AMONG the problems which Nature presents, there is none more fascinating and none more baffling than that which relates to the mechanism of gravitational attraction. Natural Philosophy deals with many recondite subjects ; but here we have an agency whose action is the common experience of everyday life. There is nothing obscure in the phenomenon itself; no special isolation, analysis or experimental skill is needed to exhibit its working. It is a force which determines nearly every act of our life.
Yet in contrast with this wealth of experience, we find that as to the ultimate nature of the mechanism which produces these actions, nothing certain is known. Many hypotheses have been put forward, some almost metaphysical in their subtlety, others thoroughly scientific in their industrious and cautious development. Yet none have brought full conviction or offered any of that substantial justification which science demands in the hypotheses that are put before her.
There is another general feature of this question, worthy of notice. Although we fail (or only partially succeed) when we attempt the construction of a process for gravitation, yet the quantitative laws which accompany that process are known with great accuracy. On the mathematical side, gravity is no insoluble riddle. We have, on the one hand, astronomical observations, and thus a continual supply of exact data which confirm the famous inductions of Kepler, and so the laws of gravitational action ; while on the other, the extensive range of pendulum experiments, torsion balance experiments, common balance experiments, give a constant verification of these laws as regards small bodies near the surface of this earth. In the hands of modern experimenters, such as Bessel, Boys, and Poynting, these  investigations have been carried to an extraordinarily high degree of accuracy; probably higher than that of any other physical measurement. For example, determinations of gravity by the pendulum have an accuracy of i part in 500,000 ; the experiments of Boys with the torsion balance (the refined form of the Cavendish experiment) have an accuracy of i in 10,000. In Poynting's experiments upon the value of the mean density of the earth, the form of common balance used was so delicate that an increase in the weight of 1 part in 100 million, could be detected. The spring gravimeter of Professor Threlfall measures variations of "g" with a precision of 1 in 500,000 ; while to the same degree of accuracy Bessel showed that attraction is independent of the nature of the attracting bodies. So in all the range of experiments, by which the laws and characteristics of gravitational attraction have been elucidated, a very high degree of accuracy has been attained ; and no natural truths are more firmly and broadly based.
Again, a very full mathematical theory of attraction has been worked out on the basis of the Newtonian laws; and in the motion of the tides, the attraction of hill ranges, the variation of gravity over the earth's surface, etc., this theory receives full confirmation. Yet against this plenitude and accuracy of observation and inference, there is very little to be set when a mechanism of the action is demanded. Even the group of corpuscular theories of which I am to speak to-night, though by far the most plausible approach to an explanation, still leave in the mind an impression of unreality hard to combat. Nevertheless it is interesting to trace the attempts at a solution of the ultimate nature of gravitation, and to exhibit the present state of opinion upon the problem.
Making first a brief review of the matter, we may conveniently begin at the time of Newton (1680). Before then, the problem presented itself in a vaguer way ; for when the exact laws to be met by any hypothesis were not known, great opportunity was left for speculations which could be neither affirmed nor denied, compared nor disproved. Still, despite the fact that many of  these theories were of a fantastic nature, it is interesting to note that in several, a dynamical explanation in terms of the intervening medium was sought for the motion of the celestial bodies. The cycles and epicycles with which the heavens were " scribbled o'er " by the Ptolemaists, the crystal spheres of Aristotle, and the whirling vortices of Descartes, all exhibit a desire to apply common experience of kinematics and kinetics to the celestial mechanism. And so far as these methods reject an occult action at a distance (which would exclude thought from the intervening medium) to that extent they are in harmony with the modern attitude.
But curiously enough this physical tendency in thought was overthrown by the advent of Newton. In 1682 he propounded his law of universal gravitation, and there at once arose a disregard for the medium, and a satisfaction with action at a distance. Gravitation came to be considered as a first cause, a prime essential of matter, as invariably connected with it as inertia or extension, and as incapable as these of further explanation. Thus there arose the doctrine of action at a distance, which, so far from being obsolete in the epoch of Faraday, Clerk Maxwell, and Hertz, has had vigorous support both here and in Germany, during the last 30 years. Newton himself was opposed to this doctrine, although his gravitation law led to its development. In the oft quoted letter to Bentley he says — "That gravity should be innate, inherent, and essential to matter, so that one body may act on another body at a distance, through a vacuum, without the mediation of anything else by, and through which, their action and force may be conveyed from one to the other, is to me, so great an absurdity, that I believe no man who has in philosophical matters, a competent faculty of thinking, can ever fall into it."
As if to enforce still further this opinion, Newton himself attempted a dynamical explanation of gravity, by means of the differential pressures of a medium in which all bodies were supposed immersed. Nevertheless with his name became associated the theory of the direct gravitational action at a distance, and his discovery of the laws of attraction upset all the physical notions of action through a medium, which were contained in the older speculations. Before Newton's time celestial mechanics, such as  it was, was physical ; thereafter it became, for a time, more exclusively mathematical. An extract from one of Voltaire's letters, quoted by Whewell, gives an interesting picture of the change — " A Frenchman," he says, "who arrives in London, finds a great alteration in philosophy, as in other matters. He left the world full, he finds it empty. At Paris you see the universe composed of vortices of subtile matter ; in London we see nothing of the kind. Among Cartesians all is done by impulsion ; with the Newtonian, it is done by an attraction of which we know the cause no better."
It was not long, however, before the " emptiness " on which Voltaire comments, became the subject of attack. Thinkers, unwilling to rest upon the physically barren (and to some views, unphilosophical) idea of direct distance action, began to put forward theories in which many kinds of mechanism were suggested. Now, however, any attempt must fit itself to the laws and characteristics enunciated by Newton. The field of legitimate hypothesis became much restricted. Yet speculation often rushed in where science feared to tread. Elastic media exerting stress, fluids of variable density, transmitting pressure, streams of particles acting by impacts, impalpable ethers conveying waves, these and many other types of action have been put forward. In the i8th century, we do not find very many theories, because the tendency during that period was, for the reasons we have seen, to give over gravity to the abstract mathematicians, and to regard action at a distance as the outpost, beyond which enquiry need not be pushed. But if there were not many, there was in that century one very notable hypothesis, which is the basis of the whole corpuscular group. This theory is worked out with an ingenuity of idea, and fulness of detail, which make it rank high among scientific works. I refer to the hypothesis of Lesage, to whom with Fatio and Redeker, belongs the credit of starting the impact view of the production of gravitational action. This theory is, in the opinion of the late Professor Tait "The only plausible answer to this problem, which has yet been propounded."
With the 19th century we have a great increase of the number of theories upon the subject ; though in some quarters it was  thought that all such theorising would tend to atheism. In a report of the Smithsonian Institution, which summarises the state of the problem up to 1876, there are mentioned about twenty accredited men of science who have taken up the problem, and put forward hypotheses for its solution. Since then about fourteen more theories have appeared in the various scientific journals. It would be quite impossible, in the time at my disposal, to give even a summary notice of all these theories. I intend therefore to confine myself to one well defined group, that of the corpuscular theories. This group is the most developed and consistent, it gives the most satisfactory explanation of the phenomena, it comes best into accord with ordinary dynamical experience. From a few simple postulates it deduces all the necessary results. It has the sanction of names like Kelvin and Tait ; and it gained from Clerk Maxwell in his Encyclopedia Britannica articles upon "Attraction," and "Atom," very serious consideration. To some minds, the mechanism suggested, seems too artificial and improbable ; and in default of any complete success, a hopeless attitude has been taken up, e.g., by du Bois Reymond. But this position is not at all inevitable ; and if electric and magnetic actions have been dealt with from the point of view of the medium ; if the phenomena of light, heat, and kindred radiations have capitulated to wave theory, there is no a priori reason why the gravitation problem should be stamped as hopeless, and devoid of all scientific importance.
Before proceeding to a detailed view of the corpuscular theories, we may form a synopsis of the laws and characteristics of gravitational action. To these any theory put forward must conform. We have : —
- 1) The radial nature of the action.
The attraction between two bodies is along the straight line joining them. This was proved mathematically by Newton for the planets in relation to the sun. From Kepler's observation that the radius vector from the sun to any planet sweeps out equal areas in equal times, the radial nature of the action follows at once by an application of the theory of central forces. When the distance between the attracting bodies becomes comparable with  their dimensions, so that they can no longer be regarded as practically at a point, the supposition is made that between every particle of the one and every particle of the other there is an action in the straight line joining the particles. The results which mathematical theory builds on this supposition are perfectly concordant with experience.
- 2) Absence of reflection, refraction, retardation, shielding or other disturbance of the action.
In all the other natural actions, which are conveyed through space from one body to another, there is a marked effect produced by varying the nature of the matter in the intervening space. Light is reflected, refracted, polarised. Sound is reflected, absorbed, retarded. Electric and magnetic actions are influenced by the nature of the medium, in ways too numerous to mention. The radiations from radioactive bodies are capable of being deflected or absorbed. The Hertzian waves and, to a less extent, the Rontgen rays are under our control in the medium. Gravitational action, alone, exists between the two bodies and cannot be affected by any known treatment of the intervening space. Only at the gravitating bodies themselves have we any control over this agency.
Experiments have been recently made upon this point. In 1897, Austin and Thwing employed a modified form of the Boys torsion apparatus to investigate the effect of screens of different material upon the gravitational action.
Layers of lead, zinc, mercury, water, alcohol, glycerine about three cms. thick were interposed between the attracting masses and the moveable arm. If a gravitational "permeability," i.e., a difference in what may be called conducting power for the gravitational effect exists, in any appreciable degree, this should have exhibited it. The authors state that to an accuracy of 1 in 500, no such effect exists. Poynting, however, suggests that there may be differences in gravitational permeability of a very small order similar to the small magnetic permeabilities observed in substances such as water, alcohol, etc. Eötöos stated at the Paris Physical Congress in 1900, that the sun's attraction is not modified by one part in 100 million by the interposition of 1  kilometre of the earth's crust. And in this connection it may be noted that astronomical observation detects no shielding effect or other alteration during eclipses.
- 3) Proportionality to the acting masses.
The intensity of the action is proportional to the product of the masses of the two bodies ; and no addition of mass impairs or cuts off the previous effect. From the small masses (about 2 grammes) used in the Boys experiments up to the great masses of the members of the solar system, this rigorous proportionality to mass is maintained.
- 4) Independence of nature of the acting masses.
The physical or chemical nature of the bodies has no effect on the action, provided the masses and distance remain constant. In this we have a striking divergence from what we find in all other physical actions. The matter was investigated by Newton who observed the times of swing of pendulums whose bobs were made of a variety of different materials. More recently Bessel examined the same point also by the pendulum method, and proved that the independence of the nature of the bobs was true to at least i part in 100,000. Still later Eötvös, in a report to the Paris Physical Congress in 1900, states that he has examined this effect by means of a special balance. He finds that for glass, brass, antimony, and some other substances, the attraction is the same (given the same masses and distance), to i part in 20 million.
- 5) Proportionality to the inverse square of the distance.
The attraction diminishes as the distance between the attracting masses increases, and the law of variation is that of the inverse square. Given a certain action at i ft., it is diminished to a fourth part at 2 ft., to a ninth part at 3 ft., and so on. Here we have the same law holding as in the case of electric and magnetic point charges, light, sound, and heat radiations from a source, etc. In the case of sound, light, heat, or in fact any radial emanation of energy from a point, the law of the inverse square is a direct consequence of the fact that the action takes place in three dimensional space. If our universe of matter were two dimensional, the law would be that of the inverse first power. In the first case the constant amount of energy, at a distance r, is spread over a spherical surface of area 4πr²; in the second, over  a line of length 2πr. There would seem to be a strong presumption that in the case of gravitational, electric and magnetic actions, the law is in some way brought about by the special space conditions under which we live. At any rate, the index is 2 to a very high degree of exactitude and not 2 ± some appreciable fraction.
Astronomical observation has shown that a change of the index from 2 to 2'000,000,16 would be sufficient to explain certain irregularities in the motion of Mercury. Even this small change, however, is not necessary as the irregularity is otherwise explicable. But it serves to indicate the exactness of the astronomical observations on which the law is based. Newton deduced this law from the induction of Kepler, that the orbits of the planets are ellipses with the sun in one focus.
The result follows easily by an application of the theory of central forces. From these immense planetary distances (Sun to Uranus, for example, 1,753 million miles), down to the small distances of about a centimetre in the Cavendish experiment, the law of the inverse square holds. Whether it holds at distances comparable with the dimensions of molecules, or the inter- molecular distances, is a question which has not been answered.
- 6) Instantaneous Propagation.
The question whether gravitational action is instantaneous or has a finite velocity, has been much debated. Suppose a body A placed suddenly in a new position. Does a finite time elapse before the mutual action between it and another body B is readjusted to the new configuration ? No such lag of the action has ever been discovered within ascertained distances and from astronomical observations, lower limits have been put to this hypothetical velocity of gravity. The number is large even in comparison with the velocity of light. For example, Laplace calculated by an "aberration" method, that a minimum value must be 10 million times that of light. Light travels at about 186,000 miles per second, so we see how great must be the velocity of gravity if it exist at all. Arago put this lower limit at 50 million times that of light. Later consideration of the subject  by Oppenheim in 1894, puts the limit at 12 million times that of light. It may be roughly said that if the gravitational influence occupied 1/10° of a second in travelling 106 miles, astronomical analysis would easily detect it.
- 7) Intensity undiminished by time.
No enfeeblement of the action has been observed in the lapse of time over which reliable astronomical observations have extended. This period is of course not large in comparison with the times of the cosmic processes ; but it gives us an idea of the exceedingly slow rate of exhaustion if any. Hence if a kinetic explanation of gravitation is sought, it must account for this maintenance of the machinery of the action.
- 8) Isotropy of all matter.
Certain bodies present along different lines, different properties in regard to certain physical actions. Thus wood conducts heat . at one rate along the grain, at another across. Crystals allow light to travel at different speeds according to the position of the line chosen in reference to the axis of the crystal. But all bodies (even the most aeolotropic to some actions) present the same property to gravitational action along every line. This suggestion as to the possibility of gravitational aeolotropy in crystals will be found in one of Kelvin's papers; and he points out that if such an aeolotropy existed, we would have means of drawing energy from the ether. But in Lesage's exposition of the corpuscular theory, this possibility is also indicated as a consequence of irregularity in molecular structure. Experiments have been made recently with great delicacy to discover whether this aeolotropy really exists. Mackenzie in 1895, using a form of the Boys apparatus and calcspar spheres 2 inches in diameter could discover no effect. Poynting and Gray in 1899 used 2 quartz crystal spheres and sought for a directive action of the one upon the other. Very delicate suspension was used and the method of sympathetic vibration was employed. But negative  results were got and the authors state that the difference of attraction in the crossed and in the parallel positions of the crystal axes cannot exceed I in 16,000. Any gravitational theory therefore, which employs a specific hypothesis as to the structure of matter must take account of this practically complete symmetry.
- 9) Absence of influence by temperature, chemical decomposition, etc.
Any such physical action upon the gravitating bodies, produces no effect. Experiments upon this point have been made by Kreichgauer, Landolt, Sandford and Ray, and these show to one part in 2 million, an absence of effect. Gravitation seems therefore, to have no tangible relation to the physical or chemical condition of the acting masses or of the intervening medium.
- 10) Duality of the action.
The mathematical theory of attraction employs the fiction of a single body creating a field of force round it, but this idea though convenient for analysis has no physical counterpart. The action is essentially a dual one requiring the presence of two bodies ; and to the attraction on the one there always corresponds ( in accordance with Newton's Third Law of Motion,) the counter and equal attraction on the other. We do not know yet whether an isolated body affects in any way the surrounding ether ; and where there are two bodies, it is impossible to say that the action emanates from the one and proceeds to the other. This should be borne in mind when considering the question of a velocity for gravity. Here we have a marked difference from such emanations as those of sound, light, heat, or energy from radio-active bodies, where a source and a recipient can be indicated, and where the idea of the single isolated source is not merely a convenient abstraction. It is a point in favour of the corpuscular theories that they are quite in harmony with this essential duality.
We come now to the consideration of the mechanism suggested for the explanation of the body of phenomena described above ; and, as stated previously, we shall confine ourselves to that group of theories which has as its essential idea, the impact of corpuscles. We have here a well defined class of speculation on the subject, and from the first suggestion in the i7th century up to the present day an interesting evolution may be traced. 
We find the germ in the writings of John Bernoulli (1734). He adopted Descartes' idea of the vortices, by which the planets were whirled along in their orbit. But to this he added what he calls the " central torrent," a stream of small and rapidly moving particles which, by their impacts from without, deflect the planets from the natural straight line into the curvilinear path. We need not consider this idea at great length, as it fails to account for the phenomena, and contains many purely arbitrary hypotheses. The memoir containing it, was, however, crowned by the French Academy. The real interest of this idea lies in the fact that the impact of corpuscles is suggested as being a possible means of explaining the mysterious attraction through apparently empty space. But it was necessary to get rid of the Cartesian vortices before the problem could come into the scope of rational dynamics. To three men we owe the development which followed — Fatio, Redeker, and Lesage. The first two had indeed the priority in this furtherance of the corpuscular system, but the step made by the last was so great, his theory was so fully developed in all its details, so carefully and ingeniously based on mathematical and dynamical results that his fame has somewhat eclipsed that of the other two. In any case, we have it on the testimony of the contemporaries of Lesage that his work was quite independent of that of his two predecessors ; and when, later, he became aware of their work on the subject, he was careful to give them all credit for it. Nicholas Fatio was a friend of Newton. He put forward his theory about the year 1694 in a letter to Leibnitz ; and also, it is said, expounded his system in a Latin poem which was never published. Francis Albert Redeker, a German physician, published, in 1736, a Latin dissertation upon the cause of gravity. An account of these works, sufficient for our purposes is given by Pierre Prevost of Geneva (1818), in a preface to an account of Lesage's work. It will, however, be more convenient to give an outline of the theory common to these three men, and then consider the points of difference. Each supposed that what we call empty space is full of streams of small particles moving with high velocities. In every conceivable straight line there is supposed to be a set of these particles at a given instant. And as the particles are supposed to be exceedingly small, we may practically assume two streams in opposite directions in  every straight thin filament of space. The particles or corpuscles as Lesage termed them, are not gross or ponderable matter ; that is to say, we are not aware of them in any way except in so far as they make their existence known by the gravitational effect.
Before going further it is important to bear in mind that there is nothing unscientific in these suppositions of exceedingly small particles and exceedingly high speeds. In what follows we shall have to deal with such postulates, and we must not be led by a priori ideas to a hasty condemnation. Consider, for example the range of magnitude 1 to 10 cms. Then consider the range 1019 to 1020; and then the range 10-19 to 10-20. In each of these three partitions we have a tract of magnitude quite definite and capable of subdivision and recognisable differentiation. Yet a hasty view of any one of the three from any other might dismiss it as a formless and meaningless affair — an infinitely great or an infinitely small so hopeless that thought need not approach it. Yet it is not so. There is a great difference between the 10-20 of our illustration and the infinitely small which defies thought. The visible spectrum contains an octave of vibrations as real, as distinguishable, as capable of analysis as an octave of notes in the human voice. Yet the one is reckoned in feet, while the other in hundred thousandths of the inch. We must not regard the "very smalls," or the "very greats," with which we shall have to deal, as meaningless exaggerations or diminutions.
We have therefore to suppose space filled with the streams of gravific corpuscles, so small that collision of one corpuscle with another rarely occurs, but so numerous that at any point on which we care to fix our attention, there will be found radiating both inwards and outwards, streams of these rapidly moving corpuscles.
An easily realised physical analogy is this : — Imagine a hollow sphere with its inner surface polished. Place a small source of light at the centre. Then every ray from the centre is reflected back again along its original course, i.e., through the centre again, so that we have at the centre an exactly similar state of affairs to that imagined in the theory. A still better illustration is obtained, if we suppose the surface of the sphere to be self luminous. Take for example a metal spherical shell heated to white heat. Then not only at the centre, but at any point in the interior, we shall have a system of rays of light proceeding in all directions, and both inwards and outwards. If we revert for the moment to the old  emission theory of light, the analogy becomes complete. In these days when ionic theories are being developed, and radio-active bodies experimented upon, it is as well not to be over contemptuous towards suppositions involving corpuscles and emissions of particles.
"Empty space " is thus to be regarded as a vast storehouse of energy full of minute corpuscles, moving in straight lines with high speeds in all possible directions. The nature of the corpuscles and their distribution, will be considered later, but to form our preliminary idea, let us now suppose a body of ordinary matter, say an isolated particle, placed in a region occupied by the streams. This body will be beaten upon by the torrent of corpuscles on all sides. Bearing in mind that the corpuscles are very small in comparison with the particles of gross matter, that the impacts follow upon each other with extreme rapidity, that the mean velocity of each stream is the same, that the mean density (i.e., the number of particles per unit length) of each stream is the same, we see that there will be no resultant tendency to move the body in any direction. That discrete impacts can produce the effect of a continuous pressure is an accredited fact of the kinetic theory of gases. The necessary condition is that the impacts shall follow each other sufficiently rapidly. We have, in fact, a complete analogy in the pressure produced on the surface of a solid sphere immersed in a gas.
We have thus an isolated body immersed in the gravific medium but as yet no apparent action or reaction. Suppose now a second body is placed in the medium near the first as in the figure. It is evident that a shielding effect of each by the other is brought about. The complete symmetry of the system of impacts is disturbed and the equilibrium of each body is destroyed. By the unbalanced pressure on the remote side each will be urged towards the other, and we thus have the effect of gravitational attraction. To revert to our illustration from light, suppose that within the self luminous hollow sphere, two opaque bodies are situated. Each will cut off from the other a certain amount of light. So in this case, we have the qualitative fact of gravitation explained. The next thing was to attack the quantitative laws. It is easily seen that this mechanism will satisfy the law of the inverse square of the distance, on account of the radial nature of the currents in respect to any point. In fact the three dimensional nature of space, here conditions the exact index 2. The area of the surface of a sphere is proportion al to  the square of the radius, and hence for any radiation from a point the action per unit area is inversely proportional to the square of the distance from the source. If we look at the figure, we see this matter illustrated for the corpuscles. Suppose A and B to be the two bodies which are producing mutual shielding from the impinging torrents. Consider the effect produced by B, in the three positions shown, upon the streams of corpuscles which would otherwise strike A. Consider the effect of the radial streams : it is clear that the impacts of the oblique streams may be resolved into radial and tangential components, the latter of which are non effective. It is evident that B in the three positions eclipses the same area on each spherical surface drawn through its position with A as centre. Also the number of particles which, in the absence of B, would impinge upon A, is, at any given instant, the same over each spherical surface. This  follows from the assumed equal density and equal velocity of the streams. Hence the number per unit area must be inversely as the square of the distance, and hence the number intercepted by the constant area 2t, must vary inversely as the square of the distance from A. The same result will hold if we regard the other aspect of the arrangement, in which A is the shielding and B the protected body. Thus the law of the inverse square holds for the mutual action.
We now come to the law of masses and here we deal with what is peculiarly Lesage's contribution. From what we have described, it is evident that the amount of shielding and consequently the attraction is, as regards bodies of different sizes, proportional to the surfaces and not to the masses. For if a body is conceived as stopping at its surface all the corpuscles which strike it, then the amount of matter inside the body has no effect on the action. This difficulty Lesage evades by a hypothesis, which may at first sight seem rather arbitrary, but which is really justified by many other phenomena in the range of Physics and Chemistry. He supposes matter to be highly porous in its structure, and this porosity, it may be said, is retained in all the subsequent modifications of the theory. To bring about the proportionality of attraction to mass, it was necessary to suppose that the rain of corpuscles beats upon every particle of the body, no matter how far in the interior that particle may be. Hence ordinary matter was conceived to be highly permeable by the corpuscles; to be in the words of Lesage "a kind of cage composed of bars very thin in comparison with the meshes of the cage, so that any body even the largest will intercept only a very small part of the currents which strike it." There can, of course, be no a priori condemnation of such a hypothesis. A mass of scaffolding seen from a great distance will appear as a small continuous body ; and it is practically certain that what we see as continuous and homogeneous matter, would in a superhuman magnification exhibit a very heterogeneous and porous appearance. Lesage himself points out that porosity is indicated by several phenomena ; and the trend of all modern science, physical as well as chemical is to admit this porous molecular structure. I need merely mention such phenomena as diffusion of gases, liquids and solids, transparency to light, and to electric  and magnetic actions, conduction of heat, etc. By assuming this cage-like structure fine enough, the reduction of effect owing to a particle's interior position may be made so small that the law of masses will be satisfied as exactly as experience requires. We must, therefore, distinguish the effective surface of a body from the apparent surface. The former is proportional to the total amount of matter in the body, and is the intercepting agency with which we are concerned in the corpuscular theory. By this supposition the law of masses is satisfied. To return to our illustration, suppose two cage structures to be placed within the self luminous sphere. It is clear that the shadow cast by one on the other will be much more nearly proportional to the mass than to the external geometrical surface.
It is to be noted as a consequence of this open structure, that only a small part of the corpuscular torrent in a given region is stopped by the interposition of a material body. By far the larger part rushes through the body without touching any part of it. It is, however, the small obstructed part that is responsible for the gravitational effect. The total strength of the torrents must then be inferred from the fact that only a small portion of the whole produces the observed effects of attraction.
Having now indicated how the laws of distance and mass are satisfied, let us turn to a few more special points. The full enunciation of the theory by Lesage was first published in a memoir to the Berlin Academy in 1782. It was entitled the "Lucrece Newtonien," and this name probably hints at the employment of corpuscles like the Lucretian hard atoms to account for the Newtonian laws of attraction. This memoir was incorporated in an amplified account which will be found in a book entitled "Deux Traites de Physique Mecanique," edited and partly written by Pierre Prevost of Geneva in 1818. It is interesting to note that in this book all fractions are written in a notation similar to the solidus notation now so popular. For example 2/3 is written 2:3 and so on. This convenient change from ordinary typographical usage was long ignored.
We may conveniently summarise the remaining points in Lesage's theory. In the first place, the lengths of the corpuscular streams are considered finite but very great ; so that a limit in time to the action of gravity is implied. Then, no explanation of the mechanism which propels the streams of corpuscles is attempted. The corpuscles are called " ultramundane," which we may take as meaning, that thought must stop short of the consideration of the beginning and the end of the streams. The same limitation, of course, occurs in our conceptions of space and time and hence of all natural phenomena. The streams are supposed to be equally dense everywhere, i.e., in the same volume of space chosen round any point, there will always be found the same number of corpuscles. The mean velocity of the streams is everywhere the same. These last two suppositions are necessary to ensure that uniform action of gravity which we find at all parts of space where observation has been possible. The dimensions of the corpuscles are very small but their number per unit volume is very great ; and the enfeeblement of energy brought about by the small size is to be compensated by the very high velocity which they possess. This high velocity (relative to all known velocities of material bodies) is also essential for two other points — first, that when two bodies begin to move towards each other, there should be no sensible diminution of the effect of the impacts, owing to the motion. If this were not so, the gravitational effect would diminish with increasing velocity. The second point is that there should be no appreciable resistance offered by the gravific medium to the motion of gross matter, for example, the planets in their orbits round the sun. If the planetary velocities, great though they are in relation to the velocities of ordinary experience, be small in comparison with those of the corpuscles, there will be practically no resistance to the motion of an isolated body in any direction; and in such motion the corpuscles will impinge as if they were striking upon a stationary body.
The question of the nature of the corpuscles is most important. Lesage supposed them to be hard bodies which after impact on gross matter rebounded with diminished velocity ; and this he supposed to be the case, even when the body struck was isolated and therefore at rest. These corpuscles are therefore imperfectly elastic, in the sense that there is a loss of energy at impact. In the language of molar dynamics, the coefficient of restitution is less than unity. There is therefore a violation of the principle of Conservation of Energy, unless this lost energy can be traced either in the body struck or in the rebounding corpuscle. Lesage did not attempt any such explanation, and it is to be remembered, of course, that the  conservation doctrine was developed long after his time. We shall see presently how the later forms of the corpuscular theory avoid this difficulty. Fatio, on the other hand supposed the corpuscles to be perfectly elastic, i.e., after impact they rebounded with the same linear momentum as before. But this, though more in accordance with modern dynamical notions, vitiates the gravity action ; for as Prevost pointed out, the shielding effect in the space between two bodies will be done away with if the linear momentum remain unimpaired after collision. There would be reflected from the nearer face of one, a set of corpuscles which would act on the nearer face of the other, so as to replace the supply cut off.
Before proceeding to consider more fully the elasticity question, we may note a few of the numbers given by Lesage, although we cannot dwell upon his reasons for establishing them. He states (p. 25) that about 3 million directions through a point are necessary for the corpuscular streams, so as to ensure a body in different positions not altering in weight by more than 1/500,000 part. Then collision of corpuscle with corpuscle must occur very rarely since the long, straight line streams are necessary for the gravitational effect. He puts it that not more than i in every 100, should meet another in several thousand years. The mean speed of rebound of a corpuscle after striking on a material body, is to be 2/3 of the speed before impact, (p. 61).
We shall discuss later on the objections to the general form of the corpuscular theory ; but, here we need only take note of those special to Lesage's type. These are (i) the destruction of energy at impact, (2) the supply of the corpuscles and the speed with which they are endowed, (3) the unnecessary expenditure of energy going on in all parts of space. The first of these objections is removed by Kelvin's form of the theory, the second by S. T. Preston's. These will both be explained later on. The third points to the fact, that the ether, instead of being the passive vehicle of energy, would be, on this theory, a hugely charged storehouse, abounding in energy in every part ; while the gross matter for whose service, this energy exists, occupies only an inconsiderable part of the region. To this it may be replied, that though we may have a prejudice in favour of economy of cosmic energy, it is questionable whether this view is justified by  a wide consideration either of inorganic or of organic nature. In any case it is impossible to reply definitely either way to such an a priori objection. Lesage seems to have suffered from such objections ; for he gives (p. 108) a list of them classifying into metaphysical, physico-metaphysical and physical. Some of the first are very curious. In another place (p. 105) he says : — " these objections arise from the principles of some particular metaphysical sect ; and before replying to them, I would ask these metaphysicians to agree first with one another. Such objectors appeal to the imagination rather than to the understanding, when they point with amazement to the extreme, extraordinary, and unheard of features in my system ; as if by our gross and finite measures, the subtlety and grandeur of nature is to be tested."
We must now pass from Lesage's theory. In 1818 Blair, Professor of Astronomy in Edinburgh University, published a theory of gravitation similar to that of Lesage. Instead, however, of dealing with ultramundane corpuscles, he supposes that space where gravitation acts, is bounded by a hard spherical surface. Upon this the corpuscles strike and are reflected back, and so the supply is maintained. With this exception, the corpuscular theory seems to have received no development for a long time after the great effort of Lesage. Many other types of kinetic theory fill up the intervening years, but it is not until 1869, that we find Leray attacking the subject from the impact point of view. In the Comptes Rendus, he puts forward ideas, essentially the same as those of Lesage, but without that careful development of detail and ingenuity of instance which give value to the earlier work. Leray bases his theory on two principles ( 1) that the corpuscular currents exist in all directions; (2) that in traversing a body the currents are enfeebled in proportion to the thickness traversed, and to the mean density of the matter in their course. He adds, in these nothing essential to Lesage's theory, but he supposes that the lost energy of the impacts (which Lesage did not account for), appeared in the light, heat, and magnetism of the celestial bodies. These cosmic energies are thus kept up by the ethereal energies, so that there is no violation of the conservation principle. This  idea was also put forward by Rysanek in 1888. No attempt is made, however, to explain the differences of condition of the planets or to justify in any substantial way this interchange of energy. We may, therefore, pass on to consider Lord Kelvin's view of the matter.
In the Philosophical Magazine for 1873, we find his contribution to the subject. It is interesting to note his approbation of Lesage's scheme in contrast with the view of Herschel. The latter in the Fortnightly Review (1865), referred to the theory as being "too grotesque to need serious consideration." But Lord Kelvin writes as follows : — " From Lesage's fundamental assumptions, it is easy to deduce the law of the inverse square and the law of proportionality to mass. Inasmuch as the law of the inverse square of the distance, for every distance, however great, would be a perfectly obvious consequence of the assumptions, were the gravific corpuscles infinitely small, and therefore incapable of coming into collision with each other, it may be extended to as great distances as we please, by giving small enough dimensions to the corpuscles relatively to the mean distance of each from its nearest neighbour. The law of masses may be extended to as great masses as those for which observation proves it (for example the mass of Jupiter), by making the diameters of the bars of the supposed cage atoms constituting heavy bodies, small enough. Thus, for example, there is nothing to prevent us from supposing that not more than one straight line of a million drawn at random towards Jupiter and continued through it, should touch one of the bars.
"Lastly, as Lesage proves, by making the velocities of the corpuscles great enough, and giving them suitably small masses, they may produce the actual forces of gravitation and not more than the amount of resistance which observation allows us to suppose that the planets experience." Later on he continues — " This much is certain, that if hard, indivisible atoms are granted at all, his (Lesage's) principles are unassailable, and nothing can be said against the probability of his assumption. The only imperfection of his theory, is that which is inherent to every supposition of hard indivisible atoms." 
Kelvin then goes on to suggest a means of avoiding this defect. He postulates perfect elasticity for the corpuscles, and so brings the scheme into accord with the doctrine of the conservation of energy. But if the effects of gravity are to be produced, the corpuscles must rebound after impact on gross matter with less linear momentum than before; for otherwise the whole effect of the shielding would be done away with, and the rebounding particles would kinetically replace those which had been diverted. Hence a diminished velocity is necessary for all those corpuscules which have just suffered collision with gross matter. What then becomes of the energy so disappearing ? Leray and Rysanek suggested heat of the celestial bodies ; but Clerk Maxwell is of opinion that if the corpuscular energy were transformed in this way, the heat produced would be sufficient to raise immediately to incandescence all matter suffering impact. This view — to be found in the article "Atom," in the Encyclopaedia Britannica — he bases upon the fact of the kinetic theory of gases, that a system of particles in free collision will tend to equalise their kinetic energies. Kelvin suggests that the corpuscles retain all their energy but that it is altered in type. Before impact, the corpuscle has a certain amount of energy of translation. By the impact, this is partly converted into energy of vibration and of rotation. The change into rotational energy might not always be possible, but that into vibrational energy is always possible, provided the corpuscle be not the hard atom of Lesage, but a body possessing internal structure, and capable of modes of vibration. On this supposition, therefore, the diminution of translational energy necessary for the gravitational effect is possible " without violating modern thermodynamics." Lastly he remarks that this loss of translational energy may be restored to the corpuscles by merely supposing that in their impacts with each other, they follow the same law as ordinary gaseous matter. According to the kinetic theory of gases, a system of particles will tend by their natural motion to adjust the ratio of translational to the rotational, and vibrational forms to a constant value ; and if this value is disturbed by any cause, it will again be restored by the natural motions. This was pointed out by Clausius who showed that the constant 2 ratio of whole energy to the translational part is given by 2/(3(γ-1)),  where γ is the usual specific heats ratio for the gas. For air this ratio of energies is 1.634. Hence if the ordinary gaseous condition be satisfied by the ultramundane corpuscles, the difficulty ot the elasticity is avoided. Kelvin concludes, — "The corpuscular theory of gravitation is no more difficult in allowance of its fundamental assumptions than the kinetic theory of gases as at present received, and it is more complete, inasmuch as from fundamental assumptions of an extremely simple character, it explains all the known phenomena of its subject."
It will be noticed that in these remarks we have the hint of the next step in the development, viz. : — the full application of the kinetic theory of gases to the system of gravific corpuscles. In fact Kelvin definitely points out that the distance through which gravity would be effective, would be dependent upon the distance through which the corpuscles move before interruption by each other. This idea was taken up independently by Picart in France, S. Tolver Preston in this country, and by Jarolimek in Austria. Picart, though first in date, gives only a short note and does not develop the idea at all. Jarolimek claims priority (1874), although his work was not published till 1883 ; but in any case, his special modification will be more conveniently treated later on. Here we will take up Preston's development given in various Philosophical Magazine papers from 1877 to 1895.</ref> The great difficulty in the theory of Lesage is the question of the supply of the corpuscles. Whence comes this huge supply of particles, sweeping through space at a high rate of speed ? To call them ultramundane, merely evades the difficulty. Further, of this vast amount of energy only an exceedingly small fraction serves the purpose for which the corpuscles are specially imagined. Preston therefore supposes the gravific fluid to exist in the following modified form. Instead of the series of arbitrary and independent assumptions as to the direction, density, and velocity of the streams, he makes the one assumption that the gravific fluid is constructed like an ordinary gas and behaves according to the kinetic theory. In accordance with this theory, a spontaneous  adjustment of the motions of the particles takes place, so that in the average the number of particles in unit volume is the same at all points, the mean velocity is equal in all parts, the mean distances of the particles are the same at all parts and the particles are moving for short distances towards all directions at all points. Here then we have the very requirements of the Lesage theory, not as so many arbitrary postulates, but as the necessary consequences from the single assumption as to the constitution of the gravific fluid. Moreover this adjustment is stable so that after any disturbance, the normal condition again asserts itself, as a consequence of the natural impinging of the particles upon each other. These facts of the kinetic theory are the results of averages taken over very large numbers of individual values. They are all capable of mathematical proof, so that if the gravific fluid be constituted as an ordinary gas, then by its own internal and natural action, it will satisfy all the requirements of the older theory. An essential of the Lesage view was that collision between corpuscles should be exceedingly rare. Here on the other hand, the large number of collisions is the phenomenon necessary for the self-acting adjustment. The frequency of these impacts of corpuscle on corpuscle tends to bring the fluid back to the uniform condition. Hence also the supply of energy does not come from and go to regions inaccessible to thought, but is self-contained in the fluid, like that of a gas in a closed vessel ; and furthermore, all the energy has an understood reason for its existence, in that it is needed to maintain the uniformity of state.
We have now to notice a very important qualification, viz. : — that the free path of the corpuscles (i.e., the distance between two successive collisions of corpuscle with corpuscle) must be great ; must have indeed a length greater than the greatest distance through which gravity is known to act. For if the corpuscles collide with each other between two material bodies, no attraction effect will be produced ; just as in an ordinary gas there is equality of pressure on all sides of an immersed body despite the presence of other bodies. If however we could immerse in an ordinary gas, two bodies sufficiently small, and at a less distance from each other than the free path of the particles of the gas, then there would be a shielding effect and a tendency to come together. We shall see the essence of Preston's view, if we take an ordinary gas and suppose its dimensions magnified until the length of the free  path becomes comparable with the greatest distances through which we know gravitation to act. Then within that range we have the equal distribution of the streams in all directions, as in Lesage's theory, but instead of the mysterious agency at the boundary propelling the streams of corpuscles, we have the ordinary dynamical action of the larger quantity of gas. The gas as a whole is in stationary motion, but in small regions — the solar system for example — the motion is that of the Lesage theory. On this view, gravity does not act between two bodies at distances greater than the free path, and hence, as Preston points out, a greater stability is given to the universe.
There is nothing inherently improbable in this assumption of the great free path. From the kinetic theory, it is known that the mean free path is inversely proportional to the square of the diameter of the particle. Hence the dimensions of the corpuscle may be chosen sufficiently small to give the desired length of path ; and this smallness accords well with the other ideas of large number per unit volume and high velocity of the corpuscle. For, as we have seen, we require a large number of corpuscles impinging in quick succession to give the effect of continuous pressure ; we require a high speed to give absence of resistance to the motion of ordinary matter and to give static attraction, the same value as kinetic ; and we need small dimensions to give impalpability, while at the same time, in virtue of the high velocity, the momentum and energy remain normal.
I quote an extract from Preston's paper in the Philosophical Magazine.
"It is an interesting fact that the distance through which gravity is effective would depend on the distance through which the gravific particles, move before being intercepted by collision with each other. By assuming the distances of the stars to be a multiple of the mean length of path of the particles, it would therefore follow that the stars do not gravitate towards each other. . . . The distance through which gravity has been observed to act is well known to be but an infinitesimal fraction of the distances of the stars. It may therefore well be that the mean length of path of the particles of the medium producing gravity, may be but an infinitesimal fraction of this distance. The column of the gravific medium intercepted between two stars, would therefore, on the whole be at rest, just as a column of gas is at rest, between two bodies a visible distance  apart, (i.e., a distance which is a large multiple of the mean length of the path of the particles of the gas). ... In order to explain all the observed facts, it is sufficient to admit that the universe is immersed in a gas (or medium constituted according to the kinetic theory), the mean length of path of whose particles is so adjusted as to cause the minor or secondary portions of the universe to gravitate towards each other. Under the simple conception of variation in the diameter of the particles of a medium, the mean length of path of the particles, (and with it the range of gravity), is capable of adjustment with precision to any range."
We now come to Jarolimek's view enunciated in 1883. While agreeing with much of Preston's theory, he points out that the free path in a gas is not an invariable quantity ; but is only an average of a large number of values which vary between wide limits. In the case of an ordinary gas, the consideration of this mean free path is sufficient, but for the gravity fluid, the absolute values must be dealt with. For in the former case, the —o mean free path is exceedingly small (5.7 x 10-6 cm. for oxygen) in comparison with the spaces dealt with, whereas, in the latter, we are essentially concerned with magnitudes less than the free path. Jarolimek considers the effect of this varying length of free path upon the attraction of two bodies. Only those corpuscles can be effective, which have a free path in excess of the distance between the bodies : all others produce no attractive tendency. The mean free path is given by F(D³/a²) where F is a constant, D is the mean distance between a the diameter of the corpuscles. Also the difference between maximum and minimum free path can be measured by D/a. Hence since the above result l= F(D³/a²) may be written (D/a)²=(1/F)(l/D) it follows that to give the necessary large value to L/D, D/a must be large and therefore l must vary between very wide limits. A consequence of this fluctuation will be that for this reason alone, the attraction between two bodies will increase as the distance is diminished because as the distance diminishes, more and more corpuscles be come effective.  But we have as well, the increase of attraction due to the cause already discussed in all the previous forms of the theory, viz., the increase of solid angle subtended. Hence, if both these causes are effective, we shall have with diminishing distance, an attraction which increases at a higher rate than the inverse square. Jarolimek thereupon supposes the second cause ineffective. He discards what is an essential point in the other forms of the theory, viz., the smallness of the corpuscle in comparison with the atom of gross matter. If these be of comparable dimensions, then it is easily seen that the mutual shielding of two particles is independent of the distance and is the same at all distances. The corpuscles flying in the line joining the two particles are diverted but no others. Hence one body (highly porous) placed near another (highly porous) would exert the same shielding effect at all distances, if the length of free path of the corpuscles were constant. But since this varies, for this reason and for this reason alone, the attraction increases with diminished distances. And hence since the law is known to be that of the inverse square a certain distribution of corpuscles is conditioned, which may be stated as follows : — there must be n² as many corpuscles of free path at least r, as there are of free path at least nr. Thus Jarolimek attacks the subject from a novel point of view ; while he retains the idea of a medium producing the gravity effect by impacts and constituted according to the kinetic theory of gases.
Isenkrahe has also, in 1879, attempted an explanation of gravity from a similar point of view. But he assumes the corpuscles to be inelastic, and so comes directly into conflict with the principle of conservation of energy. Diminution of energy at collision is only to be explained by the idea that this energy is handed on to the inner parts of the colliding bodies. But when, as in the case of the corpuscles according to Isenkrahe, the colliding bodies are structureless, there can be no such transformation. Complete elasticity is necessary in any ultimate bodies.
We have thus traced the development of the corpuscular hypothesis from its first enunciation down to the form into which modern scientific speculation has moulded it. We may now  briefly consider how far the result goes towards an explanation of the ten characteristics stated above. We have already dealt fully with the most important points, the law of masses and the law of distances (3) and (5). It may be noted in regard to the latter, that Jarolimek's objection to Preston's result is not fatal ; for we have only to take a lower value of the free path than the mean, to correspond to the known range of gravitation. As to (1) radial nature of the action; (7) undiminished intensity in time, and (10) duality of the action — the mechanism explains these conditions completely. The 6th condition — that of instantaneous propagation — is satisfied to any approximation we please, by giving the corpuscles sufficiently high velocities. Then as to (4) independence of nature of the acting masses, (8) isotropy of all matter, and (9) absence of influence by temperature, chemical decomposition, etc., we may group these together and say that either an affirmation or a denial of their consistency with the corpuscular mechanism, is impossible until more is known of the ultimate structure of matter. An agreement is quite conceivable on certain suppositions as to molecular structure, e.g., the vortex atom theory of Lord Kelvin. In this case, since all matter reduces to an aggregation of vortices of the same fluid, the above three characteristics are quite consistent with the corpuscular theory. As to the second characteristic — the absence of shielding, etc. — it must be admitted that this is the weak point in the scheme. To take a simple case, it would seem that the attraction between two spheres placed inside a hollow sphere should be less than if they were in free space. Still this difficulty was seen even by Lesage, and was avoided by making the part of the corpuscular torrent intercepted a small fraction of the whole. The highly porous nature of matter renders the diminution referred to in the above case inappreciable. In that case, the corpuscular streams are, let us suppose, thinned out a little by the external spherical surface — enough to produce attractive effects. But when the streams pass on and strike either of the inner spheres, the number of corpuscles obstructed is not appreciably less than if the original streams had arrived unimpaired.
In conclusion, we may admit that the corpuscular theories are not completely convincing, nor do they offer any special experimental justification. But on the other hand, they form the  nearest approach to an explanation that has yet been offered ; they have received the approval of a fairly representative number of natural philosophers, and they group into a very interesting speculation upon this difficult but important subject.
I have pleasure in acknowledging much kind criticism and advice from Professor Gray in the preparation of this paper.
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