Le Sage's theory of gravitation

# Le Sage's theory of gravitation

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Le Sage's theory of gravitation is a kinetic theory of gravity originally proposed by Nicolas Fatio de Duillier
Nicolas Fatio de Duillier
Nicolas Fatio de Duillier was a Swiss mathematician known for his work on the zodiacal light problem, for his very close relationship with Isaac Newton, for his role in the Newton v. Leibniz calculus controversy, and for originating the "push" or "shadow" theory of gravitation...

in 1690 and later by Georges-Louis Le Sage
Georges-Louis Le Sage
Georges-Louis Le Sage was a physicist and is most known for his theory of gravitation, for his invention of an electric telegraph and his anticipation of the kinetic theory of gases....

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, although it continued to be studied occasionally by physicists until the beginning of the 20th century, by which time it was generally considered to be conclusively discredited.

## Basic theory

The theory posits that the force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

of gravity is the result of tiny particles (corpuscles) moving at high speed in all directions, throughout the universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...

. 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
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

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
Elastic collision
An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter...

, 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 Christiaan Huygens in the spring of 1690. Two days later Fatio read the content of the letter before the Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...

in London. In the following years Fatio composed several draft manuscript
Manuscript
A manuscript or handwrite is written information that has been manually created by someone or some people, such as a hand-written letter, as opposed to being printed or reproduced some other way...

s 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
Lucretius
Titus Lucretius Carus was a Roman poet and philosopher. His only known work is an epic philosophical poem laying out the beliefs of Epicureanism, De rerum natura, translated into English as On the Nature of Things or "On the Nature of the Universe".Virtually no details have come down concerning...

, to the Paris Academy of Science, but it was dismissed. Some fragment
Fragment
Fragment may refer to:* A small part/portion broken off something; debris* Fragment , all the data necessary to generate a pixel in the frame buffer* Sentence fragment, a sentence not containing a subject or a predicate...

s of these manuscripts and copies of the poem were later acquired by Le Sage
Georges-Louis Le Sage
Georges-Louis Le Sage was a physicist and is most known for his theory of gravitation, for his invention of an electric telegraph and his anticipation of the kinetic theory of gases....

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
Karl Bopp
Karl Bopp was a German historian of mathematics.Bopp studied at the University of Strasbourg and the University of Heidelberg under Moritz Cantor. In 1906 he habilitated with a work about the Conic sections of Gregorius a Sancto Vincenti, and in 1915 he became professor extraordinarius in Heidelberg...

, 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 by 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
Pyramid
A pyramid is a structure whose outer surfaces are triangular and converge at a single point. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, meaning that a pyramid has at least three triangular surfaces...

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
Inelastic collision
An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved.In collisions of macroscopic bodies, some kinetic energy is turned into vibrational energy of the atoms, causing a heating effect, and the bodies are deformed.The molecules of a gas...

, or
• the impacts are fully elastic, but the particles are not absolutely hard, and therefore are in a state of vibration
Vibration
Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.Vibration is occasionally "desirable"...

after the impact, and/or
• due to friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

the particles begin to rotate
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

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. Fatio also 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
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

in 1738. This was the first time that a solution analogous to the similar result in kinetic theory
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

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
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of 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
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

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
David Gregory
David Gregory FRS was a Scottish mathematician and astronomer. He was professor of mathematics at the University of Edinburgh, Savilian Professor of Astronomy at the University of Oxford, and a commentator on Isaac Newton's Principia.The fourth of the fifteen children of David Gregorie, a doctor...

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
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

. 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 camisard
Camisard
Camisards were French Protestants of the rugged and isolated Cevennes region of south-central France, who raised an insurrection against the persecutions which followed the revocation of the Edict of Nantes in 1685...

s) and therefore his public reputation was ruined.

## Cramer and Redeker

In 1731 the Swiss mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

Gabriel Cramer
Gabriel Cramer
Gabriel Cramer was a Swiss mathematician, born in Geneva. He showed promise in mathematics from an early age. At 18 he received his doctorate and at 20 he was co-chair of mathematics.In 1728 he proposed a solution to the St...

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
Physician
A physician is a health care provider who practices the profession of medicine, which is concerned with promoting, maintaining or restoring human health through the study, diagnosis, and treatment of disease, injury and other physical and mental impairments...

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
Essay
An essay is a piece of writing which is often written from an author's personal point of view. Essays can consist of a number of elements, including: literary criticism, political manifestos, learned arguments, observations of daily life, recollections, and reflections of the author. The definition...

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
Rouen
Rouen , in northern France on the River Seine, is the capital of the Haute-Normandie region and the historic capital city of Normandy. Once one of the largest and most prosperous cities of medieval Europe , it was the seat of the Exchequer of Normandy in the Middle Ages...

. 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
Pierre Prévost
Pierre Prévost was a Swiss philosopher and physicist. In he showed that all bodies radiate heat, no matter how hot or cold they are.-Life:...

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
Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. It radiates uniformly in all directions from a point source sometimes called an isotropic radiator. The radiation may...

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
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

, 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
Pierre Prévost
Pierre Prévost was a Swiss philosopher and physicist. In he showed that all bodies radiate heat, no matter how hot or cold they are.-Life:...

, Charles Bonnet
Charles Bonnet
Charles Bonnet , Swiss naturalist and philosophical writer, was born at Geneva, of a French family driven into Switzerland by the religious persecution in the 16th century.-Life and work:Bonnet's life was uneventful...

, Jean-André Deluc
Jean-André Deluc
Jean-André Deluc or de Luc was a Swiss geologist and meteorologist.-Life:He was born at Geneva, descended from a family which had emigrated from Lucca and settled at Geneva in the 15th century...

, 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
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

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
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

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
Roger Joseph Boscovich
Ruđer Josip Bošković was a theologian, physicist, astronomer, mathematician, philosopher, diplomat, poet, Jesuit, and a polymath from the city of Dubrovnik in the Republic of Ragusa , who studied and lived in Italy and France where he also published many of his works.He is famous for...

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
John Playfair
John Playfair FRSE, FRS was a Scottish scientist and mathematician, and a professor of natural philosophy at the University of Edinburgh. He is perhaps best known for his book Illustrations of the Huttonian Theory of the Earth , which summarized the work of James Hutton...

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
Action at a distance (physics)
In physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. This term was used most often in the context of early theories of gravity and electromagnetism to describe how an object responds to the influence of distant...

.

Lichtenberg, Kant, and Schelling
Georg Christoph Lichtenberg's
Georg Christoph Lichtenberg
Georg Christoph Lichtenberg was a German scientist, satirist and Anglophile. As a scientist, he was the first to hold a professorship explicitly dedicated to experimental physics in Germany...

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
Immanuel Kant
Immanuel Kant was a German philosopher from Königsberg , researching, lecturing and writing on philosophy and anthropology at the end of the 18th Century Enlightenment....

, 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
Begging the question
Begging the question is a type of logical fallacy in which the proposition to be proven is assumed implicitly or explicitly in the premise....

, 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
Friedrich Wilhelm Joseph Schelling
Friedrich Wilhelm Joseph Schelling , later von Schelling, was a German philosopher. Standard histories of philosophy make him the midpoint in the development of German idealism, situating him between Fichte, his mentor prior to 1800, and Hegel, his former university roommate and erstwhile friend...

, 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
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...

undertook to determine the necessary speed of gravity
Speed of gravity
In the context of classical theories of gravitation, the speed of gravity is the speed at which changes in a gravitational field propagate. This is the speed at which a change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational...

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 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
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

.

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, P. 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
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...

and heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

. He suggested, that this might be an answer for the question of where the energy output of the stars comes from.

Kelvin and Tait

Le Sage's own theory became a subject of re-newed interest in the latter part of the 19th century following a paper published by Kelvin
William Thomson, 1st Baron Kelvin
William Thomson, 1st Baron Kelvin OM, GCVO, PC, PRS, PRSE, was a mathematical physicist and engineer. At the University of Glasgow he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging...

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 re-iterated 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
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a 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
Perpetual motion
Perpetual motion describes hypothetical machines that operate or produce useful work indefinitely and, more generally, hypothetical machines that produce more work or energy than they consume, whether they might operate indefinitely or not....

machine to accomplish this. (The flaw in Kelvin's reasoning was that Clausius's proposition would apply only if ordinary matter was in thermodynamic equilibrium with the ultramundane flux - in which case there would no net gravitational effect.)

Subsequently, Peter Guthrie Tait
Peter Guthrie Tait
Peter Guthrie Tait FRSE was a Scottish mathematical physicist, best known for the seminal energy physics textbook Treatise on Natural Philosophy, which he co-wrote with Kelvin, and his early investigations into knot theory, which contributed to the eventual formation of topology as a mathematical...

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
Samuel Tolver Preston
Samuel Tolver Preston was an English Engineer and physicist.His parents were Daniel Bloom Preston and Mary Susannah Tolver. Preston was educated as a Telegraph-engineer. He went to Munich where he attained his Ph.D in 1894 with Ludwig Boltzmann...

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
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...

. 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
Hugo von Seeliger
Hugo von Seeliger , also known as Hugo Hans Ritter von Seeliger, was a German astronomer, often considered the most important astronomer of his day....

, who proposed some sort of absorption of gravity in open space.

Maxwell

A review of the Kelvin-Le Sage theory was published by James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

in the Ninth Edition of the Encyclopaedia Britannica under the title Atom in 1875. After describing the basic concept of the theory he wrote (with sarcasm
Sarcasm
Sarcasm is “a sharp, bitter, or cutting expression or remark; a bitter jibe or taunt.” Though irony and understatement is usually the immediate context, most authorities distinguish sarcasm from irony; however, others argue that sarcasm may or often does involve irony or employs...

according to Aronson):
Maxwell commented on Kelvin’s suggestion of different energy modes of the particles that this implies the gravitational particles are not simple primitive entities, but rather systems, with their own internal energy modes, which must be held together by (unexplained) forces of attraction. He argues that the temperature of bodies must tend to approach that at which the average kinetic energy of a molecule of the body would be equal to the average kinetic energy of an ultra-mundane particle and he states that the latter quantity must be much greater than the former and concludes that ordinary matter should be incinerated within seconds under the Le Sage bombardment. He wrote:
Maxwell also argued that the theory requires "an enormous expenditure of external power" and therefore violating the conservation of energy as the fundamental principle of nature. Preston responded to Maxwell's criticism by arguing that the kinetic energy of each individual simple particle could be made arbitrarily low by positing a sufficiently low mass (and higher number density) for the particles. But this issue later was discussed in a more detailed way by Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

, who showed that the thermodynamic problem within Le Sage models remained unresolved.

Isenkrahe, Rysanik, du Bois-Reymond
Caspar Isenkrahe
Caspar Isenkrahe
Mathias Caspar Hubert Isenkrahe was a German mathematician, physicist and catholic philosopher of nature.- Life :Caspar Isenkrahe grew up without a father, who died before Caspar's birth...

presented his model in a variety of publications between 1879-1915.

His basic assumptions were very similar to those of Le Sage and Preston, but he gave a more detailed application of the kinetic theory. However, by asserting that the velocity of the corpuscles after collision was reduced without any corresponding increase in the energy of any other object, his model violated the conservation of energy. He noted that there is a connection between the weight of a body and its density (because any decrease in the density of an object reduces the internal shielding) so he went on to assert that warm bodies should be heavier than colder ones (related to the effect of thermal expansion
Thermal expansion
Thermal expansion is the tendency of matter to change in volume in response to a change in temperature.When a substance is heated, its particles begin moving more and thus usually maintain a greater average separation. Materials which contract with increasing temperature are rare; this effect is...

).

In another model A. Rysanek in 1887

also gave a careful analysis, including an application of Maxwell's law of the particle velocities in a gas. He distinguished between a gravitational and a luminiferous aether
Luminiferous aether
In the late 19th century, luminiferous aether or ether, meaning light-bearing aether, was the term used to describe a medium for the propagation of light....

. This separation of those two mediums was necessary, because according to his calculations the absence of any drag effect in the orbit of Neptune
Neptune
Neptune is the eighth and farthest planet from the Sun in the Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times...

implies a lower limit for the particle velocity of 5 · 1019 cm/s. He (like Leray) argued that the absorbed energy is converted into heat, which might be transferred into the luminiferous aether and/or is used by the stars to maintain their energy output. However, these qualitative suggestions were unsupported by any quantitative evaluation of the amount of heat actually produced.

In 1888 Paul du Bois-Reymond argued against Le Sage's model, partly because the predicted force of gravity in Le Sage's theory is not strictly proportional to mass. In order to achieve exact mass proportionality as in Newton's theory (which implies no shielding or saturation effects and an infinitely porous structure of matter), the ultramundane flux must be infinitely intense. Du Bois-Reymond rejected this as absurd. In addition, du Bois-Reymond like Kant observed that Le Sage's theory cannot meet its goal, because it invokes concepts like "elasticity" and "absolute hardness" etc., which (in his opinion) can only be explained by means of attractive forces. The same problem arises for the cohesive forces in molecules. As a result, the basic intent of such models, which is to dispense with elementary forces of attraction, is impossible.

## Wave models

Keller and Boisbaudran
In 1863 F.A.E. and Em. Keller presented a theory by using a Le Sage type mechanism in combination with longitudinal wave
Longitudinal wave
Longitudinal waves, as known as "l-waves", are waves that have the same direction of vibration as their direction of travel, which means that the movement of the medium is in the same direction as or the opposite direction to the motion of the wave. Mechanical longitudinal waves have been also...

s of the aether. They supposed that those waves are propagating in every direction and losing some of their momentum after the impact on bodies, so between two bodies the pressure exerted by the waves is weaker than the pressure around them. In 1869 L. de Boisbaudran presented the same model as Leray (including absorption and the production of heat etc.), but like Keller he replaced the particles with longitudinal waves of the aether.

Lorentz

After these attempts other authors substituted electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

for Le Sage’s particles early 20th century. This was in connection with the Lorentz ether theory
Lorentz ether theory
What is now often called Lorentz Ether theory has its roots in Hendrik Lorentz's "Theory of electrons", which was the final point in the development of the classical aether theories at the end of the 19th and at the beginning of the 20th century....

and the electron theory of that time, in which the electrical constitution of matter was assumed.

In 1900 Hendrik Lorentz
Hendrik Lorentz
Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...

wrote, that Le Sage's particle model is not consistent with the electron theory of his time. But the detection that trains of electromagnetic waves could produce some pressure in combination with the penetrating power of Röntgen rays (now called x-ray
X-ray
X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...

s), led him to the conclusion, that nothing is speaking against the possible existence of an even more penetrating radiation then x-rays, which could replace Le Sage's particles. Lorentz showed that an attractive force between charged particles (which might be taken to model the elementary subunits of matter) would indeed arise, but only if the incident energy were entirely absorbed. This was the same fundamental problem which had afflicted the particle models. So Lorentz wrote:
In 1922 Lorentz first examined Martin Knudsen
Martin Knudsen
This article is about the Danish physicist Martin Knudsen. For the Norwegian footballer, see Martin Knudsen .Martin Hans Christian Knudsen was a Danish physicist who taught and conducted research at the Technical University of DenmarkHe is primarily known for his study of molecular gas flow and the...

's investigation on rarefied gases and in connection with that he discussed Le Sage's particle model, followed by a summary of his own electromagnetic Le Sage model - but he repeated his conclusion from 1900: Without absorption no gravitational effect.

In 1913 David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

referred to Lorentz's theory and criticised it by arguing that no force in the form 1/r2 can arise, if the mutual distance of the atoms is large enough when compared with their wavelength.

J.J. Thomson
In 1904 J. J. Thomson
J. J. Thomson
Sir Joseph John "J. J." Thomson, OM, FRS was a British physicist and Nobel laureate. He is credited for the discovery of the electron and of isotopes, and the invention of the mass spectrometer...

considered a Le Sage-type model in which the primary ultramundane flux consisted of a hypothetical form of radiation much more penetrating even than x-rays. He argued that Maxwell's heat problem might be avoided by assuming that the absorbed energy is not be converted into heat, but re-radiated in a still more penetrating form. He noted that this process possibly can explain where the energy of radioactive substances is coming from - however, he stated that an internal cause of radioactivity
Radioactive decay is the process by which an atomic nucleus of an unstable atom loses energy by emitting ionizing particles . The emission is spontaneous, in that the atom decays without any physical interaction with another particle from outside the atom...

is more probable. In 1911 Thomson went back to this subject in his article "Matter" in the Encyclopædia Britannica Eleventh Edition
Encyclopædia Britannica Eleventh Edition
The Encyclopædia Britannica Eleventh Edition is a 29-volume reference work, an edition of the Encyclopædia Britannica. It was developed during the encyclopaedia's transition from a British to an American publication. Some of its articles were written by the best-known scholars of the time...

. There he stated, that this form of secondary radiation is somewhat analogous to how the passage of electrified particles through matter causes the radiation of the even more penetrating x-rays. He remarked:
Tommasina and Brush

Unlike Lorentz and Thomson, Thomas Tommasina between 1903 and 1928 suggested long wavelength radiation to explain gravity, and short wave-length radiation for explaining the cohesive forces of matter. Charles F. Brush
Charles F. Brush
Charles Francis Brush was a U.S. inventor, entrepreneur and philanthropist.-Biography:Born in Euclid Township, Ohio, Brush was raised on a farm about 10 miles from downtown Cleveland...

in 1911 also proposed long wavelength radiation. But he later revised his view and changed to extremely short wavelengths.

## Later assessments

Darwin
In 1905 George Darwin
George Darwin
Sir George Howard Darwin, FRS was an English astronomer and mathematician.-Biography:Darwin was born at Down House, Kent, the second son and fifth child of Charles and Emma Darwin...

subsequently calculated the gravitational force between two bodies at extremely close range to determine if geometrical effects would lead to a deviation from Newton’s law. Here Darwin replaced Le Sage's cage-like units of ordinary matter with microscopic hard spheres of uniform size. He concluded that only in the instance of perfectly inelastic collisions (zero reflection) would Newton’s law stand up, thus reinforcing the thermodynamic problem of Le Sage's theory. Also, such a theory is only valid if the normal and the tangential components of impact are totally inelastic (contrary to Le Sage's scattering mechanism), and the elementary particles are exactly of the same size. He went on to say that the emission of light is the exact converse of the absorption of Le Sage's particles. A body with different surface temperatures will move in the direction of the colder part. In a later review of gravitational theories Darwin briefly described Le Sage's theory and said he gave the theory serious consideration, but then wrote:
Poincaré

Partially based on the calculations of Darwin, an important criticism was given by Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

in 1908. He concluded that the attraction is proportional to , where S is earth's molecular surface area, v is the velocity of the particles, and ρ is the density of the medium. Following Laplace he argued that to maintain mass-proportionality the upper limit for S is at the most a ten-millionth of the Earth's surface. Now, drag (i.e. the resistance of the medium) is proportional to Sρv and therefore the ratio of drag to attraction is inversely proportional to Sv. To reduce drag Poincaré calculated a lower limit for v = 24 · 1017 times the speed of light. So there are lower limits for Sv and v, and a upper limit for S and with those values one can calculate the produced heat, which is proportional to Sρv3. The calculation shows that earth's temperature would rise by 1026 degrees per second. Poincaré noticed, "that the earth could not long stand such a regime." Poincaré also analyzed some wave models (Tommasina and Lorentz), remarking that they suffered the same problems as the particle models. To reduce drag, superluminal wave velocities were necessary, and they would still be subject to the heating problem. After describing a similar re-radiation model like Thomson he concluded: "Such are the complicated hypotheses to which we are led when we seek to make Le Sage's theory tenable".

He also stated that if in Lorentz' model the absorbed energy is fully converted into heat, this would raise earth's temperature by 1013 degrees per second. Poincaré then went on to consider Le Sage's theory in the context of the "new dynamics" that had been developed at the end of the 19th and beginning of the 20th centuries, specifically recognizing the relativity principle. For a particle theory he remarked that "it is difficult to imagine a law of collision compatible with the principle of relativity", and the problems of drag and heating remain.

### Matter and particles

Porosity of matter
A basic prediction of the theory is the extreme porosity of matter. As supposed by Fatio and Le Sage in 1690/1758 (and before them, Huygens) matter must consist mostly of empty space so that the very small particles can penetrate the bodies nearly undisturbed and therefore every single part of matter can take part in the gravitational interaction. This prediction has been (in some respects) confirmed over the course of the time. Indeed, matter consists mostly of empty space and certain particles like neutrino
Neutrino
A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...

s can pass through matter nearly unhindered. However, the image of elementary particles as classical entities who interact directly, determined by their shapes and sizes (in the sense of the net structure proposed by Fatio/Le Sage and the equisized spheres of Isenkrahe/Darwin), is not consistent with current understanding of elementary particles. The Lorentz/Thomson proposal of electrical charged particles as the basic constituents of matter is inconsistent with current physics as well.

Every Le Sage-type model assumes the existence of a space-filling isotropic flux or radiation of enormous intensity and penetrating capability. This has some similarity to the cosmic microwave background radiation
In cosmology, cosmic microwave background radiation is thermal radiation filling the observable universe almost uniformly....

(CMBR) discovered in the 20th century. CMBR is indeed a space-filling and fairly isotropic flux, but its intensity is extremely small, as is its penetrating capability. The flux of neutrinos, emanating from (for example) the sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

, possesses the penetrating properties envisaged by Le Sage for his ultramundane corpuscles, but this flux is not isotropic (since individual stars are the main sources of neutrinos) and the intensity is even less than that of the CMBR. Of course, neither the CMBR nor neutrinos propagate at superluminal speeds, which is another necessary attribute of Le Sage’s particles. From a more modern point of view, discarding the simple “push” concept of Le Sage, the suggestion that the neutrino (or some other particle similar to the neutrino) might be the mediating particle in a quantum field theory of gravitation was considered and disproved by Feynman.

### Gravitational shielding

Although matter is postulated to be very sparse in the Fatio-Le Sage theory, it cannot be perfectly transparent, because in that case no gravitational force would exist. However, the lack of perfect transparency leads to problems: with sufficient mass the amount of shading produced by two pieces of matter becomes less than the sum of the shading that each of them would produce separately, due to the overlap of their shadows (P10, above). This hypothetical effect, called gravitational shielding
Gravity shielding
The term gravitational shielding refers to a hypothetical process of shielding an object from the influence of a gravitational field. Such processes, if they existed, would have the effect of reducing the weight of an object. The shape of the shielded region would be similar to a shadow from the...

, implies that addition of matter does not result in a direct proportional increase in the gravitational mass. Therefore, in order to be viable, Fatio and Le Sage postulated that the shielding effect is so small as to be undetectable, which requires that the interaction cross-section of matter must be extremely small (P10, below). This places an extremely high lower-bound on the intensity of the flux required to produce the observed force of gravity. Any form of gravitational shielding would represent a violation of the equivalence principle
Equivalence principle
In the physics of general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body is actually...

, and would be inconsistent with the extremely precise null result
Null result
In science, a null result is a result without the expected content: that is, the proposed result is absent. It is an experimental outcome which does not show an otherwise expected effect. This does not imply a result of zero or nothing, simply a result that does not support the hypothesis...

observed in the Eötvös experiment
Eötvös experiment
The Eötvös experiment was a famous physics experiment that measured the correlation between inertial mass and gravitational mass, demonstrating that the two were one and the same, something that had long been suspected but never demonstrated with the same accuracy. The earliest experiments were...

and its successors — all of which have instead confirmed the precise equivalence of active and passive gravitational mass with inertial mass that was predicted by general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

. For more historical information on the connection between gravitational shielding and Le Sage gravity, see Martins, and Borzeszkowski et al.

Since Isenkrahe's proposal on the connection between density, temperature and weight was based purely on the anticipated effects of changes in material density, and since temperature at a given density can be increased or decreased, Isenkrahe's comments do not imply any fundamental relation between temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

and gravitation. (There actually is a relation between temperature and gravitation, as well as between binding energy and gravitation, but these actual effects have nothing to do with Isenkrahe's proposal. See the section below on "Coupling to Energy".) Regarding the prediction of a relation between gravitation and density, all experimental evidence indicates that there is no such relation.

### Speed of gravity

Drag
According to Le Sage's theory, an isolated body is subjected to drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

if it is in motion relative to the unique isotropic frame of the ultramundane flux (i.e., the frame in which the speed of the ultramundane corpuscles is the same in all directions). This is due to the fact that, if a body is in motion, the particles striking the body from the front have a higher speed (relative to the body) than those striking the body from behind - this effect will act to decrease the distance between the sun and the earth. The magnitude of this drag is proportional to vu, where v is the speed of the particles and u is the speed of the body, whereas the characteristic force of gravity is proportional to v2, so the ratio of drag to gravitational force is proportional to u/v. Thus for a given characteristic strength of gravity, the amount of drag for a given speed u can be made arbitrarily small by increasing the speed v of the ultramundane corpuscles. However, in order to reduce the drag to an acceptable level (i.e., consistent with observation) in terms of classical mechanics, the speed v must be many orders of magnitude greater than the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

. This makes Le Sage theory fundamentally incompatible with the modern science of mechanics based on special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

, according to which no particle (or wave) can exceed the speed of light. In addition, even if superluminal particles were possible, the effective temperature of such a flux would be sufficient to incinerate all ordinary matter in a fraction of a second.

Aberration
As shown by Laplace, another possible Le Sage effect is orbital aberration due to finite speed of gravity
Speed of gravity
In the context of classical theories of gravitation, the speed of gravity is the speed at which changes in a gravitational field propagate. This is the speed at which a change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational...

. Unless the Le Sage particles are moving at speeds much greater than the speed of light, as Le Sage and Kelvin supposed, there is a time delay in the interactions between bodies (the transit time). In the case of orbital motion this results in each body reacting to a retarded position of the other, which creates a leading force component. Contrary to the drag effect, this component will act to accelerate both objects away from each other. In order to maintain stable orbits, the effect of gravity must either propagate much faster than the speed of light or must not be a purely central force. This has been suggested by many as a conclusive disproof of any Le Sage type of theory. In contrast, general relativity is consistent with the lack of appreciable aberration identified by Laplace, because even though gravity propagates at the speed of light in general relativity, the expected aberration is almost exactly cancelled by velocity-dependent terms in the interaction.

### Range of gravity

In many particle models, such as Kelvin's, the range of gravity is limited due to the nature of particle interactions amongst themselves. The range is effectively determined by the rate that the proposed internal modes of the particles can eliminate the momentum defects (shadows) that are created by passing through matter. Such predictions as to the effective range of gravity will vary and are dependent upon the specific aspects and assumptions as to the modes of interactions that are available during particle interactions. However, for this class of models the observed large-scale structure of the cosmos constrains such dispersion to those that will allow for the aggregation of such immense gravitational structures.

### Energy

Absorption
As noted in the historical section, a major problem for every Le Sage model is the energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

and heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

issue. As Maxwell and Poincaré showed, inelastic collision
Inelastic collision
An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved.In collisions of macroscopic bodies, some kinetic energy is turned into vibrational energy of the atoms, causing a heating effect, and the bodies are deformed.The molecules of a gas...

Vaporization
Vaporization of an element or compound is a phase transition from the liquid or solid phase to gas phase. There are three types of vaporization: evaporation, boiling and sublimation....

of matter within fractions of a second and the suggested solutions were not convincing. For example, Aronson gave a simple proof of Maxwell's assertion:
Likewise Isenkrahe's violation of the energy conservation law is unacceptable, and Kelvin's application of Clausius' theorem leads (as noted by Kelvin himself) to some sort of perpetual motion
Perpetual motion
Perpetual motion describes hypothetical machines that operate or produce useful work indefinitely and, more generally, hypothetical machines that produce more work or energy than they consume, whether they might operate indefinitely or not....

mechanism. The suggestion of a secondary re-radiation mechanism for wave models attracted the interest of JJ Thomson, but was not taken very seriously by either Maxwell or Poincaré, because it entails a gross violation of the second law of thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

(huge amounts of energy spontaneously being converted from a colder to a hotter form), which is one of the most solidly established of all physical laws.

The energy problem has also been considered in relation to the idea of mass accretion in connection with the Expanding Earth theory. Among the early theorists to link mass increase in some sort of push gravity model to Earth expansion were Yarkovsky
Ivan Osipovich Yarkovsky
Ivan Osipovich Yarkovsky was a Russian-Polish civil engineer. He worked for a Russian railway company and was obscure in his own time. Beginning in the 1970s, long after Yarkovsky's death, his work on the effects of thermal radiation on small objects in the solar system was developed into the...

and Hilgenberg
Ott Christoph Hilgenberg
Ott Christoph Hilgenberg was a German engineer, scientist and a humanist. Directly after his Abitur in the year 1914 Hilgenberg was drawn to the war service, but he was discharged after a knee injury...

. The idea of mass accretion and the expanding earth theory are not currently considered to be viable by mainstream scientists. This is because, among other reasons, according to the principle of mass-energy equivalence
Mass-energy equivalence
In physics, mass–energy equivalence is the concept that the mass of a body is a measure of its energy content. In this concept, mass is a property of all energy, and energy is a property of all mass, and the two properties are connected by a constant...

, if the Earth was absorbing the energy of the ultramundane flux at the rate necessary to produce the observed force of gravity (i.e. by using the values calculated by Poincaré), its mass would be doubling in each fraction of a second.

Coupling to energy
Based on observational evidence, it is now known that gravity interacts with all forms of energy, and not just with mass. The electrostatic binding energy of the nucleus, the energy of weak interactions in the nucleus, and the kinetic energy of electrons in atoms, all contribute to the gravitational mass of an atom, as has been confirmed to high precision in Eötvös type
Eötvös experiment
The Eötvös experiment was a famous physics experiment that measured the correlation between inertial mass and gravitational mass, demonstrating that the two were one and the same, something that had long been suspected but never demonstrated with the same accuracy. The earliest experiments were...

experiments.
This means, for example, that when the atoms of a quantity of gas are moving more rapidly, the gravitation of that gas increases.
Moreover, Lunar Laser Ranging experiments
Lunar laser ranging experiment
The ongoing Lunar Laser Ranging Experiment measures the distance between the Earth and the Moon using laser ranging. Lasers on Earth are aimed at retroreflectors planted on the moon during the Apollo program, and the time for the reflected light to return is determined...

have shown that even gravitational binding energy itself also gravitates, with a strength consistent with the equivalence principle
Equivalence principle
In the physics of general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body is actually...

to high precision

which furthermore demonstrates that any successful theory of gravitation must be nonlinear and self-coupling.

Le Sage's theory does not predict any of these aforementioned effects, nor do any of the known variants of Le Sage's theory.

## Non-gravitational applications and analogies

Mock gravity
Lyman Spitzer
Lyman Spitzer
Lyman Strong Spitzer, Jr. was an American theoretical physicist and astronomer best known for his research in star formation, plasma physics, and in 1946, for conceiving the idea of telescopes operating in outer space...

in 1941 calculated, that absorption of radiation between two dust particles
Dust
Dust consists of particles in the atmosphere that arise from various sources such as soil dust lifted up by wind , volcanic eruptions, and pollution...

lead to a net attractive force which varies proportional to 1/r2 (evidently he was unaware of Le Sage's shadow mechanism and especially Lorentz's considerations on radiation pressure and gravity). George Gamow
George Gamow
George Gamow , born Georgiy Antonovich Gamov , was a Russian-born theoretical physicist and cosmologist. He discovered alpha decay via quantum tunneling and worked on radioactive decay of the atomic nucleus, star formation, stellar nucleosynthesis, Big Bang nucleosynthesis, cosmic microwave...

, who called this effect "mock gravity", proposed in 1949 that after the big bang the temperature of the electrons has dropped faster than the temperature of the background radiation. Absorption of the radiation lead to a Lesage mechanism between the electrons, which might had an important role in the process of galaxy formation shortly after the big bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...

. However, this proposal was disproved by Field in 1971, who showed that this effect was much too small, because electrons and the radiation were nearly in thermal equilibrium. Hogan and White proposed in 1986 that mock gravity might have influenced the galaxy formation by absorption of pregalactic starlight. But it was shown by Wang and Field that any form of mock gravity is incapable of producing enough force to influence the galaxy formation.

Plasma
The Le Sage mechanism also has been identified as a significant factor in the behavior of dusty plasma
Dusty plasma
A dusty plasma is a plasma containing nanometer or micrometer-sized particles suspended in it. Dust particles may be charged and the plasma and particles behave as a plasma, following electromagnetic laws for particles up to about 10 nm...

. A.M. Ignatov has shown that an attractive force arises between two dust grains suspended in an isotropic collisionless plasma due to inelastic collisions between ions of the plasma and the grains of dust. This attractive force is inversely proportional to the square of the distance between dust grains, and can counterbalance the Coulomb repulsion between dust grains.

Vacuum energy
In quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

the existence of virtual particles is proposed, which lead to the so called Casimir effect
Casimir effect
In quantum field theory, the Casimir effect and the Casimir–Polder force are physical forces arising from a quantized field. The typical example is of two uncharged metallic plates in a vacuum, like capacitors placed a few micrometers apart, without any external electromagnetic field...

. Casimir calculated that between two plates only particles with specific wavelengths should be counted when calculating the vacuum energy
Vacuum energy
Vacuum energy is an underlying background energy that exists in space even when the space is devoid of matter . The concept of vacuum energy has been deduced from the concept of virtual particles, which is itself derived from the energy-time uncertainty principle...

. Therefore the energy density between the plates is less if the plates are close together, leading to a net attractive force between the plates. However, the conceptual framework of this effect is very different from the theory of Fatio and Le Sage.

## Recent activity

The re-examination of Le Sage's theory in the 19th century identified several closely interconnected problems with the theory. These relate to excessive heating, frictional drag, shielding, and gravitational aberration. The recognition of these problems, in conjunction with a general shift away from mechanical based theories, resulted in a progressive loss of interest in Le Sage’s theory. Ultimately in the 20th century Le Sage’s theory was eclipsed by Einstein’s theory of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

.

In 1965 Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

examined the Fatio/Lesage mechanism, primarily as an example of an attempt to explain a "complicated" physical law (in this case, Newton's inverse-square law of gravity) in terms of simpler primitive operations without the use of complex mathematics, and also as an example of a failed theory. He notes that the mechanism of "bouncing particles" reproduces the inverse-square force law and that "the strangeness of the mathematical relation will be very much reduced", but then notes that the scheme "does not work", because of the drag it predicts would be experienced by moving bodies, "so that is the end of that theory".

Although it is not regarded as a viable theory within the mainstream scientific community, there are occasional attempts to re-habilitate the theory outside the mainstream, including those of Radzievskii and Kagalnikova (1960), Shneiderov (1961), Buonomano and Engels (1976), Adamut (1982), Jaakkola (1996), Tom Van Flandern
Tom Van Flandern
Thomas C Van Flandern was an American astronomer and author specializing in celestial mechanics. Van Flandern had a career as a professional scientist, but was noted as an outspoken proponent of non-mainstream views related to astronomy, physics, and extra-terrestrial life. He also published the...

(1999), and Edwards (2007)

A variety of Le Sage models and related topics are discussed in Edwards, et al.