Laplaces Response and Implications for Celestial Mechanics

Pierre-Simon Laplace (1749-1827) was the foremost mathematical physicist of his generation. He is best known for his Celestial Mechanics, which appeared in five volumes between 1799 and 1825 and which put this science on a new and more systematic foundation. Although Laplace never gave Le Sage's theory serious consideration, he was influenced by it to explore two effects that would represent departures from the Newtonian theory: a finite speed of propagation of gravity and a resistive force experienced by the planets in their orbits. Le Sage's friend and compatriot, Jean-André Deluc, was in Paris in 1781, where he was on friendly terms with Laplace and tried to interest him in Le Sage's theory. Laplace declined to be draw in, but did admit to an interest in exploring the resistive force implicit in Le Sage's sea of corpuscles:

Before pronouncing on this subject, I have taken the course of waiting until M. Sage has published his ideas; and then I propose to pursue certain analytical researches that they have suggested to me. As for the particular subject of the secular equations of the motion of the planets, it has appeared to me that the smallness of that of the Earth would imply in the gravific fluid a speed incomparably greater than that of light, and all the more considerable as the Sun and the Earth leave a freer passage to this fluid, which conforms to the result of M. Sage. This prodigious speed, the immense space that each fluid molecule traverses in only a century, without our knowing where it comes from or where it goes or the cause that has put it into motion—all that is quite capable of terrifying our weak imagination; but in the end, if one absolutely wants a mechanical cause of weight, it appears to me difficult to imagine one which explains it more happily than the hypothesis of M. Sage 50

Indeed, it is not difficult to show that Le Sage's hypothesis leads to an effective attractive force Fatt that conforms to Newton's law of gravitation. If M1 and M2 are the masses of two infinitesimal bodies, the force that one exerts upon the other is

r where r is the distance between the bodies and k is a constant that depends upon properties of the sea of ultramundane corpuscles. It turns out that k, the constant of universal gravitation, is given by k = ™v2 f \ 4n where m is the mass of a single ultramundane corpuscle, n is the number of corpuscles per unit volume of space, v is the speed of the corpuscles (assumed for simplicity to be the same for all) and f is a (presumably universal) constant with dimensions of area/mass. f is the cross-sectional area for collision presented to the corpuscles by a macroscopic object of unit mass. (Here it must be emphasized that Le Sage himself never published such formulas.)

As we have seen, in Le Sage's system, apparently solid objects must be made mostly of empty space. In his Mechanical Physics, Le Sage speculated that the atoms of ordinary matter are like "cages"—that is, they take up lots of space, but are mostly empty. In this way, ordinary objects block only a tiny fraction of the ultramundane corpuscles that are incident upon them. Otherwise, as Le Sage himself points out, merchants could change the weights of their stuff by arranging it in wide, thin layers (in which case it would weigh more) or in tall piles (in which case it would weigh less). More significantly for precision measurement, the gravitational attraction of the Moon toward the Earth would be diminished during a lunar eclipse because of the interposition of the Earth between the Sun and the Moon, a phenomenon that has never been noticed by the astronomers. Thus, in order to have a theory consistent with the phenomena, f must be so small that even planet-sized objects absorb a negligible fraction of the corpuscles incident upon them. The constant of universal gravitation k can be made to agree with the facts no matter how small we make f, provided that we suitably increase n or v.

There is an unwanted side effect in Le Sage's system, to which Laplace refers. Planets traveling through the sea of corpuscles will be slightly retarded. This is the effect to which Laplace refers under the rubric of secular equations. A body of mass M1 moving through the sea of ultramundane corpuscles will experience a resistive force Fres that is proportional to the speed u of the body:

The resistive force is directed oppositely to the body's velocity u. (Again, Le Sage himself did not publish such a formula.) Since no resistance of this kind had been detected, it was necessary to insist that the resistive force suffered by a planet be much smaller than the attractive force exerted by the Sun on the planet. Thus, if we let Mi denote the mass of a planet and M2 that of the Sun, we require

res I att

Upon substitution of the expressions for the forces, we obtain uJL << 1

M2, u and r (the mass of the Sun, the speed of the planet and the radius of the planet's orbit) are not adjustable. Thus we are led to the conclusion that the speed v of the corpuscles must be very large. Moreover, since we need f to be very small, we are forced to make v even greater. This is what Laplace meant when he said that the smallness of the secular equation of the Earth "would imply in the gravific fluid a speed incomparably greater than that of light, and all the more considerable as the Sun and the Earth leave a freer passage to this fluid." The prodigious velocity required for Le Sage's corpuscles appears, more than many other features of the theory, to have repelled Laplace.

Some years later, after the publication of Laplace's Exposition of the System of the World, Le Sage wrote to express his disappointment that Laplace had not discussed his mechanical theory of gravity. Laplace's reply drew a clear boundary between his generation's way of doing physics and the old mechanical philosophy espoused by Le Sage:

If I have not spoken in my work of your explanation of the principal of universal weight, it is because I wanted to avoid everything that might appear to be based upon a system. Among philosophers, some conceive of the action of bodies upon one another only by means of impulsion and, to them, action at a distance seems impossible. Your ingenious manner of explaining universal gravitation, in proportion to the masses and in reciprocal proportion to the square of the distances, should satisfy these philosophers and bring them to admit this great law of nature, which they would reject despite the observations and all the calculations of the geometers, if it were well demonstrated to them that it could result from impulsion.

Other philosophers, on the contrary, admit their ignorance on the nature of matter, of space, of force and of extension, and trouble themselves little about first causes, seeing in attraction only a general phenomenon which, being subjectable to a rigorous calculation, gives the complete explanation of all the celestial phenomena and the means of perfecting the tables and the theory of the motion of the stars. It is uniquely under this point of view that I have envisaged attraction in my work.

Perhaps I have not had enough consideration for the first philosophers of whom I have just spoken, in not presenting to them your manner, as simple as ingenious, of bringing the principle of weight back to the laws of impulsion; but this is a thing that you have done in a manner leaving nothing to be desired in this regard. However, I propose to calculate in my Treatise on Celestial Mechanics the deteriorations that must result from your hypotheses in the long run in the mean motions and the orbits of the planets and the satel-lites.51

Laplace did, indeed, include calculations in the fifth volume of his Celestial Mechanics which grappled with some of the consequences of Le Sage's theory. But even here Laplace did not see fit to mention Le Sage by name.52

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