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# Le Sage's theory of gravitation

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 Title: Le Sage's theory of gravitation Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Le Sage's theory of gravitation

Le Sage's theory of gravitation is a kinetic theory of

## Basic theory

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).

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

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

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).

## Fatio

Nicolas Fatio presented the first formulation of his thoughts on gravitation in a letter to 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)

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/r2. 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

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=ρv2zz/6. This solution is very similar to the formula known in the kinetic theory of gases p=ρv2/3, 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 << u and ρ = const. Fatio stated that the resistance is ρu2. In the case v >> u and ρ = const. 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 ρv2. 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 ρv2.

### Reception of Fatio's theory

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

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. On the other hand, Fatio himself stated that although Newton had commented privately that Fatio's theory was the best possible mechanical explanation of gravity, he also acknowledged that Newton tended to believe that the true explanation of gravitation was not mechanical. Also, Gregory noted in his "Memoranda": "Mr. Newton and Mr. Halley laugh at Mr. Fatio’s manner of explaining gravity." This was allegedly noted by him on December 28, 1691. However, the real date is unknown, because both ink and feather which were used, differ from the rest of the page. After 1694, the relationship between the two men cooled down.

Christiaan Huygens was the first person informed by Fatio of his theory, but never accepted it. Fatio believed he had convinced Huygens of the consistency of his theory, but Huygens denied this in a letter to Gottfried Leibniz. There was also a short correspondence between Fatio and Leibniz on the theory. Leibniz criticized Fatio's theory for demanding empty space between the particles, which was rejected by him (Leibniz) on philosophical grounds. Jakob Bernoulli expressed an interest in Fatio's Theory, and urged Fatio to write his thoughts on gravitation in a complete manuscript, which was actually done by Fatio. Bernoulli then copied the manuscript, which now resides in the university library of Basel, and was the base of the Bopp edition.

Nevertheless, Fatio's theory remained largely unknown with a few exceptions like Cramer and Le Sage, because he never was able to formally publish his works and he fell under the influence of a group of religious fanatics called the "French prophets" (which belonged to the camisards) and therefore his public reputation was ruined.

## Cramer and Redeker

In 1731 the Swiss mathematician Gabriel Cramer published a dissertation, 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

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. According to Le Sage, after creating and sending his essay he was informed on the theories of Fatio, Cramer and Redeker. In 1756 for the first time one of his expositions of the theory was 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

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 v2, so the ratio resistance/gravity can be made arbitrarily small by increasing v. Therefore, he suggested that the ultramundane corpuscles might move at the speed of light, but after further consideration he adjusted this to 105 times the speed of light.
• To maintain mass proportionality, ordinary matter consists of cage-like structures, in which their diameter is only the 107th part of their mutual distance. Also the "bars", which constitute the cages, were small (around 1020 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. However, by comparing the two theories and after a detailed analysis of Fatio's papers (which also were in possession of Le Sage) Zehe judged that Le Sage contributed nothing essentially new and he often did not reach Fatio's level.

### Reception of Le Sage's theory

Le Sage’s ideas were not well-received during his day, except for some of his friends and associates like Pierre Prévost, Charles Bonnet, Jean-André Deluc, Charles Mahon, 3rd Earl Stanhope and Simon Lhuilier. They mentioned and described Le Sage's theory in their books and papers, which were used by their contemporaries as a secondary source for Le Sage's theory (because of the lack of published papers by Le Sage himself) .

Euler, Bernoulli, and Boscovich

Leonhard Euler once remarked that Le Sage's model was "infinitely better" than that of all other authors, and that all objections are balanced out in this model, but later he said the analogy to light had no weight for him, because he believed in the wave nature of light. After further consideration, Euler came to disapprove of the model, and he wrote to Le Sage:

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 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, which 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