THE GENERAL PRINCIPLE OF 
 RELATIVITY 
 
MACMILLAN AND CO., LIMITED 
 
 LONDON BOMBAY - CALCUTTA ' MADRAS 
 MELBOURNE 
 
 THE MACMILLAN COMPANY 
 
 NEW YORK BOSTON CHICAGO 
 DALLAS SAN FRANCISCO 
 
 THE MACMILLAN CO. OF CANADA, LTD, 
 
 TORONTO 
 
 
 
THE 
 
 GENERAL PRINCIPLE 
 OF RELATIVITY 
 
 IN ITS PHILOSOPHICAL 
 AND HISTORICAL ASPECT 
 
 BY 
 
 H. WILDON CARR, D.LITT. 
 
 PROFESSOR OF PHILOSOPHY IN THE UNIVERSITY OF LONDON 
 
 MACMILLAN AND CO., LIMITED 
 ST. MARTIN'S STREET, LONDON 
 
 1920 
 
COPYRIGHT 
 
 GLASGOW I PRINTED AT THE UNIVERSITY PRESS 
 BY ROBERT MACLEHOSE AND CO. LTD 
 
PREFACE 
 
 IT may seem a bold undertaking on the part of 
 one who can claim no acquaintance with the 
 higher mathematics and no familiarity with the 
 experimental work of the physical laboratory, to 
 propose to interpret a principle which is the 
 practical concern only of the mathematician and 
 physicist. It needs no apology, however, for 
 though the principle of relativity has been formu- 
 lated by mathematicians and physicists purely as 
 a working principle in mathematics and physics, 
 the particular concepts with which it deals 
 space, time and movement are metaphysical, and 
 the essential concern of philosophy. 
 
 In this account of the principle of relativity I 
 have dealt only with the philosophical and 
 historical aspect of the problem. I have tried to 
 expound the reformed concepts of space and 
 time and movement which are the justification 
 and the foundation of the new working formulae. 
 
 V 
 
 435031 
 
vi PREFACE 
 
 I have not attempted to indicate or explain, even 
 in non-mathematical terms, the formulae them- 
 selves. I have not, for example, tried to show 
 how Einstein worked out the formula of the pre- 
 cession of the perihelion of Mercury, the dis- 
 placement of light from stars observed in the 
 eclipse observation, or the shift of the spectral lines. 
 'What I have tried to show is the exact meaning 
 in philosophy of the new concept of the frame- 
 work of nature. 
 
 My interest in the principle of relativity is 
 purely philosophical, but it is not casual or 
 accidental. I first became acquainted with it 
 at the International Congress of Philosophy at 
 Bologna in 1911, when M. Pierre Langevin, 
 Professor of the College de France, revealed its 
 philosophical importance in a remarkable paper 
 entitled " L'evolution de Pespace et du temps." 
 I introduced the subject to the Aristotelian 
 Society in a paper read in the Session of 
 1913-14 (Proceedings of the Aristotelian Society ', 
 Vol. XIV.), and I contributed an article, 
 "The Metaphysical Implications of the Principle 
 of Relativity," to the Philosophical Review of 
 January 1915. Since then the philosophical 
 importance of the principle has received full 
 
PREFACE vii 
 
 recognition. It was not, however, until the 
 preparation of my courses of lectures on the 
 " History of Modern Philosophy " delivered in 
 1918 and 1919 at King's College, London, led 
 me to read anew the works of Descartes and 
 Leibniz that the quite special historical interest 
 of the problem impressed me. It is this 
 historical aspect of the principle to which I have 
 tried to give expression in this study. The main 
 idea was developed in a course of lectures on 
 " Historical Theories of Space, Time and 
 Movement " delivered at King's College in 
 the spring of this year (1920). 
 
 My thanks are due to Professor T. P. Nunn 
 and Dr. C. D. Broad, who have rendered me 
 special service in reading my proofs. They are 
 not of course responsible for my views or for the 
 accuracy of any of my statements. 
 
CONTENTS 
 
 CHAPTER I 
 
 PAGE 
 
 SPACE, TIME AND MOVEMENT - - - - - i 
 
 CHAPTER II 
 THE ANTINOMY OF MOVEMENT 24 
 
 CHAPTER III 
 ATOMS AND THE VOID - - - 40 
 
 CHAPTER IV 
 THE VORTEX THEORY - - - 57 
 
 CHAPTER V 
 THE PROBLEM OF GRAVITATION - 77 
 
 CHAPTER VI 
 
 LEIBNIZ AND THE THEORY THAT SPACE is THE ORDER 
 
 OF COEXISTENCES 96 
 
x CONTENTS 
 
 CHAPTER VII 
 
 PAGE 
 
 THE MODERN SCIENTIFIC REVOLUTION AND ITS 
 
 LEADERS - - 119 
 
 CHAPTER VIII 
 
 CONCLUSION : IN WHAT SENSE is THE UNIVERSE 
 
 INFINITE?- - - 153 
 
 INDEX - -i 164 
 
CHAPTER I 
 SPACE, TIME AND MOVEMENT 
 
 THE new theory of Einstein which is known as 
 the general principle of relativity is perfectly 
 simple when once it is understood and peculiarly 
 difficult to understand. This arises from the 
 fact that the human mind, in its ordinary attitude 
 of reflection, and particularly in its well-balanced 
 moods, subject to reason and superior to emotion, 
 is always ready to revise its conclusions. When, 
 however, it is required not merely to revise 
 its conclusions but actually to amend its 
 premises, a kind of mental giddiness is experi- 
 enced, a feeling of insecurity as though the 
 firm ground on which its conclusions are based 
 and from which they derive their whole strength 
 had begun to shake and prove unstable. The 
 wonderful structure of physical science, with the 
 assurance consequent on the continual progress 
 and constant acceleration of its advance in the. 
 
RELATIVITY 
 
 last two centuries of the modern period, seems 
 in jeopardy the moment real doubt is thrown on 
 the concepts of absolute space and time and 
 movement, which appear as its conditions. It is 
 because these concepts are rejected by the new 
 principle that the revolution in science is so 
 profound and far-reaching. 
 
 Space, time and movement seem direct 
 self-revealing realities and to the ordinary 
 man the necessity of having theories about them 
 is difficult to appreciate. There are indeed, as 
 everyone knows, puzzling psychological problems 
 and even perplexing philosophical questions con- 
 cerning them, but these all seem, when we reflect 
 on them, to concern wholly and solely our 
 knowledge, and the mistakes and illusions which 
 may arise in regard to our knowledge. As to 
 the realities themselves they present themselves 
 as the simple and obvious framework of the 
 objective world of our daily experience and as 
 the subject-matter of mathematical and physical 
 science. We may know perfectly well that 
 many philosophers, following Kant, have held that 
 space and time are forms of perception which 
 the mind possesses as pure a -priori cognitions. 
 But then this is a theory of knowledge, and the 
 
BASIC CONCEPTS 3 
 
 conclusion which Kant drew from it, that know- 
 ledge is of phenomena and not of things in 
 themselves, leaves the whole reality of physical 
 science unaffected. We may know, too, that 
 some philosophers have denied the reality of 
 movement, while others have denied the reality 
 of everything which is not movement. But 
 such opinions are dismissed by us as logical 
 problems which concern meanings and which 
 leave the facts of experience unaltered. It is 
 therefore with considerable perplexity and with 
 unfeigned surprise that the scientific world 
 has received the evidence put forward, not by a 
 speculative philosopher but by a mathematician 
 and physicist, that our ordinary accepted notions 
 of space, time and movement do not correspond 
 with reality, and that the laws of nature require 
 to be all reformulated on a new principle which 
 rests primarily on the rejection of space and time 
 as constant factors. 
 
 To the metaphysician there is nothing sub- 
 versive or revolutionary in the new principle, it 
 is practically identical with principles which 
 have, time and again, been formulated in philo- 
 sophy, ancient and modern, but to the man of 
 science it seems like a sudden upheaval of the 
 
4 PRINCIPLE OF RELATIVITY 
 
 foundations on which the whole stupendous 
 structure of modern science has been reared. 
 
 Einstein's principle of relativity has two distinct 
 stages. The first formulation, in 1 905, expressed 
 the acceptance of the consistently negative results 
 of experiments contrived to determine absolute 
 velocity by reference to a fixed system at rest, such 
 as the ether of space was generally supposed to be. 
 If there is no zero system with reference to which 
 absolute velocity can be measured, we have to 
 correlate observations for systems moving rela- 
 tively to one another. The special principle of 
 relativity, or the restricted theory, is so called 
 because it applied only to uniform rectilinear 
 translations of reference-systems, and not to 
 rotations or non-uniform translations. The 
 special principle is that the velocity of the pro- 
 pagation of light in vacua is constant for every 
 observer, that it is unaffected by the translation 
 of a reference-system relatively to other systems, 
 and that the constancy of the velocity is main- 
 tained by a variation of space and time. In 
 1917 Einstein formulated the new principle of 
 generalized relativity. This was the extension 
 of the earlier principle to include the law of 
 gravitation and by implication all laws of nature. 
 
A NEW WORLD-VIEW 5 
 
 It accepted for all the laws of nature the impossi- 
 bility of any absolute standard of reference, and 
 it proposed to determine all universal laws as 
 observational facts to be deduced from the 
 movements of various systems relatively to one 
 another. It involved the rejection of Newton's 
 concept of the attraction of masses acting at a 
 distance from one another in a uniform space 
 and even _flowing._ drag, and the denial that any 
 spatial or temporal dimensions are uniform and 
 absolute for all systems of movement. It also 
 rejected the postulates of Euclid as impracticable. 
 To make the full significance of this new 
 principle appear and to show its philosophical 
 importance in the world-view it discloses to 
 us, is the aim of the present historical study. 
 It will be sufficient, before trying to follow the 
 problem from its origin, to indicate clearly the 
 two facts which the special and the general 
 principle take to be conclusively established. 
 They are both negative facts, and therefore have 
 none of the simplicity of new discovery of the 
 hitherto unknown. They do not give us new 
 notions but they upset our old notions and 
 complicate and render difficult the necessary 
 reconstruction of the world-view. 
 
6 PRINCIPLE OF RELATIVITY 
 
 The first fact is that the velocity of light is a 
 finite velocity, and yet absolutely uniform for 
 every observer, whatever the velocity of his 
 system, and whatever the direction of its move- 
 ment of translation relatively to other systems. 
 The nature of light is not in question, for 
 whether we accept the corpuscular or the undu- 
 latory theory, we know that light is propagated 
 in a movement radiating outwards in every 
 direction from its source, which is thus always 
 the centre of a sphere. The velocity of the 
 propagation of light in empty space was dis- 
 covered in the early part of the seventeenth 
 century, when the telescope revealed the moons 
 of Jupiter and enabled calculations to be made 
 by comparing the time table of the satellites for 
 the planet when at its nearest point and when at 
 its most distant. The interval of time and the 
 distance traversed, both being known, gave the 
 velocity of light. It is a velocity which for all 
 terrestrial distances is negligible, it only becomes 
 of account in the great spatial intervals which 
 separate planetary and stellar masses. We have 
 no other means than that of light signals to 
 enable us to determine the simultaneity of events, 
 and yet light signals are themselves subject to 
 
THE VELOCITY OF LIGHT 7 
 
 the interval required for their transmission between 
 the observers at distant points who are using 
 them. Now, if space and time are absolute, as 
 we ordinarily suppose, then when distances and 
 intervals are varying by reason of the movements 
 of the observers relatively to one another, it is 
 quite clear and evident that the velocity of light 
 for the observers must vary correspondingly. 
 But experiments specially designed for the purpose 
 have proved conclusively that the velocity of 
 the propagation of light does not vary, it is 
 uniform for all observers whatever the relative 
 movement of the systems in which they are 
 situated. Let us take an extreme case and 
 suppose that two observers of the same events are 
 in different systems of reference, and that each 
 observer, thinking himself at rest, sees the other 
 system moving with a translation of 100,000 
 miles a second, that is, rather more than half the 
 velocity of light in empty space. Now it would 
 be rational to conclude, and we should naturally 
 expect to find, that if these two observers com- 
 municated with one another by light signals, the 
 velocity of the propagation of the light signal 
 would be more than twice as great, in the direction 
 of the uniform movement, for the one observer 
 
8 PRINCIPLE OF RELATIVITY 
 
 as it would be for the other. This is found not 
 to accord with experimental fact. So the principle 
 of relativity declares that the velocity of light is 
 constant, however the conditions of the observer 
 vary by reason of the translation of the system, 
 and that space and time are different for different 
 systems. To ordinary reason this is a paradox. 
 Einstein has accepted the experimental proof 
 without any attempt to explain it away as appear- 
 ance or illusion. He formulated the principle of 
 relativity to accord with the result of the experi- 
 ments. The principle is then, that the velocity 
 of light is constant and that space and time are 
 variable. I am not at present inviting attention 
 to, or challenging criticism of, the evidence for 
 this fact, so subversive of ordinary ideas and up- 
 setting to our habits, I am trying only to state as 
 definitely as possible what the fact is. Certainly 
 in the case of the enormous velocity of light and 
 the infinitesimal fraction of it represented by any 
 known velocity of translation, the fact, if we accept 
 it, is negligible as applied to our common terrestrial 
 life, but it is very difficult indeed to reconcile with 
 our experience of velocity generally. Sound, for 
 example, is a propagated movement, but when 
 the source of sound is moving with us, as when 
 
CONSTANT VELOCITY 9 
 
 we are talking in an open motor-car, we naturally 
 adapt ourselves to the idea that the sound waves are 
 not spreading with equal velocity forward and 
 backward. We think, wrongly perhaps, that the 
 waves of sound, when we in the car are moving 
 in their direction, spread out from the car at a 
 lower velocity than when we are not moving with 
 them. The special or restricted principle of 
 relativity then is, that the velocity of light is 
 constant for all observers and independent of their 
 system of reference, and that space and time are 
 variable, dependent on the relative translation of 
 systems. 
 
 The general theory of relativity goes much 
 further. It extends the principle to all the laws 
 of nature. It rests upon a fact or rather upon 
 a negative discovery, a discovery which is not 
 due as in the case of special relativity to definite 
 test experiments but the result of the successful 
 application of the principle to the formulation of 
 a new law of gravitation. The proof of the new 
 principle rests on the fact that it has been found to 
 account for a well-known discordance between the 
 astronomical calculation for the precession of the 
 perihelion of Mercury and the actual observation, 
 which had previously baffled all attempts to 
 
io PRINCIPLE OF RELATIVITY 
 
 explain. Further, it enabled a prediction to be 
 made as to the deflection of the light from a star 
 passing near the sun during a total eclipse, which 
 was verified in the observations of the eclipse of 
 the sun in May 1919. A further prediction 
 by Einstein that the spectroscopic analysis of 
 atoms vibrating in the gravitational field of the 
 sun compared with the analysis of similar atoms 
 on the earth would show a shifting of the lines 
 towards the red end of the spectrum has at 
 present not been verified and is the subject 
 of research. It is not, however, with the 
 details of these tests of the principle which I 
 am now concerned. I want rather to make 
 plain the fact which is alleged as the basis of 
 the new theory. As applied to the new theory 
 of gravitation it is called equivalence. 
 
 If we raise an object and then release it, it 
 drops. We explain this as an instance of a law 
 of gravitation by which bodies attract one another 
 in a definite relation of their mass and distance. 
 We regard the floor as fixed in relation to the 
 earth, and the released object falls to it, drawn, we 
 say, by the attraction of the earth. But the earth, 
 to which the floor and the room are attached, 
 is rotating on its axis; it is also travelling on its 
 
EQUIVALENCE 1 1 
 
 orbit at many miles a second; and the whole 
 solar system is moving in the stellar system. 
 It is clear, therefore, that there might be an 
 observer who would say that the released object 
 remained at rest and that the floor of the room 
 moved to it. The theory of equivalence is 
 that there is no way of deciding between the 
 alternative descriptions, whether in fact the 
 object fell to the floor or the floor rose to the 
 object. If one observer had the right to decide 
 positively for the one, another observer would have 
 the equal right to decide positively for the other. 
 If the principle be accepted, it completely negatives 
 the idea that forces of attraction are exercised by 
 bodies on one another in the sense supposed in 
 Newton's law. 
 
 If this negative fact be established, namely, 
 that there is no way of determining the actual 
 line which two objects follow in their movement 
 towards one another, and that contradictory de- 
 scriptions of such movement are really equivalent, 
 it follows that space cannot have the properties 
 which Euclid required, and force cannot have 
 the nature which Newton supposed. The 
 discovery can only be compared in importance 
 with the discovery of Copernicus that the 
 
12 PRINCIPLE OF RELATIVITY 
 
 earth is not at rest but undergoing a diurnal 
 rotation on its axis and an annual revolution 
 round the sun. 
 
 Space and time are concepts of the mind. 
 They are indeed for all of us realities with which 
 we feel we are in direct relation, a relation so 
 fundamental that our whole existence depends on 
 it. But space and time in themselves, though 
 not abstractions, lack the concreteness of objects 
 and events. They are a framework of the 
 physical universe and give form and con- 
 tinuity to its content. As concepts they are 
 judged by their consistency or inconsistency. 
 The dominant place they occupy in philosophy, 
 and the persistence of the problems they give 
 rise to throughout the whole history of philosophy, 
 ancient and modern, are due to the inherent 
 logical and metaphysical difficulties they present. 
 But space and time are not only concepts, they 
 are also images. In studying the theories of 
 space and time it is very important to take into 
 account the imagery which supplies to the 
 concepts their content. It is usual to neglect 
 this completely. The reason is that philosophers 
 never reveal the imagery which lies behind and 
 supports the concepts they analyse. For imagery 
 
IMAGES AND CONCEPTS 13 
 
 we have to go to the poets. Part passu with the 
 evolution of the concepts of space and time has 
 been an evolution of their images. Homer, 
 Dante and Milton are as distinct in the imagery 
 of their expression of a world-view as Aristotle, 
 Aquinas and Leibniz are in their concepts of its 
 reality. Every philosopher starts his reflection 
 from the stand-point of his world-view. The 
 world-view is an imaginative background of his 
 thoughts, his reflections borrow their shape and 
 draw their content from it, revolve round it and 
 always return to re-form it. But when we study 
 a philosopher's theories we treat them in the 
 mathematical method, substituting signs for 
 images. We suppose there is a special advantage 
 in this power of detaching the sign completely 
 from the image in which it arose. As soon 
 as we grasp a man's concept we adapt it 
 to our own imagery, whatever it may be, 
 and proceed as though the world-view were of 
 no importance. A familiar illustration is the 
 way in which the Bible is interpreted in Christian 
 households. The concepts are detached from 
 the imagery of the writers and fitted on to the 
 homely imagery of the reader whatever it may 
 be. We study in the philosophers their logical 
 
i 4 PRINCIPLE OF RELATIVITY 
 
 principles and abstract concepts, in the poets 
 their imagery, we forget that the poets express 
 the imagery which the philosophers require to 
 embody their concepts. If we would reconstitute 
 the thought of an historical period we must 
 read its poetry in conjunction with its philosophy. 
 When we discuss to-day the theories of Newton, 
 we take no account of the world-view which 
 presented itself to him and of its complete differ- 
 ence from our ordinary world-view to-day. Our 
 world-view is continually changing, and the 
 imagery in which we clothe it becomes outworn 
 and cast aside. How completely different, for 
 example, is the world picture presented to us in 
 Mr. Wells 's Outlines of History from anything 
 which filled the imagination of a previous genera- 
 tion. I have chosen Newton as an illustration 
 because we are accustomed to accept his concepts 
 as essentially modern. Science has advanced, 
 but his concepts remain of universal application. 
 Newton 's age is so near our own, as compared 
 with the Greek and Mediaeval ages, that we 
 hardly appreciate how much its imagery has 
 changed. Yet how fantastic the world-scheme 
 of Milton's Paradise Lost appears to us to-day 
 and how inadequate his imagery to embody 
 
IMAGERY 15 
 
 modern concepts. It was, however, the familiar 
 background of Newton's thoughts. 
 
 " Now had the Almighty Father from above, 
 From the pure empyrean where he sits 
 High throned above all height, bent down his eye, 
 His own works and their works at once to view. 
 
 On Earth he first beheld 
 Our two first parents, yet the only two 
 Of mankind, in the happy garden placed, 
 Reaping immortal fruits of joy and love. 
 
 He then surveyed 
 
 Hell and the gulf between, and Satan there 
 Coasting the wall of heaven on this side night 
 In the dun air sublime, and ready now 
 To stoop with wearied wings, and willing feet, 
 On the bare outside of this world, that seemed 
 Firm land imbosomed without firmament, 
 Uncertain which, in ocean or in air." 
 
 Unlike Dante's world, heaven and hell have 
 no direct connexion with our universe, which is 
 conceived as a system of sun and planets swinging 
 in vast space, yet an ordered system with laws of 
 nature imposed upon it. It is a new creation, 
 espied from afar by Satan, and offering, in its 
 order and arrangement, rest for wearied wings 
 and a sphere for concerted action. But what 
 Strikes us particularly in such imagery, as 
 
1 6 PRINCIPLE OF RELATIVITY 
 
 compared with that which we should now deem 
 adequate, is that the distant observer surveying 
 our world sees it as it appears to us and makes 
 no allowance for systems of reference. Our 
 spatial and temporal coordinates are also those 
 of God and of Satan. This was essentially 
 Newton's view. 
 
 The importance of imagery and the way in 
 which it qualifies concepts may be illustrated 
 also in a somewhat different way. Take the case 
 of the familiar phenomenon of the ebb and flow 
 of the tide which we explain by the concept of 
 gravitation. For us the tides mean an alternate 
 rise and streaming of the water in one direction 
 and a fall and streaming in the reverse direction, 
 with all its minute and dependent circumstances. 
 To an outside observer the tide would mean only 
 the unalterable shape a plastic body in rotation 
 would assume in spite of the changing position 
 of the mass. 
 
 Throughout the whole history of human 
 thought, while imagery and concepts have been 
 changing continuously, the fundamental notions 
 of space and time and movement, both as being 
 direct data of experience, and necessary conditions 
 of experience, have withstood all change. They 
 
THE PSYCHOLOGICAL PROBLEM 17 
 
 appear to us as the framework of our universe, 
 whatever the content and the nature and the 
 history of that universe be. And yet from the 
 very beginning of our historical records of 
 human reflective thought, everyone who has 
 turned his thoughts upon them has found that 
 they present insoluble problems and offer the 
 strangest paradoxes. 
 
 Neither our images nor our concepts of space 
 and time are identical with anything spatial 
 or temporal which we perceive. It is from this 
 incongruence of percepts and concepts of space 
 and time that the psychological problems in 
 regard to them arise. Space is imaged either 
 by its negative character as the void or by its 
 positive character as extension. But neither 
 void nor extension is direct experience or a datum 
 of sense-intuition. Although space and time are 
 intimately bound up with all sense experience, 
 there is no actual sense experience of space and 
 of time. We cannot, for example, satisfy in 
 regard to the ideas of them a demand such as 
 Hume proposed for a universal test, produce the 
 impression which has given rise to the idea. Of 
 space and of time there are no impressions. A 
 still more surprising and even disconcerting 
 
1 8 PRINCIPLE OF RELATIVITY 
 
 fact is that while image and concept of space 
 and time fulfil completely the Euclidean postu- 
 lates and conform exactly to the axioms, not one 
 of our senses gives us spatial and temporal 
 experience conformable to those conditions. In 
 his New Theory of Vision Berkeley proved that 
 the sense of sight cannot yield a perception of 
 distance or give us knowledge of the third 
 dimension of space, and based on this the theory 
 that visual perceptions are a language of signs, 
 the purpose of which is to enable us to anticipate 
 tactile sensation. But tactile sensation will 
 not, any more than will visual sensation, give 
 us knowledge of distance, such knowledge depends 
 on movement, and movement involves time as 
 well as space. If so, then what is the absolute 
 standard by which we are to measure time ? 
 Try in what way we will, we can never by direct 
 perception arrive at the notions of absolute space 
 and time which yet we imagine and conceive to 
 be the basis of the reality of nature. 
 
 This is no new discovery. It is indeed a 
 commonplace of philosophy and even of the 
 modern science of psychology. One of the 
 large problems in contemporary psychology con- 
 cerns the nature and origin of the perception of 
 
THE PERCEPTION OF SPACE 19 
 
 space. There are numerous theories, which fall 
 however into two main groups. They are named 
 the genetic and the nativistic theories. The genetic 
 theories derive our notion of space from sense 
 experience which is not itself spatial, by means 
 of inference and mental construction. The 
 nativistic theories, on the other hand, derive it 
 from the mind itself and the mode of its activity 
 in experience. 
 
 A genetic theory has been held by most of the 
 older, as well as by many of the present, generation 
 of modern psychologists. An illustration of it 
 is the theory expounded by Herbert Spencer 
 (Principles of Psychology ', ii. 178), according to 
 which the perception of space is simply an 
 interpretation of the simultaneity of sensations, 
 explained physiologically in the case of sight 
 by the overlapping of successive stimuli on the 
 retina and in the case of touch by the reversibility 
 of series of tactile impressions. Another illustra- 
 tion is the well-known local-sign theory of Lotze. 
 The local sign is not a localization or extension 
 in the sensation itself, but a character belonging 
 to tactile impressions which later causes the 
 mind to locate them in particular points of the 
 body. It is from these impressions that our 
 
20 PRINCIPLE OF RELATIVITY 
 
 mind is supposed by the theory to construct the 
 perception of space. 
 
 An example of nativistic theory is the view 
 expounded by William James in Principles of 
 Psychology (vol. ii. p. 134^".) that there are 
 sensations to which the character of voluminous- 
 ness distinctly belongs and which are thereby 
 able to give the mind direct perception of space. 
 This character, called by other psychologists 
 extensity, is not extension, that term being only 
 applicable to physical objects. Extension is a 
 sensible quality, extensity is a character of sensa- 
 tions. 
 
 It is not then in philosophy nor in the science 
 of psychology that the principle of relativity is 
 revolutionary. It is only a revolution in physical 
 science, and it is a complete revolution in science, 
 because mathematics and physics have seemed 
 justified in rejecting, as outside their sphere and 
 completely indifferent to them, the problem of 
 the relation of the mind to its objects. The 
 objective character of physical science, upon 
 which it has prided itself, has therefore come to 
 mean the uncritical assumption of absolute space 
 and time. The introduction into pure mathe- 
 matics or into pure physics of a subjective 
 
SUBJECTIVISM 21 
 
 element seems not only a sacrilege but a downright 
 betrayal of the very principle on which science is 
 based. It has been supposed that in its purely 
 objective basis lies the strength of physical 
 science and that to this objective basis is due the 
 steady and rapid and continuous progress which 
 is often vaunted as presenting a favourable con- 
 trast to speculative philosophy. 
 
 When the principle of relativity was first 
 formulated it was generally put forward as a 
 methodological principle applicable only within 
 the sciences concerned and with no relation 
 whatever to any question of general philosophical 
 or metaphysical theory. It simply, it was said, 
 proposed a reform of mathematical procedure, 
 a reform which was radical indeed, for it involved, 
 not the correction or improvement of the accepted 
 equations, but a new set of equations involving 
 new constants and new variables. The general 
 principle of relativity now proposed by Einstein 
 is acknowledged, however, to concern the most 
 fundamental philosophical concepts of the nature 
 of the universe. The essence of it is to introduce 
 the bane of the physicist, subjectivism, into the 
 arcana of physical science. It shows that it is 
 impossible to abstract from the mind of the 
 
22 PRINCIPLE OF RELATIVITY 
 
 observer and treat his observations as themselves 
 absolute and independent in their objectivity. 
 It requires us to give up the assumption of an 
 absolute standard of reference for the measure- 
 ment of the velocity of a system. It rejects 
 the inference, which all our experience and all 
 our science has seemed with such increasing 
 assurance to affirm, that beneath the objects 
 we perceive, juxtaposed in the external world, there 
 is an absolute space which would be void, but not 
 abolished, if they were removed, and that behind 
 the events which succeed one another in our 
 consciousness, there is an absolute time which 
 might lose all distinction if there were no events, 
 but which would still flow. We are to reject 
 this inference not because it is found to be 
 useless, not because pure space and pure time 
 are undiscoverable, not because we can never 
 by direct perceptive means become acquainted 
 with them, but because physical experiments 
 which ought to have revealed them if they exist, 
 have uniformly failed to do so. The new 
 principle is not a belated discovery of our 
 ignorance ; it is a new advance in positive 
 knowledge. In this lies its strength. The study 
 of nature has revealed to us that the nature we 
 
SYSTEMS OF REFERENCE 23 
 
 study is not independent of the mind which 
 studies it. There is no absolute physical reality 
 which a mind may contemplate in its pure 
 independence of the contemplator and the con- 
 ditions of his contemplation. The new principle 
 is that every observer is himself the absolute, 
 and not, as has been hitherto supposed, the 
 relative, centre of the universe. There is no 
 universe common to all observers and private to 
 none. The work of physical science is to co- 
 ordinate the observations of observers, each of 
 whom uses his own co-ordinates and for whom 
 there is no common measure. 
 
CHAPTER II 
 
 \ ' '' 
 
 THE ANTINOMY OF MOVEMENT 
 
 ARISTOTLE in the Physics (vi. 14) says, that Zeno 
 committed a fallacy when he argued : " If 
 everything in order to be, must, whether moving 
 or at rest, occupy an equal space, and if a body 
 when displaced occupies at every moment an 
 equal space, then it follows that the flying arrow 
 is immobile." It is an error, Aristotle argues, 
 because time is not composed of moments, that 
 is~ of indivisibles. Neither indeed, he adds, is 
 any other magnitude/ 
 
 Whether or not Aristotle's refutation of Zeno's 
 argument is sound, it is certain that philosophy 
 generally has not found' if possible to dismiss 
 the problem of movement in. : this summary way. 
 Many philosophers indeed have been equally 
 confident, but a glance at the history of philosophy 
 shows the problem cropping up in some form in 
 
ZENO'S ARGUMENTS 25 
 
 every stage of the evolution of the concept of 
 metaphysical reality. 
 
 Zeno's famous arguments against movement 
 are four in number, and together they are so 
 compact that those who would refute them look 
 in vain for a logical loophole. The first declares 
 that it is impossible that a body can move from 
 one point to another distant from it, because, 
 before .it can traverse the whole intervening 
 space it must pass through half, and before it 
 can traverse that half, the half of the half, and 
 so on, to infinity. The second is that Achilles 
 in his race with the tortoise can never overtake 
 it, if it is allowed to have a start, for to do so he 
 must first reach the point at which the tortoise 
 is, but when he reaches it the tortoise will have 
 moved on, and Achilles, therefore, will have always 
 a step to take. The third is that the flying arrow 
 does not move because at every moment it is 
 .at rest. The fourth is that if there are three 
 processions in the stadium, each composed of 
 equal numbers and equal masses, one of which 
 remains stationary while the other two move 
 with an equal velocity but in an opposite parallel 
 direction, passing the first in mid course, then it 
 follows that each moving procession will traverse 
 
26 ' PRINCIPLE OF RELATIVITY 
 
 an identical space in a time which will be both 
 half and double of itself. 
 
 The last of these arguments can be made 
 quite clear in a diagram. Let us suppose 
 AiAtdtAt, B^BsBt and C 1 C 2 C 3 C 4 to be the 
 three processions. Let us suppose their first 
 position to be 
 
 The A' s are stationary, the B's are moving to the 
 right, the C's to the left. When then B^ reaches 
 AI , Cj will reach A , and their position will be 
 
 But in reaching this position the C's will have 
 been consecutively in line with all the B's and 
 with half the A's, and the B's will likewise have 
 been in line with all the C's and with half the 
 A's. But B's and C's and A's occupy equal 
 spatial magnitudes. The difference therefore 
 is not in the space. The time also is identical 
 for it is one and the same interval, yet it is only 
 half for the B's and C's what it is for the A's 
 
POINTS AND INSTANTS 27 
 
 the half, therefore, is identical to the whole, or 
 the time is the double of itself. 
 
 The argument may be put in another form 
 which perhaps is even more perplexing. Suppose 
 the processions to be points and the succession 
 instants, that is, suppose the divisions of the 
 movement to be units of time and space. Suppose 
 the position at a first instant to be 
 
 CiCfiCt 
 
 and at the second instant (when the B's have 
 moved one point to the right, the C's one point 
 to the left) 
 
 Then at the first instant C 4 is in line with B^ ; 
 at the second instant it is in line with B 3 , but 
 it must have passed # 2 , and there is no instant 
 in which it could have been in line with B 2 . Also 
 5 4 is at one instant in line with C\ and at the next 
 with C 3 but C 2 lies between, when was $ 4 in 
 line with C 2 ? 
 
 Aristotle's refutation of this fourth argument 
 
28 PRINCIPLE OF RELATIVITY 
 
 is of particular interest. * The fallacy consists 
 in supposing that the equal magnitude, possessing 
 the same velocity, moves in the same time both 
 relatively to a mass in movement and relatively 
 to a mass at rest ; therein lies the error " (Physics, 
 vi. 14, 10). By this he appears to mean that 
 while mass and velocity of a moving body remain 
 constant, the time it takes to pass a body at rest 
 and a similar body in movement is not the same. 
 This might be interpreted as an anticipation of 
 the principle of relativity so far as time is con- 
 cerned, but clearly the very opposite is intended. 
 Aristotle means that time is absolute and that 
 less of it is occupied in passing a mass at rest than 
 in passing an equal mass moving parallel and 
 opposite to it. This, however, leaves Zeno's 
 argument unanswered, merely affirming what 
 Zeno supposes to be affirmed. Zeno says in 
 effect that if movement is real and a body passes 
 from point to point, from moment to moment, 
 then you are committed to the contradictory and 
 absurd assertion that the same time is different. 
 Zeno lived in the fifth century before Christ, 
 the century which preceded the great philoso- 
 phical enlightenment represented by Socrates, 
 Plato and Aristotle. He was a pupil of 
 
BECOMING AND BEING 29 
 
 Parmenides, the head of the famous Eleatic 
 school of philosophy. The rival Ionic school had 
 as its founder Heracleitus of Ephesus. The two 
 schools represented opposite and contradictory 
 principles. According to Heracleitus " becom- 
 ing," according to Parmenides " being," is the 
 first principle of existence. There is a curious 
 outward resemblance between these early specu- 
 lations and those of modern transcendental 
 philosophers. The resemblance is in the con- 
 cepts, and it is a striking illustration of the way 
 concepts abide identical throughout all change 
 of imagery. Moreover, first principles present 
 themselves to reflection as essentially simple 
 and extremely general. It was, however, in 
 their successors that the doctrines of the great 
 founders developed into paradox. Thus the 
 doctrine that all things flow, that reality is uni- 
 versal becoming, was developed into complete 
 paradox by Cratylus, as related by Aristotle in 
 the following description of the Heracleiteans. 
 " And again they held these views because they 
 saw that all this world of nature is in movement, 
 and about that which changes no true statement 
 can be made ; at least, regarding that which 
 everywhere in every respect is changing, nothing 
 
30 PRINCIPLE OF RELATIVITY 
 
 could truly be affirmed. It was this belief 
 which blossomed into the most extreme of the 
 views above mentioned, that of the professed 
 Heracleiteans, such as was held by Cratylus, 
 who finally did not think it right to say anything 
 but only moved his finger, and criticized Hera- 
 cleitus for saying that it is impossible to step 
 twice into the same river ; for he thought he 
 could not do it even once " (Metaph. iv. 5). It 
 was this doctrine which Zeno combated. 
 
 No one will understand Zeno's arguments 
 who regards him as merely a skilful dialectician 
 and ignores the essential fact that he had reached 
 independently the conclusion that movement is 
 not reality but appearance and used the arguments 
 to enforce it. The arguments therefore are not 
 sophisms nor exercises in logomachy. If you seek 
 his own solution of his paradoxes, it is quite simple. 
 He held that nothing moves, that reality is one 
 and unchangeable. 
 
 It should be noticed that the four arguments 
 are cumulative in force. The first shows move- 
 ment to be impossible, the second shows it to 
 be unreal, the third, contradictory, and the fourth, 
 absurd. The first deals only with space, and 
 the infinite divisibility of space is made the 
 
THE ARGUMENTS NOT SOPHISMS 31 
 
 obstacle of movement. The second shows that 
 if movement be supposed actually in progress, 
 contradiction breaks out in the concept of velocity. 
 In the third, discrete points in space are correlated 
 with discrete instants of time, and the contra- 
 diction lies in the attempt to correlate the passage 
 from one point to the next with the passage 
 from one instant to the next. It involves the 
 paradox that the arrow is somewhere at no time 
 or nowhere at some time. The fourth combines 
 all the other three, for it takes into account the 
 space, the time and the movement, and it shows 
 that measured by points and instants velocities 
 are infinitely different and all equal. 
 
 This was Zeno's problem. It is a pro- 
 blem, therefore, which has its origin in the 
 early Greek nature speculations in which the 
 development of Western philosophy takes its 
 rise, and it is a problem which has persisted 
 throughout the whole of that development and 
 is an unsolved problem to-day. The form, 
 however, has changed. It is as an antinomy of 
 reason that it presents itself to us. No one 
 to-day, even if he argues, as Mr. F. H. Bradley 
 does, that movement is appearance and not 
 reality, is content with the simple denial 
 
32 PRINCIPLE OF RELATIVITY 
 
 of movement and the affirmation of the un- 
 changeable one. The antinomy in the concept 
 of movement consists in the fact that the thesis 
 which affirms it, and the antithesis which denies 
 it, present themselves to the mind as equally 
 valid ; yet they are mutually self-contradictory. 
 The thesis is : There are movements, for reality, 
 the reality of life in particular, denotes activity ; 
 a thing is what it does. The antithesis is : 
 There are no movements, for a condition of 
 movement is that a thing which moves shall 
 endure unchanged throughout the movement ; 
 but if nothing changes nothing moves. 
 
 The antinomies of reason were made by Kant 
 the central point of interest in the modern philo- 
 sophical problem, so far as it concerns the 
 basis of physical science. According to Kant's 
 theory antinomies arise when the mind makes 
 an object of the whole series of conditions which 
 constitute the system of the world. It is the 
 nature of the mind to present to itself such an 
 object, but the world so presented is an object 
 of reason, not an object of sense intuition nor 
 of understanding. The object of reason is an 
 idea of the unconditioned, it transcends any 
 possible experience and as thing in itself is 
 
THE ANTINOMIES OF KANT 33 
 
 unknowable. The objects of reason give rise to 
 Ideas (the soul, the world, and God), which have 
 an important function in theory of knowledge, 
 but they are not objects of which we can possess 
 any empirical knowledge. Our interest in them, 
 and their value to us, is practical not speculative. 
 We only know phenomena, not things in them- 
 selves. 
 
 The antinomies of Kant give us, then, in modern 
 form, the contradictions which lie concealed, or 
 which if known are consciously ignored, in our 
 ordinary common-sense concepts of space, time 
 and movement. Two of the four antinomies, 
 which Kant distinguished as mathematical from 
 the other two as dynamical, are directly concerned 
 with these concepts. The first deals with the self- 
 contradiction involved in thinking of the world 
 as limited or as unlimited in space and time. 
 The thesis is : The world has a beginning 
 in time, and is also limited in regard to space. 
 And the antithesis is : The world has no beginning 
 and no limits in space, but is in relation both 
 to time and space infinite. This antinomy ex- 
 presses a difficulty which occurs to everyone in 
 moments of reflection. It is impossible to think 
 that the world had no first moment, for in that 
 
34 PRINCIPLE OF RELATIVITY 
 
 case how are we to represent the actuality of 
 the present moment, for that moment is a now 
 which ends a series, and its reality therefore 
 seems to depend on a now which began the 
 series ? But then, how on the other hand can 
 we present to the mind a moment to which 
 there is only an after and not a before ? Similarly 
 in regard to space. There is a point " here " 
 which has definite relations to the whole extended 
 universe. The reality of these relations limits 
 the universe. Yet how can we think limits to 
 the universe without in the very thought suppos- 
 ing an extension outside the limits ? 
 
 There are two contemporary philosophers, 
 Mr. Bertrand Russell and M. Bergson, who 
 have analysed Zeno's arguments in their original 
 simplicity as the denial of the reality of movement. 
 Mr. Bertrand Russell (Principles of Mathematics, 
 chap, xlii., and Our Knowledge of the External 
 World) chap, v.) holds that Zeno is right, but 
 that the paradoxical character of the arguments 
 entirely disappears when they are expressed in 
 terms of the modern mathematical theory of 
 infinity. M. Bergson (Creative Evolution, pp. 
 325-330 and Time and Freewill, chap, ii.) holds 
 that Zeno's conclusion is wrong in so far as it 
 
MATHEMATICAL CONTINUITY 35 
 
 denies the reality of movement, and that his 
 paradox is due to confusion between a reality in 
 its essential nature indivisible and the intellectual 
 device of a scheme, created and contrived for the 
 practical purpose of division and articulation. 
 The two modes of analysing the old argu- 
 ment and the antithetical conclusions they reach 
 reveal that two principles are contending in 
 philosophy to-day, recalling in a striking way 
 the principles which divided the ancient world, 
 the principle of the unchangeable one and the 
 principle of the universal flow. 
 
 Mr. Russell maintains that the paradox is 
 completely solved by the philosophical theory of 
 mathematical continuity. According to this 
 theory space and time actually consist of discrete 
 points and instants, but in any finite portion of 
 space and interval of time the number of points 
 and instants is infinite. In an infinite series no 
 two members are next one another, for between 
 any two there is always another. When accord- 
 ingly space is conceived as infinitely divisible, 
 this means that the series of points is compact, 
 there is no interstice between one and another. 
 Yet, though there is nothing between the points 
 but points, the points are not next one another, 
 
3 6 PRINCIPLE OF RELATIVITY 
 
 there is no next point to any point. The infinite 
 divisibility of time implies the same of the 
 instants. Having defined continuity in this way 
 it is claimed that all the supposed contradictions 
 in a continuum composed of elements are com- 
 pletely swept away and the foundation laid bare 
 of a reality on which a firm constructive philosophy 
 can be built. The answer then to Zeno is as 
 follows. Zeno asks how can you go from one 
 position at one moment to the next position 
 at the next moment without in the transition 
 being at no position at no moment ? The answer 
 is that there is no next position to any position, 
 no next moment to any moment because between 
 any two there is always another. If there were 
 infinitesimals movement would be impossible, 
 but there are none. Zeno therefore is right in 
 saying that the arrow is at rest at every moment 
 of its flight, wrong in inferring that therefore it 
 does not move, for there is a one-one corres- 
 pondence in a movement between the infinite series 
 of positions and the infinite series of instants. 
 According to this doctrine then it is possible to 
 affirm the reality of space, time and movement, 
 and yet avoid the paradox in Zeno's arguments. 
 Bergson's way of escape from the paradox is 
 
THE TRUE DURATION 37 
 
 entirely different, for it rests on a metaphysical 
 concept of life and a philosophical theory of the 
 nature of the intellect. It does not depend on 
 the mathematical definition of continuity, for 
 mathematical continuity has no relevance to the 
 problem. I mean that as Bergson presents the 
 problem it is indifferent how we describe, or 
 in what terms we define, the continuity of space 
 and time, because it is space and time them- 
 selves which are wrongly apprehended. They 
 belong essentially to the intellectual view of 
 reality, while movement as true duration or 
 change is the fundamental reality of life. 
 Take the points and instants of space and 
 time as the elements composing the movement 
 and you will be forced to the conclusion that 
 there is no movement, for the elements are 
 immobilities and movement cannot be generated 
 out of immobilities. But there are real move- 
 ments, and the immobilities into which we seem 
 able to decompose them are not constituents of 
 the movement, they are views of it. 
 
 There are thus two solutions of the antinomy 
 offered to us in contemporary philosophy. I 
 have not included Mr. Bradley's argument in 
 Appearance and Reality because it can hardly be 
 
38 PRINCIPLE OF RELATIVITY 
 
 classed as a solution. It founds an important 
 philosophical doctrine on the antinomy of move- 
 ment, but it does so by accepting the contra- 
 diction and not by resolving it. There is, 
 however, now offered to us a third and more 
 complete way of escape in the new principle of 
 relativity. This is in effect a reform of the 
 foundational concept of physical reality, and it 
 gives us a new world-view from which the 
 antinomy has disappeared without violence done 
 to reason, or to science, or to common-sense. 
 
 If we accept the terms of Zeno's argument 
 there is no escape from the conclusion, 
 and the only salvation from the antinomy lies 
 in successfully attacking the premises. This is 
 what Einstein's theory does. It rejects the 
 concept of absolute space and time. Space and 
 time are not independent of the observer, and 
 there exists no abstract spatio-temporal system 
 by reference to which the velocity, direction and 
 duration of a movement can be absolutely deter- 
 mined. Space and time are variable, and they 
 vary for each observer with his system of reference 
 and with every change in the acceleration of the 
 movement of that system relatively to other 
 systems. Our four-dimensional world preserves 
 
VARIABLE SPACE-TIME 39 
 
 its uniformity because our units of length, breadth 
 and depth and our unit of time, alter continually, 
 adapting themselves to the standpoint of an 
 observer at rest, or rather to the standpoint of 
 a system at rest relatively to the translation of 
 other systems. 
 
CHAPTER III 
 
 ATOMS AND THE VOID 
 
 WE have seen that in the speculations of the 
 early Greeks in nature-philosophy, two opposite 
 and contradictory principles emerged and divided 
 the schools into rival camps. One took " becom- 
 ing," the other " being " as the first principle 
 of existence. The conflict of these two principles 
 issued in the ancient world in the synthetic 
 construction of a system which has ever since 
 held sway over the human mind. This is the 
 atomic theory of Democritus, of Abdera in 
 Thrace, an older contemporary of Socrates, and 
 the first formulator of philosophical materialism. 
 In so far as the atomic theory is a science of 
 nature there is at every point, despite the enormous 
 advance of physical science in modern times, and 
 the development of means of extending our know- 
 ledge by experiment, a most striking consistency 
 
 between the old atomic theory and the new. 
 
 4 o 
 
PHILOSOPHICAL MATERIALISM 41 
 
 The theory of Democritus is the first attempt 
 of Western thought to present nature as a complete 
 self-contained system. It is a pure materialism 
 for it deduces the whole of the phenomena of the 
 universe, psychical as well as physical, mental 
 as well as bodily, internal and spiritual as well 
 as external and objective, from the concept of an 
 eternal and indestructible matter. There would 
 seem to be a bias towards materialism in the 
 nature of human intelligence, for nothing is 
 able to exorcise completely the hold which it 
 maintains over ordinary experience. Its prin- 
 ciple seems eminently rational, and it demands, 
 it would seem, continual and sustained effort 
 to maintain against it what we may have come 
 to regard as stronger reason. Yet although 
 materialism has always appealed to the human 
 intellect as rational and indeed as enforced in 
 some measure by every practical concern of life, 
 it has never held sway for long. Humanity 
 has revolted against it, sometimes with contempt, 
 generally with loathing, too often with passionate 
 hatred. The reason is not that it is irrational, 
 but that it has always seemed to destroy morality 
 at its roots and to sap the foundations of religion. 
 Yet to reject materialism on moral and religious 
 
42 PRINCIPLE OF RELATIVITY 
 
 grounds so far from serving philosophy is 
 disastrous to it. If materialism is condemned it 
 ought to be on philosophical grounds alone, and 
 if it be philosophically untenable, everything is 
 unfortunate which tends to conceal its weakness. 
 For my own part, I frankly confess, materialism 
 seems consistent with the highest ethical principles 
 and with the purest religion. I reject it solely on 
 philosophical grounds. Its essential principle not 
 only fails to satisfy me but stands opposed to what 
 appears to me the most obvious truth. Mind is 
 more than matter. In every respect and from every 
 standpoint mind is richer, fuller, greater, more 
 comprehensive. Any principle which proposes 
 to deduce that more from the less stands self- 
 condemned. Yet this is the essential principle 
 of materialism. Given something absolutely self- 
 identical and deprived of difference, materialism 
 declares that by mere external combination and 
 relation there will be produced the variety of the 
 universe including the spiritual values. According 
 to the ancient theory, indivisible atoms, identical 
 in everything but quantity and shape, by their 
 combinations and movements, were held to be 
 able to produce, and in fact had produced, the 
 infinite complexity of the universe. According 
 
LUCRETIUS 43 
 
 to the modern theory, simple elements, 
 reducible ultimately to single electrical charges, 
 by mere external combinations in atoms and 
 molecules, are thought to be able to give 
 rise to every form of reality, natural and 
 spiritual. 
 
 Our knowledge of Democritus is derived only 
 from references to him in the classical writings, 
 but a very complete account of his atomic theory 
 is enshrined in the great poem, De Rerum Natura, 
 of the Roman philosopher-poet Lucretius. In that 
 poem Lucretius has presented to us the philo- 
 sophy of Epicurus, a philosopher regarded by his 
 followers as divinely inspired and revered as the 
 founder of a religion, or at least of a philosophy 
 practised as a religious duty. Lucretius lived 
 in the first half of the century before Christ, and 
 therefore belongs to the last period of the Roman 
 Republic. Epicurus taught in Athens at the 
 end of the fourth and beginning of the third 
 century B.C. ; Democritus was a century earlier 
 still. The philosophy of Epicurus was an ethical 
 theory. He accepted and adopted the atomic 
 theory of Democritus as the scientific basis of 
 his ethics. Lucretius is a true poet, and the 
 science of nature which he has expounded in 
 
44 PRINCIPLE OF RELATIVITY 
 
 his poem is not directly intended for instruction 
 but to support a moral and religious argument. 
 He is moved by a deep love of nature and profound 
 pity for human misery and by a firm belief in 
 the power of philosophy to dispel the principal 
 evils in man's lot. The greatest misery which 
 humanity endures is not physical evil but mental 
 torture due to superstitious fear. Could a man 
 be convinced that the Gods have no interest in 
 human affairs and cannot intervene in the concerns 
 of his earthly life, could he moreover be assured 
 that death is a release and not the beginning of 
 imagined terrors, the two great hindrances to 
 human happiness would be removed. The 
 pleasure which every living creature craves for 
 as part of its nature could at least be enjoyed 
 unspoilt by the poison of superstition. For this 
 purpose he unfolds the philosophy of his almost 
 divine master, and the poem, from the invocation 
 to Venus, not only as goddess of love but as 
 the goddess who has some influence over the 
 cruel God of war, to the close with its terrible 
 description of the plague in Athens, is inspired by 
 a melancholy and deep yearning to alleviate the 
 miserable lot of mankind by an effective deliver- 
 ance from superstitious fears. The thought 
 
NIHIL EX NIHILO 45 
 
 that runs through the poem may be gathered 
 from a few examples. 
 
 " When we shall have seen that nothing 
 can be produced from nothing, we shall then 
 be able to ascertain correctly what the elements 
 are out of which everything can be produced 
 and the manner in which all things are done 
 without the hand of the gods." 
 
 " If things come from nothing, any kind of 
 thing might be born of anything, no seed would 
 be required. Men might rise out of the sea, 
 fish out of the earth, birds out of the sky. Fruits 
 would not be constant to the trees which produce 
 them, any tree might bear any fruit. But instead 
 we see that the rose blooms in spring, the corn 
 ripens in summer, the vintage comes in autumn. 
 If things came from nothing there would be no 
 certain seasons and no time required for growth. 
 Infants would grow at once to men and trees 
 spring in a moment from the ground. But 
 none of these events happen ; all things grow 
 step by step and in growing preserve their kind." 
 
 " Moreover nature dissolves everything back 
 into its primitive elements and does not annihilate 
 things." 
 
 " If infinite time has not destroyed things it 
 
46 PRINCIPLE OF RELATIVITY 
 
 can only be that things are indestructible/* 
 
 These passages are from the beginning of 
 the first book, and introduce the theory of the 
 atoms. Another passage may be quoted t<p 
 show the argument for the void. " The waters, 
 some say, make way for the fish which swim in 
 them, they open liquid paths to them because 
 the fish leave room behind them into which the 
 yielding waters may stream. Thus things may 
 move and change among themselves although 
 the whole seem to be full. But, I ask, how can 
 the fish move forwards unless the waters have 
 first made room ? And on what side can the 
 displaced water go, so long as the fish has not 
 moved ? You must therefore deny motion or 
 admit that the void is mixed up with things in 
 order that motion may get a start. " 
 
 The science which Lucretius offers us rests 
 on the theory that all things are composed of 
 atoms, that atoms can move by reason of the 
 surrounding void and that all phenomena are 
 produced by the movements of the atoms. It is 
 not exact science as the moderns conceive science, 
 for the ancient philosophers, however acute their 
 observations and precise their descriptions, and 
 ingenious their hypotheses, had neither devised 
 
THE LOGIC OF ATOMISM 47 
 
 nor developed the experimental method which is 
 the distinguishing feature of modern scientific 
 research. 
 
 The ancient atomic theory arose directly from 
 the paradoxes of the rival principles of the old 
 nature-philosophy. It was the great and profound 
 synthetic work of a man of genius. It was 
 worked out into a complete system, and as a 
 perfect expression of materialism it has exerted 
 a continuous influence throughout the whole 
 history and development of Western thought. 
 
 We can see how the system of Democritus 
 arose. If all things flow, some simple unchange- 
 able being must support the movement. If 
 this being moves, non-being cannot be nought. 
 Movement is impossible if everything is divisible, 
 therefore if there is movement there is a limit 
 to divisibility. Movement is unreal if the 
 indivisible atoms fill all space, for then movement 
 is blocked ; therefore, besides the atoms there 
 is void. Movement is contradictory if the 
 moving body is at every moment at rest, there- 
 fore there is some force which causes the 
 atoms to move. There must be persistence 
 of matter throughout the infinite variety of 
 changing form. The atoms are the identity 
 
48 PRINCIPLE OF RELATIVITY 
 
 unchanged beneath the varying wealth of sensory 
 appearance. 
 
 The principle of materialism is that the 
 simplest explanation is the best, and the atomic 
 theory deduces all the wealth of existence from 
 absolutely simple beginnings. It is helped by 
 an analogy. Just as the infinite variety of 
 literature is produced by means of the letters 
 of the alphabet, which undergo no change 
 throughout all their combinations, so we may 
 suppose that the phenomena of the universe with 
 their infinite variety of colour and form can be 
 reduced in the last analysis to very simple elements, 
 almost identical, yet able to produce variety in 
 profusion simply by combinations. These simple 
 elements are the atoms ; by uniting and com- 
 bining they form material objects ; by changing 
 their place they bring about the continual shifting 
 of phenomenal change. 
 
 What can we know positively about atoms ? 
 We cannot see them, nor by any conceivable 
 means make them evident to the senses. Not 
 only has no one seen an atom, but we can be 
 certain that no one ever will ; for anything large 
 enough to be an object of sense-perception would 
 not be indivisible. The concept of indivisibility 
 
THE ATOMS 49 
 
 places them far below the limit of perception, 
 and the fact of indivisibility assures us of their 
 reality. Collected into a group of sufficient 
 members they form a body which can be seen 
 and touched. The fact of their existence is 
 thus derived from the necessity of denying the 
 infinite divisibility of being. 
 
 The quantity of the atoms is infinite, for there 
 is no limit to the bodies which the universe 
 contains. Also they are eternal and indestructible. 
 This also follows from the concept of them. 
 They have no other quality than their form or 
 shape. In this alone is their difference from one 
 another. Colour, odour, weight, resistance are 
 due to their combinations and movements. The 
 unchangeableness of the atom follows therefore 
 from its nature. It has been and will be what 
 it is throughout eternity. How can it change 
 seeing that it is indivisible ? How can it alter 
 its quality, seeing that it has none ? 
 
 Bodies which are composed of the atoms 
 appear to us coloured, resistant, sonorous, hot 
 or cold, but this is illusion, for these qualities 
 are the impressions on our sense organs 
 and therefore appearances. Dissipate the illu- 
 sion, think of bodies as they must be in 
 
50 PRINCIPLE OF RELATIVITY 
 
 themselves, and it will be seen that they 
 must consist of atoms and that atoms cannot 
 themselves possess the sense qualities. But just 
 as atoms have different forms so the bodies 
 composed of them will be different according 
 to the arrangement and direction of the atoms 
 in them. When, then, one identical body appears 
 different at different moments it is because its 
 atoms have changed place or because some of 
 its atoms have been lost or gained. It is analogous 
 to the case of words which alter and change in 
 both sound and meaning by the addition or 
 subtraction of a letter or the alteration of the 
 arrangement of the letters. 
 
 Such then are the atoms how have they come 
 to form the world in which we live ? We must 
 suppose that the atom left to itself in the void 
 would have a natural movement, a movement 
 inherent in it, a weight or gravity which would 
 cause it to fall for ever in the infinite void. From 
 this it will come about that from time to time 
 atoms will clash, will block one another and form 
 conglomerations or heaps. Our world must be 
 conceived as such a heap, and by the clashing, 
 sorting, collecting and dispersing of its atoms, 
 there have successively formed themselves, the 
 
THE OLD MATERIALISM 51 
 
 earth floating in the air, the moon and the sun 
 which are bodies like the earth, the stars, and also 
 the living beings on the earth. The soul which 
 appears to animate the organized bodies must 
 be supposed to consist of more subtile atoms, 
 very mobile, which we may imagine to be round 
 and polished and so comparatively frictionless. 
 
 The thoughts which succeed one another in 
 the soul are the movements of its component 
 atoms. Democritus seems to have explained 
 the perception of material objects by a theory 
 that those objects are at every moment emitting 
 on all sides extremely small images of themselves 
 which strike on the organs of sense. It is to 
 this theory that Aristotle probably alludes when 
 he says (Metaph. iv. 5), " Democritus at any 
 rate says that either there is no truth or to us at 
 least it is not evident. And in general it is 
 because these thinkers suppose knowledge to be 
 sensation, and this to be a physical alteration, 
 that they say what appears to our senses must 
 be true/ 1 
 
 Such then is the materialistic naturalism of 
 the ancient philosophy. Bodies and souls, objects 
 and worlds, are composed of atoms, the phenomena 
 of nature and the acts of thought are movements 
 
52 PRINCIPLE OF RELATIVITY 
 
 of atoms. There is not, never has been, never 
 can be, anything but atoms, void and time, these 
 are the conditions of movement, and movement 
 is the reality of the phenomenal world. 
 
 There remained, however, one problem un- 
 solved ; it led to an important and somewhat 
 inconsistent modification by Epicurus of the theory 
 which he adopted. This was the problem of direc- 
 tion. Bodies fall. Their natural direction is 
 downwards. If bodies seem to rise it is either 
 because their fall is relatively slower than that of 
 heavier bodies or, if the direction of their upward 
 movement is absolute, it is due to a rebound from 
 the clash of colliding bodies. Apparently this 
 difficulty was met by supposing that the void 
 is infinite, that atoms are indestructible, that 
 worlds are for ever being formed and unformed, 
 and that their number is infinite. In such 
 a world-view absolute direction could be accepted 
 as fact without introducing direct contradiction. 
 But a new difficulty occurred to Epicurus. 
 If atoms are falling perpendicularly by an 
 inherent natural movement in the infinite void, 
 they will pursue parallel courses from which 
 there is nothing to turn them aside and no 
 heaps or conglomerations will be formed. He 
 
FREE .WILL 53 
 
 introduces therefore a new notion. From time 
 to time he supposes the atoms show a slight 
 inclination, imperceptible and capricious, which 
 Lucretius named their clinamen. It draws them 
 from time to time out of the perpendicular and 
 brings them into collision, causing them to 
 form masses. The interest of such a theory, 
 however, is not its physical importance, for in 
 that respect it is quite arbitrary, but that it is 
 inspired by the need of the philosopher to find 
 in nature some basis for the free action of the 
 human soul. 
 
 This, then, became the accepted form in which 
 space, time and movement entered into the 
 ancient nature-philosophy. As a philosophical 
 concept atoms and the void could not withstand 
 criticism. On what principle could a limit to 
 divisibility be fixed ? To appeal to perception is 
 impossible for by the hypothesis the perceptible 
 is divisible. Is the appeal to conception any 
 more successful ? Shape or form itself involves 
 the notion of whole and part. It is not difficult 
 indeed to show that the concept of the atom is 
 riddled with contradiction, and moreover possesses 
 no principle by which a synthesis of contradictions 
 can be effected on the Hegelian method. It is a 
 
54 PRINCIPLE OF RELATIVITY 
 
 whole without parts. It is a quantity with no 
 quality but its quantity. The void is even more 
 difficult to conceive. It is a pure negation posited 
 as the very basis of reality. It is an absence which 
 forms the absolute condition of presence. Finally, 
 its occupancy supposes a matter which is ulti- 
 mately indivisible filling a portion of a space 
 which is divisible to infinity. 
 
 On the other hand, the atomic theory is not 
 a baseless speculation ; it is grounded in the 
 reality of experience. Moreover, it is not a 
 rough and ready practical solution of an insoluble 
 theoretical problem. It is based on a sound 
 intellectual principle which we may even describe 
 as an intellectual instinct ; the principle that 
 from nothing there is nothing, and the applica- 
 tion of this principle to points and instants. 
 Extension is not composed of extensionless 
 points, duration is not composed of durationless 
 instants. The very same intuition which makes 
 the philosopher of mind affirm the moment of 
 experience to be a specious present makes the 
 natural philosopher affirm the atom to be the 
 spatial unit. 
 
 The ancient atomic theory has little but a 
 merely outward resemblance to the modern 
 
THE MODERN ANALOGY 55 
 
 atomic theory. The latter is not a philosophical 
 theory though of immense philosophical interest. 
 It is a purely scientific theory and not an effort 
 of the mind to conceive the ultimate constitution 
 of matter based on deductions from the logical 
 principle of non-contradiction. It is scientific in 
 the meaning that it is based on discovery,and that 
 the mathematical formulae by which it is expressed 
 are submitted to the test of experiment, and 
 corrected continually by the results of experiment. 
 It is only in the sense that the atom of modern 
 science is a conceptual object which can never 
 be brought to a direct perceptual test, for the 
 reason that its size is below the amplitude of the 
 waves of light and therefore can never be made 
 visible to our ordinary illumination, that it is 
 permissible to indicate it by the same name. In 
 contrast to the concept of the ancient philosophy 
 the atom of modern physical theory is not simple 
 and undifferentiated but infinitely complex. The 
 discovery of the x rays, and their application to 
 the analysis of crystal structure, with the conse- 
 quent increase in the range of our direct perceptual 
 penetration of matter, have indeed revealed in a 
 positive manner the nature of molecular and atomic 
 structure, and have to that extent given a surer 
 
56 PRINCIPLE OF RELATIVITY 
 
 basis to physical science. Yet even this vast 
 extension of perception which our modern world 
 possesses, as compared with the ancient, does not 
 relieve it from the necessity of conception for its 
 idea of the ultimate nature of physical reality. 
 
 The value of the ancient theory is that it shows 
 us the form in which space, time and movement 
 provided the scheme of a philosophy of nature. 
 Space was the void, a concept of pure infinite 
 emptiness. Time was the other formal expanse 
 which reality required, but it seems to have been 
 taken for granted and not conceptually analysed 
 as space was. Movement seems ultimately to 
 have been explained by the principle that some- 
 thing is more than nothing and therefore that 
 the something occupying space must, by the very 
 fact that it is something, fall through the void 
 which is nothing. It is clear that to the ancient 
 mind there is one fundamental empirical fact 
 which is accepted as ultimate, apparently uncon- 
 sciously, and this is the fact which we now call 
 gravitation, and which to them meant the weight 
 which made everything sink in the void. 
 
CHAPTER IV 
 
 THE VORTEX THEORY 
 
 THE atomic theory of Democritus supported, 
 and indeed for the most part represented, 
 rationalistic and materialistic opinion throughout 
 the whole of the pre-Copernican period. The 
 theory was essentially atheistic, but the reason 
 of its atheism is not immediately evident. It 
 is difficult at first to see why the constitution 
 of nature should have any relation whatever to 
 the question of the origin of nature. As a 
 matter of fact also those who accepted the atomic 
 theory, even Epicurus himself, did not on that 
 account deny the existence of gods. The atheistic 
 character of the theory lay entirely in the fact 
 that its argument dispensed with the necessity of 
 God. The ultimate constituents of reality, the 
 atoms, were by their very concept absolute. 
 Creation and annihilation could only have mean- 
 ing in respect of the grouping of the atoms, the 
 
 57 
 
58 PRINCIPLE OF RELATIVITY 
 
 creation or annihilation of an atom would involve 
 self-contradiction. The annihilation of the atom 
 contradicts the notion of a limit to its divisibility ; 
 the creation of the atom contradicts the notion of 
 its simplicity. Clearly then if there are atoms 
 uncreated and imperishable whose combinations 
 (like the letters of the alphabet) produce infinite 
 variety, we have in them a self-sufficient ground 
 of nature. The world may have arisen by 
 chance ; there is no necessity to postulate a 
 creator. So when Dante sees Democritus among 
 the ancient sages in the first circle of the Inferno^ 
 he refers to him as " Democrito, che il mondo 
 a caso pone," Democritus who ascribes the world 
 to chance. 
 
 A curious glimpse is afforded to us of the 
 medieval mind, and the form which materialism 
 and rationalism assumed for it in the scholastic 
 period, in another passage of Dante (Canto X. of 
 the Inferno)^ where he describes the punishment 
 of the heretics. Who are the heretics ? They 
 are not the adversaries against whom Athanasius 
 and Augustine struggled in the formation and 
 interpretation of the creeds, and Dante is some 
 centuries before the Reformation and the institu- 
 tion by the Holy Inquisition of the Auto-da-fe. 
 
THE EPICUREANS 59 
 
 The heretics we find are the followers of a 
 materialistic philosophy, not teachers of false 
 doctrine. Dante names them the Epicureans. 
 They include some famous Florentines of Dante's 
 time, and also the Emperor Frederic II., who 
 gathered to his court at Naples and Sicily erudite 
 Grecians and Saracens and revived the classical 
 learning. They lie in their tombs on the fiery 
 plain surrounding the fortified walls of the city of 
 Hell. As Dante, guided by Virgil, passes along, 
 they push up the covering stone of the sepulchre 
 anxious for news of the living. They are the 
 rationalists who thought this life is all, and that 
 the tomb is the end. * There the wicked cease 
 from troubling and the weary are at rest." Alas ! 
 they discover that " their worm dieth not and 
 their fire is not quenched." 
 
 Apart, however, altogether from the religious 
 and ethical questions involved, the concept of 
 atoms and the void, furnished to pre-Copernican 
 thought the type of physical reality. The void 
 was Euclidean space in its purest uncomplicated 
 form. It was absolute in the sense of perfect 
 emptiness. The puzzling fact in regard to the 
 atoms was what we now call gravitation. It 
 could be determined empirically and its law 
 
60 PRINCIPLE OF RELATIVITY 
 
 stated, and for the conglomeration of atoms which 
 forms our earth its direction could be fixed ; 
 but the falling of the atoms in the void was 
 evidently embarrassing. Putting ourselves at the 
 standpoint of the ancient world-view we can see 
 that no means exist to decide whether the atoms 
 are falling continuously through eternal time in 
 infinite void or whether they are at rest. If, 
 however, it is not in the nature of the atom 
 to fall in the void it is difficult to understand 
 why there is movement any when or anywhere. 
 Movement would have to be impressed from 
 without and the ground of the self-sufficiency of 
 the atomic theory would be gone. 
 
 The whole of this ancient world-view was 
 changed by the Copernican discovery. This 
 discovery brought about a most profound and 
 complete revolution in human thought, turned 
 science and philosophy in a new direction, and 
 with a new world-view opened to the human mind 
 new problems, new methods and reformed con- 
 cepts. No such tremendous effect in determining 
 the intellectual development of our race has 
 approached in importance that which followed 
 this discovery. If we would classify scientifically 
 the historical stages in the evolution of philosophy 
 
THE COPERNICAN DISCOVERY 61 
 
 and distinguish them by a central epoch we ought 
 to mark all theories as being pre-Copernican or 
 post-Coper nican. That is in fact what we do 
 when we name Descartes as the founder of modern 
 philosophy, for Descartes was the systematizer of 
 that discovery, the philosopher of that revolution, 
 and his principles, his method and his system are 
 completely determined by it. 
 
 Yet the Copernican discovery seems a simple 
 enough matter and we are generally inclined to 
 wonder how it could have been possible for 
 mankind to continue so long without someone 
 suspecting that the celestial movements were an 
 appearance consequent on our own translation. 
 We understand the shock to the religious faith of 
 those who had pictured this earth as the scene 
 of a tragedy, prepared from eternity by the divine 
 source and sustainer of the universe, and for whom 
 human history led up to, and followed -from, that 
 unique event. But so easily have we come to 
 adapt ourselves to the new world-view that we are 
 unconscious of the change, and indeed our 
 difficulty in reading the ancient philosophers is 
 to remember that their concepts were con- 
 cerned with an imagery totally different from 
 ours. 
 
62 PRINCIPLE OF RELATIVITY 
 
 Suppose one had been born in a smoothly 
 running railway-carriage, and brought up to find 
 in it all the conveniences and necessities of life, 
 able to look out on the world through which it is 
 continually journeying. It would and it must 
 seem to one so circumstanced, that the panorama 
 without is in ceaseless movement. Every alter- 
 ation of relative position of the moving system 
 would appear as a movement of the panorama. 
 This was the condition of the human race. It 
 developed intellectually through continuously 
 successive ages without ever suspecting that the 
 movement it looked out upon in the panorama 
 of the heavens might be an appearance due to its 
 own translation. The discovery came suddenly 
 and with something of a shock. But the dis- 
 covery having been made, the evidence for 
 it accumulated with such force that the world- 
 view adapted itself to it, and we are no more able 
 to-day to return to the old world-view than we 
 are able when we take a railway journey to believe 
 that our carriage is at rest and the landscape 
 moving. 
 
 The philosopher of this new world-view was 
 Descartes. It is no mere chance coincidence that 
 Descartes was philosophizing and elaborating a 
 
DESCARTES'S NEW METHOD 63 
 
 new system of the universe when Galileo was 
 experimenting to prove the earth's movement. 
 A new concept of truth and reality on which the 
 new aspect of the universe could be rationalized 
 and harmonized was imperatively called for. It 
 must be a return to absolutely first principles. 
 
 Descartes laid down two principles of philo- 
 sophy, one subjective and one objective, and both 
 the direct outcome of the Copernican discovery 
 and its revolution in the world-view. 
 
 The first principle is that the intellect alone by 
 the clearness and distinctness of its ideas can 
 furnish a criterion of truth. The senses are 
 deceptive, the source of confused and obscure ideas. 
 The senses do indeed induce belief, and seem 
 to furnish an assurance of truth, but their purpose 
 is not to lead to truth but to preserve the body. 
 It is then not to sense but to the clear and 
 distinct ideas of the mind, to reason, that we must 
 turn for true knowledge. Why the clear and 
 distinct ideas of understanding should possess 
 superiority over, and greater validity than, the 
 obscure and confused ideas of sense was indis- 
 coverable in their nature. Descartes fell back 
 on the proof of the existence of God and the 
 impossibility of our conceiving that in the case 
 
64 PRINCIPLE OF RELATIVITY 
 
 of clear and distinct ideas God could deceive 
 a principle which no longer appeals to us. 
 On this distinction between sense and intellect 
 was founded the well-known method of Descartes. 
 He proposed to doubt everything which could 
 possibly be doubted in order to discover, as a 
 starting point, some fact which expressed, in the 
 clearness and distinctness of its idea, a truth which 
 it was not possible to doubt. Such fundamental 
 truth he claimed to have discovered in the famous 
 Cogito ergo sum. It is easy to see the connexion 
 of this with the Copernican discovery. Had not 
 that discovery clearly demonstrated that mankind, 
 universally and continually, trusting the inter- 
 pretation of direct sense experience, had lived in 
 age-long error ? 
 
 The second principle of Descartes concerns the 
 objective reality of the universe. The universe 
 is a mechanistic not a materialistic system. It 
 is not the outcome of the behaviour of atoms in a 
 void, it is the mechanical disposition of matter 
 resulting from the imparting to it of movement. 
 The essence of material substance is extension 
 alone and there is no void. Movement is not 
 change of place but relative change of neighbour- 
 hood. Movement is only possible in a plenum ; 
 
DESCARTES'S MECHANISM 65 
 
 movement in the void is unmeaning and self- 
 contradictory. Movement in a plenum is a vortex 
 movement, that is, a movement which involves 
 simultaneously every part of the system and is not 
 propagated from part to part. The universe is a 
 system of vortices, each vortex determining 
 vortices within it, and determined by relations to 
 vortices without it. The solar system is a vortex, 
 the fixed stars are similar vortices, and the 
 planets and their satellites are all vortices within 
 the vortex, and all movement down to the beating 
 of the heart and the circulation of the blood is one 
 in principle, having its part in the universal 
 mechanism. This constitutes the first great 
 systematization of the universe in accordance with 
 the revolution in astronomy. 
 
 These two principles, the subjective principle 
 or new method and the objective principle or 
 mechanistic interpretation, have had a diverse 
 fate in the history of thought. While the t first 
 has been accepted as marking the beginning of a 
 new period of philosophical speculation so that 
 we regard Descartes as the founder of modern 
 philosophy, the second, the cosmological and 
 physical theory, is neglected and forgotten, or 
 read, when it is read, as an intellectual curiosity 
 
66 PRINCIPLE OF RELATIVITY 
 
 with no relation to present physical or meta- 
 physical science. But in Descartes's own time 
 and during the development of Cartesianism 
 in the half century which followed, there was no 
 such dissociation of metaphysics and physics, 
 philosophy and science. It was the vortex theory 
 which established the fame of Descartes. 
 
 " Give me matter and movement and I will 
 make a world," was the famous challenge which 
 he threw down to the theory of atoms and the void. 
 Keeping in mind that for Descartes matter is 
 extension, we can translate it to mean that the 
 variety and the uniformity of the universe are a 
 function of systems of movement. To under- 
 stand it we must examine a little more closely the 
 three distinct doctrines, interconnected and inter- 
 dependent, on which it rests. These are (i) that 
 the essence of matter is extension ; (2) that move- 
 ment is relative not absolute, not change of place 
 in an independent expanse but the relative change 
 of neighbourhood of extended systems ; and 
 (3) that nature is a plenum, there is no void and 
 movement in a plenum is a vortex. 
 
 The first of these doctrines is fundamental. It 
 is the ground of Descartes's rejection of the void, 
 Extension is not the empty place in which there 
 
REJECTION OF THE VOID 67 
 
 is or is not matter, it is the essential attribute of 
 matter. The direct argument for this is that 
 extension is the only attribute which is inseparable 
 and indistinguishable from material substance. 
 Every other attribute colour, weight, sonority, 
 resistance, shape can be thought absent, but 
 if we abstract from its extension, material 
 substance itself is annihilated. The apparent 
 contradiction that the extension of any matter 
 is variable, as instanced by rarefaction and con- 
 densation, is easily explained as the disposition of 
 a matter's extension in relation to other material 
 extensions. Extension being the essence of 
 material substance, if and when matter moves its 
 extension moves. Extension is not the quantity 
 of emptiness matter fills. To say of empty space 
 that it is extended is to endow it with the essential 
 attribute of material substance, and so to deny 
 that it is a void. The rejection of the atoms is 
 still more direct. They are geometrically im- 
 possible, not on account of self-contradiction 
 in the concept of indivisible particles having form 
 or shape but no parts, but on account of their 
 unchangeability. Movement would be im- 
 possible if the constituents of matter had un- 
 alterable shapes. 
 
68 PRINCIPLE OF RELATIVITY 
 
 The second doctrine concerned the relativity 
 of movement. The Copernican theory had merely 
 substituted a heliocentric for a geocentric astro- 
 nomical hypothesis. Descartes saw that it raised 
 the metaphysical problem of movement. Nothing 
 is at rest in the whole system of nature if being at 
 rest means being in a moving system and not being 
 carried along in its movement. But this is not 
 what we mean by rest in ordinary experience. 
 We say we are at rest and not moving when the 
 members of our system keep their relative posi- 
 tions notwithstanding that the whole system 
 may be in movement of translation or may be 
 itself not moving but borne along in a movement. 
 We are at rest, for example, in the cabin of our 
 ship when wind and stream are transporting the 
 ship to France. The earth may be considered at 
 rest if we mean that it is carried along its path 
 through the solar system like a ship on the ocean. 
 It is no longer possible then, Descartes argued, to 
 regard anything as in its nature at rest. There 
 are no fixed immovable points. Nothing has a 
 permanent place except in so far as it is fixed by 
 our thought. The common notion is that a body 
 moves when it changes its place in a void. Strictly 
 defined, movement is not change of place but 
 
MOVEMENT IN A PLENUM 69 
 
 change of neighbourhood, it is the translation of a 
 part of matter or of a body from the neighbour- 
 hood of bodies with which it is in contact into the 
 neighbourhood of others. We can only define 
 it relatively. When I push a boat off a beach 
 it is merely convenience which makes me express 
 it as a movement of the boat relatively to the 
 beach at rest and not as a movement of the beach 
 relatively to the boat at rest. 
 
 The third doctrine had for its main argument 
 the defence, against the atomists, who denied it, 
 of the concept of the possibility of movement in 
 a plenum. The argument of the atomists had 
 been that there must of necessity be the void, for 
 without it movement is impossible. Where, 
 they asked, is the place into which to move if every 
 place is already occupied ? Movement, Descartes 
 contended, is possible in a plenum if the chain of 
 moving members is complete so that the last of 
 the series moves into the place of the first. Such 
 movement is really changing place and not passing 
 through a void which does not change. 
 
 Indeed, if in moving we did not carry our 
 extension with us, how could we have a science 
 of geometry? In geometry we are not measuring 
 we are measuring extension. The 
 
70 PRINCIPLE OF RELATIVITY 
 
 figures we construct and study in geometry 
 circles, squares, triangles, cubes, spheres are 
 measurements and constructions of the extension 
 which moves with the earth, not of a supposed 
 vacuum independent of that movement and 
 indifferent to it. Endless complexities and con- 
 tradictions, actual as well as logical, arise if we 
 attempt to interpret geometry in terms of vacuum. 
 We say, for example, that moving is the opposite 
 of resting. Now suppose that, following the 
 common notion, we define movement as change of 
 place, and rest as remaining in the same place, 
 then we see at once that for anything on the earth 
 to be at rest, it must be parting company with the 
 earth at a prodigious velocity. Descartes had 
 therefore the choice of two alternatives. Either 
 extension is an attribute of material substance and 
 accompanies it in all its changes, or it is vacuum 
 existing independently of substance. If he chose 
 the latter he must sacrifice geometry, for no means 
 exist of measuring vacuum. His philosophical 
 theory, though opposed to the universally accepted 
 notion, was a necessity of thought and a great 
 advance in mathematical and physical method. 
 When I move about a ship I am really moving 
 notwithstanding that to the observer on shore I 
 
CELESTIAL MECHANISM 71 
 
 may be at rest ; the beating of my heart, the 
 revolving wheels of the watch in my pocket, are 
 regular movements, though the tracing of them 
 on a chart, against an absolute background, 
 would be hopelessly complicated and different 
 for different observers. 
 
 The application of this principle led to the 
 construction of the magnificent scheme of celes- 
 tial or rather cosmical mechanism which amazed 
 and held spell-bound the intellectual world of the 
 latter half of the seventeenth century the vortex 
 theory. The solar system is a vortex with 
 the sun as its centre, extending beyond the 
 orbits of the distant planets. It is bounded by 
 other vortices. These are the fixed stars which 
 like our own sun are centres of revolving systems. 
 
 There are two laws of nature which Descartes 
 formulates. They are rational deductions empiri- 
 cally verifiable. The first is that everything 
 remains as it is till something changes it. The 
 second is that every body which is moved tends to 
 continue moving in a straight line. The rationale 
 of the second law is that the straight line being 
 the shortest line measures the force. The first 
 law explains how bodies get involved in vortex 
 movements. By these two laws he accounted for 
 
72 PRINCIPLE OF RELATIVITY 
 
 the planets in the solar vortex. The planets, he 
 said, are at rest so far as their sky is concerned. 
 They are not careering through space but being 
 carried along with the moving system, that is, 
 the solar vortex. They have been caught up in 
 it, projected it may have been from other vortices 
 and sent travelling in a straight line until they 
 became involved in our system. 
 
 Such is the mechanism which Descartes sub- 
 stituted for the old materialism. I have not 
 dealt with details but tried to bring into relief its 
 essential features. It was the constructive work 
 of a single genius. It enjoyed a brilliant vogue, 
 capturing the imagination of more than one 
 intellectual generation. Yet it has passed away 
 so completely that it is hardly remembered even 
 as a stage in the evolution of scientific theory. 
 The picture of the physical universe as a system of 
 vortices, described with such mastery of minute 
 detail, and with such assurance, in the Principia, is 
 no doubt as far removed from our present imagery 
 of physical reality as the description of the organism 
 controlled by the animal spirits in Descartes's 
 Les passions de Fame is far removed from our 
 modern physiological concepts. Nevertheless, in 
 the one case as in the other there are important 
 
THOUGHT AND EXTENSION 73 
 
 principles insisted on from the neglect of which 
 science has distinctly suffered. In outward 
 resemblance Descartes's world-view is extra- 
 ordinarily like that which is presented to us by 
 the general principle of relativity. So much 
 so that it appears at first as though in rejecting 
 Newton's concepts we are simply returning to 
 those of Descartes. We have only to remember 
 however that the whole development of physical 
 science has, in recent times, come to centre round 
 the electro-magnetic theory, and that this concerns 
 a continuity of experimental discovery in a realm 
 of phenomena entirely unexplored by the mathe- 
 maticians, astronomers, and mechanicians of 
 the seventeenth and eighteenth centuries, to see 
 that there can be no simple reversion. The 
 principle of relativity is in reality the rationaliza- 
 tion of the electrical theory of matter. It is 
 interesting to note how it was anticipated in the 
 principles which suggested to Descartes the vortex 
 theory. The concept of the vortex is itself a 
 quite striking anticipation of the modern concept 
 of the "field of force." 
 
 Descartes distinguished two ideas as intel- 
 lectual and therefore not subject to the deceptive 
 appearance which characterizes the ideas depen- 
 
74 PRINCIPLE OF RELATIVITY 
 
 dent on sense perception. These are thought and 
 extension. By these concepts he distinguished 
 the essence of the two substances which we refer 
 to familiarly as mind and matter. So far as 
 physical theory is concerned the important con- 
 cept is extension. In Descartes's theory that 
 extension is the essential attribute of matter it is 
 denied that there is any void or pure space within 
 which matter moves. Extension is not something 
 moving matter leaves behind it or of which it 
 exchanges one quantity for another. The moving 
 mass or system carries its extension within its 
 movement. From this it follows that all movement 
 is relative and concerned only with the relations 
 of material systems to one another. This accords 
 completely with the modern theory of relativity. 
 
 It is curious that the duration of the universe 
 did not impress Descartes as having anything like 
 the importance which he attributes to its extension. 
 He recognizes that the universe endures, but it is 
 not, he thinks, by reason of anything in its essence, 
 the fact of duration simply shows its dependence 
 on God. If duration is the essential attribute of 
 a substance, we must conclude that this substance 
 is God, but Descartes does not himself draw this 
 conclusion. Time is as necessary as space for a 
 
EXTENSION AND DURATION 75 
 
 mechanism. Without time the machine cannot 
 work, but time plays the part of independent 
 variable, it has no grip on the reality of the 
 machine, any more than the time the clock 
 measures is part of the contrivance. To us 
 extension and duration are correlative and inter- 
 dependent. From the historical standpoint this 
 is of peculiar significance. The great Copernican 
 revolution brought in a new concept of the 
 celestial mechanism, and incidentally it re- 
 formed our view of the spatial universe. It 
 was not until three centuries later that a reform 
 of our concept of the duration of the universe, 
 parallel to the Copernican concept of its 
 extension took place. It followed the great 
 biological discoveries of the nineteenth century. 
 The Darwinian theory brought as complete a 
 revolution in our conception of time as the 
 Copernican theojy had produced in our conception 
 of space. To Descartes, therefore, duration 
 appears not as the essence of the universe, but only 
 as that which is necessary to its existence. Its 
 continuity from moment to moment depended on 
 a creating and sustaining power. 
 
 It is, then, in the concept of matter as extension, 
 and in the concept that movement and rest are 
 
76 PRINCIPLE OF RELATIVITY 
 
 mutually dependent on systems of reference, and 
 in the concept of translation as relative to the 
 members and parts of the system, together with 
 the rejection of any absolute zero, such as the 
 void, affording a standard for the measurement 
 of absolute velocity, that Descartes's vortex- 
 system anticipates the principle of relativity. The 
 doctrine that material substance consists in ex- 
 tension alone does not mean that pure extension 
 exists materially without any other quality or 
 character whatever. It means that extension is 
 constant and that no other attribute of matter is. 
 Any other attribute which matter may have, or 
 any attribute that it may need in order that we 
 may apprehend it, is variable. It follows that 
 all the diversity and endless variety of the material 
 universe must be due to movement and a direct 
 function of movement. This follows simply from 
 the fact that extension is not stuff and there- 
 fore cannot harbour occult properties essences 
 or forces. 
 
CHAPTER V 
 
 THE PROBLEM OF GRAVITATION 
 
 IT is a curious thing that Descartes who proposed 
 a new method of philosophy, the distinctive form 
 of which is universal doubt, and the principle of 
 which is that nothing must be accepted as true 
 unless its evidence is presented to the mind with 
 a clearness and distinctness which excludes doubt, 
 should have worked out a hypothesis of the system 
 of the universe complete down to the minutest 
 detail. There is no greater contrast in the 
 history of western intellectual development than 
 his system presents to the method and philo- 
 sophical system of Newton which completely 
 supplanted it. Newton's method was experiment, 
 and his philosophy he described as the experi- 
 mental philosophy. This does not mean that 
 Newton himself was an experimentalist. He 
 conducted no investigations in the manner, for 
 example, of Galileo. He was a mathematician 
 
 77 
 
7 8 PRINCIPLE OF RELATIVITY 
 
 like Descartes, and all that he insisted on in 
 his natural philosophy was that nothing should 
 be accepted as true on the clearness and 
 evidence of the pure idea, unless it had first been 
 submitted to the test of positive experimental 
 proof. It is also curious, in comparing these 
 two great minds and the work they accomplished, 
 to observe that while the speculative philosophy 
 of Descartes has secured a permanent place 
 in literature, his physical system, which was 
 the supremely important thing to his contem- 
 poraries and his successors, is entirely rejected and 
 studied only, if it is studied, for its antiquarian 
 interest. Newton, on the other hand, who was 
 equally famous to his contemporaries and imme- 
 diate successors as a speculative philosopher, is 
 now remembered as a great mathematician and as 
 the discoverer of the universal law of gravitation, 
 while his philosophy is entirely neglected. Even 
 our knowledge of his great discovery is not first- 
 hand. The Philosophiae Naturalis Principia 
 Mathematica is not a classic which takes a place with 
 Hobbes's Leviathan, Locke's Essay on the Human 
 Understanding, Spinoza's Ethics, and the other 
 great works of his contemporaries, it is a con- 
 cealed book for all but advanced mathematicians. 
 
NEWTON'S PRINCIPIA 79 
 
 and our knowledge of Newton's discovery is 
 enshrined in a moral tale concerning his reflection 
 on the fall of the apple. It is strange there should 
 be so little direct study of the work of one of 
 the greatest geniuses our country has produced. 
 It is further of interest to note that though 
 Newton in his life-time won immediate and un- 
 questioned recognition he was elected President 
 of the Royal Society twenty-five years in succes- 
 sion his views never obtained wide or important 
 acceptance. It was after his death (1727) and in 
 consequence of the publication of Voltaire's Element 
 de la Philosophic de Newton in 1738 and the trans- 
 lation of the Principia Philosophiae into French 
 some years later that Newton's " Philosophy " 
 triumphantly deposed the Cartesian vortex theory 
 and became the accepted basis of physical science. 
 Newton was born in 1642, and was therefore 
 eight years old when Descartes died, and his years 
 of study and research were those during which the 
 Cartesian system reigned unchallenged. The 
 Principia Philosophiae, the work of many years, 
 written in Latin and bearing the same title as 
 Descartes's famous work, was published in 1686-7. 
 It presented his theory of gravitation and his 
 formula, of the universal law. The famous story 
 
8o PRINCIPLE OF RELATIVITY 
 
 that this was a sudden discovery following a 
 meditation set going by seeing an apple fall in 
 the orchard of Woolsthorpe (from a tree which 
 was shown to visitors for about a century, and 
 which after it had fallen through decay was carved 
 up into souvenirs), is generally rejected as 
 legendary and mythical. Certainly if we suppose, 
 as is usually implied, that Newton at the time was 
 of a poetical and impressionable turn of mind and 
 awakened by a simple occurrence to the thought of 
 the mystery underlying natural processes, nothing 
 can well be more improbable than the story. 
 When, however, we take the anecdote in its 
 historical setting we see at once that, whether it 
 be legendary or not, something very like it must 
 have happened. What is certain is that the 
 circumstance, falling apple or whatever it may 
 have been, did not originate the meditation but 
 broke in upon it. The authority for the story 
 is Voltaire, who had heard it from Madame 
 Conduit, a niece of Newton, married to a Fellow 
 of the Royal Society who was one of Voltaire's 
 intimate friends. The story is : " One day 
 in the year 1666, Newton in the retirement of 
 the country " (he had withdrawn from Cambridge 
 to Woolsthorpe on account of the plague), " seeing 
 
THE APPLE STORY 81 
 
 the fruit falling from a tree in the orchard, accord- 
 ing to the story which his niece Madame Conduit 
 told me, fell into deep meditation as to the cause 
 which thus draws all bodies in a line which if 
 prolonged would pass almost through the earth's 
 centre. What, he asked himself, is this force ? 
 It acts on all bodies in proportion to their mass 
 and not to their surfaces ; it would act on this 
 fruit now falling from this tree were it so removed 
 that it had three thousand or ten thousand fathoms 
 to fall. If this be so, then this force must be 
 acting from where the moon is right to the centre 
 of the earth. If so, then, whatever this power 
 be, is it not the same as that which keeps the 
 planets moving round the sun and the satellites of 
 Jupiter in their orbit round that planet ? Now 
 it has been shown, by the inferences drawn from 
 Kepler's laws, that these secondary planets are 
 weighted towards the centre of their orbits, 
 more in proportion as they are near, less in 
 proportion as they are distant, that is reci- 
 procally according to the square of their 
 distances. A body in the moon's position 
 and a body close to the earth must both 
 weigh on the earth in exact accordance with 
 that law." 
 
 F 
 
82 PRINCIPLE OF RELATIVITY 
 
 This story shows clearly the subject of 
 Newton's meditation. He was pondering 
 over Descartes's system of celestial movement. 
 But why should that be disturbed by the falling 
 apple ? It flashed across him that here was a 
 phenomenon in flagrant violation of Descartes's 
 vortex principle. By every reason alleged by 
 Descartes the apple ought to have flown upward 
 and outward and not to have fallen downward. 
 The centrifugal force of a vortex causes the heavy 
 body held by it, when released, to leave it at right 
 angles to the axis of rotation, as when for example 
 the stone in a sling is released. This at once 
 brings to mind the laws of Kepler, in particular 
 the third law, that the cube of the distance of each 
 planet from the sun is proportional to the square 
 of its time of revolution. But what strikes Newton 
 is that weight must be determined by mass and 
 distance, and not, as Descartes's principle required, 
 by surfaces in changing relations consequent on 
 movement. It is quite certain therefore that if 
 it was not a falling apple it was some analogous 
 circumstance which originated in Newton's mind 
 the doubt concerning Descartes's principle and 
 led to the formulation of the new law and to the 
 return to the concept of absolute space and time 
 
ORIGIN OF THE VORTEX 83 
 
 as the basis and framework of the physical 
 universe. 
 
 To make this clearer it will be well to look a 
 little closely at the way in which Descartes con- 
 ceived the vortex to have worked in bringing 
 about the distribution of matter in the solar 
 system. In rejecting the concept of atoms and 
 the void Descartes had no need to endow matter 
 with weight as an essential attribute. As there is 
 no void, and therefore no distinction between space 
 and matter, that is to say, as matter is not con- 
 ceived as in space or space as containing matter, 
 the one and only essence of matter is extension. 
 Without movement this is pure homogeneity. 
 If we conceive matter at rest we must of necessity 
 conceive its parts, if it consist of separate parts 
 uniform in character, as regular figures, such as 
 cubes or hexagons, for only such will fit to- 
 gether to compose a plenum. Suppose now that 
 movement is originated, it must, on the theory, 
 involve relative change throughout the mass and 
 the effect must be that the shape of the figures will 
 alter and continue to alter till they become entirely 
 spherical. The more and the finer the spheres 
 become the more easily will they pass one another 
 in moving and their velocity will tend to increase 
 
84 PRINCIPLE OF RELATIVITY 
 
 continually. There will eventually be formed 
 very minute and perfectly spherical particles, a 
 very fine dust through which the particles 
 will cleave an easy way, and denser particles 
 which will be driven outwards and form a dust 
 on the circumference of the vortex sphere. 
 This, according to Descartes's natural history 
 of the world, is what has actually happened 
 in our system. The fine spherical matter has 
 collected at the centre, it is the luminous or 
 fire element and composes the sun, the finer dust 
 fills the firmament and is the air element or 
 transparent matter, lastly there is opaque matter 
 which composes the crust of the earth. The 
 place of this matter is somewhat unsatisfactorily 
 explained in the system of Descartes. He sug- 
 gests that the sun spots are a kind of scum which 
 collects on the surface of the mass of spherical 
 matter, forming the denser masses which are 
 thrown off and become celestial bodies travelling 
 in a straight line until they are captured by 
 some vortex into which they enter. He supposes 
 that our planets may have come from other 
 systems and been caught up in the solar vortex. 
 And he also supposes that the earth is itself a' 
 vortex within the solar vortex with a similar 
 
THE INVERSE SQUARE 85 
 
 structure, the fire element at its centre, the 
 air element surrounding this, and the opaque 
 crust on the circumference. The obvious diffi- 
 culty of this theory, which Descartes never saw, 
 and which is only obvious to us since Newton's 
 discovery, is that according to it the heavier the 
 matter on the circumference the greater should be 
 its tendency to obey centrifugal force and fly off. 
 Descartes's theory therefore offered an explana- 
 tion why the planets do not fall into the sun, but 
 it had no explanation to offer as to why they keep 
 a constant position in the solar vortex. It could 
 explain why movement is centrifugal or centri- 
 petal at the equator, not why it is radial to the 
 centre from every point on the circumference. 
 And lastly, it made no attempt to explain the 
 behaviour of heavy objects on the surface of the 
 globe. 
 
 Let us now turn to Newton's theory of gravi- 
 tation. There is an active power which impresses 
 on all bodies a tendency to move towards one 
 another. In the celestial bodies this power acts 
 in inverse ratio of the squares of the distances of 
 the centres of the masses and in direct ratio of the 
 masses. The power is called attraction when we 
 refer to the centre, gravitation when we refer to 
 
86 PRINCIPLE OF RELATIVITY 
 
 the bodies which fall to the centre. It is this 
 power which causes bodies, liberated on the 
 surface of the earth, to fall to the earth in direct line 
 to the centre. The same force probably acts on 
 the light emitted by luminous bodies, but we are 
 ignorant in what proportion and have no means of 
 discovering. It was not, however, till long after 
 Newton had conceived the idea, that he was able 
 to confirm the theory and formulate the universal 
 law. The account of his difficulties, which 
 Voltaire has given us, is of special interest for the 
 light it throws on his character as well as on his 
 method. To determine the amount of the 
 attraction of the earth on the moon he must know 
 the radius of the earth and the distance of the 
 moon. Here is Voltaire's account. " Newton 
 was at the disadvantage that the only data he 
 possessed on which to base his measurement of 
 the earth were the faulty calculations of English 
 pilots who reckoned sixty English miles to a 
 degree of latitude, the true amount being more 
 nearly seventy. And yet there had been a more 
 correct measurement of the earth. An English 
 mathematician, Norwood, had in 1636 measured 
 fairly accurately a degree of the meridian and had 
 discovered it to be about seventy miles. But 
 
NEWTON'S DIFFICULTIES 87 
 
 although this measurement had been made thirty 
 years before, it was unknown to Newton. The 
 civil wars which had afflicted England (always as 
 disastrous for science as for general prosperity), 
 had buried in oblivion the only exact measurement 
 of the earth which then existed, and there was 
 nothing available but the vague estimate of the 
 pilots. Employing this measurement it was 
 found that the moon was too near the earth and 
 the expected equations did not come out right. 
 Newton did not try to supplement his theory by 
 forcing nature to accord with his ideas. He gave 
 up his great discovery, notwithstanding that the 
 analogy with the other stars had appeared to make 
 it so probable, and though it seemed to come so 
 little short of demonstration. This example of 
 good faith alone deserves to give great weight to 
 his opinions." 
 
 Later on more exact measurements were made 
 in France giving twenty-five leagues as a degree 
 of the earth. This gave the distance of the moon 
 as sixty racii of the earth and was exactly what was 
 required to give Newton the demonstration of his 
 theory. 
 
 Newton's theory is that the moon is travel- 
 ling by its own inertia in a straight line and at the 
 
88 PRINCIPLE OF RELATIVITY 
 
 same time being attracted by the force of gravity 
 towards the earth's centre, the influence being 
 mutual, the composition of the forces giving the 
 moon's orbit. By applying Kepler's formula 
 he could calculate the exact amount of the gravi- 
 tational force, or the weight of the moon on the 
 earth at its distance from the earth's centre. 
 
 Why was this theory unacceptable to the 
 Cartesians ? The main reason is dear. It 
 proposed the very thing which to their principle 
 was most abhorrent. Gravitation was an occult 
 influence, and the Cartesians would accept no 
 explanation which was not wholly intelligible and 
 explicable by the aid of simple mechanics. 
 Further, it raised the whole problem of action at 
 a distance. According to the Cartesians a body 
 might exercise pressure on another body, and in 
 fact this was a constant phenomenon, but in every 
 case the propagation was instantaneous by reason 
 of the continuity of matter in the vortex. The 
 bell rings at the same moment at which the cord 
 is pulled. In this way the Cartesians explained 
 how the sun, though distant, could be the source 
 of the sensation of light. No emanation passes 
 from the sun to our organs of sight, but a pressure 
 is exerted and propagated through the transparent 
 
ACTION AT A DISTANCE 89 
 
 matter of the firmament. In like manner they 
 explained the tides. The pressure of the moon, 
 propagated through the intervening medium, 
 depresses the liquid element on the surface of the 
 earth causing it to bulge in the tidal wave. Action 
 at a distance was puzzling and even disconcerting 
 to Newton, and although if it take place in fact 
 it has to be accepted, on the principle that nothing 
 is impossible to God, yet he inclined to the hypo- 
 thesis that the space, in which the masses in the 
 solar system move, is pervaded with a subtile 
 substance, an ether, through which the influence 
 of attraction and gravitation is conveyed. 
 
 It will be seen then that Newton's discovery 
 demanded a complete revision not only of the 
 special hypothesis of the Cartesian vortex but of 
 the whole philosophical concept on which the 
 Cartesian mechanism was based. Attracting and 
 gravitating masses, forces determined by the 
 inverse ratio of the squares of the distances 
 between the centres of the masses, were clearly 
 incompatible with the concept of a material 
 substance whose essence was extension alone and 
 whose form was determined by relative movement 
 within a self-contained and self-sufficing system. 
 The new system required a framework of absolute 
 
90 PRINCIPLE OF RELATIVITY 
 
 space and time space which was independent of, 
 and indifferent to, the masses acting on one another 
 within it, and a time flowing indifferently to the 
 changes which it measured. 
 
 Newton therefore takes us back to the old atomic 
 theory. He re-affirms the void and a matter which 
 is atomic in the sense that it may be impressed 
 with movement (for to deny the atom would be 
 in effect, as we have seen, to deny the possibility 
 of movement). But it is not a simple return. 
 There is a notable advance in the fact that the 
 theory of gravitation has completely overcome the 
 ancient difficulty regarding absolute direction. 
 The problem which exercised Democritus and 
 produced Epicurus's arbitrary hypothesis of an 
 " inclination " is replaced with this law of uni- 
 versal gravitation. The mysterious force of 
 attraction, which observation and experiment have 
 required us to accept, remains a mystery in natural 
 philosophy. 
 
 From the standpoint of physical science the 
 gain seems enormous. The whole scheme of the 
 universe is simplified and settled on a basis of 
 common-sense. We are provided with absolute 
 standards of reference and appear to have no 
 obstruction to the unlimited advance, through 
 
THE SENSORIUM 91 
 
 experimentation, of exact knowledge. But the 
 new concepts raised very serious difficulties in 
 philosophy. To Newton himself the materialism 
 of his system presented no difficulty. He held 
 firmly to the necessary existence of God, and the 
 creation or annihilation of matter seemed to him 
 to come easily within his conception of the power 
 of God. The old difficulty of the atomists that 
 matter in its very concept was a necessary exis- 
 tence, and therefore eternal and indestructible, did 
 not trouble him. " It seems probable to me," 
 he says, " that God in the beginning formed 
 matter in solid, massy, hard, impenetrable, move- 
 able particles ; of such sizes and figures, and with 
 such properties, and in such proportions to space, 
 as most conduced to the end for which he formed 
 them." But space and time themselves what 
 was their relation to God ? All existence 
 depends upon them and they depend on nothing 
 and their non-existence is unthinkable. At the 
 close of his questions in the Of tics he says: " Do 
 not these phenomena of nature make it clear, 
 that there is a being- incorporeal, living, omni- 
 present, who in infinite space as in his sensorium 
 sees, discerns and understands everything most 
 intimately and with absolute perfection ? " 
 
92 PRINCIPLE OF RELATIVITY 
 
 By the term sensorium Newton expressed his 
 idea of the relation of God to infinite space. God 
 is not in space and space does not contain God, 
 it is part of God's nature. It was a difficult, 
 and ambiguous expression and of course not 
 allowed to go unchallenged. It was the subject, 
 of a correspondence between Leibniz and Newton's 
 most famous disciple, Clarke, in 1715-16, a 
 correspondence which was only interrupted by 
 Leibniz's death. What Newton intended by the 
 doctrine was that allowing, as we must, for the 
 impossibility of expressing God's nature in human 
 terms, we have to admit that nothing can act, 
 know or see where it is not, and when therefore 
 we affirm of God that he is omnipresent we mean 
 that he acts, knows and sees at every point and in 
 all points of space. Space therefore is God's 
 sensorium. Leibniz criticized this as being tanta- 
 mount to making God's relation to the world 
 analogous to the relation of the soul to the body. 
 The sensorium would then represent in God's 
 nature what the pineal gland represented in 
 human nature, in Descartes's theory of the 
 relation of the soul to the body. The real 
 difficulty for Newton was, that however he 
 might wish to conceive -God's relation to nature. 
 
EXPERIMENTAL PHILOSOPHY 93 
 
 the absolute existence of space and time as 
 physical realities was fundamental for his con- 
 ception of gravitation as the attraction of masses. 
 Space is not where God places masses, and 
 time is not when God elects to create them, for 
 then we should have to say that if God had 
 chosen a different place or a different time 
 for creation, the different places and times would 
 have been not different but identical, and this 
 involves absurdity. Physical science, however, 
 has ignored these philosophical difficulties and 
 framed its concepts on the supposition of absolute 
 space and time, and this pure assumption has 
 come to be a common-place of every day 
 thought. 
 
 To go back to the distinctive work of Newton 
 and the formulation of the universal law of gravi- 
 tation as a consequence of his discovery, great 
 light is shed on his own conception of the nature 
 of his work and on its relation to philosophical 
 theory by the closing passage of the Principia 
 Philosophiae which I will quote. It also brings 
 out the beauty of his personal character. 
 
 " So far I have expounded the force of gravitation 
 by celestial phenomena and by those of the sea, but 
 I have in no way attempted to assign the cause. 
 
94 PRINCIPLE OF RELATIVITY 
 
 That force comes from a power which penetrates 
 to the centre of the sun and of the planets, without 
 any diminution of activity ; and it acts not in 
 proportion to the quantity of the surfaces of the 
 particles of matter (as mechanical causes do), but 
 according to the quantity of solid matter ; and its 
 action extends on all sides to immense distances, 
 diminishing always in exact ratio according to the 
 square of the distances." 
 
 " I have not tried to deduce the cause of these 
 properties of gravitation from the phenomena 
 and I make no hypotheses. For whatever is not 
 deduced from phenomena is a hypothesis ; and 
 hypotheses, whether they are metaphysical or 
 whether they are physical, whether they presup- 
 pose occult qualities or mechanical qualities, have 
 no place in experimental philosophy. In this 
 philosophy propositions are deduced from pheno- 
 mena and general propositions are obtained by 
 inference. Thus the inpenetrability, the mobility 
 and the impetus of bodies and the laws of their 
 movements of gravity have been set down. And 
 it is shown that gravity really exists, and that it 
 acts according to the laws I have expounded and 
 that it applies to all movements of the celestial 
 bodies and of our sea." 
 
HTPOTHESES NON FINGO 95 
 
 " It is not now possible to add anything 
 concerning the very subtile spirit pervading heavy 
 bodies and latent in them ; by whose force and 
 actions the particles of bodies are mutually drawn 
 together at minimal distances, and the contiguous 
 cohere ; and concerning electrical bodies acting 
 at greater distances, now attracting now repelling 
 neighbouring bodies ; and how light is emitted, 
 reflected, refracted, inflected, and also how it 
 warms bodies ; and how all sensation is excited and 
 how in animals the limbs are moved by volition, 
 to wit, by the vibrations of this spirit propagated 
 through the solid threads of the nerves from the 
 external organs of sense to the brain and from the 
 brain to the muscles. These cannot be expounded 
 in a few words, and at present there are not 
 sufficient experiments by which to determine 
 accurately and demonstrate the laws of action of 
 this spirit." 
 
CHAPTER VI 
 
 LEIBNIZ AND THE THEORY THAT SPACE 
 IS THE ORDER OF COEXISTENCES 
 
 THE effect of Newton's discovery of the law of 
 gravitation was to reinstate the old theory of 
 atoms and the void in the scientific conception of 
 nature-philosophy. Newton could offer no ex- 
 planation of gravitation, but he definitely estab- 
 lished the fact. There actually is an influence or 
 force in masses, whatever their nature, which draws 
 them out of their path, whatever its direction, 
 and whatever the velocity of their movement in it, 
 and this force is proportional to the distances of 
 the masses. But though it restored the old 
 concept of atoms and the void it seemed in doing 
 so to supply just that principle, ignorance of 
 which had constituted the gravest defect in the 
 ancient theory. Instead of the purely fanciful 
 theory of the clinamen proposed ad hoc by Epicurus, 
 
 we now have a principle of attraction which 
 
 96 
 
DISPROOF OF THE VORTEX 97 
 
 however mysterious, can be formulated as an 
 invariable law, and tested by actual experiment. 
 It lent itself to theological interpretation also, for 
 in giving matter weight God was endowing the 
 creation with the principle of its order and 
 arrangement. 
 
 The success of the new physical discovery in 
 upsetting the mechanical theory of the vortices 
 was complete. The vortex movement is incon- 
 sistent with the gravitation phenomenon in two 
 distinct particulars. The vortex will explain 
 gravitation at the equator, for there the centrifugal 
 line offeree is at right angles to the axis and passes 
 through the centre, but if the revolution of the 
 sphere round its axis is the cause of this force, 
 then nowhere else but at the equator should the 
 falling body be directed to the centre, and also 
 weight if it be centripetal force should decrease 
 to zero at the poles. The vortex theory there- 
 fore is inconsistent with the fact that a body 
 falling freely towards any point on the surface of 
 the sphere pursues a line which if continued 
 would carry it to or near the centre. 
 
 A second fact was also plainly inconsistent with 
 the vortex theory, viz. the fact that the planetary 
 orbits are ellipses and not circles. The vortex 
 
9 8 PRINCIPLE OF RELATIVITY 
 
 will explain circular movement but not eccen- 
 tricity. 
 
 A theoretical difficulty was also pointed out in 
 Descartes's theory of the origin of motion and the 
 possibility of its origin and development in a 
 plenum. Matter at rest according to this theory 
 must, if movement is to be possible, be conceived 
 as disintegrated. Being a plenum it will consist 
 of closely packed figures with plane surfaces, let 
 us say cubes. Now Descartes supposed that 
 movement set going in this plenum would 
 cause the cubes to become spherical by grad- 
 ually wearing off the angles. But how could this 
 process start without first creating a void ? The 
 movement of the cubes cannot alter their relative 
 position without creating void. Before the move- 
 ment alters the shape of the cubes by fracturing 
 the angles, it must cause their displacement, and 
 the slightest displacement destroys the plenum. 
 
 The new philosophy, as it was called by con- 
 temporaries, rapidly and completely superseded the 
 old. It did not correct it or supplement it. It 
 did not borrow and incorporate some of its 
 principles while adding new ones of its own. 
 The Principia Philosophiae of Newton became 
 what th.e Princifia Philosophiae of Descartes had. 
 
THE NEW PHYSICS 99 
 
 been. It came to pass that Descartes died and 
 Newton reigned in his stead. The new principle 
 was an active force resident in masses, but measur- 
 able in its effects and capable of developing a 
 complete celestial mechanism. Nature was a 
 system of forces residing in masses, distributed 
 in an infinite and absolute void. The forces 
 manifested themselves by mutual influences 
 causing the masses to move and successively 
 change their relative positions in an even 
 flowing, absolute, time. 
 
 While Newton was working out the great 
 discovery which was to give a physical system 
 destined to supersede the apparently firmly 
 established system of Descartes, a contemporary 
 philosopher and rival mathematician was opposing 
 the philosophy of Descartes on purely meta- 
 physical grounds. Leibniz (1649-1716) con- 
 centrated his criticism on Descartes's conception 
 of substance and in particular strove to present 
 a rational theory of the relation of the soul to the 
 body, of mind to matter, of God to the world. 
 It is especially in their concepts of the relation 
 of God to the world, that the philosophical 
 principles of Leibniz and Newton are antagonistic, 
 and on this point Newton was most sensitive. 
 
ioo PRINCIPLE OF RELATIVITY 
 
 Newton did not recognize any theological 
 difficulty in his nature concept such as that 
 which seemed to make the old atomism essentially 
 materialistic and atheistic. He could find no 
 ground for regarding God's power over matter 
 as limited to its disposition, God could create it or 
 annihilate it at his pleasure. But what was God's 
 relation to an infinite and absolute space and 
 time ? Were they not in their very nature 
 limitations ? The only alternative was to make 
 space and time the nature of God, and Newton 
 took refuge in the attribute of omnipresence. 
 Wherever God is acting he is. This was the 
 meaning of the theory that space is God's 
 sensorium. 
 
 The theological problem which seems to us 
 to engage almost exclusively the philosophical 
 speculation of the seventeenth century has so 
 changed its form in our philosophy to-day that 
 we incline to regard it as an outgrown mythology. 
 It is, however, just as vital an issue to-day. All that 
 we have discarded is the anthropological concept, 
 but in the issue between idealism and naturalism 
 we have kept all that was essential in the old theistic 
 problem. The challenge of the old theology to 
 physical and metaphysical theories was whether 
 
THE THEISTIC PROBLEM ibi" 
 
 God was conceived in them as a possibility or as 
 a necessity. Both Newton and Leibniz believed 
 in God, believed not only that a supreme and 
 infinite being exists, but that the necessity of 
 such an existence can be deduced from the fact 
 of the universe. But while Newton argued from 
 
 what the universe is to what God's attributes 
 
 
 
 must be, Leibniz argued from what God is to 
 what the universe must be. Newton was con- 
 vinced that there is a living God, but no more 
 than Laplace, who a hundred years later carried 
 out and developed his principles, had he any 
 need of that hypothesis. Leibniz on the contrary 
 could not move one step without it. 
 
 Voltaire, who spread the fame of Newton's 
 discovery and who, as I have tried to show, did 
 more than anyone to secure its triumphant 
 acceptance, pointed his advocacy by the wit and 
 mirth-provoking satire with which he treated 
 Leibniz. Leibniz is the Dr. Pangloss of Candide. 
 Voltaire knew well the intellectual strength of 
 the philosopher he satirized. When Leibniz 
 argued against the void that the concept of such 
 a reality was inconsistent with the attributes 
 of power and goodness in God, for it would imply 
 a shortcoming or defect in creation ; that it would 
 
\6i-V- Ml-NOIPLE OF RELATIVITY 
 
 imply room for a possibility of creation which had 
 been unfulfilled and would mean that the world 
 was not the best possible ; the argument was 
 received with laughter. To-day we find it 
 difficult to think it could ever have been put 
 forward in seriousness. Yet a study of Leibniz's 
 thought will show how his whole concept of the 
 monads rests upon it. It is his principle of 
 continuity. The world God has created is not a 
 patchwork of stuffs but a city of active workers. 
 The creation is not an abiding-place for souls to 
 dwell in, or a stage whereon to display their 
 activities, the world is souls, and only souls 
 exist. In the perfect city every citizen realizes 
 his own individuality and in so doing fulfils 
 the higher life of the community, the com- 
 pound or composite existence. To suppose 
 a void then is to suppose that something 
 essential to the perfection of the system is left 
 unprovided for or unfulfilled. This is incon- 
 sistent with the idea of a perfect God. Voltaire's 
 biting satire of Dr. Pangloss, preserving his 
 philosophical conviction unshaken when over- 
 whelmed with accumulated disasters, made Leib- 
 niz appear a kind of Don Quixote in the popular 
 philosophy of the eighteenth century. It is 
 
LEIBNIZ AND NEWTON 103 
 
 evident enough to us that whatever force the 
 satire may have against particular theological or 
 even religious theories, it has absolutely none 
 against the conception in its scientific and philo- 
 sophical application. 
 
 Leibniz inaugurated the most momentous 
 movement in modern philosophy. He is the 
 founder of modern idealism. If we divest his 
 doctrine of the peculiar theological form in 
 which he clothed it we see the origin of his 
 theory in the profound dissatisfaction with the 
 materialist attempts to give a rational explanation 
 of the universe. The revolution which his 
 concept of substance marks, is to some extent 
 analogous to the revolution which Newton's 
 discovery produced in natural philosophy. It 
 substituted the concept of a dynamic for that of 
 a static reality, it replaced the notion of stuff with 
 the notion of force. The atoms of Leibniz are 
 forces, activities, just as the masses of Newton 
 attract and repel. But there the analogy ends. 
 No greater contrast is presented in philosophy than 
 the divergency of these contemporary philosophers. 
 
 In one instance only they appeared as rivals, 
 and on one side at least the recriminations were 
 bitter. This was the dispute concerning the 
 
io 4 PRINCIPLE OF RELATIVITY 
 
 differential calculus which each claimed to have 
 discovered. When Leibniz published his work 
 he was charged with having stolen it from un- 
 published writings of Newton which he was 
 known to have seen, and Newton gave substance 
 to the charge, which he evidently believed, in an 
 innuendo in the Preface to the Optics. Leibniz 
 resented the charge bitterly and defended him- 
 self against it. The sad thing is that one generous 
 word from Newton would have closed the matter, 
 for Leibniz addressed to him a personal appeal. 
 The letter remained unanswered. 
 
 Leibniz's philosophy was a doctrine of the true 
 atoms of nature, the mode of their activity, 
 and the way in which they combine to form the 
 universe. The old atomic theory stood con- 
 demned by its unsolved contradictions and 
 constituted therefore a continual offence against 
 human reason. The fundamental principles of the 
 intellect are the law of identity and the law of 
 sufficient reason, and both are flagrantly violated 
 in the theories of atoms and the void. Divisi- 
 bility is part of the concept of extension. In 
 declaring anything extended we are predicating 
 of it an infinite divisibility. To say that the atom 
 is indivisible and impenetrable and yet extended 
 
THE MONADS 105 
 
 is a self-contradiction. There are atoms, but 
 they are not extended or parts of extension, for 
 they are indivisible and without parts. These 
 atoms are the monads. They are spiritual 
 entities. They are the subjects of experience and 
 their activity is perception. The universe is 
 mirrored into each monad. To this doctrine 
 Leibniz was led by considering the contradictions 
 in the old concept of atoms and the void. I will 
 quote his own account written in the last year 
 of his life. 
 
 1 When I was a youth I accepted the void and 
 the atoms, but reason saved me from the imagina- 
 tion which makes sport of us. Imagination 
 limits our researches, fastens our meditation as 
 it were with a nail, makes us think we have found 
 the ultimate first element, the non-plus ultra. We 
 want nature to stop where our imagination reaches 
 its limit ; we want nature to be finite like our 
 mind ; but this is to fail to rise to a knowledge 
 of the greatness and majesty of the author of 
 nature. The minutest corpuscle is actually sub- 
 divisible to infinity, and contains a world of new 
 creatures which would be absent from the universe 
 if the corpuscle were the atom, that is, a body all 
 in one piece and with no subdivisions." 
 
106 PRINCIPLE OF RELATIVITY 
 
 It is important to notice here that Leibniz 
 conceives the monads to be in their number 
 actually infinite whereas the atom was conceived 
 as in its nature finite. It is interesting to see also 
 how this concept of the nature of the monad is 
 connected with the rejection of the other article 
 of the old theory, the void. I have already referred 
 to Leibniz's argument against the void as being 
 derogatory to the perfection of God. Here is an 
 allusion to that argument which I quote because 
 it connects it with his general doctrine. 
 
 " Leaving aside other reasons against the void 
 and atoms, here are those I base on the perfection 
 of God and on the principle of sufficient reason. 
 I assume that every perfection which God could 
 put into things without derogation to the other 
 perfections he has put there. Now imagine to 
 yourself a space completely empty, surely God 
 could put some matter there without derogating 
 at all from other things. It follows then that he 
 would do so, consequently there is no completely 
 empty space : all is full. The same argument 
 proves that there is no corpuscle which is not 
 subdivisible." 
 
 Such was Leibniz's attitude to the old atomic 
 theory. What then was his position with regard 
 
THE REAL ATOMS 107 
 
 to the doctrine of Descartes ? From his earliest 
 period his interest and his research was turned 
 inward on the mind rather than outward on nature. 
 The main point of the Cartesian system which 
 exercised him was the theory of the relation of 
 mind and body. In that relation the irreconcilable 
 nature of the dualism of the two substances, each 
 distinguished by an essential and mutually exclu- 
 sive attribute, is most pronounced. The logical 
 development of Descartes's principle in the 
 monistic philosophy of Spinoza only served to 
 emphasize the poverty of the concept. Leibniz's 
 train of thought was probably along some such 
 line as this. It started with the concept of God 
 and creation not as an assumption introduced to 
 explain the existence and origin of the world, but 
 as presenting the problem of existence and origin 
 in its most complete and definite form. Give 
 God matter and movement, could he then, as 
 Descartes supposed, create a world ? Clearly 
 not, for what God has created are finite individuals 
 active subjects, moral agents. These individual 
 activities are the real existences, and the universe 
 is wholly composed of them. What we have to 
 study then is the nature of these active substances, 
 the monads, the real atoms of nature, their 
 
io8 PRINCIPLE OF RELATIVITY 
 
 relations to one another and how they come to 
 form a world. Take then my own existence. I 
 am a monad, an active centre, an agent, the whole 
 universe is mirrored into that centre, focussed 
 there, and my activity consists in perceiving. The 
 whole universe consists of my perceptions, but 
 only an infinitesimal portion of these are clear and 
 distinct perceptions, the rest are massed together, 
 confused, obscure and undiscerned. I am also 
 self-conscious, aware of myself as perceiving. My 
 monad, the monad which is me, is apperceptive. 
 But then I am in relation to a body, my mind is a 
 dominant monad, and it works in complete har- 
 mony with the body, and yet this body is totally 
 different and distinct in its nature from the mind. 
 What is it ? It. consists of monads but of inferior 
 monads. They are infinite in number, for no 
 principle exists which imposes a limitation on 
 them, and yet each is individual, an active centre 
 mirroring the universe from its own point of view. 
 In this relation of mind and body there is clearly 
 a harmony, and it is an original harmony. It 
 cannot have been brought about by chance for it 
 is of the essence of the relation. Here then in 
 this fact of mind and body I have a reality which 
 is a compound, decomposable to infinity, and yet 
 
THE MONAD'S PERCEPTIONS 109 
 
 consisting of simple elements which are individual 
 and therefore indivisible. But this is typical of 
 the whole universe and reveals its nature and 
 origin. The harmony is pre-established, for it is 
 the essence of the reality. If the universe came 
 into existence by an act of creation the harmony 
 entered into the creative design and was brought 
 into existence with the universe. 
 
 Such with its necessarily theistic form seems to 
 me to be the train of thought which produced in 
 Leibniz's mind the idealist principle which he 
 proposed to substitute for the materialistic prin- 
 ciple, condemned on the ground first of self- 
 contradiction and secondly of failure to satisfy 
 the principle of sufficient reason. Though all 
 the monads are alike in their nature they differ in 
 their degree. This difference of degree is wholly 
 concerned with the ideas of the monad. Each 
 monad is in a necessary relation to all parts of the 
 universe for its ideas are relative to the whole 
 universe. And further, the monads do not differ 
 from one another in the number of their ideas, 
 for this number is infinite, they differ only in the 
 degree of clearness which their ideas possess. 
 Accordingly Leibniz supposes a hierarchy of the 
 monads based on the clearness or obscurity of 
 
no PRINCIPLE OF RELATIVITY 
 
 their ideas, that is, the perceptions in which their 
 activity wholly consists. There are three classes 
 of created monads : (i) Simple monads, the 
 elements of matter which have no clear thought ; 
 (2) the souls of animals, which have some clear 
 but no distinct ideas ; (3) finite minds, which 
 have confused and also some clear and distinct 
 ideas. The supreme or uncreated monad, God, 
 has only adequate ideas. 
 
 The meaning is that, taking my own mind, 
 for example, as the monad, the whole universe 
 consists in its perceptions, there is no passing 
 beyond perception, or as Leibniz said, there are 
 no windows by means of which anything can come 
 in or pass out, each mind contains the universe. 
 But whereas I have certain clear ideas and certain 
 distinct ideas, the great mass of my perceptions 
 outside these are a confused, obscure, blended 
 heap. Just as, to take one of his illustrations, the 
 sound of the waves on the seashore consists of an 
 infinite number of small sounds, which are not 
 heard by me as each clear and distinct, but in 
 the blur of one agglomerated undifferentiated 
 mass. There are degrees therefore of confusion 
 and obscurity ranging from its upper limit in the 
 absolute adequacy of the perceptions of God, all 
 
THE DOCTRINES COMPARED in 
 
 of whose ideas are clear and distinct, and its lower 
 limit in the indefinite, possibly complete, confusion 
 of perceptions in the simplest monad. 
 
 The difference between Newton and Leibniz 
 as to the nature of the elements of matter is 
 brought out with clear decision by Voltaire. 
 
 " The opinion of Newton is perhaps as modest 
 as human opinion can be. It is limited to 
 believing that the elements of matter are material, 
 that is, there is an extended and impenetrable 
 existing thing into the inner nature of which we 
 enter ; God can divide it to infinity or he can 
 annihilate it : he has not done so, and he pre- 
 serves its extended and indivisible parts as the 
 basis of all the products of the universe. 
 
 " Perhaps on the other hand no bolder theory 
 than that of Leibniz has ever been put forward. 
 Setting out from the principle of sufficient reason 
 he tries to penetrate even into the deepest origins 
 and into the inexplicable nature of the elements. 
 Each body, he says, is composed of extended parts, 
 but of what are these extended parts composed ? 
 They are actually divisible and divided to infinity, 
 therefore you can never find anything which is not 
 extension. Now to say that extension is the 
 sufficient reason of extension is simply to argue 
 
ii2 PRINCIPLE OF RELATIVITY 
 
 in a circle and affirm nothing at all. The ground 
 or cause of extended beings must be in non- 
 extended beings, that is, in simple beings or 
 monads. Matter is therefore an assemblage of 
 simple beings." 
 
 If I have succeeded in making the doctrine of 
 the monads understood, it will be seen that it 
 is quite as inconsistent with the concept of 
 absolute space or void as it is with the concept 
 that extension is the essence of material substance. 
 The monads being non-extensive but all-inclusive 
 activities, their relations, if we describe them as an 
 assemblage, cannot be juxta-positions in a space 
 external to them and indifferent to them. The 
 universe mirrored into each active centre is not a 
 particular part of some vast expanse conceived as 
 a container. Space therefore is a reality which 
 must fall within the universe as the monad per- 
 ceives it mirrored, and it cannot fall outside the 
 monad, even were the concept of outside consis- 
 tent with the affirmation of the monad. For 
 Leibniz therefore space is neither a clear nor a 
 distinct idea, and consequently space does not 
 exist for God. It belongs to our confused and 
 obscure perception, and in fact denotes the 
 obscurity and confusion which is inherent in 
 
THE NEED OF CREATION 113 
 
 our view of the universe. It is the mode in which 
 we present to our mind as co-existelit with our 
 clear and distinct ideas the infinite residue of 
 indistinct perceptions. Space therefore is not 
 a thing, not something which is, it is an order. 
 In this respect it holds precisely the same rank 
 as time. Space is the order of co-existences, 
 time the order of succession. Neither is a reality, 
 both are names for the confused blur of per- 
 ceptions against which, as against a background, 
 our clear and distinct ideas stand out. 
 
 There is then a complete contrast between the 
 two conceptions the idealist universe of Leibniz, 
 the materialist universe of Newton. The ultimate 
 principle of the first is universal mind, represented 
 as a supreme God, perfect in wisdom and infinite 
 in power. His need of creation is not a mechan- 
 ical need, founded in some impulse, rational or 
 irrational, to construct complex systems or direct 
 simple movements to complex effects, on the con- 
 trary it is purely a spiritual need, and creation is 
 the bringing into existence of active subjects, so 
 perfectly harmonized in their range, so fitted into 
 the scheme by their degree, that no one is redun- 
 dant and no possibility unrealized. God can 
 create or annihilate the universe, but not in part, 
 
n 4 PRINCIPLE OF RELATIVITY 
 
 only as a whole. This was the famous concept 
 of " the best of all possible worlds," which since 
 Voltaire has mainly served as a theme for jest. 
 Divested of its theistic setting, and studied as it 
 should be, that is, as a metaphysical research 
 into the nature of reality and the origin of our 
 concept of external nature, it will answer any 
 logical challenge. 
 
 Newton's conception on the contrary has 
 commended itself to the scientific mind, and it has 
 in its pre-suppositions unquestionable advantages 
 in practice. Logically and metaphysically it 
 is riddled with contradictions, and these appear 
 most strikingly in its theological consequences. 
 Newton could not be indifferent to the theistic 
 difficulty, however little importance it may seem 
 to have had for his successors. Hence the theory 
 of the sensorium. It was in the last years of 
 Leibniz's life (1715-16) and in Newton's old age 
 that this theistic problem was discussed in the 
 correspondence between Leibniz and Clarke, one 
 of the most valuable of the philosophical remains 
 included in the editions of Leibniz's works. It 
 was an unfinished correspondence, for it was 
 interrupted by death. 
 
 The occasion of this correspondence was a 
 
NATURE AND GRACE 115 
 
 letter which Leibniz wrote to the Princess of 
 Wales, afterwards Queen Caroline, wife of 
 George II., in 1715. In it he said : " It seems 
 that even in England natural religion has grown 
 very weak. There are many who hold that souls 
 are corporeal and others who even hold that God 
 himself is corporeal. Newton says that space is 
 the organ which God uses in order to be conscious 
 of things. But if God is in need of means in 
 order to be conscious of things, it follows that 
 these things cannot wholly depend on God. 
 They cannot be his production. Newton and 
 his followers have a still odder notion of God's 
 work, for according to them God has every now 
 and again to wind up his work as we wind up a 
 watch which would otherwise stop. God has 
 not, it seems, had enough foresight to give his work 
 perpetual motion. Indeed the machine which 
 God has made is so imperfect, according to them, 
 that it requires polishing up every now and again 
 by a special effort, and even needs regulating. 
 Like a watchmaker he reveals the defects of his 
 watch by the number of times he has to correct 
 and retouch it. In my view the same force and 
 vigour is everywhere in evidence, passing from 
 one thing to another according to laws of nature 
 
n6 PRINCIPLE OF RELATIVITY 
 
 and the perfect order pre-established. If God 
 performs miracles it is not because nature requires 
 them, it is on account of grace. To judge other- 
 wise is to entertain a very low idea of God's 
 wisdom and power." 
 
 The sting of this letter so far as it concerns 
 Newton is the reference to the theory of the 
 sensorium. The Princess asked Newton to reply 
 to it and he commissioned his disciple Clarke to 
 defend him. It is in the correspondence which 
 followed that Leibniz expounds with great clear- 
 ness his theory of space. Real absolute space, 
 he declares, is the idol of some modern English. 
 He uses the word idol, he explains, not in its 
 theological but in its philosophical meaning, and 
 he quotes Lord Bacon's idola trMs, idola pec As. 
 If space is a real thing, as these writers contend, 
 then it is eternal and infinite, and they must 
 identify it with God. Either it is God himself 
 or it is an attribute of God, his immensity. But 
 then space has parts how does that apply to 
 God ? " Space for me," he says, " is purely 
 relative as also time is. Space is an order of 
 co-existences as time is an order of successions. 
 Space marks in terms of possibility an order of 
 things which exist in the same time so that they 
 
THE OPPOSING PRINCIPLES 117 
 
 exist together. The manner of their existing 
 is not in question.'* When we see several things 
 together we perceive the order in which they 
 stand to one another. Were it otherwise there 
 would be no sufficient reason for the world being 
 here not there, now not then. 
 
 In Clarke's reply there occurs a remarkable 
 passage intended as a reductio ad absurdum y but 
 which sounds to us almost as an anticipation of 
 the negative result of the Michelson-Morley 
 experiment which led to the first theory of relati- 
 vity. "If space were no more than the order of 
 co-existent things, it would follow that should 
 God make the whole world move in a straight line 
 with any velocity he liked to choose, it would 
 still be always in the same place, and when the 
 movement ceased nothing would sustain a shock." 
 
 We have then in the contrast between the 
 principles of Newton and Leibniz, the distinction, 
 in its full intensity and most emphatic expression, 
 between a nature-philosophy based on a material- 
 istic principle and a nature-philosophy based on a 
 spiritualistic or idealistic principle. Both prin- 
 ciples are directed to mathematical and physical 
 researches, both are conceived as the trye prin- 
 ciples which underlie the science of nature. 
 
ii8 PRINCIPLE OF RELATIVITY 
 
 Neither has stopped dead, neither has ceased to 
 undergo development, but each has chosen a 
 separate and a widely divergent path. So far as 
 experimental science is concerned the principle 
 of Leibniz has been rejected absolutely, at times 
 with contumely and scorn, while Newton's prin- 
 ciple has seemed to be confirmed by the sure 
 advance of science during the last two hundred 
 years. It is with a shock of surprise that we are 
 receiving to-day a challenge to that principle 
 which seems to question the very foundations of 
 the concept on which it is based. Still more 
 strange is it that physical science itself is seeking 
 a principle which will enable it to co-ordinate 
 observations from individual centres of experience 
 (monads), without the aid of, and recognizing 
 the impossibility of having, absolute standards 
 of reference independent of the observers. In 
 effect we are proposing in mathematical and 
 physical science to abandon Newton's philosophy 
 and adopt that of Leibniz. 
 
CHAPTER VII 
 
 THE MODERN SCIENTIFIC REVOLUTION 
 AND ITS LEADERS 
 
 IT is curious that we should associate the maxim 
 hypotheses non Jingo with the scientific method of 
 experiment, and suppose that the making of 
 hypotheses is a particular vice of speculative 
 metaphysics. The exact contrary is true. It 
 is the scientific experimenter who makes hypo- 
 theses, and it is no reproach to his method that he 
 does so, it is its essential feature and by no other 
 means can he advance. On the other hand it is 
 the philosopher who vitiates his method the 
 moment he proceeds to make hypotheses, for the 
 assumptions he introduces into his premises will 
 assuredly re-appear in his conclusions. It is the 
 philosophers, of whom we may take Descartes 
 and Hume as types, who are entitled to inscribe 
 
 on their banners Newton's often quoted motto. 
 
 119 
 
120 PRINCIPLE OF RELATIVITY 
 
 This does not mean that Newton was vainly 
 boasting. He was indeed a true philosopher in his 
 simple and direct acceptance of fact, his disregard 
 of its conflict with accepted theory, and his attempt 
 to interpret it without any respect for precon- 
 ceived opinion. I shall endeavour to show that 
 Einstein, in his formulation of a new theory of 
 gravitation, in this respect follows his great 
 predecessor, for the new theory is not a hypothesis, 
 it is the acceptance of paradoxical facts in spite 
 of the paradox, because they are based on experi- 
 ence and confirmed by experiment, and the 
 formation of a principle in conformity with them. 
 
 It will be well to begin by stating in definite 
 and positive terms what the new principle is 
 without reference to the experiments which have 
 called for it. 
 
 The relativity principle of the classical mechanics 
 supposes an absolute space, an absolute time 
 and an infinite, that is, an infinitely variable, 
 velocity. Transformations of measurements for 
 different systems of reference, spatial or temporal, 
 can be simply and arithmetically calculated and 
 expressed as variations of velocity. Euclid's 
 postulates are applicable and universally valid. 
 Observational deformations of Euclidean figures 
 
THE CLASSICAL MECHANICS 121 
 
 are apparent, not real, and easily explicable 
 as perspectives, due to the conditions of the 
 observer. Let me illustrate it by supposing that 
 I travel, say from London to Edinburgh, in a 
 slow train while you travel the same route by an 
 express, the space and the time have not varied, 
 they are identical for each of us, but the apparent 
 stretching of the space and time for me, and their 
 apparent contraction for you, are simply adjusted 
 by taking into account the velocity of the train, 
 say thirty miles an hour for me, sixty miles an 
 hour for you. We express this relativity by say- 
 ing that we have moved through the same space, 
 you in half the time lived through by me. This 
 is the commonly accepted principle. 
 
 The new principle of relativity has two stages, 
 the special principle and the general principle. 
 The difference is not intensive but extensive. 
 The first was confined to a definite phenomenon, 
 the velocity of the propagation of light in vacua. 
 The second is the application of the same principle 
 to all laws of nature. The special relativity 
 principle is based on the fundamental concept that 
 space is variable and that time is variable, and that 
 there is one finite velocity which is constant and 
 also a limiting velocity, that is, a velocity which 
 
122 PRINCIPLE OF RELATIVITY 
 
 cannot be exceeded, but to which the approach is 
 infinite. This constant velocity is that of the 
 propagation of light. The concept of this 
 velocity as constant and as a limit is not as 
 arbitrary as it at first appears to the uninitiated, for 
 as matter of fact we are absolutely dependent 
 on light signals for sending messages, or measur- 
 ing distances, or estimating intervals between 
 events. The special principle of relativity is 
 that observers in systems moving uniformly in 
 relation to one another do not use the same 
 standards in measuring space and time dimensions, 
 but each observer employs a standard which 
 supposes his own system at rest. When then 
 there are observers of the same events, in different 
 systems of reference, moving uniformly relatively 
 to one another (as in the two trains), each observer 
 sees the light signals propagated with the same 
 velocity, but the difference due to the movements 
 of the two systems, relatively to one another, is 
 in the space and time which are different. It is 
 easy to illustrate. Let us go back to the two 
 railway journeys. According to the classical 
 mechanics one is double the velocity of the 
 other. According to the principle of relativity 
 the velocity of each is identical because in each 
 
SPECIAL RELATIVITY 123 
 
 train the observer is at rest. The difference 
 is in the space and the time. These are 
 elongated for the traveller in the slow train, 
 shortened for the traveller in the express. To 
 common-sense this appears contradictory, but 
 reflection will show that it is a simple alternative 
 to the common-sense view and logically an exact 
 equivalent. It is simply equal to saying, what 
 is also fact, that in our two journeys neither I 
 nor you moved at all, but our destination moved 
 to us, and in doing so traversed double the space 
 in double the time in coming to me that it did 
 in coming to you. 
 
 The special relativity principle was confined 
 to the consideration of observers in systems of 
 reference in uniform movement of translation 
 relatively to one another as, for example, in the 
 two trains. The general relativity principle is 
 the extension of the special principle to non- 
 uniform systems, in particular to rotational 
 systems, and it affects the concept of the laws of 
 nature. It declares that when the movement of 
 a system relatively to other systems is non- 
 uniform, the peculiar quality which the observer 
 experiences as influence or force can find an 
 equivalent expression in the motion of the system 
 
i2 4 PRINCIPLE OF RELATIVITY 
 
 for an observer in another system. The rota- 
 tional movement of a system which gives rise 
 to the phenomenon which an observer at rest 
 within it describes as weight or gravitation, that 
 is, the falling of loose bodies towards the 
 centre of rotation, might appear to an observer 
 in another system to be simply the movement 
 of the rotating system itself towards other 
 systems moving non-uniformly in relation to 
 it. It further deduces that for all non-uniform 
 systems the spatial coordinates (length, breadth, 
 depth) are not Euclidean. In a gravitational 
 field space for all observers is warped or curved. 
 For the moment I am trying to state what the 
 principle is without explaining the grounds for 
 adopting it. Thus, to return to the illustration 
 of the train journeys, in the two previous uses of 
 the illustration I supposed the movement uniform, 
 and there was consequently little difficulty in 
 showing that it is exactly equivalent to regard 
 the train as not moving and the destination 
 moving, or to regard the train as moving and the 
 destination as fixed. In the one case we shall say 
 that the space and time vary and the velocity is 
 constant, in the other that space and time do not 
 change but velocity does. But now suppose 
 
GENERAL RELATIVITY 125 
 
 that the train is brought to a stop by the sudden 
 application of the brakes and that the traveller is 
 thrown heavily from his seat. There will be 
 difficulty in making him take the view that he 
 was all the while at rest, that it was the seat which 
 moved from him and the floor which moved 
 towards him. Yet this is exactly how it might 
 appear to an observer in another system, and in 
 so describing it he would have an equal right to 
 declare it true. 
 
 Suppose then we understand the principle and 
 accept it to the extent of admitting that we can, if 
 it is worth our while, always find an equivalent 
 way of describing the phenomena of movement 
 in place of the one to which we are accustomed. Is 
 it then, we shall ask, merely a matter of choice ? 
 If not, what is the necessity for disturbing our 
 ingrained common-sense principles ? The reply 
 is not an argument, but an experiment. 
 
 The experiment which has occasioned the 
 revolution in our fundamental concepts and called 
 for the formulation of a new basis of a philosophy 
 of nature was carried out by Michelson and 
 Morley in 1887 an d is described in the Philoso- 
 phical Magazine for December of that year. It 
 has been so often described since that it will be 
 
126 PRINCIPLE OF RELATIVITY 
 
 sufficient to allude to it here and go straightway 
 to the principle of it. Since Newton, and on the 
 foundations he laid down, there has been a steady 
 and continuous evolution of scientific knowledge, 
 marked by an ever widening range, an ever 
 deepening insight, and an ever surer inclusiveness. 
 The theoretical objection to his theory of gravi- 
 tation that the concept of action at a distance 
 which it involved is a concept of something 
 wholly unimaginable, was made by Leibniz. 
 Newton himself felt the force of the objection, 
 and to meet it supposed that space might be filled 
 with a subtile etherial matter. The theory of the 
 ether of space became a necessary hypothesis, 
 however, only when the undulatory theory of light 
 was formulated. With the extension of the dis- 
 coveries of electro-magnetic phenomena this ether 
 tended to become one of the fundamental concepts 
 of science. It was, however, purely conceptual, 
 and the famous Michelson-Morley experiment 
 was contrived with the primary intention of testing 
 its reality as a physical existence. The principle 
 is easily explained. The earth is in a movement of 
 translation relatively to the sun amounting to about 
 30 kilometres a second. If then we consider the 
 earth as at rest we must suppose the ether to be 
 
THE EXPERIMENT 127 
 
 streaming past it at the equivalent velocity. Now 
 it is easy to calculate the difference in time 
 required for any given uniform propagation to 
 reach a point and return in a direction which 
 crosses the stream, and to reach and return from 
 an equally distant point in the direction of the 
 stream. Thus, let us suppose, for instance, that 
 the stream is flowing 30 kilometres a second, and 
 that our signal unretarded would travel 50 kilo- 
 metres a second, and that we project it to a 
 measured point in each direction, up the stream 
 and across it, from which points it is immediately 
 reflected and returns to us. Across the stream 
 the flow will be equal to a retardation of 10 kilo- 
 metres a second, our projectile will therefore 
 travel 40 kilometres in a second and will return 
 in the same time, that is, it will go and return in 
 two seconds. Now suppose it is directed up 
 stream, it will have the full retardation and there- 
 fore will travel only 20 kilometres in the first 
 second and to arrive at the measured point, 40 
 kilometres, it will require two seconds, the same 
 time in which, across the stream, it went that 
 distance and back. It is true it will return in half 
 a second, for it will have the 30 kilometres a 
 second added to its own velocity of 50, but it will 
 
128 PRINCIPLE OF RELATIVITY 
 
 be that half-second later than the other. In the 
 actual experiment a beam of light was sent to a 
 mirror and reflected back along each axis. Light 
 has a velocity approaching 300,000 kilometres a 
 second, but the calculation was worked out to an 
 accuracy, and within a margin of error, which 
 made it still quite certain that the retardation 
 would be revealed. To the great surprise and 
 deep disappointment of the experimenters the 
 result was negative. Along each axis the return 
 was simultaneous, ^ 
 
 This was the famous experiment ; let us now 
 look at its consequences. In the first place it 
 negatived the ether hypothesis. There is no 
 ether if by ether is meant something occupying 
 space and at rest relatively to the matter which 
 moves through it. If there be ether it must move 
 with the system which is in relative translation 
 (exactly as Descartes supposed). If to suppose 
 an ether is to suppose infinite ethers, then the 
 hypothesis is useless and may be dismissed. 
 
 A second consequence is that we must suppose 
 the velocity of light in vacua to be unaffected by 
 the movement of the source. This is more 
 fundamental still, for if there be no ether or if 
 the ether be perfectly frictionless, the velocity, 
 
THE CONTRACTION THEORY 129 
 
 when the source is moving, ought to be different 
 from the velocity when the source is at rest, yet 
 the experiment proved it to be the same in each 
 case. 
 
 A third consequence is that as the laws of 
 nature remain constant for observers in moving 
 systems, the space and time dimensions in order 
 to keep the constant ratio must themselves vary. 
 
 A hypothesis to account for the negative result 
 of the experiment was put forward independently 
 by the late G. F. Fitzgerald, Professor in Trinity 
 College, Dublin, and H. A. Lorentz, Professor 
 of Physics in the University of Leiden. It is 
 generally known as the Lorentz-Fitzgerald con- 
 traction. It assumed that the dimensions of a 
 solid body moving through the ether undergo 
 slight changes, that the moving body is auto- 
 matically contracted in the line of the direction in 
 which it is moving, and that this contraction is 
 exactly equivalent to and counterbalances the 
 difference which would otherwise be manifested 
 in the velocity of light. It will be seen that such 
 hypothesis is primarily nothing but a simple ad 
 hoc device for giving expression to the negative 
 result of the experiment, and is little more than 
 an acknowledgment and way of stating positively 
 
1 30 PRINCIPLE OF RELATIVITY 
 
 the fact observed. It has, however, been possible 
 to submit the question whether there is in reality, 
 or as physical fact, such a contraction, to laboratory 
 test, by measuring the electrical resistance of 
 metal rods both across and in the direction of 
 their translation. These experiments have proved 
 negative and have caused the abandonment of the 
 hypothesis by physicists, in so far at least as it 
 purports to be the actual explanation of the 
 phenomenon. 
 
 A much more radical interpretation was given 
 by Hermann Minkowski in 1908, shortly before 
 his lamented death at the age of forty-five, in an 
 address on " Space and Time/* which has since 
 become historical. In it he formulated a new 
 mathematical theory which embodied in a 
 complete form the principle, the necessity of 
 which had been revealed in the negative result 
 of the experiment. Minkowski proposed a 
 new mathematical scheme of the universe in 
 which time entered as a dimension. The 
 universe is to be conceived not, as hitherto, 
 as a three-dimensional continuum enduring in a 
 one-dimensional time, to which it is indifferent, 
 but as a four-dimensional continuum in which the 
 three dimensions of space and the dimension of 
 
THE FOURTH DIMENSION 131 
 
 time are the axes of coordination by which the 
 observers in systems of reference moving relatively 
 to one another relate the constituent factors of the 
 universe, that is, the events the assemblage of 
 which is the universe. The beauty of this theory 
 is that its apparent strangeness when first pro- 
 pounded tends to give way to familiarity and 
 obviousness ; for when we come to think of it 
 we recognize that the world of our living experi- 
 ence is four-dimensional. The opening sentences 
 of the address (delivered at Cologne on Sep- 
 tember 21, 1908) have been often quoted as 
 marking the beginning of a new epoch in physical 
 theory. " The observations concerning space and 
 time which I am about to expound are the results 
 of experiments in physics. Therein lies their 
 strength. They go to the root of matters. Hence- 
 forward space and time as independent things 
 must sink to mere shadows and the only thing 
 which can preserve some sort of subsistence is a 
 kind of union of the two." He then developed 
 his argument. The universe must be conceived 
 as an assemblage of events. An event is deter- 
 mined for every observer from the standpoint of 
 his system of reference and in relation to that 
 system, by four coordinates, three for space and 
 
132 PRINCIPLE OF RELATIVITY 
 
 one for time. Suppose two events to occur, 
 observers in one system of reference will measure 
 a certain distance separating the two events in 
 space and a certain interval separating them in 
 time. These measurements will not accord with 
 those made by observers in other systems of 
 reference who will measure the same events from 
 their own standpoint and will find different 
 distances and different intervals. Each observer 
 will keep constant the ratio between his co- 
 ordinates of measurement in passing from one 
 system to another, but the four axes themselves 
 will each undergo variation. 
 
 The simplicity of Minkowski's scheme won for 
 it general admiration and made the adoption of 
 the principle of relativity easy in mathematics. 
 A fourth dimension of space had hitherto been 
 always associated with the attempt to rationalize 
 the claims of spiritualistic phenomena to actuality, 
 but Minkowski showed that the fourth dimension 
 was necessary to explain experience on the 
 ordinary common-sense plane. Einstein, refer- 
 ring to this four dimensional theory of Minkowski, 
 writes : "A mystical shudder seizes the non- 
 mathematician when a fourth dimension is spoken 
 of, a creepy emotion, something like that produced 
 
ABSOLUTE RELATIVITY 133 
 
 by a stage effect. Yet there is no more obvious 
 commonplace than that our world of everyday 
 experience is a four dimensional space-time 
 continuum." 
 
 There was, however, a deduction of the experi- 
 ment which presented the aspect of complete 
 paradox. This was the constancy of the velocity 
 of light for all observers and its independence 
 of all variations in' the relative velocity of a 
 system. The velocity of light in vacuo is 300,000 
 kilometres a second, it is a finite velocity, and no 
 velocity is known to exceed it. Some of the 
 radio-active substances give off particles which 
 approach it the/3 particles but though they attain 
 to 99 per cent, of the velocity they are not known 
 to reach it. Now suppose that relatively to our- 
 selves there is a system being translated at the rate 
 of 1 50,000 kilometres a second, for observers in 
 that system as for us light is propagated at the 
 uniform finite velocity of 300,000 kilometres 
 a second. This appears a direct self-contra- 
 diction. How is it to be reconciled ? However 
 surprising it may be to have to acknowledge that 
 we can find no evidence of the absolute movement 
 of a material system nor discover the direction 
 of that movement by experiments made within 
 
134 PRINCIPLE OF RELATIVITY 
 
 the system, it is more than surprising, for it 
 seems downright absurd, to affirm that a finite 
 velocity is identical for all observers in all systems 
 of translation, uniform or non-uniform in relation 
 to one another. It is in regard to this problem 
 that the work of Einstein is particularly important. 
 His philosophical attitude towards this paradox 
 is specially deserving of notice. 
 
 Einstein from the first accepted the negative 
 result of the experiment. It did not for him 
 indicate an agnostic position. It did not merely 
 mean that an absolute movement, or movement 
 measured from an absolute or fixed zero, is 
 unknowable. He accepted the negative evidence 
 as definite proof of the non-existence of an absolute 
 standard, and he proposed to reject the postulate 
 Iff 4 J^ that an absolutejtandard is a necessity of thought. 
 
 h 11* But how is the rejection of absolute movement 
 compatible with the affirmation of a constant finite 
 velocity ? According to Einstein this incom- 
 patibility is purely apparent, in fact it is com- 
 pensated by the deformation of the axes of co- 
 ordination used by one observer as seen by another. 
 Thus to an observer in a system moving relatively 
 and uniformly to us at half the speed of light 
 our proportions are foreshortened to /halfj what 
 
THE THREE TESTS 135 
 
 they appear to us, so that measuring the propaga- 
 tion of light our unit is double that of his, and 
 his is correspondingly half that of ours. Each 
 observer therefore finds the light propagated 
 at the same velocity of 300,000 kilometres a 
 second, but the kilometres used by the one appear 
 to the observer in the rapidly moving system 
 elongated to double their length, and those used 
 by the observer in the rapidly moving system 
 appear halved in their proportion to the observer 
 in the slow moving system. 
 
 The speciaj_ principle_of_rela.tivity which Fin- 
 
 stein formulated in 1905 had in view this problem 
 but only in regard to the velocity of light and its 
 independence of the movement of the source. 
 Already, however, it had seemed to him that if 
 this principle be true it is not limited in its 
 application and it probably implies an entirely 
 new conception of physical reality. The general 
 principle of relativity, which formulates a new law 
 of gravitation in place of the law of Newton, is the 
 result. It is not based on a new experiment but 
 on an application of the principle of the original 
 experiment to all the laws of nature. In 1917 
 Einstein had thought out the three means by 
 which his new theory could be brought to the 
 
136 PRINCIPLE OF RELATIVITY 
 
 test. First that it would account for the dis- 
 cordance of the motion of the perihelion of 
 Mercury, as calculated on the Newtonian formula 
 and as it is found to be in fact. Its cause had 
 been sought for in vain in the supposed pre- 
 sence of a mass of matter between the orbit 
 and the sun. It had also been suggested that 
 the Newtonian law may require an infini- 
 tesimal alteration in its mathematical formula to 
 make it bring out the result correctly. But the 
 idea of the existence of an unknown minor planet 
 is now generally agreed to be so improbable as 
 almost to amount to the certainty that there is 
 none ; and an alteration of the mathematical 
 formula to make the calculation of Mercury's 
 period come right would make other calculations 
 work out wrong. In this case Einstein's theory 
 based on the principle of Relativity has been so 
 triumphantly vindicated that to many mathema- 
 ticians and physicists it is sufficient of itself to 
 establish the principle. Einstein's formula gave 
 the exact result without upsetting the calculations 
 in any other case. Th9 second means was the 
 displacement of star$ in the gravitational field. 
 He predicted that the light of stars near the sun 
 would be deflected and that the deflection would 
 
FIXED AND MOVING SYSTEMS 137 
 
 be twice the amount which Newton's law, calcu- 
 lated for the mass and velocity, would give, if 
 light, as Newton thought possible, had mass. 
 This prediction was verified in the observations 
 of the Eclipse Expedition of May 1919, and was 
 the occasion of the extraordinary public interest 
 in the new theory. The third means proposed 
 has so far not passed the test, and is the sub- 
 ject of research and of much discussion. The 
 vibration of atoms on the surface of the sun 
 compared with the vibrations of atoms on the 
 earth ought, Einstein says, by reason of the 
 difference of the gravitational field, to show a 
 shifting of spectral lines towards the red end of 
 the spectrum. 
 
 I will now try and explain by illustrations what 
 the new principle is. Let us suppose, then, that 
 from the window of a smoothly travelling railway 
 carriage a stone is dropped, and that an observer 
 in the train watches its fall. For him it follows 
 a straight course to the ground. An observer 
 watches the same event from a position which is 
 fixed in respect of the moving system of the train. 
 For him it follows a curve. Which is right ? 
 Is there a real straight line which is the shortest 
 distance between two points ? Is there any way 
 
138 PRINCIPLE OF RELATIVITY 
 
 by which each observer can so correct his observa- 
 tion that he can discover the real as distinct from 
 the apparent line ? The principle of relativity 
 declares that there is no way of deciding between 
 the two observers, that each must use the co- 
 ordinates he carries with him and that these adapt 
 themselves to accord with every system of refer- 
 ence he enters, and that each observer therefore 
 measures an event from the standpoint of his own 
 system regarded as at rest. From this it follows 
 that an event (dropping the stone from the 
 carriage window) which for one observer occurs 
 in one and the same place, for another observer 
 has its beginning separated from its end by a 
 distance in space. If we consider the whole 
 event as single, then the point at which it is 
 observed from one system of reference will not 
 correspond with its place as observed from 
 another. This is the principle of relativity as 
 it applies to space. 
 
 Now suppose that time not space is in question, 
 and that observers take positions equidistant 
 from the point where an event is to take place 
 for the purpose of recording it simultaneously. 
 Let two observers be placed at an equal distance 
 from a point on an electric railway at which it is 
 
SIMULTANEITY 139 
 
 arranged that a break in the circuit shall be indi- 
 cated by a flash. Each observer will see the flash 
 at the same instant. Suppose now that two other 
 observers are similarly situated, but on the moving 
 train instead of on the permanent way. It is true 
 that in such case, as the timing is with light signals, 
 it will not be possible to make any difference 
 appreciable. The velocity of light is so dispro- 
 portionate to any conceivable velocity of the train 
 that the effect due to that difference could only be 
 expressed in thousands of millionths of a second 
 in time or of an inch in space, but, all the same, 
 the principle can be made clear by the illustration. 
 If the precise moment of the emission of the signal 
 (the electric spark) is the same for two observers 
 equidistant on the permanent way, it cannot be 
 the same for two observers equidistant in the 
 moving train, because during the propagation 
 one observer will have advanced to meet the 
 signal, the other will have receded, giving it a 
 correspondingly longer route to travel. The 
 ^^ndplj^ofrelativity^ is^ th^rrfor^hat^simultaneity 
 has no absolute meaning. No two events which 
 happen at the same moment for observers on one 
 system of reference can be simultaneous for ob- 
 servers in another system. Therefore, two events 
 
1 40 PRINCIPLE OF RELATIVITY 
 
 which for one observer are simultaneous, for 
 another are before and after. 
 
 So far we have been following the special 
 principle of relativity. It teaches us that, con- 
 trary to the notion of classical mechanics^ according 
 to which all differences in the observations of 
 events, and all difference in the appearance of 
 events to observers, are calculable in terms of a 
 constant space and time and a variable velocity, 
 there is no distance in space and no interval of 
 time which is invariable and independent of the 
 system of relative movement to which the observer 
 is attached. It takes us a long way in the direction 
 of a new coordination of nature on a philosophical 
 principle, but there is nothing in it peculiarly 
 subversive of our ordinary concepts. Indeed to 
 many mathematicians it commended itself at 
 once as seeming to offer a much better 
 scheme on which to undertake the organization 
 of the science of nature. / When, however, the 
 principle is applied not only to systems moving 
 uniformly in relation to one another, and to 
 phenomena inappreciable in ordinary experience, 
 such as electro-magnetic propagations, but also to 
 gravitation and the ordinary laws of nature, it 
 touches the science and mathematics of common 
 
EQUIVALENCE 141 
 
 life. It is then that it disturbs our feeling of 
 at-home-ness in the universe, brings over us a 
 feeling of giddiness and makes it seem impossible 
 at once to attain a new equilibrium. 
 
 There is something fundamental in our experi- 
 ence of weight. Without it the world would seem 
 to have no substance. Without it we should feel 
 like the man in the folk tale who was induced 
 to barter his shadow. Now it is easy enough 
 to imagine that the phenomena of gravitation 
 may be unknown to observers in other systems, 
 but to suppose that they may observe the identical 
 phenomenon which we experience as weight and 
 yet observe it not as weight but as the movement 
 of the system, and that this movement is the exact 
 equivalent of what we experience as force this is 
 very difficult to accept. It is precisely this that 
 the general principle of relativity affirms. 
 
 Suppose, to take one of Einstein's illustrations, 
 a room, such as that in which the reader may be 
 sitting, to be detached and transported bodily into 
 some distant region so remote from masses of 
 matter that it is entirely free from any force of 
 attraction. Let us suppose no gravitational field. 
 We shall have to think of it as artificially held 
 together and the objects in it as fastened by cords 
 
i 4 2 PRINCIPLE OF RELATIVITY 
 
 or such like means, for the loose objects it might 
 contain would follow the direction of any chance 
 movement which might be imposed on them. If 
 we suppose ourselves still sitting or standing on 
 the floor it will not be by reason of our weight but 
 by attachment or holding on. Let us now 
 imagine that this room is attached by a hook to a 
 cord outside which is being pulled, thereby 
 drawing the room in a definite direction. The 
 hook being attached to the ceiling that definite 
 direction of the ceiling will be upwards. At once 
 all loose objects will lie and remain lying on the 
 floor, and free suspended objects will hang down- 
 wards, and we, if we would rise from the floor, will 
 have to put forth an effort sufficient to produce a 
 movement in excess of the dragging movement. 
 To us in the room it will seem then that objects 
 have suddenly become heavy and that they fall 
 downwards by their weight, but to observers 
 outside the whole phenomenon will be a simple 
 consequence of the dragging movement and 
 explicable in terms of it. If an object is detached 
 it will not, as seen by the outside observer, fall on 
 the floor, it will remain immobile till the floor 
 reaches it. This is what Einstein means by 
 equivalence. What appears to the observer in a 
 
THE GRAVITATIONAL FIELD 143 
 
 rotating system as weight, that is as the attraction 
 of the object to the centre of the rotating mass, 
 will appear to an observer in another system as 
 the movement of the rotating mass to the object 
 at rest. 
 
 Very curious consequences follow, completely 
 subversive of our ordinary ideas. Gravitation is 
 a phenomenon which is connected with a rota- 
 tional system, and the gravitational field is the 
 space which surrounds this rotational system^ 
 To an observer attached to the system, regarding 
 the firmament from the standpoint of his system at 
 rest, the firmament is in movement. Consequently 
 for this observer an object (Newton's apple) 
 detached from his system moves with the firma- 
 ment. But to an observer on another system at 
 rest, for whom the first system is rotating, the 
 detached object (the apple) ceases to move with 
 the rotating system and remains at rest. There- 
 fore to the first observer the movement of the apple 
 will be its fall towards the centre of the rotating 
 earth, but to the second it will be the movement 
 of the earth towards the apple. The gravitational 
 field is then the moving firmament, as observed 
 from the rotating system, by an observer on the 
 system taking the standpoint of himself at rest. 
 
144 PRINCIPLE OF RELATIVITY 
 
 It is clear that the nearer the object in the field 
 is to the rotating system the greater is the velocity 
 of its observed movement. What we call an 
 attractive force decreasing inversely with the 
 distance is the exact equivalent of the move- 
 ment of bodies in a gravitational field, as 
 observed from a standpoint of a rotational 
 system at rest. 
 
 This enables us to see what is meant when it is 
 affirmed that space in the gravitational field is 
 non-Euclidean. In the gravitational field the 
 'observer is either at rest in the firmament relatively 
 to the rotational system or else at rest on the 
 rotational system relatively to the moving firma- 
 ment. In neither case can Euclid's postulates 
 be fulfilled or Euclid's axioms hold true. Con- 
 sider, for example, the postulate that a straight 
 line is the shortest distance between two points. 
 The shortest distance for one observer will not be 
 the shortest distance for the other, or, what is the 
 same thing, the straight line drawn by the one 
 will appear curved to the other. But when we 
 realize this relativity can we not make allowance 
 for the appearance to different observers and 
 work out the perspective law for each ? This is 
 what we imagine we are always engaged in doing. 
 
FIXED CENTRES 
 
 Is it not a commonplace of psychology that 
 visual space is not real space ? The principle I . , 
 of relativity shows us that this supposed power of r ^ -' 
 correcting^ appeajjUic^ J^^xference-- toan mtel- \ 
 lectual absolute is illusion, - 
 
 The illusion is persistent. It is so easy to 
 represent our system of reference in movement 
 by means of our imagination, that we have come 
 to cease to think of the firmament as in move- 
 ment and, though it still appears so, we translate 
 the appearance automatically into the reality 
 of our own movement. Can anyone really 
 think the smoothly running railway carriage 
 is at rest while the landscape is in movement ? 
 We fail therefore to see that though our imagina- 
 tion aids us in representing our own movement 
 as relative it never brings us to a system at absolute 
 rest. JThat necessary standpoint of a system at 
 jest must be within the phsprver birnsp.lf andL is 
 the^condition of his observation^ It is from that 
 standpoint that our measurements are made, and 
 we carry with us, attached to us, inseparable from 
 us, the axes of coordination which we apply to the 
 universe. We are therefore perpetually under 
 the illusion that the absolute criterion lies without 
 us, whereas it is indissolubly part of our nature. 
 
146 PRINCIPLE OF RELATIVITY 
 
 It is just as if we supposed the centre of a circle 
 to be independent of its circumference and free 
 to move about within it, even indeed to come 
 outside and survey it. 
 
 Common experience offers us an example of a 
 persistent illusion in regard to space. We think 
 that we know by actual experience real space. 
 But the space we think real is not the space we 
 perceive by any of our senses. We do not think, 
 for example, that the sun and the moon are in reality 
 the same size as a threepenny bit, though they 
 cover the same amount of visual extension as this 
 object held at arm's length. The space we think 
 real is purely conceptual and ideal, and this space 
 is Euclidean. We do not encounter it in experience, 
 nevertheless we are convinced it really exists and 
 suppose it must underlie and condition our 
 experience, and we distinguish it as noumenal 
 from the phenomenal. It is this persistent 
 illusion of a real space, noumenal not phenomenal, 
 which makes it so disconcerting to us and so 
 difficult to accept the notion of a space which 
 may be curved, warped or distorted, not in its 
 appearance only, but in reality. It is affirmation 
 of a space with such properties that makes the 
 general principle of relativity appear paradoxical. 
 
WARPED SPACE 147 
 
 If, however, we accept the negative result of the 
 physical experiments we are bound to reject as 
 pure illusion the notion that Euclidean space 
 exists beneath the world of sense perception. 
 
 We may make the concept of curvature in space 
 clear by simplifying for the purpose of illustration 
 the scheme of the principle. Let us suppose then 
 that instead of manifold gravitational fields there 
 exists one only. Let us abstract, that is to say, 
 from everything in nature except the rotating 
 earth and the surrounding extension relative to it 
 which we will call the firmament. The gravi- 
 tational field will now consist of two and only two 
 systems of reference. An observer at rest on the 
 earth sees the firmament in a continuous move- 
 ment from east to west. An observer in the firma- 
 ment sees the earth rotating on its axis from 
 west to east. The surface of the earth will then be ^/vV, 
 the centre of a gravitational field, and the limits of 
 this field will be the earth's axis on the one side, 
 and an external imaginary limit of a space purely 
 Euclidean on the other. It is clear that within 
 this gravitational field space must be warped, 
 because every observer will be at rest on one of 
 the two systems. Any movement, therefore, be 
 it of propagation or of translation, which takes 
 
148 PRINCIPLE OF RELATIVITY 
 
 place in the gravitational field must be occurring 
 in the space of one of the systems. Suppose then 
 any object to be detached from the rotating earth, 
 it at once takes its place in the circulating firma- 
 ment. Similarly, any object detached from the 
 firmament takes its place in the rotating earth. 
 The universal space simply denotes the continuity 
 of these two systems in the relation of movement 
 and rest for one another. Space is not a third 
 somewhat in which these movements are taking 
 place. To affirm it would simply be to deny the 
 reality of one of the movements, and what possible 
 ground can there be for that ? The essence of 
 the principle of relativity is the explicit denial of 
 such an absolute space and the recognition of the 
 impossibility of interpreting facts on the sup- 
 position of its existence. The curvature of space 
 is therefore physical fact, not subjective appear- 
 ance. Now let us suppose a straight line drawn 
 from within the earth system to any point in the 
 firmament system, the course of that straight line 
 must by its spatial condition curve in its path 
 through the earth system and curve in the reverse 
 direction when it passes through the firmament 
 system. Any line which is the shortest between 
 two points for an observer in one system will 
 
GALILEO'S DISCOVERY 149 
 
 not be the shortest for an observer in the 
 other. 
 
 Someone may still object. He may demur that 
 my conclusion that there is real curvature of 
 space in the gravitational field follows only because 
 I have started with an arbitrary hypothesis, viz. 
 that the universe is a single gravitational field. 
 The reply is that the conclusion does not depend 
 on this hypothesis and it is only introduced for 
 simplicity and to enable attention to be directed 
 on the essential point. Admit that the universe 
 is infinitely complicated, that it consists of infinite 
 systems moving uniformly and non-uniformly 
 relatively to one another, this only increases the 
 difficulty of correlating observations. No doubt 
 this difficulty has been the guiding motive in the 
 evolution of the concept of absolute space, but it 
 is not a proof that absolute space exists. 
 
 Three hundred years ago it was discovered that 
 a heavy body such as lead and a light body such 
 as wood follow the same identical course in the 
 same identical time in the gravitational field, when 
 allowed to fall with the same initial velocity and 
 with all frictional obstructions removed, as for 
 example in the vacuum produced by an air pump. 
 The significance of this discovery was never 
 
150 PRINCIPLE OF RELATIVITY 
 
 thought out. It is precisely what is affirmed by the 
 theory of equivalence. Why does a stone fall to 
 the ground when let go ? The usual answer is 
 because it was lifted off the ground, and the hypo- 
 thesis in that answer is that the earth exercises 
 a direct influence on the stone, called the force of 
 attraction. The strength of this force was found 
 to vary with the distance according to the well- 
 known Newtonian law. As an explanation this 
 has not satisfied modern physics because it admits 
 the notion of action at a distance. The advance 
 in the study of electro-magnetic phenomena has 
 led to the conclusion that there is no action at a 
 distance. When we see a magnet attract a piece 
 of iron, science will not let us be satisfied with the 
 simple explanation that the magnet draws the iron 
 across empty space, science requires us to see in 
 the phenomenon a property or character of the 
 intervening space which it names the magnetic 
 field. It is .this and not the magnet which 
 influences the iron, causing it to move toward 
 the magnet. In gravitation we have the exact 
 analogy and at the same time an important 
 contrast. The contrast is that in the gravita- 
 tional field (as Galileo's experiments proved), 
 bodies undergo an acceleration which does not 
 
SPACE AND INFINITY 151 
 
 depend on their material or constitution. And in 
 this way the principle of relativity explains gravi- 
 tation, showing it to be a phenomenon which de- 
 pends on relative systems of movement, and on 
 the position or standpoint of rest, necessarily 
 assumed by the observer for his own system. 
 
 What kind of world is it then that we live in ? 
 A world which is finite in so far as the concepts 
 of space and time determine it, and which yet is 
 not circumscribed. The principle of relativity 
 enables us without doing violence to the laws of 
 thought and without contradicting experience to 
 dispute the Euclidean axioms. Space and time, 
 which throughout the whole history of philosophy 
 have been the stubborn realities of the framework 
 of the universe, baffling the mind in every effort 
 to form a consistent scheme of nature, are deposed, 
 at least from their dominating position in the 
 mathematical and physical sciences. It is a 
 triumph of philosophy, for the principle of 
 relativity is a return to the concept which Leibniz 
 indicated and which was abandoned by the scien- 
 tific successors of Newton. In nothing is the 
 contrast more striking than in the concept of 
 space. For Newton space is an infinite, absolute, 
 immensity, which can only be present to the 
 
152 PRINCIPLE OF RELATIVITY 
 
 perception of an infinite God. For Leibniz God 
 alone of intelligent beings is wholly without the 
 perception of space, because God is conceived as 
 an intelligence with adequate ideas and with no 
 obscure perception. Let us drop the theological 
 expression and state the same contrast in scientific 
 terms. We shall then say that for the materialist 
 space is the fundamental reality and the universe 
 presupposes it ; for the relativist, on the other 
 hand, space is a limitation of the observer's appre- 
 hension of the universe. Infinity is not the 
 affirmation of space but its disappearance. 
 
CHAPTER VIII 
 
 CONCLUSION : IN WHAT SENSE IS THE 
 UNIVERSE INFINITE ? 
 
 EVERY revolution in the world-view has profoundly 
 affected mankind in those aspects of life which 
 depend upon reason. So far as most of us are 
 concerned the principle of relativity may seem a 
 matter of small importance, dealing with infini- 
 tesimals which in the ordinary business of life are 
 entirely inappreciable. It disturbs our general 
 scientific methods no more than the Copernican 
 theory disturbed the practical adjustments of the 
 human mind. For mankind the sun continues 
 to rise and set. We reckon the times and the 
 seasons, as men have always done, and will do, 
 irrespective of any change which has taken 
 place, or which may take place, in astronomical 
 theory. Newton's law of the inverse square will 
 not cease to be a practical rule for engineers 
 
 and mechanicians for all economic projects, 
 
 153 
 
154 PRINCIPLE OF RELATIVITY 
 
 nor will it cease to commend itself by its 
 simplicity, if Einstein's formula comes to be 
 recognized as theoretically perfect. In religion, 
 however, and in philosophy of life philosophy 
 as it concerns mankind generally and not as 
 technical metaphysics or theory of knowledge 
 its effect will be profound and far-reaching. 
 I will conclude therefore with an indication 
 of some of these higher interests in the new 
 principle. 
 
 It seems to me that the new world-view must 
 take the form of, and find the imagery for, a new 
 concept of the nature of the continuity and infinity 
 of the universe. In the world-view as it has found 
 expression in religion and philosophy hitherto the 
 concept of infinity has been inseparably associated 
 with the ideas of space and time. "As it was in 
 the beginning, is now, and ever shall be, world 
 without end," is the liturgical expression of this 
 idea. It depends on the notion of the absolute- 
 ness of the spatio-temporal order. The mathe- 
 matical definition of continuity and infinity is, as 
 we have seen, simply a precise form of expression 
 for the spatial and temporal concepts, depends 
 upon them for its applicability, and appeals 
 exclusively to the pragmatic test. It is justified 
 
MATHEMATICAL INFINITY 155 
 
 because it works, but it only works in so far as 
 we accept in advance the postulate of an external 
 world in space and time. The new principle 
 of relativity goes behind and beneath the mathe- 
 matical definition of infinity, for it rejects the 
 postulate on which it is based. So far as the 
 mathematical principle rests on the Euclidean 
 postulates and so far as infinity means, when 
 applied to Euclidean space and its imagery, 
 boundlessness and absence of limits, we have 
 seen that the new principle definitely rejects the 
 concept of infinity. It gives us in fact what to 
 common-sense is a new paradox a world which 
 is finite and yet not circumscribed. So far, 
 however, we have to do only with the negative 
 aspect of the principle. What has it to offer 
 on the positive side ? 
 
 The answer to me seems clear and manifest. 
 We are offered in place of the contradictory 
 pseudo-concepts of endless extension and limitless 
 duration the concept of a truly infinite universe. 
 The infinity of the universe is based on the nature 
 of life and consciousness. The principle of 
 relativity declares that there is no absolute magni- 
 tude, that there exists nothing whatever which 
 can claim to be great or small in its own nature, 
 
156 PRINCIPLE OF RELATIVITY 
 
 also there is no absolute duration, nothing what- 
 ever which in its own nature is short or long. I 
 coordinate my universe from my own standpoint 
 of rest in a system of reference in relation to which 
 all else is moving. That system may change, 
 and there is no limit to the change it may undergo, 
 but however great the change, measured by its 
 relation to other systems, its dimensions remain 
 constant. I, the observer, am not a point 
 at an instant. Space and time dimensions 
 do not apply to mind. A monad has no dimen- 
 sions, so that one mind or monad can be in a relation 
 of magnitude to another ; one monad does not 
 occupy more or less space than another. Space 
 and time are not containers, nor are they contents, 
 they are variants. Consequently, whatever my 
 system of reference, as I pass into it or out of it, 
 that is, as it changes, so my spatial points and 
 temporal instants change ; my units of measure- 
 ment vary, so keeping the dimensions of my 
 universe constant. 
 
 I may illustrate my meaning if I now give with 
 as much imagery as the concept will admit my 
 idea of the nature of the infinity and continuity of 
 the universe. In doing so I will set aside all 
 questions of detail and all special problems in 
 
REAL CONTINUITY 157 
 
 order to set out the scheme with as much genera- 
 lity as possible. 
 
 I start, then, from the monadic concept. We 
 all belong to the order of self-conscious minds. 
 Everyone has his own system of reference within 
 which from his standpoint of rest therein he may 
 correlate events which happen for him with events 
 which happen for his fellows. The systems of 
 reference appear to us to have practically every- 
 thing in common and we are able consequently 
 to have intercourse with one another, to refer to 
 the actual events as common. A philosophical 
 problem of fundamental importance is involved 
 in this fact of intercourse, but it need not interrupt 
 our attempt to form a world-view. Whether the 
 world is the condition of intercourse or inter- 
 course the condition of the existence of the 
 world, we think of the world as common to 
 all. 
 
 Our world then, or to be precise let each of us 
 say, my world, appears as a certain range of 
 activity and a certain possibility of experience. 
 This world is definitely limited, its limits have 
 been pushed out by our advancing science, but it 
 is on one side limited by our concept of the atomic 
 system, on the other by our concept of the stellar 
 
158 PRINCIPLE OF RELATIVITY 
 
 system. Within these limits there are infinite 
 systems of reference, using the word infinite here 
 in its ordinary discursive meaning. For example, 
 there is the animal world, the insect world, the 
 vegetable world, the protozoan world, and the 
 microbian world, each containing within it count- 
 less different ranges of activity and innumerable 
 possibilities of experience. Now imagine, in the 
 manner of folk-tales, that we have the power to 
 pass from our own system into any of these, that 
 is, imagine that our mind can enter the living 
 organism of bird or insect or microbe, possess its 
 range of activity and enjoy its experience, still 
 continuing by memory our former experience. 
 We have then only to reflect on the nature of life 
 and consciousness to see that the change in the 
 system of reference cannot be a change in the 
 subject of experience, and can only be a change 
 in the object of experience. The subject passing 
 to the new system of reference must therefore 
 necessarily bring with it its norm, the standard of 
 its measurement in reference to which it judges 
 objects to be great or small. It is obvious that a 
 creature, small in size as judged by me, and to 
 whom I am a Brobdingnagian, does not feel its 
 smallness, it feels my greatness, for its norm is 
 
CHANGING SYSTEMS 159 
 
 supplied by its own instrument of activity, its 
 living body. There are in fact within our 
 ordinary perspective myriads of subjects of 
 experience, each of which finds in its bodily 
 organization the norm of its dimensions. If then 
 I pass from one system to another it is certain that 
 my space and time units must vary, for, unless they 
 do, there is no conceivable way of effecting the 
 exchange of standpoint. Now we have admitted 
 that there are limits to our universe. We are 
 bounded, we have said, by the atomic system on 
 one side, the stellar system on the other. Even 
 the minutest microbe is far removed from the 
 atomic limit, and the mythical beings whom the 
 poets have created are well within the stellar limit. 
 Let us, however, boldly imagine that the change 
 of system carries us right to the limit. It is 
 easily conceivable. We have to imagine our 
 proportions reduced to the ten thousand trillionth 
 and we are within the atom, or, increased to the 
 same amount and the earth is as far below the 
 limit of perception as an electron. So that in the 
 first case the electron of the atom has become for 
 us a planet which will appear to us, at least we 
 can imagine it, as a universe precisely like the 
 present, and its limits will be as now, an atomic 
 
160 PRINCIPLE OF RELATIVITY 
 
 system on the one hand, a stellar system on the 
 other. And in the other case the present stellar 
 system will have become an atomic system. This 
 is the way in which the infinity of the universe 
 presents itself to me when space and time are 
 recognized as variable and not constant. 
 
 I shall be challenged however. It will be said 
 that I have not escaped from the dilemma of the 
 old mathematical infinity, because though I may 
 carry my norm of measurement as the inseparable 
 adjunct of consciousness and vary my space and 
 time through infinite change of system, it is after 
 all only for me that the standard is constant, to an 
 independent external observer I become larger or 
 smaller absolutely. But it is precisely this idea 
 that there can be an independent observer in an 
 absolute system of reference that the principle of 
 relativity negatives and rejects. 
 
 It seems to me, therefore, that the principle of 
 relativity is a philosophical principle which is not 
 only called for by the need of mathematical and 
 physical science for greater precision in the new 
 field of electro-magnetic theory in which it is 
 continually advancing, but is destined to give us 
 a new world-view. It will be found, as it has 
 always been found, that the poets with their 
 
THE NEW WORLD-VIEW 161 
 
 mythical interpretations, and the philosophers 
 with their speculative hypotheses, have led the 
 way in this new advance. The continuity of the 
 universe can only be a continuity of consciousness, 
 and the mode of this continuity is imaginatively 
 presented to us in the old eastern myth of the 
 transmigration of the soul and (may we not 
 also say ?) in the Christian mystery of the 
 Incarnation. 
 
 I conclude, then, that in every reflection on our 
 actual experience we are directly conscious of an 
 objectivity which we distinguish from our sub- 
 jective activity of knowing. Whether we approach 
 the problem of that objectivity from the abstract 
 standpoint of physical science or from the concrete 
 standpoint of philosophy, the result is the same. 
 Ultimately, in spite of its claim to independence, 
 all that an object or event is, in substance or in 
 form, it derives from the activity of the life or 
 mind for which alone it possesses the meaning 
 which makes it an object or event. This is not a 
 mystical doctrine, nor is it esoteric. If we adopt 
 in mathematics and physics the principle of 
 relativity (and have we any choice ?) the obstinate, 
 resistant form of the objectivity of the physical 
 world dissolves to thin air and disappears. Space 
 
1 62 PRINCIPLE OF RELATIVITY 
 
 and time, its rigid framework, sink to shadows. 
 Concrete four-dimensional space-time becomes a 
 system of world-lines, infinitely deformable. And 
 these world-lines, do not they at last bring us in 
 sight of an irreducible minimum of self-subsistent 
 objectivity ? No. The world-lines are not things- 
 in-themselves, they are only an expression for 
 what is or may become common to different 
 observers in the relations between their stand- 
 points. Carried to its logical conclusion the 
 principle of relativity leaves us without the image 
 or the concept of a pure objectivity. The ultimate 
 reality of the universe, as philosophy apprehends it, 
 is the activity which is manifested in life and 
 mind, and the objectivity of the universe is not a 
 dead core serving as the substratum of this 
 activity, but the perception-actions of infinite 
 individual creative centres in mutual relation. 
 
 A closing illustration will perhaps serve better 
 than argument to bring home to the reader the 
 philosophical meaning of the principle. On a 
 frosty morning we may see the watery vapour in 
 the air we breathe condense into a small cloud 
 and then rapidly disappear, reabsorbed into the 
 atmosphere. Imagine that at such a moment we 
 should undergo a sudden transformation of all our 
 
WHAT DOES NOT CHANGE 163 
 
 proportions so that our new dimensions become 
 infinitesimal in comparison with our present state. 
 Would it appear to us that we ourselves had 
 changed ? The principle of relativity declares 
 that the change could not possibly be experienced 
 by us as change in ourselves because with the 
 alteration in proportions the ratios remain con- 
 stant. The change would express itself in the 
 new dimensions of objects. The little globules 
 of water which composed the cloud would now 
 appear as stars and planets at immense distances 
 from one another, undergoing a slow age-long 
 evolution and obeying the law of the, inverse 
 square. The change would be a new space and 
 a new time. 
 
INDEX 
 
 Aristotle, 24, 27, 29, 51 
 
 Berkeley's Theory of Vision, 
 
 18 
 Bergson on Zeno's Problem, 
 
 34> 36 
 
 Bradley, F. H., on Zeno's 
 Problem, 31 
 
 Clarke, The Correspondence 
 with Leibniz, 92, 114 
 
 Conduit, Madame, The Story 
 of Newton and the Apple, 
 80 
 
 Contraction Theory of Fitz- 
 gerald and Lorentz, 129 
 
 Copernican Discovery, n, 
 60 S. 
 
 Cratylus, the Disciple of 
 Heracleitus, 29 
 
 Dante, 15, 58, 59 
 
 Darwinian theory compared 
 with the Copernican dis- 
 covery, 75 
 
 Democritus, 40 ff., 58 
 
 Descartes, 61 ff. 
 
 Dispute concerning the dis- 
 covery of the Calculus, 
 104 
 
 Eclipse expedition of 1919, 
 10, 136 
 
 Einstein, I, 4, 38, 133 ff. 
 
 Epicurus, 43, 52 
 
 Epicureans, Dante's descrip- 
 tion, 59 
 
 Equivalence, 10, 141 
 
 Euclid's Postulates, 144 
 
 Fitzgerald, Professor G. F., 
 129 
 
 Galileo, 149, 150 
 Genetic Theories of Space- 
 perception, 19 
 
 Heracleitus, 29 
 
 Hume, 17 
 
 Hypotheses non fingo, 94, 119 
 
 James, William, 20 
 
 Kant, 2, 3 ; The Anti- 
 nomies of Reason ', 32, 33 
 Kepler, 82 
 
 Laplace, 101 
 
 Leibniz, 99 ff. ; Correspond- 
 ence with Clarke, 92 ; 
 satirized by Voltaire, 101, 
 114 ff. 
 
 164 
 
INDEX 
 
 165 
 
 Lorentz, 129 
 
 Lotze, Local-sign theory, 19 
 
 Lucretius, 43 ff. 
 
 Michelson-Morley experi- 
 ment, 125 ff. 
 Milton, 14, 15 
 Minkowski, 130 ff. 
 Monads, 105 
 
 Nativistic theories of Space- 
 perception, 20 
 
 Newton, 77 ff. ; The Apple 
 story, 80 ; Theory of the 
 sensorium, 91, 116 
 
 Parmenides, 29 
 
 Perihelion of Mercury, 10, 136 
 Russell, Hon. Bertrand, 34 
 
 Sensorium, Theory of the, 
 
 91, 116 
 Spencer, Herbert, Theory of 
 
 Space-perception, 19 
 Spinoza, 107 
 
 Velocity of Light, 6, 121 ff. 
 Voltaire, 79, 86, 101, in 
 Vortex theory, 66 ff., 97 ff. 
 
 Warping of Space, 147 
 World-lines, 161 
 
 Zeno, 24 ff. 
 
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