Le Sage's theory of gravitation is a kinetic theory of gravity originally proposed by Nicolas Fatio de Duillier in 1690 and later by Georges-Louis Le Sage in 1748. The theory proposed a mechanical explanation for Newton's gravitational force in terms of streams of tiny unseen particles (which Le Sage called ultra-mundane corpuscles) impacting all material objects from all directions. According to this model, any two material bodies partially shield each other from the impinging corpuscles, resulting in a net imbalance in the pressure exerted by the impact of corpuscles on the bodies, tending to drive the bodies together. This mechanical explanation for gravity never gained widespread acceptance.

Basic theory

left|thumb|150px|<div class="center">P1: Single body – no net directional force</div>The theory posits that the force of gravity is the result of tiny particles (corpuscles) moving at high speed in all directions, throughout the universe. The intensity of the flux of particles is assumed to be the same in all directions, so an isolated object A is struck equally from all sides, resulting in only an inward-directed pressure but no net directional force (P1).

right|thumb|150px|<div class="center">P2: Two bodies "attract" each other</div>

With a second object B present, however, a fraction of the particles that would otherwise have struck A from the direction of B is intercepted, so B works as a shield, i.e. from the direction of B, A will be struck by fewer particles than from the opposite direction. Likewise B will be struck by fewer particles from the direction of A than from the opposite direction. One can say that A and B are "shadowing" each other, and the two bodies are pushed toward each other by the resulting imbalance of forces (P2). Thus the apparent attraction between bodies is, according to this theory, actually a diminished push from the direction of other bodies, so the theory is sometimes called push gravity or shadow gravity, although it is more widely referred to as Lesage gravity.

Nature of collisions

left|thumb|150px|<div class="center">P3: Opposite streams</div>

If the collisions of body A and the gravific particles are fully elastic, the intensity of the reflected particles would be as strong as of the incoming ones, so no net directional force would arise. The same is true if a second body B is introduced, where B acts as a shield against gravific particles in the direction of A. The gravific particle C which ordinarily would strike on A is blocked by B, but another particle D which ordinarily would not have struck A, is re-directed by the reflection on B, and therefore replaces C. Thus if the collisions are fully elastic, the reflected particles between A and B would fully compensate any shadowing effect. In order to account for a net gravitational force, it must be assumed that the collisions are not fully elastic, or at least that the reflected particles are slowed, so that their momentum is reduced after the impact. This would result in streams with diminished momentum departing from A, and streams with undiminished momentum arriving at A, so a net directional momentum toward the center of A would arise (P3). Under this assumption, the reflected particles in the two-body case will not fully compensate the shadowing effect, because the reflected flux is weaker than the incident flux.

Inverse square law

right|thumb|150px|<div class="center">P4: Inverse square relation</div>

Since it is assumed that some or all of the gravific particles converging on an object are either absorbed or slowed by the object, it follows that the intensity of the flux of gravific particles emanating from the direction of a massive object is less than the flux converging on the object. We can imagine this imbalance of momentum flow – and therefore of the force exerted on any other body in the vicinity – distributed over a spherical surface centered on the object (P4). The imbalance of momentum flow over an entire spherical surface enclosing the object is independent of the size of the enclosing sphere, whereas the surface area of the sphere increases in proportion to the square of the radius. Therefore, the momentum imbalance per unit area decreases inversely as the square of the distance.

Mass proportionality

From the premises outlined so far, there arises only a force which is proportional to the surface of the bodies. But gravity is proportional to the masses. To satisfy the need for mass proportionality, the theory posits that a) the basic elements of matter are very small so that gross matter consists mostly of empty space, and b) that the particles are so small, that only a small fraction of them would be intercepted by gross matter. The result is, that the "shadow" of each body is proportional to the surface of every single element of matter. If it is then assumed that the elementary opaque elements of all matter are identical (i.e., having the same ratio of density to area), it will follow that the shadow effect is, at least approximately, proportional to the mass (P5).

center|frame|<div class="center">P5: Permeability, attenuation and mass proportionality</div>

Fatio

thumb|right|150px|<div class="center">[[Nicolas Fatio de Duillier|Nicolas Fatio</div>]]

Nicolas Fatio presented the first formulation of his thoughts on gravitation in a letter to Christiaan Huygens in the spring of 1690. Two days later Fatio read the content of the letter before the Royal Society in London. In the following years Fatio composed several draft manuscripts of his major work De la Cause de la Pesanteur, but none of this material was published in his lifetime. In 1731 Fatio also sent his theory as a Latin poem, in the style of Lucretius, to the Paris Academy of Science, but it was dismissed. Some fragments of these manuscripts and copies of the poem were later acquired by Le Sage who failed to find a publisher for Fatio's papers. So it lasted until 1929, when the only complete copy of Fatio's manuscript was published by Karl Bopp, and in 1949 Gagnebin used the collected fragments in possession of Le Sage to reconstruct the paper. The Gagnebin edition includes revisions made by Fatio as late as 1743, forty years after he composed the draft on which the Bopp edition was based. However, the second half of the Bopp edition contains the mathematically most advanced parts of Fatio's theory, and were not included by Gagnebin in his edition. For a detailed analysis of Fatio's work, and a comparison between the Bopp and the Gagnebin editions, see Zehe. The following description is mainly based on the Bopp edition.

Features of Fatio's theory

Fatio's pyramid (Problem I)

left|thumb|150px|<div class="center">P6: Fatio's pyramid</div>

Fatio assumed that the universe is filled with minute particles, which are moving indiscriminately with very high speed and rectilinearly in all directions. To illustrate his thoughts he used the following example: Suppose an object C, on which an infinite small plane zz and a sphere centered about zz is drawn. Into this sphere Fatio placed the pyramid PzzQ, in which some particles are streaming in the direction of zz and also some particles, which were already reflected by C and therefore depart from zz. Fatio proposed that the mean velocity of the reflected particles is lower and therefore their momentum is weaker than that of the incident particles. The result is one stream, which pushes all bodies in the direction of zz. So on one hand the speed of the stream remains constant, but on the other hand at larger proximity to zz the density of the stream increases and therefore its intensity is proportional to 1/r<sup>2</sup>. And because one can draw an infinite number of such pyramids around C, the proportionality applies to the entire range around C.

Reduced speed

In order to justify the assumption, that the particles are traveling after their reflection with diminished velocities, Fatio stated the following assumptions:

  • Either ordinary matter, or the gravific particles, or both are inelastic, or
  • the impacts are fully elastic, but the particles are not absolutely hard, and therefore are in a state of vibration after the impact, and/or
  • due to friction the particles begin to rotate after their impacts.

These passages are the most incomprehensible parts of Fatio's theory, because he never clearly decided which sort of collision he actually preferred. However, in the last version of his theory in 1742 he shortened the related passages and ascribed "perfect elasticity or spring force" to the particles and on the other hand "imperfect elasticity" to gross matter, therefore the particles would be reflected with diminished velocities. Additionally, Fatio faced another problem: What is happening if the particles collide with each other? Inelastic collisions would lead to a steady decrease of the particle speed and therefore a decrease of the gravitational force. To avoid this problem, Fatio supposed that the diameter of the particles is very small compared to their mutual distance, so their interactions are very rare.

Condensation

Fatio thought for a long time that, since corpuscles approach material bodies at a higher speed than they recede from them (after reflection), there would be a progressive accumulation of corpuscles near material bodies (an effect which he called "condensation"). However, he later realized that although the incoming corpuscles are quicker, they are spaced further apart than are the reflected corpuscles, so the inward and outward flow rates are the same. Hence there is no secular accumulation of corpuscles, i.e., the density of the reflected corpuscles remains constant (assuming that they are small enough that no noticeably greater rate of self-collision occurs near the massive body). More importantly, Fatio noted that, by increasing both the velocity and the elasticity of the corpuscles, the difference between the speeds of the incoming and reflected corpuscles (and hence the difference in densities) can be made arbitrarily small while still maintaining the same effective gravitational force.

Porosity of gross matter

left|thumb|150px|<div class="center">P7: Crystal lattice ([[icosahedron)</div>]]

In order to ensure mass proportionality, Fatio assumed that gross matter is extremely permeable to the flux of corpuscles. He sketched 3 models to justify this assumption:

  • He assumed that matter is an accumulation of small "balls" whereby their diameter compared with their distance among themselves is "infinitely" small. But he rejected this proposal, because under this condition the bodies would approach each other and therefore would not remain stable.
  • Then he assumed that the balls could be connected through bars or lines and would form some kind of crystal lattice. However, he rejected this model too – if several atoms are together, the gravific fluid is not able to penetrate this structure equally in all direction, and therefore mass proportionality is impossible.
  • At the end Fatio also removed the balls and only left the lines or the net. By making them "infinitely" smaller than their distance among themselves, thereby a maximum penetration capacity could be achieved.

Pressure force of the particles (Problem II)

Already in 1690 Fatio assumed, that the "push force" exerted by the particles on a plain surface is the sixth part of the force, which would be produced if all particles are lined up normal to the surface. Fatio now gave a proof of this proposal by determination of the force, which is exerted by the particles on a certain point zz. He derived the formula p&nbsp;=&nbsp;ρv<sup>2</sup>zz/6. This solution is very similar to the formula known in the kinetic theory of gases p&nbsp;=&nbsp;ρv<sup>2</sup>/3, which was found by Daniel Bernoulli in 1738. This was the first time that a solution analogous to the similar result in kinetic theory was pointed out – long before the basic concept of the latter theory was developed. However, Bernoulli's value is twice as large as Fatio's one, because according to Zehe, Fatio only calculated the value mv for the change of impulse after the collision, but not 2mv and therefore got the wrong result. (His result is only correct in the case of totally inelastic collisions.) Fatio tried to use his solution not only for explaining gravitation, but for explaining the behaviour of gases as well. He tried to construct a thermometer, which should indicate the "state of motion" of the air molecules and therefore estimate the temperature. But Fatio (unlike Bernoulli) did not identify heat and the movements of the air particles – he used another fluid, which should be responsible for this effect. It is also unknown, whether Bernoulli was influenced by Fatio or not.

Infinity (Problem III)

In this chapter Fatio examines the connections between the term infinity and its relations to his theory. Fatio often justified his considerations with the fact that different phenomena are "infinitely smaller or larger" than others and so many problems can be reduced to an undetectable value. For example, the diameter of the bars is infinitely smaller than their distance to each other; or the speed of the particles is infinitely larger than those of gross matter; or the speed difference between reflected and non-reflected particles is infinitely small.

Resistance of the medium (Problem IV)

This is the mathematically most complex part of Fatio's theory. There he tried to estimate the resistance of the particle streams for moving bodies. Supposing u is the velocity of gross matter, v is the velocity of the gravific particles and ρ the density of the medium. In the case v&nbsp;≪&nbsp;u and ρ&nbsp;=&nbsp;constant Fatio stated that the resistance is ρu<sup>2</sup>. In the case v&nbsp;≫&nbsp;u and ρ&nbsp;=&nbsp;constant the resistance is 4/3ρuv. Now, Newton stated that the lack of resistance to the orbital motion requires an extreme sparseness of any medium in space. So Fatio decreased the density of the medium and stated, that to maintain sufficient gravitational force this reduction must be compensated by changing v "inverse proportional to the square root of the density". This follows from Fatio's particle pressure, which is proportional to ρv<sup>2</sup>. According to Zehe, Fatio's attempt to increase v to a very high value would actually leave the resistance very small compared with gravity, because the resistance in Fatio's model is proportional to ρuv but gravity (i.e. the particle pressure) is proportional to ρv<sup>2</sup>.

Reception of Fatio's theory

Fatio was in communication with some of the most famous scientists of his time.

right|thumb|350px|<div class="center">P8: Signatures of [[Isaac Newton|Newton, Huygens and Halley on Fatio's manuscript</div>]]

There was a strong personal relationship between Isaac Newton and Fatio in the years 1690 to 1693. Newton's statements on Fatio's theory differed widely. For example, after describing the necessary conditions for a mechanical explanation of gravity, he wrote in an (unpublished) note in his own printed copy of the Principia in 1692:The unique hypothesis by which gravity can be explained is however of this kind, and was first devised by the most ingenious geometer Mr. N. Fatio. at the end of which appeared a sketch of a theory very similar to Fatio's – including net structure of matter, analogy to light, shading – but without mentioning Fatio's name. It was known to Fatio that Cramer had access to a copy of his main paper, so he accused Cramer of only repeating his theory without understanding it. It was also Cramer who informed Le Sage about Fatio's theory in 1749. In 1736 the German physician Franz Albert Redeker also published a similar theory. Any connection between Redeker and Fatio is unknown.

Le Sage

thumb|left|150px|<div class="center">Georges-Louis Le Sage</div>

The first exposition of his theory, Essai sur l'origine des forces mortes, was sent by Le Sage to the Academy of Sciences at Paris in 1748, but it was never published. and in 1758 he sent a more detailed exposition, Essai de Chymie Méchanique, to a competition to the Academy of Sciences in Rouen. In this paper he tried to explain both the nature of gravitation and chemical affinities. The exposition of the theory which became accessible to a broader public, Lucrèce Newtonien (1784), in which the correspondence with Lucretius' concepts was fully developed. Another exposition of the theory was published from Le Sage's notes posthumously by Pierre Prévost in 1818.

Le Sage's basic concept

right|thumb|350px|<div class="center">P9: Le Sage's own illustration of his ultramundane corpuscles</div>

Le Sage discussed the theory in great detail and he proposed quantitative estimates for some of the theory's parameters.

  • He called the gravitational particles ultramundane corpuscles, because he supposed them to originate beyond our known universe. The distribution of the ultramundane flux is isotropic and the laws of its propagation are very similar to that of light.
  • Le Sage argued that no gravitational force would arise if the matter-particle-collisions are perfectly elastic . So he proposed that the particles and the basic constituents of matter are "absolutely hard" and asserted that this implies a complicated form of interaction, completely inelastic in the direction normal to the surface of the ordinary matter, and perfectly elastic in the direction tangential to the surface. He then commented that this implies the mean speed of scattered particles is 2/3 of their incident speed. To avoid inelastic collisions between the particles, he supposed that their diameter is very small relative to their mutual distance.
  • That resistance of the flux is proportional to uv (where v is the velocity of the particles and u that of gross matter) and gravity is proportional to v<sup>2</sup>, so the ratio resistance/gravity can be made arbitrarily small by increasing&nbsp;v. Therefore, he suggested that the ultramundane corpuscles might move at the speed of light, but after further consideration he adjusted this to 10<sup>5</sup> times the speed of light.
  • To maintain mass proportionality, ordinary matter consists of cage-like structures, in which their diameter is only the 10<sup>7</sup>th part of their mutual distance. Also the "bars", which constitute the cages, were small (around 10<sup>20</sup> times as long as thick) relative to the dimensions of the cages, so the particles can travel through them nearly unhindered.
  • Le Sage also attempted to use the shadowing mechanism to account for the forces of cohesion, and for forces of different strengths, by positing the existence of multiple species of ultramundane corpuscles of different sizes, as illustrated in Figure 9.

Le Sage said that he was the first one, who drew all consequences from the theory and also Prévost said that Le Sage's theory was more developed than Fatio's theory.

Daniel Bernoulli was pleased by the similarity of Le Sage's model and his own thoughts on the nature of gases. However, Bernoulli himself was of the opinion that his own kinetic theory of gases was only a speculation, and likewise he regarded Le Sage's theory as highly speculative.

Roger Joseph Boscovich pointed out that Le Sage's theory is the first one that actually can explain gravity by mechanical means. However, he rejected the model because of the enormous and unused quantity of ultramundane matter. John Playfair described Boscovich's arguments by saying:

A very similar argument was later given by Maxwell (see the sections below). Additionally, Boscovich denied the existence of all contact and immediate impulse at all, but proposed repulsive and attractive actions at a distance.

Lichtenberg, Kant, and Schelling

Georg Christoph Lichtenberg's knowledge of Le Sage's theory was based on "Lucrece Newtonien" and a summary by Prévost. Lichtenberg originally believed (like Descartes) that every explanation of natural phenomena must be based on rectilinear motion and impulsion, and Le Sage's theory fulfilled these conditions. In 1790 he expressed in one of his papers his enthusiasm for the theory, believing that Le Sage's theory embraces all of our knowledge and makes any further dreaming on that topic useless. He went on by saying: "If it is a dream, it is the greatest and the most magnificent which was ever dreamed..." and that we can fill with it a gap in our books, which can only be filled by a dream.

He often referred to Le Sage's theory in his lectures on physics at the University of Göttingen. However, around 1796 Lichtenberg changed his views after being persuaded by the arguments of Immanuel Kant, who criticized any kind of theory that attempted to replace attraction with impulsion. Kant pointed out that the very existence of spatially extended configurations of matter, such as particles of non-zero radius, implies the existence of some sort of binding force to hold the extended parts of the particle together. Now, that force cannot be explained by the push from the gravitational particles, because those particles too must hold together in the same way. To avoid this circular reasoning, Kant asserted that there must exist a fundamental attractive force. This was precisely the same objection that had always been raised against the impulse doctrine of Descartes in the previous century, and had led even the followers of Descartes to abandon that aspect of his philosophy.

Another German philosopher, Friedrich Wilhelm Joseph Schelling, rejected Le Sage's model because its mechanistic materialism was incompatible with Schelling's very idealistic and anti-materialistic philosophy.

Laplace

Partly in consideration of Le Sage's theory, Pierre-Simon Laplace undertook to determine the necessary speed of gravity in order to be consistent with astronomical observations. He calculated that the speed must be "at least a hundred millions of times greater than that of light", in order to avoid unacceptably large inequalities due to aberration effects in the lunar motion. This was taken by most researchers, including Laplace, as support for the Newtonian concept of instantaneous action at a distance, and to indicate the implausibility of any model such as Le Sage's. Laplace also argued that to maintain mass-proportionality the upper limit for Earth's molecular surface area is at the most the ten-millionth of Earth's surface. To Le Sage's disappointment, Laplace never directly mentioned Le Sage's theory in his works.

Kinetic theory

Because the theories of Fatio, Cramer and Redeker were not widely known, Le Sage's exposition of the theory enjoyed a resurgence of interest in the latter half of the 19th century, coinciding with the development of the kinetic theory of gases.

Leray

Since Le Sage's particles must lose speed when colliding with ordinary matter (in order to produce a net gravitational force), a huge amount of energy must be converted to internal energy modes. If those particles have no internal energy modes, the excess energy can only be absorbed by ordinary matter. Addressing this problem, Armand Jean Leray proposed a particle model (perfectly similar to Le Sage's) in which he asserted that the absorbed energy is used by the bodies to produce magnetism and heat. He suggested, that this might be an answer for the question of where the energy output of the stars comes from.

Kelvin and Tait

thumb|150px|<div class="center">[[William Thomson, 1st Baron Kelvin|Lord Kelvin</div>]]

Le Sage's own theory became a subject of renewed interest in the latter part of the 19th century following a paper published by Kelvin in 1873. Unlike Leray, who treated the heat problem imprecisely, Kelvin stated that the absorbed energy represents a very high heat, sufficient to vaporize any object in a fraction of a second. So Kelvin reiterated an idea that Fatio had originally proposed in the 1690s for attempting to deal with the thermodynamic problem inherent in Le Sage's theory. He proposed that the excess heat might be absorbed by internal energy modes of the particles themselves, based on his proposal of the vortex-nature of matter. In other words, the original translational kinetic energy of the particles is transferred to internal energy modes, chiefly vibrational or rotational, of the particles. Appealing to Clausius's proposition that the energy in any particular mode of a gas molecule tends toward a fixed ratio of the total energy, Kelvin went on to suggest that the energized but slower moving particles would subsequently be restored to their original condition due to collisions (on the cosmological scale) with other particles. Kelvin also asserted that it would be possible to extract limitless amounts of free energy from the ultramundane flux, and described a perpetual motion machine to accomplish this.

Subsequently, Peter Guthrie Tait called the Le Sage theory the only plausible explanation of gravitation which has been propounded at that time. He went on by saying:

Kelvin himself, however, was not optimistic that Le Sage's theory could ultimately give a satisfactory account of phenomena. After his brief paper in 1873 noted above, he never returned to the subject, except to make the following comment:

Preston

Samuel Tolver Preston illustrated that many of the postulates introduced by Le Sage concerning the gravitational particles, such as rectilinear motion, rare interactions, etc.., could be collected under the single notion that they behaved (on the cosmological scale) as the particles of a gas with an extremely long mean free path. Preston also accepted Kelvin's proposal of internal energy modes of the particles. He illustrated Kelvin's model by comparing it with the collision of a steel ring and an anvil – the anvil would not be shaken very much, but the steel ring would be in a state of vibration and therefore departs with diminished velocity. He also argued, that the mean free path of the particles is at least the distance between the planets – on longer distances the particles regain their translational energy due collisions with each other, so he concluded that on longer distances there would be no attraction between the bodies, independent of their size. Paul Drude suggested that this could possibly be a connection with some theories of Carl Gottfried Neumann and Hugo von Seeliger, who proposed some sort of absorption of gravity in open space.

Maxwell

thumb|150px|<div class="center">[[James Clerk Maxwell</div>]]

A review of the Kelvin-Le Sage theory was published by James Clerk Maxwell in the Ninth Edition of the Encyclopædia Britannica under the title Atom in 1875. After describing the basic concept of the theory he wrote (with sarcasm according to Aronson):