IBRARY 
 
On an Inversion of Ideas 
 as to the Structure of 
 the Universe 
 
3L0ntJon: C. J. CLAY AND SONS, 
 
 CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, 
 AVE MARIA LANE. 
 
 : 50, WELLINGTON STREET. 
 
 ILetpjig: F. A. BROCKHAUS. 
 
 efa gorfc: THE MACMILLAN COMPANY. 
 
 Bombag ant Calcutta: MACMILLAN AND CO., LTD. 
 
 [All Rights reserved.} 
 
On an Inversion of Ideas 
 as to the Structure of 
 the Universe 
 
 (The Rede Lecture, June 10, 1902) 
 
 by 
 
 Osborne Reynolds 
 
 M.A., F.R.S., LL.D., Mem. Inst. C.E. 
 Professor of Engineering in the 
 
 Owens College, Manchester ; 
 Honorary Fellow of Queens' College, Cambridge 
 
 CAMBRIDGE 
 
 at the University Press 
 
 1903 
 
PRINTED BY J. AND C. F. CLAY, 
 AT THE UNIVERSITY PRESS. 
 
 First Edition 1902. 
 Reprinted 1903. 
 
CONTENTS. 
 
 PAGE 
 
 1. Evidence afforded by the outward facts of nature . i 
 
 2. Phenomena not hitherto explained ... 3 
 
 3. Theories of transmission of light .... 3 
 
 4. The granular structure of the universe ... 4 
 
 5. The degradations of light and normal waves . 16 
 
 6. The permanence of the inequalities . . . 17 
 
 7. Mobility of the medium ""V . . . . 18 
 
 8. Singular surfaces are wave-surfaces . . . 22 
 
 9. The molecules are individuals .... 23 
 
 ro. Negative inequalities ...... 23 
 
 n. The principal stresses in the medium ... 24 
 
 12. The cause of gravitation ..... 24 
 
 13. Positive inequalities -39 
 
 14. The cause of light, and general conclusions . . 42 
 
 231 
 
ON AN INVERSION OF IDEAS AS 
 TO THE STRUCTURE OF THE 
 UNIVERSE. 
 
 i. Evidence afforded by the outward facts of 
 nature. 
 
 The general problem of the universe, as 
 hitherto presented by the phenomena which 
 Tyndall, in this house, called " the outward facts 
 of nature," demands that matter, besides being 
 continuous in time and occupying space to 
 exclusion of other matter shall have such 
 physical properties as admit : 
 
 (i) of conditions in space which allow of 
 motions of matter, like those of the earth 
 and planets round the sun, at velocities of 
 upwards of 20 miles a second, with scarcely 
 any diminution after thousands of years ; i.e. they 
 must admit of a perfect vacuum of matter, 
 such as would be obtained by a perfect air- 
 pump ; 
 
 R. i 
 
2 The outward facts of nature 
 
 (2) they must also allow of the transmission 
 of light, such as is being transmitted through 
 these windows, to be reflected or absorbed by 
 the opposite wall ; 
 
 (3) of the gravitation of matter, as when 
 I drop this ball; 
 
 (4) of the limited cohesion of matter, on 
 which the strength of our structures depend. 
 Thus I can break this stick of sealing-wax, 
 and when I warm the ends and bring them 
 together, when cold it is as strong as before ; 
 
 (5) of the elasticity of matter, as shown 
 by the continued vibration of this spring ; 
 
 (6) of the limited friction of matter, as is 
 shown by the weight resting on the inclined 
 plane until the inclination reaches a certain 
 angle, when it slides down at an accelerating 
 rate ; 
 
 (7) of the viscosity of matter, as is shown 
 by putting oil on the inclined plane, when the 
 weight slides down slowly and at a steady rate ; 
 
 (8) of the electric and magnetic properties 
 of matter, shown by the absence of any affinity 
 of the stick of sealing-wax for the paper until it 
 is rubbed by silk, when it at once picks up the 
 paper ; 
 
 (9) of the freedoms and mutual constraints 
 
Theories of transmission 
 
 of the molecules of matter, shown by the uniform 
 pressure of the air in this room ; 
 
 (10) of the combination and dissociations 
 of molecules, as shown respectively by any 
 combustion and any electrolytic decompositions. 
 
 2. Phenomena not hitherto explained. 
 
 That the physical properties demanded for 
 the mechanical explanation of the ten phenomena 
 illustrated, as well as others, exist, is certain. 
 
 But it is equally certain, that, hitherto, they 
 had not been found, in spite of all attempts. 
 
 3. Theories of the transmission of light. 
 
 In place of explanations there have been the 
 theories of Huygens and Newton, two hundred 
 years ago, put forward as explaining, in some 
 measure, the transmission of light ; and again, 
 the modification of Huygens' theory, by Dr 
 Young, a hundred years ago, which latter up 
 to the present time has carried all before it. 
 
 Thus for the last hundred years the idea of 
 the structure of the universe, or the luminiferous 
 ether, which has prevailed, is that of space oc- 
 cupied by an incompressible elastic jelly yielding 
 to tangential stress, having a density which is all 
 but indefinitely small. 
 
 And so the idea which has alone prevailed 
 
Granular structure 
 
 as to the structure of the universe is such as 
 approximates to empty space. 
 
 There have been other ideas not so much as 
 to the structure of the universe, which have been 
 strongly held ; these will however come in at a 
 later stage, while our attention is turned to the 
 ideas, as to the structure, which follow as the 
 results of an exhaustive research " On the sub- 
 mechanics of the universe." 
 
 4. The granular structiire of the universe. 
 
 This research has occupied 20 years, and is 
 just now completed. 
 
 It has revealed \hzprime cause of the physi- 
 cal properties of matter. 
 
 And, notwithstanding the time it has taken 
 to find them, the results are for the most part of 
 marvellous simplicity ; although, on the other 
 hand, so contrary to previous conceptions as to 
 entail an inversion of ideas hitherto advanced. 
 
 Before, however, we proceed to sketch the 
 results of the research it seems necessary to give 
 some account as to the manner in which these 
 results have been obtained. 
 
 Certain steps, as it now appears, were taken 
 for objects quite apart from any idea that they 
 would be steps towards the mechanical solution 
 
Steps taken 
 
 of the problem of the universe. The first of 
 these steps was taken in 1874 with the object of 
 finding a mechanical explanation of the sudden 
 change in the rate of flow of the gas in the tubes 
 of a boiler when the velocity reached a certain 
 limit ; perhaps this would be better described as 
 a step towards a step*. 
 
 The second step was the discovery of the 
 thermal transpiration of gas, together with the 
 analytical proof of the dimensional properties of 
 matterf. 
 
 The third step was the discovery of the cri- 
 terion of the two manners of motion of fluids {. 
 And it was only on taking the fourth step, 
 namely, the study of the action of sand, which 
 revealed dilatancy as the ruling property of all 
 granular media which directed attention to the 
 possibility of a mechanical explanation of gravi- 
 tation. In spite of the apparent possibility all 
 attempts to effect the necessary analysis failed 
 at the time. 
 
 There was, however, a fifth step ; the effect- 
 ing of the analysis for viscous fluids, and the 
 determination of the criterion ||, which led to the 
 
 * ' Manchester Lit. and Phil. Soc.,' 1874-5, p. 7. 
 t 'Phil. Trans.,' 1879. t 'Phil. Trans.,' 1883. 
 
 'Phil. Mag.,' 1885. || 'Phil. Trans.,' 1895. 
 
Steps taken 
 
 recognition of the possibility of the analytical 
 separation of the general motion of a fluid into 
 mean varying motion, displacing momentum, 
 and relative motion, without mean momentum ; 
 and this suggested the possibility that the 
 medium of space might be granular, the grains 
 being in relative motion, and at the same time 
 being subject to varying mean motion. And 
 this has proved to be the case. 
 
 At the same time it became evident that it 
 was not to be attacked by any method short of 
 the general equations of a conservative system 
 starting from the very first principles ; and it is 
 from such study that this purely mechanical 
 account of the physical evidence has been ob- 
 tained. 
 
 Apart from the introduction the analysis is 
 effected in sections II to XV. 
 
 II. The general equations of motion of any 
 entity axiomatic. 
 
 III. The general equations of motion in a 
 purely mechanical medium, i.e. a medium in 
 which the energy is purely kinetic, which can 
 only be, 
 
 1. empty space. 
 
 2. perfect fluid. 
 
 3. perfect solid. 
 
Analysis effected 
 
 IV. The equations of continuity for compo- 
 nent systems of motion. 
 
 V. The mean and relative motions of a 
 medium. 
 
 VI. The approximate equations of compo- 
 nent systems of mean and relative motion. 
 
 VII. The general condition for the continu- 
 ance of component systems of mean and relative 
 motion. 
 
 VIII. The conducting properties of the 
 absolutely rigid granule-ultimate atom or prim- 
 ordium. 
 
 IX. The probable ultimate distribution of 
 the members of granular media, as the result of 
 encounters, when there is no mean motion. 
 
 X. Extensions of the kinetic theory to 
 include rates of conduction of momentum and 
 energy through the grains, when the medium 
 is in ultimate condition and under no mean 
 strains. 
 
 XI. The redistribution of angular inequali- 
 ties in the relative system. 
 
 XII. The linear dispersion of mass, and of 
 momentum and energy of relative motion by 
 convection. 
 
8 Mean and relative motions 
 
 XIII. The exchanges between the mean and 
 relative systems. 
 
 XIV. The conservation of inequalities in 
 the mean mass and their motions about local 
 centres. 
 
 XV. The determination (i) of the relative 
 quantities a", X", <r, and G which define the state 
 of the medium by the results of experience ; 
 (2) the general integration of the equation. 
 
 From the research it appears that the motion 
 of the medium must be such as admits of ana- 
 lytical separation into two systems : 
 
 (1) A system of mean varying motion, dis- 
 placing momentum : 
 
 (2) A system of relative motion without 
 mean momentum, within a certain space. 
 
 It also appears : 
 
 (3) that it is only media consisting of ab- 
 solutely rigid parts that can satisfy the condition 
 of being mean and relative systems of motion. 
 
 (4) that the rigid parts must be uniform 
 spheres. 
 
 The actions between such spheres is outside 
 all experience. But it follows, since there is no 
 elasticity, that for the conservation of energy, 
 when these spheres, or grains, as they will now 
 be called, come into collision, they will exchange 
 
Conduction 
 
 their momenta, on the instant, and thus displace 
 momentum and energy by instantaneous con- 
 duction. 
 
 This displacement by instantaneous conduc- 
 tion may be illustrated on billiard-tables ; for 
 when one ball strikes another at rest, if in the 
 same line, the first stops while the other takes 
 up its motion a full diameter in front, whence 
 the instantaneous displacement of momentum 
 and energy. 
 
 The arrangement, or piling, of the grains, is 
 
 T^^ . ^P^ 
 
 FIG. i. 
 
10 
 
 Arrangement of grains 
 
 normal when the mean position of each of the 
 grains is such that every grain is at the same 
 mean distance from each of its twelve neighbours ; 
 as shown in models I and 2. Model I shows the 
 normal piling, model 2 shows the twelve neigh- 
 bours of the grain in normal piling. 
 
 FIG. 7. 
 
Arrangement of grains 
 
 1 1 
 
 Then, if the grains are in relative motion, 
 under pressure, from the outside, such relative 
 motion would entail the separation of the mean 
 positions of the grains, according to the outside 
 pressure and the relative velocity of the grains 
 the piling remaining normal. 
 
 FIG. 3. 
 
 The curvature as shown in model 3, in which the grains are 
 in a bowl in two layers, the bowl being somewhat tilted, is 
 very evident in the lines of grains in the lower layer which 
 extend beyond the grains in the upper layer. And the con- 
 formation of the lines of grains in the upper layer may be traced. 
 
12 
 
 Arrangement of grains 
 
 Strained normal piling implies that, although 
 the shape of the medium is strained, so that the 
 distances of the grains from their twelve neigh- 
 bours are no longer equal since the successive 
 layers of grain in the normal piling instead of 
 being flat are subject to slight spherical curva- 
 ture the strains are such as do not allow any 
 change of neighbours ; so that when the strain 
 is removed each grain will find itself in normal 
 piling with the same neighbours. 
 
 FIG. 
 
Elasticity 
 
 Abnormal piling implies any disarrangement 
 which displaces the grains in the medium from 
 their neighbours and so affects the gearing of the 
 grains, as shown in models 4 and 5 ; in 4 the 
 piling is approximately normal, the edges being 
 free, while in 5 the piling is so abnormal as to 
 prevent, entirely, the gearing in several lines. 
 
 The relative motion of the grains renders the 
 medium elastic : and is thus the only cause of 
 
 FIG. 5. 
 
14 Size of grains 
 
 elasticity in the universe, and, hence, is the prime 
 cause of the elasticity of matter. 
 
 The grains which are of definite size are 
 inconceivably small ; their diameters being as 
 measured by the wave-length of violet light 
 only the seven hundred thousand millionth part 
 of the wave-length. 
 
 While the mean path of the grains, as com- 
 pared with the diameter of the grain, is only the 
 four hundred thousand millionth part of the 
 diameter of the grain. 
 
 The mean relative velocity of the grains is 
 about one and one-third feet per second. 
 
 These three quantities define the state of the 
 medium in spaces where the piling is normal ; 
 and also define the mean density of the medium 
 as being ten thousand times greater than that of 
 water, or, as being four hundred and eighty times 
 greater than that of the densest matter on the 
 earth. 
 
 The mean pressure of the medium is nearly 
 seven hundred and fifty thousand tons on the 
 square inch, being more than three thousand 
 times greater than the strongest material can 
 sustain. 
 
 The coefficient of the transverse elasticity, 
 resulting from the pressure and gearing of the 
 
Size of grains 15 
 
 grains, in C.G.S. units is nine multiplied by ten to the 
 power of twenty -four-, and from this it follows that 
 the rate of the transverse wave is that of light ; 
 while that of the normal wave is two and four- 
 tenths greater than that of the transverse wave. 
 
 Then as the general result of the research, 
 in spaces in which the piling is normal, which 
 spaces are almost indefinitely greater than those 
 occupied by matter, it is shown, that there is 
 one and only one conceivable purely mechanical 
 system capable of accounting for all the physical 
 evidence, as we know it, in the universe. 
 
 The system being neither more nor less 
 than an arrangement of indefinite extents of 
 spherical grains in normal piling, so close that the 
 grains cannot change their neighbours, although 
 continually in relative motion with one another ; 
 the grains being of changeless shape and size. 
 Thus constituting, to a first approximation, an 
 elastic medium, with six axes symmetrically 
 placed, see Fig. 2, p. 10. This, then, is the struc- 
 ture of the universe, except in the comparatively 
 indefinitely small spaces in which there is 
 matter ; grains in normal piling, of which the 
 size, mean path, relative velocity and mean 
 pressure are defined, and extend to infinity. 
 Could anything be more simple? 
 
1 6 Degradation of light 
 
 In place of an elastic medium, of density 
 approximating to empty space, we have this 
 definite granular structure of the universe, elastic 
 and of density ten thousand times that of water. 
 
 5. The degradation of light. 
 
 Up to the present time it has been a moot 
 question whether there is any loss of light in 
 transmission. Herschel held that there was 
 none. But it is now shown that, owing to the 
 angular redistribution or viscosity of the grains, 
 there is degradation, such as would require fifty- 
 six millions of years to reduce the total initial 
 energy of the light to one-eighth, while the 
 rate of the degradation of the normal wave, 
 owing to the linear redistribution of the grains, 
 is such as would be reduced, in the same ratio, 
 in the two hundred and fifty thousandth part of 
 a second ; thus accounting for the absence of 
 any evidence of . normal waves except such 
 evidence as might be obtained within some 
 thousands of metres from its origin ; as in the 
 case of Rontgen rays. 
 
 And we thus have an explanation of the 
 blackness of the sky on a clear dark night as 
 a consequence of the dissipation of the mean 
 motion of light to increase the relative motion 
 of the grains. 
 
Permanence of inequalities 1 7 
 
 6. The permanence of the inequalities. 
 
 In spaces in which there are inequalities, 
 owing to the presence or absence of a number 
 of grains, in deficiency or excess of the number 
 which would render the piling normal about 
 local centres in the medium, these are per- 
 manent, as is now shown. And these are 
 attended by inward or outward displacements 
 and strains in the normal piling, as the case 
 may be, extending indefinitely through the 
 medium, causing everywhere dilatation equal to 
 the strain but of opposite sign. 
 
 And, according to the evidence, the diameters 
 of such inequalities are about loo-ioooth of the 
 wave-length. 
 
 When the arrangement of the grains about 
 the centres is that of a nucleus of grains in 
 normal piling on which the grains in strained 
 normal piling rest, the nucleus, in normal piling, 
 cannot gear with the grains in strained normal 
 piling; so that there is a singular surface of 
 misfit between the nucleus and the grains in 
 strained normal piling. 
 
 Such singular surfaces are surfaces of weak- 
 ness, and may be surfaces of freedom or surfaces 
 of limited stability with the neighbouring grains. 
 R. 2 
 
i8 
 
 Mobility of medium 
 
 7. Mobility of the medium. 
 
 The singular surfaces, when their limited 
 stability is overcome, are free to maintain their 
 motions through the medium by a process of 
 propagation in any direction the number of 
 grains forming up on the grains already in 
 position being the same as those left behind ; 
 so that the grains in the nucleus have no mean 
 motion whatever. 
 
 Any conception of this external propagation 
 is difficult. But the process admits of such 
 
 FIG. 6. 
 
External propagation 
 
 experimental demonstration as explains the 
 apparent paradox presented by a nucleus 
 within the surface of which the grains are at 
 rest, in the normal piling, propagating through 
 the medium in one direction, while the only 
 mean motion, that of the grains in strained 
 normal piling, outside the nucleus, are moving 
 in the opposite direction. 
 
 The apparatus is shown in Figs. 6 and 7. 
 
 In this apparatus, Fig. 6, the six balls in 
 close order in the lower chase constitute the 
 
 Fir, 7. 
 
 2 2 
 
2O External propagation 
 
 nucleus, and the other balls, higher up, represent 
 the medium, which is strictly limited. The gap 
 between these two sets of balls represents the 
 inequality which is to propagate through the 
 medium for a distance of 18", as soon as the 
 inequality receives an impulse by the removal 
 of the block on the upper chase, when the 
 ball runs down, and, as you see, the nucleus has 
 advanced 18". 
 
 The progress of the action is shown in 
 Fig. 7. The downward slope in the upper chase 
 being uniform for a small distance beyond 
 the position of the stop ; then varying, as an 
 inverted s, so as to be horizontal at the level of 
 the lower chase when the stop is removed the 
 balls start with nearly equal acceleration, then 
 having accelerated, when they reach the steep 
 slope they are separated. And thus the first ball 
 from the upper chase reaches the first ball in the 
 lower chase somewhat ahead of the second ball 
 from the upper chase, and strikes the first ball 
 in the lower chase when it comes to rest without 
 disturbing any of the balls which were in the 
 lower chase except the last, which takes up the 
 motion and runs on towards the basket : the 
 momentum having been conducted through the 
 six balls. Thus the nucleus has advanced in 
 
External propagation 2 1 
 
 the opposite direction to that of the motion of 
 the grains, or balls, in strained normal piling by 
 the diameter of a ball. 
 
 And in the same way for each of the remain- 
 ing five balls, each grain as it comes down 
 causes the nucleus to advance by a diameter 
 in the opposite direction to that ball. Fig. 7 
 shows the action in progress when there have 
 been four encounters. Three balls have dis- 
 appeared into the basket, and one is on its way 
 to the basket ; the nucleus has advanced four 
 diameters. One ball is still on the upper chase, 
 while another is on the steep slope. 
 
 Then, when the inequalities are such as 
 result from an absence of grains, the singular 
 surfaces correspond to the molecules of matter. 
 
 It may help in the formation of a conception 
 if we recal Lord Kelvin's theory of vortex atoms 
 which promised so much, and afforded the 
 first conception of matter passing through a 
 space completely occupied by matter without 
 resistance. In that theory the vortex ring, in 
 which the displacement is from the inside, was 
 the instrument, so to speak, that was to secure 
 the free motion of matter through the medium. 
 This theory has been found intractable, and is 
 now shown to be impossible. But in its place we 
 
22 Waves in the medium 
 
 have the external propagation, which presents 
 none of the difficulties of its predecessor. 
 
 Nor can we pass this stage without calling 
 attention to the startling conclusion to which 
 this external propagation leads. 
 
 8. Singular surfaces are wave surfaces. 
 
 It is shown that the matter of the molecules 
 passes freely through the medium or vice versa. 
 What does this imply ? 
 
 That the singular surface has all the charac- 
 teristics of a wave boundary. 
 
 If the medium is stationary and the molecules 
 are moving with the earth, the grains within the 
 surfaces do not partake of the mean motion 
 of these surfaces, being continuously replaced 
 by other grains by the action of propagation, by 
 which the singular surfaces in their motion are 
 continually absorbing the grains in front and 
 leaving those behind without any mean effect 
 on the motion of the grains. And thus there is 
 perfect freedom of motion of the molecules or 
 aggregate matter, although the grains which 
 constitute the nuclei are changing at the rates 
 expressed by 20 miles a second. 
 
 To be standing on a floor that is running 
 away at a rate of 20 miles a second without 
 
Molecules are individuals 23 
 
 being conscious of any motion, is our continual 
 experience ; but to realize that such is the case 
 is, certainly, a tax on the imagination. 
 
 Such motion has all the character of a wave 
 in the medium : and that is what the singular 
 surfaces, which we call matter, are waves. We 
 are all waves. 
 
 g. The molecules are individuals. 
 
 The singular surfaces which we call molecules 
 are individuals, which, although they may co- 
 here, cannot pass through each other ; and thus 
 although the only mass, that of the medium, is 
 changing every instant, at the extreme rates 
 already mentioned, these singular surfaces or 
 molecules preserve their individuality, the realiza- 
 tion of which is a further tax on the imagination. 
 
 10. Negative inequalities. 
 
 When the singular surface of a negative 
 inequality is propagating through the medium, 
 which is at rest, the grains forming the nucleus 
 will have no mean motion, whatever may be 
 the motion of the surface. But the strained 
 normal piling which surrounds the surface, and 
 propagates with the surface, being of less density 
 than the mean density of the medium, represents 
 
24 Negative inequalities 
 
 a displacement of the negative mass of the in- 
 equality, i.e., of the grains absent, and in what- 
 ever new direction the surface is propagating, the 
 motion of the medium, outside the surface, is 
 such as represents equal and opposite momentum, 
 as when a bubble rises in water. 
 
 And in the same way for the inequality 
 resulting from an excess of grains, the mo- 
 mentum would be positive. 
 
 1 1. The principal stresses in the medium. 
 
 The principal stresses outside the singular 
 surface of a negative inequality are to a first 
 approximation, 
 
 Two equal tangential pressures equal to f of 
 the mean pressure in the medium, and a normal 
 pressure equal to f of the mean pressure in the 
 medium. The resultant of the pressures being 
 everywhere equal to the mean pressure of the 
 medium. 
 
 1 2. The cause of gravitation. 
 
 We have now sketched the structure of the 
 inequalities, where inequalities exist, as resulting 
 from the presence or absence of grains, the 
 permanence of the singular surfaces, and their 
 freedom of motion by propagation, and have 
 
Cause of gravitation 25 
 
 thus arrived at what may be called the historical 
 problem, as to the cause of gravitation. 
 
 From Kepler's discoveries of the relation be- 
 tween the distances and periodic times of the 
 planets, Newton was able to show that the 
 law of gravitation is as the inverse square of the 
 distance. But this in no way explained the 
 cause of this law. And in spite of all attempts 
 in the mean time, up to the year 1885 no clue 
 had been found. . 
 
 In that year, however, as the result of an 
 attempt to discover what properties a medium 
 possesses, that it might fulfil the functions of the 
 ether, including gravitation, the property of all 
 granular matter in close piling to expand under 
 strain was recognized, I believe for the first 
 time. 
 
 This property was at once recognized as an 
 obvious clue to the cause of gravitation, however 
 intractable it might prove. 
 
 The experiments made in 1885 will serve 
 to illustrate the cause of gravitation, besides 
 affording me the gratification, I cannot but 
 feel, in first exposing the verification of my con- 
 viction in this house and before this audience. 
 
 Hitherto in this lecture it has been found 
 necessary to refer to granular medium as con- 
 
26 Cause of gravitation 
 
 tinuous in space. But it is doubtful whether 
 the significance of this continuity has been 
 realized. 
 
 We have defined the pressure in such media 
 as being enormous, and it now becomes necessary 
 to realize that, in order to get granular material 
 under pressure, one or other of two conditions 
 must be satisfied. 
 
 If, as in the universe, the grains in normal 
 piling extend indefinitely there can be no mean 
 motion of the boundaries, whatever the pressure 
 may be ; and thus the grains are virtually within 
 a closed surface. 
 
 But if the number of grains is limited and 
 we can only experiment on a finite number of 
 grains it is necessary, in order to realize the 
 dilatation, that the grains should be in a closed 
 envelope of some kind which constrains them ; 
 and it may be mentioned here that the reason 
 why dilatancy had not been discovered pre- 
 viously was undoubtedly the general absence of 
 any constraining surface. 
 
 Without such a surface, in order to illustrate 
 the dilatation, we may join the horizontal layers 
 of grains in normal piling together, as in 
 model (i), then subjecting these layers of grains 
 in normal piling to transverse strain, the dilata- 
 
Dilatancy 
 
 27 
 
 tion which results from parallel distortional 
 strain in the normal piling becomes evident, as 
 
 FIG. 8. 
 
 is shown by comparing model i, p. 9, in which 
 the grains are in closest order, with model 8, 
 in which the grains are shown in maximum 
 dilatation. 
 
 But this only affords a very partial illustra- 
 tion of the possibilities of the general dilatation, 
 
28 Dilatancy 
 
 when all the grains are separate, and only 
 constrained by the pressure outside the con- 
 straining surface. 
 
 I have in my hand the first experimental 
 model universe, a soft indiarubber bag with a 
 small aperture to admit of its being filled with 
 small shot ; which aperture is partly closed, 
 sufficiently to prevent the shot from coming out, 
 by a glass tube which also serves the purpose of 
 a gauge to measure the dilatation. After filling 
 the bag with small shot, the interstices are filled 
 with highly coloured water, and subjecting the 
 bag to small distortional strains to get the shot 
 in normal piling: then in order to render 
 apparent the inverse behaviour of bag with shot 
 and water under distortional strains we take 
 another similar bag with a similar tube filled 
 only with water and subject both to similar 
 distortional squeezing. When, as was obvious, 
 the water in the tube of the bag without shot 
 rises in the tube ; while on the contrary the 
 water in the tube with shot sinks, drawing water 
 from the tube into the bag, as is shown 
 respectively in figs. 9 and 10, and 1 1 and 12. 
 
 It is thus shown that any deformation from 
 the normal piling causes the interstices between 
 the grains to increase, expanding the bag. This 
 
Dilatancy 
 
 29 
 
Dilatancy 
 
Dilatancy 3 1 
 
 experiment, which you see is on a very small 
 scale, was not designed to show to an audience, 
 being the original experiment made for my own 
 satisfaction when the idea of dilatancy first 
 presented itself. 
 
 Without knowledge of dilatancy the results 
 thus obtained must appear paradoxical, not to 
 say magical. 
 
 Taking a larger apparatus, a bag which holds 
 six pints of sand, the interstices being filled with 
 water, without any air, the glass neck being 
 graduated, so as to measure the water drawn 
 in and the sand and water subjected to such 
 distortion as would cause the grains to be in 
 normal piling. 
 
 Then, if the neck be closed so that it cannot 
 draw more water, we come upon the startling 
 fact, that this indiarubber bag, filled with sand 
 and water, cannot have its shape changed with- 
 out a sufficient distorting pinch, as by these 
 pincers, to cause a vacuum inside the bag. 
 
 The pinch is now some 200 Ibs., but the bag 
 does not flinch. It cannot change its shape 
 without drawing water into the bag and this is 
 prevented. 
 
 To show that there is an effort to expand the 
 bag, while the pinch is on, it is only necessary 
 
Dilatancy 
 
 to bring the bag into communication with a 
 pressure-gauge, when the mercury shows that 
 the pressure in the bag is only half of that of 
 the atmosphere outside the bag. 
 
 FIG. 13. 
 
 Had the pinch been greater a vacuum would 
 have been caused within the bag by the expan- 
 sion of the interstices. As it is, however, the 
 bag has maintained its rigidity. 
 
 Then, if the pressure-gauge is closed and the 
 neck opened, so that water may be drawn in, the 
 
Dilatancy 
 
 33 
 
 bag at once begins to change its shape, the water 
 sinks in the neck until a pint of water has been 
 drawn into the bag. 
 
 FIG. 14. 
 
 This is the maximum dilatation, the grains 
 are now in the most open order into which they 
 can be brought by squeezing. Further squeezing 
 causes the grains to take closer order, so that 
 the interstices diminish and the water rises in 
 the neck. 
 
 R. 3 
 
34 Dilatation of sand 
 
 The dilatation of sand is well shown by 
 india-rubber balloons, which can be expanded 
 by the mouth, these afford an almost transparent 
 
 FIG. 15. 
 
 envelope. Taking one which holds six pints of 
 sand and water, closed without air, there being 
 more water than will fill the interstices at maxi- 
 mum density, but not enough to allow the full 
 extension of the interstices. When standing on 
 a table the elasticity of the envelope gives it a 
 
Dilatation of sand 35 
 
 rounded shape ; the sand has now settled down 
 to the bottom, and the excess of water appears 
 above the sand, the surface of which is free. 
 The bag may now be squeezed and its shape 
 altered, apparently without resistance ; but this 
 is only as long as the surface is free. 
 
 FIG. 16. 
 
 Putting the bag between two vertical plates 
 and slightly shaking and squeezing, so as to 
 keep the sand at its densest, it still has a free 
 surface and it can, in this way, be spread out until 
 
36 Dilatancy of sand 
 
 it is a broad flat plate ; it is still soft, as long as 
 it is squeezed, but when the pressure is removed 
 the elasticity of the bag tends to draw it back to 
 its rounded form, changing the shape, enlarging 
 the interstices and absorbing the excess of water. 
 This is soon gone and the bag remains a flat 
 cake with peculiar properties ; to pressures on its 
 
 FIG. 17. 
 
 sides it yields at once, such pressures having 
 nothing to overcome but the elasticity of the 
 bag since change of shape in that direction 
 diminishes the interstices. But to pressures on 
 
Clue to the cause of gravitation 37 
 
 its edge it is perfectly rigid, as such pressures 
 tend to increase the interstices ; when placed on 
 its edge it sustains 2 cwt. 
 
 These experiments made on tangible matter 
 fall short of what would be the dilatations if 
 made with absolutely rigid uniform spheres. 
 But after seeing the evidence that dilatation is 
 the ruling property of granular matter, I venture 
 to think that you will agree with me that with- 
 out this clue the cause of gravitation would not 
 have been discovered. 
 
 It now appears, however, as a complement 
 to dilatation, in causing gravitation, we have the 
 relative motion of the grains, which has already 
 been shown to impart the elasticity necessary 
 for the transmission of light, and this affords the 
 complement for the complete explanation. 
 
 But, even so, the explanation is not what 
 was expected ; for gravitation is not the result 
 of that dilatation which results from uniform 
 parallel strains in the medium in normal piling, 
 but results solely from those components of the 
 dilatations caused by the space variation of the 
 inward strains. 
 
 Thus, as long as the dilatation strains are 
 parallel there is no attraction; but if there is 
 curvature in the strains there will be efforts, 
 
 33 
 
38 Curvature in the strains necessary 
 
 proportional to the inverse square of the distance, 
 to cause the negative inequalities to approach 
 from a finite distance. 
 
 Thus gravitation is the result of those com- 
 ponents of the dilatations (taken to a first 
 approximation) which are caused by the varia- 
 tions of the components of the inward strains, 
 caused by curvature in the normal piling of the 
 medium. 
 
 The other components of the strains, being 
 parallel distortions, which satisfy the conditions 
 of geometrical similarity, do not affect the efforts. 
 
 Then, since if the grains were indefinitely 
 small, while the curvature in the normal piling 
 was finite, there would be no effort. And thus 
 the diameter of the grains becomes the parameter 
 of the effort. And multiplying this parameter 
 by the curvature of the medium, and again by 
 the mean pressure of the medium, the product 
 measures the intensity of the efforts to approach. 
 
 The dilatation diminishes as the centres 
 of the negative inequalities approach, and work 
 is done by the pressure, outside the singular 
 surfaces, to bring the singular surfaces of the 
 negative inequalities together. 
 
 The efforts to cause the approach of the 
 centres correspond, exactly, to the gravitation 
 
Matter absence of mass 39 
 
 of matter if matter represents the absence of 
 mass, and thus the inversion of preconceived 
 ideas is complete. Matter is measured by the 
 absence of the mass necessary to complete the 
 normal piling. And the effort to bring the 
 negative inequalities together is also an effort 
 on the mass to recede ; and since the actions 
 are those of positive pressure, there is no attrac- 
 tion involved, the efforts being the result of the 
 virtual diminution of the pressures inwards, and 
 in this inversion we have a complete, quantitative, 
 purely mechanical explanation of the cause of 
 gravitation. 
 
 13. Positive inequalities. 
 
 The explanations of the transmission of light 
 and the cause of gravitation do not complete the 
 explanation necessary to satisfy all the evidence. 
 
 Thus far we have sketched the efforts of 
 negative inequalities only. 
 
 The efforts of the positive inequalities are the 
 reverse of the negative inequalities, tending to 
 separate the positive centres, and cause the 
 positive inequality to scatter through the medium, 
 thus dissipating any effects throughout the 
 medium. Then, since the space occupied by 
 inequalities is almost indefinitely small com- 
 
4O Positive inequalities 
 
 pared to the space in normal piling, it appears, 
 even if there are as many positive inequalities 
 as there are negative inequalities, the positive 
 will present no evidence, being scattered, while 
 the negative inequalities being brought together 
 by gravitation, are in evidence. 
 
 Besides the positive and negative inequali- 
 ties, there is another inequality which can easily 
 be conceived. 
 
 Whatever may be the cause it is possible 
 to conceive that a number of grains may be 
 removed from one position, in the medium, to 
 another position, the medium being otherwise 
 uniform ; thus instituting a complex inequality, 
 as between two inequalities, one positive and the 
 other negative, the number of grains in excess 
 in the one being exactly the same as the number 
 absent in the other. 
 
 Such complex inequalities differ fundamen- 
 tally from the gravitating inequalities, inasmuch 
 as the former involve the absolute displacement 
 of mass, while the latter have no effect on the 
 position of the mean mass in the medium : and 
 in respect of involving absolute displacement of 
 mass, the complex inequality corresponds with 
 electricity. 
 
 Apart from the displacement of mass, the 
 
Complex inequalities 41 
 
 complex inequality differs from the gravitating 
 inequalities. 
 
 In the complex inequalities the parameter of 
 the dilatation is not the diameter of the grain, 
 but is one half the linear dimension of the 
 volume of the grain displaced taken as spherical. 
 
 The intensity of the efforts to revert in the 
 case of a complex inequality is the product of 
 the pressure multiplied by the product of the 
 volumes of the positive and negative inequalities, 
 and again by the parameter. 
 
 Thus we have a purely mechanical explana- 
 tion of electricity. And not only so, for we have 
 also a determination of what has been a moot 
 question is electricity one thing or two ? 
 
 For since the absolute displacement of mass 
 is one displacement, and the expression for the 
 inequality is negative, it is such as cannot be 
 resolved, as hitherto assumed, into two rational 
 factors. 
 
 It is interesting to compare the efforts of 
 gravitation with the efforts to revert of electri- 
 city. 
 
 If we consider the case of two gravitating 
 inequalities, at unit distance, and compare with 
 these a complex inequality, the number of 
 grains and distances being the same, it is found 
 
42 Electricity 
 
 that the effort to revert of the complex in- 
 equalities, is more than one thousand billion 
 times greater than that of the gravitating in- 
 equalities. 
 
 The cohesions and surface tensions, on which 
 the strength of our structures depend, are the 
 result of the manner in which the efforts to 
 approach vary, when within distances which are 
 small, as compared to the dimensions of the 
 molecules, and are thus the complements of 
 gravitation. 
 
 14. The cause of light. 
 
 Although we have already sketched the 
 transmission of light it is only now that we 
 come to the cause of light. 
 
 The institution of the waves of light follows 
 as the result of the disruptive reversion of the 
 complex inequalities, or electric-discharges ; the 
 recoil from which sets up a vibration, in the 
 medium, which is exhausted in initiating waves 
 of light and heat. 
 
 Thus far the sketch of the new ideas has 
 included only those for which there exists 
 sufficient evidence to admit of definite quantita- 
 tive analysis. 
 
 Nevertheless these quantitative results show 
 
Aberration of light 43 
 
 that the granular medium, as already defined, 
 accounts by purely mechanical considerations 
 for the evidence ; and also affords the only 
 purely mechanical explanations possible. 
 
 If then the substructure of the universe is 
 mechanical, all the evidence, not already ad- 
 duced, is such, as may be accounted for, by an 
 extension of the analysis ; and, thus far, this has 
 been found to be the case. 
 
 Thus we have the purely mechanical explan- 
 ation of the absorption of light by the molecules 
 of matter ; 
 
 The dispersion of the spectrum ; 
 
 The association and dissociation of the mole- 
 cules ; 
 
 The refraction of light ; 
 
 The polarization of light by reflection, and as 
 it now appears this is caused only by that 
 component of the motion of the transverse 
 wave which is in the plane of incidence ; also 
 
 The metallic reflection of light ; 
 
 And of the aberration of light, as resulting 
 from the absence of any appreciable resistance 
 to the passage of matter through the medium. 
 
 Then, considering that not one of these phe- 
 nomena had previously received a mechanical 
 explanation, it appears how indefinitely small 
 
44 These ideas will prevail 
 
 must be the probability that there should be 
 another structure for the universe which would 
 satisfy the same evidence. 
 
 And thus we may have the fullest confidence 
 that the structure is purely mechanical, and that 
 ideas, such as I have endeavoured to sketch, 
 will ultimately prevail, displacing for ever such 
 metaphysical conceptions as that of action at a 
 distance, and accomplishing that ideal which, 
 from the time of Thales and Plato, has excited 
 the highest philosophical interest. 
 
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