CORNELL UNIVERSITY LIBRARY ENGINEERING -nell University Library The science of mechanics; a c";"'" ' ^"'' 3 1924 004 010 504 DATE DUE worV?^ ^ ^r rfi^ fciirr? 11*7 inn7 PRINTED IN U S A The original of tliis bool< is in tlie Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924004010504 THE SCIENCE OF MECHANICS WORKS BY ERNST MACH. Contributions to the Analysis Of the Sensations. Translated by C. M. Williams. With Notes and New Additions by the Author. Pages xi -)- 208. 37 Cuts. Price, Cloth, $1.25 net (6s. 6d.). Popular Scientific Lectures. Translated by T. J. McCormack. Third Revised and Enlarged Edi- tion. 411 pages. 59 Cuts. Price, Cloth, $1.50 net (7s. 6d.). The Science of Mechanics. Translated by T. J. McCormack. Second Revised and Enlarged Edition, 259 Cuts and Illustrations. Pages, 605 -|- XX. Price, Cloth, $2.00 net {9s. 6d.). THE SCIENCE OF MECHANICS A CRITICAL AND HISTORICAL ACCOUNT OF ITS DEVELOPMENT DR. ERNST MACH PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN THE UNIVERSITY OF VIENNA TRANSLATED FROM THE GERMAN BY THOMAS J. McCORMACK SECOND REVISED AND ENLARGED EDITION WITH 259 CUTS AND ILLUSTRATIONS CHICAGO THE OPEN COURT PUBLISHING COMPANY LONDON Kegan Paul, Trench, Trubner & Co., Ltd. ig02 Copyright, 1893 BY The Open Court Publishing Co. Chicago All Rights Reserved 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 philoso- phy of science have found wide and steadily increas- ing acceptance. Many fruitful and elucidative con- troversies 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 countries. 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 ex- emplary 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 made by the author to the * Die Mechanik in ihrer Entvjickelung historisch-kritisch dargestellt , Von Dr. Ernst Mach, Professor an der Universitat zu Wien. Mit 257 Abbildungen. First German edition, 1883. Fourth German edition, 1901. First edition of the English translation, Chicago, The Open Court Publishing Co., 1893. vi TRANS LA TOR' S PRE FA 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 trans- lator'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 predomi- nantly under the aspect of their development {Entwicke- lung). To avoid cqnfusion 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 refer- ences in other works to the pages of the first edition thus hold good for the present edition also, Snd sec- ondly, because with few exceptions the additions are either supplementary in character, or in answer to criticisms. A list of the subjects treated in these ad- ditions 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, ab- solute motion, etc., on pp. 542-547, 555-574, and 579 TRANSLA TOR' S PRE FA CE. vii ^583- The remarks here made will be found highly elucidative, -while the references given to the rich lit- erature 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 his- tory 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 in- applicable to the system commonly taught in this country. Thomas J. McCormack. La Salle, III., February, 1902. AUTHOR'S PREFACE TO THE TRANS- LATION. Having read the proofs of the present translation of my work, Die Mechanik in ihrer Entwickelung, 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 dis- covery of many errors, heretofore overlooked. I may, thus, confidently hope, that the rise and growth of the ideas 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 FIRST EDITION. The present volume is not a treatise upon the ap- plication 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 ex- isting treatises completely buried and concealed be- neath a mass of technical considerations. The gist and kernel of mechanical ideas has in al- most 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- 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 comprehen- sion of the general upshot of mechanics is to be at- tained. I have framed my exposition of the subject agree- ably 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 abbrevi- ated 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 suf- ficiently 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 Geschichte und die Wurzel des Satzes von der Erhaltung der Arbeit (Prague, Calve, 1872). At a later date nearly the same views were expressed by Kirchhoff {Vorlesungen tiber mathematische Physik: Mechanik, Leipsic, 1874) and by Helmholtz (Z>zV Thatsachen in der Wahrnehmung, 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 sci- ence as Economy of Thought, — a view which I in- dicated both in the treatise above cited and in my PREFACE TO THE FIRST EDITION. xi pamphlet, Die Gestalten der Flussigkeit (Prag-ue, Calve, 1872), and which I somewhat more extensively devel- oped in my academical memorial address, Die okono- mische Natur der physikalischen Forschung (Vienna, Ce- roid, 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 {Philoso- phic als Denken der Welt, gemass dem Princip des klein- sten 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 meta- physical 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 Mechanik (Stuttgart, 1852). If Duehring's esti- mable work, Kritische Geschichte der Principien der Me- chanik (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 sub- ject, here figured and described, were designed entirely xii PREFACE TO THE FIRST EDITION. by me and constructed by Mr. F. Hajek, the mechani- cian of the physical institute under my control. In less immediate connection with the text stand the fac-simile reproductions of old originals in my pos- session. The quaint and naive traits of the great in- quirers, which find in them their expression, have al- ways 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 ex- hausted in less than five years. This circumstance and the treatises that have since then appeared of E. Wohl- will, H. Streintz, L. Lange, J. Epstein, F. A. Miiller, 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 my- self, 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, 1888. 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, Lon- don, 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 pseudo- scientific tendencies in science. Mechanics appears at present to be entering on a new relationship to physics, as is noticeable particularly in the publica- tion 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 Gravitations- gesetz," Sitzungsbericht der Munchener Akademie, 1896), who has shown the incompatibility of a rigorous inter- pretation of Newton's law with the assumption of an unlimited mass of the universe. Vienna, January, 1897. E. Mach. 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 wri-tings of Boltzmann, Foppl, 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 kin- dred aims and points of view. New books and criticisms touching on my discus- sions have received attention in special additions, which in some instances have assumed considerable proportions. Of these strictures, O. 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 dis- pute that rigorous demonstrations are as possible in mechanics as in mathematics. But with respect to XV i PREFACE TO THE FOURTH 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 in- sertions 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, igoi. V viii ix . xiii xiv XV 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 ... . . xvii Introduction . i CHAPTER I. THE DEVELOPMENT OF THE PRINCIPLES OF STATICS. I. The Principle of the Lever ... g II. The Principle of the Inclined Plane . . -24 III. The Principle of the Composition of Forces . . -^3 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 yil. The Principles of Statics in Their Applicalicn to Gas- eous Bodies . ... . . . . no 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 Ihe Con- cept of Mass .... 2i6 VI. Newton's Views of Time, Space, and Moticn 222 xviii THE SCIENCE OF MECHANICS. 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 DEVELOP- MENT 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 ... 33i 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 DEPART- MENTS 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, 510. — 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, Gali- leo on the Law of Inertia, 523. — X. Galileo on the Motion of Projec- tiles, 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.— XlV. Newton's Predecessors, 531. — XV. The Explanations of Gravi- tation, 533 —XVI. Mass and Quantity of Matter, 536.— XVII. Gali- leo 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 Sci- ence [Mass, Inertia, etc.), 555. — XXIII. Descartes's Achievements in Physics, 574. — XXIV. Minimum Principles, 575. — XXV. Grass- mann's Mechanics, 577.— XXVI. Concept of Cause, 579.— XXVII. Mach's Theory of the Economy of Thought, 579.— XXVIII. Descrip- tion of Phenomena by Differential Equations, 583.— XXIX. Mayer and the Mechanical Theory of Heat, 584.— XXX. Principle of En- ergy, 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 est 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 sim- ple and instructive example of the processes by which natural science generally is developed. An instinctive, irreflective knowledge of the processes instinctive of nature will doubtless always precede the scientific, conscious apprehension, or investigation, of phenom- ena. The former is the outcome of the relation in which the processes of nature stand to the satisfac- tion of our wants. The acquisition of the most ele- mentary 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- Meciianicai ,,,.,. 1 • 1 • J 1 experiences tmction between mechanical experience and mechan- ical science, in the sense in which the latter term is at present employed. Mechanical experiences are, un- questionably, 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 ac- cnanical jsnowiedge counts of the scientific knowledge of these peoples of antiquity . . . ^ ^ are either totally lacking, or point conclusively to a very inferior grade of attainment. By the side of highly ingenious ap- pliances, we behold the crudest and rough- est expedients em- ployed — as the use of sleds, for instance, for the transportation of enormous blocks of stone. All bears an instinctive, unperfec- ted, accidental char- acter. So, too, prehistoric graves contain imple- ments whose construc- tion and employment imply no little skill and much mechanical experience. Thus,long before theory . was dreamed of, imple- ments, machines, me- chanical experien- ces, and mechanical knowledge were abun- dant. 3. The idea often suggests itself that perhaps the incom- plete accounts we pos- INTRODUCTION. 3 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. Ill, 6 : " The voice is a flowing breath, made sensible to a passage "the organ of hearing by the movements it produces v^iu™ ^'"^" "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 ter- "mination ; 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 Controvert- 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 lit- erature, 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 con- taining such crude and patent errors as cannot be con- ceived to exist in any high stage of scientific culture. (See Appendix, I., p. 509.) of science. 4 THE SCIENCE OF MECHANICS. The origin 4. When, where, and in what manner the develop- ' ment of science actually began, is at this day difficult historically to determine. It appears reasonable to assume, however, that the instinctive gathering of ex- periential 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 strug- gles 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. Thefunc- The transition from this stage to the classified, d™^°iasses Scientific knowledge and apprehension of facts, first be- leiop^ment comes possible on the rise of special classes and pro- of science, fgggj^jjg ^jjQ make the satisfaction of definite social wants their lifelong vocation. A class of this sort oc- cupies 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 of knowi- determined beforehand. The communication of knowl- edge is thus the first occasion that compels distinct re- flection, as everybody can still observe in himself. Further, that which the old members of a guild me- chanically pursue, strikes a new member as unusual introduction: 5 and strange, and thus an impulse is given to fresh re- flection and investigation. When we wish to bring to the knowledge of a per- involves , r ■. description. son any phenomena or processes oi 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 per- sonally making anew each experiment. Description, however, is only possible of events that constantly re- cur, or of events that are made up of component parts that constantly recur. That only can be de- scribed, 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 ordinary a unitary "^ ■'■'■' conception events occur; while others appear uncommon, per- of nature, plexing, astonishing, or even contradictory to the or- dinary 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 com- munication are rendered possible ; and on the other. The nature when once a person has acquired the skill of recog-edge. nising these permanent elements throughout the great- est range and variety of phenomena, of seeing them in the same, this ability leads to a comprehensive, compact, consistent, and facile conception of the facts. When once we have reached the point where we are everywhere 6 THE SCIENCE OF MECHANICS. The adap- able to detect the same few simple elements, combin- thoughts to ing in the ordinary manner, then they appear to us as things that are familiar ; we are no longer surprised, there is nothing new or strange to us in the phenom- ena, we feel at home with them, they no longer per- plex 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 apprehen- th^ught. 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 his- torical 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 knowl- edge of a given province of natural phenomena — a knowledge that is attained with the least intellectual expenditure — naturally becomes in itself an econom- ical 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 phe- ofth^™e^° nomena of nature, to discover the elements thereof and the mode of their interconnection and interdepend- ence — this is the business of physical science. It en- deavors, by comprehensive and thorough description, to make the waiting for new experiences unnecessary ; it seeks to save us the trouble of experimentation, by making use, for example, of the known interdepend- ence 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- IXTRODUCTIOX. 7 scribing the greatest possible number of different ob- Their pres- 1-1 - « 1. 1 • -11 en'discus- jects at once and m the concisest manner. All this will sion merely preparatory be made clearer by the examination of points of detail 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 subject of our inquiries, and, at the same time, without fr^tmem. 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 pres- ent 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 The incen- epochs, incentives which taken as a whole are more rfved from . . , _ contact fruitful than the greatest men of the present day are with the . , , , , great intel- able to offer, there is no grander, no more intellectually lects of the world. elevating spectacle than that of the utterances of the fundamental investigators in their gigantic power. Possessed as j-et of no methods, for these were first created by their labors, and are only rendered compre- hensible to us b}' 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 in- . crease of more correctly of the significance of an\- present scien- power . " . . which such tific movement than the}', who limited in their views a contact lends. to the age in which their own lives have been spent, 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. 1. THE PRINCIPLE OF THE LEVER. The earliest I. The earliest investigations concerning mechan- researches ics of which we have any account, the investigations statics. of the ancient Greeks, related to statics, or to the doc- trine 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 an- cient writings that the fugitive Greeks brought with them, it was investigations in statics, principally evoked by the works of Archimedes, that occupied the fore- most investigators of the period. (See p. 510.) Archimedes 2. Archimedes of Syracuse (287-212 B. C.) left (287-212 B. behind him a number of writings, of which several have come down to us in complete form. We will first employ ourselves a moment with his treatise De ^quiponderantibus, which contains propositions re- specting the lever and the centre of gravity. In this treatise Archimedes starts from the follow- ing assumptions, which he regards as self-evident : Axiomatic a. Magnitudes of equal weight acting at equal assump- . '.J o n tions of Ar- distances (from their point of support") are in eaui- chimedes. jt jr y -1 librium. THE PRINCIPLES OF STATICS. g b. Magnitudes of equal weight acting at une- Axiomatic qual distances (from their point of support) are tionTof Ar- 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 equilib- rium when they are inversely proportional to their distances (from the point of support). It would seem as if analysis could hardly go be- hind 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 dis- tances from the fulcrum we ap- pend two equal weights. That 1 jr ■ the two weights, thus circum- A-, X-. stanced, are in equilibrium, is ^ — ' — ' Fig. -i. the assumption from which Archi- medes starts. We might suppose that this was self- Analysis of evident entirely apart from any experience, agreeably to mldean as- the so-called principle of sufficient reason ; that in view ^°™p"°°^- 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 vi- cinity, and the like, exercise no influence ; the positive, on the other hand, (as it appears in the second as- sumption,) that not only the weights but also their dis- tances from the supporting point are decisive factors in the disturbance of equilibrium, that they also are cir- 10 THE 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* de- termined, or defined, by the determinative conditions of the case.f Character Now we are entitled to regard our knowledge of and value of,,., ,.. ^ i rr • the Archi- the decisive conditions of any phenomenon as suiticient me^MQ re ^^^^ .^ ^^^ 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 evi- dence and is eminently qualified to be made the foun- dation of further investigations. If the spectator place himself in the plane of symmetry of the arrangement in question, the first proposition manifests itself, more- over, as a highly imperative instinctive perception, — a result determined by the symmetry of our own body. The pursuit of propositions of this character iSj fur- thermore, an excellent means of accustoming ourselves in thought to the precision that nature reveals in her processes. Thegenerai 3. We wiU now reproduce in general outlines the of the lever train of thought by which Archimedes endeavors to re- the simple duce the general proposition of the lever to the par- uiarcase. 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 de- scended, 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. THE PRINCIPLES OF STATICS. bar, will have to support the weight 2. The equal The general • 1 - - r 1 1 proposition weights at the extremities of the bar supply accor- of^the lever dingly the place of the double weight at the centre a h ■ 2 a ^ reduced to the simple and partic- ular case. m [i ^ Fig. 3- F'g- 4- On a lever (Fig. 4), the arms of which are in the proportion of i to 2, weights are suspended in the pro- portion 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 TIT Fig. 5 ~\.^ r^T h r^ L_J L-J L.J 4 and 3 pairs of symmetrically attached weights J, in the manner indicated in the figure. Now again we have perfect symmetry. The preceding reasoning, The gener- which we have here developed with specific figures, is 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. THE SCIENCE OF MECHANICS, Galileo's mode of treatment. ■im 2» Fig. 6. Galileo imagines (Fig. 6) a heavy horizontal prism, 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 attach- ^„ „ I m n ment. In. this case equi- librium will obtain ; this we perceive at once. But in this case is contained every other case, — which Galileo shows in the following manner. Let us suppose the whole length of the bar or the prism to be i{mA^ ri). Cut the prism in two, in such a manner that one portion shall have the length im and the other the length in. 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 at- tached to the bar by their centres. Since the whole length of the bar is i{m. -f ri), the length of each half \& 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 mag- nitude ■zm be suspended at the distance n on the one side and any weight of the magnitude in be suspended at the distance in on the other. The instinctive elements of our perception of this phenomenon are even more THE PRINCIPLES OF STATICS. 13 prominently displayed in this form of the deduction than in that of Archimedes. We may discover, moreover, in this beautiful pre- sentation, a remnant of the ponderousness which was particularly characteristic of the investigators of an- tiquity. How a modern physicist conceived the same prob- Lagrange's lem, maybe learned from the following presentation of tion. Lagrange. Lagrange says : Imagine a horizontal ho- mogeneous prism suspended at its centre. Let this prism (Fig. 7) be conceived divided into two prisms of the lengths im and 2«. If now we consider the centres of gravity of these two parts, at which we may imagine weights to act proportional to 2ot and in, the 2n I X I Fig. 7. two centres thus considered will have the distances n and m from the point of support. This concise dis- posal of the problem is only possible to the practised mathematical perception. 5. The object that Archimedes and his successors object of sought to accomplish in the considerations we have here andhissuc- presented, consists in the endeavor to reduce the more complicated case of the lever to the simpler and ap- parently 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 THE SCIENCE OF MECHANICS. Critique of tion, nevertheless spring up. From the mere assump- oa". "^ ' tion of the equilibrium of equal weights at equal dis- tances 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 ! Thestaticai As a matter of fact, the' assumption that the equi- moment in- . . . . voived in librium- disturbing effect of a weight P at the distance all their de- . . ductions. 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 ie itself 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 im- demonstra- ^ r ^i j r. -r tion is im- port 01 the product P.L, no one can prove (Fig. 8> that a bar, placed in any way on the ful- crum S, is supported, with the help of a string attached to its possible. a ^J centre of gravity and Fig- 8. . , -'„ carried over a pulley, by a weight equal to its own weight. But this is con- tained in the deductions of Archimedes, Stevinus, Galileo, and also in that of Lagrange. THE PRINCIPLES OF STATICS. 15 6. HuYGENS, indeed, reprehends this method, and gives a different deduction, in which he behaves he has avoided the error. If in the presentation of La- grange we imagine the two portions into which the prism is divided turned ninety degrees about two vertical axes passing through the cen- tres of gravity s,s' of the prism-portions (see Fig. ga), and it be shown that under these circum- stances equilibrium still D continues to subsist, we shall obtain the Huygenian Huygens's criticism. and simplified, it is as follows. Fig. 9. deduction. Abridged In a rigid weightless Fig 9a Fig ga plane (Fig. 9) through the point S we draw a straight line, on which we cut off on the one side the length i i6 THE SCIENCE OF MECHANICS. His own duction. Apparently unimpeach- able. de- and on the other the length 2, at A and B respectively. 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 AG = \A C) and, parallel to it, the line CF, and translating the prism-portion CG by par- allel displacement to FH, everything about the axis GH is symmetrical and equilibrium obtains. But equihbrium also obtains for the axis AB ; obtains con- sequently 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, nothing else is assumed here than that equal weights /,/ (Fig. 10) 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, o- M A' Fig, 10. 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 proposi- tion. 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 STATICS. 17 The error first arises in the inference : if equilib- Yet invoiv- num obtains tor two axes of the plane, it also obtains final infer- r 1 . . , , , . f . ence an er- lor every other axis passing through the point of inter- ror. section of 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 as- cribed to the weights proportional 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 coordinates (Fig. 11). The co- ordinates of the centre of gravity of a system of masses m m m!' . . . having the coordinates x x' x" . . . y y' y" ■ ■ ■ are, as we know, Mathemat- ^ 2mx 2my 'cai discus- ^ =^ , T) =^ — . sion of 2m 2m Huygens's inference. If we turn the system through the angle a, the new co- ordinates of the masses will be x^=^ X cosa — y sina, y^ ^ycosa -{- xs\na and consequently the coordinates of the centre of gravity 2m (x cosa — y sma) 2mx . 2my $ , =^ ^ ^ = cosa ^^ sina ^=; — '■ 2m 2m 2m = S cosa — r) sina and, similarly, ri^ = ri cosa -|- ^ sina. We accordingly obtain the coordinates of the new centre of gravity, by simply transforming the coordi- nates of the first centre to the new axes. The centre of gravity remains therefore the self-same point. If we select the centre of gravity itself as origin, then 2mx^^2my=:Q. On turning the system of axes, this relation continues to subsist. If, accordingly, equi- i8 THE SCIENCE OF MECHANICS. librium obtains for two axes of a plane that are per- pendicular 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 in- tersection of the two. The inter- These conclusions, however, are not deducible if enceadmis- . , . sibie only the coordinates of the centre of gravity are determined on one con- . dition. by some other, more general equation, say ■ _ mfix) + mj{x-) + ot'/Cx") + . . . ■m -\- ni -\- m" -|- . . . The Huygenian mode of inference, therefore, is in- admissible, and contains the very same error that we remarked in the case of Archimedes, seif-decep- Archimedes's self-deception in this his endeavor to chimedes. reduce the complicated case of the lever to the case instinctively grasped, probably consisted in his uncon- scious employment of studies previously made on the centre of gravity by the help of the very proposition he sought Jo 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 verifi- cation 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 Archi- medes, in his Grecian mania for demonstration, strives to get around this, his deduction is defective. But re- garding the import of P.L as given, the Archimedean THE PRINCIPLES OF STATICS. 19 deductions still retain considerable value, in so far as Function ot the modes of conception of different cases are supported medean de- 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. Im- agine (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 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 5. THE SCIENCE OF MECHANICS. 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 verifica- tion. (See Appendix, IV., p. 514.) Treatment 7. The manner in which the laws of the lever, as of the lever ,- , , - . a i • 1 • 1 • • ■ 1 by modern handed down to us from Archimedes m their original simple form, were further generalised and treated by modern physicists, is very interesting and instructive. Leonardo DA Vinci (1452-15 19), the famous painter and investigator, appears to have been the first to rec- ognise the importance of the general notion of the so- Leonardo Da Vinci .1452-1519) Fig. 13. called statical moments. In the manuscripts he has 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 ob- tain? The lever-arm for the weight P is not AD, but the "potential" lever AB. The lever-arm for the weight ^ is not ^Z?, 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 STATICS. 21 nised the essential circumstances by which the effect of the weight is determined. Considerations similar to those of Leonardo da Guido 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 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 F's- h- to the plane of the cord and bisects {EE) the angle made by its two parts. The motion that might be supposed a method ... - , ■ , ■ , 1 ■ , bywhich possible cannot in triis case be precisely determined or the notion of the stat- deiined by any rule whatsoever : no motion will there- icaimo- ™ent might fore take place. If we note, now, further, that the mate- have been 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 tan- gential 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. THE SCIENCE OF MECHANICS. This notion derived from the considera- tion of a wheel and azle. Let US examine a so-called wheel and axle (Fig. 15) of wheel-radius 2 and axle-radius i, provided re- spectively with the -cord-hung loads i and 2 ; an appa- 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 Fig. 15. Fig. 16. the two pulls marked in Fig. 16 as being in equilib- rium, by leaving the two others, as mutually destruc- tive, out of account. But we arrive in so doing, dis- missing 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 deter- minative of motion. The decisive factors are, then, the products of the weights into the respective per- pendiculars let fall from the axis on the directions of the pulls ; in other words, the so-called statical mo- ments. 9. What we have so far considered, is the devel- opment of our knowledge of the principle of the lever, expiain'the Quite independently of this was developed the knowl- chines. edge of the principle of the inclined plane. It is not necessary, however, for the comprehension of the ma- The princi- ple of the lever all- sufficient to THE PRINCIPLES OF STATICS. 23 Fig. 17. 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. 17) an inclined plane, on which rests the weight Q, held in equilibrium by the weight P. Gali- leo, now, points out the 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 shght rotation about the point C, the weight will °i 'be in- • ^ clined move in the element of an arc coincident with the in- plane by the lever. chned 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 in- stant alone is decisive. Reverting, however, to the observation before mentioned of Leonardo da Vinci, we readily perceive the validity of the theorem Q. CB = P.CA or Q/F—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. 24 THE SCIENCE OF MECHANICS. THE PRINCIPLE OF THE INCLINED PLANE. stevinus i. SxEviNUS, or Stevin, (1548-1620) was the first first invest;- who investigated the mechanical properties of the in- mechanics clined plane ; and he did so in an eminently original of the in- Tx ■ 1 1 1- /'T-'- clined manner. If a weight lie (t ig. 18) on a horizontal table, we perceive at once, since the pressure is directly perpendic- ular to the plane of the table, by the principle of symmetry, ^'S' '^- 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 neces- sary to preserve it on the vertical wall. The ascertain- ment 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. follows. He imagines a triangular prism with horizon- tally placed edges, a cross-section of which ABC is represented in Fig. ig. For the sake of illustration we will say that AB = ■zBC ; also that AC\s horizon- tal. 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 re- place 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 STATICS. 25 the conditions of the event are not altered by its mo- tion, must, when once actually in motion, continue to move for ever, that is, it must present a perpetual mo- tion, which Stevinus deems absurd. Consequently only stevinus's the first case is conceivable. The chain remains in equi- olthe I'aw librium. The symmetrical portion ADC may, there- dined '" fore, without disturbing the equilibrium, be removed. '' *°^' 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. Fig. 19. F'g- 20. In the cross-section of the prism in Fig. 20 let us imagine AC horizontal, ^C vertical, and AB ^= ■zBC; furthermore, the chain-weights Q and P on AB and BC proportional to the lengths ; it will follow then that 26 THE SCIENCE OF MECHANICS. QjP — ABIBC—i. The generalisation is self-evi- dent. The as- 2. Unquestionably in the assumption from which of sfiii-"' Stevinus starts, that the endless chain does not move, dSclioli^' 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 character, when We reflect that all results of experiment are ob- scured 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 ob- servation 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 individ- ual fallibility of investigators, the explanation is not far Their cog- to Seek. We feel clearly, that we ourselves have con- tributed nothing to the creation of instinctive knowl- edge, that we have added to it nothing arbitrarily, but that it exists in absolute independence of our partici- pation. Our mistrust of our own subjective interpre- tation of the facts observed, is thus dissipated. Stevinus's deduction is one of the rarest fossil in- ency. THE PRINCIPLES OF STATICS. 27 dications that we possess in the primitive history of Highhistor- 1 ■ 1 r 1 1- 1 ical value of mecnanics, and throws a wonderful hght on the pro- stevinus's [ ^ r . r • • ■ deduction. 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 fre- quently taken as the starting-point of investigations. Every experimenter can daily observe in his own per- son 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 im- portant advance in science. Stevinus's procedure is no error. If an error were The trust- ... , 11111 ■ XI 1-. worthiness contained in it, we should all share it. Indeed, it isotinstinc- - , . - . . ^ , . tive linowl 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 mysti- cism 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 em- ployed 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 car- ried along in a parallel direction above it. The instinc- tive 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 tive°knowi- and what are its contents? Everything which we ob- * ^^' serve in nature imprints itself wicomprehended and un- analysed in our percepts and ideas, which, then, in their turn, mimic the processes of nature in their most gen- eral 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 circum- stance that it is far easier to resort to these experi- ences 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 hap- pen, 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 knowl- and extern- edge may be, for discovery, we must not, from our mutually point of view, rest content with the recognition of its 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 fun- damental 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 corrobo- rated by the existence of that knowledge, which is a separate fact by itself. This we will find on close ex- amination is the state of things in Stevinus's case. THE PRINCIPLES OF STATICS. 29 3. The reasoning of Stevinus impresses us as so The ingen- highly ingenious because the result at which he arrives vinus's rea- 1 - 11 ■ r soning. apparently contains more than the assumption from which he starts. While on the one hand, to avoid con- tradictions, 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 ingen- ious ; but we should have obtained a much more satis- factory 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, how- ever, 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 everybody that could possibly de- Critique uf 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 30 THE SCIENCE OF MECHANICS. The merit readers, consists, therefore, in the contrast and com- "/vs^sTrice- parison of knowledge that is instinctive with knowledge ''""• that is clear, in the bringing the two into connection and accord with one another, and in the supporting Fig. 31. 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 THE PRINCIPLES OF STATICS. 31 work Hypotnnemata Mathematica (Leyden, 1605).* As a fact, every enlightening progress made in science is accompanied with a certain feeling of disillusionment. We discover that that which appeared wonderful to 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- clined plane, it was easy for Stevinus to apply that principle to the other machines and to explain by it 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. Stevinus, now, proceeds by a method similar to that later taken by Galileo. He remarks that it is not ne- cessary that the load Q should lie directly on the inclined plane. Provided only the form of the machine's motion be preserved, the proportion between force and load will in all cases re- main 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 ]Visconstige Gedachienissen, Leyden, 1605. — Trans. Enlighten- ment in science al- ways ac- companied with disillu- sionment. Explana- tion of the other ma- chines by Stevinus' s principle. Fig 22 32 THE SCIENCE OF MECHANICS. fhe funicu- lar machine And the special case of the paral- lelogram of forces. The general form of the last-men- tioned prin- ciple also employed. inclined plane. If we carry out this alteration, we 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 be, we have P/Q^AC/AB=^ac/ab. Therefore ac represents the tension of the string aA. Nothing pre- vents us, now, from making the two strings change functions and from imagining the load Q to lie on the dotted inclined plane EDF. Similarly, here, we ob- 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 for-ces) ac, ad at right angles to one another. Subsequently, indeed, Stevinus employs the prin- ciple of the composition and resolution of forces in a more general form ; yet the method by which he Fig- 23- Fig. 24. reached the principle, is riot 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 F is suspended from the first, the tensions may be determined in the following manner. We produce (Fig. 23) AB to -Sfand cut off on it a portion AE. Drawing from the point E, EF parallel to AD and EG paral- lel to A C, the tensions of AB, AC, AD are respectively pro- portional to AE, AF, AG. With the assistance of this principle of construction Ste- vinus solves highly compli- cated problems. He determines, for instance, the solution of tensions of a system of ramifying strings like that putted™ illustrated in Fig. 24; in doing which of course he^" ^™^' 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 man- ner indicated in Fig. 25. We may therefore, by means of the principle of the General re- inclined plane, seek to elucidate the conditions of op- eration of the other simple machines, in a manner sim- ilar to that which we employed in the case of the prin- ciple of the lever. III. THE PRINCIPLE OF THE COMPOSITION OF FORCES. I. The principle of the parallelogram of forces, at The princi- which Stevinus arrived and employed, (yet without ex- paraiieio- pressly formulating it,) consists, as we know, of the forces, 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 THE SCIENCE OF MECHANICS. 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 exactly proportional to the lengths AB, AC he sup- posed to act, a single weight acting on the string '''^' ^^' AD exactly proportional to the length AD will produce the same effect as the first two. The forces AB and A C are called the compo- nents, 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- mvestigations of Stevinus, to give ourselves some idea tionofthe ° ' o paraiieio- of the manner in which the gram of k^ W .... forces \ f general proposition of the might have \ ^^\ ° ^ ^ beenar- \ ^-■ '^ \ parallelogram of forces rived at. ^ 5i| \ . , might have been arrived at. The relation, — dis- covered by Stevinus, — that exists between two mutually perpendicular forces and a third force that equilibrates them, we shall assume as (indi- rectly) given. We sup- pose now (Fig. 27) that there act on three strings OX, OY, OZ, pulls which balance each other. Let us endeavor to determine the nature of these pulls. Each pull holds the two rernain- ing ones in equilibrium. The pull CFwe will replace \ ^ t s \ ^ Y ^^-^ w H m V A Fig. 27. THE PRINCIPLES OF STATICS. 35 (following Stevinus's principle) by two new rectangular pulls, one in the direction Ou (the prolongation of OX'), and one at right angles thereto in the direction Ov. And let us similarly resolve the pull OZ in the directions Ou and Ow. The sum of the pulls in the di- rection 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 com- ponents Op and Oq parallel to Ou, as well also as the pulls Or, Os. Now the sum Op -f- Oq is equal and op- posite to the pull in the direction of OX ; and if we draw st parallel to OY, or r/ parallel to OZ, either line will cut off the portion Ot z= Op -\- Oq : with which re- sult the general principle of the parallelogram of forces is reached. The general case of composition may be deduced in still another way from the special composition of rectangular forces. Let OA and OB be the two forces acting at O. For 0£ substitute a force OC acting parallel to OA and a force OD acting at right angles to OA. There then act for OA and OB the two forces Oil = OA + OC and OD, the resultant of which forces OJ^ is at the same time the diagonal of the parallelogram OAFB con- structed on OA and OB as sides. 3. The principle of the parallelogram of forces, when reached by the method of Stevinus, presents it- self as an indirect discovery. It is exhibited as a con- sequence and as the condition of known facts. We perceive, however, merely that it does exist, not, as yet The deduc- tion of the general principle from the special case of Stevinus. A different mode of the same de- duction. o D Fig. 28. The prin- ciple here presents it- self as an indirect discovery. 36 THE SCIENCE OF MECHANICS. And is first clearly enunciated by Newton and Varig- The geo- metrical theorem employed by Varig- why it exists ; that is, we cannot reduce it (as in dy- namics) to still simpler propositions. In statics, in- deed, the principle was not fully admitted until the time of Varignon, when dynamics, which leads directly to the principle, was already so far advanced that its adoption therefrom presented no difficulties. The prin- ciple of the parallelogram of forces was first clearly enunciated by Newton in his Principles of Natural Phi- losophy. In the same year, Varignon, independently of Newton, also enunciated the principle, in a work sub- mitted to the Paris Academy (but not published un- til after its author's death), and made, by the aid of a geometrical theorem, extended practical application of it.* The geometrical theorem referred to is this. If we consider (Fig. 29) a parallelogram the sides of which are/ and q, and the diagonal is r, and from any point m in the plane of the par- allelogram we draw per- pendiculars on these three straight lines, which perpendiculars we will designate as u, V, w, then p . u -\- q . V ^= r . w. This is easily proved by draw- ing straight lines from m Fig. 29. Fig. 30. 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 ia be taken within the parallelogram and perpendiculars then be * In the same year, 1687, Father Bernard Lami published a little appendix to his Train de michanique, developing the same principle. — Trans, THE PR/NAPLES OF STA TICS. 37 drawn, the theorem passes into the form p . u — q .v =^r . w. Finally, if m 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 ^^ or/, u = q . V. With the assistance of the observation that forces The deduc- 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 parallelo- gram, are replaceable by a single force, similarly rep- resented 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 °^'^^^- 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 m 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 now in a position to treat the machines in a much simpler manner than were his predecessors. Let us consider, for example, (Fig. 31) a rigid body capable of rotation about an axis passing through O. Perpendicular to the axis we conceive a plane, and select therein two '^' ^'' 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 law of the of appHcation be displaced along their line of action, lever from . ,, . ., i-,- ''ji the parai- Since all points in the same direction are rigidly con- princliOe. 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 C is a point on the resultant, and if we drop the per- pendiculars u and V from O on the directions of the forces/, q, we shall have, in conformity with the the- orem before mentioned, p ■ u ^= q ■ 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 adynamic™ cases of equilibrium by the equilibration of the result- statics. ^^^ force by some obstacle or restraint. On the in- clined plane, for example, equilibrium exists if the re- sultant 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 dy- namics. The more general dynamical case constantly hovers before him and he restricts himself in his inves- tigation 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 THE PRINCIPLES OF STATICS. 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- erations also lead to the proposition of the parallel- ogram of forces. In special cases, in fact, 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 suc- cessive lines make equal angles, are in equilibrium. If, for exam- ple, (Fig. 32) the three equal forces OA, OB, OC act on the point O at angles of 120°, each two of the forces holds the third in equilibrium. We see imme- diately that the resultant of OA 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 opposite directions, the resultant is equal to the sum or the difference of the components. We rec- — 4 i ognise both cases with- out any difficulty as particular cases of the principle of the paral- lelogram of forces. If in the two drawings of Fig. 33 we imagine the angle A OB to be gradually reduced to the value 0°, and the angle A' O' B' increased to the Special statical con- siderations also lead to the prin- ciple. Fig. 32- B a O' c A' The case of coincident forces merely a particular case of the general principle. Fig. 33. 40 THE SCIENCE OF MECHANICS. value 1 80°, we shall perceive that CC passes into OA -(- AC=^ OA-{- OB And O' C into O' A' — A' C = O' A' — O' 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 sitionde- the form in which it was set forth by Newton and rived from . , ^ ... experience. Vangnon, clearly discloses itself as a proposition de- rived 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 Ber- noulli, however, was of opinion that the proposition of the parallelogram of forces was a geometrical truth, in- dependent 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 BernouUian view has not, even at the present day, entirely disappeared. Daniel Ber- If two equal forces, at right angles to each other noulli's at- ,—«. . , . -. . . . tempted (rig- 34), act On a point, there can be no doubt, ac- demonstra- ^ cording to Bernoulli, that the line truth. ^' ~/\\ ' °^ bisection of the angle (con- formably to the principle of sym- metry) is the direction of the re- sultant r. To determine geomet- rically also the magnitude of the resultant, each of the forces / is ^'S 34- decomposed into two equal forces g, parallel and perpendicular to r. The relation in respect of magnitude thus produced between / and q is consequently the same as that between r and /. We have, accordingly : p = jj.. q and r = )j.. p; whence r = pfiq. THE PRINCIPLES OF ST A TICS. 41 Since, however, the forces q acting at right angles to r destroy each other, while those parallel to r con- stitute the resultant, it further follows that r ■= 2q; hence fi = "l/2, and r = t/2 . p. The resultant, therefore, is represented also in re- spect of magnitude by the diagonal of the square con- structed on J) 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- hand concerning the direction of the resultant r. If we decompose the components /, q (Fig. 35), parallel and perpendicular to the yet undetermined direction r, into the forces u, s and v, t, the new forces will form with the compo- nents p, q the same angles that p, q form with r. From which fact the following relations in respect of magnitude are determined : r p ,r q r p ,r a — = — and — ^ — , — ^ - and — = — , p u q V q s P t from which two latter equations follows s ^ i r^pqjr. On the other hand, however, -^or r'^=zp^ -f q^. The diagonal of the rectangle constructed on / and q represents accordingly the magnitude of the result- ant. Therefore, for all rhombs, the direction of the re- General re- sultant is determined ; for all rectangles, the magni- tude; and for squares both magnitude and direction. Bernoulli then solves the problem of substituting for r . suits. 42 THE SCIENCE OF MECHANICS. 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, there- fore, is to simulate towards himself a complete 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 pur- pose. On the contrary, we discover by such proce- dures, 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 over- look no experience which we involuntarily employ. What are the assumptions, then, contained in Bernoul- li's deduction? The as- 9. Statics, primarily, is acquainted with force only sumptions „ i r -1 ot hfs de- as a pull or a pressure, that from whatever source it rived from may come always admits of being replaced by the pull experience. ^^ ^^ pressure of a weight. All forces thus may be re- garded 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 it? 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 STATICS. 43 such as temperature, potential function, but not direc- tion. 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 unob- trusive experience. Granting, then, that the magnitude and direction Magnitude 1 ■ i-'--ii and direc- of 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 de- termine a motional effect. But if a force / is inclined at an an- gle to another direction s s' (Fig. 36), it is able to determine a mo- tion in that direction. Yet ex- ^' perience alone can inform us, that the motion is determined in the direction of s' s and not in that oi ss' ; that is to say, in the direction of the side of the acute angle or in the direction of the projection ol p on s's. Now this latter experience is made use of by Ber-The«if«rfot „, 1 r ii direction noulli at the very start. The sense, namely, of the re- derivable . • 1 1 i °o'y bom sultant of two equal forces acting at right angles to one experience. another is obtainable only on the ground of this expe- rience. 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 sur- render this latter determination, our whole proof is ex- ploded 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 MECHANICS. So also solely obtainable from experience, still less then shall must the , , . . . ^ . , i form of the We beliBve it in our power to ascertain by any other way thusde- th& form of this effect. It is utterly out of our power, to divine, that a force / 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 expe- riences that involve by implication this very mathe- matical fact. The man- A person already familiar with the composition ner in which the and resolution of forces is well aware that several forces assump- tions men- actingf at a point are, as regards their effect, replaceable, tionedenter. o f^ ... ' r > into Ber- in everv respect and in every direction, by a sintrle force. nouUi's de- . , , " duotion. This knowledge, in Bernoulli's mode of proof, is ex- pressed 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 equiv- alent of / and q. It is further a,ssumed as wholly in- different, whether we estimate s, u, t, v first in the directions of/, q, and then/, 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 knowl- edge of the fact referred to, by starting from the knowl- edge of another fact, namely that a force / 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 THE PRINCIPLES OF STATICS. 45 one and the same point at the angles a, a', a" . . . with Mathemat- . ' ical analy- a given direction X. These forces, let us suppose, are sis of the results of replaceable by a single force TI, which makes with Xthe true and . , . necessary an angle /<. By the familiar principle we have then assumption 2P cosa = 77 cospi. If II 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 2F cos {a -\- 6) ^ n cos {ij. -\- d), or {'S F cosa— n cos fX) cos8 — {2F sma — Ilsmfj) svad^d. If we put 2P cosa — n cosyU = A, — (2/' since — n sinju) = B, B tanr = -— , A it follows that A cosS + B sv^d = t/^2 + ^ sin ((J + t) = 0, which equation can subsist for every S only on the con- dition that A = 2F cosa — II cos ju = and B = {2F sina — 77 sin//) = ; whence results 77 coS/< = 2F cosa 77sinyu = 2 F sina. From- these equations follow for 7T and jj. the deter- minate values n=^l/li2Fsmay + {SFcosayi and 2F sina tanu = =-= . '^ 2Fcosa 46 , THE SCIENCE OF MECHANICS. The actual Granting, therefore, that the effect of a force in every results not - . . , - , . . . , deducibie direction can be measured by its projection on that di- othersup- rcction, then truly every system of forces acting at a 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 func- tion of an angle, cp (a). Yet if this be done, and we still regard the resultant as determinate, we shall obtain for 9>(a), as may be seen, for example, from Poisson's deduction, the form cos a. The experience that several forces acting at a point are always, in every respect, replaceable by a single force, is therefore mathemat- ically equivalent to the principle of the parallelogram of forces or to the principle of projection. The prin- ciple of the parallelogram or of projection is, how- ever, much easier reached by observation than the General re- more general experience above mentioned by statical observations. And as a fact, the principle of the par- allelogram was reached earlier. It would require in- deed an almost superhuman power of perception to deduce mathematically, without the guidance of any further knowledge of the actual conditions of the ques- tion, the principle of the parallelogram from the gen- eral principle of the equivalence of several forces to a single one. We criticise accordingly in the deduction of Bernoulli this, that that which is easier to observe is reduced to that which is more difficult to observe. This is a violation of the economy of science. Bernoulli is also deceived in imagining that he does not proceed from any fact whatever of observation. An addi- We must further remark that the fact that the forces tional as- sumption of are independent of one another, which is involved in Dernoulli, u i r i . the law of their composition, is another experience which Bernoulli throughout tacitly employs. As long THE PRINCIPLES OF STATICS. 47 as we have to do with uniform or symmetrical systems of forces, all equal in magnitude, each can be affected by the others, even if they are not independent, only to the same extent and in the same way. Given but three forces, however, of which two are symmetrical to the third, and even then the reasoning, provided we admit that the forces may not be independent, pre- sents considerable difficulties. 11. Once we have been led, directly or indirectly. Discussion to the principle of the parallelogram of forces, once we aoter of the have perceived it, the principle is just as much an ob- '"^'°"'' ^" servation as any other. If the observation is recent, it of course is not accepted with the same confidence as old and frequently verified observations. We then seek to support the new observation by the old, to demon- strate their agreement. By and by the new observa- tion acquires equal standing with the old. It is then no longer necessary constantly to reduce it to the lat- ter. Deduction of this character is expedient only in cases in which observations that are difficult directly to obtain can be reduced to simpler ones more easily obtained, as is done with the principle of the parallel- ogram of forces in dynamics. 12. The proposition of the parallelogram of forces Exijenmen- has also been illustrated by experiments especially tion of the instituted for the purpose. An apparatus very well I contriv- '' adapted to this end was contrived by Cauchy. The Cauohy. centre of a horizontal divided circle (Fig. 37) is marked by a pin. Three threads/,/',/", tied together at a point, are passed over grooved wheels r, r', r", which can be fixed at any point in the circumference of the circle, and are loaded by the weights /, p', p". If three equal weights be attached, for instance, and the wheels placed at the marks of division o, 1 20, 240, the point at THE SCIENCE OF MECHANICS. Experimen- which the Strings are knotted will assume a position tal illustra- , . ^ . , _,, , tionof the lust above the centre of the circle. Ihree equal forces principle. . t r n i- i acting at angles of 120 , accordingly, are in equilib- rium. Fig- 37- If we wish to represent another and different case, we may proceed as follows. We imagine any two forces /, q acting at any angle a, represent (Fig. 38) them by lines, and construct on them as sides a paral- lelogram. We supply, further, a force equal and opposite to the resultant r. The three forces p, q, — r hold each other in equilibrium, at the angles vis- ible from the construction. We now place the wheels of the divided circle on the points of division o, a, a -\- ft, and load the appropriate strings with the weights p, q, r. The point at which the strings are knotted will come to a position exactly above the middle point of the circle. Fig. 38. THE PRINCIPLES OF STATICS. 49 THE PRINCIPLE OF VIRTUAL VELOCITIES. I. We now pass to the discussion of the principle The truth . . , , . ^ ^ of the prin- of Virtual (possible) displacements.* The truth of cipie first . . remarked this principle was first remarked by Stevinus at the by stevmus close of the sixteenth century in his investigations on - the equilibrium of pulleys and combinations of pulleys. Stevinus treats combinations of pulleys in the same way they are treated at the present day. In the case *Termed in English the principle of " virtual velocities," this being the original phrase {vttesse viriuelle) introduced by John Bernoulli. See the text, page 56. The word virtualis seems to have been the fabrication of Duns Scotus (see the Century Dictionary^ under virtual) ; but virtualiter was used by Aquinas, and virtus had been employed for centuries to translate dvvafiig, and therefore as a synonym for potentia. Along with many other scholastic terms, virtual passed into the ordinary vocabulary of the English language. Everybody remembers the passage in the third book of Paradise Lost, " Love not the heav'nly Spirits, and how thir Love Express they, by looks onely, or do they mix Irradiance, vir(uai or immediate touch? " — Miilon. So, we all remember how it was claimed before our revolution that America had " z/zV/wtj:/ representation " in parliament. In these passages, as in Latin, virtual means : existing in effect, but not actually. In the same sense, the word passed into French ; and was made pretty common among philosophers by Leibnitz. Thus, he calls innate ideas in the mind of a child, not yet brought to consciousness, "des connoissances virtuelles.*' The principle in question was an extension to the case of more than two forces of the old rule that "what a machine gains \nPower, it loses in velocity.'^ Bernoulli's modification reads that the sum of the products of the forces into their virtual velocities must vanish to give equilibrium. He says, in effect : give the system any possible and infinitesimal motion you please, and then the simultaneous displacements of the points of application of the forces, resolved in the directions of those forces, though they are not exactly velocities, since they are only displacements in one time, are, nevertheless, virtually velocities, for the purpose of applying the rule that what a machine gains in power, it loses in velocity. Thomson and Tait say : "If the point of application of a force be dis- placed through a small space, the resolved part of the displacement in the di- rection of the force has been called its Virtual Velocity. This is positive or negative according as the virtual velocity is in the same, or in the opposite, direction to that of the force." This agrees with Bernoulli's definition which may be found in Varignon's Nouvelle mecanique^ Vol. II, Chap. \%.— Trans, 5° THE SCIENCE OF MECHANICS. stevinus's a (Fig. 39) equilibrium obtains, when an equal weight P dons on the is suspended at each side, for reasons already familiar. equilibrium . . 111 n 1 1 of pulleys. In b, the weight P is suspended by two parallel cords. each of which accordingly supports the. weight Pji, with which weight in the case of equilibrium the free end of the cord must also be loaded. In c, P is sus- pended by six cords, and the weighting of the free ex- tremity with PIb will accordingly produce equilibrium. In d, the so-called Archimedean or potential pulley,* P in the first instance is suspended by two cords, each of which supports Pji ; one of these two cords in turn is suspended by two others, and so on to the end, so that the free extremity will be- held in equilibrium by the weight /"/S. If we impart to these assemblages of pulleys displacements corresponding to a descent of the weight P through the distance h, we shall observe that as a result of the arrangement of the cords the counterweight P " " Pji P/b P/S will ascend a distance .^ in a " " 2/1 " b " " 6/4 " c " " %h " d ♦These terms are not in use in English. — Trans. THE PRINCIPLES OF STATICS. 51 Fig. 40. In a system of pulleys in equilibrium, therefore, the products of the weights into the displacements they sustain are respectively equal. (" Ut spatium agentis ad spatium patientis, sic potentia patientis ad potentiam agentis." — Stevini, Hypomnemata, T. IV, lib. 3, p. 172.) In this remark is contained the germ of the principle of virtual displacements. 2. Galileo recognised the truth of the principle in another case, and that a somewhat more general one ; namely, in its application to the inclined plane. On an inclined plane (Fig. 40), the length of which A£ is double the height BC, z. load Q placed on AB is held in equilibrium by the load F act- ing along the height BC, if P = Q/2.. If the machine be set in motion, F = Q/2 will descend, say, the vertical distance A, and Q will ascend the same distance k along the incline AB. Galileo, now, allowing the phenom- enon to exercise its full effect on his mind, perceives, that equilibrium is determined not by the weights alone but also by their possible approach to and reces- sion from the centre of the earth. Thus, while Qji de- scends along the vertical height the distance h, Q as- cends h along the inclined length, vertically, however, only h/i ; the result being that the products Q^h/i) and {Q/2)h come out equal on both sides. The eluci- dation that Galileo's observation affords and the light it diffuses, can hardly be emphasised strongly enough. The observation is so natural and unforced, moreover, that we admit it at once. What can appear simpler than that no motion takes place in a system of heavy His conclu- sions the germ of the principle. Galileo's recognition of the prin- ciple in the case of the inclined plane. Character of Galileo's observation 52 THE SCIENCE OF MECHANICS. bodies when on the whole no heavy mass can descend. Such a fact appears instinctively acceptable. Comparison Galileo's conception of the inclined plane strikes of it with . _ . . thatofste- US as much less ingenious than that of Stevmus, but vinus. we recognise it as more natural and more profound. It is in this fact that Galileo discloses such scientific great- ness : that he had the intellectual audacity to see, in a subject long before investigated, more than his prede- cessors had seen, and to trust to his own perceptions. With the frankness that was characteristic of him he unreservedly places before the reader his own view, together with the considerations that led him to it. TheTorri- 3. ToRRiCELLi, by the employment of the notion of cellian j r j form of the "centre of gravity," has put Galileo's principle in a principle. , . . . . . form in which it appeals still more to our instincts, but in which it is also incidentally applied by Galileo him- self. According to Torricelli equilibrium exists in a machine when, on a displacement being imparted to it, the centre of gravity of the weights attached thereto cannot descend. On the supposition of a displacement in the inclined plane last dealt with, P, let us say, de- scends the distance h, in compensation wherefor Q vertically ascends h . sin a. Assuming that the centre of gravity does not descend, we shall have P.h — 0.hsva.a „ „ , „ , . p-^Q = 0, or f. /z — (2 . /^ sm a- — 0, or /'=<2sina = (24^. If the weights bear to one another some different pro- portion, then the centre of gravity can descend when a displacement is made, and equilibrium will not obtain. We expect the state of equilibrium instinctively, when the centre of gravity of a system of heavy bodies can- THE PRINCIPLES OF STA TICS. 53 not descend. The Torricellian form of expression, how- ever, contains in no respect more than the Galilean. 4. As with systems of pulleys and with the inclined The appii- plane, so also the validity of the principle of virtual the prmci- displacements is easily demonstrable for the other ma- other ma- chines : for the lever, the wheel and axle, and the rest. In a wheel and axle, for instance, with the radii R, r and the respective weights P, Q, equilibrium exists, as we know, when FJi = Qr. If we turn the wheel and axle through the angle a, P will descend Ra, and Q will ascend ra. According to the conception of Stevinus and Galileo, when equilibrium exists, P. Ra = Q . ra, which equation expresses the same thing as the preceding one. 5. When we compare a system of heavy bodies inThecnte- ... . . , . , . , • -1 rionofthe which motion is taking place, with a similar system state of ..... -1-1 ■ ^ • r • ir equilibrium which is m equilibrium, the question forces itself upon us : What constitutes the difference of the two cases? What is the factor operative here that determines mo- tion, the factor that disturbs equilibrium, — the factor that is present in the one case and absent in the other? Having put this question to himself, Galileo discovers that not only the weights, but also the distances of their vertical descents (the amounts of their vertical displacements) are the factors that determine motion. Let us call P, P' , P" . . . the weights of a system of heavy bodies, and h, h' , h" . . . their respective, simul- taneously possible vertical displacements, where dis- placements downwards are reckoned as positive, and displacements upwards as negative. Galileo finds then, that the criterion or test of the state of equilib- rium is contained in the fulfilment of the condition Ph + P' h' -Y P" h" -\- . . = 0. The sum Ph -\- P'h' _|_ p"h"-\- ... is the factor that destroys equilibrium, 54 THE SCIENCE OF MECHANICS. the factor that determines motion. Owing to its im- portance this sum has in recent times been character- ised by the special designation work. There is no 6. Whereas the earher investigators, in the compari- our choice SOD of cases of equiHbrium and cases of motion, directed teria^ °" their attention to the weights and their distances from the axis of rotation and recognised the statical mo- ments as the decisive factors involved, Galileo fixes his attention on the weights and their distances of de- scent and discerns -work as the decisive factor involved. It cannot of course be prescribed to the inquirer what mark or criterion of the condition of equilibrium he shall take account of, when several are present to choose from. The result alone can determine whether his choice is the right one. But if we cannot, for rea- And all are sons already stated, regard the significance of the stat- from the ical moments as given independently of experience, as source. Something self-evident, no more can we entertain this view with respect to the import of work. Pascal errs, and many modern inquirers share this error with him, when he says, on the occasion of applying the principle of virtual displacements to fluids: "Etant clairque c'est la meme chose de faire faire un pouce de chemin a cent livres d'eau, que de faire faire cent pouces de chemin a une livre d'eau. " This is correct only on the suppo- sition that we have already come to recognise work as the decisive factor ; and that it is so is a fact which experience alone can disclose. Illustration If we have an equal-armed, equally-weighted lever of the pre- . , ceding re- beiore US, we recognise the equilibrium of the lever as the only effect that is uniquely determined, whether we regard the weights and the distances or the weights and the vertical displacements as the conditions that determine motion. Experimental knowledge of this THE PRINCIPLES OF STATICS. 55 or a similar character must, however, in the necessity of the case precede any judgment of ours with regard to the phenomenon in question. The particular way in which the disturbance of equilibrium depends on the conditions mentioned, that is to say, the significance of the statical moment {_PL) or of the work {Ph'), is even less capable of being philosophically excogitated than the general fact of the dependence. 7. When two equal weights with equal and op- Reduction posite possible displacements are opposed to each erai case of other, we recognise at once the subsistence of equilib- pie to the . -11, simpler and num. We might now be tempted to reduce the more special case general case of the weights P, P' with the capacities of displacement^,^', where Ph = P'h', to the sim- pler case. Suppose we have, for example, (Fig. 41) the weights 3 P and 4 jP on a wheel and axle with the radii 4 and 3. We divide the weights into equal portions of the definite magnitude P, which we designate by a, b, c, ^' ^> /■> S- We then transport a, b, c to the level + 3, and d, e, f to the level — 3. The weights will, of themselves, neither enter on this displacement nor will they resist it. We then take simultaneously the weight g at the level and the weight a at the level + 3, push the first upwards to — i and the second downwards to -|- 4, then again, and in the same way, ^ to — 2 and i5 to + 4, ^ to — 3 and ^ to -|- 4- To all these displacements the weights offer no resistance, nor do they produce them of themselves. Ultimately, however, a, b, c (or j,P^ appear at the level -|- 4 and Fig. 41. 56 THE SCIENCE OF MECHTlNICS. The gen- d, e, f. z (ox 4F) at the level — 3. Consequently, eralisation. .,,, , . j^^ij-i With respect also to the last-mentioned total displace- ment, the weights neither produce it of themselves nor do they resist it ; that is to say, given the ratio of displacement here specified, and the weights will be in equilibrium. The equation 4.3-^ — 3 . 4/^= is, therefore, characteristic of equilibrium in the case as- sumed. The generalisation (Fk — F'k' =^ 0) is ob- vious. Thecondi- If we Carefully examine the reasoning of this case, character we shall quite readily perceive that the inference in- ence.° '° ^"^ volved Cannot be drawn unless we take for granted that the order of the operations performed and the /atA by which the transferences are effected, are indifferent, that is unless we have previously discerned that work is determinative. We should commit, if we accepted this inference, the same error that Archimedes com- mitted in his deduction of the law of the lever ; as has been set forth at length in a preceding section and need not in the present case be so exhaustively dis- cussed. Nevertheless, the reasoning we have pre- sented is useful, in the respect that it brings palpably home to the mind the relationship of the simple and the complicated cases. Theuniver- 8. The universal applicability of the principle of bTiUy of fhe virtual displacements to all cases of equilibrium, was Sfstper-^ perceived by John Bernoulli ; who communicated his ceived by j . ^ tt ■ • 1 • ■ ^Tr John Ber- Qiscovery to Varignon m a letter written in 1717. We will now enunciate the principle in its most general form. At the points A, B, C . . (Fig. 42) the forces P, P' , P" ■ ■ . are applied. Impart to the points any infinitely small displacements v, v' , v" . . . compatible with the character of the connections of the points (so- called virtual displacements), and construct the pro- noulli. THE PRINCIPLES OF STATICS. 57 jections/, /', J>" of these displacements on the direc- General tions of the forces. These projections we consider of the prin- ciple. PB positive when they fall in the direction of the force, and negative when they fall in the opposite direction. The products Fp, F'p', P" p", . . are called virtual moments, and in the two cases just mentioned have Fig. 42. contrary signs. Now, the principle asserts, that for the case of equilibrium Pp + P' p' + P" p" -|- . = 0, or more briefly '2Pp,=^ 0. 9. Let us now examine a few points more in detail. Detailed Previous to Newton a force was almost universally tion of the conceived simply as the pull or the pressure of a heavy body. The mechanical researches of this period dealt almost exclusively with heavy bodies. When, now, in the Newtonian epoch, the generalisation of the idea of force was effected, all mechanical principles known to be applicable to heavy bodies could be transferred at once to any forces whatsoever. It was possible to replace every force by the pull of a heavy body on a string. In this sense we may also apply the principle of virtual displacements, at first discovered only for heavy bodies, to any forces whatsoever. F?W««/ displacements are displacements consistent Definition ■ 1 1 1 r 1 - r- 1 of virtual With the character of the connections of a system and dispiace- with one another. If, for example, the two points of a system, A and B, at which forces act, are connected (Fig. 43, i) by a rectangularly bent lever, free to re- volve about C, then, if CB = iCA, all virtual dis- placements of B and A are elements of the arcs of cir- cles having C as centre ; the displacements of B are 1 B 58 THE SCIENCE OF MECHANICS. always double the displacements of A, and both are in every case at right angles to each other. If the points A, B (Fig. 43, 2) be connected by a thread of the length /, adjusted to slip through ^ r stationary rings at C and D, I \ then all those displacements "^ 2 ^ °f ^ ^'^^ ^ ^^^ virtual in Fig. 43. which the points referred to move upon or within two spherical surfaces described with the radii r^ and r^ about C and D as centres, where r j^ -(- r^ -\- CD = /. The reason The use of infinitely small displacements instead of for the use /..i-i 1 r^ ^•^ j*.- of m&ra.ie\y finite displacements, such as Cjalileo assumed, is justi- piacements. fied by the following consideration. If two weights are in equilibrium on an inclined plane (Fig. 44), the equilibrium will not be disturbed if the inclined plane, at points at which it is not in immediate contact with the bodies considered, passes into a surface of a different form. The essential condition is, therefore, the momentary possibility of dis- Fig. 44. placement in the momentary con- figuration of the system. To judge of equilibrium we must assume displacements vanishingly small and such only ; as otherwise the system might be carried over into an entirely different adjacent configuration, for which perhaps equilibrium would not exist. A limita- That the displacements themselves are not decisive but only the extent to which they occur in the direc- tions of the forces, that is only their projections on the lines of the forces, was, in the case of the inclined plane, perceived clearly enough by Galileo himself.- With respect to the expression of the principle, it will be observed, that no problem whatever is presented tion. ■ THE PRINCIPLES OF STA TICS. 59 if all the material points of the system on which forces General re- act, are independent of each other. Each point thus conditioned can be in equilibrium only in the event that it is not movable in the direction in which the force acts. The virtual moment of each such point vanishes separately. If some of the points be independent of each other, while others in their displacements are de- pendent on each other, the remark just made holds good for the former ; and for the latter the fundamental proposition discovered by Galileo holds, that the sum of their virtual moments is equal to zero. Hence, the sum-total of the virtual moments of all jointly is equal to zero. lo. Let us now endeavor to get some idea of the Examples, significance of the principle, by the consideration of a few simple examples that cannot be dealt with by the ordinary method of the lever, the inclined plane, and the like. The differential pulley of Wes- ton (Fig. 45) consists of two coax- ial rigidly connected cylinders of slightly different radii r .^ and r^ <,r.^. A cord or chain is passed round the cylinders in the manner indicated in the figure. If we pull in the direction of the arrow with the force/', and rotation takes place through the angle tp, the weight Q attached below will be raised. In the case of equilibrium there will exist between the two virtual moments involved the equa- tion Fig. 45- '^cp^Pr^ cp, or P= q'- •2r, 6o THE SCIENCE OF MECHANICS. A suspend- ed wheel and azle. A double cylinder on a horizon- tal surface. Roberval's balance. A wheel an d axle of weight Q (Fig. 46), which on the unrolling of a cord to which the weight P is at- tached rolls itself up on a second cord wound round the axle and rises, gives for the virtual moments in the case of equilibrium the equation P{R-r~)- r • 1 - fundamen- mass of water A (rig. 60), immersed in water, is intaiprinci- pl6. equilibrium in all its parts. If A — A Fig. 60. were not supported by the sur- rounding water but should, let us say, descend, then the portion of water taking the place of A and placed thus in the same circum- stances, would, on the same as- sumption, also have to descend. This assumption leads, therefore, to the establishment of a perpetual motion, which is contrary to our ex- perience and to our instinctive knowledge of things. Water immersed in water loses accordingly its The second whole weight. If, now, we imagine the surface of the tai princi- submerged water solidified, the vessel formed by this surface, the vas superficiarium as Stevinus calls it, will still be subjected to the same circumstances of pres- sure. If empty, 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 90 THE SCIENCE OF MECHANICS. Stevinus's deductions. Galileo, in the treat- ment of this subject, em- ploys the principle of virtual dis- placements base is equal to the weight of the liquid. The pressure is 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 so- lidified, 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. 5. Galileo treats the equilibrium of liquids in com- municating vessels and the problems connected there- with by the help of the principle of virtual displace- ments. NN (Fig. 61) being the common level of a liquid in equilib- rium in two communicating vessels, Galileo explains the equilibrium here presented by observing that in the case of any disturbance the dis- placements of the columns are to each other in the inverse proportion of the areas of the transverse sec- tions and of the weights of the columns — that is, as with machines in equilibrium. But this is not quite cor- rect. The case does not exactly, correspond to the cases of equilibrium investigated by Galileo in ma- chines, 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 B A ^ ■ S N ^ - _ _ — — - - - - _ . _ L^ Fig. 61. THE PRINCIPLES OF STATICS. 91 by Pascal. 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 displacements, but in a more correct manner, leaving made use of 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 surface- areas, equilibrium will obtain, because in consequence of the invariability of the volume of the liquid the displace- ments in every disturbance are in- versely proportional to the weights. Fig. 62. For Pascal, accordingly, \t follows, as a necessary con- sequence, from the principle of virtual displacements, that in the case of equilibrium every pressure on a su- perficial portion of a liquid is propagated with undi- minished effect to every other superficial portion, how- ever and in whatever position it be placed. No objec- tion 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 examine the most important cases of liquid equilibrium, tion of tiie and from such different points of view as may be con- venient. 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 our- selves an element of volume of a liquid, the gravity of which we disregard^say a tiny cube. If the slightest 92 THE SCIENCE OF MECHANICS. The funda- excess of pressure be exerted on one of the surfaces of property o( this cube, (which we now conceive, for the moment, moMifty o^f as a fixed geometrical locus, containing the fluid but parts. ^^^ ^^ .^^ substance) the liquid (supposed to have pre- viously 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 polyhe drons. In this conception, as thus geometrically eluci- dated, is contained nothing but the crude experience that the particles of a liquid yield to the slightest pres- sure, and that they retain this property also in the in- terior of the liquid when under a high pressure ; it being observable, for example, that under the condi- tions cited minute heavy bodies sink in fluids, and so on. A second With the mobility of their parts liquids combine the com- still another property, which we will now consider. Li- pressibility . , .. , , ... . . . . of their vol- quids sufter through pressure a diminution oi volume 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 STATICS. 93 -.Ji Fig. 63. of water was demonstrated. A thermometer glass is filled with water, boiled, and then sealed. (Fig. 63.) The liquid reaches to a. But since the space above a is airless, the liquid supports no atmospheric pres- sure. 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 compression of the liquid by atmospheric pres- sure. 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 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 com- pression of the liquid by the pressure of the atmos- phere. The first to institute exact experiments on the com- pressibility of water, was Oersted, who employed to this end a very ingenious method. A thermometer glass A (Fig. 64) is filled with boiled water and is inverted, with 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 ap- paratus is then placed in a vessel filled with water, which is compressed by the aid of a pump. By this means the water in A is also compressed, and the filament of quicksilver which rises in the capillary tube of the thermometer- The first demonstra- tion of the compressi- bility of liquids. The experi- ments of Oersted on this subject. B Fig. 64. 94 THE SCIENCE OF MECHANICS. glass indicates this compression. The alteration of capacity which the glass A suffers in the present in- stance, is merely that arising from the pressing to- gether of its walls by forces which are equal on all sides. The experi- The most delicate experiments on this subject have Grassi. been conducted by Grassi with an apparatus con- structed by Regnault, and computed with the assist- ance of Lamp's correction-formulae. To give a tan- gible 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 100,000 parts.- If we imagine, accordingly, the vessel A to have the capacity of one litre (1000 ccm.), and affix to it a cap- illary tube of I sq. mm. cross-section, the quicksilver filament will ascend in it 5 cm. under a pressure of one atmosphere. Surface- g. Surface-pressure, accordingly, induces a physical pressure in- , . . , . . , , , . . , , , . i duces in alteration in a liquid (an alteration in density), which alteration can be detected by sufficiently delicate means — even optical. We are always at liberty to think that por- tions of a liquid under a higher pressure are more dense, though it may be very slightly so, than parts under a less pressure. The impii- Let US imagine now, we have in a liquid (in the in- cations of . .,., ^ .,,., this fact, tenor of which no forces act and the gravity of which we accordingly neglect) two portions subjected to un- equal 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 equilib- rium at the bounding surface and both portions are equally compressed. THE PRINCIPLES OF STATICS. 95 If we endeavor, now, quantitatively to elucidate our The state- mental conception of these two facts, the easy mobility these impU- 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 prin- ciple have, perhaps, never been instituted with the re- quisite degree of exactitude. But the proposition has by our experience of liquids been made very familiar, and readily explains it. lo. If a liquid be enclosed in a vessel (Fig. 65) Preiimi- nary re- which is supplied with a piston A, the cross-section marks to • -., . _,,., the discuss- of which is unit m area, and with a piston J3 which ion of Pas- cal's deduc- for the time being is made station- cta^ '""' ary, and on the piston A a load / z^^^^mit. be placed, then the same pressure ^I-??_-_ ^^ ^ p, gravity neglected, will prevail l ^££-^- ^r^^ throughout all the parts of the vessel, ^ ^g^^ g^^ The piston will penetrate inward and ^^^^^ ^^^ ^ the walls of the vessel will continue ^^q^'^'mm^ to be deformed till the point is reached P'S- ^s- 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/, to be mov- able, a. force /./ alone will keep it in equilibrium. Concerning Pascal's deduction of the proposition before discussed from the principle of virtual displace- ments, it is to be remarked that the conditions of dis- 96 THE SCIENCE OF MECHANICS. Criticism of placement which he perceived hinge wholly upon the ducfkin! ^"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 dis- placements are assured by the elastic forces of the lever-body, which do not permit of any great devia- tion from these relations. Thebehav- j i. We shall now consider the action of liquids un- louror 11- _ _ ^ * quids under (Jer the influence of gravity. The upper surface of a the action o J jr j- of gravity. liquid in equilibrium is horizontal, NN (^\g. 66). This fact is at once rendered intelligible when we re- flect that every alteration of the sur- face in question elevates the centre of gravity of the liquid, and pushes the liquid mass resting in the shaded space beneath NN and having the centre of gravity 5 into the shaded space above NN having the centre of gravity S' . Which alteration, of course, is at once re- versed by gravity. Let there be in equilibrium in a vessel a heavy liquid with a horizontal upper surface. We consider N N Fig. 66. THE PRiyCIPLES OF STATICS. 97 ity. (Fig. 67) a small rectangular parallelepipedon in the The con- interior. The area of its horizontal base, we will say, is eqi?ubriu a, and the length of its vertical edges dh. The weight subjecled of this parallelepipedon is therefore adhs, where s istfonofgrav- its specific gravity. If the paral- lelepipedon do not sink, this is possible only on the condition that a greater pressure is exerted on the P+dp lower surface by the fluid than on the upper. The pressures on the upper and lower surfaces we will ^^^- ^7- respectively designate as ap and a {p -\- dp). Equi- librium obtains when adh.s =^ adp or dpjdh^ s, where h in the downward direction is reckoned as posi- tive. We see from this that for equal increments of h vertically downwards the pressure / must, correspond- ingly, also receive equal increments. So that / = hs-\'q; and if q, the pressure at the upper surface, which is usually the pressure of the atmosphere, be- comes = 0, we have, more simply, p ^^hs, that is, the pressure is proportional to the depth beneath the sur- face. If we imagine the liquid to be pouring into a ves- sel, and this condition of affairs not yet attained, every liquid particle will then sink until the compressed par- ticle 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 ther apparent, that the increase of pressure in a liquid tions exist takes place solely in the direction in which gravity ilSeofthe^ 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, gS THE SCIENCE OF 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 points of the liquid at which the same pressure / 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 dis- place 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 ob- servations are admissible if the liquid particles be acted on by other forces than gravity, magnetic forces, for example. Theirfuno- The level surfaces afford, in a certain sense, a dia- 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 be- low the surface of a heavy liquid may be illustrated by a series of experiments which we chiefly owe to Pas- cal. 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 //, to which a string is attached, and the whole plunged into a vessel of water. When immersed to a sufHcient depth we may let the string go, without the metal disc, which is THE PRINCIPLES OF STATICS. 99 c supported by the pressure of the liquid, falling. In 2, the metal disc is replaced by a tiny column of mer- cury. If (3) we dip an open siphon tube filled with mercury into the water, we shall see the mercury, in consequence of the pressure at a, rise into the longer arm. In 4, we see a tube, at the lower extremity of which a leather bag filled with mercury is tied : continued im- mersion forces the mercury higher and higher into the tube. In 5, a piece of wood h is driven by the pressure of the water into the small arm of an empty siphon tube. In 6, a piece of wood H immersed in mercury adheres to the bottom of the vessel, and is pressed firmly against it for as long a time as the mercury is kept from working its way un- derneath it. 13. Once we have made quite clear to ourselves that the pres- sure in the interior of a heavy liquid increases proportionally to the depth below the surface, the law that the pressure at the base of a vessel is independent of its form will be readily perceived. The pressure increases as we de- scend at an equal rate, whether the vessel (Fig. 69) has the form abed or ebcf. In both cases the walls of the vessel where they meet the liquid, go on deforming Pascal's ex- periments on the pressure of liquids. e\i The pres- sure at the base of a vessel inde- pendent of its form. THE SCIENCE OF MECHANICS. Elucida- tion of this fact. Fig. 69. The princi- ple of vir- tual dis- placements applied to the consid- eration of problems of this class. 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 a direct justification of Ste- vinus's fiction of the solidi- fied fluid supplying the place of the walls of the vessel. The pressure on the base always remains Pz^Ahs, 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. The fact that, the walls of the vessel being neg- lected, the vessels i, 2, 3 of Fig. 70 of equal base- area and equal pressure-height weigh differently in the / '> 3 balance, of course in no wise con- ^\-* tradicts the laws of pressure men- 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. 14. The principle of virtual displacements is ad- mirably adapted to the acquisition of clearness and comprehensiveness in cases of this character, and we shall accordingly make use of it. To begin with, how- 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 STATICS. I Prelimi- "^ nary re- 2 to 3, the work performed in this operation is g h, qh^ =? (-^1 + /zj)' *^^ same, that is, as if the weight ■"^'^'^^ q passed directly from i to 3 and the weight at 2 re- mained in its original position. The observation is easily generalised. / k, 2 3 tu Fig. 71. Let us consider a heavy homogeneous rectangular parallelepipedon, with vertical edges of the length h, 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 Ahs .dh, or, also, A dhs .h. In the first expres- sion we conceive the whole weight Ahs displaced the vertical distance dh ; in the second we conceive the weight A dhs 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. f Both methods of concep- ! tion are admissible and f equivalent. 15. With the aid of this observation we shall obtain a clear insight into 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, Fig. 73- I02 THE SCIENCE OF MECHANICS. which, by means of a string passing through the axis of the cyhnders, 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 consider- able size, the sum of which will be A lis, 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 expia- Let US look to the virtual displacements of the two nation of _,. ^ t i /> - i - the paradox cases (rig. 74). In the first case, supposing the pis- ton to be lifted a distance dh, the virtual moment is Adhs .h ox Ahs.dh. It thus comes to the same thing, \dh whether we consider the m ass that the motion of the piston / dmimm •= — ¥ii displaces to be lifted to the upper surface of the fluid ^'^' ''''■ through the entire pressure- height, or consider the entire weight Ahs lifted the distance of the piston-displacement dh. In the second case, the mass that the piston displaces is not lifted to the upper surface of the fluid, but suffers a displace- ment which is much smaller— the displacement, namely, of the piston, li A, a are the sectional areas respect- ively of the greater and the less cylinder, and k and / their respective heights, then the virtual moment of the present case is Adhs . k -\-' adhs . I ^ {A k -\- al) s. dh; which is equivalent to the lifting of a much smaller weight (^A k -\- al) s, the distance dh. 16. The laws relating to the lateral pressure of liquids are but slight modifications of the laws of ba,sal THE PRINCIPLES OF STATICS. 103 pressure. If we have, for example, a cubical vessel The laws of of I decimetre on the side, which is a vessel of litre pressure, capacity, the pressure on any one of the vertical lateral walls A BCD, 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 rep- resented by a wedge of water AB CD HI resting upon the wall horizontally y] £ placed, where ID is at right angles to BD and ID = IICz=AC. The lateral pressure accor- ^^ dingly is equal to half a kilogramme. To determine the point of application of the resultant pressure, conceive A BCD again horizontal with the water-wedge resting upon it. We cut o^ AK =z BL =^IAC, draw the straight line KL and bisect it at M; M 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 pres- surG on £L filled with a liquid, is divided into the elements a, a , plane in- a' . . with the depths h, h' , h" . . . below the level of the liquid. The pressure on the base is {ah-\- oi 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 + a'h' + a"h" -f . . . ah-\- a'h' + Fig- 75- a + a' + a" + . . . A whence the pressure on the base is AHs. -=H, I04 THE SCIENCE OF MECHANICS. The deduc- tion of the principle of Archime- des maybe effected in various ways. One meth- od. Another method in- volving the principle of virtual dis- placements. 17. The principle of Archimedes can be deduced in various ways. After the manner of Stevinus, let us conceive in the interior of the liquid a portion of it solidified. This portion now, as before, will be sup- ported by the circumnatant liquid. The resultant of the forces of pressure acting on the surfaces is accor- dingly applied at the- centre of gravity of the liquid dis- placed by the solidified body, and is equal and opposite to its weight. If now we put in the place of the solid- ified 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 up- ward 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. If we immerse a rectangular parallelepipedon of al- titude h and base a, 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 {It -\- h) s. As the lateral pressures destroy each other, an excess of pressure ahs upwards re- mains ; or, where v denotes the volume of the paral- lelepipedon, an excess v . s. We shall approach nearest the fundamental con- ception from which Archimedes started, by recourse to the principle of virtual displacements. Let a paral- lelepipedon (Fig. 76) of the specific gravity ff, base a, and height A sink the distaiice d/i. The virtual mo- ment of the transference from the upper into the lower shaded space of the figure will he adk. ff/i. But while THE PRINCIPLES OF STA TICS. 105 this is done, the liquid rises from the lower into the up- per space, and its moment is adhsh. The total vir- tual moment is therefore ah {a — s)dh^{^p — q) dh, where / denotes the weight of the body and q the weight of the displaced liquid. B Fig. 76. Fig. 77. 18. The question might occur to us, whether the upward pressure of a body in a liquid is affected by the. immersion of the latter in another liquid. As a fact, this very question has been proposed. Let therefore (Fig. 77) a body K be submerged in a liquid A and the 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- ments, we easily comprehend the more complicated cases of this character. If a body be first gradually immersed in B, then partly in B and partly in A, finally in A wholly ; then, in the second case, consider- ing 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- Is the buoy- ancy of a body in a liquid af- fected by the immer- sion of that liquid in a second liquid? The eluci- dation of more com- plicated cases of this class. io6 THE SCIENCE OF MECHANICS. The Archi- medean principle il- lustrated by an experi- ment. The coun- ter-experi- ment, Remarks on the experi- ment. placement no longer rises, and therefore B is no longer of consequence. 19. Archimedes's principle maybe illustrated by a pretty experiment. From the one extremity of a scale- beam (Fig. 78) we hang a hollow cube H, and beneath it a solid cube M, 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 beneath it, the equilibrium will be dis- turbed ; but it will be immediately re- stored if H, the hollow cube, be filled with water. A counter-experiment is the follow- ing. H is left suspended alone at the one extremity of the balance, and into the opposite pan is placed a vessel of water, above which on an independent support J/hangs by a thin wire. The scales are brought to equilibrium. If nowJ/be lowered until it is im- mersed in the water, the equilibrium of the scales will be disturbed ; but on filling H with water, it will be restored. At first glance this. experiment appears a little para- doxical. We feel, however, instinctively, that M can- not be immersed in the water without exerting a pres- sure 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 van- ishes. Fig. 78. THE PRINCIPLES OF STATICS. 107 20. The most important statical principles have The gene- been reached in the investigation of solid bodies. This pies of stat- ■j 111 7. -7 1 • • -i ics might course is accidentally the historical one, but it is by no have been means the only possible and necessary one. The dif- the investi- 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 princi- ples, might, with the assistance of some very simple propositions from the statics of rigid bodies, have been reached in the investigation of liquids. Stevinus cer- tainly 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 dis- lect. Let this liquid be enclosed in a vessel and sub- illustration jected to a definite pressure. A portion of the liquid, statement. let us suppose, solidifies. On the closed surface nor- mal 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 non- closed 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 plane curve as the perimeter of its base, solidifies. We may neglect the two basal sur- faces, perpendicular to the axis. And instead of the cylindrical surface the closed curve simply may be con- sidered. From this method follow quite analogous io8 THE SCIENCE OF MECHANICS. The dis- propositions for normal forces proportional to the ele- cussion and . , illustration ments ot a plane curve. statement. If the closed curve pass into a triangle, the con- sideration 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 magni- tude by straight lines (Fig. 80). The _ lines mentioned intersect at a point — the centre of the circle described about the triangle. It will further be noted. Fig. 80. 'CnsX by the simple parallel displace- ment of the lines representing the forces a triangle is constructible which is similar and congruent to the original triangle. Tiiededuc- Thence follows this proposition : tion of the Air i • i ■ triangle of Any three forces, which, actmg at a pomt, are pro- forces bv this method portional and parallel in direction to the sides of a tri- angle, 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 n a portion solidified. On the element a of the closed encompassing surface there acts a normal force anz, where %\& 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 anz, duction of , . another im- where a denotes an element ot area and z its perpen- portant pro- , . , , . . . , ^ , , position, dicular distance from a given plane E, act on a closed surface inwards, the resultant will be V. h, in which ex- pression V represents the enclosed volume. The THE PRINCIPLES OF STATICS. 109 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 propo- be bounded by a plane curve, which encloses on the deduced, a plane the area ^. The resultant of the forces acting of Greens . Theorem. on the curved surface is R, where E^ ^ {AZ uy + {Vny — AZVk'^ cos V, 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 mathe- matically practised readers will have recognised a par- ticular case of Green's Theorem, which consists in the reduction of surface-integrations to volume-integra- tions or vice versa. We may, accordingly, see into the force-system of aTheimpii- fluid in equilibrium, or, if you please, see out of it, sys- the view tems of forces of greater or less comple'xity, and thus 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 re- ,. .. ri--i T-i -1 [ suits of the the investigation of liquids, supplied a stimulus to fur- investiga- . 1T11J1 LTX tionsofthis ther reflection and research. It should also not be left domain, unnoticed, that the conception of a physico-mechanical continuum was first formed on the occasion of the in- vestigation of liquids. A much freer and much more fruitful mathematical mode of view was developed thereby, than was possible through the study even of no 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 de- . . . , . . , 1-1 partment of in the investigation of liquids are applicable with but 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 eius- Although the ordinary man has abundant oppor- its subject- tunity, by his experience of the resistance of the air, by 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 popu- lar 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 . __.. . ^ . , . i disclosures accounts 01 Vitruvius, possessed instruments which, ince. like the so-called hydraulic organs, were based on the condensation of air, although the invention of the air- gun 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 THE PRINCIPLES OF STATICS. njiivitf 'iii"'ii 'I'" OTTO De GUERICKE Serenifs.-i. Potentifs; Elector : Brandebi Confilmrius « Civitat:Magdel5.Coimu- 112 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 ex- periments is readily comprehended, if we transport our- selves back into the epoch of these discoveries. What indeed could be more wonderful than the sudden dis- covery 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 gi- gantic mechanical phenomena. It was on this occa- sion, perhaps, first made manifest by a great and strik- ing 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 , r ^ ■ . i . • r ■ i on this sub- phenomenon oi suction, the action oi syringes and leo's'time.' pumps by the so-called horror vacui — nature's abhor- rence 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 phe- nomenon. The person competent to enunciate it must actually have discerned some principle in the phenom- enon. 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 suc- tion he called altezza limitaiissima. He sought, more- over, 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 Torriceiii's . experimeDL resistance to a vacuum by a column of mercury mstead of a column of water, and he expected to obtain a col- umn of about yij of the length of the water column. His expectation was confirmed by the experiment per- formed 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. Re- moving the finger, the column of mercury falls and re- mains 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 weigh air. glass bottle containing nothing but air and then again weighing the bottle after the air had been partly ex- pelled 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 Torriceiii'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 "^HE SCIENCE OF MECHANICS. Atmospher- logical manner by the pressure exerted by the weight disOTTC^ed of a fluid column — a column of air. Torricelli discov- ceiii""'" 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 circu- lated in France by Mersenne, and came to the knowl- edge of Pascal in the year 1644. The accounts of the theory of the experiment were presumably so imper- fect that Pascal found it necessary to reflect indepen- dently thereon. {Pesanteur de Pair. Paris, 1663.) Pascal's ex- He repeated the experiment with mercury and with periments. . , . . . , a tube of water, or rather of red wme, 40 feet m 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 them- selves, to stand a high pressure in fluids, provided only the pressure is equal on all sides ; and he applies this THE PRINCIPLES OF STATICS. "5 Fig. 8i. at once to the case of fishes and of animals that live in The anai- the air. Pascal's chief merit, indeed, is to have estab- Uquid "nl" lished a complete analogy between the phenomena con- fc "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 ves- sel 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 to the constantly increasing pressure of the water, as- cend 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 raoun- this very phenomenon that led Pascal to the idea that tains deter- the barometer-column must necessarily stand lower atthebarom- eter. 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 re- sistance experienced when a large hat lying flat on a table is suddenly lifted. The cleaving of wood to the Il6 THE SCIENCE OF MECHANICS. A siphon which acts by water- pressure. bottom of a vessel of quicksilver is a phenomenon of the same kind. Pascal imitated the flow produced in a siphon by atmospheric pressure, by the use of water-pressure. The two open unequal arms a and ^ of a three-armed tube ab 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 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 Fig. 83. Pascal's modifica- tion of the Torricelli- an experi- ment. to the air. d The Torricellian experiment was modi- fied by Pascal in a very ingenious manner. A tube of the form abed (Fig. 83), of double the length of an ordinary barom- eter-tube, is filled with mercury. The openings a and b are closed with the fin- gers and the tube placed in a dish of mercury with the end a downwards. If now a be opened, the mercury va. cd will all fall into the expanded portion at c, and the mercury 'v\ ab will sink to the height of the ordinary barometer-column. A vac- uum is produced at b which presses the finger closing the hole painfully inwards. If b also be opened the column 'v!\ ab will sink completely, while the mercury in the expanded portion c, being now exposed to the pressure of the K:y Fig. 83. THE PRINCIPLES OF STATICS. 117 atmosphere, will rise in c d to the height of the barom- eter-column. Without an air-pump it was hardly pos- sible to combine the experiment and the counter- experiment in a simpler and more ingenious manner than Pascal thus did. 6. With regard to Pascal's mountain-experiment, Supple- mentary re- we shall add the following brief supplementary remarks, marks on Pascal's Let b„ be the height of the barometer at the level of mountain- T 1 - r 11 • <• experiment the sea, and let it fall, say, at an elevation of m metres, to kb^, where /J is a proper fraction. At a further eleva- tion of m metres, we must expect to obtain the barom- eter-height k .kb^, since we here pass through a stratum of air the density of which bears to that of the first the proportion oi k -.X. If we pass upwards to the altitude h=^ n . m metres, the barometer-height corresponding thereto will be b,, = k".b^or„=^^^}^^^or 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 experi. 1 1 . r .. • . /^ r^ ments of the domain of aerostatics we owe to Utto von (jUE- otto von RiCKE. His experiments appear to have been suggested in the main by philosophical speculations. He pro- ceeded 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 dem- onstrated the experimental discoveries made by him about 1650. This statement is confirmed by his method ii8 THE SCIENCE OF MECHANICS. of constructing a water-barometer which was entirely different from that of Torricelli. Thehistori- Guericke's book (Experiment a nova, ut vocantur, cal value oE r •. \ i- Guericke's Magdeourgica. Amsterdam. 1672) makes us reahse book. , . , . . rr\-\ r the narrow views men took m 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 lon- ger marvel that the barbarism of the social order still so oppresses us. its specula- In the introduction to this book and in various other live charac- , „.. . ,. . ter. places, (jruericke, in the midst of his experimental in- vestigations, 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. Disquisi- tions on empty space occupy a considerable portion of the work. Guericke's Guericke regards the air as the exhalation or odor notion of ^ , . . i • i ■ i 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 Fi'/a Heronis \ 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 fire- engine was fastened to its lower end. The water, it was thought, in following the piston and the action of THE PRINCIPLES OF STATICS. 119 Guericke's First Experiments. {Experim, Magdeb.) THE SCIENCE OF MECHANICS. His at- gravity, would fall and be pumped out. Guericke ex- produce a pected that empty space would remain. The fastenings vacuum. success. 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 power- ful 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 unsuit- able material for the purpose, and Guericke having re- marked 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 de- scribes the great force with which the air rushed in on the opening of the cock. g. After these experiments Guericke constructed an independent air-pump. A great glass globular re- ceiver 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 THE PRINCIPLES OF STATICS. 121 y- in the receiver. To secure more perfect closure the Guericke's 1 T • 1 ■ 1 J 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 THE SCIENCE OF MECHANICS. The curious The phenomena which Guericke observed with this observed by apparatus are manifold and various. The noise which the air- water in a vacuum makes on striking the sides of the glass receiver, the violent rush of air and water into exhausted vessels suddenly opened, the escape on ex- haustion of gases absorbed in liquids, the liberation of their fragrance, as Guericke expresses it, were imme- diately 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 vacui 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 Gue- ricke a definite ratio of weight between air and water does not exist. The experi- But the deepest impression on the contemporary mentsrelat- -^ -' ingto at- world was made by the experiments relating to atmos- mospheric . at 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 PRINCIPLES OF STATICS. 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 huge weight is lifted. Guericke mentions the compressed-air gun as some- Guericke's Hirudin thing already known, and constructs independently an instrument that might appropriately be called a rari- fied-air gun. A bullet is driven by the external atmos- pheric 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 consider- able 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. ID. The investigations of Guericke were continued The investi- . gations of by an Englishman, Robert Boyle.* The new experi- Robert . Boyle. ments which Boyle had to supply were few. He ob- serves 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 re- ceiver 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 ot which Confirms in a simple manner the view of GaHleo vacuum, that when the resistance of the air has been ehminated light and heavy bodies both fall with the same velo- city. In an exhausted glass tube a leaden bullet and a piece of paper are placed. Putting the tube in a ver- tical position and quickly turning it about a horizontal axis through an angle of i8o°, both bodies will be seen to arrive simultaneously at the bottom of the tube. Quantita- Of the quantitative data we will mention the fol- live data. . . ,^, , . lowing. The atmospheric pressure that supports a column of mercury of 76 cm. is easily calculated from the specific gravity 13 -So of mercury to be i 0336 kg. to 1 sq.cm. The weight of 1000 cu.cm. of pure, dry air at 0° C. and 760 mm. of pressure at Paris at an ele- vation 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- 1 1 . Guericke knew of only one kind of air. We ery of other . . gaseous may imagine therefore the excitement it created when substances. . - , , . , m 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 physical properties of gases are very strik- ing. Faraday has il- lustrated their great inequality of weight by a beautiful lecture- experiment. If from a balance in equilib- rium, 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 Fig. 84. 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 mercu- made 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 cen- tury. The mercurial air-pumps now in common use are really barometers of which the extremities are sup- plied 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's law. served by Guericke, was more accurately investigated by Boyle, and, later, by Mariotte. The law which both found is as follows. If Vh& 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 Marietta'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 ascertain- ment of the law was very simple. He partially filled 126 THE SCIENCE OF MECHANICS. experi- ments. His appa- ratus. Mariotte's Torricellian tubes with mercury, measured the volume of the air remaining, and then performed the Torricel- lian experiment. The new volume of air was thus obtained, and by subtract- ing the height- of the column of mer- cury from the barometer-height, also the new pressure to which the same quantity of air was now subjected. To condense the air Mariotte em- ployed a siphon^tube with vertical 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 upper end. The volume of the air was read off on the graduated tube, and to the difference of level of the mercury in the two arms the barometer- height was added. At the present day 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 and connecting it by a caoutchouc tube kk with a second open glass tube r' r' , which is movable up and down the scale. If the tubes be partly filled with mercury, any difference of level whatsoever of the two surfaces of mer- cury may be produced by displacing / /, and the corresponding variations of volume of the air enclosed in r r observed. It struck Mariotte on the occasion of his investiga- tions that any small quantity of air cut off completely Fig. 86. THE PRINCIPLES OF STATICS. 127 from the rest of the atmosphere and therefore not The expan- directly affected by the latter's weight, also supported isolated the barometer-column ; as where, to give an instance, the atmos- the 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 ar- rangement 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 dis- coveries of aerostatics furnished so much that was new and wonderful that a valuable intellectual stimulus pro- ceeded from the science. CHAPTER II. THE DEVELOPMENT OF THE PRINCIPLES OF DYNAMICS. Dynamics wholly a modern science. Galileo's achievements. I. We now pass to the discussion of the funda- mental principles of dynamics. This is entirely a mod- ern science. The mechanical speculations of the an- cients, 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 li- quids 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 pro- jectile. 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 THE PRINCIPLES OF DYNAMICS. i2g times to lay the true foundations of this department of inquiry. (See Appendix, VII., p. 520.) 2. The treatise Discorsi e dimostrazioni inatematiche, in which Galileo communicated to the world the first I30 THE SCIENCE OF MECHANICS. Galileo's dynamical investigation of the laws of falling bodies, investiga- . . , /^ ti j- tionof the appeared in i6s8. The modern spirit that Cjanieo dis- laws of fall- '^ . . , , , , , , r ing bodies, covers IS evidenced here, at the very outset, by the lact that he does not ask why heavy bodies fall, but pro- pounds the question, Hoiv 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 ■ , . , -111 r 1 r n- 1 1 theory. It seems m 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 veloci- ties acquired in the descent increase proportionally to the distances descended through. Before he pro- ceeds 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 origi- nal simple distance. If, accordingly, in the case of the double distance we conceive the first half trav- ersed, no time will, it would seem, fall to the account of the second half. The motion of a falling body ap- pears, therefore, to take place instantaneously; which not only contradicts the hypothesis but also ocular evi- dence. We shall revert to this peculiar fallacy of Galileo's later on. THE PRINCrPLES OF DYNAMICS. 131 3. After Galileo fancied he had discovered this as- Hissecond, sumption 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 dif- ficult 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 is simple, distinct, and per- fectly correct. He draws (Fig. 87) a straight line, and on it cuts off successive por- O^ tions that represent to him Fig- 87. the times elapsed. At the extremities of these por- tions he erects perpendiculars (ordinates), and these represent the velocities acquired. Any portion OG 6i the line OA denotes, therefore, the time of descent elapsed, and the corresponding perpendicular GIT 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 oi 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 MECHANICS. Uniformly equally distant in opposite directions from the instant motion. C, we shall observe that the velocity HG exceeds the mean velocity CD by the same amount that EP falls short of it. For every instant antecedent to C there exists a corresponding one equally distant from it sub- sequent to C. Whatever loss, therefore, as compared with uniform motion with half the final velocity, is suf- fered 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 uniformly de- scribed with half the final velocity. If, accordingly, we make the final velocity v proportional to the time of descent t, we shall obtain v=gt, where ^ denotes the final velocity acquired in unit of time — the so-called acceleration. The space s descended through is there- fore given by the equation s = i^gtj'i) t or s =^ gf^ ji. Motion of this sort, in which, agreeably to the assump- tion, equal velocities constantly accrue in equal inter- vals of time, we call uniformly accelerated motion. Tabieofthe If wc collect the times of descent, the final veloci- locities.and ties, and the distances traversed, we shall obtain the distances of ^ ,, . , , descent, following table : t. V. s. 1. ^S- 1 X 1 ■ g 2 2. 2^- 2X2 g 2 3. 3."-. 3X3 g o .-1 4. H- 4X4. g 2 • • a : : tg- /X i ■ g 2 THE PRINCIPLES OF DYNAMICS. 133 4. The relation obtaining between / and j admits Experimen- of experimental proof ; and this Galileo accomplished tion of the 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 dif- ferent 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 un- modified. If, beginning from the upper extremity, thejhearti- ° ° ^ '^ ■' fices em- distances 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 in- genious 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 bottoiii a minute orifice which was closed with the ^^°'^^' finger. As soon as the ball began to roll down the in- clined plane Galileo removed his finger and allowed the water to flow out on a balance ; when the ball had ar- rived 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, percept- ibly 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 con- firmed by experiment, and with it the assumption itself. The reia- c. To form some notion of the relation which sub- tion of mo- . . . , . j , j i j* tion on an sists between motion on an inclined plane and that oi plane to free descent, Galileo made the assumption, that a body 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 be- low, 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 DYNAMICS. 135 zero. We perceive, therefore, that a body will rise, justifica- -. ,, ,. ..... . tion of the in Virtue of the velocity acquired in its descent, just as assumption high as it has fallen. If, accordingly, a body falling final veioc- down an inclined plane could acquire a velocity which motions are 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, accord- ingly, 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 ap- prehension and recognition of \h&fact that heavy bodies do not possess the tendency to rise, but only the ten- dency to fall. If we should assume that a body fall- ing 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 in- clined or vertical plane to make it rise to a greater ver- tical height than it had fallen from. And if the velo- city attained on the inclined plane were less, we should only have to reverse the process to obtain the same re- sult. In both instances a heavy body could, by an ap- propriate 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 expe- rience. He took a simple filar pendulum (Fig. 88) with a heavy ball attached. Lifting the pendulum, while 136 THE SCIENCE OF MECHANICS. Galileo's elongated its full length, to the level of a given altitude, experimen- tai 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 exactly, 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 Effected by in the case of a heavy metal one. However, this neg- partially ,■,,,,, impeding lected, the body ascends to the same altitude on the the motion . . . - . . . ., , of apendu- Opposite Side. Now it is permissible to regard the mo- lum string. . ^ j , . , . . tion 01 a pendulum m 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 driv- ing in at one side of the thread, as it vertically hangs, a nail / or ^, 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 z.xcam or an. Now if the inclination of the planes had any influence THE PRINCIPLES OF DYNAMICS. 137 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, al- ways 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 as- sumption attained on an inclined plane whether the body fall leads to the '^ . . , law of rela- through the height or the length of the plane, — in which tive aocei- 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 de- scent 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 ^ the length the proportion of the length to the height. Let AB (Fig. 89) be the height and ACB\- the length of the inclined plane. Fig. 89. Both will be descended through in uniformly accel- erated motion in the times t and t.^ with the final ve- locity V. Therefore, V , . ^ V AB t AB=-,i^r.AAC^^t,,-^-^=- 138 THE SCIENCE OF MECHANICS. If the accelerations along the height and the length be called respectively g and g.^, we also have V ^ gt and V ^ g^ i^, whence ^A = AB = sma. In this way we are able to deduce from the accel- eration on an inclined plane the acceleration of free descent. A corollary From this proposition Galileo deduces several cor- of the pre- . ^ , ceding 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 dis- tances are that are traversed by the two in equal inter- vals of time, the solution of the problem will be readily found (Fig. 90) by simply letting fall from B a perpen- dicular on the length. The part AD, thus cut off, will be the distance traversed by the one body on the in- clined plane, while the second body is freely falling through the height of the plane. A F'S. 90. Fig. 91. Relative If We describe (Fig. 91) a circle on AB as diame- tiraesofde- , . , .„ scription of ter, the Circle will pass through D, because Z> is a anddiame- right angle. It will be seen thus, that we can imagine ters of cir- ,-.,.,, ° cies. any number of inclined planes, A£, AF, of any degree of inclination, passing through A, and that in every THE PRINCIPLES OF DYNAMICS. 139 circles. 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 diame- ter itself. Since, obviously, only the lengths and in- clinations 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 pretty form in which Galileo gave it, is usually no b™Is fa^ii- longer incorporated in elementary expositions. We chords of imagine gutters radiating in a vertical plane from a common point A at 3. 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 simultaneous- ly their motion of des- cent. The bodies will always form at any one 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 pro- portionally to the squares of the times. This will be I40 THE SCIENCE OF MECHANICS. perceived by imagining the figure revolved about the vertical A V. Character^ 8. We see thus, — as deserves again to be briefly i°nqufriel° ^ noticed, — that Galileo did not supply us with a theory of the falling of bodies, but investigated and estab- lished, 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 ttie light of a new law. In all his reasonings, Galileo followed, to the greatest advantage of science, a prin- ciple which might appropriately be called the principle The prin- of continuity. Once we have reached a theory that ap- ciple of . J J 11 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 concep- tion originally reached. There is no method of pro- cedure more surely calculated to lead to that compre- hension of all natural phenomena which is the simplest and also attainable with the least expenditure of men- tality and feeling. (Compare Appendix, IX., p. 523.) A particular instance will show more clearly than any general remarks what we mean. Galileo con- Fig. 93. siders (Fig. 93) a body which is falling down the in- clined plane AB, and which, being placed with the THE PRINCIPLES OF DYNAMICS. 141 velocity thus acquired on a second plane BC, for ex- Galileo's Qiscovcrv ample, ascends this second plane. On all planes BC, oftheso- BD, and so forth, it ascends to the horizontal plane of inenia. 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 s.-^- proach 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 hori- zontal 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 ad- vancing 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 re- . tain forever its velocity (and direction). We shall presently revert to this subject. 9. The motion of falling that Galileo found actually The deduc- ... . _ , . , , , . tion of the to exist, IS, accordmgly, a motion of which the velocity idea of uni- 11 1 . 111. f ormly ac- increases proportionally to the time — a so-called uni- ceierated f , . , . motion. formly accelerated 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 QdXiX&o force was known solely as dons. pressure. 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 accelera- tion, is determined by it. This cannot be philosophi- cally 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 tem- perature not compensatory accelerations are deter- mined, but compensatory velocities. The fact That it is accelerations which are the immediate ef- determine fccts of the circumstances that determine motion, that tionsisan is, of the forces, is a fact which Galileo /^r(r«'z;^(/ in the taifact, natural phenomena. Others before him had also per- ceived many things. The assertion that everything seeks its place also involves a correct observation. The ob- servation, 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 Ga- lileo perceived, is still present. His observation always remains correct ; it holds true more generally ; it em- braces in one mental effort much more. 11. We have already remarked that Galileo dis- covered 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. 143 preserves its direction and velocity unaltered. The History of fortunes of this law of inertia have been strange. It called law appears never to have played a prominent part in Gali- leo'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, how- ever, that the law of inertia is not at all an indepen- dent law, but is contained implicitly in Galileo's per- ception that all circumstances determinative of motion, or forces, produce accelerations. In fact, if a force determine, not position, not velo- The law a ... r 1 • ■ J simple in- city, but acceleration, change of velocity, it stands toference reason that where there is no force there will be no leo's funda, . . . mental ob- change 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 1 . . r 1 1 ■ ■ 1 1 r methods of to derive it from the general proposition that "the ei- deducing it, feet 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 pres- ent 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 cor- rect ; if we call the acceleration "effect," then the sec- ond proposition holds. 12. We shall now examine Galileo's researches from another side. He began his investigations with the 144 THE SCIENCE OF MECHANICS. Galileo's time Notion of notions familiar to his time — notions developed mainly vclocitv'^s it existed in in the practical arts. One notion of this kind was that of velocity, which is very readily obtained from the con- sideration 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 h^ s ^= ct. The distance c traversed in a second of time we call the ve- locity, and obtain it from the examination of any por- tion of the distance and the corresponding time by the help of the equation c = s jt, 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. ,2 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 in- crement of the time, in all cases we obtain for the ve- locity 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 in 2 approaches to a straight line, we may regard the tion of this increase as uniform. The velocity in this element of the motion we may then define as the quotient, A s/A t, of the element of the time into the corresponding ele- ment 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 in- finitely 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 reason- ings. 13. An entirely new notion to which Galileo was The notion 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 velo- city acquired in time /, then v ^^ gi- Here g denotes the increment of the velocity in unit of time or the ac- celeration, which we also obtain from the equation g ^= v/i. When the investigation of variably accel- erated motions was begun, this notion of accelera- tion 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 pre- ceding reasoning and define the acceleration as dv/di, 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 MECHANICS. Graphic representa- tion of these ideas. have for the acceleration of a rectilinear motion, q} =■ dv/dt — d'^s/di^. The ideas here developed are susceptible, moreover, of graphic representation. If we lay off the times as abscissse and the distances as ordinates, we shall per- ceive, 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 Fig. 95- The curve usual manner (Fig. 95), a uniform motion represented of distance. ^ ° ^•'■" ^ by a straight Ime 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 £Z> = 2 AC; in the second case, an ordinate less than BD. We see thus, without difficulty, that a curve of dis- tance convex to the axis of the time-abscissae corre- sponds to accelerated motion, and a curve concave thereto to retarded motion. If we imagine a lead-pen- cil to perform a vertical motion of any kind and in THE PRINCIPLES OF DYNAMICS. 147 front of it during its motion a piece of paper to be uni- formly 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 mo- tion. At a the velocity of the pencil was directed up- wards, at b it was greater, at c it was = 0, ■aX d it was directed downwards, at e it was again = 0. At a, b, d, e, the acceleration was directed upwards, at c down- wards ; at c and e it was greatest. 14. The summary representation of what Galileo Tabular discovered is best made by a table of times, acquired meut of Ga- ■' ^ lileo'sdis- V . s . - If 3^ 94 covery. velocities, and traversed distances. But the numbers The table follow so simple a law, — one immediately recognisable, placed by , , . . . ... rules for its — that there is nothing to prevent our replacing theconstruc- table by a rule for its construction. If we examine the relation that connects the first and second columns, we shall find that it is expressed by the equation v ^ gt, which, in its last analysis, is nothing but an abbrevi- ated direction for constructing the first two columns of the table. The relation connecting the first and third columns is given by the equation J = ^ f^ I1. The con- nection of the second and third columns is represented by J = v"^ jig. 148 THE SCIENCE OF MECHANICS. The rules. Of the three relations V = gt strictly, the first two only were employed by Galileo. Huygens was the first who evinced a higher apprecia- tion of the third, and laid, in thus doing, the founda- tions of important advances. A remark 1 5. We may add a remark in connection with tionofthe this table that is very valuable. It has been stated tfe tfmes. previously that a body, by virtue of the velocity it has acquired in its fall, is able to rise again to its origi- nal 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 ac- quires 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 ver- tically upward double velocity will ascend twice as long a time, hv\ 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 tesians and is inherent in the velocity of a body a something that Sanson the corresponds to a force — a something, that is, by which measure of , . -y, ,, . , 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 propor- tional to the velocity or to the square of the velocity. The Cartesians held the former, the Leibnitzians the l3.tter. But it will be perceived that the question in- volves no dispute whatever. The body with the double velocity overcomes a given force through double the THE PRINCIPLES OF D YNAMICS. 149 time, but through four times the distance. With re- spect to tirfie, therefore, its efficacy is proportional to the velocity ; with respect to distance, to the square of the velocity. D'Alembert drew attention to this mis- understanding, although in not very distinct terms. It is to be especially remarked, however, that Huygens's thoughts on this question were perfectly clear. 1 6. The experimental procedure by which, at the The present present day, the laws of falling bodies are verified, istaimeansof vcrifvins somewhat different from that of Galileo. Two methods the laws of may 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 ob- served ; or the motion of falling is not altered at all, but our means of observation are improved in deli- cacy. On the first principle Galileo's inclined gutter and Atwood's machine rest. Atwood's machine consists (Fig. gy) of an easily run- ning pulley, over which is thrown a thread, to whose extremities two equal weights P are attached. If upon one of the weights /" we lay a third small weight p, a uniformly accel- \^p erated motion will be set up by the added F'b- 97- weight, having the acceleration (^p j i P -\- f) g — a result that will be readily obtained when we shall have dis- cussed 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 i, 4, 9, 16 . . . . are traversed. The final velocity corresponding to any given time of descent is investigated by catching the small additional weight,/, 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. P*f! I50 THE SCIENCE OF MECHANICS. The appa- The apparatus of Morin is based on a different prin- Morin, La- ciple. A body to which a writing pencil is attached pich, and describes on a vertical sheet of paper, which is drawn 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 dis- tances of descent described. For the abscissae i, 2, 3, 4 .... we obtain the ordinates i, 4, 9, 16 ... . 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 horizon- tally 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 in- creasing velocity of the descent, the undulations traced by the rod become longer and longer. Thus (Fig. 98) bc:^'>,ab, cd=:^^ab, de^yab, and so forth. The law of falling bodies is clearly exhibited by this, since ab -\- cb = iSfab, ab-\-bc-\-cd^qab, 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 DYNAMICS. 151 nute portions of time a rapidly operating clock-work The de- vices o£ called a chronoscope, which is set in motion at the be- wheat- ginning of the time to be measured and stopped at the Hipp., termination of it. Hipp has advantageously modified 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 THE SCIENCE OF MECHANICS. Galileo's minor in- vestiga- tions. capement. The throwing in and out of gear is effected by an electric current. Now if, as soon as the body be- gins 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. 17. Among the further achievements of Galileo we have yet to mention his ideas concerning the motion 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 an- other 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 pro- posed its use in astronomical observations and to a cer- tain extent employed it therein himself. The motion 18. Of Still greater importance are his investiga- of projec- . . , • r ■ M A r 1 1 tiles. tions concerning the motion 01 projectiles. A tree 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 „ velocity c, its velocity at the end of the time t will be z/ ^ c-\gt. An initial velocity up- wards would have to be reck- oned negative here. The dis- V tance described at the end of time t is represented by the where ct and Ar^^ are the X Fig. 99. equation s ^^ a -\- c t -\- \g f^ portions of the traversed distance that correspond re- spectively to the uniform and the uniformly accelerated motion. The constant a is to be put = when we reckon THE PRINCIPLES OF DYNAMICS. 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 inde- pendent motions, a horizontal uniform motion, and a vertical uniformly accelerated motion. He thus intro- duced into use the principle of the parallelogram of mo- tions. Even oblique projection no longer presented the slightest difficulty. If a body receives a horizontal velocity c, it de- The curve scribes in the horizontal direction in time t the distance tion a par- y =:z c t, while simultaneously it falls in a vertical direc- tion the distance X ^ gf^ ji. Different motion-deter- minative 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 em- ploying the two equations above given, the expression y ^V {p.c ^ /g) X. It is the parabola of ApoUonius hav- ing its parameter equal to ir^/^and its axis vertical, as Galileo knew. We readily perceive with Galileo, that oblique pro- oblique jection involves nothing new. The velocity c imparted 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 5 154 THE SCIENCE OF MECHANICS. The range of projec- tion. Fig. 100. onward (Fig. loo) it continues its motion as a horizon- tal projection. If we examine any two epochs equally distant in time, before and after the transit through S, we shall see that the body at these two epochs is equally distant from the perpendicu- lar through 5 and situated the same distance below the hori- zontal line through 6'. The curve is therefore symmet- rical with respect to the vertical line through ^. It is a parabola with vertical axis and the parameter ((T cos a) '^ jg. To find the so-called range of projection, we have 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, t^=c sina/g; and the same for the descent. With the horizontal velocity c . cos a, therefore, the distance is traversed — 2 sin a cos a = — sin 2 a. S g c cos a . c sma g The range of projection is greatest accordingly when a = 45°, and equally great for any two angles « = 45° ± >S°. The mutual ig. The recognition of ihe: voMtVLsX independence oi indepen- i r • dence of the forccs, or motion-determinative circumstances oc- forces. curring in nature, which was ^ reached and found expression in the investigations relating to ■^ projection, is important. A body may move (Fig. loi) in the di- rection AB, while the space in which this motion oc- curs is displaced in the direction A C. The body then Fig, loi. THE PRINCIPLES OF D YATAMICS. 155 goes from A to D. Now, this also happens if the two circumstances that simultaneously determine the mo- tions AB and AC, have no influence on one another. It is easy to see that we may compound by the paral- lelogram not only displacements that have taken place but also velocities and accelerations that simultane- ously take place. (See Appendix, X., p. 525.) II. THE ACHIEVEMENTS OF HUYGENS. I. The next in succession of the great mechanical in- Huygens's quirers is Huygens, who in every respect must beas\n?n- ranked as Galileo's peer. If, perhaps, his philosophical ''""'^"'' 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 him- self to the dynamics of a single body. The plenitude of Huygens's achievements is bestEnumera- seen in his HorologiumOscillatorium, which appeared ingens's 1673. The most important subjects there treated of ferments, the first time, are : the theory of the centre of oscilla- tion, the invention and construction of the pendulum- clock, the invention of the escapement, the determina- tion of the acceleration of gravity, g, by pendulum- observations, 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 THE 'SCIENCE OF MECHANICS. 2. With respect to the form of presentation of his work, it is to be remarked that Huygens shares with 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. 157 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 dif- ferent manner from that which we pursued in the case of Galileo. Galileo's views, in their classical sim- plicity, 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 be- come inadequate and cum- brous. For reasons of brev- ity, therefore, we shall re- produce all the conceptions of which we treat, in mod- ern form, retaining, how- ever, Huygens's essential and characteristic ideas. Huygens's Pendulum Clock. iS8 THE SCIENCE OF MECHANICS. Centrifugal 3. We begin with the investigations concerning petal force. Centrifugal force. When once we have recognised with GaHleo that force determines acceleration, we are im- pelled, 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, rep- resents 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. Fig. 103. Fig. 103. Let us suppose that we have a body to which a ve- locity 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 investi- gate. We imagine (Fig. 102) two equal circles uni- THE PRINCIPLES OF DYNAMICS. 159 formly travelled round by two bodies ; the velocities in Uniform the circles I and II bear to each other the proportion eguai circlGS. 1:2. If in the two circles we consider any same arc- element corresponding to some very small angle «, 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 cp.^ and cp^ the re- spective accelerations, and t and t/2 the time-elements for the angle a, we find by Galileo's law 2s ,1s ^ . cp^ = ~, (^2 = 4 — , that is to say ^^ = 't'Pi- 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. 9^2 = ri' that is to say ; the ele- ments of the distance described AB = A'B' = s ; and the I .-:ll of the ac- celeration. are times are respectively rand t'. We obtain, then, T = 1/2 s/cp, Fig. 109. r' = 1/2 j'/4 (p = t/2. The element A'£' is accordingly trav- elled through in one-half the time the element AB is. The final velocities v and v' at B and B' are found by the equations v = cpr and »' = 4 (p{T/2) = 2v. Since, therefore, the initial velo- cities at B and B' are to one another as 1:2, and the accelerations are again as 1:4, the element of II suc- ceeding the first will again be traversed in half the time of the corresponding one in I. Generalising, we get : For equal excursions the time of oscillation is in- versely proportional to the square root of the accelera- tions. 9. The considerations last presented may be put in a very much abbreviated and very obvious form by a method of conception first employed by Newton. New- ton calls those material systems similar that have geo- metrically similar configurations and whose homolo- i66 THE SCIENCE OF MECHANICS. The princi-gous masses bear to one another the same ratio. He pleotsimil- , , , r i ■ i ■ i • -i itude. says further that systems of this kind execute similar movements when the homologous points describe simi- lar paths in proportional times. Conformably to the geometrical terminology of the present day we should not be permitted to call mechanical structures of this kind (of five dimensions) similar unless their homolo- gous linear dimensions as well as the times and the masses bore to one another the same ratio. The struc- tures might more appropriately be termed affined to one another. We shall retain, however, the name phoronomically similar structures, and in the consideration that is to follow leave entirely out of account the masses. In two such similar motions, then, let the homologous paths be s and as, the homologous times be t and fit; whence the homologous velo- ... s ^ a s cities are w = — and yv = -^ —, the homologous accel- „ ^ „ ^. ^ 2s , a 2s erations ' = m' g' . If, now, we were able to prove, that, independently of the material (chemical) compo- sition of bodies, g=g' at every same point on the earth's surface, we should obtain m/m' ^ p/p'; that is to say, on the same spot of the earth's surface, it would be possible to measure mass by weight. Now Newton established this fact, that g is inde- pendent of the chemical composition of bodies, by experiments with pendulums of equal lengths but dif- ferent material, which exhibited equal times of oscilla- tion. He carefully allowed, in these experiments, for the disturbances due to the resistance of the air ; this last factor being eliminated by constructing from differ- ent materials spherical pendulum-bobs of exactly the same size, the weights of which were equalised by ap- propriately hollowing the spheres. Accordingly, all bodies maybe regarded as affected with the same g, and THE PRINCIPLES OF DYNAMTCS. 197 their quantity of matter or mass can, as Newton pointed out, be measured by their weight. If we imagine a rigid partition placed between an Suppie- assemblage of bodies and a magnet, the bodies, if the considera- 1 r 1 1 1 1 ■ • tions. magnet be powerful enough, or at least the majority of the bodies, will exert a pressure on the partition. But it would occur to no one to employ this magnetic pressure, in the manner we employed pressure due to weight, as a measure of mass. The strikingly notice- able inequality of the accelerations produced in the different bodies by the magnet excludes any such idea. The reader will furthermore remark that this whole argument possesses an additional dubious feature, in that the concept of mass which up to this point has simply been named and felt as a necessity, but not de- fined, is assumed by it. 10. To Newton we owe the distinct formulation ofThedoc- ,. .. 4.Trii trine of the the prmciple of the composition of forces.* If a bodycomposi- is simultaneously acted on by two forces (Fig. 128), forces, of which one would produce the motion AB and the other the '^V"--^." ^ motion AC in the same interval \ ~~\ d of time, the body, since the two forces and the motions produced '^' ' by them are independent of each other, will move in that interval of time to AD. This conception is in every respect natural, and distinctly characterises the essen- tial point involved. It contains none of the artificial and forced characters that were afterwards imported into the doctrine of the composition of forces. We may express the proposition in a somewhat * Roberval's (1668) achievements with respect to the doctrine of the com- position of forces are also to be mentioned here. Varignon and Lami have al- ready been referred to. (See the text, page 36.) 198 THE SCIENCE OF MECHANICS. Discussion different manner, and thus bring it nearer its modern trine of the form. The accelerations that different forces impart composi- . . r tion of to the same body are at the same time the measure ot these forces. But the paths described in equal times are proportional to the accelerations. Therefore the latter also may serve as the measure of the forces. We may say accordingly : If two forces, which are propor- tional to the lines AB and A C, act on a body A in the directions A£ and AC, a. motion will result that could also be produced by a third force acting alone in the direction of the diagonal of the parallelogram con- structed on AB and A C and proportional to that di- agonal. The latter force, therefore, may be substituted for the other two. Thus, if ip and ip are the two ac- celerations set up in the directions AB and AC, then for any definite interval of time /, AB = (pi^/2, AC^ (/,'/2^2. If, now, we imagine^^Z* produced in the same interval of time by a single force determining the accel- eration X, we get AB = x/V2> and ^^ : .^ C : AJD ^

Fig- 135a- Fig. 135b. by a weight /" (Fig. 135a) is passed over a pulleys, attached to the end of a scale-beam. A weight / is laid on one of the weights first mentioned and tied by a fine thread to the axis of the pulley. The pulley now supports the weight iP -\- p. Burning away the thread that holds the over-weight, a uniformly accel- erated motion begins with the acceleration y, with which P -\- p descends and P rises. The load on the pulley is thus lessened, as the turning of the scales in- dicates. The descending weight P is counterbalanced by the rising weight P, while the added over-weight, instead of weighing/, now weighs {p/g){g — y). And since y = (//a P -\- p) g, we have now to regard the load on the pulley, not as/, but asp{2. P/2P-\-p). The THE PRINCIPLES OF DYNAMICS. 207 descending weight, only partially impeded in its motion of descent, exerts only a partial pressure on the pulley. We may vary the experiment. We pass a thread a variation ,,, .... .,„ , of the last loaded at one extremity with the weight P over the experiment pulleys a, b, d, of the apparatus as indicated in Fig. Fig. 135c. 1 35(5. , tie the unloaded extremity at m, and equilibrate the balance. If we pull on the string at m, this can- not directly affect the balance since the direction of the string passes exactly through its axis. But the side a immediately falls. The slackening of the string causes a to rise. An unacceleraied raotion of the weights would 2o8 THE SCIENCE OF MECHANICS, Thesuspen- son af mi- nute bodies in liquids of different 5-pecific gravity. f Fig. 136. Do such suspended particles af- fect the specific gravities of the support- ing liquids? not disturb the equilibrium. But we cannot pass from rest to motion without acceleration. 6. A phenomenon that strikes us at first glance is, that minute bodies of greater or less specific gravity than the liquid in which they are immersed, if suffi- ciently small, remain suspended a very long time in the liquid. We perceive at once that particles of this kind have to over- come the friction of the liquid. If the cube of Fig. 136 be divided into 8 parts by the 3 sections indicated, and the parts be placed in a row, their mass and over-weight will re- main the same, but their cross-sec- tion and superficial area, with which the friction goes hand in hand, will be doubled. Now, the opinion has at times been advanced with respect to this phenomenon that suspended particles of the kind described have no influence on the specific gravity indicated by an areometer immersed in the liquid, because these particles are themselves areo- meters. But it will readily be seen that if the sus- pended particles rise or fall with constant velocity, as in the case of very small particles immediately occurs, the effect on the balance and the areometer must be the same. If we imagine the areometer to oscillate about its position of equilibrium, it will be evident that the liquid with all its contents will be moved with it. Applying the principle of virtual displacements, therefore, we can be no longer in doubt that the areo- meter must indicate the mean specific gravity. We may convince ourselves of the untenability of the rule by which the areometer is supposed to indicate only the specific gravity of the liquid and not that of the sus- THE PRINCIPLES OF D YNAMICS. 209 pended particles, by the following consideration. In a liquid A a smaller quantity of a heavier liquid B is in- troduced and distributed in fine drops. The areometer, let us assume, indicates only the specific gravity of A. Now, take more and more of the liquid B, finally just as much of it as we have of A: we can, then, no longer say which liquid is suspended in the other, and which specific gravity, therefore, the areometer must indicate. 7. A phenomenon of an imposing kind, in which The phe- the relative acceleration of the bodies concerned is the tides, seen to be determinative of their mutual pressure, is that of the tides. We will enter into this subject here only in so far as it may serve to illustrate the point we are considering. The connection of the phenomenon of the tides with the motion of the moon asserts itself in the coincidence of the tidal and lunar periods, in the augmentation of the tides at the full and new moons, in the daily retardation of the tides (by about 50 minutes), corresponding to the retardation of the culmination of the moon, and so forth. As a matter of fact, the connection of the two occurrences was very early thought of. In Newton's time people imagined to themselves a kind of wave of atmospheric pressure, by means of which the moon in its motion was sup- posed to create the tidal wave. The phenomenon of the tides makes, on every one its impas- that sees it for the first time in its full proportions, an ter. overpowering impression. We must not be surprised, therefore, that it is a subject that has actively engaged the investigators of all times. The warriors of Alex- ander the Great had, from their Mediterranean homes, scarcely the faintest idea of the phenomenon of the tides, and they were, therefore, not a little taken aback 2IO THE SCIENCE OF MECHANICS. by the sight of the powerful ebb and flow at the mouth of the Indus ; as we learn from the account of Curtius Rufus (Z>« Rebus Gestis Alexandri Magni), whose words we here literally quote : Extract "34- Proceeding, now, somewhat more slowly in from Cur- , . . , ^ ; ii • u • tiusRutus. "their course, owmg to the current of the river being " slackened by its meeting the waters of the sea, they " at last reached a second island in the middle of the "river. Here they brought the vessels to the shore, "and, landing, dispersed to seek provisions, wholly "unconscious of the great misfortune that awaited "them. DescribiDg " 35. It was about the third hour, when the ocean, the effect on the army "in its Constant tidal flux and reflux, began to turn der the ' ' and press back upon the river. The latter, at first Great of the ,,11,1 1 , 1, 1 tides at the " merely checked, but then more vehemently repelled, the Indus. " at last Set back in the opposite direction with a force "greater than that of a rushing mountain torrent. "The nature of the ocean was unknown to the multi- "tude, and grave portents and evidences of the wrath "of the Gods were seen in what happened. With "ever- increasing vehemence the sea poured in, com- " pletely covering the fields which shortly before were " dry. The vessels were lifted and the entire fleet dis- " persed before those who had been set on shore, ter- " rifled and dismayed at this unexpected calamity, "could return. But the more haste, in times of great "disturbance, the less speed. Some pushed the ships " to the shore with poles ; others, not waiting to adjust "their oars, ran aground. Many, in their great haste "to get away, had not waited for their companions, "and were barely able to set in motion the huge, un- " manageable barks ; while some of the ships were too "crowded to receive the multitudes that struggled to THE PRINCIPLES OF DYNAMICS. 211 " get aboard. The unequal division impeded all. TheThedisas- r , ■ , , , 1 r 1 terto Alex- ' ' cries 01 some clamoring to be taken aboard, 01 others ander's "crying to put off, and the conflicting commands of "men, all desirous of different ends, deprived every one "of the possibility of seeing or hearing. Even the ' ' steersmen were powerless ; for neither could their "cries be heard by the struggling masses nor were their "orders noticed by the terrified and distracted crews. ' ' The vessels collided, they broke off each other's oars, " they plunged against one another. One would think ' ' it was not the fleet of one and the same army that "was here in motion, but two hostile fleets in combat. "Prow struck stern; those that had thrown the fore- "most in confusion were themselves thrown into con- " fusion by those that followed; and the desperation "of the struggling mass sometimes culminated in "hand-to-hand combats. "36. Already the tide had overflown the fields sur- " rounding the banks of the river, till only the hillocks "jutted forth from above the water, like islands. "These were the point towards which all that had given "up hope of being taken on the ships, swam. The "scattered vessels rested in part in deep water, where "there were depressions in the land, and in part lay ' ' aground in shallows, according as the waves had "covered the unequal surface of the country. Then, "suddenly, a new and greater terror took possession "of them. The sea began to retreat, and its waters "flowed back in great long swells, leaving the land "which shortly before had been immersed by the salt "waves, uncovered and clear. The ships, thus for- "saken by the water, fell, some on their prows, some "on their sides. The fields were strewn with luggage, "arms, and pieces of broken planks and oars. The 212 THE SCIENCE OF MECHANICS. The dismay " soldiers dared neither to venture on the land nor to of the army. . . , , . ^ , ^i . j "remain in the ships, for every moment they expected "something new and worse than had yet befallen "them. They could scarcely believe that that which " they saw had really happened — a shipwreck on dry "land, an ocean in a river. And of their misfortune "there seemed no end. For wholly ignorant that the " tide would shortly bring back the sea and again set "their vessels afloat, they prophesied hunger and dir- " est distress. On the fields horrible animals crept "about, which the subsiding floods had left behind. Theeftorts "^7. The night fell, and even the king was sore of the king •" ° ' ° , . and the re- "distressed at the slight hope of rescue. But his so- turn of the tide. ' ' licitude could not move his unconquerable spirit. He "remained during the whole night on the watch, and "despatched horsemen to the mouth of the river, that, " as soon as they saw the sea turn and flow back, they " might return and announce its coming. He also "commanded that the damaged vessels should be re- " paired and that those that had been overturned by "the tide should be set upright, and ordered all to be " near at hand when the sea should again inundate the "land. After he had thus passed the entire night in "watching and in exhortation, the horsemen came " back at full speed and the tide as quickly followed. "At first, the approaching waters, creeping in light "swells beneath the ships, gently raised them, and, "inundating the fields, soon set the entire fleet in mo- "tion. The shores resounded with the cheers and "clappings of the soldiers and sailors, who celebrated " with immoderate joy their unexpected rescue. 'But " whence,' they asked, in wonderment, 'had the sea " so suddenly given back these great masses of water? "Whither had they, on the day previous, retreated? THE PRiyCIPLES OF DYXAMICS. 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 expia- j-1 -J- 1 1 1 •-11T ration of tion of the tides is, that the earth as a rigid bodj' can the phe- , _ . . nozDena 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. Fig. 137. Let us consider (Fig. 1 37) on the earth E, 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 respect- ively cp-\- A q), (p, cp — J q). The earth as a whole, however, has, as a rigid body, the acceleration ), -\-g — g or ^g— Acp)~cp, —{g—A^)—ep or g' — 1- .1 ulationof in all respects, are placed opposite each other, we ex- the con- pect, 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 an)- difference, however slight, of form, of chemical constitution, or are in any other re- spects different, the principle of S3'mmetr3' forsakes us, unless we assume or know hcforchand that sameness of form or sameness of chemical constitution, or whatever else the thing in question may be, is not determina- tive. If, however, mechanical experiences clearly and indubitably point to the existence in bodies of a special and distinct property determinative of accelerations. 2iS THE SCIENCE OF MECHANICS. nothing stands in the way of our arbitrarily estabhsh- ing the following definition : Definition All those bodies are bodies of equal mass, which, mu- masles. 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 pro- ceed similarly. The bodies A and B receive respec- tively as the result of their mutual action (Fig. 140 b") the accelerations — cp and -\- cp' , where the senses of the accelerations are indicated by the signs. We say then, B has

), that is, when the accel- erations of the masses at the base and the ver- tex are given by 2//5 and f/e,. At the com- mencement of the dis- tortion (p increases, and simultaneously the accelera- tion of the mass at the vertex is decreased by double that amount, until the proportion subsists between the two of 2 : I. We have yet to consider the case of equilibrium of a schematic lever, consisting (Fig. 148) of three masses w , , m Fig. 147- 2> and M, of which the last is again supposed Fig. 148. to be very large or to be elastically connected with very large masses. We imagine two equal and oppo- site forces s, — s applied to m^ and m^in the direction m^m^, or, what is the same thing, accelerations im- pressed inversely proportional to the masses m^, m.^. The stretching of the connection m^m.^ also generates THE EXTEiVSIOiY OF THE PRIXCIPLES. 267 1' accelerations inversely proportional to the masses m m^, which neutralise the first ones and produce equi- librium. Similarly, along m^M \va.z%va& the equal and contrary forces /, — t operative ; and along m^ M Xh& forces u, — u. In this case also equilibrium obtains. If AT he elastically connected with masses sufficiently large, — u and — f 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, accord- ingly, for the two equal and opposite forces s, — j as well as for the wholly arbitrary forces /, u. As a matter of fact J, — s destroy each other and /, u pass through the fixed mass Af, that is, are destroyed on distortion setting in. The condition of equilibrium readily reduces itself The reduc- to the common form when we reflect that the mo- preceding ments of t and u, forces passing through M, are with common respect to Jlf zero, while the moments of s and — j are equal and opposite. If we compound / and j- to p, and u and — sto ^, then, by Varignon's geometrical princi'ple of the parallelogram, the moment of / is equal to the sum of the moments of j and /, and the moment of ^ is equal to the sum of the moments of z/ and — s. The moments of/ and ^ are therefore equal and opposite. Consequently, any two forces J> and ^ will be in e^ui- librium if they produce in the direction m^ m^ equal and opposite components, by which condition the equal- ity of the moments with respect to 71/ is posited. That then the resultant of f and q also passes through M, is likewise obvious, for j and — j destroy each other and / and u pass through M. 6. The Newtonian point of view, as the example just developed shows us, includes that of Varignon. 268 THE SCIENCE OF MECHANICS. Newton's We were right, therefore, when we characterised the point of ■ 7 ■ T. • i_ i. 4. view in- statics of Varignon as a dynamical statics, which, start- 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 opera- tions, as for example that of the translation of the forces in their own directions, is not so distinctly ex- hibited as in the instance just treated. The econ- The Considerations here developed will convince amy and , i -kt • • • i wealth of us that we can dispose by the Newtonian principles ian 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 be- hold the masses actually impressed with the accelera- tions 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 it- the modern, self ; it'h'is, ilso p7-oblems to solve XhdX touch the needs methods, of practical life and affect the furtherance of other sci- ences. Those problems are now for the most part ad- vantageously solved by other methods than the New- tonian, — methods whose equivalence to that has already been demonstrated. It would, therefore, be mere im- practical pedantry to contemn all other advantages and insist upon always going back to the elementary New- tonian ideas. It is sufficient to have once convinced ourselves that this is always possible. Yet the New- THE EXTENSION OF THE PRINCIPLES. 269 tonian conceptions are certainly the most satisfactory 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. THE FORMULA AND UNITS OF MECHANICS. 1. All the important formulae of modern mechanics History of the formu- were discovered and employed in the period of Galileo i^ and . , ... . units of and Newton. The particular designations, which, mechanics, owing to the frequency of their use, it was found con- venient to give them, were for the most part not fixed upon until long afterwards. The systematical mechan- ical 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 orig- stantaneous velocity, and cp the acceleration of a uni- tionsof formly accelerated motion. From the researches of Huygens. Galileo and Huygens, we derive the following equa- tions : V = q)t s = %f^ (1) (ps Multiplying throughout by the mass m, these equa- Theintro- 1 r 11 • duction tions give the following : of "mass and "mov- jn V =^ mcpt '"g force," "I V ms =^ —^t 2 Li mi////cvV/i'.v«i/and the Troy /•aiinJ. Con- gress has since established a standard Troy pound, which is kept in the Mint in Philadelphia. It was a copv of the old Imperial Troy pound which had been adopted in England after American independence. It * The sD-called standard of 175S had not been legalised. 284 THE SCIENCE OF MECHANICS. TheAmeri- is a hollow brass Weight of unknown volume ; and no canunit of . . . . , j j j i_ mass. accurate comparisons of it with modern standaras nave 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, be- cause they are not used in mechanics. It may, how- ever, 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 its one 86400th part, which is called a second. Side- real 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 physi- cists in the United States and Great Britain, is called THE EXTENSION OF THE PRINCIPLES. 285 the Centimetre-Gramme-Second system. In this sys-Theabso- . . , r^ r ^"*® system tem, wntmg C for centimetre, G for gramme mass, of the and S for second, states and Great Brit- ain. 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 Gali- leo Galilei first measured an accele- ration) is C/S 2 the unit of density is . . G/C ^ the unit of momentum is G C/S the unit of force (called a dyne) is . . G C/S - the unit of pressure (called one mil- lionth of an absolute atmosphere) is . . G/CS^; the unit of energy {vis viva, or work, called an ^r^) is ^GCs/S^; etc. The gravitational system of measurement of me- The Gravi- chanical quantities, takes the kilogramme or pound, or sy°\£^. rather the attraction of these towards the earth, com- pounded with the centrifugal force, — which is the ac- celeration called gravity, and denoted by g, and is dif- ferent 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 SCIENCE OF MECHANICS. Compari- son of the absolute aad gravi- tational systems. 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 g8o ■ 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 pur- poses the more convenient ; but the former is the more familiar. In either system, the formula p^^mg is retained; but in the former g retains its absolute value, while in 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 centi- metre, and the unit of time a second, then the unit of mass will be 1/981-0 kilogramme, and the unit of energy will be a kilogramme-centimetre, or (1/2)- (1000/981 -o)GC2/S 2. Then, at Washington the gravity of a kilogramme will be, not i, as at Paris, but 980-1/981-0=0-99907 units or Paris kilogramme- weights. Consequently, to produce a force of one Paris kilogramme-weight we must allow Washington gravity to act upon 981-0/980-1 = 1-00092 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 exposition, see Nature, in the issue for November 15, 1894, THE EXTENSION OF THE PRINCIPLES. 287 III. THE LAWS OF THE CONSERVATION OF MOMENTUM, OF THE CONSERVATION OF THE CENTRE OF GRAVITY, AND OF THE CONSERVATION OF AREAS. 1. Although Newton's principles are fully adequate Speciaiisa- , , . , ... , , - . tion of the to deal with any mechanical problem that may arise, mechanical it IS yet convenient to contrive lor cases more frequently occurring, particular rules, which will enable us to treat problems of this kind by routine forms and to dis- pense with the minute discussion of them. Newton and his successors developed several such principles. Our first subject will be Newton's doctrines concern- ing freely movable material systems. 2. If two free masses m and m' are subjected in Mutual ac- ,,. . r-.,. p . . , . - tion of free the direction of their line of junction to the action 01 masses. forces that proceed from other masses, then, in the in- terval of time t, the velocities v, v' will be generated, and the equation (/ -f- /') t ^=tnv -\- m'v' will subsist. This follows from the equations pt = mv and p'f = m'v'. The sum mv -\- m'v' is called the momentum bi the system, and in its computation oppositely directed forces and velocities are regarded as having opposite signs. If, now, the masses m, m' in addition to being subjected to the action of the external forces /, /' are also acted upon by internal forces, that is by such as are mutually exerted by the masses on one another, these forces will, by Newton's third law, be equal and op- posite, q, — q. The sum of the impressed impulses is, then, (/ + / -f ? — ?) ^ = (/ + /') A the same as before ; and, consequently, also, the total momentum of the system will be the same. The momentum of a Conserva- tion of Mo- mentum. 288 THE SCIENCE OF MECHANICS. system is thus determined exclusively by external iorc&s, that is, by forces which masses outside of the system exert on its parts. Law of the Imagine a number of free masses m, m , m" distributed in any manner in space and acted on by external forces/, /, /'. . . . whose lines have any di- rections. 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 a:-direction will be equal to the momentum generated in the .r-direction ; and so with the rest. If we imagine additionally in action between the masses f/i, m' , in" . . ., pairs of equal and opposite internal forces^, — q, r, — r, s, — s, etc., these forces, resolved, will also give in every direction pairs of equal and opposite components, and will con- sequently have on the sum-total of the impulses no in- fluence. Once more the momentum is exclusively de- termined by external forces. The law which states this fact is called the law of the conservation of momen- tum. Law of the 3. Another form of the Same principle, which New- tionotthe ton likewise discovered, is called the law of the conser- Centre of . ^ , . Gravity. vation Of the Centre of grav- —j^ — -3 — J g ^ ity. Imagme m A and B (Fig. 149) two masses, im ^'^' '*'■ and m, in mutual action, say that of electrical repulsion ; their centre of gravity is situated at S, where BS z= 2 AS. 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, 2 m describes a dis- tance A£>, III will necessarily describe a distance BC = THE EXTENSION OF THE PRINCIPLES. 289 lAD. The point 5' will still remain the position of the centre of gravit)-, as CS = Q.DS. Therefore, two masses cannot, by mutual action, displace theii: common centre of gravity. If our considerations involve several masses, dis- This law •13- - 1 1-111 applied to tributed m 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 mu- tual 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 m, m', m". acted on by external forces of any kind. We refer the forces to a system of rectangular coordi- nates and call the coordinates respectively x, y, z, x', y , z, and so forth. The coordinates of the centre of gravity are then ^ 'Smx ISmy 2mz 2im 2,m ^ m in which expressions x, y, z may change either by uni- form 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 aU these cases a different motion, and in the first may even be at rest. If now internal forces, acting be- tween every two masses, wz' and m" , come into play in the system, opposite displacements w' , u" will thereby be produced in the direction of the lines of junction of the masses, such that, allowing for signs, in'w ■\- ni'it/' = 0. Also with respect to the components x^ and *2 of these displacements the equation m'x^ -\- m"x, = will hold. The internal forces consequently 290 THE SCIENCE OF MECHANICS. produce in the expressions 5, rj, 8, 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. Acceiera- If we wish to know the acceleration of the centre of centre o£^ gravity of the system, the accelerations of the system's gravity of a , • -i i , , i rr in j system. parts must be similarly treated. \.i cp, cp , cp . . . . de- note the accelerations of m, rr^ , m". ... in any direc- tion, and cp the acceleration of the centre of gravity in the same direction, (p='2m

, that is the pressure of reaction upwards on I is equal to the momentum imparted to the fluid jet in unit of time. We will select here the unit of weight as our unit of force, that is, use gravitation measure. We obtain for Factor (2) the expression [a,'v{s/g)]v =-p, (where the expression in brackets denotes the mass which flows out in unit of time,) or aV'i'gh . — . V'2'gh — '2ahs. Similarly we find the pressure on II to be [av . l.\'w = q, or factor 3 : as. g THE EXTENSION OF THE PRINCIPLES. 311 j- Matheraat- a — V2gh y2g(h + k). i"' devel- o- a • 6 V I y opment of * the result. The total variation of the pressure is accordingly — Vigh P^-m as lahs -^~V2ghV2g{h + k-) o or, abridged, — 2as\yh{h-\-k) — h'\ — 2ahs + 2asVh{h-\- k), — which three factors covipletely destroy each other. In the very necessity of the case, therefore, Galileo could only have obtained a negative result. We must supply a brief comment respecting Fac- a comment tor (2). It might be supposed that the pressure on the by the ex- basal orifice which is lost, \^ ahs and not o.ahs. But this statical conception would be totally inadmissible in the present, dynamical case. The velocity v is not generated by gravity instantaneously in the effluent particles, but is the outcome of the mutual pressure between the particles flowing out and the particles left behind ; and pressure can only be determined by the momentum generated. The erroneous introduction of the value ahs would at once betray itself by self-con- tradictions. If Galileo's mode of experimentation had been less elegant, he would have determined without much diffi- culty the pressure which a continuous fluid jet exerts. But he could never, as he soon became convinced, have counteracted by a pressure the effect of an instan- taneous impact. Take — and this is the supposition of 312 THE SCIENCE OF MECHANICS. Galileo's reasoning. Compari- son of the ideas im- pact and pressure. Galileo— a freely falling, heavy body. Its final veloc- ity, we know, increases proportionately to the time. The very smallest velocity requires a definite portion of time to be produced in (a principle which even Mari- otte contested). If we picture to ourselves a body moving vertically upwards with a definite velocity, the body will, according to the amount of this velocity, ascend a definite time, and consequently also a definite distance. The heaviest imaginable body impressed in the vertical upward direction with the smallest im- aginable velocity will ascend, be it only a little, in opposition to the force of gravity. If, therefore, a heavy body, be it ever so heavy, receive an instan- taneous upward impact from a body in motion, be the mass and velocity of that body ever so small, and such impact impart to the heavier body the smallest imagin- able velocity, that body will, nevertheless, yield and move somewhat in the upward direction. The slightest impact, therefore, is able to overcome the greatest pres- sure ; or, as Galileo says, the force of percussion com- pared with the force of pressure is infinitely great. This result, which is sometimes attributed to intellectual ob- scurity on Galileo's part, is, on the contrary, a bril- liant proof of his intellectual acumen. We should say to-day, that the force of percussion, the momentum, the impulse, the quantity of motion m v, is a quantity of different dii7iensions Irom the pressure /. The dimen- sions of the former are mlt~^ , those of the latter w//~2 In reality, therefore, pressure is related to momentum of Impact as a line is to a surface. Pressure is/, the momentum of impact is//. Without employing mathe- matical terminology it is hardly possible to express the fact better than Galileo did. We now also see why it is possible to measure the impact of a continuous fluid THE EXTENSION OF THE PRINCIPLES. 313 jet by a pressure. We compare the momentum de- stroyed per second of time with the pressure acting per second of time, that is, homogeneous quantities of the form p t. 4. The first systematic treatment of the laws ofThesyste- 11-1 /-/-<-.! r 1 matic treat- impact was evoked in the year 1668 by a request of the ment of the T-» 1 r> • r X 1 rr>.t • 1 • • laws of im- Koyal Society of London. Three eminent physicists pact. Wallis (Nov. 26, 1668), Wren (Dec. 17, 1668), and HuYGENS (Jan. 4, 1669) complied with the invitation of the society, and communicated to it papers in which, independently of each other, they stated, without de- ductions, the laws of impact. Wallis treated only of the impact of inelastic bodies. Wren and Huygens only of the impact of elastic bodies. Wren, previously to publication, had tested by experiments his theorems, which, in the main, agreed with those of Huygens. These are the experiments to which Newton refers in the Principia. The same experiments were, soon after this, also described, in a more developed form, by Ma- riotte, in a special treatise, Sur le Choc des Corps. Ma- riotte also gave the apparatus now known in physical collections as the percussion-machine. According to Wallis, the decisive factor in impact waiiis's re- is momentum, or the product of the mass {J>ondus) into the velocity {celeritas). By this momentum the force of percussion is determined. If two inelastic bodies which have equal momenta strike each other, rest will ensue after impact. If their momenta are unequal, the difference of the momenta will be the momentum after impact. If we divide this momentum by the sum of the masses, we shall obtain the velocity of the mo- tion after the impact. Wallis subsequently presented his theory of impact in another treatise, Mechanica sive de Motu, London, 1671. All his theorems may be 314 THE SCIENCE OF MECHANICS. brought together in the formula now in common use, u = {mv + m'v')/(m + w'), in which w, m' denote the masses, v, v' the velocities before impact, and u the velocity after impact. Huygens's 5. The ideas which led Huygens to his results, are and results, to be found in a posthumous treatise of his, De Moiu Corporum ex Percussione, 1703. We shall examine these in some detail. The assumptions from which Huygens Fig. 158. FiE 159- An Illustration from De Percussione (Huygens). proceeds are : (i) the law of inertia ; (2) that elastic bodies of equal mass, colliding with equal and oppo- site velocities, separate after impact with the same ve- locities ; (3) that all velocities are relatively estimated ; (4) that a larger body striking a smaller one at rest imparts to the latter velocity, and loses a part of its own ; and finally (5) that when one of the colliding bodies preserves its velocity, this also is the case with the other. THE EXTENSION OF THE PRINCIPLES. 315 Huygens, now, imagines two equal elastic masses, First, equal which meet with equal and opposite velocities v. After masses ex- the impact they rebound from each other with exactly locitiec the same velocities. Huygens is right in assuming and not deducing this. That elastic bodies exist which re- cover their form after impact, that in such a transac- tion no perceptible -vis viva is lost, are facts which ex- perience alone can teach us. Huygens, now, conceives the occurrence just described, to take place on a boat which is moving with the velocity v. For the specta- tor in the boat the previous case still subsists ; but for the spectator on the shore the velocities of the spheres before impact are respectively iv and 0, and after im- pact and 2 V. An elastic body, therefore, impinging on another of equal mass at rest, communicates to the latter its entire velocity and remains after the impact itself at rest. If we suppose the boat affected with any imaginable velocity, u, then for the spectator on the shore the velocities before impact will be respectively u -\- V and u — - v, and after impact u — v and u -\- v. But since u -\- v and u — v may have any values what- soever, it may be asserted as a principle that equal elastic masses exchange in impact their velocities. A body at rest, however great, is set in motion Second, the relative ve- by a body which strikes it, however small : as Ga-iodty of ap- proach and lileo pointed out. Huygens, now, recession is w the same. shows, that the approach of the w M^ bodies before impact and their ^^ \ ) recession after impact take place with the same relative velocity. A "S- ' ■ body m impinges on a body of mass M at rest, to which it imparts in impact the velocity, as yet undetermined, w. Huygens, in the demonstration of this proposition, supposes that the event takes place on a boat moving 3i6 THE SCIENCE OE MECHANICS. from J/ towards m with the velocity 7i!'/2. The initial velocities are, then, v — wji and — w jz ; and the final velocities, x and + wji. But as M has not altered the value, but only the sign, of its velocity, so m, if a loss of vis viva is not to be sustained in elastic impact, can only alter the sign of its velocity. Hence, the final velocities are — {v — w/a) and -|- w/2. As a fact, then, the relative velocity of approach before impact is equal to the relative velocity of separation after im- pact. Whatever change of velocity a body may suffer, in every case, we can, by the fiction of a boat in mo- tion, and apart from the algebraical signs, keep the value of the velocity the same before and after impact. The proposition holds, therefore, generally. Third,;; the If two masses M and m collide, with velocities V otapproachand V inversely proportional \.o the masses, yJ/ after im- ly proper- pact will rebound with the velocity Fand m with the masses, so velocity V. Let us suppose that the velocities after locitiesof impact are Fj and v^ ; then by the preceding proposi- tion we must have V-\-v^=V^-\-v.^, and by the prin- ciple of vis viva MV^ mv"^ _MV.^^ TOW, 2 ~2 ^ ~2~ ~ ~"2 ^ 1 ' Let us assume, now, that Pj ^ v -\- w; then, neces- sarily, F, = V — w ; but on this supposition And this equality can, in the conditions of the case, only subsist if to = ; wherewith the proposition above stated is established. Huygens demonstrates this by a comparison, con- structively reached, of the possible heights of ascent of the bodies prior and subsequently to impact. If recession. The extension of the principles. 317 the velocities of the impinging bodies are not inversely This propo- 11 1 1111 sition, by proportional to the masses, they may be made such by the fiction 1 r - r 1 • • ^T-11 - • 1 of a moving the nction ot a boat m motion. ihe proposition thus boat, made includes all imaginable cases. all cases. The conservation of vis viva in impact is asserted by Huygens in one of his last theorems (11), which he subsequently also handed in to the London Society. But the principle is unmistakably at the foundation of the previous theorems. 6. In taking up the study of any event or phenom- Typical ■ 1 1 J r • '^°^^^ °f enon A, we may acquire a knowledge of its component natural in- quiry. elements by approaching it from the point of view of a different phenomenon B, which we already know ; in which case our investigation of A will appear as the application of principles before familiar to us. Or, we may begin our investigation with A itself, and, as na- ture is throughout uniform, reach the same principles originally in the contemplation of A. The investiga- tion of the phenomena of impact was pursued simul- taneously with that of various other mechanical pro- cesses, and both modes of analysis were really pre- sented to the inquirer. To begin with, we may convince ourselves that the impact in ° -' the New- problems of impact can be disposed of by the New- tonian tonian principles, with the help of only a minimum of view. new experiences. The investigation of the laws of im- pact contributed, it is true, to the discovery of New- ton's laws, but the latter do not rest solely on this foun- dation. The requisite new experiences, not contained in the Newtonian principles, are simply the informa- tion that there are elastic and inelastic bodies. Inelastic bodies subjected to pressure alter their form without recovering it ; elastic bodies possess for all \}a.€\x forms definite systems of pressures, so that every alteration 3i8 THE SCIENCE OF MECHANICS. First, in- elastic masses. Impact in an equiva- lent point of view. of form is associated with an alteration of pressure, and vice versa. Elastic bodies recover their form ; and the forces that induce the form-alterations of bodies do not come into play until the bodies are in contact. Let us consider two inelastic masses M and m mov- ing respectively with the velocities Fand v. If these masses come in contact while possessed of these un- equal velocities, internal form-altering forces will be set up in the system M, m. These forces do not alter the quantity of motion of the system, neither do they displace its centre of gravity. With the restitution of equal velocities, the form-alterations cease and in in- elastic bodies the forces which produce the alterations vanish. Calling the common velocity of motion after impact u, it follows that Mu -\- mu=: MV-\- Mv, or u = (MF-{- mv)/(M-\- m), the rule of Wallis. Now let us assume that we are investigating the phenomena of impact without a previous knowledge of Newton's principles. We very soon discover, when we so proceed, that velocity is not the sole determina- tive factor of impact ; still another physical quality is decisive — weight, load, m.a.ss, pondus, moles, massa. The moment we have noted this fact, the simplest case is easily dealt with. If two bodies of equal weight or equal mass collide with equal and opposite velocities ; if, further, the bodies do not separate after impact but retain some common velocity, plainly the sole uniquely deter- mined velocity after the collision is the velocity 0. If, further, we make the observation that only the dif- ference of the velocities, that is only relative velocity, determines the phenomenon of impact, we shall, by imagining the environment to move, (which experience 3U o o Fig. l6l. THE EXTENSION OF THE PRINCIPLES. 319 tells us has no influence on the occurrence,) also readily perceive additional cases. For equal inelastic masses with velocities v and or z; and v' the velocity after impact is v fi or {v -\- v')li. It stands to reason that we can pursue such a line of reflection only after ex- perience has informed us what the essential and de- cisive features of the phenomena are. If we pass to unequal masses, we must not only The expe- know from experience that mass generally is of conse- conditions 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 F collide, we might reason V o 2^ V Fig. 162. Fig. 163. thus. We cut out of the mass 3 the mass i (Fig. 162), and first make the masses 1 and i collide : the result- ant velocity is {v + f^)/2. There are now left, to equalise the velocities {v -\- V")/2 and V, the masses 1-1-1=2 and 2, which applying the same principle gives v-Sr {^ _ V+3V v+SF + y 2 ~" 4 1 + 3 ■ Let us now consider, more generally, the masses m and m', which we represent in Fig. 163 as suitably proportioned horizontal lines. These masses are af- fected with the velocities v and v', which we represent by ordinates erected on the mass-lines. Assuming that 320 THE SCIENCE OF MECHANICS. Its points of w < m', we cut off from m' a portion m. The offsetting whh^the of m and m gives the mass 2 m with the velocity [v + Newtonian. ^,^^^^ ^^^ dotted line indicates this relation. We proceed similarly with the remainder ;;/ — m. We cut off from 2 m a portion m' — m, and obtain the mass ■2m — {in — m) with the velocity (?; + z/)/2 and the mass 2{m'—m) with the velocity [(w + v')/-2 + w']/2. In this manner we may proceed till we have obtained for the whole mass m + m' the same velocity u. The constructive method indicated in the figure shows very plainly that here the surface equation (m 4- m') u = mv -\- m'v subsists. We readily perceive, however, that we cannot pursue this line of reasoning except the sum 'mv \ m'v , that is \!a.% form of the influence of m and V, has through some experience or other been pre- viously suggested to us as the determinative and de- cisive factor. If we renounce the use of the Newtonian principles, then some other specific experiences con- cerning the import of 711 v which are equivalent to those principles, are indispensable. Second, the 7. The impact of ^/aj'/'zV masses may also be treated impact of . ... ,^, , elastic by the Newtonian principles. 1 he sole observation Newton's here required is, that a deformation of elastic bodies calls into play forces of restitution, which directly de- pend on the deformation. Furthermore, bodies pos- sess impenetrability ; that is to say, when bodies af- fected with unequal velocities meet in impact, forces which equalise these velocities are produced. If two elastic masses M, m 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 re- main unchanged, the common equahsed velocity will be MC^ mc u =^ . M -\- m Consequently, up to this time, M^s velocity has suf- fered a diminution C — u; and ot'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 in- verse 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, M will a second time sustain a diminution of velocity C — u, and m will a second time receive an increase of velocity u — c. Hence, we obtain for the velocities V, v after impact the expressions F= 2u — C and v^2u — c, or MC-^mi^c—C) _mc-\- M(2C—c) M -{- m ' M -\- m If in these formulae w& ■pxit M ^^ m, it will follow The deduc- ^ tion by this that V=2C and v=C; or, if the impinging masses are ™w of all equal, the velocities which they have will be inter- changed. Again, since in the particular case M/m = — c/C or MC -\- mc = ^ also ji = 0, it follows that V^='2u — C= — C and v =^2u — 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, m affected with the velocities C, c, estimated as positive when in the same direction, takes place with the velocity C — c\ their separation with the velocity V — V. But it follows at once from V=2ti — C, v^2u—c, that V—v^^ — {C—c); that is, the rela- tive velocity of approach and recession is the same. view. 322 THE SCIENCE OF MECHANICS. By the use of the expressions F=2« — C and v = 2u — c, we also very readily find the two theorems MV -\- /iiv ^= MC -\- mc and 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 Huy- gens's results. The impii- 8. If wc Consider the laws of impact from Huygens's Huygens's point of view, the following reflections immediately 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, ^2mv^. 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 done upon 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»2&2 also remains unchanged. Huygens now, observing that this first property of ma- terial 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 vires viva must be the same before and after im- pact. For in the mutually effected alteration of the forms of the colliding bodies the material system con- sidered has the same amount of work done on it as, on THE EXTENSION OF THE PRINCIPLES. 323 the reversal of the alteration, is done by it, provided al- ways 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 per- fectly elastic bodies. Contemplated from this point of view, the majority The deduc- of the Huygenian laws of impact follow at once. Equal laws of im- masses, which strike each other with equal but oppo- notion of ... , , . , , ... i-r^i ^^^ 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 same before and after impact. Fur- ther 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 in before and after impact, and let them be of any value and have any sign. We imagine the whole system to receive a velocity u of such magnitude that « -(- f = — (« -|- <:') or « = (c — c")!'!. 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 ve- locity-ratios of the impinging masses were not origin- 324 THE SCIENCE OF MECHANICS. Huygens's tacit appro- priation of the idea of mass. Construc- tive com- parison of the special and general case of im- pact. ally known, to draw on the Galileo-Newtonian system for certain conceptions, as was pointed out above. Such an appropriation of the concepts mass and mo- mentum, is contained, although not explicitly ex- pressed, in the proposition according to which the ve- locity of each impinging mass simply changes its sign when before impact JZ/w = — cjC. If Huygens had wholly restricted himself to his own point of view, he would scarcely have discovered this proposition, al- though, 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 oppo- site, the equalised velocity of the masses on the com- pletion of the change of form will be « = 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. If we imagine the entire system affected with a ve- locity of translation, this particular case will simulta- neously present \}a.& general c^sQ. Let the impinging masses be represented in the figure by M=BC and m = AC (Fig. 164), and their respective velo- cities by C=AD and c = BE. On AB erect the perpendicular CF, and through F draw IK parallel to AB. Then ID = [m. C~-^)/{M-\- ni) and KB = {M.C — c')II^M -\- m). 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 It = AI=^B = C— {in . C^^c)/{M-{- ni) = c+ (M. C'.^^c)/{M+ m) = (MC+mc)/{M+ m), Fig. 164. THE EXTENSION OF THE PRINCIPLES. 325 the spectator who is moving forwards with the velocity u will see the particular case presented, and the spec- tator who is at rest will see the general case, be the velocities what they may. The general formulae of im- pact, above deduced, follow at once from this concep- tion. We obtain : M -{- m M-\- m ^ M{C—c) _mc-\- M{2C^ c) v^^BH^c^ 2 M -\- m M -\- m Huygen's successful employment of the fictitious signifi- cance 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 differences of velocity. With re- spect 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 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 signifi- cance 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 veloci- ties, and in the second the square of the difference that is decisive. Velocity is a physical level, like tempera- ture, potential function, and the like. 326 THE SCIENCE OF MECHANICS. Possible It remains to be remarked, that Huygens could odgl^of have reached, originally, in the investigation of the ideaf™^ ^ phenomena of impact, the same results that he pre- viously 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- g. Let US close our examination of this part of the mentum in-subject with a few general remarks. The sum of the 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 in- creased ; 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 mo- mentum. 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. Quan- tity of motion, conceived in this sense, is always pre- served. The vis viva of a system of inelastic masses is al- tered in impact ; that of a system of perfectly elastic masses is preserved. The diminution of vis viva pro- duced in the impact of inelastic masses, or produced generally when the impinging bodies move with a com- mon velocity, after impact, is easily determined. Let M, m be the masses, C, c their respective velocities be- THE EXTENSION OF THE PRINCIPLES. 327 fore impact, and u their common velocity after impact ; Conserva- V . -1 1 1- . . . tion of vis tnen tne loss 01 vis viva is vi-va in im- pact inter- \MC'^ -\-\mc-^ — \{M -^m)U^, (l)preted. which in view of the fact that u =^ {MC + m c)/{M-\- m) may be expressed in the form ^{Mm/M-\- ni) (C — whence the values of the two accelerations are ob- tained PR—Qr„ , , PR—Qr These last determine the motion. Employ- It will be seen at a glance that the same result can 'ideas Stat- be obtained by the employment of the ideas of statical ical mo- ,. . . ^^^ , , . mentand moment and moment of inertia. We get by this method moment of . _ . inertia, to for the angular acceleration obtain this ■^«=""- _ PR—Qr _ PR~Qr g g and as y = R

ds = i2mliv^ — 7>J) (1) Theprinci- 2. In illustration of the principle of vis viva we pleillus- . ,,,,., trated by shall first Consider the simple problem which we treated the motion . . , of a wheel by the principle of D'Alembert. On and axle. a wheel and axle with the radii R, r hang the weights P, Q. When this machine is set in motion, work is per- formed by which the acquired vis viva is fully determined. For a rotation of the machine through the angle a, the work is P.Ra—Q. ra = a{PR—Qr). Calling the angular velocity which corresponds to this angle of rotation, cp, the vis viva generated will be P {RcpY Q {rcpy _ m2 Fig. 173. {FR'^ + Qr^). Consequently, the equation obtains a{PR— Qr). {PR^ -f 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 cp, and the THE EXTENSION OF THE PRINCIPLES. 345 angular acceleration ^, as obtains in free descent be- tween s, V, g. If in free descent s = v"^ /ig, then here Introducing this value of a in equation (i), we get for the angular acceleration of F, %j:^{PR — Qr/ PR"^ -\- Qr^')g, and, consequently, for its absolute ac- celeration Y = {-P-R — Qr/RR^ + Qr^) Rg, exactly as in the previous treatment of the problem. As a second example let us consider the case of a a roiling , -,, cylinder on massless cylinder of radius r, m 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 Fig. 174. these masses rolls without sliding down an inclined plane of the elevation a. First, we must convince our- selves, 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 m make with the velocity of progres- sion u the angles 6 and 0' (Fig. 175), where 6 -\- 0' = 180°. The compound velocities w and z satisfy therefore the equations ze)2 ^ 2/2 -)- z;2 — 2uvcos6 z^ =u^ -{- v^ — 2uvcos6'. 72_ 346 THE SCIENCE OF MECHANICS. The law of But since COS 6-= — COS d', it follows that motion of such a 7£/^ + 2;2 =- L! //2 _j_ ^! 7,a^ or, cylinder. ' ^mw"^ + Imz'^ = ),iirlii~ -\ lin'lv'^ = inn'' -\- wv^ If the cylinder moves through the angle cp, m describes in consequence of the rotation the space r cp, and the axis of the cylinder is likewise displaced a distance rep. As the spaces traversed are to each other, so also are the velocities v and //, which therefore are equal. The total vis viva may accordingly be expressed by 2mu^. If /is the distance the cylinder travels along the length of the inclined plane, the work done is 2img. Isma = 2mu'^; whence « = l/^/. sin a. If we compare with this result the velocity acquired by a body in sliding down an inclined plane, namely, the velocity 1/2^/ sin a, it will be observed that the contrivance we are here considering moves with only one-half the ac- celeration of descent that (friction neglected) a sliding body would under the same circumstances. The rea- soning of this case is not altered if the mass be uni formly distributed over the entire surface of the cylin- der. Similar considerations are applicable to the case of a sphere 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 nt uniformly over preceding the surface of a cylinder of radius Ji, which is coaxal with and rigidly joined to a massless cylinder of radius r, and let the latter roll down the inclined plane. Since here v/u = Ji/r, the principle of vis viva gives mgl sina = ^w«2(l _|_ Ji'i jr'^^, whence case. THE EXTENSION OF THE PRINCIPLES. -ii^-j For Rjr = I the acceleration of descent assumes its previous value g]i. For very large values of Rjr the acceleration of descent is very small. When Rjr = oo 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 chain, whose total length is /, and which lies partly on on an in- 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. mg over on the in- clined plane will draw the remainder after it. If // 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 ulv'^ X . x^ . — ^— =Mxg -^ sm a = /^^ — sm a, or V ^= X vg sin a /I. In the present case, therefore, the velocity acquired is proportional to the space de- scribed. The very law holds that Galileo first con- jectured 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 be employed, to solve problems of moving bodies, cipie of"£ when the total distance traversed and the force that """'' acts in each element of the distance are known. It was disclosed, however, by the labors of Euler, Daniel Ber- noulli, and Lagrange, that cases occur in which the 348 THE SCIENCE OF MECHANICS. principle of vis viva can be employed without a knowl- edge of the actual path of the motion. We shall see later on that Clairaut also rendered important services in this field. There- Galileo, even, knew that the velocity of a heavy searches of ^^jjj^^ -^^^^ depended solely on the vertical height de- scended 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 heights of the masses of the system. Euler was able to make a further step in advance. If a body K (Fig. 177) is at- tracted towards a fixed centre C in obedience to some given law, the increase of the vis viva in the case of rectilinear ap- proach is calculable from the initial and terminal distances Pig j^^ (r^, r,). But the increase is the same, if K passes at all from the position r^ to the position r,, independently of the forin 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 ulti- mately the same as before. The re- If K is attracted towards several fixed centres C, Daniel Ber- C , C" . . . ., the increase of its vis viva depends on the nouUi and ..... t n 1 i - i Lagrange, mitial distances r^, r^, r^ .... and on the terminal distances r,, r,', r,". . . ., that is on the initial and ter- minal 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 coordinates a, b, c with a point hav- ing the coordinates x, y, z, and denote by r the length of the line of junction and by a, ft, y the angles that line makes with the axes of x, y, z, then, according to Lagrange, because ;-2 = (.V _ a)2 + (_,,_ ^)2 + (z — 0^ X — a dr „ y — b dr cos a = = —-, cos p = = -;-, r dx r dy z — c dr cos Y = = -r- r dz dF(r) Accordingly, if fir) = — j-^— is the repulsive force, or The force dr compo- • r 1 • r • 1 1 nents, par- the negative of the attractive force acting between the tiai differ- . ... . ential coef- two points, the components will be ficientsof the same ,, , dI'Mdr dFir') function of X =/{r) cos or = ^ ' — ^ ^ '=°»"-'i'- coSrdi- dr dx dx ' n^'^^- y=/(.)COS/J=-^^- = -^J, Z=/WcosK = ^9^^^ = '^^^. •^ ^ ■' ' dr dz dz The force-components, therefore, are the partial differential coefficients of one and the same function of r, or of the coordinates of the repelling or attracting points. Similarly, if several points are in mutual ac- tion, the result will be X = ~ dx - dj Z — -- dz' dU , . dU ^ . 350 TffE SCIENCE OF MECHANICS. The force- where CA is a function of the coordinates of the points. function. ^^.^ 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 apphcable to rectangular coordinates, we obtain :2C{Xdx + Ydy + Zdz) = :2\m (w^ — w„2) or, since the expression to the left is a complete differen- tial, ^( CdU , dU , ^, 2JdU=:S(,U,— U:) = 2im(_v' — v,^), where [/^ is a function of the terminal values and U^ the same function of the initial values of the coordi- nates. This equation has received extensive applica- tions, 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 posi' tions, 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 CreW&'s Journal fur Mathe- matik, 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 Dynavtih, edited by Clebsch, 1866. THE EXTENSION OF THE PRINCIPLES. 351 the form which mechanics has historically assumed, dy- History of ■ r J J • / r 1 T~v » A 1 *^® princi- namics IS lounded upon statics, (tor example, DAlem- pie of least , , ... , ... .. ,,., constraint. bert s prmciple on the prmciple of virtual displace- 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 new principle, Gauss remarks, can now be established in mechanics ; but this does not exclude the discovery of new points of view, from which mechan- ical phenomena may be fruitfully contemplated. Such a new point of view is afforded by the principle of Gauss. 2. Let «, »2, . . . . be masses, connected in any man- statement ., / of the prin- ner with one another. These masses, ii_?9'^«', would, under cipie. the action of the forces im- pressed on them, describe in a very short element of time the spaces a b, a, b , . . . .; but in consequence of their connec- tions they describe in the same element of time the spaces a c, a, c, . . . . Now, Gauss's principle asserts, that the mo- tion 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 »2(i5,r) 2 -|- m, {b,c,y + ■ • • ■=^ ^ m{b cy , is a minimum, that is, is smaller for the actual motion 352 THE SCIENCE OF MECHANICS. than for any other conceivable motion in the same con- nections. If this sum, '2m(bcy, is less for rest than for any motion, equilibrium will obtain. The principle includes, thus, both statical and dynamical cases. Definition The Sum 2m{l>cy is called the "constraint."* In of " con- . , , . . , , . . straint." formmg this sum it IS plam that the velocities present ill the system may be neglected, as the relative posi- tions 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 m in an element of time through the space ab, the effective forces carry the same mass in the same time in consequence of the con- nections through the space ac. We resolve ab into ac and cb; and do the same for all the masses. It is thus evident that forces corresponding to the dis- ^ tances cb, c,b, . . . . and propor- tional to mcb, m^Cfb,..., do not, Fig. 179. owing to the connections, become effective, but form with the connections an equilibrat- ing system. If, therefore, we erect at the terminal posi- tions c, Ci, c,i the virtual displacements cy, c, y,-.--, forming with cb, c, b,.... the angles 6, 6,.... we may apply, since by D'Alembert's principle forces propor- tional to mcb, m, c, b,.... are here in equilibrium, the principle of virtual velocities. Doing so, we shall have * Professor Mach's term is Abweichungssumme . The Abweickung is the declination or departure from free motion, called by Gauss the Ablenkung. (See Duhring, Principien der Mechanik, g§ i68, l6g ; Routh, Rigid Dynamics. Part 1, §§ 390-394.) The quantity ^ m \pcY 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, THE EXTEXSIOX OF THE PRINCIPLES. 353 2m ci .CV COsd'P^O (1") The deduc- ' ^ ^ tion of the But principle of least (iyy = (icy + {cyy —2bc.CyCOS.e, constraint. (J>yy — (J>cy = {cyY — 2bc . cycosd, and 2m{6yy — 2m(l>cy=2m{cyy — 22m6c.cycosd (2) Accordingly, since by (i) the second member of the right-hand side of (2) can only be ^ or negative, that is to say, as the sum 2m(cyy can never be dimin- ished by the subtraction, but only increased, therefore the left-hand side of (2) must also always be positive and consequently '2m{byy always greater than '2m (bey, which is to say, every conceivable constraint from unhindered motion is greater than the constraint for the actual motion. 4. The declination, b c, for the very small element various forms in of time r, may, for purposes of practical treatment, be which the principle designated by s, and following Scheffler (Schlomilch's may be ex- pressed. Zeitschrift fiir Mathematik und Physik, 1858, Vol. Ill, p. igy), we may remark that .f = yr"^ ji, where y de- notes acceleration. Consequently, 2ms'^ may also be expressed in the forms 72 7-2 74 'Sm .s.s = ^2my.s = -y 2p .s = --2my^, where/ 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 2ms-^ (1) or :sps (2) or :Smy^ (3) is a minimum. 354 THE SCIENCE OF MECHANICS. The motion of a wheel and axle. 5. We will first employ, in our illustrations, the third form. Here again, as our first example, we se- lect the motion of a wheel and axle by the overweight of one of its parts and shall use the designations above frequently employed. Our problem is, to so determine the actual accel- erations y oi P and y, of Q, that shall be a minimum, or, since y, = — yir/R), so that Pig— yY + C(^+ y-rjRY^N shall assume its smallest value. Putting, to this end, Fig. 180. dN 7;7 = -^(^- ■r) + e[.°-+ri)i = o, we get y — {PR — Qr/PR^ + Qr^) Rg, exactly as in the previous treatments of the problem. Descent on As Our second example, the motion of descent on plane. an inclined plane may be taken. In this case we shall employ the first form, 2ms^. Since we have here only to deal with one mass, our in- quiry will be directed to find- ing that acceleration of de- scent y for the plane by which the square of the de- clination (i-2) is made a minimum. By Fig. 181 we have smar. Fig. 181. J2=U + r -^g-j. 2 / ' Y 2 j ~"v*y ^T and putting d{s '^)/dy = 0, we obtain, omitting all constant factors, 2y — 2g sin or = or y^g. sin a, the familiar result of Galileo's researches. THE EXTENSION OF THE PRINCIPLES. 355 The following example will show that Gauss's prin- a case of ciple also embraces cases of equilibrium. On the arms rium.' a, a' of a lever (Fig. 182) are hung the heavy masses m, m' . The principle requires that m{^g — y)^ -{- m'{g — ^')2 shall be a minimum. But^'=; — y{a'/a). Further, if the masses are in- versely proportional to the I A 1 lengths of the lever-arms, that «[_| f-i is to say, if m/m' = a' /a, then y' ^ — y {m/m"). Conse- 'S-^^- quently, m {g — K) ^ + ^'(^ + V ■ mjm'Y = N must be made a minimum. Putting dN/dy =^ 0, we get m{i -\- m/m')y = or ;^ = 0. Accordingly, in this case equilibrium presents the least constraint from free mo- tion. Every new cause of constraint, or restriction upon New causes the freedom of motion, increases the quantity of con- straint in- , , . . , , - ... crease the stramt, but the increase is always the least possible, departure T r 1 1 1 • r from free It 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 di- rection of its path, the acceleration g sin a. Denoting, therefore, by y sin a the acceleration corresponding to this excur- p. ^g sion at the axial distance i on the com- pound pendulum, .2w {g sin a — ry sin a) 2 or '2m {g — ryy^ will be the quantity to be made a minimum. Conse- quently, 2m{g — ryy=zQ, and y = g(2mr/2mr^). 356 THE SCIENCE OF MECHANICS. The problem is thus disposed of in the simplest man- ner. But this simple solution is possible only because the experiences that Huygens, the BernouUis, and oth- ers long before collected, are implicitly contained in Gauss's principle, iiiustra- 6. The increase of the quantity of constraint, or tions of the . .. . ^ „ , ■ i £ preceding dcchnation, from free motion by new causes oi con- emen . ^j.^.^^^^^ ^^^ ^^ exhibited by the following examples. Over two stationary pulleys A, B, and beneath a movable pulley C (Fig. 184), a cord is passed, each Fig. 185. extremity of which is weighted with a load P; and on Ca load -zF -\- j> is placed. The movable pulley will now descend with the acceleration (J>/^P -\- f) 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 in- creased. The load suspended from B, since it now moves with double the velocity, must be reckoned as possessing four times its original mass. The mova- ble pulley accordingly sinks with the acceleration (//S/" -\- f) g. A simple calculation will show that the constraint in the latter case is greater than in the former. THE EXTENSION OF THE PRINCIPLES. 357 A number, n, of equal weights, /, 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 /. According as all the pulleys are movable or all except one ^xe. fixed, we obtain for the motive weight/, allowing for the relative velocities of the masses as re- ferred to/, respectively, the accelerations (4«/i -\- \n)g and (.4/5) g- If all the n-\- 1 masses are movable, the deviation assumes the value/^/4« + i, which increases as «, the number of the movable masses, is decreased. Fig. 186. 7. Imagine a body of weight Q, movable on rollers Treatment . . , . , of a me- on a horizontal surface, and having an inclined plane chanicai' , . 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 7"s descent more hindered. To any distance of descent h of /' a horizontal velocity v and a vertical velocity u of P and a horizontal velocity w oi Q correspond. Owing to the conservation of the quantity of horizontal mo- tion, (for here only internal forces act,) we have Pv = Qw, and for obvious geometrical reasons (Fig. 186) also M = (z' -|- ^) tan a The velocities, consequently, are : =^ U 3S8 THE SCIENCE OF MECHANICS. Q First.bythe V = „ 7 ^ COi a . U, principles ' ~r Cl of the con- servation of J> momentum ^ —- COt ff . U. and of V2S jp I Q villa.. ^^ "^ ■uiva. For the work Ph performed, the principle of vis viva gives -,, Fu^ P( Q , Y^^ , gXJ'+Q I 2- Multiplying by -— , we obtain / Q cos2ar\«2 To find the vertical acceleration y with which the space h is described, be it noted that h = u"^ jo. y. In- troducing this value in the last equation, we get (7'+ 0sin2a V = ■ — —- . s: '^ Psm^a+ Q ^ For Q = CD, y = g sin ^ a, the same as on a sta- tionary inclined plane. For ^ = 0, y =:g, s.s in free descent. For finite values of Q^mF, we get, smce ^-- ■ > 1, sm^a -(- m (1 -1- m)sm.^a . „ "' — .g'^gsm^a. ' m -\- sin^a 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 THE EXTENSION OF THE PRINCIPLES. 359 principle of vis viva. Employing Gauss's principle, Second, by we should proceed as follows. To the velocities de- cipie of . Gauss. noted as u, v, w the accelerations y> "> ^ correspond. 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 P O o 00 a minimum. As the problem possesses significance only when the bodies P and Q touch, that is only when y = {d ■\- e) tan a, therefore, also P P O N^— lg—iS+ £)tana]2 -\- - 6'^ -\- - e^ . Forming the differential coefficients of this expression with respect to the two remaining independent vari- ables 8 and e, and putting each equal to zero, we ob- tain — [^_((y_|_ e)tana] i'tana-f P8 = Q and _ [^_ ((J -I- e) tan or] /'tan a -\- Qe^O. From these two equations follows immediately PS — Qs^^O, 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 /3 with the horizon the space s, of which the horizontal and vertical components are v and u, while simulta- neously Q describes the horizontal distance w. The force-component that acts in the direction of s is Psin /?, consequently the acceleration in this direction, allow- ing for the relative velocities of P and Q, is P.sin^ p~q7^' 7 .f U/ 36o THE SCIENCE OF MECHANICS. Third, by Employing the following equations which are di- the ex- ,,,.,, tended con- rectly Qeaucible, cept of mo- ^^ „ mentofin- Qw=^FV ertia. ^ V = s cosp u ^^v tan /J. the acceleration in the direction of s becomes Qsi-a.fi Q + Pco^^ and the vertical acceleration corresponding thereto is _ (2sin2^ '^ ~ Q + P^^^i' ^' an expression, which as soon as we introduce by means of the equation u = {v ^ -w) tan a, the angle-func- tions of a for those of yS, again assumes the form above given. By means of our extended conception of mo- ment of inertia we reach, accordingly, the same result as before. Fourth, by Finally we will deal with this problem in a direct cipies.''"" manner. The body/' does not descend on the mova- ble 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 coun- terforce {P/g){g — y). But as F and Q, friction neglected, can only act on each other by means of a pressure S, normal to the inclined plane, therefore P {g — y) ^ 6' cos a and n ■ <2 -P o sm a = — s ^■— From this is obtained -z(g—r)=-~ecota, o S THE EXTENSION OF THE PRINCIPLES. 361 and by means of the equation y =^{S -\- s) tan a, ulti- mately, as before, _(7'+0sin2a Psin2a+ Q Q sin a cos a g (1) Fsm^a-\- Q I' sin. a cos a g (2) ' Fsm^a -\- Q" (3) If we put P^Q and a^=A.K , we obtain for this Discussion • , „ T, of there- particular case y = \g, o = \g, e = ^g. For F/g = suits. Q/g= I we find the "constraint," or declination from free motion, to be^^^^. If we make the inclined plane stationary, the constraint will be g^/'2. If jP moved on a stationary inclined plane of elevation /J, where tan /3 = y/S, that is to say, in the same path in which it moves on the movable inclined plane, the constraint would only be g^ /S- 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 • 1 ■ • rr 1 1 1 principle no substantially new insight or perception is afforded by affords no • 1 T^ 1 ■ r / r ^ • newinsight Gauss's principle. Employing form (3) of the prin- ciple and resolving all the forces and accelerations in the mutually perpendicular coordinate-directions, giv- ing here the letters the same significations as in equa- tion (i) on page 342, we get in place of the declination, or constraint, '2, my''', the expression N^^n m I \m 7 \in (4) and by virtue of the minimum condition dN=t'2m m J \m ' 362 THE SCIENCE OF MECHANICS. m j or :2l{X—mS)dS+iY—mrf)dri+{Z—mZ)dZ'] = Q. Gauss's and If no connections exist, the coefficients of the (in berfs prin- that case arbitrary) dS, dr), dS,, severally made = 0, ciples com- . . , . .,-^ .^ . ' - mutable, give the equations of motion, cut it connections do exist, we have the same relations between d^, drj, dS, as above in equation (i), at page 342, between Sx, Sy, 8z. 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 differentia- tion D'Alembert's equations, we are led perforce to the principle of Gauss. The principle, therefore, is new only in form and not in matter. Nor does it, further, possess any advantage over the Lagrangian form of D'Alembert's principle in respect of competency to com- prehend both statical and dynamical problems, as has been before pointed out (page 342). The phys- There is no need of seeking a mystical or metaphys- ical basis . c r- •■i^i • of the prin- teal reason for Gauss's principle. The expression ' ' least constraint" may seem to promise something of the sort ; but the name proves nothing. The answer to the question, "/« what does this constraint consist ? " can- not 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. THE EXTENSION OF THE PRINCIPLES. 363 Once we have recognised work as the factor deter- Raie of the - . factor work minative 01 motion, once we have grasped the mean- ing 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 in- volve no difficulty, namely, that all the work that can be performed in an element of time actually is per- formed. 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 counter-work 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 foun- . dationsot Let there be two masses m and m at A, the one im- the princi- ple recog- pressed with a force /, the other with nisabie in „ the sim- the force ^. If we connect the two, we , „ ^o piest cases. shall have the mass 2 m acted on by a resultant force r. Supposing the spaces described in an element of time by the free masses to be represented by A C, A£, the space described by the con- joint, or double, mass will he AO ^ ^AD. The deviation, or constraint, is m{OB^ + OC^). It is less than it would be if the mass arrived at the end of the ele- ment of time in M or indeed in any point lying out- side of B C, say JV, as the simplest geometrical con- siderations will show. The deviation is proportional to the expression p^ -\- g'^ -{- '2.pq cos 6/2, which in the case of equal and opposite forces becomes 2/2^ and in the case of equal and like-directed forces zero. Two forces / and q act on the same mass. The force q we resolve parallel and at right angles to the Fig. 187. 364 THE SCIENCE OF MECHANICS. Even in the direction of / in r and s. The work done in an element the compo- of time is proportional to the squares of the forces, and forces its if there be no connections is expressible by/^ -|- ^2 __ properties , , . . . , , i arefound. ^2 _|_ 7- 2 _|_ j2 _ jf now ^ act directly counter to the force p, a diminution of work will be effected and the sum mentioned becomes (/ — r)^ -f j2_ 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 simul- taneously 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 ex- ample, 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. Theorig- I. Maupertuis enunciated, in 1747, a principle scur'eform which he called ^^ le principe de la moindre quantite d'ac- cipieof tion," the principle of least action. He declared this leastaction. ... . i - i . i -, , . , , principle to be one which eminently accorded with the 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 THE 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 prin- ciple of virtual velocities. Its indistinctness will be more saliently displayed by the details. 2. Let us see how Maupertuis applies his principle. Determina- . . . tion of the If M, m be two inelastic masses, C and c their velocities laws of im- , ... ... pact 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- — v)"^ -\- m{c — u)'^ is a minimum; that is, M{C — ii)-\- m(^c — u)^^\ or J/C+ VIC M ^ VI For the impact of elastic masses, retaining the same designations, only substituting V and v for the two ve- locities after impact, the expression M(^C — ^)^ + vi{c — z;)2 is a minimum; that is to say, M{C—V)dV^vi{c — v)dv = ^ (1) In consideration of the fact that the velocity of ap- proach before impact is equal to the velocity of reces- sion after impact, we have C—c = — (^V—v') or C+ F— (f+P) = (2) and d V— r/p = (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 SCIENCE 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 auction of the lever. Two masses M and m (Fig. 188) rest on a the law of _ . . ^ . , the lever by bar a, which the fulcrum divides into the portions this prin- . . , , . cipie. X and a — x. If the bar be set tn rotation, the veloci- ties and the spaces described will be proportional to the lengths of the lever-arms, and Mx''' -\- m{a — xY is the quantity to be made a minimum, that is Mx — m.{a — a^) = ; whence x =: majM -\- m, — a condition that in the case of equilib- , rium is actually fulfilled. In M ■ m criticism of this, it is to be '^ ^_^ remarked, first, that masses Fig. 188, not subject to gravity or other forces, as Maupertuis here tacitly assumes, are always in equilibrium, and, secondly, that the inference from Maupertuis's deduc- tion 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 ap- pertuis's proximate accord with the preceding, we should have to assume that the heavy masses M ^.-aA m 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 ex- pression, M(^g — w)2 -f- m{g — vY; whence M{g — u) du -\- m{g — v)dv = 0. But in view of the connection of the masses as lever. deduction. THE EXTENSION OF THE PRINCIPLES. 367 — = r . and a du = T dv; o whence these equations correctly follow Ma — mb , , Ma — mb ■u — a P" ^} — h p* Ma'^ ^mb'^^' Ma^-\-mb^^' and for the case of equilibrium, where « = z/ ^ 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 ■n/r '1 ii' -til °* '^^ ™°" mtz, Maupertuis also treats by his method the motion tionoiy\%ht r T 1 Tx -1 by the prin- of light. Here agam, however, — '- -' he employs the notion "least ac- tion" in a totally different sense. The expression which for the case of refraction shall be a min- imum, is m . AR -\- n . RB, where AR and RB denote the paths described by the light in the first and second media re- spectively, and m and n the corresponding velo- cities. True, we really do obtain here, if R be de- termined in conformity with the minimum condition, the result sin a /sin /J = njin = const. But before, the ' ' action " consisted in the change of the expressions mass X velocity X distance ; now, however, it is con- stituted 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. AR — n. RB or (^m — n){AR — RB) be taken as a minimum, and if not, why not ? But Fig. i8g. 368 THE SCIENCE OF MECHANICS. even if we accept Maupertuis's conception, the recip- rocal values of the velocities of the light are obtained, and not the actual values. Character;- It will thus be Seen that Maupertuis really had no Mauper- principle, properly speaking, but only a vague form- ciple.^"" 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 per- formance, 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 endeavor 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 at- tempt of Maupertuis. Euief'scon- c. Euler's view is, that the purposes of the phe- tributions "^ „ y , , . , , to this sub- nomena of nature afford as good a basis of explana- ject. tion as their causes. If this position be taken, it will be presumed a priori that all natural phenomena pre- sent 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 pro- ceeds much more scientifically than Maupertuis. He seeks an expression whose variation put = gives the ordinary equations of mechanics. For a single body moving under the action of forces THE EXTENSION OF THE PRINCIPLES. 369 Euler finds the requisite expression in the formula Cv ds, where ds denotes the element of the path and V the corresponding velocity. This expression is sm aller for the path actually taken than for any other infinitely 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 makes Cv ds a minimum, we can also determine the path. The problem of minimising Cv ds is, of course, as Euler assumed, a permissible one, only when v de- pends 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 coordinates. For a mo- tion in a plane the expression would accordingly as- sume the form The form which the principle assumed in Euler's hands. A '+(ir dx Jcp (x, y) ■ In the simplest cases Euler's principle is easily veri- fied. If no forces act, v is constant, and the curve of motion becomes a straight line, for which Cv ds = V C ds is unquestionably shorter than for any other curve between the same terminal points. Also, a body moving on a curved surface without the action of forces or friction, preserves its velocity, and describes on the surface a shortest line. The consideration of the motion of a projectile in a parabola ^^C (Fig. 190) will also show that the quantity Cv ds is smaller for the parabola than for any other neighboring curve ; smaller, even, than for the straight line ^^C between the same ter- minal points. The velocity, here, depends solely on the Fig. igo 370 THE SCIENCE OF MECHANICS. Mathemat- vertical space described by the body, and is therefore ical devel- '^ . ' opmentot 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 »'s, though in the upper portions smaller for the straight line AD than iox A B, are in the lower portions just the reverse ; and as it is here that the larger v's come into play, the sum upon the whole is smaller iox A B C than for the straight line. Putting the origin of the coordinates at A, reckon- ing the abscissas x vertically downwards as positive, and calling the ordinates perpendicular thereto y, we obtain for the expression to be minimised /^2,.(.+.)^l+(|l)^y,. dx, 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 V V(. + -)|J V- = C or <^ dy ■s, Cdx or V2g(a + x)—C^ and, ultimately, C ^ THE EXTENSION OF THE PRINCIPLES. 371 where C and C denote constants of integration that pass into C= V iga and C"= 0, if for a: = 0, dx/dy = and jc = be taken. Therefore, y = 21^ ax. By this method, accordingly, the path of a projectile is •shown to be of parabolic form. 6. Subsequently, Lagrange drew express attention The addi- ... -.-, .. tions of La- to 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 Cv ds for the ac- tual motion is a minimum, but simply that the variation of this expression, in its passage to an infinitely adjacent neighboring path, is = 0. Generally, indeed, this con- dition 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 180°, 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 1-11 expression that the ordinary equations of motion are obtained by but one of . . f ^ -n . 1 manywhich equating the variation of ivds to zero. But since the give the *^ 1 1 - r 1 • 1 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 expres- sions may be devised that give by variation the ordi- nary equations of motion, without its following that the 372 THE SCIENCE OF MECHANICS. 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 . « , i .1 . must po5- an expression as Cv ds does possess the property men- icai import, tioned, and we will now endeavor to ascertain its phys- ical import. To this end the analogies that exist be- tween the motion of masses and the motion of light, as well as between the motion of masses and the equilib- rium- of strings — analogies noted by John Bernoulli and by Mobius — will stand us in stead. A body on which no forces act, and which there- fore preserves its velocity and direction constant, de- scribes 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, as- sumes the shape of a straight line. Elucidation A body that moves in a curved path from a point of this im- . ^ , , . port by the ^ to a point B and whose velocity v = (p{x, y, z) is a motion of a . ., . ^ t -n mass, the function of Coordinates, describes between A and B a ray of light, curve for which generally Cv ds is a minimum. A ray and the r 1- i . r a ^ n i equilibrium of light passing Irom A to B describes the same curve, of a string. if the refractive index of its medium, n^^ (p {x, y, z), is the same function of coordinates ; and in this case Cnds is a minimum. Finally, a string passing from ^ to ^ will assume this curve, if its tension S = cp {x, y, z) is the same above-mentioned function of co- ordinates ; and for this case, also, CSds is a minimum. The motion of a mass may be readily deduced from the equilibrium of a string, 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 PRINCIPLES. 373 BC, BD (Fig. 191), are in equilibrium. If now, a body, The motion with a velocity v represented in magnitude and direc- deduced tion by AB, enter the element of the path ds, and re- equilibrium , , . . , , . „ „ of a string. ceive withm the same the velocity component BF = — BD, the body will proceed on- ward with the velocity v' = BC. Let Q be an accelerating force whose action is directly opposite to that of F; then for unit of time the acceleration of this force will be Q, for unit of length of the string Q/v, and for the element of the string {Q/v)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 Fig. 191. P: = S: The minus sign indicates that the directions of P and Q are opposite. A closed circular string is in equilibrium when be- The equi- ° ^ librium of tween the tension 5 of the string, everywhere constant, closed . strmgs. and the force P falling radially outwards on unit of length, the relation P = S/r obtains, where r is the radius of the circle. A body will move with the con- stant velocity z; in a circle, when between the velocity and the accelerating force Q acting radially inwards the relation V P^ — = — or Q =: — obtains. V r r A body will move with constant velocity v in any curve when an accelerating force Q = v'^/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 = Sjr 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 motion of to the motion of light. Consequently, the deduction of light from ./ o T. ./ / the motions the motiou of light from the equilibrium of a string or of masses ^ -^ _ ° and the the motion of a mass must be differently effected. A equilibrium •' of strings, mass, let US say, is moving with the velocity AB = v. (Fig. 192.) A force in the direction BD is impressed on the mass which produces an increase of velocity BE, so that by the composition of the ve- locities BC^=AB and BE the new velocity BF = v' is produced. If we resolve the velocities v, v' into com- ponents parallel and perpendicular to the force in question, we shall per- ceive that the parallel components alone are changed by the action of the force. This being the case, we get, denoting by k the perpendicular component, and by a and a' the angles v and v' make with the direction of the force, k =^ V sin a k ^v' sin a' or sin a v' sin a' V ' If, now, we picture to ourselves a ray of light that penetrates in the direction of » 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- Fig. 192. THE EXTENSION OF THE PRINCIPLES. 375 tion n into a medium having the index of refraction n' , Deveiop- . , . ... mentofthis where njn = vjv , this ray of hght will describe the illustration. same path as the body in the case above. If, there- fore, we wish to imitate the motion 0/ a mass by the motion of a ray of light (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 d I — j = Pdq, and for the index of refraction, by analogy, d{^ = Pdq, 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 /'cos a. ds rf ( .-, ) = -^ cos a . ds. For the path of a projectile, under the conditions above assumed, we obtained the expression jy = 2 1^ a x. This same parabolic path will be described by a ray of light, if the law n = \/2g(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 , . 1 1 , . . . . J the mini- manner in which this minimum property is related to mum prop- . erty to the the form of the curve. Let us take, first, (Fig. 193) a form of curves. broken straight line ABC, which intersects the straight line MN, put A£ =^ s, BC=^ s', and seek the condition that makes vs -f- v's' a minimum for the line that passes 376 THE SCIENCE OF MECHANICS. First, de- duction of the mini- mum Condi tion. through the fixed points A and B, where v and v are supposed to have different, though constant, values 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 vs -\- v's' is increased hereby by an amount — vm sin a -\- v' m sin a, where m^DB. The alteration is accordingly propor- tional to — V sva. a -\- v' zva. a' , and the condition of minimum is that — z)sina-|- z/sin or'^O, or sin a sin a Fig. 193. Fig. 194. If the expression sjv -\- s' fv' is to be made a minimum, we have, in a similar way, sma sin a' Second, the If, next, we consider the case of a string stretched application . , , . . ° of this con- m the direction ABC, the tensions of which S and S' ditiontothe equilibrium are different above and below MN, in this case it is of a string. , . . the mmmium of ^j- + 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 motion of a ray of light. String stretched once between A and B and thrice be- tween B and C, and finally a weight P attached. Then S:= F and S' = ^B. If we displace the point B a dis- tance w, 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 Of ' =: 0, no work is performed. Hence, the mini- mum of ^Sj + 6"j'' corresponds to a maximum of work. In the present case the principle of least action is sim- ply a different form of the principle of virtual displace- ments. Now suppose that ABC is a ray of light, whose ve- Third, the application locities V and v' above and below MN are to each other of this con- . r 1- 1 ditiontothe as 3 to I. The motion of light be- tween two points A and B is such that the light reaches B va. z. mini- mum of time. The physical reason of this is simple. The light travels from A to B, in the form of ele- mentary waves, by different routes. Owing to the periodicity of the light, the waves generally destroy each other, and only those that reach the designated paint in equal times, that is, in equal phases, produce a result. But this is true only of the waves that arrive by the minimum path and its adjacent neigh- boring paths. Hence, for the path actually taken by the light sfv -\- s' jv is a minimum. And since the in- dices of refraction n are inversely proportional to the velocities v of the light, therefore also ns A^ n's' is a minimum. In the consideration of the motion of a mass the con- dition that f'j- -|- v's' shall be a minimum, strikes us as something novel. (Fig. 195.) If a mass, in its passage Fig. 195. 378 THE SCIENCE OF MECHANICS. Fourth, its through a plane MN, receive, as the result of the action to the mo- of a force impressed in the direction DB, an increase of tion of a , , i • i • ■ • i i • • j / mass. velocity, by which v, its original velocity, is made v , we have for the path actually taken by the mass the equa- tion V sin a ^ v' sin a' ^= k. This equation, which is also the condition of minimum, simply states that only the ve- locity-component parallel to the direciio?i of the force is altered, but that the component k at right angles thereto re- mains unchanged. Thus, here also, Euler's principle simply states a familiar fact in a new form. (See p. 575.) Form of the I o. The minimum condition — v sin a -\- v' sin a'= minimum - , . . . condition may also be written, 11 we pass from a finite broken applicable . , ., . to curves, straight line to the elements of curves, m the form — V sin a -\- (v -\- dv) sin(a -|- doi) = or d(v sin a) = or, finally, V sin a = const. In agreement with this, we obtain for the motion of light d {n sin «■) ^ 0, n sin a = const, /sinaX sin a a := 0, = const, \ V J V and for the equilibrium of a string ^ (6* sin a) = 0, .S sin a = const. To illustrate the preceding remarks by an ex- ample, 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 ve- locity h&v = V-zgia -\- X), and let the axis of theji^-or- dinates be horizontal. The condition v . sin or = const, or Vng(^a -\- x) . dy/ds = const, is identical with that which the calculus of variation gives, and we now know THE EXTENSION OF THE PRINCIPLES. 379 Fig. ig6. its simple physical ^\%miiC'&.T\c&. If we picture to ourselves illustration / of the three a string whose tension \s S ^:=V i g (a -\- x), an arrange- typical . ... cases by ment which might be effected by fixing frictionless curvilinear 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 phys- ical significance of which is now ob- vious. When the distances between the rods are made infinitely small the string assumes the parabolic form. In a medium, the refractive index of which varies in the vertical direction by the law n = V'igia -\- x), or the velocity of light in which similarly varies by the law v = l/l/2^(a -\- x), a ray of light will describe a path which is a parabola. If we should make the velocity in such a medium v=^V' '2'g{a-\-x), the ray would describe a cycloidal path, for which, not f'\/2g{a -\- x^ . ds, but the expression Cds l\/ 7,g{a -\- x) would be a minimum. II. 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 that the systems of forces that make the tension, or, as the case may be, the ve- locity, the same function of coordinates, are differ- ent. If-,we consider, for example, the force of gravity. Fig. 197. 38o THE SCIENCE OF MECHANICS. The condi- V = V 2g(a + x). A String, however, subjected to the conse^° action of gravity, forms a catenary, the tension of ?he priced- which IS given by the formula 5= m — nx, where m mg^ana o- ^^^ ^^ ^^^ Constants. The analogy subsisting between the equihbrium 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 force- function [/, there obtains in the case of equilibrium the easily demonstrable equation U'-\- S=: const. This physical 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 La- grange, who presented it in the form 62m Ci'ds = 0. If we reflect that the principle of vts 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 THE EXTENSION OF THE PRINCIPLES. 381 dJ{U-\- T)di = Q, or j{6U-\- ST)dt = 0, The points of identity of Hamil- ton's and D'Alem- bert's prin- ciples. where (JC^and dy denote 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 differ- ence 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 example — of a wheel ■^ -^ -^ -^ and axle. the same that served to illustrate the prin- ciple 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 inter- val of time, a different motion, varying in- finitely little from the actual motion, but coinciding exactly with it at the beginning and end. There are thus produced in every element of time dt, variations of the work iSU^ and of the vis viva (dTy, variations, that is, of the values £/'and T realised in the actual motion. But for the actual mo- tion, 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 di an amount a from the angle of the actual motion, the variation of the work corre- sponding to such an alteration will be oU= {FJi — Qr) a = Ma. * Compare, for example, Kirchhoff, Vorlesungen tiber mathematische Phy- sik, Meckanik, p. 25 et segg., and Jacobi, Vorlesungen titer Dynamik, p. 58. Fig. 198. 382 THE SCIENCE OF MECHANICS. Mathemat- The vis viva, for any given angular velocity go, is ical devel- „ opment of ,„ 1 , ^ ^^ ^ «v ^ tiiis case. r = - {FH^ -\- Qr^)^r' g ^ and for a variation dw of this velocity the variation of the vis viva is But if the angle of rotation varies in the element dt an amount a, 00^^—^ and at g^ ^ ^ ' dt dt The form of the integral expression, accordingly, is I ///v dt = ^. j^Ma^N'^" dt to " Btit as d ,,, . dN , da therefore, f(M- ^-^] a.dt+ (7V«)'' ^ 0. 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 -J an expression which, since a in every element of time is arbitrary, cannot subsist unless generally M T- = 0. dt THE EXTENSION OF THE PRINCIPLES. 383 Substituting for the symbols the values they represent, we obtain the familiar equation dco PR— Qr (it PR-^ + Qr ■g- D'Alembert's principle gives the equation The same results ob- / d N\ tained bv I M -J— \a ^= 0, the use o£ dt \ "' D'Alem- bert's prin- which holds for every possible displacement. We might, '^'''^' in the converse order, have started from this equation, have thence passed to the expression to and, finally, from the latter proceeded to the same re- sult J' Ma + N~\ dt — (iVa)' = dt I to f{Ma + Jv'-^)di=0. io 3. As a second and more simple example let us illustration . -, , . r - 1 1 x^ of this point consider the motion of vertical descent. For every by the mo- - r-1 111-1 1 ■ 1- t^°" °^ '^^^' innnitely small displacement s the equation subsists ticai de- [mg — m(d7i/di')'\s =^0, in which the letters retain their conventional significance. Consequently, this equation obtains mg — m -j-\s . di = 0, to which, as the result of the relations limits') dv , ds , a = m -i- s -\- mv ^- ana dt dt at - 1 384 THE SCIENCE OF MECHANICS. t. / d (jii V j) t^ dt = {mvs) = 0, dt provided s vanishes at both hmits, passes into the form jLgs+mv^\dt=Q, to ^ ' that is, into the form of Hamilton's principle. Thus, through all the apparent differences of the mechanical principles a common fundamental same- ness 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. SOME APPLICATIONS OF THE PRINCIPLES OF MECHANICS TO HYDROSTATIC AND HYDRODYNAMIC QUESTIONS. Method of I. We will now supplement the examples which fhe™ac«o'n^ we have given of the application of the principles on liquid of mcchauics, as they applied to rigid bodies, by a masses. 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 Pla- teau.* It is effected by immersing olive oil in a mix- ture 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 counterbal- anced, and the liquid really acts as if it were devoid of weight. * Statique experimentaU et thiorique des liquides, 1873! THE EXTENSION OF THE PRINCIPLES. 385 2. First, let us imagine a weightless liquid mass The work of free in space. Its molecular forces, we know, act only forces de- at very small distances. Taking as our radius the dis- a change in tance at which the molecular forces cease to exert a superldaf measurable influence, let us describe about a particle a, b, c in the interior of the mass a sphere — the so- called sphere of action. This sphere of action is regu- larly and uniformly filled with other particles. The resultant force on the central particles a, b, c is there- fore 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 Fig. igg. 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 posi- tion 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 / o 386 THE SCIENCE OF MECHANICS. 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 through- out 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 perform- ance 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 super- . ... , . ficiai area transportation of particles into the interior effects a due to posi- J... tive work, diminution of the surface-area is positive or negative, that is, whether work is performed or work is ex- pended. If we put two fluid drops in contact, they will coalesce of their own accord; and as by this action the area of the surface is diminished, it follows that the work that pro- duces a diminution of superfi- Fig. 201. ' '^i^-l 3.rea in a liquid mass is posi- tive. 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 punc- tured, the film outside the loop will contract till the thread bounds a circle in the middle of the liquid sur- face. 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 of liquid liquid, the forces acting on which are molecular forces, equilibrium . . ' Will 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 PRINCIPLES 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 va- riation ^ 0. For a given volume a minimum of super- ficial area gives stable equilibrium ; a maximum un- stable 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 equi- librium, 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 de- pendent on various collateral conditions, which render the problem more complicated. 3. The connection between the size and i}D.&form of Mode of de- termining the liquid surface may be investigated as follows. We the connec- .... tion of the imagine the closed outer sur- size and r 1 1 - * 1 • ^<^^^^ ^'^' ' '■'/'7^ '^'^=asy>^ form of a face of the liquid to receive /:?<^ '''' % ^^^^^rf» liquid sur- A~Jy' ^-' 'M ' ' '\\ face. without alteration of the li- quid'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 ele- ments. 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 clO 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 MECHANICS. The mathe- sides dp' , dq of the element dO' , then, these relations matical de- velopment obtain of this , c* N method. / , . 6 fl dp' = dp\ 1 + - dq ^dq\\^ - where r and / are the radii of curvature of the princi- pal sections touching the elements of the lines of cur- vature/, q, or the so-called principal radii of curva- ture.* 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, S.dO = dO' — dO = dpdq[l-\- dn 1 + S n - dp dq. Neglecting the higher powers of Snwe get S.dO = (^- + -,]sn.dO. The variation of the whole surface, then, is expressed by do ^Si}^^ Sn.dO (1) Furthermore, the normal displacements must be so chosen that Fig. 203. fd//.dO = (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 neigh- boring 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. THE 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 = only if i /r -(- i A has the same value the generai- . , -,i , iM ityoftheex- lor all points of the surface. This will be readily seen pressions 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 dn he. erected as ordinates thereon in the plane E, the outward displacements up- wards as positive and the inward displacements down- wards as negative. Join the extremities E' of these ordinates so as to form a curve, and take the quadra- ture of the curve, reckoning the sur- face above AX as positive and that below it as nega- tive. For all systems oi Sn for which this quadra- ture = 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/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 // 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 Sn that the expression (i) shall be different from zero. Only on the condition that i A + i/r' has the same value for all the elements, is expres- Fig. 204. 390 THE SCIENCE OF MECHANICS. sion (i) necessarily and universally equated to zero with expression (2). The sum Accordingly, from the two conditions (i) and (2) it "qShrium follows that 1/r 4- l/r'= const ; that is to say, the sum cmstantforof the reciprocal values of the principal radii of curva- Imilce!^ ture, or of the radii of curvature of the principal nor- mal 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 superfLcial form 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 gen- 1 1 - 1 r 1 , erai condi- we have Seen, the spherical form, and presents an ab- interrupted solutc minimum of superficial area. The equation ses. V'' "l~ 1/r' = const is here visibly fulfilled in the form 2/J? = 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 mid- dle 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 Xjr -\- 1/r' for the surface of revolu- tion. When r^ r' r^ry, R' the surface of revolution becomes a cylindrical surface. For 1/r -|- l/r':= 0, where one normal section is con- ♦ Principia Generalia Tkeorice Fierce Fluidoruvi in Statu JEquilihrii^ GOttingen, 1830; Werke, Vol. V, 29, GOttingen, 1867. THE EXTENSION OF THE PRINCIPLES. 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 men- tioned. Now let us picture to ourselves a liquid mass Liquid mas- ses whose bounded by surface-parts for which the expression surfaces are 7 . Ill partly con- 1/r -|- 1/r' has a positive value, and by other parts cave and for which the same expression has a negative value, vex or, more briefly expressed, by convex and concave sur- faces. 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 performed when concave surfaces move outwards and convex surfaces inwards. Work also is performed when a superficial portion moves outwards for which 1/r -(- 1/r' = -)- a, while simulta- neously an equal superficial portion for which 1/;- -f- \jr' > a moves inwards. Hence, when differently curved surfaces bound a liquid mass, the convex parts are forced inwards and the concave outwards till the condition l/r -{- 1/r' = const is fulfilled for the entire surface. Similarly, when a connected liquid mass has several isolated surface- parts, bounded by rigid bodies, the value of the ex- pression 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 Experi- ^ ' ^ mental 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 MECHANICS. ingly fulfil the condition XjR + l/oo = 1/p + l/p, or p = %R, 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 ing ahoi- encloses a hollow space. The condition that Xjr -(- 1/r' ^ '^' 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 sur- face, 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 occu- pied by a fluid or gaseous substance subjected to a de- terminate pressure, the work done in the last-men- tioned process can be counteracted by the work ex- pended to produce the compression, and thus equilib- rium may be produced. Theme- Let US picture to ourselves a liquid mass confined properties between two similar and similarly situated surfaces of bubbles, A I T f 7 ■ 1 very near each other. A bubble is such a system. Its primary condition of equi- librium is the exertion of an excess of pressure by the inclosed gaseous con- tents. If the sum 1/r + 1/;-' has the value -\- a for the exterior surface, it will Fig. 205. have for the m tenor surface very nearly the value — a. A bubble, left wholly to itself, will al- ways 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 ibrndr. If, therefore, in the diminution of the surface by unit of area the work A is performed, then A . ibrndr will THE EXTENSION OF THE PRINCIPLES. 393 be the total amount of work to be compensated for by the work of compression /.4r 2 ;r(/;' expended by the pressure / on the inclosed contents. From this follows ifAjr =p ; from which A may be easily calcu- lated if the measure of r is obtained and / is found by means of a manometer introduced in the bubble. An open spherical bubble cannot subsist. If an Open open bubble is to become a figure of equilibrium, the sum 1/r + 1/;' 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 sur- faces, then, Xjr -\- 1/r' = 0. Consequently, r = — ;' for all points. Such a surface is called a minimal sur- face ; that is, it has the smallest area consistent with its containing certain closed contours. It is also a sur- face of zero-sum of principal curvatures ; and its ele- ments, as we readily see, are saddle-shaped. Surfaces of this kind are obtained by constructing closed space- curves 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 films, possess a peculiar property. The work of the uresofequi- forces of gravity affects the entire mass of a liquid ; that of the molecular forces is restricted to its super- ficial film. Generally, the work of the forces oi grav- ity 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 MECHANICS. one another at the edges of the framework. When thin plane films are so joined that they meet at a hol- low 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 Hquid 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 physical reasons, which are, as it appears, not yet perfectly known, a state of equilib- rium is effected. Yet, notwithstanding the fact that the fundamental equation l/r-)-l/r' ^= const is not fulfilled in these fig- ures, 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 minimum 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 1 20°, and by fours in corners at approxi- mately equal angles. And it is geometrically demon- strable that these relations correspond to a minimum of superficial area. In the great diversity of phenom- ena 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 ob- equiUbrfum tained by dipping wire polyhedrons in solutions of metrical, soap, form systems of liquid films presenting a re- markable symmetry. 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 sup- plied by symmetry. Regularity is successive symme- try. 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 . - . . , , 1 ^ , ^ of the earth m connection with a special problem — that of the figure 1 -2 -T Fig. 2o6. 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 fila- ments 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 [Theorie 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 MECHANICS. Clairaut's point of view. Conditions of equilib- rium of Clairaut's canals. Clairaut's starting-point is this. If the fluid earth is in equilibrium, we may, without disturbing its equi- librium, imagine any portion of it solidified. Accord- ingly, let all of it be solidified but a canal AB, of any form. The liquid in this canal must also be in equilib- rium. But now the conditions which control equilib- rium are more easily investigated. If equilibrium exists in every imaginable canal of this kind, then the entire mass will be in equilibrium. Incidentally Clairaut re- marks, 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). But the kernel of the problem, according to Clai- raut, lies in a different view. In all imaginable canals, Z M N Fig- «>7. Fig. 308. 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 po- sition of its terminal points. Imagine in the fluid in question a canal MN ol any form (Fig. 208) referred to a system of rectangular co- THE EXTENSION OF THE PRINCIPLES. 397 ordinates. Let the fluid have the constant density p Mathemat- 1 1 1 r ^' -^^ rr • ■ ^ Jcal expres- and let the lorce-components A, 1, Z acting on unit oisionot these con- mass of the fluid in the coordinate directions, be f unc- ditions, and f 1 J • r 1 ■ T 1 *^® conse- tions of the coordinates .r, y, z of this mass. I^et the quent gen- element of length of the canal be called ds, and let its tion of 1 IT J rr^i c liquid equi- projections on the axes be ax, ay, dz. The force-corn- librium. ponents acting on unit of mass in the direction of the canal are then X{dx jds), Yijiy/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 This force must be balanced by the increment of pres- sure 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 pres- sure (/) between the two extremities M and N is found by integrating this expression from J/ to N. But as this difference is not dependent on the form of the canal but solely on the position of the extremities M and N, it follows that p{Xdx-\- Ydy-\- Zdz), or, the density being constant, Xdx -\- Ydy -\- Zdz, must be a com- plete differential. For this it is necessary that ,- dU ,,_dU ^_dU dx dy dz where t/' is a function of coordinates. Hence, according to Clairaut, the genei-al conditioti of liquid equilibrium is, that the liquid be controlled by forces which can be ex- pressed as the partial differential coefficients of one and the same function of coordinates. 9. The Newtonian forces of gravity, and in fact all central forces, — forces that masses exert in the direc- tions of their lines of junction and which are functions 398 THE SCIENCE OF MECHANICS. Character of the distances between these masses, ^ — possess this otlhe . . forces property. Under the action of forces of this character requisite to produce the equilibrium of fluids is possible. If we know U. equilibrium i i /- - i we may replace the first equation by IdU , , dU , . dU , ■ or dp = pdU and p = pU-j- const. The totality of all the points for which U ^ const is a surface, a so-called level surface. For this surface also / = 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. ciairaut's In the theory of Clairaut, here presented, is con- germ of the tained, beyond all doubt, the idea that underlies the doctrine of potential, doctrme of force-function or potential, which was after- wards developed with such splendid results by La- place, 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 recog- nised 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 the forces on unit of mass of the fluid in the displacement ds, whose projections are dx, dy, dz. Consequently, if we trans- port unit mass from a point for which Ur=z C to an- THE EXTENSION OF THE PRINCIPLES. 399 other point, indifferently chosen, for which U^ C^, character- " istics of the or, more generally, from the surface U=C.^ to the force-func- surface £/"= 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 A— /i=P(<^2-Ci), where the quantities designated by the same indices belong to the same surface. 10. Let us picture to ourselves a group of such character- very closely adjacent surfaces, of which every two sue- level, or cessive ones differ from each other by the same, very tiai, sur- small, amount of work required to transfer a mass from one to the other ; in other words, imagine the surfaces U= C, U= C+ dC, U= C-\-idC, and so forth. A mass moving on a level surface evidently per- forms no work. Hence, every component force in a direction tangential to the surface is = ; and the di- rection of the resultant forceis everywhere normal to the surface. If we call dn the element of the normal intercepted between two consecutive surfaces, andy the force requisite to con- vey unit mass from the one surface to the other through this element, the work done is/. dn = dC. KsdC'is by hypothesis every- where constant, the force /= dC/dn is inversely pro- portional to the distance between the surfaces consid- 400 THE SCIENCE OF MECHANICS. ered. If, therefore, the surfaces U are known, the directions of the forces are given by the elements of a system of curves everywhere at right angles to these surfaces, and the inverse distances between the sur- faces measure the magnitude of the forces. * These sur- faces and curves also confront us in the other depart- ments of physics. We meet them as equipotential surfaces and lines of force in electrostatics and mag- netism, 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 Clai- raut'sdoc- Clairaut's doctrine by another, very simple example. trine by a ^ . ,/ ' ^ simple Imagine two mutually perpendicular planes to cut the example. . , . , . , paper at right angles m the straight lines CX and OY (Fig. 210). We assume that a force-function exists 6^"= — xy, where x andjc are the distances from the two planes. The force-components parallel to OX and OKare then respectively X = ^=- dx and 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. But 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 consecutive 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 equi- lateral hyperbolas. The lines of force are obtained by turning the first mentioned system of curves through an angle of 45° in the plane of the paper about O. If a unit of mass pass from the point rtoO ^ by the route rpO, or rgO, or by any other route, the work done is always Op Y^ Oq. If we imagine a closed canal OprqO filled with a liquid, the liquid in the ca- nal will be in equi- librium. If transverse sections be made at any two points, each Fig. 210. section will sustain at both its surfaces the same pressure. We will now modify the example slightly. Let the a modifica- - tion of this forces be -X':= — y, Y= — a, where a has a constant example, value. There exists now no function (7 so constituted that Jf = dUjiix and F= dU/dy ; for in such a case it would be necessary that dX/dy = dV/dx, which is ob- viously not true. There is therefore no force-function, and consequently no level surfaces. If unit of mass be transported from r to C by the way of /, the work done is a y, Oq. If the transportation be effected by the route rqO, the work done is a X O q -\- Op X Oq. 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 MECHANICS. rotate constantly in the direction OprqO. Currents of this character, which revert into themselves but con- tinue their motion indefinitely, strike us as something quite foreign to our experience. Our attention, how- ever, 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 coordinates. Whenever exceptions to this principle are observed, we are disposed to regard them as appa- rent, and seek to clear up the difficulties involved. Torriceiii's 12. We shall now examine a few problems of liquid researches .,, riii • on the veio- motion. The founder of the theory of hydrodynamics is quid efflux. ToRRiCELLi. Torricelli,* by observations on liquids dis- charged through orifices in the bottom of vessels, dis- covered 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 «"", interval be taken as the unit, there will be dis- charged in the {n — l)"" , the {n — 2)* , the {n — 3)'^ . . . . 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 sug- gested. 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 dis- charged through an orifice in the bottom of a vessel is connected with the height h of the surface of the liquid by the equation v ^= V'2g/i ; that is to say, the velocity * De Motu Graviutn 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 sur- face. * Torricelli's theorem consorts excellently with the Varignons deduction rest of our knowledge of natural processes; but weottheveio- city 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^^mv gives, if by a we designate the area of the basal orifice, by h 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 T a small interval of time, this result ahs . r = . w or z»2 := gh. S Here ahs represents the pressure acting during the time T on the liquid mass avrs/g. Remembering that » is a final velocity, we get, more exactly, V a-5- . rs ahs . r = 2 .v. g and thence the correct formula z;2 =2gh. 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 ren- dering the idea more modern. The equation which we employ is ps = mv'^ fi. In a vessel of transverse sec- tion q (Fig. 211), into which a liquid of the specific * The early inquirers deduce their propositions in the incomplete form of proportions, and therefore usually put » proportional to ^gh or ^ h. 404 THE SCIENCE OF MECHANICS. Daniel Ber- noulli's treatment of the same problem. gravity j is poured till the head h is reached, the surface sinks, say, the small distance dh, and the liquid mass q. dh. 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 is discharged directly through the basal orifice, or passes, say, to a position a, while the liquid at a is displaced to b, that at b displaced to 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 v"^ S>' dh Fig. 211. q . dh . s . h ■■ g or V2.i The law of liquid efBux when pro- duced by the pres- sure of pistons. 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. Let us neglect the gravity of the liquid in the ves- sel, and imagine it loaded by a movable piston, on whose surface-unit the pressure p falls. If the piston 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 q.p.dh^q.dh.p -—-, or v = J1. \ 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 sup- posed that this theorem is directly applicable to gases. Its form, indeed, is correct ; but the deduction fre- quently employed involves an error, which we shall now expose. 14. Two vessels (Fig. 212) of equal cross-sections are placed side by side and connected with each other by a small aperture in the base of their dividing walls. For the velocity of flow through this aperture we ob- tain, under the same suppositions as before, d h s v"^ q.dh.s {h^ — ^2) = ^_— _, or V = V2,g{h^—h^~). If we neglect the gravity of the liquid and imagine the pressures p^ and p^ produced by pistons, we shall similarly have v = l^2{p^ — p2)lP- For example, if the pistons employed be loaded with the weights P and Pji, the weight P will sink the distance h and Pfri will rise the distance h. The work {P/'2)h is thus left, to generate the vis viva of the effluent fluid. A gas under such circumstances would behave dif- ferently. Supposing the gas to flow from the vessel containing the load Pinto that contain- ing the load P/2, the first weight will fall a distance h, the second, however, since under half the pressure a gas dou- bles its volume, will rise a distance 2.h, so that the work Ph — (P/2) 2/^ = would be performed. In the case of 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- The appli- cation of this last re- sult to the flow of gases. fi, S=^ dJ, The behav- iour of a gas under the as- sumed con- ditions. Fig. 212, 4o6 THE SCIENCE OF MECHANICS. The result sion & pressure. The expansive force / and the volume ^^rmZV^'w of a gas stand to each other in the familiar relation toflnftude^./w^^, where A, so long as the temperature of the gas remains unchanged, is a constant. Supposing the volume of the gas to expand under the pressure / by an amount dw, the work done is f^dw = kf^^. For an expansion from w^ to w, or for an increase of pressure from /^ to /, we get for the work Conceiving by this work a volume of gas w^ of density p, moved with the velocity v, we obtain /; -V 2/o log 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 pletenessof , r ,i i r i this view, changes of the volumes of gases are always accom- panied 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 /leat [Nor can even a thermo- dynamical 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 THE 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, comjiressed rn rr • 1 • • • -I • 1 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. Imag- ine, in Fig. 213, a gas and a mobile liquid of the same volume, measured by OA, subjected to the same pres- sure, a pressure of one atmosphere, designated b}' 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 AK=^o-ooooie^OA. If the pressure decrease till it become zero, the total work of the liquid is represented by the surface ABI, where AI ^o-oooo^O A, and the total work of the gas by the surface contained between AB, the infinite straight line ACEG . . . ., and the infinite hyperbola branch BDFH . . . . Ordinarily, therefore, the work of expansion of liquids may be neglected. There are however phenomena, for ex- ample, the soniferous vibrations of liquids, in which work of this very order plays a pirincipal 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 4oS THE SCIENCE OF MECHANICS. The hydro- dynamic principle of Daniel Bernoulli. approximation to the truth as a mere matter of molar mechanics. i6. We now come to the idea which Daniel Ber- noulli sought to apply in his work Hydrodynamica, sive de Viribus et Motibus Fluidorum Commentarii (1738). When a liquid sinks, the space through which its cen- tre 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 ac- quired in the fall can ascend {ascensus potentialis). 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 of the basal orifice, be called /(x). Let the liquid move and its surface descend a distance dx. The centre of gravity, then, descends the distance xf{x') . dx/M, where M =: f/(x) dx. If k is the space of potential ascent of the liquid in a cross- section equal to unity, the space of po- tential ascent in the cross-section _/"(.«) will be i/f^x)^, and the space of potential ascent of the centre of gravity will be Fig. 214. ^^ dx fix) M M' where JV -X dx 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 EXTENSION OF THE PRINCIPLES. 409 This equation was employed by Bernoulli in the solu- The parai- r • 11 T -11 1 -1 1 lelism 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 assump- tion 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 in- verse 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 The water- pendulum treated by Newton and John Bernoulli. We shall of Newton. consider here one to which a familiar law is directly applic- able. A cylindrical U-tube with vertical branches is filled with a liquid (Fig. 215). The length 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 the other branch will rise the distance x, and the difference of level corresponding to the excursion x will be ^x. \i 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 lasx, which, since it must move a mass a /j/^ will determine the acceleration (2 asx)/{als/g) — {2g/l) x, or, for unit Fig. 215. 4IO THE SCIENCE OF MECHANICS. excursion, the acceleration igjl. We perceive that pendulum vibrations of the duration "^27 Bernoulli. /2^ will take place. The liquid column, accordingly, vi- brates the same as a simple pendulum of half the length of the column. The liquid A similar, but somewhat more general, problem was of John treated by Tohn Bernoulli. The two branches of a cylindrical tube (Fig. 216), curved in any manner, make with the horizon, at the points at which the surfaces of the liquid move, the angles a and /?. Displacing one of the surfaces the dis- tance X, the other sur- face suffers an equal displacement. A difference of level is thus produced X (sin a + sinyS), and we obtain, by a course of reason- ing similar to that of the preceding case, employing the same symbols, the formula Fig. 216. -aI, / »^(sin a -(- sin/3) ' 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 n s tion is only apparent. One example is 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 £, and this pressure forces the water in 5 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- parent exception to the principle in question is that of Montgol- fier's hydraulic ram, in which the liquid by its own gravitational work appears to rise considerably above its original level. The liquid flows (Fig. 217) from a cistern A 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 m affected with the velocity v is suddenly arrested in RR, which must Hero's fountain. Fig. 217. 412 THE SCIENCE OF MECHANICS. be deprived of its momentum. If this be done in the time /, the liquid can exert during this time a pressure g = mv/t, to which must be added its hydrostatical pressure /. The liquid, therefore, will be able, during this interval of time, to penetrate with a pressure/ + g 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 56' than that corresponding to its simple pressure /. It is to be observed here, that a considerable portion of the liquid must first flow off into B, before a velocity requi- site to close V\& 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 iiiustra- 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 ^ i . t otthehy-^ Smaller one, which \0 thus acquires a great vis viva, may be illus- Z. [ trated in the following ■-* \ very simple manner. \ Close the narrow V" opening O of a funnel \ and plunge it, with its Fig.j,8. ■^^'^^ opening down- wards, 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 draulic ram 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 fun- nel. 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 ^. 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 vio- lently separated from the parts below and thrown out through the neck of the funnel high above the original surfa(5e. The remainder, however, are left considerably below that point, and the centre of grav- ity of the whole never as much as reaches the original level of S. 19. One of the most important achievements of Hydrostatic •1 ii--i*ji--- f7 1 and hydro- Darnel Bernoulli is his distinction of hydrostatic and dynamic pressure. hydrodynamic pressure. The pressure which liquids exert is altered by motion ; and the pressure of a liquid in motion may, according to the circumstances, be 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 ^, which has the form of a body of revolution with vertical axis, is kept 's-^'s- constantly filled with a frictionless liquid, so that its surface a.t mn does not change during the discharge at kl. We will reckon the vertical distance of a particle 414 THE SCIENCE OF MECHANICS. Determina- from the surface m n downwards as positive and call tion of the . ^ r 1 1 i r • • i r pressures it z. Let US follow the course ot a prismatic element of acting in li- volume, whose horizontal base-area is a and height /S, quids in . , , . . , motion. in its downward motion, neglecting, on the assump- tion 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 dis- tance dz, we have by the principle of vis viva (v'^\ dp a§ pd\-^\^ afi pgdz — a~-(idz (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 \s a\_p-\- {dp/dz')P'\. The element sustains, therefore, if the pressure in- crease downwards, an upward pressure a {dp/dz)fi ; and for any displacement dz of the element, the work a{dpldz')^dz must be deducted. Reduced, equation (i) assumes the form P-\^ = PSdz-'^-ldz and, integrated, gives P • Y = Pg^ —P + const (2) If we express the velocities in two different hori- zontal cross-sections a^ and a^ at the depths Zj and z^ below the surface, by v^, v^, and the corresponding pressures by /j, p^, we may write equation (2) in the form ^.{v\-vD^pg{z^-z^)^{p^-p^) . (3) THE EXTENSION OF THE PRINCIPLES. 415 Taking for our cross-section a^ the surface, Zj = 0, The Uydro- ^j =:= ; and as the same quantity of liquid flows through pressure ,, .. . ,, . ... varies with all cross-sections m the same mterval of time, a^v^ =thecircuin- ^2 v^. Whence, finally, the motion. The pressure p^ of the liquid in motion (the hydro- dynamic pressure) consists of the pressure pgz^ of the liquid at rest (the hydrostatic pressure) and of a pres- sure (p/2)w2[(ffl| — a2)/fl!|] 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 illustration of these re- principle may be obtained by imagining the liquid in suits by the the vessel A unacted on by gravity, and its outflow quids under , . pressures produced by a constant pressure p^ on the surface, produced Equation (3) then takes the form 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 ve- locity 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 of pressure. 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 4i6 THE SCIENCE OF MECHANICS. 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 direc- tion 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 ele- ments in the narrower section will not take place ; the elements will not escape fast enough ; they will accumu- late before the narrower section ; and at the entrance to it the requisite augmentation of pressure will be im- mediately 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 vis- cosity and friction are considered. 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 al- though 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 THE EXTEN'SION OF THE PRINCIPLES. 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 =^V'2gh.^,^Nhsx&h^:- ..... , position of generally are divisible into constant and variable quan- theprinci- . . . .... . pies of the titles : the latter being subdivided into independent Caicuius of VariaticDS. 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 con- nects them is termed the form of the function. Now, quite analogous to this division of quantities into con- stant and variable, is the division of the forms of func- tions into determinate (constant) and indeterminate (vari- able). If the form of a function, y = (pix'), is inde- terminate, or variable, the value of the function y can change in two ways : (i) by an increment dx of the • An Elementary Treatise on the Calculus of Variations, By the Rev, John Hewitt Jellett. Dublin, 1850. 438 THE SCIENCE OF MECHANICS. independent variable x, or (2) by a change oiform, by a passage from cp to cp^. The first change is the dif- ferential dy, the second, the variation Sy. Accord- ingly. dy= cp{x -\- dx) — cp {x), and 8y=^(p^{pc)—(p{pc'). The object The change of value of an indeterminate function cQius of va- due to a mere change of form involves no problem, lustrated. just as the change of value of an independent variable involves none. We may assume any change of form we please, and so produce any change of value we please. A problem is not presented till the change in value of a determinate function {F") of an indetermi- nate function q), due to a change of form of the included indeterminate function, is required. For example, if we have a plane curve . of the indeterminate ioxxa. y^ (p {pi), the length of its arc between the abscissae x^ and jCj is a determinate function of an indeterminate function. The moment a definite form of curve is fixed upon, the value of S can be given. For any change of form of the curve, the change in value of the length of the arc, SS, is determinable. In the example given, the func- tion 6' does not contain the function y directly, but through its first differential coefficient dy/dx, which is itself dependent on y. Let u = F{y') be a determinate function of an indeterminate function y =^

+ • which now contains only 6y under the integral sign. The terms in the first line of this expression are independent of any change in the form of the function and depend solely upon the variation of the limits. 442 THE SCIENCE OF MECHANICS. The inter- The terms of the two following lines depend on the fhl're'sSu?. change in the form of the function, for the limiting values of x only ; and the indices i and 2 state that the actual limiting values are to be put in the place of the general expressions. The terms of the last line, finally, depend on the general change in the form of the function. Collecting all the terms, except those in the last line, under one designation a^ — a^, and calling the expression in parentheses in the last line §, we have ^ a J — ^0 + fP-^y- ^^• But this equation can be satisfied only if «,-«o = (1) and J^fiSydx = (2) For if each of the members were not equal to zero, each would be determined by the other. But the in- tegral of an indeterminate function cannot be expressed in terms of its limiting values only. Assuming, there- fore, that the equation J^/3dydx = 0, The equa- holds generally good, it^ conditions can be satisfied, solves the since Sy is throughout arbitrary and its generality of problem, or , -iii ^ • y^ n t-^ makes the form Cannot be restricted, only by making p =0. By function in question a the equation maximum „ , „ , ^ or mini- ,^ dF^ , d^F^ d^P. ^ therefore, the form of the function y == , -^ . . riations for ical part of mechanics. But the history of the isoperi- mechanics, metrical problems and of the calculus of variations had to be touched upon, because these researches have ex- ercised a very considerable influence on the develop- ment of mechanics. Our sense of the general prop- erties of systems, and of properties of maxima and minima in particular, was much sharpened by these investigations, and properties of the kind referred to were subsequently discovered in mechanical systems with great facility. As a fact, physicists, since La- grange's time, usually express mechanical principles in a maximal or minimal form. This predilection would be unintelligible without a knowledge of the historical development. 446 THE SCIENCE OF MECHANICS. II. THEOLOGICAL. ANIMISTIC, AND MYSTICAL POINTS OF VIEW IN MECHANICS. I. If, in entering a parior in Germany, we happen to hear something said about some man being very pious, without having caught the name, we may fancy that Privy Counsellor X was spoken of, — or Herr von Y ; we should hardly think of a scientific man of our acquaintance. It would, however, be a mistake to sup- pose that the want of cordiality, occasionally rising to embittered controversy, which has existed in our day between the scientific and the theological faculties, always separated them. A glance at the history of science suffices to prove the contrary. The con- People talk of the ' ' conflict " of science and the- ence and ology. Or better of science and the church. It is in ' truth a prolific theme. On the one hand, we have the long catalogue of the sins of the church against pro- gress, on the other side a "noble army of martyrs," among them no less distinguished figures than Galileo and Giordano Bruno. It was only by good luck that Descartes, pious as he was, escaped the same fate. These things are the commonplaces of history ; but it would be a great mistake to suppose that the phrase "warfare of science" is a correct description of its general historic attitude toward religion, that the only repression of intellectual development has come from priests, and that if their hands had been held off, grow- ing science would have shot up with stupendous velo- city. No doubt, external opposition did have to be fought ; and the battle with it was no child's play. FORMAL DEVELOPMENT. 447 Nor was any engine too base for the church to handle The strue- ds of SClBIl' in this struggle. She considered nothing but how to tists with J ... , , their own conquer ; and no temporal policy ever was conducted precon- so selfishly, so unscrupulously, or so cruelly. But in- ideas, vestigators have had another struggle on their hands, and by no means an easy one, the struggle with their own preconceived ideas, and especially with the notion that philosophy and science must be founded on the- ology. It was but slowly that this prejudice little by little was erased. 2. But let the facts speak for themselves, while we Historical J , , , - . . , examples. mtroduce the reader to a few historical personages. Napier, the inventor of logarithms, an austere Puri- tan, who lived in the sixteenth century, was, in addi- tion to his scientific avocations, a zealous theologian. Napier applied himself to some extremely curious speculations. He wrote an exegetical commentary on the Book of Revelation, with propositions and mathe- matical demonstrations. Proposition XXVI, for ex- ample, maintains that the pope is the Antichrist ; propo- sition XXXVI declares that the locusts are the Turks and Mohammedans ; and so forth. Blaise Pascal (i 623-1 662), one of the most rounded geniuses to be found among mathematicians and phys- icists, was extremely orthodox and ascetical. So deep were the convictions of his heart, that despite the gen- tleness of his character, he once openly denounced at Rouen an instructor in philosophy as a heretic. The healing of his sister by contact with a relic most seri- ously impressed him, and he regarded her cure as a miracle. On these facts taken by themselves it might be wrong to lay great stress ; for his whole family were much inclined to religious fanaticism. But there are plenty of other instances of his religiosity. Such was 448 THE SCIENCE OF MECHANICS. Pascal. Otto von Guericke. his resolve, — which was carried out, too, — to abandon altogether the pursuits of science and to devote his life solely to the cause of Christianity. Consolation, he used to say, he could find nowhere but in the teachings of Christianity ; and all the wisdom of the world availed him not a whit. The sincerity of his desire for the conversion of heretics is shown in his Lettres provin- ciales, where he vigorously declaims against the dread- ful subtleties that the doctors of the Sorbonne had devised, expressly to persecute the Jansenists. Very remarkable is Pascal's correspondence with the theo- logians of his time ; and a modern reader is not a little surprised at finding this great "scientist" seriously discussing in one of his letters whether or not the Devil was able to work miracles. Otto von Guericke, the inventor of the air-pump, occupies himself, at the beginning of his book, now little over two hundred years old, with the miracle of Joshua, which he seeks to harmonise with the ideas of Copernicus. In like manner, we find his researches on the vacuum and the nature of the atmosphere in- troduced by disquisitions concerning the location of heaven, the location of hell, and so forth. Although Guericke really strives to answer these questions as ra- tionally as he can, still we notice that they give him considerable trouble, — questions, be it remembered, that to-day the theologians themselves would consider absurd. Yet Guericke was a man who lived after the Reformation ! The giant mind of Newton did not disdain to employ itself on the interpretation of the Apocalypse. On such subjects it was difficult for a sceptic to converse with him. When Halley once indulged in a jest concerning theological questions, he is said to have curtly repulsed FORMAL DEVELOPMENT. 449 him with the remark : "I have studied these things ; Newtonand Leibnitz. you have not ! " We need not tarry by Leibnitz, the inventor of the best of all possible worlds and of pre-established har- mony — inventions which Voltaire disposed of in Can- dide, a humorous novel with a deeply philosophical pur- pose. But everybody knows that Leibnitz was almost if not quite as much a theologian, as a man of science. Let us turn, however, to the last century. Euler, in Euier. his Letters to a German PrincesSy deals with theologico- philosophical problems in the midst of scientific ques- tions. He speaks of the difficulty involved in explaining the interaction of body and mind, due to the total diversity of these two phenomena, — a diversity to his mind undoubted. The system of occasionalism, devel- oped by Descartes and his followers, agreeably to which God executes for every purpose of the soul, (the soul it- self not being able to do so,) a corresponding movement of the body, does not quite satisfy him. He derides, also, and not without humor, the doctrine of pre- established harmony, according to which perfect agree- ment was established from the beginning between the movements of the body and the volitions of the soul, — although neither is in any way connected with the other, — just as there is harmony between two different but like-constructed clocks. He remarks, that in this view his own body is as foreign to him as that of a rhinoceros in the midst of Africa, which might just as well be in pre-established harmony with his soul as its own. Let us hear his own words. In his day, Latin was almost universally written. When a German scholar wished to be especially condescending, he wrote in French : "Si dans le cas d'un ddreglement "de mon corps Dieu ajustait celui d'un rhinoceros, 4SO THE SCIENCE OF MECHANICS. "en sorte que ses mouvements fussent tellement d'ac- " cord avec les ordres de mon ame, qu'il levat la patte " au moment que je voudrais lever la main, et ainsi " des autres operations, ce serait alors mon corps. Je "me trouverais subitement dans la forme d'un rhino- "ceros au milieu de I'Afrique, mais non obstant cela "mon ame continuerait les meme operations. J'aurais "^galement I'honneur d'^crire a V. A., mais je ne sais " pas comment elle recevrait mes lettires." Euier's One would almost imagine that Euler, here, had been theological . ,^^,. at, proclivities tempted to play Voltaire. And yet, apposite as was his criticism in this vital point, the mutual action of body and soul remained a miracle to him, still. But he extricates himself, however, from the question of the freedom of the will, very sophistically. To give some idea of the kind of questions which a scientist was per- mitted to treat in those days, it may be remarked that Euler institutes in his physical "Letters" investiga- tions concerning the nature of spirits, the connection between body and soul, the freedom of the will, the influence of that freedom on physical occurrences, prayer, physical and moral evils, the conversion of sin- ners, and such like topics ; — and this in a treatise full of clear physical ideas and not devoid of philosophical ones, where the well-known circle-diagrams of logic have their birth-place. Character 3. Let these examples of religious physicists suffice. logical We have selected them intentionally from among the leanings of ...„_. the great in- foremost of Scientific discoverers. The theological pro- clivities which these men followed, belong wholly to their innermost private life. They tell us openly things which they are not compelled to tell us, things about which they might have remained silent. What they utter are not opinions forced upon them from without ; FORMAL DEVELOPMENT. 451 they are their own sincere views. They were not con- scious of any theological constraint. In a court which harbored a Lamettrie and a Voltaire, Euler had no rea- son to conceal his real convictions. According to the modern notion, these men should characier 11 11 • 1 1- of their age. at least have seen that the questions they discussed did not belong under the heads where they put them, that they were not questions of science. Still, odd as this contradiction between inherited theological beliefs and independently created scientific convictions seems to us, it is no reason for a diminished admiration of those leaders of scientific thought. Nay, this very fact is a proof of their stupendous mental power : they were able, in spite of the contracted horizon of their age, to which even their own aper^us were chiefly limited, to point out the path to an elevation, where our genera- tion has attained a freer point of view. Every unbiassed mind must admit that the age in which the chief development of the science of mechan- ics took place, was an age of predominantly theological cast. Theological questions were excited by everything, and modified everything. No wonder, then, that me- chanics took the contagion. But the thoroughness with which theological thought thus permeated scientific inquiry, will best be seen by an examination of details. 4. The impulse imparted in antiquity to this direc- Galileo's ' ^ researclies tion of thought by Hero and Pappus has been alluded on the to in the preceding chapter. At the beginning of the materials. seventeenth century we find Galileo occupied with prob- lems concerning the strength of materials. He shows that hollow tubes offer a greater resistance to flexure than solid rods of the same length and the same quantity of material, and at once applies this discovery to the explanation of the forms of the bones of animals, which 452 THE SCIENCE OF MECHANICS. are usually hollow and cylindrical in shape. The phe- nomenon is easily illustrated by the comparison of a flatly folded and a rolled sheet of paper. A horizontal beam fastened at one extremity and loaded at the other may be remodelled so as to be thinner at the loaded end without any loss of stiffness and with a consider- able saving of material. Galileo determined the form of a beam of equal resistance at each cross-section. He also remarked that animals of similar geometrical con- struction but of considerable difference of size would comply in very unequal proportions with the laws of resistance. Evidences The forms of bones, feathers, stalks, and other or- of design . i i i • i ■ • in nature, ganic Structures, adapted, as they are, in their minut- est details to the purposes they serve, are highly cal- culated to make a profound impression on the thinking beholder, and this fact has again and again been ad- duced in proof of a supreme wisdom ruling in nature. Let us examine, for instance, the pinion-feather of a bird. The quill is a hollow tube diminishing in thick- ness as we go towards the end, that is, is a body of equal resistance. Each little blade of the vane re- peats in miniature the same construction. It would require considerable technical knowledge even to imi- tate a feather of this kind, let alone invent it. We should not forget, however, that investigation, and not mere admiration, is the office of science. We know how Darwin sought to solve these problems, by the theory of natural selection. That Darwin's solution is a complete one, may fairly be doubted ; Darwin him- self questioned it. All external conditions would be powerless if something were not present that admitted of variation. But there can be no question that his theory is the first serious attempt to replace mere ad- FORMAL DEVELOPMENT. 453 miration of the adaptations of organic nature by seri- ous inquiry into the mode of their origin. Pappus's ideas concerning the cells of honeycombs The cells of f ' IT • 1 *^® honey- were the subject of animated discussion as late as the comb. eighteenth century. In a treatise, published in 1865, entitled Homes Without Hands (p. 428), Wood substan- tially relates the following : " Maraldi had been struck with the great regularity of the cells of the honey- comb. He measured the angles of the lozenge-shaped plates, or rhombs, that form the terminal walls of the cells, and found them to be respectively 109° 28' and 70° 32'. Reaumur, convinced that these angles were in some way connected with the economy of the cells, requested the mathematician Konig to calculate the form of a hexagonal prism terminated by a pyramid composed of three equal and similar rhombs, which would give the greatest amount of space with a given amount of material. The answer was, that the angles should be 109° 26' and 70° 34'. The difference, accord- ingly, was two minutes. Maclaurin,* dissatisfied with thisagreement,repeatedMaraldi'smeasurements,found them correct, and discovered, in going over the calcu- lation, an error in the logarithmic table employed by Konig. Not the bees, but the mathematicians were wrong, and the bees had helped to detect the error ! " Any one who is acquainted with the methods of meas- uring crystals and has seen the cell of a honeycomb, with its rough and non-reflective surfaces, will question whether the measurement of such cells can be executed with a probable error of only two minutes, f So, we must take this story as a sort of pious mathematical * Philosophical Transactions for 1743.— TVaKJ. t But see G. F. Maraldi in the Memoires de Vacademie for 171Z. It is, how- ever, now well known the cells vary considerably. See Chauncey Wright, Philosophical Discussions, iSy;, p. ill.— Trans. 454 THE SCIENCE OF MECHANICS. fairy-tale, quite apart from the consideration that noth- ing would follow from it even were it true. Besides, from a mathematical point of view, the problem is too imperfectly formulated to enable us to decide the ex- tent to which the bees have solved it. Other The ideas of Hero and Fermat, referred to in the instances. . , , . ^ , . , previous chapter, concernmg the motion ot light, at once received from the hands of Leibnitz a theolog- ical coloring, and played, as has been before mentioned, a predominant role in the development of the calculus of variations. In Leibnitz's correspondence with John Bernoulli, theological questions are repeatedly dis- cussed in the very midst of mathematical disquisitions. Their language is not unfrequently couched in biblical pictures. Leibnitz, for example, says that the problem of the brachistochrone lured him as the apple had lured Eve. The theo- Maupertuis, the famous president of the Berlin nei of the Academy, and a friend of Frederick the Great, gave principle of . ,,,.., p , . , least ac- a new impulse to the theologising bent of physics by the enunciation of his principle of least action. In the treatise which formulated this obscure principle, and which betrayed in Maupertuis a woeful lack of mathe- matical accuracy, the author declared his principle to be the one which best accorded with the wisdom of the Creator. Maupertuis was an ingenious man, but not a man of strong, practical sense. This is evidenced by the schemes he was incessantly devising : his bold prop- ositions to found a city in which only Latin should be spoken, to dig a deep hole in the earth to find new substances, to institute psychological investigations by means of opium and by the dissection of monkeys, to explain the formation of the embryo by gravitation, and so forth. He was sharply satirised by Voltaire in the FORMAL DEVELOPMENT. 455 Histoire du docteur Akakia, a work which led, as we know, to the rupture between Frederick and Voltaire. Maupertuis's principle would in all probability soon Euier's re- have been forgotten, had Euler not taken up the sug- the theoioE- T-»i • -tiri ••!• ical basis of gestion. tuler magnanimously left the prmciple its this prin- name, Maupertuis the glory of the invention, and con- verted it into something new and really serviceable. What Maupertuis meant to convey is very difficult to ascertain. What Euler meant may be easily shown by simple examples. If a body is constrained to move on a rigid surface, for instance, on the surface of the earth, it will describe when an impulse is imparted to it, the shortest path between its initial and terminal positions. Any other path that might be prescribed it, would be longer or would require a greater time. This principle finds an application in the theory of atmospheric and oceanic currents. The theological point of view, Euler retained. He claims it is possible to explain phenomena, not only from their physical causes, but also from their purposes. ' ' As the construction of the universe is the "most perfect possible, being the handiwork of an " all- wise Maker, nothing can be met with in the world "in which some maximal or minimal property is not "displayed. There is, consequently, no doubt but "that all the effects of the world can be derived by "the method of maxima and minima from their final "causes as well as from their efiScient ones."* 5. Similarly, the notions of the constancy of the quantity of matter, of the constancy of the quantity of * " Quum enim mundi universi f abrica sit perfectissima, atque a creators sapientissimo absoluta, nihil omnino in mundo contingit, in quo non maximi minimive ratio quaepiam eluceat; quam ob rem dubium prorsus est nullum, quin omnes mundi eflfectus ex causis finalibus, ope methodi maximorum et minimorum, aeque feliciter determinari quaeant, atque ex ipsis causis efl&cien- tibus." {Methodus invenzendi lineas curvas maximi minimive proprietaU gaudentes, Lausanne, 1744.) 456 THE SCIENCE OF MECHANICS. The central motion, of the indestructibility of work or energy, con- m°od°ern° ceptioHS which completely dominate modern physics, mata'iy%f all arose under the influence of theological ideas. The orfgin.^"^^ notions in question had their origin in an utterance of Descartes, before mentioned, in the Principles of Philos- ophy, agreeably to which the quantity of matter and mo- tion originally created' in the world, — such being the only course compatible with the constancy of the Crea- tor, — is always preserved unchanged. The conception of the manner in which this quantity of motion should be calculated was very considerably modified in the progress of the idea from Descartes to Leibnitz, and to their successors, and as the outcome of these modifi- cations the doctrine gradually and slowly arose which is now called the "law of the conservation of energy." But the theological background of these ideas only slowly vanished. In fact, at the present day, we still meet with scientists who indulge in self-created mys- ticisms concerning this law. Gradual During the entire sixteenth and seventeenth centu- transition . , , . , , , ., from the ries, down to the close of the eighteenth, the prevail- theological ..... . . . _...,,., point of ing inclination of inquirers was, to find m all physical laws some particular disposition of the Creator. But a gradual transformation of these views must strike the attentive observer. Whereas with Descartes and Leibnitz physics and theology were still greatly inter- mingled, in the subsequent period a distinct endeavor is noticeable, not indeed wholly to discard theology, yet to separate it from purely physical questions. Theo- logical disquisitions were put at the beginning or rele- gated to the end of physical treatises. Theological speculations were restricted, as much as possible, to the question of creation, that, from this point onward, the way might be cleared for physics. view. FORMAL DEVELOPMENT. 457 Towards the close of the eighteenth century a re- Ultimate markable change took place, — a change which wasemancipa- apparently an abrupt departure from the current trend physics of thought, but in reality was the logical outcome of ogy. the development indicated. After an attempt in a youthful work to found mechanics on Euler' s principle of least action, Lagrange, in a subsequent treatment of the subject, declared his intention of utterly disre- garding theological and metaphysical speculations, as in their nature precarious and foreign to science. He erected a new mechanical system on entirely different foundations, and no one conversant with the subject will dispute its excellencies. All subsequent scientists of eminence accepted Lagrange's view, and the pres- ent attitude of physics to theology was thus substan- tially determined. 6. The idea that theology and physics are two dis- The mod- °-^ '^ ■' ern ideal tinct branches of knowledge, thus took, from its first always the ° attitude of germination in Copernicus till its final promulgation t^e greatest . . inquirers. by Lagrange, almost two centuries to attain clearness in the minds of investigators. At the same time it cannot be denied that this truth was always clear to the greatest minds, like Newton. Newton never, de- spite his profound religiosity, mingled theology with the questions of science. True, even he concludes his Optics, whilst on its last pages his clear and luminous intellect still shines, with an exclamation of humble contrition at the vanity of all earthly things. But his optical researches proper, in contrast to those of Leib- nitz, contain not a trace of theology. The same may be said of Galileo and Huygens. Their writings con- form almost absolutely to the point of view of La- grange, and may be accepted in this respect as class- ical. But the general views and tendencies of an age 458 THE SCIENCE OF MECHANICS. must not be judged by its greatest, but by its average, minds. The theo- To comprehend the process here portrayed, the gen- logical con- . ^.. .., . ^i ., ception of eral condition of affairs in these times must be consid- the world , , j. . ., . . natural and ered. It Stands to reason that in a stage ot civilisation able. in which rehgion is almost the sole education, and the only theory of the world, people would naturally look at things in a theological point of view, and that they would believe that this view was possessed of compe- tency in all fields of research. If we transport ourselves back to the time when people played the organ with their fists, when they had to have the multiplication table visibly before them to calculate, when they did so much with their hands that people now-a-days do with their heads, we shall not demand of such a time that it should critically put to the test its own views and the- ories. With the widening of the intellectual horizon through the great geographical, technical, and scien- tific discoveries and inventions of the fifteenth and six- teenth centuries, with the opening up of provinces in which it was impossible to make any progress with the old conception of things, simply because it had been formed prior to the knowledge of these provinces, this bias of the mind gradually and slowly vanished. The great freedom of thought which appears in isolated cases in the early middle ages, first in poets and then in scientists, will always be hard to understand. The en- lightenment of those days must have been the work of a few very extraordinary minds, and can have been bound to the views of the people at large by but very slender threads, more fitted to disturb those views than to re- form them. Rationalism does not seem to have gained a broad theatre of action till the literature of the eigh- teenth century. Humanistic, philosophical, historical. FORMAL DEVELOPMENT. 459 and physical science here met and gave each other mutual encouragement. All who have experienced, in part, in its literature, this wonderful emancipation of the human intellect, will feel during their whole lives a deep, elegiacal regret for the eighteenth century. 7. The old point of view, then, is abandoned. Its The en- history is now detectible only in the form of the me- ment o£ the IIGW VlGWSi chanical principles. And this form will remain strange to us as long as we neglect its origin. The theological conception of things gradually gave way to a more rigid conception ; and this was accompanied with a considerable gain in enlightenment, as we shall now briefly indicate. When we say light travels by the paths of shortest time, we grasp by such an expression many things. But we do not know as yet why light prefers paths of shortest time. We forego all further knowledge of the phenomenon, if we find the reason in the Creator's wis- dom. We of to-day know, that light travels by all paths, but that only on the paths of shortest time do the waves of light so intensify each other that a per- ceptible result is produced. Light, accordingly, only appears to«travel by the paths of shortest time. After Extrava- gance as the prejudice which prevailed on these questions had well as been removed, cases were immediately discovered in nature. which by the side of the supposed economy of nature the most striking extravagance was displayed. Cases of this kind have, for example, been pointed out by Jacobi in connection with Euler's principle of least ac- tion. A great many natural phenomena accordingly produce the impression of economy, simply because they visibly appear only when by accident an econom- ical accumulation of effects take place. This is the same idea in the province of inorganic nature that Dar- 460 THE SCIENCE OF MECHANICS. win worked out in the domain of organic nature. We facilitate instinctively our comprehension of nature by applying to it the economical ideas with which we are familiar. Expiana- Often the phenomena of nature exhibit maximal imai and or minimal properties because when these greatest or effects, least properties have been established the causes of all further alteration are removed. The catenary gives the lowest point of the centre of gravity for the simple reason that when that point has been reached all fur- ther descent of the system's parts is impossible. Li- quids exclusively subjected to the action of molecular forces exhibit a minimum of superficial area, because stable equilibrium can only subsist when the molecular forces are able to effect no further diminution of super- ficial area. The important thing, therefore, is not the maximum or minimum, but the removal of work ; work being the factor determinative of the alteration. It sounds much less imposing but is much more elucida- tory, much more correct and comprehensive, instead of speaking of the economical tendencies of nature, to say : "So much and so much only occurs as in virtue of the forces and circumstances involved can occur." Points of The question may now justly be asked, If the point the theoiog- of view of theology which led to the enunciation of the ical and ..,.,. . - scientific principles 01 mechanics was utterly wropg, how comes tions. it that the principles themselves are in all substantial points correct ? The answer is easy. In the first place, the theological view did not supply the contents of the principles, but simply determined \h&vs: guise; their mat- ter was derived from experience. A similar influence would have been exercised by any other dominant type of thought, by a commercial attitude, for instance, such as presumably had its effect on Stevinus's thinking. In FORMAL DEVELOPMENT. 461 the second j)lace, the theological conception of nature itself owes its origin to an endeavor to obtain a more comprehensive view of the world ; — the very same en- deavor that is at the bottom of physical science. Hence, even admitting that the physical philosophy of theology is a fruitless achievement, a reversion to a lower state of scientific culture, we still need not repudiate the sound root from which it has sprung and which is not differ- ent from that of true physical inquiry. In fact, science can accomplish nothing by the con- Necessity "■ 1 ■ r r r ■ - - °^^ con- sideration of indtvidual facts ; from time to time it must stant con- *^ 1 -1 sideration cast its glance at the world as a whole. Galileo's of the ah, --,..,,. ... in research laws 01 falling bodies, Huygens's principle of vis viva, the principle of virtual velocities, nay, even the con- cept of mass, could not, as we saw, be obtained, ex- cept by the alternate consideration of individual facts and of nature as a totality. We may, in our ijien- tal reconstruction of mechanical processes, start from the properties of isolated masses (from the elementary or differential laws), and so compose our pictures of the processes ; or, we may hold fast to the properties of the system as a whole (abide by the integral laws). Since, however, the properties of one mass always in- clude relations to other masses, (for instance, in ve- locity and acceleration a relation of time is involved, that is, a connection with the whole world,) it is mani- fest that purely differential, or elementary, laws do not exist. It would be illogical, accordingly, to exclude as less certain this necessary view of the All, or of the more general properties of nature, from our studies. The more general a new principle is and the wider its scope, the more perfect tests will, in view of the possi- bility of error, be demanded of it. The conception of a will and intelligence active in 462 THE SCIENCE OF MECHANICS. Pagan ideas nature is by no means the exclusive property of Chris- dces'rife in tian monotheism. On the contrary, this idea is a quite world. familiar one to paganism and fetishism. Paganism, however, finds this will and intelligence entirely in in- dividual phenomena, while monotheism seeks it in the All. Moreover, a pure monotheism does not exist. The Jewish monotheism of the Bible is by no means free from belief in demons, sorcerers, and witches ; and the Christian monotheism of mediaeval times is even richer in these pagan conceptions. We shall not speak of the brutal amusement in which church and state indulged in the torture and burning of witches, and which was undoubtedly provoked, in the mp.jority of cases, not by avarice but by the prevalence of the ideas mentioned. In his instructive work on Primitive Culture Tylor has studied the sorcery, superstitions, and miracle-belief of savage peoples, and compared them with the opinions current in mediaeval times con- cerning witchcraft. The similarity is indeed striking. The burning of witches, which was so frequent in Europe in the sixteenth and seventeenth centuries, is to-day vigorously conducted in Central Africa. Even now and in civilised countries and among cultivated people traces of these conditions, as Tylor shows, still exist in a multitude of usages, the sense of which, with our altered point of view, has been forever lost. 8. Physical science rid itself only very slowly of these conceptions. The celebrated work of Giambatista della Porta, Magia naturalis, which appeared in 1558, though it announces important physical discoveries, is yet filled with stuff about magic practices and demono- logical arts of all kinds little better than those of a red- skin medicine-man. Not till the appearance of Gil- bert's work, De magnete (in 1600), was any kind of re- FORMAL DEVELOPMENT. 463 striction placed on this tendency of thought. When we Animistic . notions in reflect that even Luther is said to have had personal science, encounters with the Devil, that Kepler, whose aunt had been burned as a witch and whose mother came near meeting the same fate, said that witchcraft could not be denied, and dreaded to express his real opinion of astrology, we can vividly picture to ourselves the thought of less enlightened minds of those ages. Modern physical science also shows traces of fetish- ism, as Tyler well remarks, in its "forces." And the hobgoblin practices of modern spiritualism are ample evidence that the conceptions of paganism have not been overcome even by the cultured society of to-day. It is natural that these ideas so obstinately assert themselves. Of the many impulses that rule man with demoniacal power, that nourish, preserve, and propagate him, without his knowledge or supervision, of these impulses of which the middle ages present such great pathological excesses, only the smallest part is accessible to scientific analysis and conceptual knowledge. The fundamental character of all these instincts is the feeling of our oneness and sameness with nature ; a feeling that at times can be silenced but never eradicated by absorbing intellectual occupa- tions, and which certainly has a sound basis, no matter to what religious absurdities it may have given rise. 9. The French encyclopaedists of the eighteenth century imagined they were not far from a final ex- planation of the world by physical and mechanical prin- ciples ; Laplace even conceived a mind competent to foretell the progress of nature for all eternity, if but the masses, their positions, and initial velocities were given. In the eighteenth century, this joyful overestimation of the scope of the new physico-mechanical ideas is par- 464 THE SCIENCE OF MECHANICS. Overesti- mation of the me- chanical view. Pretensions and atti- tude of physical science. donable. Indeed, it is a refreshing, noble, and ele- vating spectacle ; and we can deeply sympathise with this expression of intellectual joy, so unique in history. But now, after a century has elapsed, after our judg- ment has grown more sober, the world-conception of the encyclopaedists appears to us as a mechanical mythology in contrast to the animistic of the old religions. Both views contain undue and fantastical exaggerations of an incomplete perception. Careful physical research will lead, however, to an analysis of our sensations. We shall then discover that our hunger is not so essen- tially different from the tendency of sulphuric acid for zinc, and our will not so greatly different from the pressure of a stone, as now appears. We shall again feel ourselves nearer nature, without its being neces- sary that we should resolve ourselves into a nebulous and mystical mass of molecules, or make nature a haunt of hobgoblins. The direction in which this en- lightenment is to be looked for, as the result of long and painstaking research, can of course only be sur- mised. To anticipate the result, or even to attempt to introduce it into any scientific investigation of to-day, would be mythology, not science. Physical science does not pretend to be a complete view of the world ; it simply claims that it is working toward such a complete view in the future. The high- est philosophy of the scientific investigator is precisely this toleration of an incomplete conception of the world and the preference for it rather than an apparently per- fect, but inadequate conception. Our religious opin- ions are always our own private affair, as long as we do not obtrude them upon others and do not apply them to things which come under the jurisdiction of a differ- ent tribunal. Physical inquirers themselves entertain FORMAL DEVELOPMENT. 465 the most diverse opinions on this subject, according to the range of their intellects and their estimation of the consequences. Physical science makes no investigation at all into things that are absolutely inaccessible to exact investi- gation, or as yet inaccessible to it. But should prov- inces ever be thrown open to exact research which are now closed to it, no well-organised man, no one who cherishes honest intentions towards himself and others, will any longer then hesitate to countenance inquiry with a view to exchanging his opinion regarding such provinces for positive knowledge of them. When, to-day, we see society waver, see it change Results of the incom- its views on the same question according to its mood and pieteness oi our view of the events of the week, like the register of an organ, when the world. we behold the profound mental anguish which is thus produced, we should know that this is the natural and necessary outcome of the incompleteness and transi- tional character of our philosophy. A competent view of the world can never be got as a gift ; we must ac- quire it by hard work. And only by granting free sway to reason and experience in the provinces in which they alone are determinative, shall we, to the weal of man- kind, approach, slowly, gradually, but surely, to that ideal of a monistic view of the world which is alone compatible with the economy of a sound mind. III. ANALYTICAL MECHANICS. I. The mechanics of Newton are purely geometrical. The geo- metrical He deduces his theorems from his initial assumptions mechanics . of Newton. entirely by means of geometrical constructions. His procedure is frequently so artificial that, as Laplace 466 THE SCIENCE OF MECHANICS. remarked, it is unlikely that the propositions were dis- covered in that way. We notice, moreover, that the expositions of Newton are not as candid as those of Galileo and Huygens. Newton's is the so-called syn- thetic method of the ancient geometers. Analytic When we deduce results from given suppositions, . ^^ procedure is called synthetic. When we seek the conditions of a proposition or of the properties of a fig- ure, the procedure is analytic. The practice of the latter method became usual largely in consequence of the application of algebra to geometry. It has become customary, therefore, to call the algebraical method generally, the analytical. The term ' ' analytical me- chanics," which is contrasted with the synthetical, or geometrical, mechanics of Newton, is the exact equiva- lent of the phrase "algebraical mechanics." Euier and 2. The foundations of analytical mechanics were rin'scon- laid by EuLER {Mechanica, sive Motus Scientia Analytice Exposita, St. Petersburg, 1736). But while Euler's method, in its resolution of curvilinear forces into tan- gential and normal components, still bears a trace of the old geometrical modes, the procedure of Maclaurin {A Complete System of Fluxions, Edinburgh, 1742) marks a very important advance. This author resolves all forces in three fixed directions, and thus invests the computations of this subject with a high degree of symmetry and perspicuity. Lagrange's 3. Analytical mechanics, however, was brought to of the its highest degree of perfection by Lagrange. La- grange's aim is {Mecanique anatytique, Paris, 1788) to dispose once for all of the reasoning necessary to resolve mechanical problems, by embodying as much as pos- sible of it in a single formula. This he did. Every case that presents itself can now be dealt with by a very science. FORMAL DEVELOPMENT. 467 simple, highly symmetrical and perspicuous schema ; and whatever reasoning is left is performed by purely mechanical methods. The mechanics of Lagrange is a stupendous contribution to the economy of thought. In statics, Lagrange starts from the principle of statics 1 1 • ■ r~\ 1 r . 1 • founded on Virtual velocities. On a number of material points the princi- m^, m^, OTg. . . ., definitely connected with one another, tuai veioci- are impressed the forces F^, P^, F^. . . . If these points receive any infinitely small displacements /j^, p2> Pz- ■ ■ ■ compatible with the connections of the sys- tem, then for equilibrium 2Fp = ; where the well- known exception in which the equality passes into an inequality is left out of account. Now refer the whole system to a set of rectangular coordinates. Let the coordinates of the material points be jTj, y^, Zj, x^, jCg? Zg • • ■ ■ Resolve the forces into the components X^^, Y^, Z^, X^, Y^, Z^. . . . parallel to the axes of coordinates, and the displacements into the displacements Sx^, Sy^, Sz^, 6x^, Sy^, Sz^. . . ., also parallel to the axes. In the determination of the work done only the displacements of the point of appli- cation in the direction- of each force-component need be considered for that component, and the expression of the principle accordingly is 2(Xdx+ Y6y-{- ZSz) = . . . . (1) where the appropriate indices are to be inserted for the points, and the final expressions summed. The fundamental formula of dynamics is derived Dynamics on the prin- from D'Alembert's principle. On the material points cipie of m^, OTj, »«3 . . ., having the coordinates x^, y^, z^, x^, ^ert. jCg, Zg • • • • the force-components X^, Y^, Z^, X^^, Y^, Z3 . . . . act. But, owing to the connections of the 468 THE SCIENCE OF MECHANICS. system's parts, the masses undergo accelerations, which are those of the forces. d'^x^ d^y, d^z^ These are called the effective forces. But the impressed forces, that is, the forces which exist by virtue of the laws of physics, X, Y, Z. . . . and the negative of these effective forces are, owing to the connections of the system, in equilibrium. Applying, accordingly, the principle of virtual velocities, we get Discussion A. Thus, Lagrange conforms to tradition in making of La- grange's Statics precede dynamics. He was by no means com- method. . . 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 6g. If X^'Z^ Y^Z,Z = '''- dx dy' dz' FORMAL DEVELOPMENT. 469 that is to say, if X, V, Z are the partial differential co- efficients of one and the same function of the coordi- nates of position, the whole expression under the sign of summation is the total variation, S V, of V. If the latter is = 0, Fis in general a maximum or a minimum. ■;. We will now illustrate the use of equation (i~) by indication . , . . . ofthegen- a simple example. If all the points of application of the erai steps ^ . ^ ^ -^-^ . forthesolu- forces are independent of each other, no problem is 'ion of stat- ical prob- presented. Each point is then in equilibrium onlyiems. when the forces impressed on it, and consequently their components, are = 0. All the displacements Sx, dj, Sz. . . . are then wholly arbitrary, and equation (i) can subsist only provided the coefficients of all the displacements dx, dy, 6z. . . . are equal to zero. But if equations obtain between the coordinates of the several points, that is to say, if the points are sub- ject to mutual constraints, the equations so obtaining will be of the form -Fix^, y^, z^, x^, j^jj ^2- ■ ■ •) ^ ^' or, more briefly, of the form J^^O. Then equations also obtain between the displacements, of the form dF . , dF . , dF . , dF _^,-:>>) Their em- A^ -, A = -, ployment in X y the deter- mination of ajld from these by simple reduction the iinal -' ^ equation. V \ y v X^ A^ ^X ,g Y^Y^ y ^ ' that is to say, the resultant of the forces applied at M and JV acts in the direction OM. * The /our force-components are accordingly subject to only /wo conditions, (7) and (8). The problem, con- sequently, 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 coordinates, we treat equations (6) in exactly the same manner. Only, we can now make use, in addi- tion, of equations (4). Accordingly, we have, upon the elimination of A and ix, equations (7) and (8) and two equations (4). From these the following, which fully solve the problem, are readily deduced • * The mechanical interpretation of the indeterminate coefficients \, fi may be shown as follows. Equations (6) express the equilibrium of tv/o/ree points on which in addition to X, V, X^ , V-^ other forces act which answer to the re- maining expressions and just destroy .V, y',Xx, I'l. The point N", for example, is in equilibrium if X^ is destroyed by a force // [x^ — x), undetermined as yet in magnitude, and Vx by a force // {y^ — 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 that is to say, the force due to the connections acts in the direction of t. FORMAL DEVELOPMENT. 473 a(X+X,) b X^ Character of the pres- '^(y+J^i ) ^_ '^K, i/(x+ x,)2 + (y + y,)2 v'Jf ? + Y\ 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 par- ticular cases scarcely any additional thinking is re- quired. 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 1 • 1 • X If f r Steps for tion (2), which IS Lagrange s form 01 statement of the solution D'Alembert's principle. There is no problem when icai prob- the masses move quite independently of one another. Each mass yields to the forces applied to it ; the va- riations d X, Sy, d z . . . . are wholly arbitrary, and each coefficient may be singly put = 0. For the motion of n masses we thus obtain 3 n simul- taneous differential equations. But if equations of condition {F^ 0) obtain between the coordi- nates, these equations will lead to others {DJ^=0) between the dis- placements or variations. With the ^ . Fig. 233. latter we proceed exactly as in the application of equation (i). Only it must be noted here that the equations ^^0 must eventually be em- ployed in their undifferentiated as well as in their dif- ferentiated form, as will best be seen from the follow- ing example. 474 THE SCIENCE OF MECHANICS. A dynam- A heavv material point m, lying in a vertical plane ical exam- . , ,. . ,. , pie. XY, IS tree to move on a straight line, y ■:^ ax, inclmed at an angle to the horizon. (Fig. 233.) Here equa- tion (2) becomes and, since Jf =: 0, and F= — mg, also S^' + (^+g)^^=» « The place of i^^ is taken by y = ax (10) and for DF = we have Sy ^ aS X. Equation (9), accordingly, since Sy drops out and Sx is arbitrary, passes into the form d^x / , d^y\ By the differentiation of (10), or {_F^ 0), we have d'^y d'^x 'dt^~^ Hl^' and, consequently, liT^^-\s+-iiji) = ^ (11). Then, by the integration of (11), we obtain and — a2 /2 ■'' = r+^ -^ 2 + "^ ^ ^ + "^ "■' 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. FORMAL DEVELOPMENT. 475 Some care is necessary in the application of equa- a modifica- f -e r\ - 1 - AT^i 1 • ^^'^ of this tion (i) if ^=:: contains the time. The procedure m example, such cases may be illustrated by the following example. Imagine in the preceding case the straight line on which m descends to move vertically upwards with the acceleration y. We start again from equation (9) i^^ is here replaced by y^ax-{-y^ (12) To form DF^^ 0, we vary (12) only with respect to * and y, for we are concerned here only with the possible displacement of the system in its position at any given instant, and not with the displacement that actually takes place in time. We put, therefore, as in the pre- vious case, Sy =^ aSx, and obtain, as before, S?+(^+Sf>=« <>') But to get an equation in x alone, we have, since x andjc are connected in (13) by the actual motion, to differentiate (12) with respect to t and employ the re- sulting equation d^ y d^x , — — = a \- V dt^ df^ ^ ' for substitution in (i 3). In this way the equation d^x f ^ , d^x\ _ is obtained, which, integrated, gives 476 THE SCIENCE OF MECHANICS. Discussion of the mod' iiied exam- ple. ig^Y')%^bt^c y = Y— 1 + ^2 /2 -\- abt -\- ac. 1 + «2 If a weightless body m lie on the moving straight line, we obtain these equations /2 ■ a 1 + ^2 y^+bt + c -\- abt -\- ac, m -'— l_|_a2 2 — results which are readily understood, when we re- flect 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. 7. The procedure with equation (12) in the preced- ing example may be rendered somewhat clearer by the following consideration. Equation (2), D'Alembert's principle, asserts, that all the work that can be done in the displacement of a system is done by the irnpressed forces and not by the connections. This is evident, since the rigidity of the con- nections allows no changes in the rela- tive positions which would be neces- sary for any alteration in the potentials of the elastic forces. But this ceases to be true when the connec- tions undergo changes in time. In this case, the changes of the connections perform work, and we can then ap- ply 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 OF (Fig. 234.) Let this line be subject to Fig. 234. FORMAL DEVELOPMENT. 477 a forced acceleration in the direction of x, such that illustration of the mod- the equation y^= becomes ified exam- pie, =^^y% (14) D'Alembert's principle again gives equation (g). But since from DF= it follows here that Sx ^=^, this equation reduces itself to ^ + S>^ = ° ^''^ in which Sy is wholly arbitrary. Wherefore, and y = ^-+a^ + i. to which must be supplied (14) or It is patent that (15) does not assign the total work of the displacement that actually takes place, but only that of some possible 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 mechan- ism moved by a force m y, equation (2) will be re- placed by [■my — m ~^\ dx -f f — mg — m -j^j^y = 0, and since d x, Sy are wholly arbitrary here, we obtain the two equations y-^ = Q 478 THE SCIENCE OF MECHANICS. 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 subse- quently. Deduction 8. As the different mechanical principles only ex- cipie^o£"?j press different aspects of the same fact, any one of Lagrange's them is easily deducible from any other ; as we shall tafdynam- now illustrate by developing the principle of vis viva tion.^''"^ from equation (2) of page 468. Equation (2) refers to instantaneously possible displacements, that is, to "vir- tual " displacements. But when the connections of a system are independent of the time, the motions that actually take place ^xe. "virtual" displacements. Conse- quently the principle may be applied to actual motions. For Sx, 6y, 8z, we may, accordingly, write cix, dy, dz, the displacements which take place in time, and put 2 {Xdx + Ydy + Zdz) = Id^x , d^y , d-^z ' 2M[^_ax+-^^-^-dy + ^^-^dz The expression to the right may, by introducing for dx, (dx/dt) di and so forth, and by denoting the velo- city by V, also be written ^ (d'^x dx ,^ , d'^y dy , ^ , d'^z dz ,\ M-di^ dt'^+d it'^+dt^ Tt 'n= ^d'Sni dxV (dyy , {dz\i Tt) "'"VZ^j '^XTt] ld:2 2 mv' FORMAL DEVELOPMENT. 479 Also in the expression to the left, {dx/df) dt may be Force- written for dx. But this gives ^'2 {Xdx + Ydy + Zdz^ = 2 1 /« (»2 — »2), where z/^ 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, V, Z are the partial differ- ential coefficients of the same function Uoi coordinates, if, that is to say, X=^— Y^~ Z= — dx ' dy' dz' as is always the case when only central forces are in- volved, this reduction is unnecessary. The entire ex- pression to the left is then a complete differential. And we have 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 vires vivce at the be- ginning and the end of the motion. The vires vivce are in such case also functions of the coordinates. In the case of a body movable in the plane of X and F suppose, for example, X= — y, Y= — x\ we then have r( — ydx — xdy") = — Cd{xy') =z But if Jf == — a, F= — X, the integral to the left is — C{a dx -\- X dy"). This integral can be assigned the moment we know the path the body has traversed, that 48o THE SCIENCE OF MECHANICS. Essential character of analyt- ical me- chanics. is, if jc is determined a function of x. If, for example, y ^px"^, the integral would become , 2;>(^„ — xY —J (a + 2/;c2) dx = a (x^ — a:) + ^^ °g ^-. The difference of these two cases is, that in the first the work is simply a function of coordinates, that a force-function exists, that the element of the work is a complete differential, and the work consequently is de- termined by the initial and final values of the coordi- nates, while in the second case it is dependent on the entire path described. g. These simple examples, in themselves present- ing no difficulties, will doubtless suffice to illustrate the general nature of the operations of analytical mechan- ics. 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 q)!\zx&.cX&x. Poin- sot did not altogether escape this error. It remains to be mentioned that as the result of the labors of Mobius, 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 geomet- rical ideas than do the conceptions of common analyt- ical geometry ; and the advantages of analytical gene- rality 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.) FORMAL DEVELOPMENT. 481 THE ECONOMY OF SCIENCE. I. It is the object of science to replace, or save, ex- The basis of SC16I1C6 periences, by the reproduction and anticipation of facts economy of 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 mys- ticism 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 him- self ; and thus, to spare posterity, the experiences of whole generations are stored up in libraries. Language, the instrument of this communication, The eco- is itself an economical contrivance. Experiences are chS-aiTter analysed, or broken up, into simpler and more familiar guage! 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 fu- ture ; 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 THE SCIENCE OF MECHANICS. Possibility we have an actual example of a true ideographic lan- saA™-"'^'^ guage, pronounced diversely in different provinces, yet ^"*s®- 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 gram- mar, as English mostly drops them, would be quite requisite to the adoption of such a system. But uni- versality would not be the sole merit of such a char- acter ; 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 pre- cluded from reading. EcoDom- 2. In the reproduction of facts in thought, we terofair° never reproduce the facts in full, but only that side of semalfons them which is important to us, moved to this directly world. or indirectly by a practical interest. Our reproductions are invariably abstractions. Here again is an econom- ical tendency. Nature is composed of sensations as its elements. Primitive man, however, first picks out certain com- pounds of these elements — those namely that are re- latively permanent and of greater importance to him. The first and oldest words are names of "things." Even here, there is an abstractive process, an abstrac- tion from the surroundings of the things, and from the continual small changes which these compound sensa- tions 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 reasoa 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 DEVELOPMENT. 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 com- pound sensation of relative fixedness. Properly speak- ing the world is not composed of "things" as its ele- ments, 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 bev- eled edges, expressions involving contradictions, un- less 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 give relief to those elements to whose connection we effect, have to attend in the reproduction of a fact in the re- spect in which it is important to us. There is no cause nor effect in nature ; nature has but an individual exis- tence ; 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 abstrac- tion which we perform for the purpose of mentally re- producing the facts. Let a fact become familiar, and we no longer require this putting into relief of its con- necting 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 THE SCIENCE OF MECHANICS. 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 Htmus ; 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- thmg A act on another thmg j9? Hume, m fact, re- hauer's ex- , . , pianations lects causahtv and recognises only a wonted succes- of cause ' •' ° , , , and eflect. siou in time. Kant correctly remarked that a necessary 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, distin- guishes four forms of the "principle of sufficient rea- son" — 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 ap- economical parcntly this. The ideas of cause and effect originally 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 con- nection between M 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 fa- miliar 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 nov- ice. 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 experi- ences are subsumed ; but that idea has itself been da- FORMAL DEVELOPMENT. 485 veloped from experience. The notion of the necessity 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 con- tributed 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 develop- ment of the race. Cause and effect, therefore, are things of thought, having an economical office. It can- not 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 Econom- ical fea- is 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 fea- tures 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 embod- ied in a single expression. Thus, instead of noting indi- vidual cases of light-refraction, we can mentally recon- struct 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 or /sin fi^^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 MECHANICS. fraction. The law of refraction is a concise compen- dious rule, devised b}' us for the mental reconstruction of a fact, and only for its reconstruction in part, that is, on its geometrical side. Theecon- 5. The sciences most highly developed economically matiiemaf- are those whose facts are reducible to a few numerable ences!^' elements of like nature. Such is the science of mechan- ics, in which we deal exclusively with spaces, times, and masses. The whole previously established econ- omy of mathematics stands these sciences in stead. Mathematics may be defined as the economy of count- ing. Numbers are arrangement-signs which, for the sake of perspicuity and economy, are themselves ar- ranged 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. ^ ^^ 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, substitut- ing relations for values, symbolises and definitively fixes all numerical operations that follow the same rule. For example, we learn from the equation — , — = ^ — yt FORMAL DEVELOPMENT. 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 centu- ries ago. Often operations involving intense mental effort The theory may 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 «i •»^+ -^1 J' + ^1 =0 «2 ■«^+ -^2 :'' + ^2 = 0' from which result ^_ ^1 b^ — c^b^ _ P a^ c^ — ^^2 ^1 Q '^1 ^2—^2 ^i~ ^' 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 me- chanically. Thus, 1^1 ''1 ^2 ^2 ■■ a ^b^-a^b^=N and similarly \c^ b^\ \a^ c„\ 488 THE SCIENCE OF MECHANICS. Calculating Even a total disburdening of the mind can be ef- maohmes. fg^.^^^ jjj j^j^thg^^tical operations. This happens where operations of counting hitherto performed are symbol- ised 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 hke 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 ma- chines 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 pre- sented, other ab- A numerical result is not always reached by the methods of actual solution of the problem ; it may also be reached 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 mx'"~^dx of the quadrature for the increment dx of the abscissa. But we then also know that ('7?ix'"~' dx ^=x"'; that is, we recognise the quan- tity 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, al- ways goes that way, he undoubtedly saves himself the difference of the work. But memory is really not work. FORMAL DEVELOPMENT. 489 It only places at our disposal energy within our present or future possession, which the circumstance of igno- rance 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 out clear views of this matter, must often have the views on this sub- uncomfortable feeling that his paper and pencil sur- ject. pass him in intelligence. Mathematics, thus pursued as an object of instruction, is scarcely of more educa- tional" 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 this economy of thought, altogether similar to those omy of we have just examined. A brief reference here will suf- physics.'" fice. The moment of inertia saves us the separate con- sideration of the individual particles of masses. By the force-function we dispense with the separate in- vestigation of individual force-components. The sim- plicity 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 THE SCIENCE OF MECHANICS. Science a of fact, within the short span of a human Hfe and with problem, man's limited powers of memory, any stock of knowl- edge 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 thought. 7. The function of science, as we take it, is to re- place experience. Thus, on the one hand, science must remain in the province of experience, but, on the other, must hasten beyond it, constantly expecting con- firmation, 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 ob- servation supplies in the study of the larger portions. The comparison of theory and experience may be far- ther and farther extended, as our means of observation increase in refinement. The princi- Experience alone, without the ideas that are asso- ple of con- . - . , . 1 1 r tinuity, the ciated With it, would forever remain strange to us. entific Those ideas that hold good throughout the widest do- mains of research and that supplement the greatest amount of experience, are the most scientific. The prin- ciple of continuity, the use of which everywhere per- vades 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 method. FORMAL DEVELOPMENT. 491 graphically recorded. If the rod be shortened, the Example ii- ., . -11 ' • - 1- 1 11- lustrative Vibrations will increase in rapidity and cannot be di- of the rectly seen ; the rod will present to the sight a blurred science. image. This is a new phenomenon. But the sensa- tion 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 n'ot all change at once ; only this or that phenomenon changes ; conse- quently 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 vibra- tions 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 ex- perience intelligible to us ; it supplements and sup- plants 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 SCIENCE OF MECHANICS. 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 origi- ories'no't nated so naturally and artlessly. Thus, chemical, elec- the p^-fncl" trical, and optical phenomena are explained by atoms. finuTty!^™ But the mental ' artifice atom was not formed by the principle of continuity ; on the contrary, it is a pro- duct 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 provisional 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 tai artifices, to that of Certain auxiliary conccp ts in mathematics; it is a mathematical model for facilitating the mental reproduction of facts. Although we represent vibra- tions by the harmonic formula, the phenomena of cool- ing by exponentials, falls by squares of times, etc., no one will fancy that vibrations in themselves have any- thing to do with the circular functions, or the motion of falling bodies with squares. It has simply been ob- served that the relations between the quantities inves- tigated were similar to certain relations obtaining be- tween familiar mathematical functions, and these more FORMAL DEVELOPMENT. 493 familiar ideas are employed as an easy means of sup- plementing experience. Natural phenomena whose re- lations are not similar to those of functions with which we are familiar, are at present very difficult to recon- struct. But the progress of mathematics may facilitate the matter. As mathematical helps of this kind, spaces of more Muiti- , dimen- than 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 arti- fices.* *As the outcome of the labors of Lobatchevski, Bolyai, Gauss, and Rie- mann, the view has gradually obtained currency in the mathematical world, that that which we call space is a particular, acttial 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 manifold- ness. 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 experience, and the pseudo-theories of geometry that seek to excogitate these properties by meta- physical 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 ex- perience 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 re- ferred to, has, primarily, nothing to do with the question whether things really exist which correspond to these possibilities; and we must not hold mathe- maticians responsible for the popular absurdities which their investigations have given rise to. The space of sight and touch is Mr^^-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 dis- cussed whether it would facilitate and promote our insight into things to con- ceive experiential space as part of a four-dimensional or multi-dimensional 494 THE SCIENCE OF MECHANICS. Hypotheses This IS the casc, too, with all hypothesis formed ^ ^^ for the explanation of new phenomena. Our concep- tions of electricity fit in at once with the electrical phe- nomena, 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 mentioned were not forthcoming until after the new views were published, and were then ex- hibited in the presence of certain persons at spiritualistic seances. The fourth dimension was a very opportune discovery for the spiritualists and for theo- logians who were in a quandary about the location of hell. The use the spiri- tualist 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 dimen- sion ; 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 halo 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 there is absolutely nothing at stake. All is purpose- less jugglery. We have not yet found an ^rccoMcA^Kr who has accomplished parturition through the fourth dimension. If we should, the question would at once become a serious one. Professor Simony's beautiful tricks in rope- tying, which, 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 the end, 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 space as a mathematico-physical help even prior to the appearance of Riemann's memoir. But I trust that no one will employ what I have thought, said, and written on this subject as a basis for the fabrication of ghost stories, (Compare Mach, Die Gesckichte und die Wurzel des Satzes -von der Erkaltung der Arbeit.) CHAPTER V. THE RELATIONS OF MECHANICS TO OTHER DE- PARTMENTS OF KNOWLEDGE. I. THE RELATIONS OF MECHANICS TO PHYSICS. 1. Purely mechanical phenomena do not exist. The The events •' ■*■ ^ ^ ^ of nature production of mutual accelerations in masses is, to all ^° °.°' f^- ^ ^ cmsively appearances, a purely dynamical phenomenon. But belong to with these dynamical results are always associated «°':e- thermal, magnetic, electrical, and chemical phenom- ena, and the former are always modified in proportion as the latter are asserted. On the other hand, thermal, magnetic, electrical, and chemical conditions also can produce motions. Purely mechanical phenomena, ac- cordingly, 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 be- ing 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 world. 496 THE SCIENCE OF MECHANICS. The me- gained. As more and more facts are discovered and aspects of classified, entirely new ideas of general scope can be necessarily formed. We have no means of knowing, as yet, which mental " of the physical phenomena go deepest, whether the aspec s. jj^gj-i^anical phenomena are perhaps not the most super- ficial 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 me- .,. .^ , chanicai in an historical view, both intelligible and pardonable ; conception . . . . , of the and it may also, for a time, have been of much value. 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, re- stricts 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 hy- potheses 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 Geschickie und die Wurzel des Satzes von der Erhaltung der Arbeit. ITS RELATIONS TO OTHER SCIENCES. 497 values of the potential function V and the dielectric Science 1 T rr e 1 should be constants. If we assume the differences of the values based on of J^to be measured (on the electrometer) by the forces, on hypoth- and regard Fand not the quantity of electricity Q as the primary notion, or measurable physical attribute, we shall have, for any simple insulator, for our quan- tity of electricity — 1 rfd^V d^V d^V^ ^ ~ T^ J V ^^ ~dy^ ~'~ dz^ (where x, y, z denote the coordinates and dv the ele- ment of volume,) and for our potential* Here Q and W appear 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 quan- titative conceptual expression of actual facts. All su- perfluous and futile notions are eliminated, and the imaginary problems to which they have given rise fore- stalled. (See Appendix XXVIII, p. 583.) The removal of notions whose foundations are his- torical, conventional, or accidental, can best be fur- thered 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 tem- peratures 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 THE SCIENCE OF MECHANICS. Desirabil- ity of a compara- tive phys- ics. Circum- stances which fa- vored the develop- ment of the mechanical view. know, only differences come into consideration, the significance of temperature is not only contained in its difference with respect to other temperatures. Thermal capacities correspond to masses, the potential of an electric charge to quantity of heat, quantity of elec- tricity to entropy, and so on. The pursuit of such re- semblances and differences lays the foundation of a comparative physics, which shall ultimately render pos- sible the concise expression of extensive groups of facts, without arbitrary additions. We shall then possess a homogeneous physics, unmingled with artificial atomic theories. It will also be perceived, that a real economy of scientific thought cannot be attained by mechanical hypotheses. Even if an hypothesis were fully com- petent to reproduce a given department of natural phe- nomena, say, the phenomena of heat, we should, by accepting it, only substitute for the actual relations be- tween the mechanical and thermal processes, the hy- pothesis. The real fundamental facts are replaced by an equally large number of hypotheses, which is cer- tainly no gain. Once an hypothesis has facilitated, as best it can, our view of new facts, by the substitu- tion of more familiar ideas, its powers are exhausted. We err when we expect more enlightenment from an hypothesis than from the facts themselves. 3. The development of the mechanical view was favored by many circumstances. In the first place, a connection of all natural events with mechanical pro- cesses is unmistakable, and it is natural, therefore, that 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 gen- eral and extensive scope were discovered. A law of ITS RELATIONS TO OTHER SCIENCES. 499 this kind is the principle of vis viva "ZiU^ — U^ = 2^m (v^ — ?'§), 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 Spannkraft, S,* then the work actually performed, 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^-(- ^'2mv'^ ^^ const, that is to say, every diminution of the Spannkraft, is The Con- compensated for by an increase of the vis viva. In this Energy, form the principle is also called the law of the Conser- vation 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 only 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 mechanical action as the basis of all natural actions. However, nothing is contained in the expression but the fact of an invariable quantitative connection between mechani- cal 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 Abhandlungen, II, 965) he expressly discards it in favor of the English " potential energy." He even (p. 968) pre- fers Clausius's word Ergal to Spannkraft, which is quite out of agreement with modern terminology. — Trans. 500 THE SCIENCE OF MECHANICS. Compre- insight was possessed at all times by the foremost ness of inquirers and even entered into the construction of VI GW tll6 condition, mechanics itself, and was, accordingly, not first created suit, of me- by the latter. Galileo and Huygens constantly alter- 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 indi- vidual 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 gen- erally, 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. Exempiifl- Take the fictions of Stevinus — say, that of the end- this in ste- less chain on the prism. Here, too, a deep, broad searches, 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 Ste- vinus a motion of descent that is not a descent, a mo- tion 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, gener- ally, 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 = Vigh that is here displayed, though of course in not so definite a form. A contradiction exists in this ITS RELATIONS TO OTHER SCIENCES. 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 individual with the universal, is also displayed, only in of camot this instance not restricted to mechanics, in the per- Mayer. formances of Sadi Camot. 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 t' , can only depend on the tempera- tures 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 equiva- lence 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 pro- fessional physicists Joule, Helmholtz, Clausius, and Thomson, who have done very much, perhaps all, to- wards the detailed establishment and perfection 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 THE SCIENCE OF MECHANICS. The inter- 5. We shall now attempt to show that the broad depend- . , . , . . , - , ence of the View expressed m the principle oi the conservation ture. of energy, is not peculiar to mechanics, but is a condi- tion of logical and sound scientific thought generally. The business of physical science is the reconstruction of facts in thought, or the abstract quantitative expres- sion of facts. The rules which we form for these recon- structions 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 em- phasis 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 phe- nomena are dependent on one another. The laws of nature are equations between the meas- urable elements a^yS . . . . oooi phenomena. As na- ture is variable, the number of these equations is al- ways less than the number of the elements. If we know a// the values of a j3yd . . ., by which, for example, the values oi X /j.v . . . are given, we may call the group a/3yd . . . the cause and the group Xj^v . . . the effect. In this sense we may say that the effect is uniquely determined by the cause. The prin- ciple 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 mutual interdependence is met with ITS RELATIONS TO OTHER SCIENCES. 503 in Stevinus, Galileo, Huygens, and other great inquir- Sense of ers. The idea is also at the basis of the discovery of depend- ence at tbe counier-phenoTaena. Thus, a change in the volume of basis of aii . . , J great dis- a gas due to a change of temperature is supplemented covenes. by the counter-phenomenon of a change of tempera- ture on an alteration of volume ; Seebeck's phenome- non by Peltier's effect, and so forth. Care must, of course, be exercised, in such inversions, respecting the form 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 X not necessarily by a change of a. The relations be- tween electromagnetic and induction phenomena, dis- covered by Faraday, are a good instance of this truth. If a set of circumstances a 6yS . . ., by which a various ' ' -^ forms of ex- second set Xuv . . . is determined, be made to pass P5.e"'°n °f ^ ' -^ this truth. from its initial values to the terminal values a' fi'y' 6'. . ., then X fiv . . . also will pass into X' ju' 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 permanent ones. The fertile methods of thought of Galileo, Huygens, S. Carnot, Mayer, and their peers, are all reducible to the simple but sig- nificant perception, that purely periodical alterations of one set of circumstances can only constitute the source of similarly periodical alterations of a second set of circum- stances, not of continuous and permanent alterations. Such maxims, as "they effect is equivalent to the cause," tion. 504 THE SCIENCE OF MECHANICS. "work cannot be created out of nothing," "a per- petual motion is impossible," are particular, less defi- nite, 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 mystic- ism that may still adhere to the principle of the con- servation of energy* is dissipated. (See p. 585.) Purpose of All ideas of conservation, like the notion of sub- the ideas of .,.,,., conserva- stance, have a solid foundation m the economy of thought. A mere unrelated change, without fixed point of support, or reference, is not comprehensible, not mentally reconstructible. We always inquire, accord- ingly, what idea can be retained amid all variations as permanent, 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, ,;,'• remains ^= g-Siow in all cases of the motion of heavy bodies, the energy remains con- stant in every isolated system, all our assertions have one and the same economical function, namely that of facilitating our mental reconstruction of facts. THE RELATIONS OF MECHANICS TO PHYSIOLOGY. Conditions I- AH science has its origin in the needs of life. detefop""^ However minutely it may be subdivided by particular ^?ence, 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 the whole. 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 OTHER SCIENCES. 505 its true maturity, and be insured against lop-sided and monstrous growths. The division of labor, the restriction of individual Contusion , . . J . , . , . _ of the inquirers to limited provinces, the investigation of means and 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 in- struments, 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 made^the development of physics. The majority of natural in- phys1o°iogy. 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 expe- riences, 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 mechan- ism 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 intellectual machinery, employed in the representation of the world on the stage of thought, be regarded as the basis of the real world. 3. A philosophy is involved in any correct view of 5o6 THE SCIENCE OF MECHANICS. The at- the relations of special knowledge to the great body of p^infeer knowledge at large, — a philosophy that must be de- motions, 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 physi- ology, such a mistaken conception of the true relations of the two sciences, is displayed in the inquiry whether it is possible to explain 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 confidence 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 sen- sations of time and space are just as much sensations 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 quan- tities 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 RELATIONS TO OTHER SCIENCES. 507 combination of touch, sight, light, and time sensations. They possess intelHgible meaning only by virtue of . the sensations they involve, the contents of which maj- of course be very complicated. It would be equivalent, accordingly, to explaining Mode of the more simple and immediate by the more compli- such cr- eated and remote, if we were to attempt to derive sen- sations 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 psycho- logical facts. If the means and aims of research were properly distinguished, and our expositions were re- stricted to the presentation of actual facts, false prob- lems of this kind could not arise. 4. All physical knowledge can only mentally repre- The princi- sent and anticipate compounds of those elements we chanics not call sensations. It is concerned with the connection of tion but these elements. Such an element, say the heat of a body aspect of A, is connected, not only with other elements, say with 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 iVis not essentially, but only conventionally, different from A and B. The connection of A and .5 is a problem of physics, that of A and N a problem of physiology. 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 mechani- cal 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.. (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. Schia- parelli 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 Optics of Ptolemy. The view that the Greeks were especially neglectful of experiment can no longer be maintained unqualifiedly. The most ancient ex- periments 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 corporeal- ity 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 ob- servations in physiological optics are still full of in- terest to-day. Aristotle (^Meteorology) describes phe- nomena 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 MECffANICS. 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, more- over, rendered quite intelligible when we consider the difficulties and the expense attending the production of ancient books and their consequent limited circula- tion. The conditions here involved are concisely dis- cussed by J. Mueller in his paper, "Ueber das Ex- periment in den physikalischen Studien der Grie- chen," Naturwiss. Verein zulnnsbruck, XXIII., 1896- 1897. (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 me- chanical 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 pul- leys. 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. 511 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, char- acteristic of the early history of mechanics that atten- tion should have been first directed to its practical advantages and to the construction of automata de- signed 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 ma- terially changed. So, too, during the decadence of civilisation in the Middle Ages, the same tendency as- serts itself. The artificial automata and clocks of this period, the construction of which popular fancy as- cribed 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 mechan- ics loom up before the minds of inquirers. Aristotle's tract. Mechanical Problems (German trans, by Poselger, Hannover, 1881) is characteristic in this regard. Aris- totle is quite adept in detecting and in formulating problems ; he perceived the principle of the parallel- ogram of motions, and was on the verge of discover- ing centrifugal force ; but in the actual solution of problems he was infelicitous. The entire tract par- takes more of the character of a dialectic than of a scientific treatise, and rests content with enunciating the "apories," or contradictions, involved in the prob- lems. But the tract upon the whole very well illus- trates the intellectual situation that is characteristic of the beginnings of scientific investigation. 512 THE SCIENCE OF MECHANICS. "If a thing take place whereof the cause be not apparent, even though it be in accordance with na- ture, it appears wonderful. . . . Such are the instances in which small things overcome great things, small weights heavy weights, and incidentally all the prob- lems that go by the name of 'mechanical.' . . . To the aperies (contradictions) of this character belong those that appertain to the lever. For it appears con- trary 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 pri- mordial 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 won- derful should proceed out of something else that is wonderful. The combination of contradictory prop- erties, however, into a single unitary product is the most wonderful of all things. Now, the circle is ac- tually 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 evi- dence of a capacity for detecting and enunciating prob- lems, but are far from conducting the investigator to their solution. III. (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 sym- metrical in every respect, equilibrium obtains on the assumption of any form of dependence whatever of the disturbing factor on L, or, generally, on the as- sumption P.fi^L) ; and that consequently \h& 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 as- sumes that distance from the fulcrum is determina- tive, 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 symmet- rically 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 proce- dure, 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 he, proportional X.0 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 MECHANICS. 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 compres- sions of the prisms may now be carried out with per- fect certainty. A knife edge may be introduced at any point un- der a prism suspended from its center without dis- turbing the equilib- rium (see Fig. 236), and several such ar- rangements may be rigidly combined to- ' ~ zsj gether so as to form Fig. 336. apparently new cases of equilibrium. The conversion and disintegration of the case of equi- librium into several other cases (Galileo) is possible only by taking into account the value of PL. I can- not agree with O. Holder who upholds the correct- ness of the Archimedean deductions against my criti- cisms in his essay Denken und Anschauung in der Geo- meirie, 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 cequiponderantibus) regarded it as a general experience that two equal weights may under all circumstances be replaced by one equal to their APPENDIX. 515 combined weight at the center (Theorem 5, Corrol- ary 2). Ih such an event, his long deduction (Theo- rem 6) would be necessary, for the reason sought fol- lows 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 re- garded as a priori evident ; and the views advanced on pp. 14, 513 appear to me to be still uncontro- verted. (See page 2g.) Stevinus's procedure may be looked at from still another point of view. If it is a fact, for our mechan- ical instinct, that a heavy endless chain will not ro- tate, 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 exper- iments, 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 per- fect rigor. We shall have still many instances to ad- duce, among the foremost inquirers in the domain of mechanics, of this tentative adaptation of special 5i6 THE SCIENCE OF MECHANICS. quantitative conceptions to general instinctive im- pressions. 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 Principles of Heat, 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 concep- tion or ideal of an entire province of phenomena, their constant consideration of the whole in their treatment of parts, may be characterised as a genuinely philo- sophical 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. Stevinus 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 hand, is the most special conceivable. He cannot possibly deduce from his special case in an unassailable man- ner 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 de- viates somewhat from the theoretical truth. The de- viation 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 dis- turbing elements. It appears in the Archimedean and Stevinian procedures as an hypothesis without which the individual facts of experience would at once be- come involved in logical contradictions. Not until we have possessed this hypothesis can we by operat- ing 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 ob- jects 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. An- axagoras proves the corporeality of air from its resist- ance 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). Ac- cording to Empedocles, the air prevents the water 5i8 THE SCIENCE OF MECHANICS. from penetrating into the interior of a vessel immersed with its aperture downwards (Gomperz, Griechische Denker, I., p. igi). Philo of Byzantium employs for the same purpose an inverted vessel having in its bot- tom 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 per- formed, in almost the precise form customary in the schools to-day {Philonis lib. de ingeniis spiritualibus, in V. Rose's Anecdota grceca et latino). 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 intermedi- ate 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 rare- fying and compressing bodies, including air (inrush- ing and outrushing of the air in Hero's ball). An ar- gument 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 in- creasing a vacuum, according to Hero and his prede- cessors, is always the attraction and solicitation of adjacent bodies. A light vessel with a narrow aper- ture 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 cup- ping-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 inconceiv- able." 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 con- ceives of the flow of water as analogous to the move- ment of a chain hanging with unequal lengths over a pulley. The union of the two columns, which for us is preserved by the 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 in- genious 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 auto- matic sounding of trumpets and the opening of tem- ple 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, Hero's Werke, Leipsic, 1899, 520 THE SCIENCE OF MECHANICS. and Diels, System des Strata, Sitzungsberichie der Ber- liner Akademie, 1893). VII. {See page 129.) It has often been asserted that Galileo had prede- cessors of great prominence in his method of think- ing, 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. Leo- nardo knew the ratio of the times of descent down the slope and the height of an inclined plane. Fre- quently also a knowledge of the law of inertia is at- tributed to him. Indeed, some sort of instinctive knowledge of the persistence of motion once begun will not be gainsaid to any normal man. But Leo- nardo 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 be- lieves 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 alto- gether removed. (Compare Wohlwill, Bibliotheca Ma- thematica, Stockholm, 1888, p. 19). Benedetti (1530- 1590) knows that falling bodies are accelerated, and explains the acceleration as due to the summation APPENDIX. 521 of the impulses of gravity {Divers, speculat. math, et physic, liber, Taurini, 1585). He ascribes the progres- sive 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 actu- ally 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 ac- cording 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 Be- nedetti's investigations {Atti della R. Acad, di Torino, Vol. XXXIII. , 1898), finds the chief merit of Bene- detti 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 un- tenable and possible only in special cases. Let the velocity of descent be proportional to p — q, where/ is the weight of the body and q the upward impulsion due to the medium. If only half the velocity of de- scent is set up in a medium of double the density, the equation/ — ^:=2(/ — 2^) must exist, — a relation which is possible only in case / = 3^. 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 THE SCIENCE OF MECHANICS. conceiving equal bodies falling alongside each other first disconnected and then connected, where the con- nexion 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 rotat- ing 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 posses- sion of instinctive experiences prior to his resorting to experiment. Freely falling bodies are followed with more diffi- culty 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 scien- tific purposes our mental representations of the facts of sensual experience must be submitted to conceptual APPENDIX. 523 formulation. Only thus may they be used for discov- ering by abstract mathematical rules unknown prop- erties 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 neg- lecting what is subsidiary, by abstracting, by idealis- ing. The experiment determines whether the form chosen is adequate to the facts. Without some pre- conceived opinion the experiment is impossible, be- cause 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 experi- ment 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 » a function? What function of / is z;? Galileo asks, in his ingenu- ous and primitive way : is v proportional to s, is v proportional to /? Galileo, thus, gropes his way along synthetically, but reaches his goal nevertheless. Sys- tematic, routine methods are the final outcome of re- search, and do not stand perfectly developed at the disposal of genius in the first steps it takes. (Com- pare the article " Ueber Gedankenexperimente, " Zeit- schrift fUr denphys. und chem. Unterricht, 1897, I.) IX. (See page 140.) In an exhaustive study in the Zeitschrift fiir Volker- psychologie, 1884, Vol. XIV., pp. 365-410, and Vol. 524 THE SCIENCE OF MECHANICS. 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 cir- cular motion and uniform horizontal motion occupy distinct places. Wohlwill's researches are very ac- ceptable and show that Galileo had not attained per- fect 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 univer- sality 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 velo- city could not be destroyed, — a fact which Wohlwill himself points out (p. 112). It is not at all surpris- ing 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 mus- ket-ball possessing no weight would continue rectiline- arly on its path in the direction of the barrel. {Dia- logues on the two World- Systems, German translation, Leipsic, 1891, p. 184.) His hesitation in enunciating APPENDIX. 525 in its most general terms a law that at first blush ap- pears 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' endeav- ors 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 dis- tinctly 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 impres- sion 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 (^Philosophical Magazine, 1878), in the locomotive which rushes safely across a defective 526 THE SCIENCE OF MECHANICS. bridge, through which it would have crashed if at rest, but which, owing to the insufficient time of des- cent 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. Essentially 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 curios- ity 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 epistemologi- cal 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 compo- sition of the vibrations of strings by Sauveur, the com- position of thermal disturbances by Fourier. Through this latter inquirer the method of compounding a phe- nomenon out of mutually independent partial phe- nomena by means of representing a general integral APPENDIX. 527 as the sum of particular integral^ has penetrated into every nook and corner of physics. The decomposi- tion of phenomena into mutually independent parts has been aptly characterised by P. Volkmann as iso- lation, 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 major- ity 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 phenom- enon as an entirety do we acquire a command over the whole. ..." (Cf. P. Volkmann, Erkenntnisstheo- retische GrundzUge der Naturwissenschaft, 1896, p. 70. Cf. also my Principles of Heat, German edition, pp. 123, 151, 452). XI. (See page i6l.) 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 velo- city V 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 magni- tude and direction all the velocities the body succes- 528 THE SCIENCE OF MECHANICS. sively acquires, these lines will represent the sum of the radii v of the circle. For OM to be trans- formed into ON, the perpendicular component to it, MN, must be added. During the period of revolu- tion 7" the velocity is uniformly increased in the direc- tions of the radii r by an amount inv. The numeri- M H Fig. 237- Fig 238. Fig. 239- cal measure of the radial acceleration is therefore r '77' 7/ 7J m=:—=-, and since vT^^2nr, therefore also w^ — . T ^ r If to OM=^v the very small component w is added (Fig. 239), the resultant will strictly be a greater w / iXJ velocity y v'' -\- ■uP' =»+ h-> as the approximate ex- traction of the square root will show. But on contin- uous deflection ^r- vanishes with respect to v ; hence, only the direction, but not the magnitude, of the velocity changes. XII. (See page 162.) Even Descartes thought of explaining the centri- petal impulsion of floating bodies in a vortical me- dium, after this manner. But Huygens correctly re- APPENDIX. 529 marked that on this hypothesis we should have to assume that the lightest 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 phe- nomena 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 exam- ple, 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 heavier than water and hence touch the bot- tom of the vessel. If the vessel be rotated, the par- ticles 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 move- ment, 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, mo- tions 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 MECHANICS. 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 at- tention was drawn, in fact, to precisely those features of luminous phenomena that had escaped Newton. With respect to physics he took up with great enthu- siasm 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 theo- retical and practical dioptrics advanced these depart- ments very considerably. As a technicist he is the inventor of the powder-machine, the idea of which APPENDIX. 531 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 Huy- gens grows as his labors are made better known by the complete edition of his works. A brief and reveren- tial sketch of his scientific career in all its phases is given by J. Bosscha in a pamphlet entitled Christian Huyghens, Rede am 200. Gedachtnis stage seines Lebens- endes, German trans, by Engelmann, Leipsic, 1895. XIV. (See page 190.) Rosenberger is correct in his statement {Newton und seine physikalischen Principien, 1895) that the idea of universal gravitation did not originate with New- ton, but that Newton had many highly deserving pred- ecessors. 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 compre- hensively 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 men- tioned 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 in- herent also in the sun, the moon, and the other plan- 532 THE SCIENCE OF MECHANICS. ets." Similarly, Kepler (1609), like Gilbert before him (1600), 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 no- tions, 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 revolu- tion 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 imma- terial geometric point can exercise no physical action, and he did very much toward demonstrating the ter- restrial 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 experi- ment very similar to that described by us in Fig. io6, 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 grav- ity with the universal gravitation that determined the motions of the celestial bodies, as follows. He con- ceived a stone to be hurled with successive increases of horizontal velocity from the top of a high moun- tain. Neglecting the resistance of the air, the para- bolas successively described by the stone will increase in length until finally they will fall clear of the earth 534 THE SCIENCE OF MECHANICS. 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 ap- parent 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 gravi- tation, by some such means as actions of contact. Yet the great success which Newton achieved in astron- omy with forces acting at a distance as the basis of deduction, soon changed the situation very consider- ably. 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 par- ticles separated by vacuous interstices and thus acting on one another at a distance. Finally even, the re- sistance 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 compli- cated than those representing the latter. The doctrine of forces acting at a distance doubt- less 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 New- ton's law, and slight quantitative variations of it were looked for. After it had been demonstrated, however, that electricity travelled with finite velocity, the ques- tion 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. Foppl ("Ueber eine Erweiterung des Gravitations- gesetzes," 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 an- other but repel positive masses, and assume therefore also 7?«2V^ 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," Zeitschrift f. Math. u. Phys., i8g8, II.), from the peri- helial motion of Mercury, forty-one seconds in a cen- tury, 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 MECHANICS. pareW. Wien, "Ueber die Moglichkeit einer elektro- magnetischen Begriindung der Mechanik," Archives N^erlandaises, 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 develop- ment. Critical inquiries as to the origin of the con- cept 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 per- fectly complete, and is adopted with absolute ingenu- ousness. 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, Newton 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 sensa- tions we have been led to the assumption of an abso- lute constancy, which we call substance, the most ob- vious and prominent example being that of a moveable body distinguishable from its environment. And see- ing that such bodies are divisible into homogeneous parts, of which each presents a constant complexus APPENDIX. 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, how- ever, we are concerned with precisely as many sub- stantial 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. (Com- pare 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 men- tioned. Galileo explains the tides as due to the rela- tive 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- pernican system. If the earth (Fig. 240) rotates from the west to the east, and is affected at the same time with a pro- gressional motion, the parts of 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 fol- low 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 rele- gated to the category of explanations by "sympathy" and "antipathy," and that it admits as easily of refu- tation 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 atten- tion. But he deceived himself with regard to the difificulties involved, believing himself able to explain the daily, monthly, and yearly periods by considering the natural oscillations of the water and the altera- tions 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 ex- tremely 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 corres- ponding oblateness we should not expect. Now, sup- pose 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 pne 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. thje 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 re- spect 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 ex- tremely 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 expe- riences, 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 forced more and more toward the center of the earth, with just enough of its centripetal acceleration counter- acted 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 antici- pate the great intellectual achievements of Huygens and Newton. XVIII. (See page 218.) H. Streintz's objection {JDie physikalischen Grund- lagen der Mechanik, Leipsic, 1883, p. 117), that a com- 540 THE SCIENCE OF MECHANICS. parison of masses satisfying my definition can be ef- fected only by astronomical means, I am unable to admit. The expositions on pages 202, 218-221 amply refute this. Masses produce in each other accelera- tions 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 pop- ular manner. The criticism in question, therefore, may be regarded as refuted. My definition is the outcome of an endeavor to establish the interdependence of phenomena and to re- move all metaphysical obscurity, without accomplish- ing 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- sago, 1898), "temperature," "quantity of heat" {Zeit- schrift fiir den physikalischen und chemischen Unterricht, Berlin, 1888, No. i), and so forth. With the view here taken of the concept of mass is associated, how- ever, another difificulty, which must also be carefully noted, if we would be rigorously critical in our analy- sis of other concepts of physics, as for example the concepts of the theory of heat. Maxwell made refer- ence to this point in his investigations of the concept of temperature, about the same time as I did with re- spect to the concept of heat. I would refer here to the discussions on this subject in my Principles of Heat (German edition, Leipsic, 1896), particularly page 41 and page 190. APPENDIX. 541 XIX. (See page 226.) My views concerning physiological time, the sen- sation of time, and partly also concerning physicaf 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 simi- lar reasons, we select as our measure of time an arbi- trarily 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 inter- dependence of phenomena, as I pointed out as early as 1865 {JJeber den Zeitsinn des Ohres, Sitzungsberichte der Wiener Akademie) and 1866 (Fichte's Zeitschrift fUr Fhilosophie), all metaphysical obscurities disappear. (Compare J. Epstein, Die logischen Principien der Zeit- messung, Berlin, 1887.) I have endeavored also {Principles of Heat, German edition, page 51) to point out the reason for the natu- ral 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 ad- duced 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 connex- ion 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 filr wissenschaftliche Philosophic, 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 la;w of inertia, all of which furnish welcome evidence of a heightened interest in this question, I can here only briefly mention that of Streintz {Fhysi- kalische Grundlagen der Mechanik, Leipsic, 1883) and that of L. Lange (^Die geschichtliche Entwicklung des Bewegungsbegriffes, Leipsic, 1886). The expression " absolute motion of translation" Streintz correctly pronounces as devoid of meaning and consequently declares certain analytical deduc- tions, to which he refers, superfluous. On the other hand, with respect to rotation, Streintz accepts New- ton's position, that absolute rotation can be distin- guished 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 ex- pression of the law of inertia. I cannot share this view. For me, only relative motions exist {Erhaltung 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 objec- tion 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 mean- ingless, for the two cases are not, in sense-perception, distinguishable from each other. I accordingly re- gard these two cases as the same case and Newton's distinction as an illusion {Science of Mechanics, 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, orienta- tion. 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 inde- pendent 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, re- jected. 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 methodi- cal movement wins at once the reader's sympathy. Its careful analysis and study, from historical and criti- cal 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 (i 865-1 866), 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 arbi- trary ; but that these ratios are independent of the APPENDIX. 545 kind and of the order of combination of the bodies is a result of inquiry. 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 forth, the law of inertia, in Lange's view, would read as follows : Three material points, Px, P^, P^, are simultane- ously 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 respec- tively call G\, Gi, Gz, 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 Px shall always move on the line G\, P^ on the line G'i, Pg on the line Gs, these lines may be regarded as the axis of a coordinate or inertial system, with respect to which every other material point, left to it- self, will move in a straight line. The spaces de- scribed by the free points in the paths so determined will be proportional to one another. A system of coordinates with respect to which three material points move in a straight line is, ac- cording 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 re- sults 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 MECHANICS. 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 espe- cially 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 (^Mechanics, pp. 234-235) are left. I abandoned these attempts, be- cause 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 opin- ion, is also true of the forms in which Streintz and Lange express the law. In point of fact, it was precisely by the considera- tion 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 (Mechanics, 232-233). The considera- tion of a small number of isolated points, to the ex- clusion of the rest of the world, is in my judgment in- admissible (^Mechanics, pp. 229-235). It is quite questionable, whether a fourth material point, left to itself, would, with respect to Lange's "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 in- quirer must still be, to regard the law of inertia pri- marily 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 APPENDIX. 547 our knowledge from future experience, as I have ex- plained 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. Fried- lander {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 Beharrungs- gesetz, Berlin, 1896), although the question remains 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, Socidtd scientifique de Bruxelles, i8gj, as well as the in- vestigations of J. G. MacGregor, Royal Society of Can- ada, iSgSt 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 «(/ ^(?