- [From Edinb. Roy. Soc. Proc. Vol. VII. 1872, pp. 577-589 [read Dec. 18, 1871]; Phil. Mag. Vol. XLV. May, 1873, pp. 321-332.]
LE SAGE, born at Geneva in 1724, devoted the last sixty-three years of a life of eighty to the investigation of a mechanical theory of gravitation. The probable existence of a gravific mechanism is admitted, and the importance of the object to which Le Sage devoted his life pointed out, by Newton and Rumford in the following statements:
"It is inconceivable that inanimate brute matter should, without the mediation of something else, which is not material, operate upon, and affect other matter without mutual contact; as it must do, if gravitation, in the sense of Epicurus, be essential and inherent in it. And this is the reason why I desired you would not ascribe innate gravity to me. That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of any thing else, by and through which their action and force may be conveyed from one to another, 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. Gravity must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial, I have left to the consideration of my readers." NEWTON'S Third Letter to Bentley, February 25th, 1692-3.
"Nobody surely, in his sober senses, has ever pretended to understand the mechanism of gravitation; and yet what sublime discoveries was our immortal Newton enabled to make, merely by the investigation of the laws of its action."
Le Sage expounds his theory of gravitation, so far as he had advanced it up to the year 1782, in a paper published in the Transactions of the Royal Berlin Academy for that year, under the title "Lucrece Newtonien." His opening paragraph, entitled "But de ce mémoire," is as follows:-
"Je me propose de faire voir: que si les premiers Épicuriens avoient eu; sur la Cosmographie des idées aussi saines seulement, que plusieurs de leurs contemporains, qu'ils negligeoient d'ecouter; et sur la Géométrie, une partie des connoissances qui étoient deja communes alors: ils auroient, très probablement, découvert sans effort; les Loix de la Gravite universelle, et sa Cause mécanique. Loix; dont l'invention et la démonstration, font la plus grande gloire du plus puissant génie qui ait jamais existé: et Cause, qui après avoir fait pendant longtems, l'ambition des plus grands Physiciens; fait à présent, le désespoir de leurs successeurs. De sorte que, par exemple, les fameuses Règles de Kepler; trouvées il y a moins de deux siècles, en partie sur des conjectures gratuites, et en partie après d'immenses tâtonnemens; n'auroient te que des corollaires particuliers et inévitables, des lumières generates que ces anciens Philosophes pouvoient puiser (comme en se jouant) dans le mécanisme proprement dit de la Nature. Conclusion; qu'on peut appliquer exactement aussi, aux Loix de Galilee sur la chiite des Graves sublunaires; dont la découverte a et plus tardive encore, et plus contestée: joint a ce que, les expériences sur lesquelles cette découverte toit tablie; laissoient dans leurs resultats (nécessairement grossiers),  une latitude, qui les rendoit également compatibles avec plusieurs autres hypothèses; qu'aussi, Ton ne manqua pas de lui opposer: au lieu que, les conséquences du choc des Atoms; auroient été absolument univoques en faveur du seul principe véritable (des Accélérations égales en Tempuscules égaux)."
If Le Sage had but excepted Kepler's third law, it must be admitted that his case, as stated above, would have been thoroughly established by the arguments of his "memoire"; for the epicurean assumption of parallelism adopted to suit the false idea of the earth being flat, prevented the discovery of the law of the inverse square of the distance, which the mathematicians of that day were quite competent to make, if the hypothesis of atoms moving in all directions through space, and rarely coming into collision with one another, had been set before them, with the problem of determining the force with which the impacts would press together two spherical bodies, such as the earth and moon were held to be by some of the contemporary philosophers to whom the epicureans would not listen." But nothing less than direct observation, proving Kepler's third law Galileo's experiment on bodies falling from the tower of Pisa, Boyle's guinea and feather experiment, and Newton's experiment of the vibrations of pendulums composed of different kinds of substance could give either the idea that gravity is proportional to mass, or prove that it is so to a high degree of accuracy for large bodies and small bodies, and for bodies of different kinds of substance. Le Sage sums up his theory in an appendix to the "Lucrece Newtonien," part of which translated (literally, except a few sentences which I have paraphrased) is as follows:
- Constitution of Heavy Bodies
1. Their indivisible particles are cages; for example, empty cubes or octahedrons vacant of matter except along the twelve edges.
2. The diameters of the bars of these cages, supposed increased each by an amount equal to the diameter of one of the gravific corpuscles, are so small relatively to the mutual distance of the parallel bars of each cage, that the terrestrial globe does not intercept even so much as a ten-thousandth part of the corpuscles which offer to traverse it. 
3. These diameters are all equal, or if they are unequal, their inequalities sensibly compensate one another [in averages].
- Constitution of Gravific Corpuscules
1. Conformably to the second of the preceding suppositions, their diameters added to that of the bars is so small relatively to the mutual distance of parallel bars of one of the cages, that the weights of the celestial bodies do not differ sensibly from being in proportion to their masses.
2. They are isolated. So that their progressive movements are necessarily rectilinear.
3. They are so sparsely distributed, that is to say, their diameters are so small relatively to their mean mutual distances, that not more than one out of every hundred of them meets another corpuscule during several thousands of years. So that the uniformity of their movements is scarcely ever troubled sensibly.
4. They move along several hundred thousand millions of different directions; in counting for one same direction all those which are [within a definite very small angle of being] parallel to one straight line. The distribution of these straight lines is to be conceived by imagining as many points as one wishes to consider of different directions, scattered over a globe as uniformly as possible, and therefore separated from one another by at least a second of angle; and then imagining a radius of the globe drawn to each of those points.
5. Parallel, then, to each of those directions, let a current or torrent of corpuscules move; but, not to give the stream a greater breadth than is necessary, consider the transverse section of this current to have the same boundary as the orthogonal projection of the visible world on the plane of the section.
6. The different parts of one such current are sensibly equidense; whether we compare, among one another, collateral portions of sensible transverse dimensions, or successive portions of such lengths that their times of passage across a given surface are sensible. And the same is to be said of the different currents compared with one another.
7. The mean velocities, defined in the same manner as I have just defined the densities, are also sensibly equal. 
8. The ratios of these velocities to those of the planets are several million times greater than the ratios of the gravities of the planets, towards the sun, to the greatest resistance which secular observations allow us to suppose they experience. For example, [these velocities must be] some hundredfold a greater number of times the velocity of the earth, than the ratio of 190,000 times the gravity of the earth towards the sun, to the greatest resistance which secular observations of the length of the year permit us to suppose that the earth experiences from the celestial masses.
- CONCEPTION, which facilitates the Application of Mathematics to determine the mutual Influence of these Heavy Bodies and these Corpuscules.
1. Decompose all heavy bodies into molecules of equal mass, so small that they may be treated as attractive points in respect to theories in which gravity is considered without reference to its cause; that is to say, each must be so small that inequalities of distance and differences of direction between its particles and those of another molecule, conceived as attracting it and being attracted by it, may be neglected. For example, suppose the diameter of the molecule considered to be a hundred thousand times smaller than the distance between two bodies of which the mutual gravitation is examined, which would make its apparent semi-diameter, as seen from the other body, about one second of angle.
2. For the surfaces of such a molecule, accessible but impermeable to the gravific Huid, substitute one single spherical surface equal to their sum.
3. Divide those surfaces into facets small enough to allow them to be treated as planes, without sensible error [&c., &c.].
1. It is not necessary to be very skilful to deduce from these suppositions all the laws of gravity, both sublunary and universal (and consequently also those of Kepler, &c.), with all the accuracy with which observed phenomena have proved those laws. Those  laws, therefore, are inevitable consequences of the supposed constitutions.
2. Although I here present these constitutions crudely and without proof, as if they were gratuitous hypotheses and hazarded fictions, equitable readers will understand that on my own part I have at least some presumptions in their favour (independent of their perfect agreement with so many phenomena), but that the development of my reasons would be too long to find a place in the present statement, which may be regarded as a publication of theorems without their demonstrations.
3. There are details upon which I have wished to enter on account of the novelty of the doctrine, and which will readily be supplied by those who study it in a favourable and attentive spirit. If the authors who write on hydro-dynamics, aerostatics, or optics, had to deal with captious readers, doubting the very existence of water, or air, or light, and therefore not adapting themselves to any tacit supposition regarding equivalencies or compensations not expressly mentioned in their treatises, they would be obliged to load their definitions with a vast number of specifications which instructed or indulgent readers do not require of them. One understands "a demi-mot" and " sano sensu" only familiar propositions towards which one is already favourably inclined.
Some of the details referred to in this concluding sentence of the appendix to his " Lucrece Newtonien," Le Sage discusses fully in his Deux Traites de Physique Mecanique, edited by Pierre Provost, and published in 1818 (Geneva and Paris).
This treatise is divided into four books.
I. " Exposition sommaire du système des corpuscules ultramondains."
II. " Discussion des objections qui peuvent s'é1ever contre le système des corpuscules ultramondains."
III. « Des fluides élastiques ou expansifs."
IV. "Application des theories précédentes à certaines affinités." It is in the first two books that gravity is explained by the impulse of ultramundane corpuscules, and I have no remarks at present to make on the third and fourth books. 
From Le Sago's fundamental assumptions, given above as nearly as may be in his own words, it is, as he says himself, easy to deduce the law of the inverse square of the distance, and the law of proportionality of gravity to mass. The object of the present note is not to give an exposition of Le Sage's theory, which is sufficiently set forth in the preceding extracts, and discussed in detail in the first two books of his posthumous treatise. I may merely say that 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 corpuscules infinitely small, and therefore incapable of coming into collision with one another, it may be extended to as great distances as we please, by giving small enough dimensions to the corpuscules relatively to the mean distance of each from its nearest neighbour. The law of masses maybe 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 Le Sage proves, the resistance of his gravific fluid to the motion of one of the planets through it, is proportional to the product of the velocity of the planet into the average velocity of the gravific corpuscules; 'and hence by making the velocities of the corpuscules 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. It will be a very interesting subject to examine minutely Le Sage's details on these points, and to judge whether or not the additional knowledge gained by observation since his time requires any modification to be made in the estimate which he has given of the possible degrees of permeability of the sun and planets, of the possible proportions of diameters of corpuscules to interstices between them in the "gravific fluid," and of the possible velocities of its component corpuscules. This much is certain, that if hard indivisible atoms are granted at all, his principles are unassailable; and nothing can be said against the probability of his assumptions. The only imperfection of his theory is that which is inherent to every supposition of hard, indivisible atoms. 
They must be perfectly elastic or imperfectly elastic, or perfectly inelastic. Even Newton seems to have admitted as a probable reality hard, indivisible, unalterable atoms, each perfectly inelastic.
Nicolas Fatio is quoted by Le Sage and Prévost, as a friend of Newton, who in 1689 or 1690 had invented a theory of gravity perfectly similar to that of Le Sage, except certain essential points; had described it in a Latin poem not yet printed; and had written, on the 80th March 1694, a letter regarding it, which is to be found in the third volume of the works of Leibnitz, having been communicated for publication to the editor of those works by Le Sage. Redeker, a German physician, is quoted by Le Sage as having expounded a theory of gravity of the same general character, in a Latin dissertation published in 1736, referring to which Prevost says, "Où l'on trouve l'expose d'un système fort semblable a celui de Le Sage dans ses traits principaux, mais dépourvu de cette analyse exacte des phénomènes qui fait le principal mérite de toute espèce de théorie." Fatio supposed the corpuscules to be elastic, and seems to have shown no reason why their return velocities after collision with mundane matter should be less than their previous velocities, and therefore not to have explained gravity at all. Redeker, we are told by Prevost, had very limited ideas of the permeabilities of great bodies, and therefore failed to explain the law of the proportionality of gravity to mass; " he enunciated this law very correctly in section 15 of his dissertation; but the manner in which he explains it shows that he had but little reflected upon it. Notwithstanding these imperfections, one cannot but recognise in this work an ingenious conception which ought to have provoked examination on the part of naturalists, of whom many at that time occupied themselves with the same investigation. Indeed, there exists a dissertation by Segner on this subject. But science took another course, and works of this nature gradually lost appreciation. Le Sage has never failed on any occasion to call attention to the system of Redeker, as also to that of Fatio."
Le Sage shows that to produce gravitation those of the ultramundane corpuscules which strike the cage-bars of heavy bodies must either stick there or go away with diminished velocities. He supposed the corpuscules to be inelastic (durs), and points  out that we ought not to suppose them to be permanently lodged in the heavy body (entasses), that we must rather suppose them to slip off; but that being inelastic, their average velocities after collision must be less than that which they had before collision.
That these suppositions imply a gradual diminution of gravity from age to age was carefully pointed out by Le Sage, and referred to as an objection to his theory. Thus he says, "...Donc, la durée de la gravité seroit finie aussi, et par conséquent la durée du monde."
" Reponse. Concedo; mais pourvu que cet obstacle ne contribue pas à faire finir le monde plus promptement qu'il n'auroit fini sans lui, il doit être considère comme nul."
Two suppositions may be made on the general basis of Le Sage's doctrine:
1. (Which seems to have been Le Sage's belief.) Suppose the whole of mundane matter to be contained within a finite space, and the infinite space round it to be traversed by ultramundane corpuscules; and a small proportion of the corpuscules coming from ultramundane space to suffer collisions with mundane matter, and get away with diminished gravific energy to ultramundane space again. They would never return to the world were it not for collision among themselves and other corpuscules. Le Sage held that such collisions are extremely rare; that each collision, even between the ultramundane corpuscules themselves, destroys some energy; that at a not infinitely remote past time they were set in motion for the purpose of k',eping gravitation throughout the world in action for a limited period of time; and that both by their mutual collisions, and by collisions with mundane atoms, the whole stock of gravific energy is being gradually reduced, and therefore the intensity of gravity gradually diminishing from age to age.
2. Or, suppose mundane matter to be spread through all space,  but to be much denser within each of an infinitely great number of finite volumes (such as the volume of the earth) than elsewhere. On this supposition, even were there no collisions between the corpuscules themselves, there would be a gradual diminution in their gravific energy through the repeated collisions with mundane matter which each one must in the course of time suffer. The secular diminution of gravity would be more rapid according to this supposition than according to the former, but still might be made as slow as we please by pushing far enough the fundamental assumptions of very small diameters for the cage-bars of the mundane atoms, very great density for their substance, and very small volume and mass, and very great velocity for the ultramundane corpuscules.
The object of the present note is to remark that (even although we were to admit a gradual fading away of gravity, if slow enough), we are forbidden by the modern physical theory of the conservation of energy to assume inelasticity, or anything short of perfect elasticity, in the ultimate molecules, whether of ultramundane or of mundane matter; and, at the same time, to point out that the assumption of diminished exit velocity of ultramundane corpuscules, essential to Le Sage's theory, may be explained for perfectly elastic atoms, consistently both with modern thermodynamics, and with gravity.
If the gravific corpuscules leave the earth or Jupiter with less energy than they had before collision, their effect must be to continually elevate the temperature throughout the whole mass. The energy which must be attributed to the gravific corpuscules is so enormously great, that this elevation of temperature would be sufficient to melt and evaporate any solid, great or small, in a fraction of a second of time. Hence, though outward-bound corpuscules must travel with less velocity, they must carry away the same energy with them as they brought. Suppose, now, the whole energy of the corpuscules approaching a planet to consist of translatory motion; a portion of the energy of each corpuscule which has suffered collision must be supposed to be converted by the collision into vibrations, or vibrations and rotations. To simplify ideas, suppose for a moment the particles to be perfectly smooth elastic globules. Then collision could not generate any rotatory motion; but if the cage-atoms constituting mundane matter be  each of them, as we must suppose it to be, of enormously great mass in comparison with one of the ultramundane globules, and if the substance of the latter, though perfectly elastic, be much less rigid than that of the former, each globule that strikes one of the cage-bars must (Thomson & Tait's Natural Philosophy, § 301), come away with diminished velocity of translation, but with the corresponding deficiency of energy altogether converted into vibration of its own mass. Thus the condition required by Le Sage's theory is fulfilled without violating modern thermo-dynamics; and, according to Le Sage, we might be satisfied not to inquire what becomes of those ultramundane corpuscules which have been in collision either with the cage-bars of mundane matter or with one another; for at present, and during ages to come, these would be merely an inconsiderable minority, the great majority being still fresh with original gravific energy unimpaired by collision. Without entering on the purely metaphysical question, Is any such supposition satisfactory? I wish to point out how gravific energy may be naturally restored to corpuscules in which it has been impaired by collision.
Clausius has introduced into the kinetic theory of gases the very important consideration of vibrational and rotational energy. He has shown that a multitude of elastic corpuscules moving through void, and occasionally striking one another, must, on the average, have a constant proportion of their whole energy in the form of vibrations and rotations, the other part being purely translational. Even for the simplest case, that, namely, of smooth elastic globes, no one has yet calculated by abstract dynamics the ultimate average ratio of the vibrational and rotational, to the translatiorial energy. But Clausius has shown how to deduce it for the corpuscules of any particular gas from the experimental determination of the ratio of its specific heat, pressure constant, to its specific heat, volume constant. He found that
if γ be the ratio of the specific heats, and β the ratio of the whole energy to the transiational part of it. For air, the value of γ found by experiment, is 1.408, which makes β= 1.634. For steam, Maxwell says, on the authority of Rankine, β "may be as much  as 2.19, but this is very uncertain." If the molecules of gases are admitted to be elastic corpuscules, the validity of Clausius' principle is undeniable; and it is obvious that the value of the ratio β must depend upon the shape of each molecule, and on the distribution of elastic rigidity through it, if its substance is not homogeneous. Farther, it is clear that the value of β for a set of equal and similar corpuscules will not be the same after collision with molecules different from them in form or in elastic rigidity, as after collision with molecules only of their own kind. All that is necessary to complete Le Sage's theory of gravity in accordance with modern science, is to assume that the ratio of the whole energy of the corpuscules to the translational part of their energy is greater, on the average, after collisions with mundane matter than after inter-collisions of only ultramundane corpuscules. This supposition is neither more nor less questionable than that of Clausius foi gases, which is now admitted as one of the generally recognised truths of science. The corpuscular theory of gravity 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, which cannot be said of the kinetic theory of gases so far as it has hitherto advanced.
- Postscript, April 1872.
In the preceding statement I inadvertently omitted to remark that if the constituent atoms are aeolotropic in respect to permeability, crystals would generally have different permeabilities in  different directions, and would therefore have different weights according to the direction of their axes relatively to the direction of gravity. No such difference has been discovered, and it is certain that if there is any it is extremely small. Hence, the constituent atoms, if aeolotropic as to permeability, must be so, but to an exceedingly small degree. Le Sage's second fundamental assumption given above, under the title " Constitution of Heavy Bodies," implies sensibly equal permeability in all directions, even in an aeolotropic structure, unless much greater than Jupiter, provided that the atoms are isotropic as to permeability.
A body having different permeabilities in different directions would, if of manageable dimensions, give us a means for drawing energy from the inexhaustible store laid up in the ultramundane corpuscules, thus: First, turn the body into a position of minimum weight; Secondly, lift it through any height; Thirdly, turn it into a position of maximum weight; Fourthly, let it down to its primitive level. It is easily seen that the first and third of those operations are performed without the expenditure of work; and, on the whole, work is done by gravity in operations 2 and 4. In the corresponding set of operations performed upon a moveable body in the neighbourhood of a fixed magnet, as much work is required for operations 1 and 3 as is gained in operations 2 and 4; the magnetisation of the moveable body being either intrinsic or inductive, or partly intrinsic and partly inductive, and the part of its aeolotropy (if any), which depends on inductive magnetisation, being due either to magne-crystallic quality of its substance, or to its shape.
- On the other hand, by the middle of last century the mathematical naturalists of the Continent, after half a century of resistance to the Newtonian principles (which, both by them and by the English followers of Newton, were commonly supposed to mean the recognition of gravity as a force acting simply at a distance without mediation of intervening matter), had begun to become more "Newtonian" than Newton himself. On the 4th February, 1744, Daniel Bernoulli wrote as follows to Euler: " Uebrigens glaube ich, dass der Aether sowohl gravis versus solem, als die Luft versus terram sey, und kann Ihnen nicht bergen, dass ich uber diese Puncte ein volliger Newtonianer bin, und verwundere ich mich, dass Sie den Principiis Cartesianis so lang adhariren; es mochte wohl einige Passion vielleicht mit unterlaufen. Hat Gott konnen eine animam, deren Natur uns unbegreiflich ist, erschaffen, HO hat er auch konnen eine attractionem universalem materiae imprimiren, wenn gleich solche attractio supra captum ist, da hingegen die Principia Cartesiana allzeit contra captum etwas involviren. "
- "An Inquiry concerning the Source of the Heat which is excited by Friction." By Count Rumford, Philosophical Transactions, 1798.
- Vobis (Epicureis) minus notum est, quemadmodum quidque dicatur. Vestra enirn soliim legitis, vestra amatis; caeteros, causa incognita, condemnatis." Ciceron, De natura Deorum u. 29.
- To render the sentence more easily read, I have substituted this number in place of the following words:" le nombre de fois que le firmament contient le disque apparent du soleil."
- De Causa gravitatis Redekeriana.
- Le Sage was remarkably scrupulous in giving full information regarding all who preceded him in the development of any part of his theory.
- Le Sage estimated the velocity after collision to be two-thirds of the velocity before collision.
- Posthumous. Traite de Physique MGcanique, edited by Pierre Provost. Geneva and Paris, 1818.
- Newton (Optics, Query 30, edn. 1721, p. 373) held that two equal and similar Atoms, moving with equal velocities in contrary directions, come to rest when they strike one another. Le Sage held the same; and it seems that writers of last century understood this without qualification when they called atoms hard.
- Maxwell's Elementary Treatise on Heat, Chapter xx. Longmans, 1871.
- Note by Joseph Larmor: [It has been generally considered to be a consequence of the Maxwell-Boltzmann doctrine of equipartition of energy among molecular freedoms that there can be only one ultimate distribution of energy in the molecule however it is produced, and that there remains only the question of the time it takes for the distribution to become established. Thus Maxwell has objected that on the Le Sage hypothesis the thermal energy of the universe would become transferred to his corpuscles. Lord Kelvin here in effect puts the question whether the trend to uniform distribution would be disturbed, if the corpuscles collided only with the atoms and not with one another. For his objections to the general law of equipartition cf. Vol. iv. pp. 484 seq.: for subsequent literature cf. Larmor, Bakerian Lecture, Proc. Roy. Soc. 83 A. (1909) pp. 82.95.
The interest of the present paper, mainly historical, is somewhat enhanced by the obvious incidental analogies to the behaviour of free electrons that are presented by Le Sage's particles.]
- "Theory of magnetic induction in crystalline and non-crystalline substances/ 1 Phil. Mag., March 1851; "Forces experienced by inductively magnetized ferromagnetic and dia-magnetic non-crystalline substances," Phil. Mag., Oct. 1850; "Reciprocal action of dia-magnetic particles/' Phil. Mag., Dec. 1855; all to be found in a collection of reprinted and newly written papers on electrostatics and magnetism, nearly ready for publication (Macmillan, 1872).