape fast enough; they will accumulate before the narrower section; and at the entrance to it the requisite augmentation of pressure will be immediately produced. The converse case is obvious. Treatment 20. In dealing with more complicated cases, the of a.liquid problem in problems of liquid motion, even though viscosity be which viscosity and ___ friction are - ccnsidered. - _--:: Fig. 220. neglected, present great difficulties; and when the enormous effects of viscosity are taken into account, anything like a dynamical solution of almost every problem is out of the question. So much so, that although these investigations were begun by Newton, we have, up to the present time, only been able to master a very few of the simplest problems of this class, and that but imperfectly. We shall content ourselves with a simple example. If we cause a liquid contained in a vessel of the pressure-head h to flow, not through an orifice in its base, but through a long cylindrical tube fixed in its side (Fig. 220), the velocity of efflux TIE EXTENSION' OF THE7 PILVCIPLES. 417 v will be less than that deducible from Torricelli's law, as a portion of the work is consumed by resistances due to viscosity and perhaps to friction. We find, in fact, that v 7- 1 /2 g /1, whereh h1 < /. Expressing by / h the 'cl/ci/y-head, and by/z2 the rcsisfance-head, we may put h hI+ -+/2" If to the main cylindrical tube we affix vertical lateral tubes, the liquid will rise in the latter tubes to the heights at which it equilibrates the pressures in the main tube, and will thus indicate at all points the pressures of the main tube. The noticeable fact here is, that the liquid-height at the point of influx of the tube is h/2, and that it diminishes in the direction of the point of outflow, by the law of a straight line, to zero. The elucidation of this phenomenon is the question now presented. Gravity here does not act direclly on the liquid in The condiSon t tions of the the horizontal tube, but all effects are transmitted to it performance of by the pressure of the surrounding parts. If we imag-work in Sd. f w * such cases. mine a prismatic liquid element of basal area a and length f3 to be displaced in the direction of its length a distance dz, the work done, as in the previous case, is -a -/3dz a--3fdz. For a finite displacement we have /s/ f,-dzz "13 /(P2P,--..).....(1) 12 Work is donze when the element of volume is displaced from a place of /hirher to a place of lower pressure. The amount of the work done depends on the size of the element of volume and on the differenzce of pressure at the initial and terminal points of the motion, and not on the length and the form of the path traversed. 418 THE SCIENCE OF MlECIALNICS. If the diminution ot pressure were twice as rapid in one case as in another, the difference of the pressures on the front and rear surfaces, or the force of the work, would be doubled, but the space through which the work was done would be halved. The work done would remain the same, whether done through the space ab or ac of Fig. 221. The conse- Through every cross-section q of the horizontal tube quences of these con- the liquid flows with the same velocity v. If, neglectditions. ing the differences of velocity in the same cross-section, we consider a liquid element which exactly fills the section q and has the length /3, the vis viva q/3p(v2/2) of such an element will persist unchanged throughout its entire course in the tube. This is possible only provided the 'vis viva consumed by frictioni AA is replaced by the work of the? liquid's forces of pressure. Hence, in the direction of the motion Fig. 22. of the element the pressure must diminish, and for equal distances, to which the same work of friction corresponds, by equal amounts. The total work of gravity on a liquid element q/3p issuing from the vessel, is q 3 pgh. Of this the portion q/p (v2 /2) is the vis viva of the element discharged with the velocity v into the mouth of the tube, or, as v = Vg/z, the portion q/3pgh1. The remainder of the work, therefore, q/3pg/2,, is consumed in the tube, if owing to the slowness of the motion w- neglect the losses within the vessel. If the pressure-heads respectively obtaining in the vessel, at the mouth, and at the extremity of the tube, are A, ]/2, 0, or the pressures are_ = hgp,p 2 /igp,O, then by equation (i) of page 417 the work requisite to TIHE EXTENSION OF TIHE PRIACIPLES. 419 generate the vis viva of the element discharged into the mouth of the tube is q/3P7 -~ (/-1'2) q/gp~q (iig - p( 2)h - g p/1, and the work transmitted by the pressure of the liquid to the element traversing the length of the tube, is q315 2 --q1/3 "P /2, or the exact amount consumed in the tube. Let us assume, for the sake of argument, that the Indirect demonstrapressure does not decrease from p2 at the mouth to tion of these conzero at the extremity of the tube by the law of a straight sequences. line, but that the distribution of the pressure is different, say, constant throughout the entire tube. The parts in advance then will at once suffer a loss of velocity from the friction, the parts which follow will crowd upon them, and there will thus be produced at the mouth of the tube an augmentation of pressure conditioning a constant velocity throughout its entire length. The pressure at the end of the tube can only be - 0 because the liquid at that point is not prevented from yielding to any pressure impressed upon it. If we imagine the liquid to be a mass of smooth A simile under elastic balls, the balls will be most compressed at the which these phenomena bottom of the vessel, they will enter the tube in a state may be easily conof compression, and will gradually lose that state in ceived. the course of their motion. We leave the further development of this simile to the reader. It is evident, from a previous remark, that the work stored up in the compression of the liquid itself, is very small. The motion of the liquid is due to the work of gravity in the vessel, which by means of the pressure of the compressed liquid is transmitted to the parts in the tube. 420 THE SCIE NCE OF AE CIL4ANICS. A partial An interesting modification of the case just disexemplification of cussed is obtained by causing the liquid to flow through the results discussed, a tube composed of a number of shorter cylindrical tubes of varying widths. The pressure in the direction of outflow then diminishes (Fig. 222) more rapidly in the narrower tubes, in which a greater consumption of work by friction takes place, than in the wider ones. We further note, in every passage of the liquid into a Fig. 222. wider tube, that is to a smaller velocity of flow, an increase of pressure (a positive congestion); in every passage into a narrower tube, that is to a greater velocity of flow, an abrupt diminlution of pressure (a negative congestion). The velocity of a liquid element on which no direct forces act can be diminished or increased only by its passing to points of higher or lower pressure. CHAPTER IV. THE FORMAL DEVELOPMENT OF MECHANICS. I. THE ISOPERIMETRICAL PROBLEMS. i. When the chief facts of a physical science have The formal, as distinonce been fixed by observation, a new period of its guished from the dedevelopment begins-the deductive, which we treated ductive, development in the previous chapter. In this period, the facts are of physical science. reproducible in the mind without constant recourse to observation. Facts of a more genral and complex character are mimicked in thought on the theory that they are made up of simpler and more familiar observational elements. But even after we have deduced from our expressions for the most elementary facts (the principles) expressions for more common and more complex facts (the theorems) and have discovered in all phenomena the same elements, the developmental. process of the science is not yet completed. The deductive development of the science is followed by its formnal development. Here it is sought to put in a clear compendious form, or system, the facts to be reproduced, so that each can be reached and mentally pictured with the least intellectual effort. Into our rules for the mental reconstruction of facts we strive to incorporate the greatest possible zuniformity, so that these rules shall be easy of acquisition. It is to be remarked, that the three periods distinguished are not sharply 422 THE SCIENCE OF AMECHANICS. separated from one another, but that the processes of development referred to frequently go hand in hand, although on the whole the order designated is unmistakable. Theisoperi- 2. A powerful influence was exerted on the formal metrical problems, development of mechanics by a particular class of and questions of mathematical problems, which, at the close of the Iaxi1ma and minima seventeenth and the beginning of the eighteenth centuries, engaged the deepest attention of inquirers. These problems, the so-called isoperimnetrical problems, will now form the subject of our remarks. Certain questions of the greatest and least values of quantities, questions of maxima and minima, were treated by the Greek mathematiAN cians. Pythagoras is R said to have taught that lS S the circle, of all plane B figures of a given periFig. 223. meter, has the greatest area. The idea, too, of a certain economy in the processes of nature was not foreign to the ancients. Hero deduced the law of the reflection of light from the theory that light emitted from a point A (Fig. 223) and reflected at M will travel to B by the shortest route. Making the plane of the paper the plane of reflection, SS the intersection of the reflecting surface, A the point of departure, B the point of arrival, and M the point of reflection of the ray of light, it will be seen at once that the line AMB', where B' is the reflection of B, is a straight line. The line AfIB' is shorter than the line ANB', and therefore also AMB is shorter than ANB. Pappus held similar notions concerning organic nature; he ex FORAL4L DE VEL OPAMENT. 423 plained, for example, the form of the cells of the honeycomb by the bees' efforts to economise in materials. These ideas fell, at the time of the revival of the The researches of sciences, on not unfruitful soil. They were first taken Kepler,Fermat, and up by FERMAT and ROBERVAL, who developed a method Roberval. applicable to such problems. These inquirers observed,-as Kepler had already done,-that a magnitude y which depends on another magnitude x, generally possesses in the vicinity of its greatest and least values a peculiar property. Let x (Fig. 224) denote abscissas and y ordinates. If, while x increases, y pass through a maximum value, its increase, or rise, will be changed into a decrease, or fall; and if it pass through a -y minimum value its fall will be changed into a rise. The neighboring values of the maximum X or minimum value, consequently, Fig. 224. will lie very near each other, and the tangents to the curve at the points in question will generally be parallel to the axis of abscissas. Hence, to find the maximum or minimum values of a quantity, we seek the parallel tangents of its curve. The mc/tod of tangents may be put in analytical The method of form. For example, it is required to cut off from a tangents. given line a a portion x such that the product of the two segments x and a - x shall be as great as possible. Here, the product x (a - x) must be regarded as the quantity y dependent on x. At the maximum value of y any infinitely small variation of x, say a variation $, will produce no change in y. Accordingly, the required value of x will be found, by putting x (a - x) = (X + $) (a - x - 424 THE SCIENCE OF ALE CHANICS. ax - x2 = ax + a -- x2 - x -- x$ - $2 or 0 =a-2x-. As $ may be made as small as we please, we also get O=a-2x; whence x - a/2. In this way, the concrete idea of the method of tangents may be translated into the language of algebra; the procedure also contains, as we see, the germ of the di'fferential calculus. The refrac- Fermat sought to find for the law of the refraction tion of light as amini- of light an expression analogous to that of Hero for mal effect. law of reflection. He remarked AI that light, proceeding from a point A, and refracted at a PM D point M, travels to B, not by S Q I the shortest route, but in the shortest time. If the path AMB is performed in the shortest time, then a neighboring path B ANB, infinitely near the real Fig. 225. path, will be described in the same time. If we draw from N on AA and from M on NB the perpendiculars NP and MQ, then the second route, before refraction, is less than the first route by a distance MP- NiM sin a, but is larger than it after refraction by the distance NQ -- NMsinf. On the supposition, therefore, that the velocities in the first and second media are respectively v1 and v2, the time required for the path AMB will be a minimum when NM sin a NiMsin f 0 1 2 or FORMALL LDE VELOPMA/ZENT. 425 v1 sin a 12 sin /3 where n stands for the index of refraction. Hero's law of reflection, remarks Leibnitz, is thus a special case of the law of refraction. For equal velocities (v, = '2'), the condition of a minimum of timze is identical with the condition of a minimum of space. Huygens, in his optical investigations, applied and Huygens's completion further perfected the ideas of Fermat, considering, not of Fermat's researches. only rectilinear, but also curvilinear motions of light, in media in which the velocity of the light varied continuously from place to place. For these, also, he found that Fermat's law obtained. Accordingly, in all motions of light, an endeavor, so to speak, to produce results in a minimum of time appeared to be the fundamental tendency. 3. Similar maximal or minimal properties were The problem of the brought out in the study of mechanical phenomena. brachistoAs we have already noticed, John Bernoulli knew thathrone a freely suspended chain assumes the form for which its centre of gravity lies lowest. This idea was, of course, a simple one for the investigator who first recognised the general import of the principle of virtual velocities. Stimulated by these observations, inquirers now began generally to investigate maximal and minimal characters. The movement received its most powerful impulse from a problem propounded by John Bernoulli, in June, 1696*-the problem of the brachistochrone. In a vertical plane two points are situated, A and B. It is required to assign in this plane the curve by which a falling body will travel from A to B in the shortest time. The problem was very ingeniously * Acta Eruditorum, Leipsic. 426 THE SCIENCACOF AIECI/A4VICS. solved by John Bernoulli himself; and solutions were also supplied by Leibnitz, L'H6pital, Newton, and James Bernoulli. John Ber- The most remarkable solution was JOHN BERnoulli's ingenious so- NOULLI'S own, This inquirer remarks that problems lution of the problem of of this class have already been solved, not for the mothe brachis-. tochrone. tion of falling bodies, but for the motion of light. He accordingly imagines the motion of a falling body replaced by the motion of A a ray of light. (Comp. P. 379.) The two points A and B are supposed 1- to be fixed in a medium in which the velocity of Fig. 226. light increases in the vertical downward direction by the same law as the velocity of a falling body. The medium is supposed to be constructed of horizontal layers of downwardly decreasing density, such that v 1/2g/z denotes the velocity of the light in any layer at the distance /h below A. A ray of light which travels from A to B under such conditions will describe this distance in the shortest time, and simultaneously trace out the curve of qitz'cesi descend. Calling the angles made by the element of the curve with the perpendicular, or the normal of the layers, a, a', "......, and the respective velocities V, v',......., we have sin (Y sin a' sin a"' V V V or, designating the perpendicular distances below A by x, the horizontal distances from A by y, and the arc of the curve by s, FORJL4 L D)E IVLOPMIE0 NT. 427 (/] The brachistochrone a k.. cycloid. whence follows dy2 = k2 712 tIS2 = 2 2 (1X2 + dy2) and because v - 1/2-gx also / x 1 l -- dx - x where a =. This is the differential equation of a cycloid, or curve described by a point in the circumference of a circle of radius r= a/2 = I/4gk2, rolling on a straight line. To find the cycloid that passes through A and B, The construction of it is to be noted that all cycloids, inasmuch as they are the cycloid between produced by similar con- two given structions, are similar, points. and that if generated by the rolling of circles on R AD from the point A as origin, are also szimilarly Fig. 227. situalcd with respect to the point A. Accordingly, we draw through AB a straight line, and construct any cycloid, cutting the straight line in B'. The radius of the generating circle is, say, r'. Then the radius of the generating circle of the cycloid sought is r= r'(AB/AB'). This solution of John Bernoulli's, achieved entirely without a method, the outcome of pure geometrical fancy and a skilful use of such knowledge as happened to be at his command, is one of the most remarkable and beautiful performances in the history of physical science. John Bernoulli was an aesthetic genius in this field. His brother James's character was entirely different. James was the superior of John in critical power, 428 THE SCIENCE OF AIECIHAVIC.S. Compari- but in originality and imagination was surpassed by the son of the scientific latter. James Bernoulli likewise solved this problem, characters of John and though in less felicitous form. But, on the other hand, James Bersoulli. he did not fail to develop, with great thoroughness, a gencral method applicable to such problems. Thus, in these two brothers we find the two fundamental traits of high scientific talent separated from one another,-traits, which in the very greatest natural inquirers, in Newton, for example, are combined together. We shall soon see those two tendencies, which within one bosom might have fought their battles unnoticed, clashing in open conflict, in the persons of these two brothers. Vignette to Leibz3nizii et 2ohannis Rernoullii comercium ehistolicuinz. Lausanne and Geneva, Bousquet, 1745. James Ber- 4. James Bernoulli finds that the chief object of noulli's remarks on research hitherto had been to find the values of a varithe general nature of able quantity, for which a second variable quantity, the new probl-m. which is a function of the first, assumes its greatest or its least value. The present problem, however, is to find FOR/A I1 DEVELOPMIENY T.4 429 from among an infinite number of curves one which possesses a certain maximal or minimal property. This, as he correctly remarks, is a problem of an entirely different character from the other and demands a new method. The principles that James Bernoulli employed in The principles ermthe solution of this problem (Acta Eruditorum, May, ployed in James BerI697)* are as follows: nonlli's solution. (1) If a curve has a certain property of maximum or mniimum, every portion or element of the curve has the same property. (2) Just as the infinitely adjacent values of the maxima or minima of a quantity in the ordinary problems, for infinitely small changes of the independent variables, are constant, so also is the quantity here to be made a maximum or minimum for the curve sought, for infinitely contiguous curv's, constant. (3) It is finally assumed, for the case of the brachistochrone, that the velocity is v 1=/2 g, where / denotes the height fallen through. If we picture to ourselves a very small portion ABCThe essential feaof the curve (Fig. 228), and, imagining a horizontal tures of James Berline drawn through B, cause noulli's soA lution. the portion taken to pass intotion. the infinitely contiguous portion ADC, we shall obtain, by D considerations exactly similar / D to those employed in the treatment of Fermat's law, the wellFig. 228. known relation between the sines of the angles made by the curve-elements with the perpendicular and the velocities of descent. In this deduction the following assumptions are made, * See also his works, Vol. II, p. 768. 430 THE SCIENCE OF AMECHANICS. (i), that the part, or element, ABC is brachistochronous, and (2), that ADC is described in the same time as ABC. Bernoulli's calculation is very prolix; but its essential features are obvious, and the problem is solved* by the above-stated principles. The Pro- With the solution of the problem of the brachistogramma of James Ber- chrone, James Bernoulli, in accordance with the pracnoulli, or theproposi- tice then prevailing among mathematicians, proposed tion of the general iso- the following more general "isoperimetrical problem ": perimetrical prob- " Of all isoperimetrical curves (that is, curves of equal lem. "perimeters or equal lengths) between the same two "fixed points, to find the curve such that the space "included (i) by a second curve, each of whose ordi"nates is a given function of the corresponding ordi"nate or the corresponding arc of the one sought, (2) "by the ordinates of its extreme points, and (3) by the " part of the axis of abscissae lying between those ordi"nates, shall be a maximum or minimum." For example. It is required to find the curve BFN, described on the base BN such, that of all curves of the same length on BN, Z M this particular one shall make B the area BZN a minimum, L N where PZ = (PF)", LM r (LK)'", and so on. Let the H relation between the ordi/ G ^ nates of BZN and the corFig. 229. responding ordinates of BFA7 be given by the curve BH. To obtain PZ from PF, draw FGH at right angles to BG, where BG is at right angles to BN. By hypothesis, then, PZ= GH, and * For the details of this solution and for information generally on the history of this subject, see Woodhouse's Treatise on Isoperimetrical Problems and the Calculus of Variations, Cambridge, 18Io.-Trans. VFOA'L4L DEVEL OPM1ENT. 431 ~o for the other ordinates. Further, we put BPry, PF- x, PZ- x". John Bernoulli gave, forthwith, a solution of this John Ber1'noulli's soproblem, in the form lution of this probx/ dx lem. filt12 x 2n where a is an arbitrary constant. For n-1,. x dx f=a - xa2- x2 fill a2_ - 2 that is, BFN is a semicircle on BN as diameter, and the area BZNV is equal to the area BFN. For this particular case, the solution, in fact, is correct. But the general formula is not universally valid. On the publication of John Bernoulli's solution, James Bernoulli openly engaged to do three things: first, to discover his brother's method; second, to point out its contradictions and errors; and, third, to give the true solution. The jealousy and animosity of the two brothers culminated, on this occasion, in a violent and acrimonious controversy, which lasted till James's death. After James's death, John virtually confessed his error and adopted the correct method of his brother. James Bernoulli surmised, and in all probability James Bernoulli's correctly, that John, misled by the results of his re-criticism of John Bersearches on the catenary and the curve of a sail filled noulli's sowith wind, had again attempted an indib-ect solution,lution. imagining BFN filled with a liquid of variable density and taking the lowest position of the centre of gravity as determinative of the curve required. Making the ordinate PZ-=p, the specific gravity of the liquid in the ordinate PF x must be P/x, and similarly in every other ordinate. The weight of a vertical fila 432 THIE SCIENCE OF IiECILANIVCS. ment is then p. dy/x, and its moment with respect to BV is 1 X y 1. 2 x 2 2d. Hence, for the lowest position of the centre of gravity, tfP dy, or p Jdy BZNV, is a maximum. But the fact is here overlooked, remarks James Bernoulli, that with the variation of the curve BFVN the wCci/i/t of the liquid also is varied. Consequently, in this simple form the deduction is not admissible. The funda- In the solution which he himself gives, James Bermental principle of noulli once more assumes that the small portion F,,, James Bernoulli's of the curve possesses the propgeneral solution. z erty which the whole curve posB sesses. And then taking the four successive points F.r F,, F,,,, H J of which the two extreme ones F are fixed, he so varies F, and F,, that the length of the arc F Fig. 230. F, F,, F,,, remains unchanlgjd, which is possible, of course, only by a displacement of two points. We shall not follow his involved anA unwieldy calculations. The principle of the process is clearly indicated in our remarks. Retaining the designations above employed, James Bernoulli, in substance, states that when p dx dy a-- 1/a 2 -_ 2 fpdy is a maximum, and when (a a, - dx dy is a minimum. fidy is a minimum. FORMAL BDEVELOPMENT. 433 The dissensions between the two brothers were, we may admit, greatly to be deplored. Yet the genius of the one and the profundity of the other have borne, in the stimulus which Euler and Lagrange received from their several investigations, splendid fruits. 5. Euler (Problematis IsoperimetriciSolutio Generalis, Euler' general Corn. Acad. Pctr. T. VI, for 1733, published in 1738)* classitication of the was the first to give a more general method of treating isoperimetrical probthese questions of maxima and minima, or isoperimetri- lemis. cal problems. But even his results were based on prolix geometrical considerations, and not possessed of analytical generality. Euler divides problems of this category, with a clear perception and grasp of their differences, into the following classes: (1) Required, of all curves, that for which a property A is a maximum or minimum. (2) Required, of all curves, equally possessing a property A, that for which B is a maximum or miniMUM.. (3) Required, of all curves, equally possessing two properties, A and B, that for which C is a maximum or minimum. And so on. A problem of the first class is (Fig. 231) the finding Example of the shortest curve through M and ZN. A problem of the second class is the finding of a curve through Mf and N, which, having the given length A, makes the area MPN a maximum. A problem of the third class would be: of all curves of the given length A, which pass through 1, N and contain the same area MP2N=-B, to find one which describes when rotated about AMfV the least surface of revolution. And so on. * Euler's principal contributions to this subject are contained in three memoirs, published in the Commentaries of Petersburg for the years 1733, 1736, and 1766, and in the tract Methodus inveniendi Lineas Curvas Proprietate Maximi Minimive gaudentes, Lausanne and Geneva, 1744.-Trans. 434 THE SCIENCE OF MECtHANICS. We may observe here, that the finding of an absolute maximum or minimum, without collateral conditions, is meaningless. Thus, all the curves of which in the first example the shortest is sought P possess the common property of pasSsing through the points M and N The solution of problems of the first class requires the variation of two Selements of the curve or of one point. This is also sufficient. In problems of the second class three elements or Fig. 231. two points must be varied; the reason being, that the varied portion must possess in common with the unvaried portion the property A, and, as B is to be made a maximum or minimum, also the property B, that is, must satisfy two conditions. Similarly, the solution of problems of the third class requires the variation of four elements. And so on. The cor- The solution of a problem of a higher class involves, mutability of the iso- by implication,- the solution of its converse, in all its perimetrical proper- forms. Thus, in the third class, we vary four elements ties, with puler's in- of the curve, so, that the varied portion of the curve fereoces. shall share equally with the original portion the values A and B and, as C is to be made a maximum or a minimum, also the value C. But the same conditions must be satisfied, if of all curves possessing equally B and C that for which A is a maximum or minimum is sought, or of all curves possessing A and C that for which B is a maximum or minimum is sought. Thus a circle, to take an example from the second class, contains, of all lines of the same length A, the greatest area B, and the circle, also, of all curves containing the same area B, has the shortest length A. As the FORAL4L DEVELOPIMENi7 4'. 435 condition that the property A shall be possessed in common or shall be a maximum, is expressed in the same manner, Euler saw the possibility of reducing the problems of the higher classes to problems of the first class. If, for example, it is required to find, of all curves having the common property A, that which makes B a maximum, the curve is sought for which A +- mB is a maximum, where m is an arbitrary constant. If on any change of the curve, A + nmB, for any value of m, does not change, this is generally possible only provided the change of A, considered by itself, and that of B, considered by itself, are = 0. 6. Euler was the originator of still another impor- The fundamiental tant advance. In treating the problem of finding the principle of James Berbrachistochrone in a resisting medium, which was in- nouli's method vestigated by Herrmann and him, the existing meth- shown not to be uniods proved incompetent. For the brachistochrone in versally true. a vacuum, the velocity depends solely on the vertical height fallen through. The velocity in one portion of the curve is in no wise dependent on the other portions. In this case, then, we can indeed say, that if the whole curve is brachistochronous, every element of it is also brachistochronous. But in a resisting medium the case is different. The entire length and form of the preceding path enters into the determination of the velocity in the element. The whole curve can be brachistochronous without the separate elements necessarily exhibiting this property. By considerations of this character, Euler perceived, that the principle introduced by James Bernoulli did not hold universally good, but that in cases of the kind referred to, a more detailed treatment was required. 7. The methodical arrangement and the great number of the problems solved, gradually led Euler to sub 436 THE SCIENCE OF.1ECHA4NICS. Lagrange's stantially the same methods that Lagrange afterwards place in the history of developed in a somewhat different form, and which the Calculus of Var- now go by the name of the Calculus of Variations. First, ations. John Bernoulli lighted on an accidental solution of a problem, by analogy. James Bernoulli developed, for the solution of such problems, a geometrical method. Euler generalised the problems and the geometrical method. And finally, Lagrange, entirely emancipating himself from the consideration of geometrical figures, gave an analytical method. Lagrange remarked, that the increments which functions receive in consequence of a change in their form are quite analogous to the increments they receive in consequence of a change of their independent variables. To distinguish the two species of increments, Lagrange denoted the former by 6, the latter by d. By the observation of this analogy Lagrange was enabled to write down at once the equations which solve problems of maxima and minima. Of this idea, which has proved itself a very fertile one, Lagrange never gave a verification; in fact, did not even attempt it. His achievement is in every respect a peculiar one. He saw, with great economical insight, the foundations which in his judgment were sufficiently secure and serviceable to build upon. But the acceptance of these fundamental principles themselves was vindicated only by its results. Instead of employing himself on the demonstration of these principles, he showed with what success they could be employed. (Essai d'une nouvelle me'thode pour determincer les maxima et minnima des formules intdgrales indefnCies. Misc. Taur. 1762.) The difficulty which Lagrange's contemporaries and successors experienced in clearly grasping his idea, is quite intelligible. Euler sought in vain to clear up the FORMAL DEVELOPAIEN 7T. 437 difference between a variation and a differential by The misSconcepimagining constants contained in the function, with tions of Lagrange's the change of which the form of the function changed. idea. The increments of the value of the function arising from the increments of these constants were regarded by him as the variations, while the increments of the function springing from the increments of the independent variables were the differentials. The conception of the Calculus of Variations that springs from such a view is singularly timid, narrow, and illogical, and does not compare with that of Lagrange. Even Lindel6f's modern work, so excellent in other respects, is marred by this defect. The first really competent presentation of Lagrange's idea is, in our opinion, that of JELLETT.* Jellett appears to have said what Lagrange perhaps was unable fully to say, perhaps did not deem it necessary to say. 8. Jellett's view is, in substance, this. Quantities lelett's exposition of generally are divisible into constant and variable quan- the principles of the tities; the latter being subdivided into independent Calculus of Variaticns. and dependent variables, or such as may be arbitrarily changed, and such whose change depends on the change of other, independent, variables, in some way connected with them. The latter are called functions of the former, and the nature of the relation that connects them is termed the form of the function. Now, quite analogous to this division of quantities into constant and variable, is the division of the forms of functions into dectrminate (constant) and indeterminate (variable). If the form of a function, y + Z- /2Z)I 6z 0...............(2) Discussion 4. Thus, Lagrange conforms to tradition in making of Lagrange's statics precede dynamics. He was by no means cornethod. pelled to do so. On the contrary, he might, with equal propriety, have started from the proposition that the connections, neglecting their straining, perform no work, or that all the possible work of the system is due to the impressed forces. In the latter case he would have begun with equation (2), which expresses this fact, and which, for equilibrium (or non-accelerated motion) reduces itself to (i) as a particular case. This would have made analytical mechanics, as a system, even more logical. Equation (i), which for the case of equilibrium makes the element of the work corresponding to the assumed displacement = 0, gives readily the results discussed in page 69. If d( V (V dVX -dx7 Y_ ' - dz' d X' dy I dz' FORMA4L DE VtIL OPtA/INT. 469 that is to say, if X, Y, Z are the partial differential coefficients of one and the same function of the co6rdinates of position, the whole expression under the sign of summation is the total variation, d6, of V If the latter is -= 0, Vis in general a maximum or a minimum. 5. We will now illustrate the use of equation (i) by Indication Sof the gena simple example. If all the points of application of the eral steps ~. forthe soluforces are independent of each other, no problem is tion of statical probpresented. Each point is then in equilibrium only le-is. when thle forces impressed on it, and consequently their components, are _ 0. All the displacements dx, dy, 6z.... are then wholly arbitrary, and equation (I) can subsist only provided the coefficients of all the displacements dx, dv,, d6z..... are equal to zero. But if equations obtain between the coordinates of the several points, that is to say, if the points are subject to mutual constraints, the equations so obtaining will be of the form _F(x 1, )', Z1, 1, X2 Y2' 2....) --- 0, or, more briefly, of the form F-_ 0. Then equations also obtain between the displacements, of the form F X F, dF dF '1x dx1 + -- d; 1@ - dz1 @ x, H --.. 0 -d[X I,.,1Z1 X which we shall briefly designate as DF -- 0. If the system consist of in points, we shall have 3z11 co6rdinates, and equation (i) will contain 311 magnitudes dx, d6v, dz.... If, further, between the co6rdinates m1 equations of the form 17= 0 subsist, then m equations of the form DF == 0 will be simultaneously given between the variations x, 6y, d6z.... By these equations ill variations can be expressed in terms of the remainder, and so inserted in equation (i). In (i), therefore, there are left 31 n - mi arbitrary displacements; whose coefficients are put - 0. There are thus 470 77/1T SCIENCE OF 1AEIICHANICS. obtained between the forces and the co6rdinates 3 n- m equations, to which the mi equations (F= 0) must be added. We have, accordingly, in all, 3n equations, which are sufficient to determine the 31z co6rdinates of the position of equilibrium, provided the forces are given and only theform of the system's equilibrium is sought. But if the form of the system is given and the forces are sought that maintain equilibrium, the question is indeterminate. We have then, to determine 3 n forcecomponents, only 3 n - m equaY A, tions; the m equations (F= 0) not containing the force-components. A statical As an example of this manexample. X example. X ner of treatment we shall select a lever OM- a, free to rotate Fig. 232. about the origin of co6rdinates in the plane XY, and having at its end a second, similar lever MNV= b. At M and N, the coordinates of which we shall call x, y and x,, y1, the forces X, Yand X1, Y1 are applied. Equation (i), then, has the form X6x 3 X x, + yy+ Y,6y, = 0... (3) Of the form F - two equations here exist; namely, X + i2 _ (12 - 0 (X -X)2 + (j --Y) -- ........(4) The equations DF= 0, accordingly, are x6x -yy y 0 0 (xi - x) 6x1 - (x1 - x) 6x + (y -y) 6y1 -. (5) (y1 - y) dy - 0 Here, two of the variations in (5) can be expressed in terms of the others and introduced in (3). Also for FORMAL DE)/ VELOPJMENVT. 47i purposes of elimination Lagrange employed a per- Lagrange's indetermifectly uniform and systematic procedure, which maynate coefficients. be pursued quite mechanically, without reflection. We shall use it here. It consists in multiplying each of the equations (5) by an indeterminate coefficient A,,p, and adding each in this form to (3). So doing, we obtain [XAx- - (x,-x)]dx +[X, + -(X-x)]6x, ) [ +Ay--I(YO--)]y + [Y1 + i(y,-y)]6y, -O The coefficients of the four displacements may now be put directly = 0. For two displacements are arbitrary, and the two remaining coefficients may be made equal to zero by the appropriate choice of A and /--which is tantamount to an elimination of the two remaining displacements. We have, therefore, the four equations X + P( - X)=........(6) Y + A-I - - P( ', )-- We shall first assume that the co6rdinates are given, and seek the forces that maintain equilibrium. The values of A and p are each determined by equating to zero two coefficients. We get from the second and fourth equations, P - x, and p = - 1 X1-x Y -' whence... -. - 1,,, ~ Yi )' -Y that is to say, the total component force impressed at N has the direction MN. From the first and third equations we get 472 TIE SCIENCE OF ME CIIANICS. Thei e A X p ~ -x) A - (- O'L -'Y) Their em- - X I(-I 1 - = Y /('1 -ployment in X y the deteriniation of and from these by simple reduction the final equation. X +X, x - _ ^ *. *............. Y + y, y.......(8) that is to say, the resultant of the forces applied at M and N acts in the direction OM. * The four force-components are accordingly subject to only two conditions, (7) and (8). The problem, consequently, is an indeterminate one; as it must be from the nature of the case; for equilibrium does not depend upon the absolute magnitudes of the forces, but upon their directions and relations. If we assume that the forces are given and seek the four codrdinates, we treat equations (6) in exactly the same manner. Only, we can now make use, in addition, of equations (4). Accordingly, we have, upon the elimination of X and y, equations (7) and (8) and two equations (4). From these the following, which fully solve the problem, are readily deduced a(X~ X ) /(X+ X1)2 + (Y+ Y)2 ____(Y+,)_____ / (X+ X)2 + (y+Y)2 * The mechanical interpretation of the indeterminate coefficients 2, fL may be shown as follows. Equations (6) express the equilibrium of twofree points on which in addition to XA, Y, X1, Y other forces act which answer to the remaining expressions and just destroy X, Y, X1, Y1. The point N, for example, is in equilibrium if XN is destroyed by a force /I (xl- x), undetermined as yet in magnitude, and Y1 by a force A (ys -y). This supplementary force is due to the constraints. Its direction is determined; though its magnitude is not. If we call the angle which it makes with the axis of abscissas a, we shall have o,(VI c-ni') Y1 -Y tan a ==---.-= S(X 1--.a) XV-x that is to say, the force due to the connections acts in the direction of E. FORMALL DEVELOPJMENT. 473 S(X X) b X1__ Character /1 x+ (X i)2 (Y -j- Y 1) 2 X2+1 ' 2 ent problem. a(Y -Y) b Y1 Sx+ X1)2 + Y+ Yj)2 x1 + Y2 1(X X)2 j ) 1X+ IY? Simple as this example is, it is yet sufficient to give us a distinct idea of the character and significance of Lagrange's method. The mechanism of this method is excogitated once for all, and in its application to particular cases scarcely any additional thinking is required. The simplicity of the example here selected being such that it can be solved by a mere glance at the figure, we have, in our study of the method, the advantage of a ready verification at every step. 6. We will now illustrate the application of equa- General steps for tion (2), which is Lagrange's form of statement Of thesolution of dynamD'Alembert's principle. There is no problem when ical probleIns. the masses move quite independently of one another. Each mass yields to the forces applied to it; the variations dx, dy, 6z.... are wholly arbitrary, and each coefficient may be singly put = 0. For the motion of n masses we thus obtain 3 n simultaneous differential equations. But if equations of condition (F= 0) obtain between the co6rdinates, these equations will lead to others (DF--0) between the dis- O placements or variations. With the g. 233 Fig. 233. latter we proceed exactly as in the application of equation (i). Only it must be noted here that the equations F= 0 must eventually be employed in their undifferentiated as well as in their differentiated form, as will best be seen from the following example. 474 THE SCIENCE OOF MAE CIHAICA. A dynam- A heavy material point m, lying in a vertical plane ical example. XY, is free to move on a straight line, y = ax, inclined at an angle to the horizon. (Fig. 233.) Here equation (2) becomes e /2 X 2 X-m/_- 6x + Y-m Sy=0, and, since X= 0, and Y= - mg, also d2x dZ+ 2, 6dx (+ d-- -0.......... (9) d1t2 (1", "1,1/2 ) The place of F- 0 is taken by *y=- ax................... (10) and for DF- 0 we have 6Sy = adx. Equation (9), accordingly, since 6y drops out and 6x is arbitrary, passes into the form dt2 g+ a =0. By the differentiation of (io), or (F= 0), we have d2y,d2 x d7- 2 a dt2' and, consequently, d2 X d2 ' a+ a =0........ (11) Then, by the integration of (i ), we obtain _ a /2 x=--ji__ 9, -g 2-+ l+ C and -a2 /2 Y - -+aS -+ abt + ac, where b and c are constants of integration, determined by the initial position and velocity of m. This result can also be easily found by the direct method. FORMAIL DEVELOP I iEA4T.5 475 Some care is necessary in the application of equa-A modification of this tion (i) if F= 0 contains the time. The procedure in example. such cases may be illustrated by the following example. Imagine in the preceding case the straight line on which mi descends to move vertically upwards with the acceleration y. We start again from equation (9) + (-.X+ 6'Y=0. F=- 0 is here replaced by /2 y -ax+ y2................ (12) To form DF= 0, we vary (12) only with respect to x and j, for we are concerned here only with the possible displacement of the system in its position at any givenz i.s/ant, and not with the displacement that actually takes place in time. We put, therefore, as in the previous case, 6y = a63x, and obtain, as before, dlx (Xd2\ dt2 + < 7)a..............13 But to get an equation in x alone, we have, since x andy are connected in (13) by the actual motion, to differentiate (12) with respect to / and employ the resulting equation d2y _d2x for substitution in (i3). In this way the equation d2x(, d 2 Xv d,2~K wih igr+y+atd12, aive is obtained, which, integrated, gives 476 TIHE SCIENCIE OF AIIECILiLVNICS. a t/2 x-- ((+ Y)__~+ b + c 1 + 2a Y:=[Y- +a+a2 y)J 2 +abl +ac. If a weightless body m lie on the moving straight line, we obtain these equations a 2 -Y /2 rY /2 - +abt+ ac, -results which are readily understood, when we reflect that, on a straight line moving upwards with the acceleration y, m behaves as if it were affected with a downward acceleration y on the straight line at rest. Discussion 7. The procedure with equation (12) in the precedof the mnod-. itied exam- ing example may be rendered somewhat clearer by the pie. following consideration. Equation (2), D'Alembert's principle, asserts, that all the work SI that can be done in the displacement m of a system is done by the impressed S forces and not by the connections. This X is evident, since the rigidity of the connections allows no changes in the relaFig. 234. tive positions which would be necessary for any alteration in the potentials of the elastic forces. But this ceases to be true when the connections undergo changes in time. In this case, the changes of the connections perform work, and we can then apply equation (2) to the displacements that actually take place only provided we add to the impressed forces the forces that produce the changes of the connections. A heavy mass m is free to move on a straight line parallel to OY(Fig. 234.) Let this line be subject to FORLAL IE VEL OPMENVT 477 a forced acceleration in the direction of x, such that Illustration of the miodthe equation F_-_ 0 becomes ified example. 42 X t 2 Y................. (14) D'Alembert's principle again gives equation (9). But since from DF- 0 it follows here that 6'x - 0, this equation reduces itself to in which 6y is wholly arbitrary. Wherefore, +t+ -1 d 2 y and -r/2 9 + at + b to which must be supplied (14) or /2 It is patent that (15) does not assign the total work of the displacement that actuzally takes place, but only that of some possidle displacement on the straight line conceived, for the moment, as fixed. If we imagine the straight line massless, and cause it to travel parallel to itself in some guiding mechanism moved by a force viy, equation (2) will be replaced by l d X d2 - i 6'x2 +(- Il l ) r 2 - 1)' - 0, and since 63x, 6vy are wholly arbitrary here, we obtain the two equations d 2 vX 0 0- d2 478 THE SCIL'NCE OF MLE CHANICS. d2y which give the same results as before. The apparently different mode of treatment of these cases is simply the result of a slight inconsistency, springing from the fact that all the forces involved are, for reasons facilitating calculation, not included in the consideration at the outset, but a portion is left to be dealt with subsequently. Deduction 8. As the different mechanical principles only exof the principle of ipress different aspects of the same fact, any one of Lagrange's them is easily deducible from any other; as we shall tal dynam- now illustrate by developing the principle of vis viva tionequa- from equation (2) of page 468. Equation (2) refers to instantaneously possible displacements, that is, to "virtual " displacements. But when the connections of a system are independent of the time, the motions that actually take place are "virtual " displacements. Consequently the principle may be applied to actual motions. For 6x, 6y, 6z, we may, accordingly, write dx, dy, dz, the displacements which take place in time, and put 2 (Xdx + Ydy + Zdz) = (d2x dy d2z d12nm _ dx+ dy + dz / The expression to the right may, by introducing for dx, (dx/dt) dt and so forth, and by denoting the velocity by v, also be written d2 d1 dt+I dt d2t dt SdX 2 dy 2 d( /d2 I(t) d) del FORMAL DE VELOIPMENT. 479 Also in the expression to the left, (dx/d/) dt may be Forcefunction. written for dx. But this gives f2 (Xdx + Ydy + Zdz) = 21im (2 - (Z where v0 denotes the velocity at the beginning and v the velocity at the end of the motion. The integral to the left can always be found if we can reduce it to a single variable, that is to say, if we know the course of the motion in time or the paths which the movable points describe. If, however, X, Y, Z are the partial differential coefficients of the same function Uof co6rdinates, if, that is to say, dU dU (dU X _ -, Y =, - ' Z - dx dy dz as is always the case when only central forces are involved, this reduction is unnecessary. The entire expression to the left is then a complete differential. And we have 2 (U- Uo) -= 2 m (v2 - 712) which is to say, the difference of the force-functions (or work) at the beginning and the end of the motion is equal to the difference of the vzires vivre at the beginning and the end of the motion. The vi-es vivce are in such case also functions of the co6rdinates. In the case of a body movable in the plane of X and Ysuppose, for example, X = --y, Y= - x; we then have f(- ydx - xdy) = - --f (xy) x0oo - 'x y= ni (72 -,). But if X= - a, Y= - x, the integral to the left is -.(a dx + x dy). This integral can be assigned the moment we know the path the body has traversed, that 480 THE SCIENCE OF AIE CHIIANICS. is, if y is determined a function of x. If, for example, y =px2, the integral would become -(a + 2pX2) dx = (0 - x)-X 2 3 The difference of these two cases is, that in the first the work is simply a function of co6rdinates, that a force-function exists, that the element of the work is a complete differential, and the work consequently is determined by the initial and final values of the co6rdinates, while in the second case it is dependent on the entire path described. Essential 9. These simple examples, in themselves presentcharacter of analtt- ing no difficulties, will doubtless suffice to illustrate the ical menchanics. general nature of the operations of analytical mechanics. No fundamental light can be expected from this branch of mechanics. On the contrary, the discovery of matters of principle must be substantially completed before we can think of framing analytical mechanics; the sole aim of which is a perfect practical mastery of problems. Whosoever mistakes this situation, will never comprehend Lagrange's great performance, which here too is essentially of an economical character. Poinsot did not altogether escape this error. It remains to be mentioned that as the result of the labors of M6bius, Hamilton, Grassmann, and others, a new transformation of mechanics is preparing. These inquirers have developed mathematical conceptions that conform more exactly and directly to our geometrical ideas than do the conceptions of common analytical geometry; and the advantages of analytical generality and direct geometrical insight are thus united. But this transformation, of course, lies, as yet, beyond the limits of an historical exposition. (See p. 577.) FOR.JAIL DEVELOPM3IENT 48I IV. THE ECONOMY OF SCIENCE. 1. It is the object of science to replace, or save, ex- The basis of science, periences, by the reproduction and anticipation of facts ecoomly of thought. in thought. Memory is handier than experience, and often answers the same purpose. This economical office of science, which fills its whole life, is apparent at first glance; and with its full recognition all mysticism in science disappears. Science is communicated by instruction, in order that one man may profit by the experience of another and be spared the trouble of accumulating it for himself; and thus, to spare posterity, the experiences of whole generations are stored up in libraries. Language, the instrument of this communication, The economical is itself an economical contrivance. Experiences are character of lananalysed, or broken up, into simpler and more familiar gaage. experiences, and then symbolised at some sacrifice of precision. The symbols of speech are as yet restricted in their use within national boundaries, and doubtless will long remain so. But written language is gradually being metamorphosed into an ideal universal character. It is certainly no longer a mere transcript of speech. Numerals, algebraic signs, chemical symbols, musical notes, phonetic alphabets, may be regarded as parts already formed of this universal character of the future; they are, to some extent, decidedly conceptual, and of almost general international use. The analysis of colors, physical and physiological, is already far enough advanced to render an international system of color-signs perfectly practical. In Chinese writing, 482 7ItlE SCIEVCE OF MECHANICS. Possibility we have an actual example of a true ideographic lan of a universal lan- guage, pronounced diversely in different provinces, yet ue. everywhere carrying the same meaning. Were the system and its signs only of a simpler character, the use of Chinese writing might become universal. The dropping of unmeaning and needless accidents of grammar, as English mostly drops them, would be quite requisite to the adoption of such a system. But universality would not be the sole merit of such a character; since to read it would be to understand it. Our children often read what they do not understand; but that which a Chinaman cannot understand, he is precluded from reading. Econorn- 2. In the reproduction of facts in thought, we lte of la never reproduce the facts in full, but only that side of our represettios them which is important to us, moved to this directly of tl, worý'. or indirectly by a practical interest. Our reproductions are invariably abstractions. Here again is an economical tendency. Nature is composed of sensations as its elements. Primitive man, however, first picks out certain compounds of these elements-those namely that are relatively permanent and of greater importance to him. The first and oldest words are names of "things." Even here, there is an abstractive process, an abstraction from the surroundings of the things, and from the continual small changes which these compound sensations undergo, which being practically unimportant are not noticed. No inalterable thing exists. The thing is an abstraction, the name a symbol, for a compound of elements from whose changes we abstract. The reason we assign a single word to a whole compound is that we need to suggest all the constituent sensations at once. When, later, we come to remark the change FORMAL DE VELOPMENT 483 ableness, we cannot at the same time hold fast to the idea of the thing's permanence, unless we have recourse to the conception of a thing-in-itself, or other such like absurdity. Sensations are not signs of things; but, on the contrary, a thing is a thought-symbol for a compound sensation of relative fixedness. Properly speaking the world is not composed of "things " as its elements, but of colors, tones, pressures, spaces, times, in short what we ordinarily call individual sensations. The whole operation is a mere affair of economy. In the reproduction of facts, we begin with the more durable and familiar compounds, and supplement these later with the unusual by way of corrections. Thus, we speak of a perforated cylinder, of a cube with beveled edges, expressions involving contradictions, unless we accept the view here taken. All judgments are such amplifications and corrections of ideas already admitted. 3. In speaking of cause and effect we arbitrarily The ideas cause and give relief to those elements to whose connection we effect. have to attend in the reproduction of a fact in the respect in which it is important to us. There is no cause nor effect in nature; nature has but an individual existence; nature simply is. Recurrences of like cases in which A is always connected with B, that is, like results under like circumstances, that is again, the essence of the connection of cause and effect, exist but in the abstraction which we perform for the purpose of mentally reproducing the facts. Let a fact become familiar, and we no longer require this putting into relief of its connecting marks, our attention is no longer attracted to the new and surprising, and we cease to speak of cause and effect. Heat is said to be the cause of the tension of steam; but when the phenomenon becomes familiar 484 7T/ SCIENCE OF lME CI/ANICS. we think of the steam at once with the tension proper to its temperature. Acid is said to be the cause of the reddening of tincture of litmus; but later we think of the reddening as a property of the acid. Hume, Hume first propounded the question, How can a Kant, and Schopen- thing A act on another thing B? Hume, in fact, rehauer's explanations jects causality and recognises only a wonted succesof cause and effect. sion in time. Kant correctly remarked that a neccssary connection between A and B could not be disclosed by simple observation. He assumes an innate idea or category of the mind, a Verstandesbegriff under which the cases of experience are subsumed. Schopenhauer, who adopts substantially the same position, distinguishes four forms of the "principle of sufficient reason"-the logical, physical, and mathematical form, and the law of motivation. But these forms differ only as regards the matter to which they are applied, which may belong either to outward or inward experience. Cause and The natural and common-sense explanation is apeffect mere economical parently this. The ideas of cause and effect originally impleients of thought. sprang from an endeavor to reproduce facts in thought. At first, the connection of A and B, of C and D, of E and F, and so forth, is regarded as familiar. But after a greater range of experience is acquired and a connection between lM and N is observed, it often turns out that we recognise M as made up of A, C, E, and N of B, D, F, the connection of which was before a familiar fact and accordingly possesses with us a higher authority. This explains why a person of experience regards a new event with different eyes than the novice. The new experience is illuminated by the mass of old experience. As a fact, then, there really does exist in the mind an "idea " under which fresh experiences are subsumed; but that idea has itself been de FORAi'4L DE VEL OP,71MENT. 485 veloped from experience. The notion of the nececssity of the causal connection is probably created by our voluntary movements in the world and by the changes which these indirectly produce, as Hume supposed but Schopenhauer contested. Much of the authority of the ideas of cause and effect is due to the fact that they are developed instinctively and involuntarily, and that we are distinctly sensible of having personally contributed nothing to their formation. We may, indeed, say, that our sense of causality is not acquired by the individual, but has been perfected in the development of the race. Cause and effect, therefore, are things of thought, having an economical office. It cannot be said why they arise. For it is precisely by the abstraction of uniformities that we know the question "why." (See Appendix, XXVI, p. 579.) 4. In the details of science, its economical character Economical feais still more apparent. The so-called descriptive sci- tures of all laws of ences must chiefly remain content with reconstructing nature. individual facts. Where it is possible, the common features of many facts are once for all placed in relief. But in sciences that are more highly developed, rules for the reconstruction of great numbers of facts may be embodied in a single expression. Thus, instead of noting individual cases of light-refraction, we can mentally reconstruct all present and future cases, if we know that the incident ray, the refracted ray, and the perpendicular lie in the same plane and that sin a /sin /3 / n. Here, instead of the numberless cases of refraction in different combinations of matter and under all different angles of incidence, we have simply to note the rule above stated and the values of;n,-which is much easier. The economical purpose is here unmistakable. In nature there is no law of refraction, only different cases of re 486 THE SCIENCE OF IMECHANICS. fraction. The law of refraction is a concise compendious rule, devised by us for the mental reconstruction of a fact, and only for its reconstruction in part, that is, on its geometrical side. The econ- 5. The sciences most highly developed economically omy of the matheiiiat- are those whose facts are reducible to a few numerable ical sciences. elements of like nature. Such is the science of mechanics, in which we deal exclusively with spaces, times, and masses. The whole previously established economy of mathematics stands these sciences in stead. Mathematics may be defined as the economy of counting. Numbers are arrangement-signs which, for the sake of perspicuity and economy, are themselves arranged in a simple system. Numerical operations, it is found, are independent of the kind of objects operated on, and are consequently mastered once for all. When, for the first time, I have occasion to add five objects to seven others, I count the whole collection through, at once; but when I afterwards discover that I can start counting from 5, I save myself part of the trouble; and still later, remembering that 5 and 7 always count up to 12, I dispense with the numeration entirely. Arithmetic The object of all arithmetical operations is to save b a.ge direct numeration, by utilising the results of our old operations of counting. Our endeavor is, having done a sum once, to preserve the answer for future use. The first four rules of arithmetic well illustrate this view. Such, too, is the purpose of algebra, which, substituting relations for values, symbolises and definitively fixes all numerical operations that follow the same rule. For example, we learn from the equation x2 - y2 x+y FORIA4L DE VE L OPllfENT. 487 that the more complicated numerical operation at the left may always be replaced by the simpler one at the right, whatever numbers x and y stand for. We thus save ourselves the labor of performing in future cases the more complicated operation. Mathematics is the method of replacing in the most comprehensive and economical manner possible, new numerical operations by old ones done already with known results. It may happen in this procedure that the results of operations are employed which were originally performed centuries ago. Often operations involving intense mental effort The theory of determay be replaced by the action of semi-mechanical minants. routine, with great saving of time and avoidance of fatigue. For example, the theory of determinants owes its origin to the remark, that it is not necessary to solve each time anew equations of the form ax + /Y + C2 a2 x-+ b2 --+C2 0 from which result I b1 2 2 1 a1 2 -' 2-a 71 Q 1 ~2, 2 - 1. N ' Ib 2 a2 1 but that the solution may be effected by means of the coefficients, by writing down the coefficients according to a prescribed scheme and operating with them mec/hanical/'y. Thus, l a ) _ 1 1 2 a 2 (12 1 2 2 - 2 and similarly ' P, and 1 1. C,) 19d.~ ( 488 THE SCIENCE OF iMJIECH4ANICS. Calculating Even a total disburdening of the mind can be efmachins. fected in mathematical operations. This happens where operations of counting hitherto performed are symbolised by mechanical operations with signs, and our brain energy, instead of being wasted on the repetition of old operations, is spared for more important tasks. The merchant pursues a like economy, when, instead of directly handling his bales of goods, he operates with bills of lading or assignments of them. The drudgery of computation may even be relegated to a machine. Several different types of calculating machines are actually in practical use. The earliest of these (of any complexity) was the difference-engine of Babbage, who was familiar with the ideas here presented. Other ab- A numerical result is not always reached by the breviated methods of actual solution of the problem; it may also be reached attaining results, indirectly. It is easy to ascertain, for example, that a curve whose quadrature for the abscissa x has the value x"', gives an increment m x 'dx of the quadrature for the increment dx of the abscissa. But we then also know that 'i x"/t-dx x"; that is, we recognise the quantity x"' from the increment mx "'-'dx as unmistakably as we recognise a fruit by its rind. Results of this kind, accidentally found by simple inversion, or by processes more or less analogous, are very extensively employed in mathematics. That scientific work should be more useful the more it has been used, while mechanical work is expended in use, may seem strange to us. When a person who daily takes the same walk accidentally finds a shorter cut, and thereafter, remembering that it is shorter, always goes that way, he undoubtedly saves himself the difference of the work. But memory is really not work. FORMAL DE VEL OPMIENT. 489 It only places at our disposal energy within our present or future possession, which the circumstance of ignorance prevented us from availing ourselves of. This is precisely the case with the application of scientific ideas. The mathematician who pursues his studies with- Necessity of clear out clear views of this matter, must often have the views on this subuncomfortable feeling that his paper and pencil sur- ject. pass him in intelligence. Mathematics, thus pursued as an object of instruction, is scarcely of more educational value than busying oneself with the Cabala. On the contrary, it induces a tendency toward mystery, which is pretty sure to bear its fruits. 6. The science of physics also furnishes examples Examples of the econof this economy of thought, altogether similar to those omy of thought in we have just examined. A brief reference here will suf- physics. fice. The moment of inertia saves us the separate consideration of the individual particles of masses. By the force-function we dispense with the separate investigation of individual force-components. The simplicity of reasonings involving force-functions springs from the fact that a great amount of mental work had to be performed before the discovery of the properties of the force-functions was possible. Gauss's dioptrics dispenses us from the separate consideration of the single refracting surfaces of a dioptrical system and substitutes for it the principal and nodal points. But a careful consideration of the single surfaces had to precede the discovery of the principal and nodal points. Gauss's dioptrics simply saves us the necessity of often repeating this consideration. We must admit, therefore, that there is no result of science which in point of principle could not have been arrived at wholly without methods. But, as a matter 490 THEI SCIENCE OF MECHANICS. Science a of fact, within the short span of a human life and with minimal problem. man's limited powers of memory, any stock of knowledge worthy of the name is unattainable except by the greatest mental economy. Science itself, therefore, may be regarded as a minimal problem, consisting of the completest possible presentment of facts with the least possible expenditure of thougilt. 7. The function of science, as we take it, is to replace experience. Thus, on the one hand, science must remain in the province of experience, but, on the other, must hasten beyond it, constantly expecting confirmation, constantly expecting the reverse. Where neither confirmation nor refutation is possible, science is not concerned. Science acts and only acts in the domain of uncompleted experience. Exemplars of such branches of science are the theories of elasticity and of the conduction of heat, both of which ascribe to the smallest particles of matter only such properties as observation supplies in the study of the larger portions. The comparison of theory and experience may be farther and farther extended, as our means of observation increase in refinement. The princi- Experience alone, without the ideas that are assople of continuity, the ciated with it, would forever remain strange to us. norm of scientific Those ideas that hold good throughout the widest domethod. mains of research and that supplement the greatest amount of experience, are the most scienlific. The principle of continuity, the use of which everywhere pervades modern inquiry, simply prescribes a mode of conception which conduces in the highest degree to the economy of thought. 8. If a long elastic rod be fastened in a vise, the rod may be made to execute slow vibrations. These are directly observable, can be seen, touched, and FOIRAL 4L DEVELOPAMENT. 49i graphically recorded. If the rod be shortened, the Example llustrative vibrations will increase in rapidity and cannot be di- of the method of rectly seen; the rod will present to the sight a blurred science. image. This is a new phenomenon. But the sensation of touch is still like that of the previous case; we can still make the rod record its movements; and if we mentally retain the conception of vibrations, we can still anticipate the results of experiments. On further shortening the rod the sensation of touch is altered; the rod begins to sound; again a new phenomenon is presented. But the phenomena do not all change at once; only this or that phenomenon changes; consequently the accompanying notion of vibration, which is not confined to any single one, is still serviceable, still economical. Even when the sound has reached so high a pitch and the vibrations have become so small that the previous means of observation are not of avail, we still advantageously imagine the sounding rod to perform vibrations, and can predict the vibrations of the dark lines in the spectrum of the polarised light of a rod of glass. If on the rod being further shortened all the phenomena suddenly passed into new phenomena, the conception of vibration would no longer be serviceable because it would no longer afford us a means of supplementing the new experiences by the previous ones. When we mentally add to those actions of a human being which we can perceive, sensations and ideas like our own which we cannot perceive, the object of the idea we so form is economical. The idea makes experience intelligible to us; it supplements and supplants experience. This idea is not regarded as a great scientific discovery, only because its formation is so natural that every child conceives it. Now, this is 492 THE SCINL'A" OF JIL CItINICS exactly what we do when we imagine a moving body which has just disappeared behind a pillar, or a comet at the moment invisible, as continuing its motion and retaining its previously observed properties. We do this that we may not be surprised by its reappearance. We fill out the gaps in experience by the ideas that experience suggests. All scien- 9. Yet not all the prevalent scientific theories origitific theo nies not nated so naturally and artlessly. Thus, chemical, elecfounded on1 the princi- trical, and optical phenomena are explained by atoms. ple of continuity. But the mental artifice atom was not formed by the principle of continuity; on the contrary, it is a product especially devised for the purpose in view. Atoms cannot be perceived by the senses; like all substances, they are things of thought. Furthermore, the atoms are invested with properties that absolutely contradict the attributes hitherto observed in bodies. However well fitted atomic theories may be to reproduce certain groups of facts, the physical inquirer who has laid to heart Newton's rules will only admit those theories as yrovisional helps, and will strive to attain, in some more natural way, a satisfactory substitute. Atoms and The atomic theory plays a part in physics similar other mental artifices. to that of certain auxiliary concepts in mathematics; it is a mathematical model for facilitating the mental reproduction of facts. Although we represent vibrations by the harmonic formula, the phenomena of cooling by exponentials, falls by squares of times, etc., no one will fancy that vibrations in themselves have anything to do with the circular functions, or the motion of falling bodies with squares. It has simply been observed that the relations between the quantities investigated were similar to certain relations obtaining between familiar mathematical functions, and these more FORAlI L DE VEL OPMELNT. 493 faimiliar ideas are employed as an easy means of supplementing experience. Natural phenomena whose relations are not similar to those of functions with which we are familiar, are at present very difficult to reconstruct. But the progress of mathematics may facilitate the matter. As mathematical helps of this kind, spaces of more Multidimnenthan three dimensions may be used, as I have else- sioned spaces where shown. But it is not necessary to regard these, on this account, as anything more than mental artifices. * *As the outcome of the labors of Lobatchivski, Bolyai, Gauss, and Riemann, the view has gradually obtained currency in the mathematical world, that that which we call space is a farticular, actual case of a more general, conceivable case of multiple quantitative manifoldness. The space of sight and touch is a threefold manifoldness; it possesses three dimensions; and every point in it can be defined by three distinct and independent data. But it is possible to conceive of a quadruple or even multiple space-like manifoldness. And the character of the manifoldness may also be differently conceived from the manifoldness of actual space. We regard this discovery, which is chiefly due to the labors of Riemann, as a very important one. The properties of actual space are here directly exhibited as objects of erperience, and the pseudo-theories of geometry that seek to excogitate these properties by metaphysical arguments are overthrown. A thinking being is supposed to live in the surface of a sphere, with no other kind of space to institute comparisons with. His space will appear to him similarly constituted throughout. He might regard it as infinite, and could only be convinced of the contrary by experience. Starting from any two points of a great circle of the sphere and proceeding at right angles thereto on other great circles, he could hardly expect that the circles last mentioned would intersect. So, also, with respect to the space in which we live, only experience can decide whether it is finite, whether parallel lines intersect in it. or the like. The significance of this elucidation can scarcely be overrated. An enlightenment similar to that which Riemann inaugurated in science was produced in the mind of humanity at large, as regards the surface of the earth, by the discoveries of the first circumnavigators. The theoretical investigation of the mathematical possibilities above referred to, has, primarily, nothing to do with the question whether things really exist which correspond to these possibilities; and we must not hold mathematicians responsible for the popular absurdities which their investigations have given rise to. The space of sight and touch is t/ree-dimensional; that, no one ever yet doubted. If, now, it should be found that bodies vanish from this space, or new bodies get into it, the question might scientifically be discussed whether it would facilitate and promote our insight into tilings to conceive experiential space as part of a four-dimensional or multi-dimensional 494 THE SCIENCE OF ME CHANICS. Hypthieses This is the case, too, with all hypothesis formed and facts for the explanation of new phenomena. Our conceptions of electricity fit in at once with the electrical phenomena, and take almost spontaneously the familiar course, the moment we note that things take place as if attracting and repelling fluids moved on the surface of the conductors. But these mental expedients have nothing whatever to do with the phenomenon itself. (See Appendix, XXVII, p. 579.) space. Yet in such a case, this fourth dimension would, none the less, remain a pure thing of thought a mental fiction. But this is not the way matters stand. The phenomena mtentioned were not forthcoming until after the new views were published, and were then exhibited in the presence of certain persons at spiritualistic siances. The fourth dimension was a very opportune discover' for tie spiritualists and for theologians who were in a quandary about the location of hell. The use the spiritualist makes of the fourth dimension is this. It is possible to move out of a finite straight line, without passing the extremities, through the second dimension; out of a finite closed surface through the third; and, analogously, out of a finite closed space, without passing through the enclosing boundaries, through the fourth dimension. Even the tricks that prestidigitateurs, in the old days, harmlessly executed in three dimensions, are now invested with a new hialo by the fourth. But the tricks of the spiritualists, the tying or untying of knots in endless strings, the removing of bodies from closed spaces, are all performed in cases where tlere is absolutely nothing at stake. All is purposeless jugglery. We have not yet found an accoucheur who has accomplished parturition through the fourth dimension. If we should, tie question would at once become a serious one. Professor Simony's beautiful tricks in ropetying, whicli, as the performance of a prestidigitateur, are very admirable, speak against, not for, the spiritualists. Everyone is free to set up an opinion and to adduce proofs in support of it. Whether, though, a scientist shall find it worth his while to enter into serious investigations of opinions so advanced, is a question which his reason and instinct alone can decide. If these things, in tlhe eird, should turn out to be true, I shall not be ashamed of being the last to believe them. What I have seen of them was not calculated to make me less sceptical. I myself regarded multi-dimensioned spa-e as a nmathematico-physical help even prior to the appearance of Riemann's Inmemoir. IBut 1 tiust tLiat no one will employ what I have thoulght, said, and written on this subject as a basis for the fabrication of ghost stories. (Compare,Mach, Die (eschicite und die IW urzel di SatesSates von der Erhaung der Arbeit.) CHAPTER V. THE RELATIONS OF MECHANICS TO OTHER DEPARTMENTS OF KNOWLEDGE. THE RELATIONS OF MECHANICS TO PHYSICS. I. Purely mechanical phenomena do not exist. The The events ot nature production of mutual accelerations in masses is, to all 1o not exclusively appearances, a purely dynamical phenomenon. But belong to any sciwith these dynamical results are always associated ence. thermal, magnetic, electrical, and chemical phenomena, and the former are always modified in proportioca as the latter are asserted. On the other hand, thermal, magnetic, electrical, and chemical conditions also can produce motions. Purely mechanical phenomena, accordingly, are abstractions, made, either intentionally or from necessity, for facilitating our comprehension of things. The same thing is true of the other classes of physical phenomena. Every event belongs, in a strict sense, to all the departments of physics, the latter being separated only by an artificial classification, which is partly conventional, partly physiological, and partly historical. 2. The view that makes mechanics the basis of the remaining branches of physics, and explains all physical phenomena by mechanical ideas, is in our judgment a prejudice. Knowledge which is historically first, is not necessarily the foundation of all that is subsequently 496 THE SCIENCE' OF AIE CItANICS. The me- gained. As more and more facts are discovered and chanical aspects of classified, entirely new ideas of general scope can be nature not necessarily formed. We have no means of knowing, as yet, which its fundamental of the physical phenomena go deepesl, whether the aspects. mechanical phenomena are perhaps not the most superficial of all, or whether all do not go equally deep. Even in mechanics we no longer regard the oldest law, the law of the lever, as the foundation of all the other principles. Artificiality The mechanical theory of nature, is, undoubtedly, of the mechanical in an historical view, both intelligible and pardonable; conception of the and it may also, for a time, have been of much value. world.. But, upon the whole, it is an artificial conception. Faithful adherence to the method that led the greatest investigators of nature, Galileo, Newton, Sadi Carnot, Faraday, and J. R. Mayer, to their great results, restricts physics to the expression of actual facts, and forbids the construction of hypotheses behind the facts, where nothing tangible and verifiable is found. If this is done, only the simple connection of the motions of masses, of changes of temperature, of changes in the values of the potential function, of chemical changes, and so forth is to be ascertained, and nothing is to be imagined along with these elements except the physical attributes or characteristics directly or indirectly given by observation. This idea was elsewhere * developed by the author with respect to the phenomena of heat, and indicated, in the same place, with respect to electricity. All hypotheses of fluids or media are eliminated from the theory of electricity as entirely superfluous, when we reflect that electrical conditions are all given by the * Mach, Die Geschichte und die Wurzel des Satzes von der Erhallung der Arbeit. ITS RELATIONS TO OTHER SCIENCES. 497 values of the potential function V and the dielectric Science should be constants. If we assume the differences of the values based on facts, not of V to be measured (on the electrometer) by the forces, on hypothand regard Vand not the quantity of electricity Q as eses the primary notion, or measurable physical attribute, we shall have, for any simple insulator, for our quantity of electricity ( d2 V d2v V d2 V 2-4 7t,7:+ __, + Q=47rj dx2 d32 dz2 (where x, y, z denote the coordinates and dv the element of volume,) and for our potential* -1 f (d2V d2V d2V V W 7 --+ d + 87t d 2 dy2 dz 2 Here Q and Wappear as derived notions, in which no conception of fluid or medium is contained. If we work over in a similar manner the entire domain of physics, we shall restrict ourselves wholly to the quantitative conceptual expression of actual facts. All superfluous and futile notions are eliminated, and the imaginary problems to which they have given rise forestalled. (See Appendix XXVIII, p. 583.) The removal of notions whose foundations are historical, conventional, or accidental, can best be furthered by a comparison of the conceptions obtaining in the different departments, and by finding for the conceptions of every department the corresponding conceptions of others. We discover, thus, that temperatures and potential functions correspond to the velocities of mass-motions. A single velocity-value, a single temperature-value, or a single value of potential function, never changes alone. But whilst in the case of velocities and potential functions, so far as we yet * Using the terminology of Clausius. 498 7"LtE SCIENCE OF L MECHAICS. Desirabil- know, only differences come into consideration, the ity of a compara- significance of temperature is not only contained in its tive physics. difference with respect to other temperatures. Thermal capacities correspond to masses, the potential of an electric charge to quantity of heat, quantity of electricity to entropy, and so on. The pursuit of such resemblances and differences lays the foundation of a comyparative piysics, which shall ultimately render possible the concise expression of extensive groups of facts, without arbi/rary additions. We shall then possess a homogeneous physics, unmingled with artificial atomic theories. It will also be perceived, that a real cononiy of scientific thought cannot be attained by mechanical hypotheses. Even if an hypothesis were fully competent to reproduce a given department of natural phenomena, say, the phenomena of heat, we should, by accepting it, only substitute for the actual relations between the mechanical and thermal processes, the hypothesis. The real fundamental facts are replaced by an equally large number of hypotheses, which is certainly no gain. Once an hypothesis has facilitated, as best it can, our view of new facts, by the substitution of more familiar ideas, its powers are exhausted. We err when we expect more enlightenment from an hypothesis than from the facts themselves. c;r-cim- 3. The development of the mechanical view was whichtfa- favored by many circumstances. In the first place, a i-ored the connection of all natural events with mechanical prodevelopment or' the,nofaniteaT cesses is unmistakable, and it is natural, therefore, that view. we should be led to explain less known phenomena by better known mechanical events. Then again, it was first in the department of mechanics that laws of general and extensive scope were discovered. A law of ITS RELATIONS TO OTHER SCIENCE-S. 499 this kind is the principle of vis viva Y (Uc - U0) I2-m (v - v7), which states that the increase of the vis viva of a system in its passage from one position to another is equal to the increment of the force-function, or work, which is expressed as a function of the final and initial positions. If we fix our attention on the work a system can perform and call it with Helmholtz the Spannkiraft, S,* then the work actually prcformeid, U, will appear as a diminution of the Spannkraft, K, initially present; accordingly, S-- K- U, and the principle of vis viva takes the form 2 S +- I2 mv 12 -- const, that is to say, every diminution of the Sfannkraft, is The Con-o servation of compensated for by an increase of the vis 7viva. In this Energy. form the principle is also called the law of the Conservation of Energy, in that the sum of the Spannkraft (the potential energy) and the vis viva (the kinetic energy) remains constant in the system. But since, in nature, it is possible that not on/y vis viva should appear as the consequence of work performed, but also quantities of heat, or the potential of an electric charge, and so forth, scientists saw in this law the expression of a lmchanical action as the basis of all natural actions. However, nothing is contained in the expression but the fact of an invariable quantitative connection between mechanical and other kinds of phenomena. 4. It would be a mistake to suppose that a wide and extensive view of things was first introduced into physical science by mechanics. On the contrary, this * Helmholtz used this term in 1847; but it is not found in his subsequent papers; and in 1882 (Wissenschaftliche Abhandtlznen, II, 965) lie expressly discards it in favor of the English "potential energy," He even (p. 968) prefers Clausius's word Ergal to Spannkraft, which is quite out of agreement with modern terminology.-Trans. 500 TILHE SCIENCE OF AIE CIL4ANICS. Compre- insight was possessed at all times by the foremost hensiveness of inquirers and even entered into the construction of view the condition, mechanics itself, and was, accordingly, not first created not the resilt, of me- by the latter. Galileo and Huygens constantly alterchanics. nated the consideration of particular details with the consideration of universal aspects, and reached their results only by a persistent effort after a simple and consistent view. The fact that the velocities of individual bodies and systems are dependent on the spaces descended through, was perceived by Galileo and Huygens only by a very detailed investigation of the motion of descent in particular cases, combined with the consideration of the circumstance that bodies generally, of their own accord, only sink. Huygens especially speaks, on the occasion of this inquiry, of the impossibility of a mechanical perpetual motion; he possessed, therefore, the modern point of view. He felt the incompatibility of the idea of a perpetual motion with the notions of the natural mechanical processes with which he was familiar. Exemplifi- Take the fictions of Stevinus-say, that of the endcation of this in Ste- less chain on the prism. Here, too, a deep, broad vinuis's -esearches. insight is displayed. We have here a mind, disciplined by a multitude of experiences, brought to bear on an individual case. The moving endless chain is to Stevinus a motion of descent that is not a descent, a motion without a purpose, an intentional act that does not answer to the intention, an endeavor for a change which does not produce the change. If motion, generally, is the result of descent, then in the particular case descent is the result of motion. It is a sense of the mutual interdependence of v and h in the equation v = 1/2g/1 that is here displayed, though of course in not so definite a form. A contradiction exists in this ITS RELATIONS TO OTHER SCIE;NCES. 501 fiction for Stevinus's exquisite investigative sense that would escape less profound thinkers. This same breadth of view, which alternates the Also, in the researches individual with the universal, is also displayed, only in of Carnot and J. R. this instance not restricted to mechanics, in the per- Mayer. formances of Sadi Carnot. When Carnot finds that the quantity of heat Q which, for a given amount of work L, has flowed from a higher temperature / to a lower temperature i', can only depend on the temperatures and not on the material constitution of the bodies, he reasons in exact conformity with the method of Galileo. Similarly does J. R. Mayer proceed in the enunciation of the principle of the equivalence of heat and work. In this achievement the mechanical view was quite remote from Mayer's mind; nor had he need of it. They who require the crutch of the mechanical philosophy to understand the doctrine of the equivalence of heat and work, have only half comprehended the progress which it signalises. Yet, high as we may place Mayer's original achievement, it is not on that account necessary to depreciate the merits of the professional physicists Joule, Helmholtz, Clausius, and Thomson, who have done very much, perhaps all, towards the detailed establish/ment and perfec/ion of the new view. The assumption of a plagiarism of Mayer's ideas is in our opinion gratuitous. They who advance it, are under the obligation to prove it. The repeated appearance of the same idea is not new in history. We shall not take up here the discussion of purely personal questions, which thirty years from now will no longer interest students. But it is unfair, from a pretense of justice, to insult men, who if they had accomplished but a third of their actual services, would have lived highly honored and unmolested lives. (See p. 584.) 502 THEL SCIEANCE OF iE C/AANICS. The inter- 5. We shall now attempt to show that the broad dependence of the view expressed in the principle of the conservation facts of nature. of energy, is not peculiar to mechanics, but is a condition of logical and sound scientific thought generally. The business of physical science is the reconstruction of facts in thought, or the abstract quantitative expression of facts. The rules which we form for these reconstructions are the laws of nature. In the conviction that such rules are possible lies the law of causality. The law of causality simply asserts that the phenomena of nature are dependent on one another. The special emphasis put on space and time in the expression of the law of causality is unnecessary, since the relations of space and time themselves implicitly express that phenomena are dependent on one another. The laws of nature are equations between the measurable elements a/fy6.... c of phenomena. As nature is variable, the number of these equations is always less than the number of the elements. If we know all the values of a/3y6..., by which, for example, the values of Xjp... are given, we may call the group a/3y6... the cause and the group Apiv... the effect. In this sense we may say that the effect is uniquely determined by the cause. The principle of sufficient reason, in the form, for instance, in which Archimedes employed it in the development of the laws of the lever, consequently asserts nothing more than that the effect cannot by any given set of circumstances be at once determined and undetermined. If two circumstances a and A are connected, then, supposing all others are constant, a change of A will be accompanied by a change of a, and as a general rule a change of a by a change of A. The constant observance of this mulual interdependence is met with 1TS REL.ATIOOS TO OTHE/R SCIENCES. 503 in Stevinus, Galileo, Huygens, and other great inquir- sense of Zthis interers. The idea is also at the basis of the discovery of deperidence at the countelr-phenomena. Thus, a change in the volume of basis of all great disa gas due to a change of temperature is supplemented coveries. by the counter-phenomenon of a change of temperature on an alteration of volume; Seebeck's phenomenon by Peltier's effect, and so forth. Care must, of course, be exercised, in such inversions, respecting the fo-rm of the dependence. Figure 235 will render clear how a perceptible altera- ] tion of a may always be produced by an alteration of A, but a change of A not necessarily by a change of a. The relations between electromagnetic and induction phenomena, discovered by Faraday, are a good instance of this truth. If a set of circumstances a iy3..., by which a various forms of exsecond set Xiv. is determined, be made to pass, pessinot from its initial values to the terminal values a'/)'y' 6'..., then XAi,... also will pass into A''v'.. If the first set be brought back to its initial state, also the second set will be brought back to its initial state. This is the meaning of the "equivalence of cause and effect," which Mayer again and again emphasizes. If the first group suffer only periodical changes, the second group also can suffer only periodical changes, not continuous pcrnmalent ones. The fertile methods of thought of Galileo, Huygens, S. Carnot, Mayer, and their peers, are all reducible to the simple but significant perception, tIiat pure-c, periodical/ alraltions of one/ sCi of circullstancces cant onl/v co(nslitute tie source of si'milar/y eriodical a/lctraions of a sccond set of circumstanlcs, not of continuous an permalnent a/ltrat'ions. Such maxims, as "the effect is equivalent to the cause," 504 THE SCIENCE OF AIMECHANICS. "work cannot be created out of nothing," "a perpetual motion is impossible," are particular, less definite, and less evident forms of this perception, which in itself is not especially concerned with mechanics, but is a constituent of scientific thought generally. With the perception of this truth, any metaphysical mysticism that may still adhere to the principle of the conservation of energy* is dissipated. (See p. 585.) Purpose of All ideas of conservation, like the notion of subthe ideas of conserva- stance, have a solid foundation in the economy of tion. thotght. A mere unrelated change, without fixed point of support, or reference, is not comprehensible, not mentally reconstructible. We always inquire, accordingly, what idea can be retained amid all variations as permalnent, what law prevails, what equation remains fulfilled, what quantitative values remain constant? When we say the refractive index remains constant in all cases of refraction, gremains = -81iom in all cases of the motion of heavy bodies, the energy remains constant in every isolated system, all our assertions have one and the same economical function, namely that of facilitating our mental reconstruction of facts. II. THE RELATIONS OF MECHANICS TO PHYSIOLOGY. Conditions i. All science has its origin in the needs of life. ofthepne However minutely it may be subdivided by particular sciencoe vocations or by the restricted tempers and capacities of those who foster it, each branch can attain its full and best development only by a living connection with tie wholc. Through such a union alone can it approach * When we reflect that the principles of science are all abstractions that presuppose repetitions of similar cases, the absurd applications of the law of the conservation of forces to the universe as a whole fall to the ground. ITS RELATIONS TO OTHlER SCIENCES. 505 its true maturity, and be insured against lop-sided and monstrous growths. The division of labor, the restriction of individual Confusion of the inquirers to limited provinces, the investigation of means and aims of those provinces as a life-work, are the fundamental science. conditions of a fruitful development of science. Only by such specialisation and restriction of work can the economical instruments of thought requisite for the mastery of a special field be perfected. But just here lies a danger-the danger of our overestimating the instruments, with which we are so constantly employed, or even of regarding them as the objective point of science. 2. Now, such a state of affairs has, in our opinion, Physics actually been produced by the disproportionate formal madethe basis of development of physics. The majority of natural in- physiology. quirers ascribe to the intellectual implements of physics, to the concepts mass, force, atom, and so forth, whose sole office is to revive economically arranged experiences, a reality beyond and independent of thought. Not only so, but it has even been held that these forces and masses are the real objects of inquiry, and, if once they were fully explored, all the rest would follow from the equilibrium and motion of these masses. A person who knew the world only through the theatre, if brought behind the scenes and permitted to view the mechanism of the stage's action, might possibly believe that the real world also was in need of a machine-room, and that if this were once thoroughly explored, we should know all. Similarly, we, too, should beware lest the intellectua. machinery, employed in the representation of the world on tI/e stage of thought, be regarded as the basis of the real world. 3. A philosophy is involved in any correct view of 506 THLE SCIl_/NCII OF A -IL C1AVCS. The at- the relations of special knowledge to the great body of tem~pt to explaintfeel- knowledge at large,-a philosophy that must be deings by Imotionls. manded of every special investigator. The lack of it is asserted in the formulation of imaginary problems, in the very enunciation of which, whether regarded as soluble or insoluble, flagrant absurdity is involved. Such an overestimation of physics, in contrast to physiology, such a mistaken conception of the true relations of the two sciences, is displayed in the inquiry whether it is possible to ex!/I 'ain feelings by the motions of atoms? Explication Let us seek the conditions that could have impelled of this anomaly. the mind to formulate so curious a question. We find in the first place that greater confidcnce is placed in our experiences concerning relations of time and space; that we attribute to them a more objective, a more real character than to our experiences of colors, sounds, temperatures, and so forth. Yet, if we investigate the matter accurately, we must surely admit that our sensations of time and space are just as much sezslio/zns as are our sensations of colors, sounds, and odors, only that in our knowledge of the former we are surer and clearer than in that of the latter. Space and time are well-ordered systems of sets of sensations. The quantities stated in mechanical equations are simply ordinal symbols, representing those members of these sets that are to be mentally isolated and emphasised. The equations express the form of interdependence of these ordinal symbols. A body is a relatively constant sum of touch and sight sensations associated with the same space and time sensations. Mechanical principles, like that, for instance, of the mutually induced accelerations of two masses, give, either directly or indirectly, only some ITS RELA TIONS TO OTHER SCIENLACES. 507 combination of touch, sight, light, and time sensations. They possess intelligible meaning only by virtue of the sensations they involve, the contents of which may of course be very complicated. It would be equivalent, accordingly, to explaining rode of avoiding the more simple and immediate by the more conmpli- s ch errors. cated and remote, if we were to attempt to derive sensations from the motions of masses, wholly aside from the consideration that the notions of mechanics are economical implements or expedients perfected to represent mechanical and not physiological or psychlological facts. If the means and aims of research were properly distinguished, and our expositions were restricted to the presentation of actual facts, false problems of this kind could not arise. 4. All physical knowledge can only mentally repre- The principles of minesent and anticipate compounds of those elements we chanics not the foundacall sensations. It is concerned with the connection of tion but these elements. Such an element, say the heat of a body aspect of the world. A, is connected, not only with other elements, say witht such whose aggregate makes up the flame B, but also with the aggregate of certain elements of our body, say with the aggregate of the elements of a nerve N. As simple object and element Nis not essentially, but only conventionally, different from A and B. The connection of A and B is a problem of p/ysics, that of A and N a problem of phiysiology. Neither is alone existent; both exist at once. Only provisionally can we neglect either. Processes, thus, that in appearance are purely mechanical, are, in addition to their evident mechanical features, always physiological, and, consequently, also electrical, chemical, and so forth. The science of mechanics does not comprise the foundations, no, nor even a part of the world, but only an aspect of it. APPENDIX. I. (See page 3.) Recent research has contributed greatly to our knowledge of the scientific literature of antiquity, and our opinion of the achievements of the ancient world in science has been correspondingly increased. Schiaparelli has done much to place the work of the Greeks in astronomy in its right light, and Govi has disclosed many precious treasures in his edition of the Oplics of Ptolemy. The view that the Greeks were especially neglectful of experiment can no longer be maintained unqualifiedly. The most ancient experiments are doubtless those of the Pythagoreans, who employed a monochord with moveable bridge for determining the lengths of strings emitting harmonic notes. Anaxagoras's demonstration of the corporeality of the air by means of closed inflated tubes, and that of Empedocles by means of a vessel having its orifice inverted in water (Aristotle, Physics) are both primitive experiments. Ptolemy instituted systematic experiments on the refraction of light, while his observations in physiological optics are still full of interest to-day. Aristotle (Afclcorology) describes phenomena that go to explain the rainbow. The absurd stories which tend to arouse our mistrust, like that of Pythagoras and the anvil which emitted harmonic 510 THE SCIENCE OF ME CHAXICS. notes when struck by hammers of different weights, probably sprang from the fanciful brains of ignorant reporters. Pliny abounds in such vagaries. But they are not, as a matter of fact, a whit more incorrect or nonsensical than the stories of Newton's falling apple and of Watts's tea-kettle. The situation is, moreover, rendered quite intelligible when we consider the difficulties and the expense attending the production of ancient books and their consequent limited circulation. The conditions here involved are concisely discussed by J. Mueller in his paper, "Ueber das Experiment in den physikalischen Studien der Griechen," Naturwiss. Verein zu Innsbruck, XXIII., 1896 -1897. 11. (See page 8.1 Researches in mechanics were not begun by the Greeks until a late date, and in no wise keep pace with the rapid advancement of the race in the domain of mathematics, and notably in geometry. Reports of mechanical inventions, so far as they relate to the early inquirers, are extremely meager. Archytas, a distinguished citizen of Tarentum (circa 400 B. C.), famed as a geometer and for his employment with the problem of the duplication of the cube, devised mechanical instruments for the description of various curves. As an astronomer he taught that the earth was spherical and that it rotated upon its axis once a day. As a mechanician he founded the theory of pulleys. He is also said to have applied geometry to mechanics in a treatise on this latter science, but all information as to details is lacking. We are told, though, by Aulus Gellius (X. 12) that Archytas con APPENDIX. 5II structed an automaton consisting of a flying dove of wood and presumably operated by compressed air, which created a great sensation. It is, in fact, characteristic of the early history of mechanics that attention should have been first directed to its practical advantages and to the construction of automata designed to excite wonder in ignorant people. Even in the days of Ctesibius (285-247 B. C.) and Hero (first century A. D.) the situation had not materially changed. So, too, during the decadence of civilisation in the Middle Ages, the same tendency asserts itself. The artificial automata and clocks of this period, the construction of which popular fancy ascribed to the machinations of the Devil, are well known. It was hoped, by imitating life outwardly, to apprehend it from its inward side also. In intimate connexion with the resultant misconception of life stands also the singular belief in the possibility of a perpetual motion. Only gradually and slowly, and in indistinct forms, did the genuine problems of mechanics loom up before the minds of inquirers. Aristotle's tract, Mechanical Problems (German trans. by Poselger, Hannover, 1881) is characteristic in this regard. Aristotle is quite adept in detecting and in formulating problems; he perceived the principle of the parallelogram of motions, and was on the verge of discovering centrifugal force; but in the actual solution of problems he was infelicitous. The entire tract partakes more of the character of a dialectic than of a scientific treatise, and rests content with enunciating the "apories," or contradictions, involved in the problems. But the tract upon the whole very well illustrates the intellectual situation that is characteristic of the beginnings of scientific investigation. 512 TI!A' SCIENCE OF AIECI4NIAACS. " IIf a thing take place whereof the cause be not apparent, even though it be in accordance with nature, it appears wonderful.... Such are the instances in which small things overcome great things, small weights heavy weights, and incidentally all the problems that go by the name of ' mechanical.'. -. To the apories (contradictions) of this character belong those that appertain to the lever. For it appears contrary to reason that a large weight should be set in motion by a small force, particularly when that weight is in addition combined with a larger weight. A weight that cannot be moved without the aid of a lever can be moved easily with that of a lever added. The primordial cause of all this is inherent in the nature of the circle,-which is as one should naturally expect: for it is not contrary to reason that something wonderful should proceed out of something else that is wonderful. The combination of contradictory properties, however, into a single unitary product is the most wonderful of all things. Now, the circle is actually composed of just such contradictory properties. For it is generated by a thing that is in motion and by a thing that is stationary at a fixed point." In a subsequent passage of the same treatise there is a very dim presentiment of the principle of virtual velocities. Considerations of the kind here adduced give evidence of a capacity for detecting and enunciating problems, but are far from conducting the investigator to their solution. (See page 14.) It may be remarked in further substantiation of the criticisms advanced at pages 13-14, that it is very APPENDIX. 513 obvious that if the arrangement is absolutely symmetrical in every respect, equilibrium obtains on the assumption of any form of dependence whatever of the disturbing factor on L, or, generally, on the assumption P.f(L); and that consequently the particular form of dependence PL cannot possibly be inferred from the equilibrium. The fallacy of the deduction must accordingly be sought in the transformation to which the arrangement is subjected. Archimedes makes the action of two equal weights to be the same under all circumstances as that of the combined weights acting at the middle point of their line of junction. But, seeing that he both knows and assumes that distance from the fulcrum is determinative, this procedure is by the premises unpermissible, if the two weights are situated at unequal distances from the fulcrum. If a weight situated at a distance from the fulcrum is divided into two equal parts, and these parts are moved in contrary directions symmetrically to their original point of support; one of the equal weights will be carried as near to the fulcrum as the other weight is carried from it. If it is assumed that the action remains constant during such procedure, then the particular form of dependence of the moment on L is implicitly determined by what has been done, inasmuch as the result is only possible provided the form be PL, or be roportionalto L. But in such an event all further deduction is superfluous. The entire deduction contains the proposition to be demonstrated, by assumption if not explicitly. 514 THE SCIENCE OF AMECHANICS. IV. (See page 20.) Experiments are never absolutely exact, but they at least may lead the inquiring mind to conjecture that the key which will clear up the connexion of all the facts is contained in the exact metrical expression PL. On no other hypothesis are the deductions of Archimedes, Galileo, and the rest intelligible. The required transformations, extensions, and compressions of the prisms may now be carried out with perfect certainty. A knife edge may be introduced at any point under a prism suspended from its center without disturbing the equilibrium (see Fig. 236), and several such arrangements may be rigidly combined tox /S gether so as to form Fig. 236. apparently new cases of equilibrium. The conversion and disintegration of the case of equilibrium into several other cases (Galileo) is possible only by taking into account the value of PL. I cannot agree with O. H6lder who upholds the correctness of the Archimedean deductions against my criticisms in his essay Denken und Anschauung in der Grometrie, although I am greatly pleased with the extent of our agreement as to the nature of the exact sci ences and their foundations. It would seem as if Archimedes (De aequiponderantibus) regarded it as a general experience that two equal weights may under all circumstances be replaced by one equal to their APPENDIX. 515 515 combined weight at the center (Theorem 5, Corrolary 2). In such an event, his long deduction (Theorem 6) would be necessary, for the reason sought follows immediately (see pp. 14, 513). Archimedes's mode of expression is not in favor of this view. Nevertheless, a theorem of this kind cannot be regarded as apriori evident; and the views advanced On pp. 14, 513 appear to me to be still uncontrov-erted. V. (See page 29.) Stevinus's procedure may be looked at from still another point of view. If it is a fact, for our mechanical instinct, that a heavy endless chain will not rotate, then the individual simple cases of equilibrium on an inclined plane which Stevinus devised and which are readily controlled quantitatively, may be regarded as so many special experiences. For it is not essential that the experiments should have been actually carried out, if the result is beyond question of doubt. As a matter of fact, Stevinus experiments in thought. Stevinus's result could actually have been deduced from the corresponding physical experiments, with friction reduced to a minimum. In an analogous manner, Archimedes's considerations with respect to the lever might be conceived after the fashion of Galileo's procedure. If the various mental experiments had been executed physically, the linear dependence of the static moment on the distance of the weight from the axis could be deduced with perfect rigor. We shall have still many instances to adduce, among the foremost inquirers in the domain of mechanics, of this tentative adaptation of special 5P6 TRlE SCIENCE OF MIECH.ANICS. quantitative conceptions to general instinctive imnpressions. The same phenomena are presented in other domains also. I may be permitted to refer in this connexion to the expositions which I have given in my Principzles of Heal, page 151. It may be said that the most significant and most important advances in science have been made in this manner. The habit which great inquirers have of bringing their single conceptions into agreement with the general conception or ideal of an entire province of phenomena, their constant consideration of the whole in their treatment of parts, may he characterised as a genuinely philosophical procedure. A truly philosophical treatment of any special science will always consist in bringing the results into relationship and harmony with the established knowledge of the whole. The fanciful extravagances of philosophy, as well as infelicitous and abortive special theories, will be eliminated in this manner. It will be worth while to review again the points of agreement and difference in the mental procedures of Stevinus and Archimedes. Stevin'us reached the very general view that a mobile, heavy, endless chain of any form stays at rest. He is able to deduce from this general view, without difficulty, special cases, which are quantitatively easily controlled. The case from which Archimedes starts, on the other han(], is the most special conceivable. He cannot possibly deduce from his special case in an unassailable manner the behavior which may be expected under more general conditions. If he apparently succeeds in so doing, the reason is that he already knows the result which he is seeking, whilst Stevinus, although he too doubtless knows, approximately at least, what he is APPENDIX. 517 in search of, nevertheless could have found it directly by his manner of procedure, even if he had not known it. When the static relationship is rediscovered in such a manner it has a higher value than the result of a metrical experiment would have, which always deviates somewhat from the theoretical truth. The deviation increases with the disturbing circumstances, as with friction, and decreases with the diminution of these difficulties. The exact static relationship is reached by idealisation and disregard of these disturbing elements. It appears in the Archimedean and Stevinian procedures as an hypotlesis without which the individual facts of experience would at once become involved in logical contradictions. Not until we have possessed this hypothesis can we by operating with the exact concepts reconstruct the facts and acquire a scientific and logical mastery of them. The lever and the inclined plane are self-created ideal objects of mechanics. These objects alone completely satisfy the logical demands which we make of them; the physical lever satisfies these conditions only in measure in which it approaches the ideal lever. The natural inquirer strives to adapt his ideals to reality. VI. (See page no ) Our modern notions with regard to the nature of air are a direct continuation of the ancient ideas. Anaxagoras proves the corporeality of air from its resistance to compression in closed bags of skin, and from the gathering up of the expelled air (in the form of bubbles?) by water (Aristotle, Physics, IV., 9). According to Empedocles, the air prevents the water 518 THE SCIENCE OF MECHANICS. from penetrating into the interior of a vessel immersed with its aperture downwards (Gomperz, Griechische Denker, I., p. 191). Philo of Byzantium employs for the same purpose an inverted vessel having in its bottom an orifice closed with wax. The water will not penetrate into the submerged vessel until the wax cork is removed, wherupon the air escapes in bubbles. An entire series of experiments of this kind is performed, in almost the precise form customary in the schools to-day (Philonis lib. de ingeniis spiritualibus, in V. Rose's Anecdota grceca et latina). Hero describes in his Pneumatics many of the experiments of his predecessors, with additions of his own; in theory he is an adherent of Strato, who occupied an intermediate position between Aristotle and Democritus. An absolute and continuous vacuum, he says, can be produced only artificially, although numberless tiny vacua exist between the particles of bodies, including air, just as air does among grains of sand. This is proved, in quite the same ingenuous fashion as in our present elementary books, from the possibility of rarefying and compressing bodies, including air (inrushing and outrushing of the air in Hero's ball). An argument of Hero's for the existence of vacua (pores) between corporeal particles rests on the fact that rays of light penetrate water. The result of artificially increasing a vacuum, according to Hero and his predecessors, is always the attraction and solicitation of adjacent bodies. A light vessel with a narrow aperture remains hanging to the lips after the air has been exhausted. The orifice may be closed with the finger and the vessel submerged in water. "If the finger be released, the water will rise in the vacuum created, although the movement of the liquid upward is not APPENDIX. 519 according to nature. The phenomenon of the cupping-glass is the same; these glasses, when placed on the body, not only do not fall off, although they are heavy enough, but they also draw out adjacent particles through the pores of the body." The bent siphon is also treated at length. "The filling of the siphon on exhaustion of the air is accomplished by the liquid's closely following the exhausted air, for the reason that a continuous vacuum is inconceivable." If the two arms of the siphon are of the same length, nothing flows out. "The water is held in equilibrium as in a balance." Hero accordingly ccnceives of the flow of water as analogous to the movement of a chain hanging with unequal lengths over a pulley. The union of the two columns, which for us is preserved by tile pressure of the atmosphere, is cared for in his case by the "inconceivability of a continuous vacuum." It is shown at length, not that the smaller mass of water is attracted and drawn along by the greater mass, and that conformably to this principle water cannot flow upwards, but rather that the phenomenon is in harmony with the principle of communicating vessels. The many pretty and ingenious tricks which Hero describes in his Pneumatics and in his Automata, and which were designed partly to entertain and partly to excite wonder, offer a charming picture of the material civilisation of the day rather than excite our scientific interest. The automatic sounding of trumpets and the opening of temple doors, with the thunder simultaneously produced, are not matters which interest science properly so called. Yet Hero's writings and notions contributed much toward the diffusion of physical knowledge (compare W. Schmidt, HIro's fWerke, Leipsic, 1899, 520 TIHE SCIENCE OF MECIL4AWCS. and Diels, System des Siralo, Si/zungsberic/t/e der Berliner Akademie, 1893). VII. (See page 129.) It has often been asserted that Galileo had predecessors of great prominence in his method of thinking, and while it is far from our purpose to gainsay this, we have still to emphasise the fact that Galileo overtowered them all. The greatest predecessor of Galileo, to whom we have already referred in another place, was Leonardo da Vinci, 1452-1519; now, it was impossible for Leonardo's achievements to have influenced the development of science at the time, for the reason that they were not made known in their entirety until the publication of Venturi in 1797. Leonardo knew the ratio of the times of descent down the slope and the height of an inclined plane. Frequently also a knowledge of the law of inertia is attributed to him. Indeed, some sort of instinctive knowledge of the persistence of motion once begun will not be gainsaid to any normal man. But Leonardo seems to have gone much farther than this. He knows that from a column of checkers one of the pieces may be knocked out without disturbing the others; he knows that a body in motion will move longer according as the resistance is less, but he believes that the body will move a distance proportional to the impulse, and nowhere expressly speaks of the persistence of the motion when the resistance is altogether removed. (Compare Wohlwill, Pibliotheca Afathematica, Stockholm, 1888, p. 19). Benedetti (1530 -1590) knows that falling bodies are accelerated, and explains the acceleration as due to the summation APPEI'I'NDIX. 521 of the impulses of gravity (Divers. speculat. math. et physic. liber, Taurini, 1585). He ascribes the progressive motion of a projectile, not as the Peripatetics did, to the agency of the medium, but to the virtus impressa, though without attaining perfect clearness with regard to these problems. Galileo seems actually to have proceeded from Benedetti's point of view, for his youthful productions are allied to those of Benedetti. Galileo also assumes a virtus impressa, which he conceives to decrease in efficiency, and according to Wohlwill it appears that it was not until 1604 that he came into full possession of the laws of falling bodies. G. Vailati, who has devoted much attention to Benedetti's investigations (Atti della R. Acad. di Torino, Vol. XXXIII., 1898), finds the chief merit of Benedetti to be that he subjected the Aristotelian views to mathematical and critical scrutiny and correction, and endeavored to lay bare their inherent contradictions, thus preparing the way for further progress. He knows that the assumption of the Aristotelians, that the velocity of falling bodies is inversely proportional to the density of the surrounding medium, is untenable and possible only in special cases. Let the velocity of descent be proportional to p-q, wherep is the weight of the body and q the upward impulsion due to the medium. If only half the velocity of descent is set up in a medium of double the density, the equation p- q=2 (p- 2q) must exist,-a relation which is possible only in case /==3q. Light bodies per se do not exist for Benedetti; he ascribes weight and upward impulsion even to air. Different-sized bodies of the same material fall, in his opinion, with the same velocity. Benedetti reaches this result by 522 TLHE SCIENCE OF MECHAN4ICS conceiving equal bodies falling alongside each other first disconnected and then connected, where the connexion cannot alter the motion. In this he approaches to the conception of Galileo, with the exception that the latter takes a profounder view of the matter. Nevertheless, Benedetti also falls into many errors; he believes, for example, that the velocity of descent of bodies of the same size and of the same shape is proportional to their weight, that is, to their density. His reflexions on catapults, no less than his views on the oscillation of a body about the center of the earth in a canal bored through the earth, are interesting, and contain little to be criticised. Bodies projected horizontally appear to approach the earth more slowly. Benedetti is accordingly of the opinion that the force of gravity is diminished also in the case of a top rotating with its axis in a vertical position. He thus does not solve the riddle fully, but prepares the way for the solution. VIII. (See page 134.) If we are to understand Galileo's train of thought, we must bear in mind that he was already in possession of instinctive experiences prior to his resorting to experiment. Freely falling bodies are followed with more difficulty by the eye the longer and the farther they have fallen; their impact on the hand receiving them is in like measure sharper; the sound of their striking louder. The velocity accordingly increases with the time elapsed and the space traversed. But for scientific purposes our mental representations of the facts of sensual experience must be submitted to conceptual APP'L AENDIX. 523 formulation. Only thus may they be used for discovering by abstract mathematical rules unknown properties conceived to be dependent on certain initial properties having definite and assignable arithmetic values; or, for completing what has been only partly given. This formulation is effected by isolating and emphasising what is deemed of importance, by neglecting what is subsidiary, by abstracting, by idealising. The experiment determines whether the form chosen is adequate to the facts. Without some preconceived opinion the experiment is impossible, because its form is determined by the opinion. For how and on what could we experiment if we did not previously have some suspicion of what we were about? The complemental function which the experiment is to fulfil is determined entirely by our prior experience. The experiment confirms, modifies, or overthrows our suspicion. The modern inquirer would ask in a similar predicament: Of what is v a function? What function of / is v? Galileo asks, in his ingenuous and primitive way: is v proportional to s, is v proportional to /? Galileo, thus, gropes his way along synthetically, but reaches his goal nevertheless. Systematic, routine methods are the final outcome of research, and do not stand perfectly developed at the disposal of genius in the first steps it takes. (Compare the article " Ueber Gedankenexperimente," Zeitschrift fiir denphys. und chem. Unterricht, 1897, I.) IX. (See page 140.) In an exhaustive study in the Zeitschrift fzr VY/kerpsychologie, 1884, Vol. XIV., pp. 365-410, and Vol. 524 THE SCIENCEO OF lMECHAI. AICS. XV., pp. 70-135, 337-387, entitled "Die Entdeckung des Beharrungsgesetzes," E. Wohlwill has shown that the predecessors and contemporaries of Galileo, nay, even Galileo himself, only very gradually abandoned the Aristotelian conceptions for the acceptance of the law of inertia. Even in Galileo's mind uniform circular motion and uniform horizontal motion occupy distinct places. Wohlwill's researches are very acceptable and show that Galileo had not attained perfect clearness in his own new ideas and was liable to frequent reversion to the old views, as might have been expected. Indeed, from my own exposition the reader will have inferred that the law of inertia did not possess in Galileo's mind the degree of clearness and universality that it subsequently acquired. (See pp. 140 and 143.) With regard to my exposition at pages 140 -141, however, I still believe, in spite of the opinions of Wohlwill and Poske, that I have indicated the point which both for Galileo and his successors must have placed in the most favorable light the transition from the old conception to the new. How much was wanting to absolute comprehension, may be gathered from the fact that Baliani was able without difficulty to infer from Galileo's statement that acquired velocity could not be destroyed,-a fact which Wohlwill himself points out (p. 112). It is not at all surprising that in treating of the motion of heavy bodies, Galileo applies his law of inertia almost exclusively to horizontal movements. Yet he knows that a musket-ball possessing no weight would continue rectilinearly on its path in the direction of the barrel. (Dialogues on the two World Systems, German translation, Leipsic, I891, p. 184.) His hesitation in enunciating APPEND IX. 525 in its most general terms a law that at first blush appears so startling, is not surprising. x. (See page 155.) We cannot adequately appreciate the extent of Galileo's achievement in the analysis of the motion of projectiles until we examine his predecessors' endeavors in this field. Santbach (1561) is of opinion, that a cannon-ball speeds onward in a straight line until its velocity is exhausted and then drops to the ground in a vertical direction. Tartaglia (1537) compounds the path of a projectile out of a straight line, the arc of a circle, and lastly the vertical tangent to the arc. He is perfectly aware, as Rivius later (1582) more distinctly states, that accurately viewed the path is curved at all points, since the deflective action of gravity never ceases; but he is yet unable to arrive at a complete analysis. The initial portion of the path is well calculated to arouse the illusive impression that the action of gravity has been annulled by the velocity of the projection,-an illusion to which even Benedetti fell a victim. (See Appendix, vii., p. 129.) We fail to observe any descent in the initial part of the curve, and forget to take into account the shortness of the corresponding time of the descent. By a similar oversight a jet of water may assume the appearance of a solid body suspended in the air, if one is unmindful of the fact that it is made up of a mass of rapidly alternating minute particles. The same illusion is met with in the centrifugal pendulum, in the top, in Aitken's flexible chain rendered rigid by rapid rotation (Philosop/hical Magazine, 1878), in the locomotive which rushes safely across a defective 526 THE SCIENCE OF /E CH/ANICS. bridge, through which it would have crashed if at rest, but which, owing to the insufficient time of descent and of the period in which it can do work, leaves the bridge intact. On thorough analysis none of these phenomena are more surprising than the most ordinary events. As Vailati remarks, the rapid spread of firearms in the fourteenth century gave a distinct impulse to the study of the motion of projectiles, and indirectly to that of mechanics generally. Esentially the same conditions occur in the case of the ancient catapults and in the hurling of missiles by the hand, but the new and imposing form of the phenomenon doubtless exercised a great fascination on the curiosity of people. So much for history. And now a word as to the notion of "composition." Galileo's conception of the motion of a projectile as a process compounded of two distinct and independent motions, is suggestive of an entire group of similar important episitemological processes. We may say that it is as important to perceive the non dependence of two circumstances A and B on each other, as it is to perceive the dependence of two circumstances A and C on each other. For the first perception alone enables us to pursue the second relation with composure. Think only of how serious an obstacle the assumption of non-existing causal relations constituted to the research of the Middle Ages. Similar to Galileo's discovery is that of the parallelogram of forces by Newton, the composition of the vibrations of strings by Sauveur, the com position of thermal disturbances by Fourier. Through this latter inquirer the method of compounding a phenomenon out of mutually independent partial phenomena by means of representing a general integral APPENDIX. 527 as the sum of particular integrals has penetrated into every nook and corner of physics. The decomposition of phenomena into mutually independent parts has been aptly characterised by P. Volkmann as isolation, and the composition of a phenomenon out of such parts, superposition. The two processes combined enable us to comprehend, or reconstruct in thought, piecemeal, what, as a whole, it would be impossible for us to grasp. "Nature with its myriad phenomena assumes a unified aspect only in the rarest cases; in the majority of instances it exhibits a thoroughly composite character...; it is accordingly one of the duties of science to conceive phenomena as made up of sets of partial phenomena, and at first to study these partial phenomena in their purity. Not until we know to what extent each circumstance shares in the phenomenon as an entirety do we acquire a command over the whole...." (Cf. P. Volkmann, Erkenntnisstheoretische Grundziige der Naturwissenschafl, 1896, p. 7o. Cf. also my Principles of Heat, German edition, pp. 123, 151, 452). XI. (See page 161.) The perspicuous deduction of the expression for centrifugal force based on the principle of Hamilton's hodograph may also be mentioned. If a body move uniformly in a circle of radius r (Fig. 237), the velocity 7/ at the point A of the path is transformed by the traction of the string into the velocity v of like magnitude but different direction at the point B. If from O as centre (Fig. 238) we lay off as to magnitude and direction all the velocities the body succes 528 5 THE SCIENCE OF MAECHIANICS. sively acquires, these lines will represent the sum of the radii v of the circle. For OM to be transformed into ON, the perpendicular component to it, MN, must be added. During the period of revolution T the velocity is uniformly increased in the directions of the radii r by an amount 2n7y. The numeriI V/ MN P A -C () Fig. 237. Fig. 238. Fig. 239 -cal measure of the radial acceleration is therefore 2n7v v2 p = -, and since vT= 27rr, therefore also pp=-r If to OM-v the very small component w is added (Fig. 239), the resultant will strictly be a greater IU12 velocity 1v -- w2 = v,v + as the approximate extraction of the square root will show. But on continw,2 uous deflection vanishes with respect to v; hence, 2v only the direction, but not the magnitude, of the velocity changes. XII. (See page 162.) Even Descartes thought of explaining the centripetal impulsion of floating bodies in a vortical medium, after this manner. But Huygens correctly reft APPENDIX. 529 marked that on this hypothesis we should have to assume that the hlig/test bodies received the greatest centripetal impulsion, and that all heavy bodies would without exception have to be lighter than the vortical medium. Huygens observes further that like phenomena are also necessarily presented in the case of bodies, be they what they may, that do not participate in the whirling movement, that is to say, such as might exist without centrifugal force in a vortical medium affected with centrifugal force. For example, a sphere composed of any material whatsoever but moveable only along a stationary axis, say a wire, is impelled toward the axis of rotation in a whirling medium. In a closed vessel containing water Huygens placed small particles of sealing wax which are slightly heavicr than water and hence touch the bottom of the vessel. If the vessel be rotated, the particles of sealing wax will flock toward the outer rim of the vessel. If the vessel be then suddenly brought to rest, the water will continue to rotate while the particles of sealing wax which touch the bottom and are therefore more rapidly arrested in their movement, will now be impelled toward the axis of the vessel. In this process Huygens saw an exact replica of gravity. An ether whirling in one direction only, did not appear to fulfil his requirements. Ultimately, he thought, it would sweep everything with it. He accordingly assumed ether-particles that sped rapidly about in all directions, it being his theory that in a closed space, circular, as contrasted with radial, motions would of themselves preponderate. This ether appeared to him adequate to explain gravity. The detailed exposition of this kinetic theory of gravity is 530 THE SCIENCE OF IMEICHANICS. found in Huygens's tract On the Cause of Gravitation (German trans. by Mewes, Berlin, 1893). See also Lasswitz, Geschichte der Atomistik, 1890, Vol. II., p. 344 -XIII. (See page 187 ) It has been impossible for us to enter upon the signal achievements of Huygens in physics proper. But a few points may be briefly indicated. He is the creator of the wave-theory of light, which ultimately overthrew the emission theory of Newton. His attention was drawn, in fact, to precisely those features of luminous phenomena that had escaped Newton. With respect to physics he took up with great enthusiasm the idea of Descartes that all things were to be explained mechanically, though without being blind to its errors, which he acutely and correctly criticised. His predilection for mechanical explanations rendered him also an opponent of Newton's action at a distance, which he wished to replace by pressures and impacts, that is, by action due to contact. In his endeavor to do so he lighted upon some peculiar conceptions, like that of magnetic currents, which at first could not compete with the influential theory of Newton, but has recently been reinstated in its full rights in the unbiassed efforts of Faraday and Maxwell. As a geometer and mathematician also Huygens is to be ranked high, and in this connexion reference need be made only to his theory of games of chance. His astronomical observations, his achievements in theoretical and practical dioptrics advanced these departments very considerably. As a technicist he is the inventor of the powder-machine, the idea of which APPENDIX. 53I has found actualisation in the modern gas-machine. As a physiologist he surmised the accommodation of the eye by deformation of the lens. All these things can scarcely be mentioned here. Our opinion of Huygens grows as his labors are made better known by the complete edition of his works. A brief and reverential sketch of his scientific career in all its phases is given by J. Bosscha in a pamphlet entitled Christian Huyghens, Rede am 200. Gedichtnisslage seines Lebensendes, German trans. by Engelmann, Leipsic, 1895. XIV. (See page 19o.) Rosenberger is correct in his statement (Newton und seine physikalischen Principien, 1895) that the idea of universal- gravitation did not originate with Newton, but that Newton had many highly deserving predecessors. But it may be safely asserted that it was, with all of them, a question of conjecture, of a groping and imperfect grasp of the problem, and that no one before Newton grappled with the notion so comprehensively and energetically; so that above and beyond the great mathematical problem, which Rosenberger concedes, there still remains to Newton the credit of a colossal feat of the imagination. Among Newton's forerunners may first be mentioned Copernicus, who (in 1543) says: "I am at least of opinion that gravity is nothing more than a natural tendency implanted in particles by the divine providence of the Master of the Universe, by virtue of which, they, collecting together in the shape of a sphere, do form their own proper unity and integrity. And it is to be assumed that this propensity is inherent also in the sun, the moon, and the other plan 532 THE SCIENCE OF E ICHtANICS. ets." Similarly, Kepler (1609), like Gilbert before him (160o), conceives of gravity as the analogue of magnetic attraction. By this analogy, Hooke, it seems, is led to the notion of a diminution of gravity with the distance; and in picturing its action as due to a kind of radiation, he even hits upon the idea of its acting inversely as the square of the distance. He even sought to determine the diminution of its effect (1686) by weighing bodies hung at different heights from the top of Westminster Abbey (precisely after the more modern method of Jolly), by means of spring-balances and pendulum clocks, but of course without results. The conical pendulum appeared to him admirably adapted for illustrating the motion of the planets. Thus Hooke really approached nearest to Newton's conception, though he never completely reached the latter's altitude of view. In two instructive writings (Kepler's Lehre von der Gravitation, Halle, 1896: Die Gravitation bei Galileo u. Borelli, Berlin, 1897) E. Goldbeck investigates the early history of the doctrine of gravitation with Kepler on the one hand and Galileo and Borelli on the other. Despite his adherence to scholastic, Aristotelian notions, Kepler has sufficient insight to see that there is a real physical problem presented by the phenomena of the planetary system; the moon, in his view, is swept along with the earth in its motion round the sun, and in its turn drags the tidal wave along with it, just as the earth attracts heavy bodies. Also, for the planets the source of motion is sought in the sun, from which immaterial levers extend that rotate with the sun and carry the distant planets around more slowly than the near ones. By this view, Kepler was enabled to guess that the period of rotation of the sun APPENDIX. 533 was less than eighty-eight days, the period of revolution of Mercury. At times, the sun is also conceived as a revolving magnet, over against which are placed the magnetic planets. In Galileo's conception of the universe, the formal, mathematical, and esthetical point of view predominates. He rejects each and every assumption of attraction, and even scouted the idea as childish in Kepler. The planetary system had not yet taken the shape of a genuine physical problem for him. Yet he assumed with Gilbert that an immaterial geometric point can exercise no physical action, and he did very much toward demonstrating the terrestrial nature of the heavenly bodies. Borelli (in his work on the satellites of the Jupiter) conceives the planets as floating between layers of ether of differing densities. They have a natural tendency to approach their central body, (the term attraction is avoided,) which is offset by the centrifugal force set up by the revolution. Borelli illustrates his theory by an experiment very similar to that described by us in Fig. o16, p. 162. As will be seen, he approaches very closely to Newton. His theory is, though, a combination of Descartes's and Newton's. xv. (See page 191.) Newton illustrated the identity of terrestrial gravity with the universal gravitation that determined the motions of the celestial bodies, as follows. He conceived a stone to be hurled with successive increases of horizontal velocity from the top of a high mountain. Neglecting the resistance of the air, the parabolas successively described by the stone will increase in length until finally they will fall clear of the earth 534 TIlE SCIENCE OF MlECAAVXCS. altogether, and the stone will be converted into a satellite circling round the earth. Newton begins with the fact of universal gravity. An explanation of the phenomenon was not forthcoming, and it was not his wont, he says, to frame hypotheses. Nevertheless he could not set his thoughts at rest so easily, as is apparent from his well-known letter to Bentley. That gravity was immanent and innate in matter, so that one body could act on another directly through empty space, appeared to him absurd. But he is unable to decide whether the intermediary agency is material or immaterial (spiritual?). Like all his predecessors and successors, Newton felt the need of explaining gravitation, by some such means as actions of contact. Yet the great success which Newton achieved in astronomy with forces acting at a distance as the basis of deduction, soon changed the situation very considerably. Inquirers accustomed themselves to these forces as points of departure for their explanations and the impulse to inquire after their origin soon disappeared almost completely. The attempt was now made to introduce these forces into all the departments of physics, by conceiving bodies to be composed of particles separated by vacuous interstices and thus acting on one another at a distance. Finally even, the resistance of bodies to pressure and impact, this is to say, even forces of contact, were explained by forces acting at a distance between particles. As a fact, the functions representing the former are more complicated than those representing the latter. The doctrine of forces acting at a distance doubtless stood in highest esteem with Laplace and his contemporaries. Faraday's unbiassed and ingenious conceptions and Maxwell's mathematical formulation APPENDIX. 535 of them again turned the tide in favor of the forces of contact. Divers difficulties had raised doubts in the minds of astronomers as to the exactitude of Newton's law, and slight quantitative variations of it were looked for. After it had been demonstrated, however, that electricity travelled with finite velocity, the question of a like state of affairs in connexion with the analogous action of gravitation again naturally arose. As a fact, gravitation bears a close resemblance to electrical forces acting at a distance, save in the single respect that so far as we know, attraction only and not repulsion takes place in the case of gravitation. Fdppl ("Ueber eine Erweiterung des Gravitationsgesetzes," Sitzungsber. d. Miinch. Akad., 1897, p. 6 et seq.) is of opinion, that we may, without becoming involved in contradictions, assume also with respect to gravitation negative masses, which attract one another but repel positive masses, and assume therefore also finite fields of gravitation, similar to the electric fields. Drude (in his report on actions at a distance made for the German Naturforscherversammlung of 1897) enumerates many experiments for establishing a velocity of propagation for gravitation, which go back as far as Laplace. The result is to be regarded as a negative one, for the velocities which it is at all possible to consider as such, do not accord with one another, though they are all very large multiples of the velocity of light. Paul Gerber alone ("Ueber die raumliche u. zeitliche Ausbreitung der Gravitation," Zei/schrif f. Afath. u. Phys., 1898, II.), from the perihelial motion of Mercury, forty-one seconds in a century, finds the velocity of propagation of gravitation to be the same as that of light. This would speak in favor of the ether as the medium of gravitation. (Com 536 THE SCIENCE OF AMECHANICS. pare W. Wien, "Ueber die M6glichkeit einer elektromagnetischen Begriindung der Mechanik," Archives Njerlandaises, The Hague, 1900, V., p. 96.) XVI. (See page 195.) It should be observed that the notion of mass as quantity of matter was psychologically a very natural conception for Newton, with his peculiar development. Critical inquiries as to the origin of the concept of matter could not possibly be expected of a scientist in Newton's day. The concept developed quite instinctively; it is discovered as a datum perfectly complete, and is adopted with absolute ingenuousness. The same is the case with the concept of force. But force appears conjoined with matter. And, inasmuch as Newton invested all material particles with precisely identical gravitational forces, inasmuch as he regarded the forces exerted by the heavenly bodies on one another as the sum of the forces of the individual particles composing them, naturally these forces appear to be inseparably conjoined with the quantity of matter. Rosenberger has called attention to this fact in his book, NeVwton und seine physikalischen Principien (Leipzig, 1895, especially page 192). I have endeavored to show elsewhere (Analysis of the Sensations, Chicago, 1897) how starting from the constancy of the connexion between different sensations we have been led to the assumption of an absolute constancy, which we call substance, the most obvious and prominent example being that of a moveable body distinguishable from its environment. And seeing that such bodies are divisible into homogeneous parts, of which each presents a constant complexus A PPENDIX. 537 of properties, we are induced to form the notion of a substantial something that is quantitatively variable, which we call matter. But that which we take away from one body, makes its appearance again at some other place. The quantity of matter in its entirety, thus, proves to be constant. Strictly viewed, however, we are concerned with precisely as many substantial quantities as bodies have properties, and there is no other function left for matter save that of representing the constancy of connexion of the several properties of bodies, of which man is one only. (Compare my Principles of Heat, German edition, 1896, page 425.) XVII. (See page 216.) Of the theories of the tides enunciated before Newton, that of Galileo alone may be briefly mentioned. Galileo explains the tides as due to the relative motion of the solid and liquid parts of the earth, and regards this fact as direct evidence of the motion of the earth and as a cardinal argument in favor of the Co-a pernican system. If the earth (Fig. 240) rotates from the d west to the east, and is affected at the same time with a pro- b gressional motion,.the parts of Fig.240. the earth at a will move with the sum, and the parts at b with the difference, of the two velocities. The water in the bed of the ocean, which is unable to follow this change in velocity quickly enough, behaves like the water in a plate swung rapidly back and forth, or like that in the bottom of a skiff which is rowed 538 THE SCIENCE OF MECHANICS. with rapid alterations of speed: it piles up now in the front and now at the back. This is substantially the view that Galileo set forth in the Dialogue on the Two World Systems. Kepler's view, which supposes attraction by the moon, appears to him mystical and childish. He is of the opinion that it should be relegated to the category of explanations by "sympathy" and "antipathy," and that it admits as easily of refutation as the doctrine according to which the tides are created by radiation and the consequent expansion of the water. That on his theory the tides rise only once a day, did not, of course, escape Galileo's attention. But he deceived himself with regard to the difficulties involved, believing himself able to explain the daily, monthly, and yearly periods by considering the natural oscillations of the water and the alterations to which its motions are subject. The principle of relative motion is a correct feature of this theory, but it is so infelicitously applied that only an extremely illusive theory could result. We will first convince ourselves that the conditions supposed to be involved would not have the effect ascribed to them. Conceive a homogeneous sphere of water; any other effect due to rotation than that of a corresponding oblateness we should not expect. Now, suppose the ball to acquire in addition a uniform motion of progression. Its various parts will now as before remain at relative rest with respect to one another. For the case in question does not differ, according to our view, in any essential respect from the preceding, inasmuch as the progressive motion of the sphere may be conceived to be replaced by a motion in the opposite direction of all surrounding bodies. Even for the person who is inclined to regard the motion APPENDIX. 539 as an "absolute" motion, no change is produced in the relation of the parts to one another by uniform motion of progression. Now, let us cause the sphere, the parts of which have no tendency to move with respect to one another, to congeal at certain points, so that sea-beds with liquid water in them are produced. The undisturbed uniform rotation will continue, and consequently Galileo's theory is erroneous. But Galileo's idea appears at first blush to be extremely plausible; how is the paradox explained? It is due entirely to a negative conception of the law of inertia. If we ask what acceleration the water experiences, everything is clear. Water having no weight would be hurled off at the beginning of rotation; water having weight, on the other hand, would de scribe a central motion around the center of the earth With its slight velocity of rotation it would be forccd more and more toward the center of the earth, with just enough of its centripetal acceleration counteracted by the resistance of the mass lying beneath, as to make the remainder, conjointly with the given tangential velocity, sufficient for motion in a circle. Looking at it from this point of view, all doubt and obscurity vanishes. But it must in justice be added that it was almost impossible for Galileo, unless his genius were supernatural, to have gone to the bottom of the matter. He would have been obliged to anticipate the great intellectual achievements of Huygens and Newton. XVIII. (See page 218.) H. Streintz's objection (Die physikalischen Grundlagen der Mechanik, Leipsic, 1883, p. i 17), that a com 540 THE SCIENCE OF MECCHANICS. parison of masses satisfying my definition can be effected only by astronomical means, I am unable to admit. The expositions on pages 202, 218-221 amply refute this. Masses produce in each other accelerations in impact, as well as when subject to electric and magnetic forces, and when connected by a string on Atwood's machine. In my Elements of Physics (second German edition, 1891, page 27) I have shown how mass-ratios can be experimentally determined on a centrifugal machine, in a very elementary and popular manner. The criticism in question, therefore, may be regarded as refuted. My definition is the outcome of an endeavor to establish the interdependence of phnenmena and to remove all metaphysical obscurity, without accomplishing on this account less than other definitions have done. I have pursued exactly the same course with respect to the ideas, " quantity of electricity" ("On the Fundamental Concepts of Electrostatics," 1883, Popular Scientific Lectures, Open Court Pub. Co., Chi-:ago, 1898), "temperature," "quantity of heat" (Zcitschrift fir den physikalischen und chemischen Unterricht, Berlin, 1888, No. i), and so forth. With the view here taken of the concept of mass is associated, however, another difficulty, which must also be carefully noted, if we would be rigorously critical in our analysis of other concepts of physics, as for example the concepts of the theory of heat. Maxwell made reference to this point in his investigations of the concept of temperature, about the same time as I did with respect to the concept of heat. I would refer here to the discussions on this subject in my Princzples of Heat (German edition, Leipsic, 1896), particularly page 41 and page 190. A PIPE'ND IX. 541 XIX. (See page 226.) My views concerning physiological time, the sensation of time, and partly also concerning physical time, I have expressed elsewhere (see Analysis of the Sensations, 1886, Chicago, Open Court Pub. Co., 1897, pp. 109-118, 179-181). As in the study of thermal phenomena we take as our measure of temperature an arbitrarily chosen indicator of volume, which varies in almost parallel correspondence with our sensation of heat, and which is not liable to the uncontrollable disturbances of our organs of sensation, so, for similar reasons, we select as our measure of time an arbitrarily chosen motion, (the angle of the earth's rotation, or path of a free body,) which proceeds in almost parallel correspondence with our sensation of time. If we have once made clear to ourselves that we are concerned only with the ascertainment of the interdependence of phenomena, as I pointed out as early as 1865 (Uebcr den Zeitsinn des Ohres, Sitzungsberichte der Wifener Akademie) and 1866 (Fichte's Zeitschrift fiir Phlilosophie), all metaphysical obscurities disappear. (Compare J Epstein, Die logischen Principien der Zeitiessung, Berlin, 1887.) I have endeavored also (Principles of Heat, German edition, page 51) to point out the reason for the natural tendency of man to hypostatise the concepts which have great value for him, particularly those at which he arrives instinctively, without a knowledge of their development. The considerations which I there adduced for the concept of temperature may be easily applied to the concept of time, and render the origin 542 THE SCIENCE OF MECHANICS. of Newton's concept of "absolute" time intelligible. Mention is also made there (page 338) of the connexion obtaining between the concept of energy and the irreversibility of time, and the view is advanced that the entropy of the universe, if it could ever possibly be determined, would actually represent a species of absolute measure of time. I have finally to refer here also to the discussions of Petzoldt ("Das Gesetz der Eindeutigkeit," Vierteljahrsschrijt fzr wissenschaftliche Philosophie, 1894, page 146), to which I shall reply in another place. xx. (See page 238.) Of the treatises which have appeared since 1883 on the law of inertia, all of which furnish welcome evidence of a heightened interest in this question, I can here only briefly mention that of Streintz (Physikalische Grundlagen der Mechanik, Leipsic, 1883) and that of L. Lange (Die geschichiliche Entwicklung des Bewegungsbegrizfes, Leipsic, 1886). The expression I"absolute motion of translation" Streintz correctly pronounces as devoid of meaning and consequently declares certain analytical deductions, to which he refers, superfluous. On the other hand, with respect to rotation, Streintz accepts Newton's position, that absolute rotation can be distinguished from relative rotation. In this point of view, therefore, one can select every body not affected with absolute rotation as a body of reference for the expression of the law of inertia. I cannot share this view. For me, only relative motions exist (Erhallung der Arbeit, p. 48; Science of Mechanics, p. 229), and I can see, in this regard, no APPENDIX. 543 distinction between rotation and translation. When a body moves relatively to the fixed.stars, centrifugal forces are produced; when it moves relatively to some different body, and not relatively to the fixed stars, no centrifugal forces are produced. I have no objection to calling the first rotation "absolute" rotation, if it be remembered that nothing is meant by such a designation except relative rotation with respect to the fixed stars. Can we fix Newton's bucket of water, rotate the fixed stars, and then prove the absence of centrifugal forces? The experiment is impossible, the idea is meaningless, for the two cases are not, in sense-perception, distinguishable from each other. I accordingly regard these two cases as the same case and Newton's distinction as an illusion (Science of MAechanics, page 232). But the statement is correct that it is possible to find one's bearings in a balloon shrouded in fog, by means of a body which does not rotate with respect to the fixed stars. But this is nothing more than an indirect orientation with respect to the fixed stars; it is a mechanical, substituted for an optical, orientation. I wish to add the following remarks in answer to Streintz's criticism of my view. My opinion is not to be confounded with that of Euler (Streintz, pp. 7, 50), who, as Lange has clearly shown, never arrived at any settled and intelligible opinion on the subject. Again, I never assumed that remote masses only, and not near ones, determine the velocity of a body (Streintz, p. 7); I simply spoke of an influence independent of distance. In the light of my expositions at pages 222-245, the unprejudiced and careful reader 544 THE SCIENCE OF MECHANICS. will scarcely maintain with Streintz (p. 50), that after so long a period of time, without a knowledge of Newton and Euler, I have only been led to views which these inquirers so long ago held, but were afterwards, partly by them and partly by others, rejected. Even my remarks of 1872, which were all that Streintz knew, cannot justify this criticism. These were, for good reasons, concisely stated, but they are by no means so meagre as they must appear to one who knows them only from Streintz's criticism. The point of view which Streintz occupies, I at that time expressly rejected. Lange's treatise is, in my opinion, one of the best that have been written on this subject. Its methodical movement wins at once the reader's sympathy. Its careful analysis and study, from historical and critical points of view, of the concept of motion, have produced, it seems to me, results of permanent value. I also regard its clear emphasis and apt designation of the principle of "particular determination" as a point of much merit, although the principle itself, as well as its application, is not new. The principle is really at the basis of all measurement. The choice of the unit of measurement is convention; the number of measurement is a result of inquiry. Every natural inquirer who is clearly conscious that his business is simply the investigation of the interdependence of phenomena, as I formulated the point at issue a long time ago (1865-1866), employs this principle. When, for example (Mechanics, p. 218 et seq.), the negative inverse ratio of the mutually induced accelerations of two bodies is called the mass-ratio of these bodies, this is a convention, expressly acknowledged as arbitrary; but that these ratios are independent of the APPENDIX. 545 kind and of the order of combination of the bodies is a result of inquizy. I might adduce numerous similar nstances from the theories of heat and electricity as well as from other provinces. Compare Appendix II. Taking it in its simplest and most perspicuous form, the law of inertia, in Lange's view, would read as follows: Three material points, P1, P2, F3, are simultaneously hurled from the same point in space and then left to themselves. The moment we are certain that the points are not situated in the same straight line, we join each separately with any fourth point in space, Q. These lines of junction, which we may respectively call G1, G2, Gs, form, at their point of meeting, a three-faced solid angle. If now we make this solid angle preserve, with unaltered rigidity, its form, and constantly determine in such a manner its position, that P1 shall always move on the line G1, P2 on the line G2, P3 on the line G3, these lines may be regarded as the axis of a coordinate or inertial system, with respect to which every other material point, left to itself, will move in a straight line. The spaces described by the free points in the paths so determined will be proportional to one another. A system of co6rdinates with respect to which three material points move in a straight line is, according to Lange, under the assumed limitations, a simple convention. That with respect to such a system also a fourth or other free material point will move in a straight line, and that the paths of the different points will all be proportional to one another, are results of inquiry. In the first place, we shall not dispute the fact that the law of inertia can be referred to such a system 546 THE SCIENCE OF AlECIL.4NICS. of time and space coordinates and expressed in this form. Such an expression is less fit than Streintz's for practical purposes, but, on the other hand, is, for its methodical advantages, more attractive. It especially appeals to my mind, as a number of years ago I was engaged with similar attempts, of which not the beginnings but only a few remnants (Mcc/hanics, pp. 234-235) are left. I abandoned these attempts, because I was convinced that we only apparently evade by such expressions references to the fixed stars and the angular rotation of the earth. This, in my opinion, is also true of the forms in which Streintz and Lange express the law. In point of fact, it was precisely by the consideration of the fixed stars and the rotation of the earth that we arrived at a knowledge of the law of inertia as it at present stands, and without these foundations we should never have thought of the explanations here discussed (AMechanics, 232-233). The consideration of a small number of isolated points, to the exclusion of the rest of the world, is in my judgment inadmissible (Mechanics, pp. 229-235). It is quite questionable, whether a fourth material point, left to itself, would, with respect to Lange's i"inertial system," uniformly describe a straight line, if the fixed stars were absent, or not invariable, or could not be regarded with sufficient approximation as invariable. The most natural point of view for the candid inquirer must still be, to regard the law of inertia primarily as a tolerably accurate approximation, to refer it, with respect to space, to the fixed stars, and, with respect to time, to the rotation of the earth, and to await the correction, or more precise definition, of A IPE-NDDIA. 547 our knowledge from future experience, as I have explained on page 237 of this book. I have still to mention the discussions of the law of inertia which have appeared since 1889. Reference may first be made to the expositions of Karl Pearson (Grammar of Science, 1892, page 477), which agree with my own, save in terminology. P. and J. Friedlander (Absolute und relative Bewegung, Berlin, 1896) have endeavored to determine the question by means of an experiment based on the suggestions made by me at pages 217-218; I have grave doubts, however, whether the experiment will be successful from the quantitative side. I can quite freely give my assent to the discussions of Johannesson (Das Beharrungsgesetz, Berlin, 1896), although the question remaine unsettled as to the means by which the motion of A body not perceptibly accelerated by other bodies is to be determined. For the sake of completeness, the predominantly dialectic treatment by M. E. Vicaire, Socited scientifique de Bruxelles, 1895, as well as the investigations of J. G. MacGregor, Royal Society of Canada, 1895, which are only remotely connected with the question at issue, remain to be mentioned. I have no objections to Budde's conception of space as a sort of medium (compare page 230), although I think that the properties of this medium should be demonstrable physically in some other manner, and that they should not be assumed ad hoc. If all apparent actions at a distance, all accelerations, turned out to be effected through the agency of a medium, then the question would appear in a different light, and the solution is to be sought perhaps in the view set forth on page 230. 548 THE SCIENCE OF AECHANICS. XXI. (See page 255.) Section VIII., "Retrospect of the Development of Dynamics," was written in the year 1883. It contains, especially in paragraph 7, on pages 254 and 255, an extremely general programme of a future system of mechanics, and it is to be remarked that the Mechanics of Hertz, which appeared in the year 1894,* marks a distinct advance in the direction indicated. It is impossible in the limited space at our disposal to give any adequate conception of the copious material contained in this book, and besides it is not our purpose to expound new systems of mechanics, but merely to trace the development of ideas relating to mechanics. Hertz's book must, in fact, be read by every one interested in mechanical problems. Hertz's criticisms of prior systems of mechanics, with whiich he opens his work, contains some very noteworthy epistemological considerations, which from our point of view (not to be confounded either with the Kantian or with the atomistic mechanical concepts of the majority of physicists), stand in need of certain modifications. The constructive imagest (or better, perhaps, the concepts), which we consciously and purposely form of objects, are to be so chosen that the "consequences which necessarily follow from them in thought" agree with the " consequences which necessarily follow from them in nature." It is demanded of these images or concepts that they shall be logically *H. Hertz, Die PrinciPien der Mechanik in neuem Zusammenhange dargestellt. Leipzig, 1894. t Hertz uses the term Bild (image or picture) in the sense of the old English philosophical use of idea, and applies it to systems of ideas or concepts relating to any province. APPENDIX. 549 admissible, that is to say, free from all self-contradictions; that they shall be correct, that is, shall conform to the relations obtaining between objects; and finally that they shall be appropriate, and contain the least possible superfluous features. Our concepts, it is true, are formed consciously and purposely by us, but they are nevertheless not formed altogether arbitrarily, but are the outcome of an endeavor on our part to adapt our ideas to our sensuous environment. The agreement of the concepts with one another is a requirement which is logically necessary, and this logical necessity, furthermore, is the only necessity that we have knowledge of. The belief in a necessity obtaining in nature arises only in cases where our concepts are closely enough adapted to nature to ensure a correspondence between the logical inference and the fact. But the assumption of an adequate adaptation of our ideas can be refuted at any moment by experience. Hertz's criterion of appropriateness coincides with our criterion of economy. Hertz's criticism that the Galileo-Newtonian system of mechanics, particularly the notion of force, lacks clearness (pages 7, 14, i5) appears to us justified only in the case of logically defective expositions, such as Hertz doubtless had in mind from his student days. He himself partly retracts his criticism in another place (pages 9, 47); or at any rate, he qualifies it. But the logical defects of some individual interpretation cannot be imputed to systems as such. To be sure, it is not permissible to-day (page 7) "to speak of a force acting in one aspect only, or, in the case of centripetal force, to take account of the action of inertia twice, once as a mass and again as a force." But neither is this necessary, since Huygens and Newton 550 TIE SCIENCE OF AEIILCHANICS. were perfectly clear on this point. To characterise forces as being frequently "empty-running wheels," as being frequently not demonstrable to the senses, can scarcely be permissible. In any event, "forces" are decidedly in the advantage on this score as compared with "hidden masses " and "hidden motions." In the case of a piece of' iron lying at rest on a table, both the forces in equilibrium, the weight of the iron and the elasticity of the table, are very easily demonstrable. Neither is the case with energic mechanics so bad as Hertz would have it, and as to his criticism against the employment of minimum principles, that it involves the assumption of purpose and presupposes tendencies directed to the future, the present work shows in another passage quite distinctly that the simple import of minimum principles is contained in an entirely different property from that of purpose. Every system of mechanics contains references to the future, since all must employ the concepts of time, velocity, etc. Nevertheless, though Hertz's criticism of existing systems of mechanics cannot be accepted in all their severity, his own novel views must be regarded as a great step in advance. Hertz, after eliminating the concept of force, starts from the concepts of time, space, and mass alone, with the idea in view of giving expression only to that which can actually be observed. The sole principle which he employs may be conceived as a combination of the law of inertia and Gauss's principle of least constraint. Free masses move uniformly in straight lines. If they are put in connexion in any manner they deviate, in accordance with Gauss's principle, as little as possible from this APPE ND IX. 551 motion; their actual motion is more nearly that of free motion than any other conceivable motion. Hertz says the masses move as a result of their connexion in a straighitcst path. Every deviation of the motion of a mass from uniformity and rectilinearity is due, in his system, not to a force but to rigid connexion with other masses. And where such matters are not visible, he conceives hidden masses with hidden motions. All physical forces are conceived as the effect of such actions. Force, force-function, energy, in his system, are secondary and auxiliary concepts only. Let us now look at the most important points singly, and ask to what extent was the way prepared for them. The notion of eliminating force may be reached in the following manner. It is part of the general idea of the Galileo-Newtonian system of mechanics to conceive of all connexions as replaced by forces which determine the motions required by the connexions; conversely, everything that appears as force may be conceived to be due to a connexion. If the first idea frequently appears in the older systems, as being historically simpler and more immediate, in the case of Hertz the latter is the more prominent. If we reflect that in both cases, whether forces or connexions be presupposed, the actual dependence of the motions of the masses on one another is given for every instantaneous conformation of the system by linear differential equations between the co ordinates of the masses, then the existence of these equations may be considered the essential thing,-the thing established by experience. Physics indeed gradually accustoms itself to look upon the description of the facts by differential equations as its proper aim,-a point of view which was taken also in Chapter V. of the present 552 THE SCIENCE OF MIE CHANICS. work (1883). But with these the general applicability of Hertz's mathematical formulations is recognised without our being obliged to enter upon any further interpretation of the forces or connexions. Hertz's fundamental law may be described as a sort of generalised law of inertia, modified by connexions of the masses. For the simpler cases, this view was a natural one, and doubtless often forced itself upon the attention. In fact, the principle of the conservation of the center of gravity and of the conservation of areas was actually described in the present work (Chapter III.) as a generalised law of inertia. If we reflect that by Gauss's principle the connexion of the masses determines a minimum of deviation from those motions which it would describe for itself, we shall arrive at Hertz's fundamental law the moment we consider all the forces as due to the connexions. For on severing all connexions, only isolated masses moving by the law of inertia are left as ultimate elements. Gauss very distinctly asserted that no substantially new principle of mechanics could ever be discovered. And Hertz's principle also is only new in form, for it is identical with Lagrange's equations. The minimum condition which the principle involves does not refer to any enigmatic purpose, but its import is the same as that of all minimum laws. That alone takes place which is dynamically determined (Chapter III.). The deviation from the actual motion is dynamically not determined; this deviation is not present; the actual motion is therefore unique.* * See Petzoldt's excellent article " Das Gesetz der Eindeutigkeit" (Vierteljahrsschrift fir cwissenschaftlicke Philosofhie, XIX., page 146, especially page 186). R. Henke is also mentioned in this article as having approached Hertz's view in his tract Ueber die Methode der kleinsten Quadrate (Leipsic, 1894). APPENDIX. 553 It is hardly necessary to remark that the physical side of mechanical problems is not only not disposed of, but is not even so much as touched, by the elaboration of such a formal mathematical system of mechanics. Free masses move uniformly in straight lines. Masses having different velocities and directions if connected mutually affect each other as to velocity, that is, determine in each other accelerations. These physical experiences enter along with purely geometrical and arithmetical theorems into the formulation, for which the latter alone would in no wise be adequate; for that which is uniquely determined mathematically and geometrically only, is for that reason not also uniquely determined mechanically. But we discussed at considerable length in Chapter II., that the physical principles in question were not at all self-evident, and that even their exact significance was by no means easy to establish. In the beautiful ideal form which Hertz has given to mechanics, its physical contents have shrunk to an apparently almost imperceptible residue. It is scarcely to be doubted that Descartes if he lived today would have seen in Hertz's mechanics, far more than in Lagrange's "analytic geometry of four dimensions," his own ideal. For Descartes, who in his opposition to the occult qualities of Scholasticism would grant no other properties to matter than extension and motion, sought to reduce all mechanics and physics to a geometry of motions, on the assumption of a motion indestructible at the start. It is not difficult to analyse the psychological circumstances which led Hertz to his system. After inquirers had succeeded in representing electric and magnetic forces that act at a distance as the results 554 THE SCIENCE OF MECHANICS. of motions in a medium, the desire must again have awakened to accomplish the same result with respect to the forces of gravitation, and if possible for all forces whatsoever. The idea was therefore very natural to discover whether the concept of force generally could not be eliminated. It cannot be denied that when we can command all the phenomena taking place in a medium, together with the large masses contained in it, by means of a single complete picture, our concepts are on an entirely different plane from wthat they are when only the relations of these isolated masses as regards acceleration are known. This will be willingly granted even by those who are convinced that the interaction of parts in contact is not more intelligible than action at a distance. The present tendencies in the development of physics are entirely in this direction. If we are not content to leave the assumption of occult masses and motions in its general form, but should endeavor to investigate them singly and in detail, we should be obliged, at least in the present state of our physical knowledge, to resort, even in the simplest cases, to fantastic and even frequently questionable fictions, to which the given accelerations would be far preferable. For example, if a mass m is moving uniformly in a circle of radius r, with a velocity 7v, which we are accustomed to refer to a centripetal,,,2 force -- proceeding from the center of the circle, we might instead of this conceive the mass to be rigidly connected at the distance 2r with one of the same size having a contrary velocity. Huygens's centripetal impulsion would be another example of a force replaced by a connexion. As an ideal program APPE NDIX. 555 Hertz's mechanics is simpler and more beautiful, but for practical purposes our present system of mechanics is preferable, as Hertz himself (page 47), with his characteristic candor, admits.* XXII. (See page 255.) The views put forward in the first two chapters of this book were worked out by me a long time ago. At the start they were almost without exception coolly rejected, and only gradually gained friends. All the essential features of my Mechanics I stated originally in a brief communication of five octavo pages entitled On the Definition of Mass. These were the theorems now given at page 243 of the present book. The communication was rejected by Poggendorf's Annalen, and it did not appear until a year later (1868), in Carl's Reperlorium. In a lecture delivered in 1871, I outlined my epistemological point of view in natural science generally, and with special exactness for physics. The concept of cause is replaced there by the concept of function; the determining of the dependence of phenomena on one another, the economic exposition of actual facts, is proclaimed as the object, and physical concepts as a means to an end solely. I did not care now to impose upon any editor the responsibility for the publication of the contents of this lecture, and the same was published as a separate tract in 1872.t In 1874, when Kirchhoff in his Mec/anics came out with his theory of " description" and *Compare J. Classen, "Die Principien der Mechanik bei Hertz nnd Boltzmann" (j/akrbuch der Hamburgischen wissenschaftlichlcn An.\fltn, XV., p. I, Hamburg, 1898). t Erhaltung der Arbeit, Prague, 1872. 556 THE SCIENCE OF MECHANICS. other doctrines, which were analogous in part only to my views, and still aroused the "universal astonishment" of his colleagues, I became resigned to my fate. But the great authority of Kirchhoff gradually made itself felt, and the consequence of this also doubtless was that on its appearance in 1883 my Mechanics did not evoke so much surprise. In view of the great assistance afforded by Kirchhoff, it is altogether a matter of indifference with me that the public should have regarded, and partly does so still, my interpretation of the principles of physics as a continuation and elaboration of Kirchhoff's views; whilst in fact mine were not only older as to date of publication, but also more radical.* The agreement with my point of view appears upon the whole to be increasing, and gradually to extend over more extensive portions of my work. It would be more in accord with my aversion for polemical discussions to wait quietly and merely observe what part of the ideas enunciated may be found acceptable. But I cannot suffer my readers to remain in obscurity with regard to the existing disagreements, and I have also to point out to them the way in which they can find their intellectual bearings outside of this book, quite apart from the fact that esteem for my opponents also demands a consideration of their criticisms. These opponents are numerous and of all kinds: historians, philosophers, metaphysicians, logicians, educators, mathematicians, and physicists. I can make no pretence to any of these qualifications in any superior degree. I can only select here the most important criticisms, and answer them in the capacity of a man who has the liveliest and most ingenuous in* See the preface to the first edition. APPENDIX. 557 terest in understanding the growth of physical ideas. I hope that this will also make it easy for others to find their way in this field and to form their own judgment. P. Volkmann in his writings on the epistemology* of physics appears as my opponent only in certain criticisms on individual points, and particularly by his adherence to the old systems and by his predilection for them. It is the latter trait, in fact, that separates us; for otherwise Volkmann's views have much affinity with my own. He accepts my adaptation of ideas, the principle of economy and of comparison, even though his expositions differ from mine in individual features and vary in terminology. I, for my part, find in his writings the important principle of isolation and superposition, appropriately emphasised and admirably described, and I willingly accept them. I am also willing to admit that concepts which at the start are not very definite must acquire their "retroactive consolidation" by a "circulation of knowledge," by an "oscillation" of attention. I also agree with Volkmann that from this last point of view Newton accomplished in his day nearly the best that it was possible to do; but I cannot agree with Volkmann when he shares the opinion of Thomson and Tait, that even in the face of the substantially different epistemological needs of the present day, Newton's achievement is definitive and exemplary. On the contrary, it appears to me that if Volkmann's process of "consolidation" be allowed complete sway, it must necessarily lead to enunciations not differing in * Erkenntnisstheoretische Grundziige dcr Naturwvissenschaft. Leipzig, 1896, -Ueber Newton's Philoso1hkia Naturalis. K6nigsberg, 1898.-Einfidhrung in das Studium der theoretischen Physik, Leipsic, 1900. Our references are to the last-named work. 558 THE SCIEVNCE OF MAlEC/IANICS. any essential point from my own. I follow with genuine pleasure the clear and objective discussions of G. H-eymans.* The differences which I have with Hoflert and Poskel relate in the main to individual points. So far as principles are concerned, I take precisely the same point of view as Petzoldt, ~ and we differ only on questions of minor importance. The numerous criticisms of others, which either refer to the arguments of the writers just mentioned, or are supported by analogous grounds, cannot out of regard for the reader be treated at length. It will be sufficient to describe the character of these differences by selecting a few individual, but important, points. A special difficulty seems to be still found in accepting my definition of mass. Streintz (compare p. 540) has remarked in criticism of it that it is based solely upon gravity, although this was expressly excluded in my first formulation of the definition (1868). Nevertheless, this criticism is again and again put forward, and quite recently even by Volkmann (loc. cit., p. 18). My definition simply takes note of the fact that bodies in mutual relationship, whether it be that of action at a distance, so called, or whether rigid or elastic connexions be considered, determine in one another changes of velocity (accelerations). More than this, one does not need to know in order to be able to form a definition with perfect assurance and without the fear of building on sand. It is not correct as *Die Gesetze und Elemente des wisscnschaftlichen Denkens, II., Leipzig 1894. SStudien zur gegnwenwiirtigen Philosophie der nmathematischen Aechanik, Leipzig, 1900. 4 Vierteljahrsschrift fur wissenschaftliche Philosopzie, Leipzig, 1884, pag385. ~ " Das Gesetz der Eindeutigkeit (Vierteljahrssckriftfiur vissenschaftliche Philosofphie, XIX., page 146). APPENDIX. 559 H6fler asserts (loc. cit., p. 77), that this definition tacitly assumes one and the same force acting on both masses. It does not assume even the notion of force, since the latter is built up subsequently upon the notion of mass, and gives then the principle of action and reaction quite independently and without falling into Newton's logical error. In this arrangement one concept is not misplaced and made to rest on another which threatens to give way under it. This is, as I take it, the only really serviceable aim of Volkmann's "circulation" and "oscillation." After we have defined mass by means of accelerations, it is not difficult to obtain from our definition apparently new variant concepts like "capacity for acceleration," "capacity for energy of motion" (H6fler, loc. cit., page 70). To accomplish anything dynamically with the concept of mass, the concept in question must, as I most emphatically insist, be a dynamical concept. Dynamics cannot be constructed with quantity of matter by itself, but the same can at most be artificially and arbitrarily attached to it (loc. cit., pages 71, 72). Quantity of matter by itself is never mass, neither is it thermal capacity, nor heat of combustion, nor nutritive value, nor anything of the kind. Neither does "mass" play a thermal, but only a dynamical role (compare H6fler, loc. cit., pages 71, 72). On the other hand, the different physical quantities are proportional to one another, and two or three bodies of unit mass form, by virtue of the dynamic definition, a body of twice or three times the mass, as is analogously the case also with thermal capacity by virtue of the thermal definition. Our instinctive craving for concepts involving quantities of things, to which H6fler (loc. cit, page 72) is doubtless seeking to give expression, 560 THEi SCIENCE O0F MECHAJICS. and which amply suffices for every-day purposes, is something that no one will think of denying. But a scientific concept of "quantity of matter" should properly be deduced from the proportionality of the single physical quantities mentioned, instead of, contrariwise, building up the concept of mass upon "quantity of matter." The measurement of mass by means of weight results from my definition quite naturally, whereas in the ordinary conception the measurability of quantity of matter by one and the same dynamic measure is either taken for granted outright, or proof must be given beforehand by special experiments, that equal weights act under all circumstances as equal masses. In my opinion, the concept of mass has here been subjected to thorough analysis for the first time since Newton. For historians, mathematicians, and physicists appear to have all treated the question as an easy and almost self-evident one. It is, on the contrary, of fundamental significance and is deserving of the attention of my opponents. Many criticisms have been made of my treatment of the law of inertia. I believe I have shown (1868), somewhat as Poske has done (1884), that any deduction of this law from a general principle, like the law of causality, is inadmissible, and this view has now won some support (compare Heymans, loc. cit., page 432). Certainly, a principle that has been universally recognised for so short a time only cannot be regarded as a priori self-evident. Heymans (loc. cit., p. 427) correctly remarks that axiomatic certainty was ascribed a few centuries ago to a diametrically opposite form of the law. Heymans sees a supra-empirical element only in the fact that the law of inertia is referred to absolute space, and in the further fact that both in APPENDIX. 561 the law of inertia and in its ancient diametrically opposite form something constant is assumed in the condition of the body that is left to itself (loc. cit., page 433). We shall have something to say further on regarding the first point, and as for the latter it is psychologically intelligible without the aid of metaphysics, because constant features alone have the power to satisfy us either intellectually or practically,which is the reason that we are constantly seeking for them. Now, looking at the matter from an entirely unprejudiced point of view, the case of these axiomatic certainties will be found to be a very peculiar one. One will strive in vain with Aristotle to convince the common man that a stone hurled from the hand would be necessarily brought to rest at once after its release, were it not for the air which rushed in behind and forced it forwards. But he would put just as little credence in Galileo's theory of infinite uniform motion. On the other hand, Benedetti's theory of the gradual diminution of the vis imprcssa, which belongs to the period of unprejudiced thought and of liberation from ancient preconceptions, will be accepted by the common man without contradiction. This theory, in fact, is an immediate reflexion of experience, while the first-mentioned theories, which idealise experience in contrary directions, are a product of technical professional reasoning. They exercise the illusion of axiomatic certainty only upon the mind of the scholar whose entire customary train of thought would be thrown out of gear by a disturbance of these elements of his thinking. The behavior of inquirers toward the law of inertia seems to me from a psychological point of view to be adequately explained by this circumstance, and I am inclined to 562 TliE SCIENCE" OF lE CHIANICS. allow the question of whether the principle is to be called an axiom, a postulate, or a maxim, to rest in abeyance for the time being. Heymans, Poske, and Petzoldt concur in finding an empirical and a supraempirical element in the law of inertia. According to Heymans (loc. cit., p. 438) experience simply afforded the opportunity for applying an a priori valid principle. Poske thinks that the empirical origin of the principle does not exclude its a priori validity (loc. cit., pp. 401 and 402). Petzoldt also deduces the law of inertia in part only from experience, and regards it in its remaining part as given by the law of unique determination. I believe I am not at variance with Petzoldt in formulating the issue here at stake as follows: It first devolves on experience to inform us what particular dependence of phenomena on one another actually exists, what the thing to be determined is,-and experience alone can instruct us on this point. If we are convinced that we have been sufficiently instructed in this regard, then when adequate data are at hand we regard it as unnecessary to keep on waiting for further experiences; the phenomenon is determined for us, and since this alone is determination, it is uniquely determined. In other words, if I have discovered by experience that bodies determine accelerations in one another, then in all circumstances where such determinative bodies are lacking I shall expect with unique determination uniform motion in a straight line. The law of inertia thus results immediately in all its generality, without our being obliged to specialise with Petzoldt; for every deviation from uniformity and rectilinearity takes acceleration for granted. I believe I am right in saying that the same fact is twice formulated in the law of inertia and in APPENDIX. 563 the statement that forces determine accelerations (p. 143). If this be granted, then an end is also put to the discussion as to whether a vicious circle is or is not contained in the application of the law of inertia (Poske, Hofler). My inference as to the probable manner in which Galileo reached clearness regarding the law of inertia was drawn from a passage in his third Dialogue,* which was literally transcribed from the Paduan edition of 1744, Vol. III., page 124, in my tract on The Conservation of Energy (Eng. Trans., in part, in my Popular Scientific Lectures, third edition, Chicago, The Open Court Publishing Co.). Conceiving a body which is rolling down an inclined plane to be conducted upon rising inclined planes of varying slopes, the slight retardation which it suffers on absolutely smooth rising planes of small inclination, and the retardation zero, or unending uniform motion on a horizontal plane, must have occurred to him. Wohlwill was the first to object to this way of looking at the matter (see page 524), and others have since joined * " Constat jam, quod mobile ex quiete in A descendens per AB, gradus acquirit velocitatis juxta temporis ipsius incrementum: gradumn vero in B esse maximum acquisitorum, et suapte natura immutabil ter impressum, sublatis scilicet causis accelerationis novae, aut retardationis: accelerationis A C / G H B Fig. 241. inquam, si adhuc super extenso piano ulterius progrederetur; retardationis vero, dum super planum acclive BC fit reflexio: in horizontali autem GH nequabilis motus juxta gradum velocitatis ex A in B acquisitae in infinitum axtendleretur." ' It is plain now that a movable body, starting from rest at A and de 564 THE SCIENCE OF AIECC.4ANICS. him. He asserts that uniform motion in a circle and horizontal motion still occupied distinct places in Galileo's thought, and that Galileo started from the ancient concepts and freed himself only very gradually from them. It is not to be denied that the different phases in the intellectual development of the great inquirers have much interest for the historian, and so7me one phase may, in its importance in this respect, be relegated into the background by the others. One must needs be a poor psychologist and have little knowledge of oneself not to know how difficult it is to liberate oneself from traditional views, and how even after that is done the remnants of the old ideas still hover in consciousness and are the cause of occasional backslidings even after the victory has been practically won. Galileo's experience cannot have been different. But with the physicist it is the instant in which a new view flashes forth that is of greatest interest, and it is this instant for which he will always seek. I have sought for it, I believe I have found it; and I am of the opinion that it left its unmistakable traces in the passage in question. Poske (loc. cit., page 393) and H6fler (loc. cit., pages iII, 112) are unable to give their assent to my interpretation of this passage, for the reason that Galileo does not expressly refer to the limiting case of transition from the inclined to the horizontal plane; although scending down the inclined plane AB, acquires a velocity proportional to the increment of its time: the velocity possessed at B is the greatest of the velocities acquired, and by its nature immutably impressed, provided all causes of new acceleration or retardation are taken away: I say acceleration, having in view its possible further progress along the plane extended; retardation, in view of the possibility of its being reversed and made to mount the ascending plane BC. But in the horizontal plane GHits uniform motion, with the velocity Pcquircd in the descent from A to B, will be continued ad inf'-/, J"/, APPENDIX. 565 Poske grants that the consideration of limiting cases was frequently employed by Galileo, and although H6fler admits having actually tested the educational efficacy of this device with students. It would indeed be a matter of surprise if Galileo, who may be regarded as the inventor of the principle of continuity, should not in his long intellectual career have applied the principle to this most important case of all for him. It is also to be considered that the passage does not form part of the broad and general discussions of the Italian dialogue, but is tersely couched, in the dogmatic form of a result, in Latin. And in this way also the "velocity immutably impressed" may have crept in.* * Even granting that Galileo reached his knowledge of the law of inertia only gradually, and that it was presented to him merely as an accidental discovery, nevertheless the following passages which are taken from the Paduan edition of 1744 will show that his limitation of the law to horizontal motion was justified by the inherent nature of the subject treated; and the assumption that Galileo toward the end of his scientific career did not possess a full knowledge of the law, can hardly be maintained. " Sagr. Ma quando l'artiglieria si piantasse non a perpendicolo, ma inclinata verso qualche parte, qual dovrebbe esser' il moto della pa'la? andrebbe ella forse, come nel laltro tiro, per la linea perpendicolare, e ritornando anco poi per l'istessa? " ' Simpl. Questo non farebbe ella, ma nscita del pezzo seguiterebbe il suo noto per la linea retta, che continua la dirittura della canna, se non in quanto il proprio peso la farebbe declinar da tal dirittura verso terra." " Sagr. Talche la dirittura della canna e la regolatrice del moto della palla: n~ fuori di tal linea si muove, o muoverebbe, se '1 peso proprio non la facesse delinare in gih..."-Dialogo sopra i due massimli sistemi del mo n do. " Sagr. But if the gun were not placed in the perpendicular, but were inclined in some direction; what then would be the motion of tie ball? Would it follow, perhaps, as in the other case, the perpendicular, and in returning fall also by the same line? " " Simpl. This it will not do, but having left the cannon it will follow its own motion in the straight line which is a contitnuation of the axis of the barrel, save in so far as its own weight shall cause it to deviate from that direction toward the earth." ' Sagr. So that the axis of the barrel is the regulator of the motion of the ball: and it neither does nor will move outside of that line unless its own weight causes it to drop downwards....' 566 THE SCIENCE OF MECHANICS. The physical instruction which I enjoyed was in all probability just as bad and just as dogmatic as it was the fortune of my older critics and colleagues to enjoy. The principle of inertia was then enunciated as a dogma which accorded perfectly with the system. I could understand very well that disregard of all obstacles to motion led to the principle, or that it must be discovered, as Appelt says, by abstraction; nevertheless, it always remained remote and within the comprehension of supernatural genius only. And where was the guarantee that with the removal of all obstacles the diminution of the velocity also ceased? Poske (loc. cit., p. 395) is of the opinion that Galileo, to use a phrase which I have repeatedly employed, "discerned" or "Iperceived" the principle immediately. But what is this discerning? Enquiring man looks here and looks there, and suddenly catches a glimpse of something he has been seeking or even of something quite unexpected, that rivets his attention. Now, I have shown how this "discerning" came about and in what it consisted. Galileo runs his eye over several different uniformly retarded motions, and suddenly picks out from among them a uniform, in"Attendere insuper licet, quod velocitatis gradus, quicunque in mobili reperiatur, est in illo suapte natura indelebiliter impressus, dum externae causae accelerationis, aut retardationis tollantur, quod in solo horizontali piano contingit: nam in planis declivibus adest jam causa accelerationis majoris, in acclivibus vero retardationis. Ex quo pariter sequitur, rnotum in horizontali esse quoque aeternum: si enim est aequabilis, non debiliatur, aut remittitur, et multo minus tollitur."-Discorsie dimostrazioni matematiche. Dialogo terzo. " Moreover, it is to be remarked that the degree of velocity a body has is indestructibly impressed in it by its own nature, provided external causes of acceleration or retardation are wanting,-which happens only on horizontal planes: for on descending planes there is greater acceleration, and on ascending planes retardation. Whence it follows that motion in a horizontal plane is perpetual: for if it remains the same, it is not diminished, or abated, much less abolished." APPENDI X. 567 finitely continued motion, of so peculiar a character that if it occurred by itself alone it would certainly be regarded as something altogether different in kind. But a very minute variation of the inclination transforms this motion into a finite retarded motion, such as we have frequently met with in our lives. And now, no more difficulty is experienced in recognising the identity between all obstacles to motion and retardation by gravity,--wherewith the ideal type of uninfluenced, infinite, uniform motion is gained. As I read this passage of Galileo's while still a young man, a new light concerning the necessity of this ideal link in our mechanics, entirely different from that of the dogmatic exposition, flashed upon me. I believe that every one will have the same experience who will approach this passage without prior bias. I have not the least doubt that Galileo above all others experienced that light. May my critics see to it how their assent also is to be avoided. I have now another important point to discuss in opposition to C. Neumann,* whose well-known publication on this topic preceded minet shortly. I contended that the direction and velocity which is taken into account in the law of inertia had no comprehensible meaning if the law was referred to "absolute space." As a matter of fact, we can metrically determine direction and velocity only in a space of which the points are marked directly or indirectly by given bodies. Neumann's treatise and my own were successful in directing attention anew to this point, which *Die Principien der Galiiei-Newton'schen Theorie, Leipzig, 1870. t Erhaltung der Arbeit, Prague, 1872. (Translated in part in the article on "The Conservation of Energy," Popular Scientific Lectures, third edition, Chicago, 1898. 568 THE SCIENCE OF ME CHANICS. had already caused Newton and Euler much intellectual discomfort; yet nothing more than partial attempts at solution, like that of Streintz, have resulted. I have remained to the present day the only one who insists upon referring the law of inertia to the earth, and in the case of motions of great spatial and temporal extent, to the fixed stars. Any prospect of coming to an understanding with the great number of my critics is, in consideration of the profound differences of our points of view, very slight. But so far as I have been able to understand the criticisms to which my view has been subjected, I shall endeavor to answer them. H6fler is of the opinion that the existence of "absolute motion" is denied, because it is held to be "inconceivable." But it is a fact of "more painstaking self-observation" that conceptions of absolute motion do exist. Conceivability and knowledge of absolute motion are not to be confounded. Only the latter is wanting here (loc. cil., pages 120, 164).. Now, it is precisely with knowledge that the natural inquirer is concerned. A thing that is beyond the ken of knowledge, a thing that cannot be exhibited to the senses, has no meaning in natural science. I have not the remotest desire of setting limits to the imagination of men, but I have a faint suspicion that the persons who imagine they have conceptions of "absolute motions," in the majority of cases have in mind the memory pictures of some actually experienced relative motion; but let that be as it may, for it is in any event of no consequence. I maintain even more than Hbfler, viz., that there exist sensory illusions of absolute motions, which can subsequently be reproduced at any time. Every one that has repeated my / //'I' / DIX.. 569 experiments on the sensations of movement has experience d the full sensory power of such illusions. One imagines one is flying off with one's entire environment, which remains at relative rest with respect to the body; or that one is rotating in a space that is distinguished by nothing that is tangible. But no measure can be applied to this space of illusion; its existence cannot be proved to another person, and it cannot be employed for the metrical and conceptual description of the facts of mechanics; it has nothing to do with the space of geometry.* Finally, when H0fler (loc. cit., p. 133) brings forward the argument that "in every relative motion one at least of the bodies moving with reference to each other must be affected with absolute motion,"-I can only say that for the person who considers absolute motion as meaningless in physics, this argument has no force whatever. But I have no further concern here with philosophical questions. To go into details as HIfler has in some places (loc. cit., pp. 124-126) would serve no purpose before an understanding had been reached on the main question. Heymans (loc. cit., pp. 412, 448) remarks that an inductive, empirical mechanics could have arisen, but that as a matter of fact a different mechanics, based on the non-empirical concept of absolute motion, has arisen. The fact that the principle of inertia has always been suffered to hold for absolute motion which is nowhere demonstrable, instead of being re* I flatter myself on being able to resist the temptation to infuse lightness into a serious discussion by showing its ridiculous side, but in reflecting on these problems I was involuntarily forced to think of the question which a very estimable but eccentric man once debated with me as to whether a yard of cloth in one's dreams is as long as a real yard of cloth.- Is the dream-yard to be really introduced into mechanics as a standard of measurement? 570 THE SCIENCE OF MECHANICS. garded as holding good for motion with respect to some actually demonstrable system of co-ordinates, is a problem which is almost beyond power of solution by the empirical theory. Heymans regards this as a problem that can have a metaphysical solution only. In this I cannot agree with Heymans. He admits that relative motions only are given in experience. With this admission, as with that of the possibility of an empirical mechanics, I am perfectly content. The rest, I believe, can be explained simply and without the aid of metaphysics. The first dynamic principles were unquestionably built up on empirical foundations. The earth was the body of reference; the transition to the other co-ordinate systems took place very gradually. Huygens saw that he could refer the motion of impinging bodies just as easily to a boat on which they were placed, as to the earth. The development of astronomy preceded that of mechanics considerably. When motions were observed that were at variance with known mechanical laws when referred to the earth, it was not necessary immediately to abandon these laws again. The fixed stars were present and ready to restore harmony as a new system of reference with the least amount of changes in the concepts. Think only of the oddities and difficulties which would have resulted if in a period of great mechanical and physical advancement the Ptolemaic system had been still in vogue,-a thing not at all inconceivable. But Newton referred all of mechanics to absolute space! Newton is indeed a gigantic personality; little worship of authority is needed to succumb to his influence. Yet even his achievements are not exempt from criticism. It appears to be pretty much one and APPENDIX. 57I the same thing whether we refer the laws of motion to absolute space, or enunciate them in a perfectly abstract form; that is to say, without specific mention of any system of reference. The latter course is un precarious and even practical; for in treating special cases every student of mechanics looks for some ser viceable system of reference. But owing to the fact that the first course, wherever there was any real issue at stake, was nearly always interpreted as having the same meaning as the latter, Newton's error was fraught with much less danger than it would otherwise have been, and has for that reason maintained itself so long. It is psychologically and historically intelligible that in an age deficient in epistemological critique empirical laws should at times have been elaborated to a point where they had no meaning. It cannot therefore be deemed advisable to make metaphysical problems out of the errors and oversights of our scientific forefathers, but it is rather our duty to correct them, be they small people or great. I would not be understood as saying that this has never happened. Petzoldt (loc. cit., pp. 192 et seq.), who is in accord with me in my rejection of absolute motion, appeals to a principle of Avenarius,* by a consideration of which he proposes to remove the difficulties involved in the problem of relative motion. I am perfectly familiar with the principle of Avenarius, but I cannot understand how all the physical difficulties involved in the present problem can be avoided by referring motions to one's own body. On the contrary, in considering physical dependencies abstraction must *Der menschliche WIltbe-,riff, Leipzig, 1891, p. 130. 572 7TE SCIENCE OF MECHANICS. be made from one's own body, so far as it exercises any influence.* The most captivating reasons for the assumption of absolute motion were given thirty years ago by C. Neumann (loc. cit., p. 27). If a heavenly body be conceived rotating about its axis and consequently subject to centrifugal forces and therefore oblate, nothing, so far as we can judge, can possibly be altered in its condition by the removal of all the remaining heavenly bodies. The body in question will continue to rotate and will continue to remain oblate. But if the motion be relative only, then the case of rotation will not be distinguishable from that of rest. All the parts of the heavenly body are at rest with respect to one another, and the oblateness would necessarily also disappear with the disappearance of the rest of the universe. I have two objections to make here. Nothing appears to me to be gained by making a meaningless assumption for the purpose of eliminating a contradiction. Secondly, the celebrated mathematician appears to me to have made here too free a use of intellectual experiment, the fruitfulness and value of which cannot be denied. When experiment-g in thought, it is permissible to modify unimportant circumstances in order to bring out new features in a given case; but it is not to be antecedently assumed that the universe is without influence on the phenomenon here in question. If it is eliminated and contradictions still result, certainly this speaks in favor of the importance of relative motion, which, if it involves difficulties, is at least free from contradictions. *Analyse der Emhpfindungen, zweite Auflage, Jena. 1900, pp. r1, 12. 33, 38, 208; English translation, Chicago, The Open Court Pub. Co., 1897, pp. 13 et seq. A PPENDIIX. 5 573 Volkmann (loc. cit., p. 53) advocates an absolute orientation by means of the ether. I have already spoken on this point (comp. pp. 230, 547), but I am extremely curious to know how one ether particle is to be distinguished from another. Until some means of distinguishing these particles is found, it will be preferable to abide by the fixed stars, and where these forsake us to confess that the true means of orientation is still to be found. Taking everything together, I can only say that I cannot well see what is to be altered in my expositions The various points stand in necessary connexion. After it has been discovered that the behavior of bodies toward one another is one in which accelerations are determined,-a discovery which was twice formulated by Galileo and Newton, once in a general and again in a special form as a law of inertia,-it is possible to give only one rational definition of mass, and that a purely dynamical definition. It is not at all, in my judgment, a matter of taste.* The concept of force and the principle of action and reaction follow of themselves. And the elimination of absolute motion is equivalent to the elimination of what is physically meaningless. It would be not only taking a very subjective and short-sighted view of science, but it would also be foolhardy in the extreme, were I to expect that my views in their precise individual form should be incorporated without opposition into the intellectual systems of my contemporaries. The history of science teaches that the subjective, scientific philoso* My definition of mass takes a more organic and more natural place i Hertz's mechanics than his own, for it contains implicitly the germ of his "fundamental law." 574 THE SCIENCE OF MECf/ANICS. phies of individuals are constantly being corrected and obscured, and in the philosophy or constructive image of the universe which humanity gradually adopts, only the very strongest features of the thoughts of the greatest men are, after some lapse of time, recognisable. It is merely incumbent on the individual to outline as distinctly as possible the main features of his own view of the world. XXIII. (See page 273.) Although signal individual performances in science cannot be gainsaid to Descartes, as his studies on the rainbow and his enunciation of the law of refraction show, his importance nevertheless is contained rather in the great general and revolutionary ideas which he promulgated in philosophy, mathematics, and thie natural sciences. The maxim of doubting everything that has hitherto passed for established truth cannot be rated too high; although it was more observed and exploited by his followers than by himself. Analytical geometry with its modern methods is the outcome of his idea to dispense with the consideration of all the details of geometrical figures by the application of algebra, and to reduce everything to the consideration of distances. He was a pronounced enemy of occult qualities in physics, and strove to base all physics on mechanics, which he conceived as a pure geometry of motion. He has shown by his experiments that he regarded no physical problem as insoluble by this method. He took too little note of the fact that mechanics is possible only on the condition that the positions of the bodies are determined in their dependence on one another by APPENDIX. 575 a relation of force, by a function of time; and Leibnitz frequently referred to this deficiency. The mechanical concepts which Descartes developed with scanty and vague materials could not possibly pass as copies of nature, and were pronounced to be phantasies even by Pascal, Huygens, and Leibnitz. It has been remarked, however, in a former place, how strongly Descartes's ideas, in spite of these facts, have persisted to the present day. He also exercises a powerful influence upon physiology by his theory of vision, and by his contention that animals were machines,a theory which he naturally had not the courage to extend to human beings, but by which he anticipated the idea of reflex motion (compare Duhem, L'lvolution des thlories piysiques, Louvain, 1896). XXIV. (See page 378.) To the exposition given on pages 377 and 378, in the year 1883, I have the following remarks to add. It will be seen that the principle of least action, like all other minimum principles in mechanics, is a simple expression of the fact that in the instances in question precisely so much happens as possibly can happen under the circumstances, or as is determined, viz., uniquely determined, by them. The deduction of cases of equilibrium from unique determination has already been discussed, and the same question will be considered in a later place. With respect to dynamic questions, the import of the principle of unique determination has been better and more perspicuously elucidated than in my case by J. Petzoldt in a work entitled Maxima, Minima und Oekonomie (Altenburg, 1891). He says (loc. cit., page Ii): "In the case of 576 THE SCIENCE OF MfECHANICS. all motions, the paths actually traversed admit of being interpreted as signal instances chosen from an infinite number of conceivable instances. Analytically, this has no other meaning than that expressions may always be found which yield the differential equations of the motion when their variation is equated to zero, -for the variation vanishes only when the integral assumes a unique value." As a fact, it will be seen that in the instances treated at pages 377 and 378 an increment of velocity is uniquely determined only in the direction of the force, while an infinite number of equally legitimate incremental components of velocity at right angles to the force are conceivable, which are, however, for the reason given, excluded by the principle of unique determination. I am in entire accord with Petzoldt when he says: " The theorems of Euler and Hamilton, and not less that of Gauss, are thus nothing more than analytic expressions for the fact of experience that the phenomena of nature are uniquely determined." The uniqueness of the minimum is determinative. I should like to quote here, from a note which I published in the November number of the Prague Lotos for 1873, the following passage: "The static and dynamical principles of mechanics may be expressed as isoperimetrical laws. The anthropomorphic conception is, however, by no means essential, as may be seen, for example, in the principle of virtual velocities. If we have once perceived that the work A determines velocity, it will readily be seen that where work is not done when the system passes into all adjacent positions, no velocity can be acquired, and consequently that equilibrium obtains. APPE NDIX. 577 The condition of equilibrium will therefore be 8A = 0; where A need not necessarily be exactly a maximum or minimum. These laws are not absolutely restricted to mechanics; they may be of very general scope. If the change in the form of a phenomenon B be dependent on a phenomenon A, the condition that B shall pass over into a certain form will be A -= 0." As will be seen, I grant in the foregoing passage that it is possible to discover analogies for the principle of least action in the most various departments of physics without reaching them through the circuitous course of mechanics. I look upon mechanics, not as the ultimate explanatory foundation of all the other provinces, but rather, owing to its superior formal development, as an admirable prototype of such an explanation. In this respect, my view differs apparently little from that of the majority of physicists, but the difference is an essential one after all. In further elucidation of my meaning, I should like to refer to the discussions which I have given in my Princizpes of Heat (particularly pages 192, 318, and 356, German edition), and also to my article "On Comparison in Physics" (Popular Scientific Lectures, English translation, page 236). Noteworthy articles touching on this point are: C. Neumann, "Das Ostwald'sche Axiom des Energieumsatzes" (Berichte der k. sdcks. Gesel/schaft, 1892, p. 184), and Ostwald, "Ueber das Princip des ausgezeichneten Falles" (loc. cit., 1893, p. 600). XXV. xxv. (See page 480.) The Ausdeh/nungslehre of 1844, in which Grassmann expounded his ideas for the first time, is in many re 578 7'11 SCIENCE OF MECHANICS. spects remarkable. The introduction to it contains epistemological remarks of value. The theory of spatial extension is here developed as a general science, of which geometry is a special tri-dimensional case; and the opportunity is taken on this occasion of submitting the foundations of geometry to a rigorous critique. The new and fruitful concepts of the addition of line-segments, multiplication of line-segments, etc., have also proved to be applicable in mechanics. Grassmann likewise submits the Newtonian principles to criticism, and believes he is able to enunciate them in a single expression as follows: "The total force (or total motion) which is inherent in an aggregate of material particles at any one time is the sum of the total force (or total motion) which has inhered in it at any former time, and all the forces that have been imparted to it from without in the intervening time; provided all forces be conceived as line-segments constant in direction and in length, and be referred to points which have equal masses." By force Grassmann understands here the indestructibly impressed velocity. The entire conception is much akin to that of Hertz. The forces (velocities) are represented as line-segments, the moments as surfaces enumerated in definite directions, etc.,-a device by means of which every development takes a very concise and perspicuous form. But Grassmann finds the main advantage of his procedure in the fact that every step in the calculation is at the same time the clear expression of every step taken in the thought; whereas, in the common method, the latter is forced entirely into the background by the introduction of three arbitrary co-ordinates. The difference between the analytic and the synthetic method is again done away APPENDIX. 579 with, and the advantages of the two are combined. The kindred procedure of Hamilton, which has been illustrated by an example on page 528, will give some idea of these advantages. XXVI. (See page 485.) In the text I have employed the term "cause" in the sense in which it is ordinarily used. I may add that with Dr. Carus,* following the practice of the German philosophers, I distinguish "cause," or Realgrund, from Erkenntnissgrund. I also agree with Dr. Carus in the statement that "the signification of cause and effect is to a great extent arbitrary and depends much upon the proper tact of the observer." t The notion of cause possesses significance only as a means of provisional knowledge or orientation. In any exact and profound investigation of an event the inquirer must regard the phenomena as dependent on one another in the same way that the geometer regards the sides and angles of a triangle as dependent on one another. He will constantly keep before his mind, in this way, all the conditions of fact. XXVII. (See page 494.) My conception of economy of thought was developed out of my experience as a teacher, out of the work of practical instruction. I possessed this conception as early as 1861, when I began my lectures as Privat-Docent, and at the time believed that I was * See his Grund, Ursache und Zweck, R. v. Grumbkow, Dresden, 1881, and his Fundamental Problems, pp. 79-91, Chicago: The Open Court Publishing Co., 1891. t Fundamental Problems, p. 84. 580 THE SCIENCE OF MECHANICS. in exclusive possession of the principle,-a conviction which will, I think, be found pardonable. I am now, on the contrary, convinced that at least some presentiment of this idea has always, and necessarily must have, been a common possession of all inquirers who have ever made the nature of scientific investigation the subject of their thoughts. The expression of this opinion may assume the most diverse forms; for example, I should most certainly characterise the guiding theme of simplicity and beauty which so distinctly marks the work of Copernicus and Galileo, not only as aesthetical, but also as economical. So, too, Newton's Reguice }hilosophandi are substantially influenced by economical considerations, although the economical principle as such is not explicitly mentioned. In an interesting article, "An Episode in the History of Philosophy," published in The Open Court for April 4, 1895, Mr. Thomas J. McCormack has shown that the idea of the economy of science was very near to the thought of Adam Smith (Essays). In recent times the view in question has been repeatedly though diversely expressed, first by myself in my lecture Ueber die Erhaltung der Arbeit (1875), then by Clifford in his Lectures and Essays (1872), by Kirchhoff in his Mechanics (1874), and by Avenarius (1876). To an oral utterance of the political economist A. Herrmann I have already made reference in my Erhaltung der Arbeit (p. 55, note 5); but no work by this author treating especially of this subject is known to me. I should also like to make reference here to the supplementary expositions given in my Popular Scientific Lectures (English edition, pages 186 et seq.) and in my Principles of Heat (German edition, page 294). In the latter work, the criticisms of Petzoldt (Viertel APPENDIX. 581 jahrsschrift fur wissenschaftliche P/ilosophie, I891) are considered. Husserl, in the first part of his work, Logische Untersuchungen (1900), has recently made some new animadversions on my theory of mental economy; these are in part answered in my reply to Petzoldt. I believe that the best course is to postpone an exhaustive reply until the work of Husserl is completed, and then see whether some understanding cannot be reached. For the present, however, I should like to premise certain remarks. As a natural inquirer, I am accustomed to begin with some special and definite inquiry, and allow the same to act upon me in all its phases, and to ascend from the special aspects to more general points of view. I followed this custom also in the investigation of the development of physical knowledge. I was obliged to proceed in this manner for the reason that a theory of theory was too difficult a task for me, being doubly difficult in a province in which a minimum of indisputable, general, and independent truths from which everything can be deduced is not furnished at the start, but must first be sought for. An undertaking of this character would doubtless have more prospect of being successful if one took mathematics as one's subject-matter. I accordingly directed my attention to individual phenomena: the adaptation of ideas to facts, the adaptation of ideas to one another,* mental *Popular Scientific Lectures, English edition, pp. 244 et seq., where the adaptation of thoughts to one another is described as the object of theory proper. Grassmann appears to me to say pretty much the same in the introduction to his Ausdehnungslehre of 1844, page xix: "The first division of all the sciences is that into real and formal, of which the real sciences depict reality in thought as something independent of thought, and find their truth in the agreement of thought with that reality; the formal sciences, on the other hand, have as their object that which has been posited by thought and itself, find their truth in the agreement of the mental processes with one another." 582 THE SCIENCE OF MECHANICS. economy, comparison, intellectual experiment, the constancy and continuity of thought, etc. In this inquiry, I found it helpful and restraining to look upon every-day thinking and science in general, as a biological and organic phenomenon, in which logical thinking assumed the position of an ideal limiting case. I do not doubt for a moment that the investigation can be begun at both ends. I have also described my efforts as epistemological sketches.* It may be seen from this that I am perfectly able to distinguish between psychological and logical questions, as I believe every one else is who has ever felt the necessity of examining logical processes from the psychological side. But it is doubtful if any one who has carefully read even so much as the logical analysis of Newton's enunciations in my Mechanics, will have the temerity to say that I have endeavored to erase all distinctions between the "blind" natural thinking of every-day life and logical thinking. Even if the logical analysis of all the sciences were complete, the biologico-psychological investigation of their development would continue to remain a necessity for me,which would not exclude our making a new logical analysis of this last investigation. If my theory of mental economy be conceived merely as a teleological and provisional theme for guidance, such a conception does not exclude its being based on deeper foundations,t but goes toward making it so. Mental economy is, however, quite apart from this, a very clear logical ideal which retains its value even after its logical analysis has been completed. The systematic form of a science can be deduced from the same prin*Princizles of Heat, Preface to the first German edition. t Analysis of the Sensations, second German edition, pages 64-65. APPENDIX. 583 ciples in many different manners, but some one of these deductions will answer to the principle of economy better than the rest, as I have shown in the case of Gauss's dioptrics.* So far as I can now see, I do not think that the investigations of Husserl have affected the results of my inquiries. As for the rest, I must wait until the remainder of his work is published, for which I sincerely wish him the best success. When I discovered that the idea of mental economy had been so frequently emphasised before and after my enunciation of it, my estimation of my personal achievement was necessarily lowered, but the idea itself appeared to me rather to gain in value on this account; and what appears to Husserl as a degradation of scientific thought, the association of it with vulgar or "blind" (?) thinking, seemed to me to be precisely an exaltation of it. It has outgrown the scholar's study, being deeply rooted in the life of humanity and reacting powerfully upon it. XXVIII. (See page 497.) The paragraph on page 497, which was written in 1883, met with little response from the majority of physicists, but it will be noticed that physical expositions have since then closely approached to the ideal there indicated. Hertz's "Investigations on the Propagation of Electric Force" (1892) affords a good instance of this description of phenomena by simple differential equations. *Principles of Heat, German editien, page 394. 584 THE SCIENCE, OF JIL'CIANICS. XXIX. (See page 501.) In Germany, Mayer's works at first met with a very cool, and in part hostile, reception; even difficulties 9f publication were encountered; but in England they found more speedy recognition. After they had been almost forgotten there, amid the wealth of new facts being brought to light, attention was again called to them by the lavish praise of Tyndall in his book Heat a M4ode of Motion (1863). The consequence of this was a pronounced reaction in Germany, which reached its culminating point in Diihring's work Robert Mayer, the Galileo of/the Nineteenlh Century (1878). It almost appeared as if the injustice that had been done to Mayer was now to be atoned for by injustice towards others. But as in criminal law, so here, the sum of the injustice is only increased in this way, for no algebraic cancelation takes place. An enthusiastic and thoroughly satisfactory estimate of Mayer's performances was given by Popper in an article in Ausland (1876, No. 35), which is also very readable from the many interesting epistemological apergus that it contains. I have endeavored (Princi>ples of Heat) to give a thoroughly just and sober presentation of the achievements of the different inquirers in the domain of the mechanical theory of heat. It appears from this that each one of the inquirers concerned made some distinctive contribution which expressed their respective intellectual peculiarities. Mayer may be regarded as the philosopher of the theory of heat and energy; Joule, who was also conducted to the principle of energy by philosophical considerations, fur APPENDIX. 585 nishes the experimental foundation; and Helmholtz gave to it its theoretical physical form. Helmholtz, Clausius, and Thomson form a transition to the views of Carnot, who stands alone in his ideas. Each one of the first-mentioned inquirers could be eliminated. The progress of the development would have been retarded thereby, but it would not have been checked (compare the edition of Mayer's works by Weyrauch, Stuttgart, 1893). XXX. (See page 504.) The principle of energy is only briefly treated in the text, and I should like to add here a few remarks on the following four treatises, discussing this subject, which have appeared since 1883: Die physikalischen Grundstize der elektrischen Krafliibertragung, by J. Popper, Vienna, 1883; Die Lehre von der Energie, by G. Helm, Leipsic, 1887; Das Princip der Erhaltung der Energie, by M. Planck, Leipsic, 1887; and Das Problem der Conlinuitdt in der Mathematik und Mechanik, by F. A. Miiller, Marburg, 1886. The independent works of Popper and Helm are, in the aim they pursue, in perfect accord, and they quite agree in this respect with my own researches, so much so in fact that I have seldom read anything that, without the obliteration of individual differences, appealed in an equal degree to my mind. These two authors especially meet in their attempt to enunciate a general science of energetics; and a suggestion of this kind is also found in a note to my treatise Ueber die Erhaltung der Arbeit, page 54. Since then "energetics" has been exhaustively treated by Helm, Ostwald, and others. 586 THE SCIENCE OF MECHANICS. In 1872, in this same treatise (pp. 42 et seqq.), I showed that our belief in the principle of excluded perpetual motion is founded on a more general belief in the unique determination of one group of (mechanical) elements, afly..., by a group of different elements, xyz... Planck's remarks at pages 99, 133, and 139 of his treatise essentially agree with this; they are different only in form. Again, I have repeatedly remarked that all forms of the law of causality spring from subjective impulses, which nature is by no means compelled to satisfy. In this respect my conception is allied to that of Popper and Helm. Planck (pp. 21 et seqq., 135) and Helm (p. 25 et seqq.) mention the "metaphysical" points of view by which Mayer was controlled, and both remark (Planck, p. 25 et seqq., and Helm, p. 28) that also Joule, though there are no direct expressions to justify the conclusion, must have been guided by similar ideas. To this last I fully assent. With respect to the so-called "metaphysical" points of view of Mayer, which, according to Helmholtz, are extolled by the devotees of metaphysical speculation as Mayer's highest achievement, but which appear to Helmholtz as the weakest feature of his expositions, I have the following remarks to make. With maxims, such as, "Out of nothing, nothing comes," "The effect is equivalent to the cause," and so forth, one can never convince another of anything. How little such empty maxims, which until recently were admitted in science, can accomplish, I have illustrated by examples in my treatise Die Erhaltung der Arbeit. But in Mayer's case these maxims are, in my judgment, not weaknesses. On the contrary, they are with him the expression of a powerful instinctive APPENDIX. 587 yearning, as yet unsettled and unclarified, after a sound, substantial conception of what is now called energy. This desire I should not exactly call metaphysical. We now know that Mayer was not wanting in the conceptual power to give to this desire clearness. Mayer's attitude in this point was in no respect different from that of Galileo, Black, Faraday, and other great inquirers, although perhaps many were more taciturn and cautious than he. I have touched upon this point before in my Analysis of the Sensations, Jena, 1886, English translation, Chicago, 1897, p. 174 et seqq. Aside from the fact that I do not share the Kantian point of view, in fact, occupy no metaphysical point of view, not even that of Berkeley, as hasty readers of my last-mentioned treatise have assumed, I agree with F. A. Miiller's remarks on this question (p. 104 et seqq.). For a more exhaustive discussion of the principle of energy see my Principles of Heat. CHRONOLOGICAL TABLE OF A FEW EMINENT INQUIRERS AND OF THEIR MORE IMPORTANT MECHANICAL WORKS. ARCHIMEDES (287-212 B. C.). A complete edition of his works was published, with the commentaries of Eutocius, at Oxford, in 1792; a French translation by F. Peyrard (Paris, 1808); a German translation by Ernst Nizze (Stralsund, 1824). LEONARDO DA VINCI (1452-1519). Leonardo's scientific manuscripts are substantially embodied in H. Grothe's work, " Leonardo da Vinci als Ingenieur und Philosoph " (Berlin, 1874). GUIDO UBALDI(O) e Marchionibus Montis (1545-1607). Mechanicorum Liber (Pesaro, 1577). S. STEVINUS (I548-16201). Begiinselen der Weegkonst (Leyden, 1585); IHypomnemala Mathlemnatica (Leyden, 1608). GALILEO (1564-1642). Discorsi e dimostrazioni matematiche (Leyden, 1638). The first complete edition of Galileo's writings was published at Florence (1842-1856), in fifteen volumes 8vo. KEPLER (1571-1630). Astronomia XNova (Prague, 1609); Harmonice AMundi (Linz, 1619); Stereometria Doliorum (Linz, 1615). Complete edition by Frisch (Frankfort, 1858). MARCUS MARCI (1595-1667). De Proportione AMotus (Prague, 1639). DESCARTES (1596-1650). Principia Philosophiic (Amsterdam, 1644). ROBERVAL (1602-1675). Sur la composition des mouvements. Anc. Medm. de l'Acad. de Paris. T. VI. GUERICKE (1602-1686). Experimenta Nova, ut Vocantur iMagdeburgica (Amsterdam, 1672). 590 THE SCIENCE OF IMECHANICS. FERMAT (16o01-665). Varia Opera (Toulouse, 1679). TORRICELLI (1608-1647). Opera Geometrica (Florence, 1644). WALLIS (1616-1703). Mechanica Sive de Motu (London, 1670). MARIOTTE (1620-1684). CEuvres (Leyden, 1717). PASCAL (1623-1662). Recit de la grande expdrience de l'dquilibre des liqueurs (Paris, 1648); Traild de d'equilibre des liqueurs et de la pesanteur de la masse de 'air. (Paris, 1662). BOYLE (1627-1691). Experimenta Phvsico Mechanica (London, 1660). HUYGENS (1629-1695). A Summary Account of the Laws of Motion. Philos. Trans. 1669; HJorologium Oscillatorium (Paris, 1673); Opuscula Posthuma (Leyden, 1703). WREN (1632-1723). Lex Naturce de Collisione Corporum. Philos. Trans. 1669. LAMI (1640-1715). Nouvelle manibre de ddmontrer les principaux theoremes des de'mens des mdcaniques (Paris, 1687). NEWTON (1642 -1726). Philosophie Naturalis Principia Mathematica (London, 1686). LEIBNITZ (1646-1716). Acta Eruditorum, 1686, 1695; Leibnitzii et Jok. Bernoullii Comercium Epistolicum (Lausanne and Geneva, 1745). JAMES BERNOULLI (1654-1705). Opera Omnia (Geneva, 1744). VARIGNON (1654-1722). Projet d'une nouvelle mecanique (Paris, 1687). JOHN BERNOULLI (1667-1748). Acta Erudit. 1693; Opera Omnia (Lausanne, 1742). MAUPERTUIS (1698-1759). Mim. de I'Acad. de Paris, 1740; Miem. de l'Acad. de Berlin, 1745, 1747; (Euvres (Paris, 1752). MACLAURIN (1698-1746). A Complete System of Fluxions (Edinburgh, 1742). DANIEL BERNOULLI (1700-1782). Comment. Acad. Petrop., T. I. Hydrodynamica (Strassburg, 1738). EULER (1707-1783). Mechanica sive Motus Scientia (Petersburg, 1736); Methodus Inveniendi Lineas Curvas (Lausanne, 1744). CIHRONVOL OGICAL TABLE. 59' 591 Numerous articles in the volumes of the Berlin and St. Petersburg academies. CLAIRAUT (1713-1765). Thr1W ie de la figure de hi ferre (Paris, 1743). D'ALEMBERT (1717-1783). Ti-ait" ate (l/ynanliquzh (Paris, '743). LAGRANGE (1736-1813). Lssai(' line n01171r.1' noelean/hode jour; - t/erniner les ia--inia et mnainima. iMisc. Taurin. 1762; il/ceanil'quel analy/tique (Paris, 1788). LAPLACE (1749-I827). ille'anique ctieste (Paris, 1799). FOURIER (1768-1830). T/lhe'o;ie analy/lique et ie c/ia/eur (Paris, 1822). GAUSS (1777-1855). De Figaro F/uiat'd uni i n S/a/ue,-Euiiuil'wi. Coniment. Sociel. Cot//nj._~., 1828; N'eues I'rincip do- Jfechanikl (Crelle's Journal, IV, 1829); Intensi/as Vis Afagneticn- Terr;-estr-is ad il/ensurai Aso/utmlani A'cvoca/a (1833). Complete works (Gdttingen, 1863). PoINSoT (1777-1859). Iinientls ti', statique (Paris, i804). PONCELET (1788-1867). Cours dIe indcaniquc (Metz, 1826). BELANGER (1790-1874). Cours de nide'anique (Paris, 1847). M5BlUs (1790--1867). Stiatk (Leipsic, 1837). CORIOLIS (1792- 1843). [frail de nic'anique (Paris, 1829). C. G. J. JAcoBI (1804--1851)V- Vor/esungrCn ii/br Dynanilik, herausgegeben von Clebsch (Berlin, 1866). W. R. HAMILTON (1805-1865). Lectlures on Quaterlenions, 1853.Essays. GRASSMANN (1809-i877). Ausdehnungsle/ire (Leipsic, 1844). H. HERTZ (1857-1894). Principien der ecihzanic (Tleip, i, 1894). INDEX. Absolute, space, time, etc. (See the nouns.) Absolute units, 278, 284. Abstractions, 482. Acceleration, Galileo on, 131 et seq.; Newton on, 238; also 218, 230, 236, 243, 245, Action and reaction, 198-201, 242. Action, least, principle of, 364-380, 454; sphere of, 385. Adaptation,in nature, 452; of thoughts to facts, 6, 515 et seq., 581 et seq. Adhesion plates, 515 Aerometer, effect of suspended particles on, 208. Aerostatics. See Air. Affined, 166. Air, expansive force of, 127; quantitative data of, 124; weight of, 113; pressure of, 114 et seq.; nature of, 517 et seq. Air-pump, 122 et seq. Aitken, 525. Alcohol and water, mixture of, 384 et seq. Algebra, economy of, 486. Algebraical mechanics, 466. All, The, necessity of its consideration in research, 235, 461, 516. Analytical mechanics, 465-480. Analytic method, 466. Anaxagoras, 509, 517. Animal free in space, 290. Animistic points of view in mechanics, 461 et seq. Appelt, 566. Archimedes, on the lever and the centre of gravity, 8-11; critique of his deduction, 13-14, 513 et seq.; illustration of its value, io; on hydrostatics, 86-88; various modes of deduction of his hydrostatic principle, 104; illustration of his principle, io6. Archytas, 51o. Areas, law of the conservation of, 293-305. Aristotle, 509, 511, 517, 518. Artifices, mental, 492 et seq. Assyrian monuments, i. Atmosphere. See Air. Atoms, mental artifices, 492. Attraction, 246. Atwood's machine, 149. Automata, 511. Avenarius, R., x, 571, 580. Axiomatic certainties, 561. Babbage, on calculating machines, 488. Babo, Von, 150. Baliani, 524. Ballistic pendulum, 328. Balls, elastic, symbolising pressures in liquid, 419. Bandbox, rotation of, 301. Barometer, height of mountains determined by, 115, 117. Base. pressure of liquids on, 90, 99. Belanger, on impulse, 271. Benedetti, 520 et seq., 561. Berkeley, 587. Bernoulli, Daniel, his geometrical demoi stration of the parallelogram of forces, 40-42; criticism of Bernoulli s demonstration, 42-46: on the law of areas, 293; on the principle of vis viva, 343, 348; on 594 THE SCIENCE OF MEC/4HANICS. the velocity of liquid efflux, 403; his hydrodynamic principle, 408; on the parallelism of strata, 409; his distinction of hydrostatic and hydrodynamic pressure, 413. Bernoulli, James, on the catenary, 74; on the centre of oscillation, 311 et seq; on the brachistochrone, 426; on the isoperimetrical problems, 428 et seq.; his character, 428; his quarrel with John, 431; his Programma, 430. Bernoulli, John, his generalisation of the principle of virtual velocities, 56; on the catenary, 74; on centre of oscillation, 333, 335; on the prin ciple of vis viva, 343; on the analogies between motions of masses and light, 372; his liquid pendulum41o; on the brachistochrone, 425 et seq.; his character, 427; his quarrel with James, 431; his solution of the isoperimetrical problem, 431. Black, 124, 587. Boat in motion, Huygen's fiction of a, 315, 325, 570. Body, definition of, 506. Bolyai, 493. Bomb, a bursting, 293. Borelli, 533. Bosscha, J., 531. Bouguer, on the figure of the earth, 395. Boyle, his law, 125 et seq.; his investigations in airostatics, 123. Brachistochrone, problem of the, 425 et seq. Brahe, Tycho, on planetary motion, 187. Bruno, Giordano, his martyrdom, 446. Bubbles, 392. Bucket of water, Newton's rotating, 227, 232, 543. Budde, 547. Cabala, 489. Calculating machines, 488. Calculus, differential, 424; of variations, 436 et seq. Canal, fluid, equilibrium of, 396 et seq. Cannon and projectile, motion of, 291. Canton, on compressibility of liquids, 92. Carnot, his performances, 501, 585; his formula, 327. Carus, P., on cause, 516. Catenary, The, 74, 379, 425. Cauchy, 47. Causality, 483 et seq., 502. Cause and effect, economical character of the ideas, 485; equivalence of, 502, 503; Mach on, 555; Carus on, 579. Causes, efficient and final, 368. Cavendish, 124. Cells of the honeycomb, 453. Centimetre-gramme-second system, 285. Central, centrifugal, and centripetal force. See Force. Centre of gravity, 14 et seq." descent of, 52; descent and ascent of, 174 et seq., 408; the law of the conservation of the, 287-305. Centre of gyration, 334. Centre of oscillation, 173 et seq., 331 -335; Mersenne, Descartes, and Huygens on, 174 et seq.; relations of, to centre of gravity, 180-185; convertibility of, with point of suspension, 186. Centre of percussion, 327. Centripetal impulsion, 528 et seq. Chain, Stevinus's endless, 25 et seq., 500; motion of, on inclined plane, 347. Change, unrelated, 504. Character, an ideal universal, 481. Chinese language, 482. Church, conflict of science and, 446. Circular motion, law of, 169, 161. Clairaut, on vis viva, work, etc., 348; on the figure of the earth, 395; on liquid equilibrium, 396 et seq.; on level surfaces, etc., 398. Classen, J., 555. Classes and trades, the function of, in the development of science, 4. INDEX. 595 Claasius, 497, 499, 501, 585. Clifford, 580. Coefficients, indeterminate, Lagrange's, 471 et seq. Collision of bodies. See Impact. Colors, analysis of, 481. Column, a heavy, at rest, 258. Commandinus, 87. Communication, the economy of, 78. Comparative physics, necessity of, 498. Component of force, 34. Composition, of forces, see Forces; Gauss's principle and the, 364; notion of, 526. Compression of liquids and gases, 407. Conradus, Balthasar, 308. Conservation, of energy, 499 et seq. 585 et seq.; of quantity of motion, Descartes and Leibnitz on, 272, 274, purpose of the ideas of, 504. Conservation of momentum, of the centre of gravity, and of areas, laws of, 287-305; these laws, the expression of the laws of action and reaction and inertia, 303. Conservation of momentum and vis viva interpreted, 326 et seq. Constancy of quantity of matter, motion, and energy, theological basis of, 456. Constraint, 335, 352; least, principle of, 350-364, 550, 576. Continuity. the principle of, 140, 490 et seq, 565. Continuum, physico-mechanical, lo9. Coardinates, forces a function of, 397. See Force-function. Copernicus, 232, 457, 531,580. Coriolis, on vis viva and work, 272. Counter-phenomena, 503. Counter-work, 363, 366. Counting, economy of, 486. Courtivron, his law of equilibrium, 73. Ctesibius, his air-gun, iio, 511. Currents, oceanic, 302. Curtius Rufus, 210. Curve-elements, variation of, 432. Curves, maxima and minima of, 429. Cycloids, 143, 86, 379, 427. Cylinder, double, on a horizontal surface, 60; rolling on an inclined plane, 345. Cylinders, axal, symbolising the relations of the centres of gravity and oscillation, 183. D'Alembert, his settlement of the dispute concerning the measure of force, 149, 276; his principle, 331 -343. D'Arcy, on the law of areas, 293. Darwin, his theories, 452, 459. Declination from free motion, 352 -356. Deductive development of science, 421. Democritus, 518. Demonstration, the mania for, 18, 82; artificial, 82. Departure from free motion, 355. Derived units, 278. Descartes, on the measure of force, 148, 250, 270, 272-276; on quantity of motion, conservation of momentum, etc., 272 et seq.; character of his physical inquiries, 273, 528, 553, 574; his mechanical ideas, 250. Descent, cn inclined planes, 134 et seq.; law of, 137; in chords of circles, 138; vertical, motion of, treated by Hamilton's principle, 383; quickest, curve of, 426; of centre of gravity, 52, 174 et seq., 408. Description, a fundamental feature of science, 5, 555. Design, evidences of, in nature, 452. Determinants, economy of, 487. Determination, particular, 544. Determinative factors of physical processes, 76. Diels, 520. Differences, of quantities, their r61e in nature, 236; of velocities, 325. Differential calculus, 424. Differential laws, 255, 461. Dimensions, theory of, 279. Dioptrics, Gauss's economy of, 489. Disillusionment, due to insight, 77 596 THE SCIENCE OF MEChIANICS. Dream-yard, 569. Duliem, 575. Diihring, x, 352, 584. Dynamics, the development of the principles of, 128-255; retrospect of the development of, 245-255; founded by Galileo, 128; proposed new foundations for, 243; chief results of the development of, 245, 246; analytical, founded by Lagrange on the principle of virtual velocities, 467. Earth, figure of, 395 et seq. Economical character of analytical mechanics, 480. Economy in nature, 459. Economy of description, 5. Economy of science, 481-494. Economy of thought, the basis and essence of science, ix, 6, 481; of language, 481; of all ideas, 482; of the ideas cause and effect, 484; of the laws of nature, 485; of the law of refraction, 485; of mathematics, 486; of determinants, 487; of calculating machines, 488; of Gauss's dioptrics, moment of inertia, forcefunction, 489; history of Mach's conception of, 579, Efflux, velocity of liquid, 402 et seq. Egyptian monuments, i. Eighteenth century, character of, 458. Elastic bodies, 315, 317, 320. Elastic rod, vibrations of, 490. Elasticity, revision of the theory of, 496. Electromotor, Page's, 262; motion of a free, 296, et seq. Elementary laws, see Differential laws. Ellipsoid, triaxal, 73; of inertia, 186; central, 186. Empedocles, 509, 517. Encyclopaedists, French, 463. Energetics, the science of, 585. Energy, Galileo's use of the word, 271; conservation of, 499 et seq.; potential and kinetic, 272, 499; principle of, 585 et seq. See Vis viva. Enlightenment, the age of, 458. Epstein, 541. Equations, of motion, 342; of mechanics, fundamental, 270. Equilibrium, the decisive conditions of, 53; dependence of, on a maximum or minimum of work, 69; stable, unstable, mixed, and neutral equilibrium, 70-71; treated by Gauss's principle, 355; figures of, 393; liquid, conditions of, 386 et seq. Equipotential surfaces. See Level Surfaces. Ergal, 499. Error, our liability to, in the reconstruction of facts, 79. Ether, orientation by means of the, 573. Euler, on the loi de repos, 68; on moment of inertia, 179, 182, 186; on the law of areas, 293; his form of D'Alembert's principle, 337; on vis viva, 348; on the principle of least action, 368, 543, 576; on the isoperimetrical problems and the calculus of variations, 433 et seq; his theological proclivities, 449, 455; his contributions to analytical mechanics, 466; on absolute motion, 543, 568. Exchange of velocities in impact, 315. Experience, i et seq., 481, 490. Experimenting in thought, 523, 582. Experiments, 509, 514. Explanation, 6. Extravagance in nature, 459. Facts and hypotheses, 494, 496, 498. Falling bodies, early views of, 128; investigation of the laws of, 130 et seq., 520 et seq.; laws of, accident of their form, 247 et seq.; see Descent. Falling, sensation of, 206. Faraday, 124, 503, 530, 534, 587. Feelings, the attempt to explain them by motion, 506. Fermat, on the method of tangents, 423 Fetishism, in modern ideas, 463. INDEX. 597 Fiction of a boat in motion, Huygens's, 315. 325. Figure of the earth, 395 et seq. Films, liquid, 386, 392 et seq. Fixed stars, 543 et seq., 568. Flow, lines of, 400; of liquids, 416 et seq. Fluids, the principles of statics applied to, 86-11o. See Liquids. Fluid hypotheses, 496. Fippl, 535. Force, moment of, 37; the experiential nature of, 42-44; conception of; in statics, 84; general attributes of, 85; the Galilean notion of, 142, dispute concerning the measure of 148, 250, 270, 274-276; centrifugal and centripetal, 158 et seq.; Newton on, 192, 197, 238, 239; moving, 203, 243; resident, impressed, centripetal, accelerative, moving, 238, 239; the Newtonian measure of, 203, 239, 276; lines of, 400. Force-function, 398 et seq., 479, 489; Hamilton on, 350. Force-relations, character of, 237. Forces, the parallelogram of, 32, 33 -48, 243; principle of the composition and resolution of, 33-48, 197 et seq.; triangle of, io8; mutual independence of, 154; living, see Vis viva; Newton on the parallelogram of, 192, 197; impressed, equilibrated, effective, gained and lost, 336; molecular, 384 et seq.; functions of coardinates, 397, 402; central, 397; at a distance, 534 et seq. Formal development of science, 421. Formulae, mechanical, 269-286. Foucault and Toepler, optical method of, 125. Foucault's pendulum, 302. Fourier, 270, 526. Free rigid body, rotation of, 295. Free systems, mutual action of, 287. Friction, of minute bodies in liquids, 208; motion of liquids under, 416 et seq. Friedlander, P. and J., 547. Functions, mathematical, their office in science, 492. Fundamental equations of mechanics, 270. Funicular machine, 32. Funnel, plunged in water, 412; rotating liquid in, 303. " Galileo," name for unit of acceleration, 285. Galileo,his dynamical achievements, 128-155; his deduction of the law of the lever, 12, 514; his explanation of the inclined plane by the lever, 23; his recognition of the principle of virtual velocities, 51; his researches in hydrostatics, 90; his theory of the vacuum, 112 et seq; his discovery of the laws of falling bodies, 130 et seq, 522; his clock, 133; character of his inquiries, 140; his foundation of the law of inertia, 143, 524 et seq., 563 et seq; on the notion of acceleration, 145; tabular presentment of his discoveries, T47; on the pendulum and the motion of projectiles, 152 et seq., 525 et seq.; founds dynamics, 128; his pendulum, 162; his reasoning on the laws of falling bodies, 130, 131, 247; his favorite concepts, 250; on impact, 308-312; his struggle with the Church, 446; on the strength of materials, 451; does not mingle science with theology, 457; on inertia, 509; his predecessors, 520 et seq.; on gravitation, 533; on the tides, 537 et seq., 580, 587. Gaseous bodies, the principles of statics applied to, 110-127. Gases, flow of, 405; compression of, 407. Gauss, his view of the principle of virtual velocities, 76; on absolute units, 278; his principle of least constraint, 350-364, 550 et seq., 576; on the statics of liquids, 390; his dioptrics, 489. Gerber, Paul, 535. Gilbert, 462, 532, 533. Goldbeck, E., 532. Gomperz, 518. 598 THE SCIENCE OF MECHANICS. Grassi, 94. Grassmann, 480, 577 et seq., 581. Gravitation, universal, 190, 531 et seq., 533. Gravitational system of measures, 284-286. Gravity, centre of. See Centre of gravity. Greeks, science of, 509 et seq. Green's Theorem, o19. Guericke, his theological speculations, 448; his experiments in aerostatics, 117 et seq.; his notion of air, 118; his air-pump, 120: his airgun, 123. Gyration, centre of, 334. Halley, 448. Hamilton, on force-function, 350; his hodograph, 527; his principle, 380-- 384, 480, 576. Heat, revision of the theory of, 496. Helm, 585 et seq. Helmholtz, ix; on the conservation of energy, 499, 501, 585. Hemispheres, the Magdeburg, 122. Henke, R., 552. Hermann, employs a form of D'Alembert's principle, 337; on motion in a resisting medium, 435. Hero, his fountain, 411; on the motion of light, 422; on maxima and minima, 451, 511, 518 et seq. Herrmann, A.. 580. Hertz's system of Mechanics, 548 et seq., 583. Heymdns, 558 et seq., 569 et seq. Hiero, 86. Hipp, chronoscope of, 151. Hodograph, Hamilton's, 527. Htfler, 558 et seq., 568. Hblder, 0., 514. Hollow space, liquids enclosing, 392. Homogeneous, 279. Hooke, 532. Hopital, L', on the centre of oscillation, 331; on the brachistochrone, 426. Horror vacui, II2. Hume, on causality, 484. Husserl, 581 et seq. Huygens, dynamical achievements of, 155-187; his deduction of the law of the lever, 15-16; criticism of his deduction, 17-18; his rank as an inquirer, 155; character of his researches, 156 et seq.; on centrifugal and centripetal force, 158 et seq; his experiment with light balls in rotating fluids and his explanation of gravity, 162, 528 et seq.; on the pendulum and oscillatory motion, 162 et seq.; on the centre of oscillation, 173 et seq.; his principle of the descent and rise of the centre of gravity, 174; his achievements in physics, 186, 187, 530; his favorite concepts, 251; on impact, 313-327, 570; on the principle of vis viva, 343, 348; on the figure of the earth, 395; his optical researches, 425; does not mingle science and theology, 457, 575. Hydraulic ram, Montgolfier's, 411. Hydrodynamic pressure, 413. Hydrodynamics, 402-420. Hydrostatic pressure, 413. Hydrostatics, 384-402. Hypotheses and facts, 494. Images, constructive, 548 et seq. Inclined plane, the principle of the, 24-33, 515 et seq.; Galileo's deduction of its laws, 151; descent on, 354; movable on rollers. 357 et seq. Indeterminate coefficients, Lagrange's, 471 et seq. Inelastic bodies, 317, 318. Inertia, history and criticism of the law of, 141, 143, 232, 238, 520, 524 et seq., 542 et seq., 560 et seq, 563 et seq., 567 et seq.; moment of, 179, 182, 186, 489; bodies with variable moments of, 302; Newton on, 238, 243. Inertial system, 515. Impact, the laws of, 305-330; force of, compared with pressure, 312; in the Newtonian view, 317 et seq.; oblique, 327; Maupertuis's treatment of, 365. Impetus, 275. INDEX. 599 Impulse, 271. Inquirers, the great, character and value of their performances, 7; their different tasks, 76; their attitude towards religion, 457; their philosophy, 516. Inquiry, typical modes of, 317. Instinct, mechanical, importance of, 304. Instinctive knowledge, its cogency, origin, and character, I, 26-28, 83. Instincts, our animal, 463. Instruction, various methods of, 5. Integral laws, 255, 461. Intelligence, conception of, in nature, 461. Interdependence of the facts of nature, 502 et lassim. Internal forces, action of, on free systems, 289, 295. International language, 481. I'olation, 527. Isoperimetrical problems, 421-446; Euler's classification of, 433. Isothermal surfaces, 400. Jacobi, 76, 381, 459; on principle of least action, 371. Jellett, on the calculus of variations, 437 et seq. Johannesson, 547. Jolly, x, 532. Joule, 501, 584. Judgments, economical character of all, 483. Kant, on causality, 484. Kater, 186. Kepler, his laws of planetary motion, 187; possibility of his discovery of the laws of falling bodies, 248; on maxima and minima, 423; on astrology, 463; on gravity, 532; on tides, 538. Kilogramme, 281. Kilogramme-metre, 272. Kinetic energy, 272, 499. Kirchhoff, ix, 381, 555 et seq., 580. Knowledge, instinctive, 1, 26-28, 83; the communication of, the foundation of science, 4; the nature of, 5; the necessary and sufficient conditions of, 1o. Kinig, on the cells of the honeycomb, 453. Laborde, apparatus of, 15o. Lagrange, his deduction of the law of the lever, 13; his deduction of the principle of virtual velocities, 65 -67; criticism of this last deduction, 67-68; his form of D'Alembert's principle, 337; on vis viva, 349; on the principle of least action, 371; on the calculus of variations, 436 et seq.; emancipates physics from theology, 457; his analytical mechanics, x, 466, 553; his indeterminate coefficients, 471 et seq. Lami, on the composition of forces, 36. Lange, 542 et seq. Language, economical character of, 481; possibility of a universal, 482; the Chinese, 482. Laplace, 463 534. Lasswitz, 530. Lateral pressure, 103. Laws of nature, 502. Laws, rules for the mental reconstruction of facts, 83-84, 485. Least action, principle of, 364-380; its theological kernel, 454; analogies of, 577. Least constraint, principle of, 350 -364. Leibnitz, on the measure of force, 148, 250, 270, 274-276, 575; on quantity of motion, 274; on the motion of light, 425, 454; on the brachistochrone, 426; as a theologian, 449. Level surfaces, 98, 398 et seq. Lever, the principle of the, 8-25, 512 et seq.; "potential," 20; application of its principles to the explanation of the other machines, 22; its laws deduced by Newton's principles, 263-267; conditions of its rigidity, 96; Maupertuis's treatment of, 366. Libraries, stored up experience, 481. Light, motion of, 422, 424, 426; Mau 600 THE SCIENCE OF MECHANICS. pertuis on motion of, 367; motion of, in refracting media, 374-376, 377 -379; its minimal action explained, 459. Limiting cases, 565. Lindeliff, 437. Lippich, apparatus of, 150. Liquid efflux, velocity of, 402. Liquid-head, 403, 416. Liquid, rotating in a funnel, 303. Liquids, the statics of, 86-1no; the dynamics of, 402-420; fundamental properties of, 91; compressibility of, 92; equilibrium of, subjected to gravity, 96; immersed in liquids, pressure of, 105; lateral pressure of, 103; weightless, 384 et seq.; compression of, 407; soniferous, vibrations of, 407; mobile, 407; motion of viscous, 416. Living forces. See Vis viva. Living power, 272. Lobatch6vski, 493. Locomotive, oscillations of the body of, 292. Luther, 463. MacGregor, J. G., 547. Mach, history of his views, 555 et seq.; his definition of mass, 558 et seq.; his theory of the development of physical knowledge, 581 et seq.; his treatment of the law of inertia, 560 et seq. Machines, the simple, 8 et seq. Maclaurin on the cells of the honeycomb, 453; his contributions to analytical mechanics, 466. Magnus, Valerianus, 117. Manometer, statical, 123. Maraldi, on the honeycomb, 453. Marci, Marcus, 305-308. Mariotte, his law, 125; his apparatus and experiments, 126 et seq.; on impact, 313. Mass-areas, 295. Mass, criticism of the concept of, 216 -222; Newton on, and as quantity of matter, 192, 194, 217, 238, 251, 536 et seq.; John Bernoulli on, 251; as a physical property, 194; distin guished from weight, 195; measurable by weight, 195, 220; scientific definition of, 218 et seq., 243, 540 et seq., 558 et seq., 573; involves principle of reaction, 220. Mass, motion of a, in principle of least action, 372. Mathematics, function of, 77. Matter, quantity of, 216, 238, 536 et seq., 559 et seq. Maupertuis, his loi de retpos, 68 et seq; on the principle of least action, 364, 368; his theological proclivities, 454. Maxima and minima, 368 et seq.; problems of, 422 et seq. Maximal and minimal effects, explanation of, 460. Maxims, scholastic, 143. Maxwell, 271, 530, 534, 540. Mayer, J. R., 249, 503, 584, 586. McCormack, Thomas J., 580. Measures. See Units. Mechanical, experiences, i; knowledge of antiquity, 1-3; phenomena, purely, 495 et seq.; theory of nature, its untenability, 495 et seq.; phenomena not fundamental, 496; conception of the world, artificiality of, 496. Mechanics, the science of, i; earliest researches in, 8; extended application of the principles of, and deductive development of the science, 256-420; the formulae and units of, 269-286; character of the principles of, 237; form of its principles mainly of historical and accidental origin, 247 et seq.; theological, animistic, and mystical points of view in, 446-465; fundamental equations of, 270-276; new transformation of, 480; relations of, to other departments of knowledge, 495-507; relations of, to physics, 495-504; relations of, to physiology, 504-507; an aspect, not the foundation of the world, 496, 507; analytical, 465-480; Newton's geometrical, 465; Hertz's, 548 et seq. Medium, motion-determinative, hy INDtEX 6oi pothesis of, in space, 230, 547; resisting, motion in, 435. Memory, 481, 488. Mensbrugghe, Van der, on liquid films, 386. Mental artifices, 492 et seq. Mercurial air-pump, 125. Mersenne, 114, 174. Metaphysical point of view, 586. Method of tangents, 423. Metre, 280. Mimicking, of facts in thought. See Reproduction. Minima. See Maxima. Minimum of superficial area, 387. Minimum principles, 550, 575 et seq. Mixed equilibrium, 70-71. Mobile liquids, 407. Mibius, 372, 480. Models, mental, 492. Molecular forces, 384 et seq. Moment, statical, 14; of force, 37; of inertia, 179, 182, 186. Moments, virtual, 57. Momentum, 241, 244, 271; law of the conservation of, 288; conservation of, interpreted, 326. Monistic philosophy, the, 465. Montgolfier's hydraulic ram, 411. Moon, its acceleration towards the earth, 190; length of its day increased to a month, 299. Morin, apparatus of, 150. Motion, Newton's laws of, 227, 241; quantity of, 238, 271 et seq.; equations of, 342, 371; circular, laws of, 158 et seq.; uniformly accelerated, 132, relative and absolute, 227 et seq., 542 et seq., 568. Motivation, law of, 484. Mueller, J., 510. Miiller, F. A., 585 et seq. Mystical points of view in mechanics, 456. Mysticism in science, 481. Mythology, mechanical, 464. Napier, his theological inclinations, 447 -Nature, laws of, 502. Necessity, 484, 485. Neumann, C., 255, 567 et seq., 572, 577 -Neutral equilibrium, 70-71. Newton,his dynamical achievements, 187-201; his views of absolute time, space, and motion, 222-238, 543, 568, 570 et seq.; synoptical critique of his enunciations, 238-245, 557, 578; scope of his principles, 256-269; enunciates the principle of the parallelogram of forces, 36; his principle of similitude, 165 et seq; his discovery of universal gravitation, its character, and its law, 188 et seq., 533 et seq.; effect of this discovery on mechanics, 191; his mechanical discoveries,, 192; his regulce p/ilosophzandi, 193, 580; his idea of force, 193; his concept of mass, 194 et seq., 536 et seq.; on the composition of forces, 197; on action and reaction, 198; defects and merits of his doctrines, 201, 244; on the tides, 209 et seq.; his definitions, laws, and corollaries, 238-242; his water-pendulum, 409; his theological speculations, 448; the economy and wealth of his ideas, 269; his laws and definitions, proposed substitutes for, 243; his favorite concepts, 251; on the figure of the earth, 395; does not mingle theology with science, 457; on the brachistochrone, 426; his theory of light, 530; his forerunners, 531. Numbers, 486. Observation, 82. Occasionalism, the doctrine of, 449. Oersted, 93. Oil, use of, in Plateau's experiments, 384 et seq. Oscillation, centre of, 331-335. Oscillatory motion, 162 et seq. Ostwald, 577, 585. Pagan ideas in modern life, 462. Page's electromotor, 292. Pappus, 422; on maxima and minima, 451. Parallelism of strata, 409. 602 THE SCIENCE OF MECHANICS. Parallelogram of forces. See Forces. Particular determination, principle of, 544. Pascal, his application of the principle of virtual velocities to the statics of liquids, 54, 91, 96; his experiments in liquid pressure, 99; his paradox, 101-102; his great piety, 447; criticism of his deduction of the hydrostatic principle, 95-96; his experiments in atmospheric pressure, 114 et seq., 575. Pearson, Karl, 547. Peltier's effect, 503. Pendulum, motion of, 152, 163, 168; law of motion of, 168; experiments illustrative of motion of, 168 et seq.; conical, 171; determination of g by, 172; simple and compound, 173, 177; cycloidal, 186; a falling, 205; ballistic, 328; liquid, 409. Percussion, centre of, 327. See npact. Percussion-machine, 313. Perier, 115. Perpetual motion, 25, 89, 500. Petzoldt, 542, 552, 558 et seq., 562, 571 et seq., 575 et seq., 580. Philo, 518. Philosophy of the specialist, the, 506. Phoronomic similarity, 166. Physics and theology, separation of, 456. Physics, artificial division of, 495; necessity of a comparative, 498; relations of mechanics to, 495-504; disproportionate formal development of, 505. Physiology, relations of mechanics to, 504-507; distinguished from physics, 507. Pila Heronis, 118, 412. Place, 222, 226. Planck, 585 et seq. Planets, motion of, 187 et seq. Plateau, on the statics of liquids, 384 -394; Plateau's problem, 393. Pliny, 51o. Poggendorf's apparatus, 206 et seq. Poinsot, 186, 251, 269, 480. Poisson, 42, 46. Polar and parallel coBrdinates, 304. Poncelet, 251, 272, Popper, J., 584 et seq. Porta, 462. Poske, on the law of inertia, 524, 558. Potential, iio, 398 et seq.; potential function, 497; potential energy, 499. Pound, Imperial, Troy, Avoirdupois, 283. Pre-established harmony, 449. Pressure, origin of the notion of, 84; liquid, go, 99, et seq.; of falling bodies, 205; hydrodynamic and hydrostatic, 413; of liquids in motion, 414. Pressure-head, 403, 416. Principles, their general character and accidental form, 79, 83, 421. See Laws. Projectiles, motion of, 152 et seq., 525 et seq.; treated by the principle of least action, 369. Projection, oblique, 153; range of, 154. Proof, the natural methods of, 80. Ptolemy, 232, 509. Pulleys, 21, 49-51. Pump, 112. Pythagoras, 422, 509. Quantity, of matter, 216, 238, 536 et seq., 539 et seq.; of motion, 238, 271 et seq. Quickest descent, curve of, 426. Radii vectores, 294. Rationalism, 458. Reaction, discussion and illustration of the principle of, 201-216; criticism of the principle of, 216-222; Newton on, 198, 201, 242. Reaction-tubes, 301. Reaction-wheels, 299 et seq. Reaumur, 453. Reason, sufficient, principle of, 9, 484, 502. Reconstruction of facts, mental. See Reproduction. Refiguring of facts in thought. See Reproduction. Refraction, ecqnomical character of law of 485. INDEX. 603 Regulie,shilosopklandi, Newton's, 193, 580. Regularity, 395. Religious opinions, Our, 464. Retos, loi de, 68. Representation. See Reproduction. Reproduction of facts in thought, 5, 84, 421, 481-494. Research, means and aims of, distinguished, 507. Resistance head, 417. Rest, Maupertuis's law of, 68, 259. Resultant of force, 34. Richer, 161, 251. Riemann, 493. Roberval, his balance, 60; his method of maxima and minima, 423; on momenta, 305; on the composition of forces, 197. Robins. 330. Rose, V., 518. Rosenberger, 531, 536. Routh, 352. Routine methods, 181, 268, 287, 341. Rules, 83, 485; the testing of, 81. Sail filled with wind, curve of, 431. Santbach, 525. Sauveur, 526. Scheffler, 353, 364. Schiaparelli, 509. Schmidt, W., 519. Schopenhauer, on causality, 484. Science, the nature and development of, 1-7; the origin of, 4, 8, 78; deductive and formal development of, 421; physical, its pretensions and attitude, 464 et seq.; the economy of, 481-494; a minimal problem, 490; the object of, 496, 497, 502, 507; means and aims of, should be distinguished, 504, 505; condition of the true development of, 504; division of labor in, 505; tools and instruments of, 505. Science and theology, conflict of, 446; their points of identity, 460. Scientists, struggle of, with their own preconceived ideas, 447. Seebeck's phenomenon, 503. Segner, 186; Segner's wheel, 309. Sensations, analysis of, 464; the elements of nature, 482; their relative realness, 506. Shortest line, 369, 371. Similarity, phoronomic, 166. Similitude, the principle of, 166, 177. Siphon, 114 et seq. Smith, Adam, 580. Space, Newton on, 226; absolute and relative, 226, 232, 543, 568, 570 et seq.; a set of sensations, 506; multidimensioned,an artifice of thought, 493; a sort of medium, 547. Spannkraft, 499. Specific gravity, 87-88. Sphere, rolling on inclined plane, 346. Spiritism, or spiritualism, 49. Stable equilibrium, 70-71. Stage of thought, the, 505. Statical manometer, 123. Statical moment, 14; possible origin of the idea, 21. Statics, deduction of its principles from hydrostatics, 07 et seq.; the development of the principles of, 8-127; retrospect of the development of, 77-85; the principles of, applied to fluids, 86-nio; the principles of, applied to gaseous bodies, 110-127; Varignon's dynamical, 38, 268; analytical, founded by Lagrange on the principle of virtual velocities, 467. Stevinus, his deduction of the law of the inclined plane, 24-31, 515 et seq.; his explanation of the other machines by the inclined plane, 31 -33; the parallelogram of forces derived from his principle, 32-35; his discovery of the germ of the principle of virtual velocities, 49-51; his researches in hydrostatics, 88 -90; his broad view of nature, 5oo. Strata, parallelism of, 409. Strato, 518. Streintz, 542 et seq. String, equilibrium of a, 372 et seq. See Catenary. Strings, equilibrium of three-knotted, 61; equilibrium of ramifying, 33. 604 THE SCIENCE OF MECHANICS. Substance, 536. Suction, 112. Sufficient reason, the principle of, 9, 484, 502. Superposition, 527. Surface of liquids, connection of, with equilibrium, 386-390. Surfaces, isothermal, 400; level, 98, 398 et seq. Symmetry of liquid films explained, 394. Synoptical critique of the Newtonian enunciations, 238-245. Synthetic method, 466. Tangents, method of, 423. Taylor, Brook, on the centre of oscillation, 335. Teleology, or evidences of design in nature, 452. Theological points of view in mechanics, 446 et seq.; inclinations of great physicists, 450. Theology and science, conflict of, 446; their points of identity, 460. Theorems, 421. Theories, 491 et seq. Thermometers, their construction, 282. Things, their nature, 482; things of thought, 492 et seq. Thomson and Tait, their opinion of Newton's laws, 245, 557, 585. Thought, instruments of, 505; things of, 492 et seq.; experimenting in, 523; economy of. See Economy. Tides, Newton on. 209 et seq.; their effect on the army of Alexander the Great, 209; explanation of, 213 et seq; their action illustrated by an experiment 215; Kepler on, 532, 538; early theories of, 537 et seq.; Galileo on, 537 et seq. Time, sensations of, 506, 541; Newton's view of, 222-238; absolute and relative, 222, 542; nature of, 223 -226, 234. Toeppler and Foucault, optical method of, 125. Torricelli, his modification of Galileo's deductiuo of the law of the inclined plane, 52; his measurement of the weight of the atmosphere, 113; founds dynamics, 402; his vacuum experiment, 113; founds hydrodynamics, 402; on the velocity of liquid efflux, 402. Trades and classes,function of, in the development of science, 4. Trade winds, 302. Tubes, motion of liquids in, 416 et seq. Tylor, 462, 463. Tyndall, 584. Ubaldi, Guido, his statical researches, 21. Uniquely determined, o1, 502, 575 et seq. Unitary conception of nature, 5. Units, 269-286. Unstable equilibrium, 70-71. Vacuum, 112 et seq. Vailati. 521, 526. Variation, of curve-elements, 432 et seq. Variations, calculus of, 436 et seq. Varignon, enunciates the principle of the parallelogram of forces, 36; on the simple machines. 37; his statics a dynamical statics, 38; on velocity of liquid efflux, 403. rVas superficiarium of Stevinus, 89. Vehicle on wheels, 291. Velocity, 144; angular, 296; a physical level, 325. Velocity-head, 417. Venturi, 520. Vibration. See Oscillation. Vicaire, 547. View, breadth of, possessed by all great inquirers, 500 et seq. Vinci, Leonardo Da, on the law of the lever, 20, 520. Virtual displacements, definition of, 57. See also Virtual velocities. Virtual moments, 57. Virtual velocities, origin and meaning of the term, 49; the principle of, 49-77. Viscosity of liquids, 416. INDEX. 605 I Ys mortua, 272, 275. Vis viva, 272 et seq., 315; conservation of, 317; interpreted, 326; in impact, 322 et seq.; principle of, 343 -350; connection of Huygens's principle with, 178; principle of, deduced fron Lagrange's fundamental equations, 478. 499. Vitruvius, on the nature of sound, 3; his account of Archimedes's discovery, 86; on ancient air-instruments, IIo. Viviani, 113. Volkmann, P., 527, 557 et seq., 573. Voltaire, 449, 454. Volume of liquids, connection of with equilibrium, 387-390: Wallis, on impact, 313; on the centre of percussion, 327. Water, compressibility of, 93. Weightless liquids, 384 et seq. Weights and measures. See Units. Weston, differential pulley of, 59. Weyrauch, 585. Wheatstone, chronoscope of, 151. Wheel and axle, with non-circular wheel, 72; motion of, 22 et seq., 60, 337, 344, 354, 381. Wien, W., 536. Will, conception of, in nature, 461. Wire frames, Plateau's, 393. Wohlwill, on the law of inertia, 308, 520, 524 et seq. Woodhouse, on isoperimetrical problems, 430. Wood, on the cells of the honeycomb, 453. Work. 54, 67 et seq, 248 et seq., 363; definition of, 272; determinative of vis viva, 178; accidentally not the original concept of mechanics, 548; J. R. Mayer's views of, 249; Huygens's appreciation of, 252, 272; in impact, 322 et seq.; of molecular forces in liquids, 385 et seq.; positive and negative, 386; of liquid forces of pressure, 415; of compression, 407. Wren, on impact, 313. Wright, Chauncey, 453. Yard, Imperial, 281; American, 283 .P 4P UIVERSITY OF MICHIGAN 3 9015 01713 6972 SOUND42 SEP26 1947 ijNwv (0, micG4. 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