The Epicurean Systemby Le Sage
The Epicureans had the following propositions:
- The atoms which pass to one side of any central body contribute nothing to the gravity which it exercises toward other bodies.
This is because such atoms are exactly counterbalanced by direct antagonists.
Gravitation would be due solely to those atoms which are fortuitously directed toward the central body.
The resultant action of these atoms is everywhere directed toward the central body, like the rays of light converging toward a focus when assembled by a convex lens or a concave mirror.
Hence, it is proper to apply to them what has been proven in Paragraph 4 touching the terrestrial gravitation; that is to say, their gravitational effect is inversely proportional to the square of the distance of the attracted body from the central body.
- The gravitational atoms are directed not only toward the centers of the greater bodies, but toward each of their particles as well, since they move indiscriminately in all directions in space.
The atoms, moreover, act effectively in those directions in which their antagonists are intercepted; that is to say, in all directions in which there are particles of matter. Therefore they tend to move the heavy masses which they encounter not toward the heavenly bodies in gross, but toward each of their particles in detail.
Hence the gravitation of masses toward the center of a celestial body is nothing but the resultant of an imperceptible movement of the masses toward all parts of the great body (as, from certain passages of Cicero and Plutarch, it appears had been before supposed by some of the ancients).
Consequently this gravitation would be proportional to the number of the particles; that is to say, to the mass of the central body.
From these 2 propositions alone there might have been deduced synthetically the entire theory of universal gravitation without further mention of gravitational atoms.
- This is the place to insert a certain proposition which is commonly spoken of as if it were distinct from those which teach that gravitation is universal, but which appears to me to be included in that expression.
I refer to that which affirms that gravitation is mutual or reciprocal; or, in other words, that it is subject to the ancient law of mechanics, which states that action and reaction are equal.
I say that this is the place to consider this proposition, because it can equally well be proved either through the introduction of the agent of gravitation, as I have done in preceding paragraphs, or by considering gravitation abstractly, as I shall do in those which follow. This proposition therefore forms, as it were, a gradation between those which I have established by the first method and those which I shall establish by the second.
Inasmuch as one body is pushed toward another by the atoms which the second body has deprived of direct antagonists, while the latter body is pushed toward the former by these same antagonists, the two bodies are necessarily pushed toward each other with equal force, whatever be the inequality of their masses or the differences in their forms.
Since each particle of one of the two bodies tends toward every particle of the other, the first body is urged toward the second with a force proportional to the number of particles which the second contains, or, in other words, with a force proportional to the mass of the second.
Furthermore, since the impetus or momentum of the first body is the summation of the impetus of its separate particles, it is proportional to the total mass of the first body. Thus it follows that the impetus of the first body is proportional to the product of the masses of the two bodies.
By a similar train of reasoning the impetus of the second body is also proportional to this product. Therefore the usual bodies are urged together with equal forces.
What are the other consequences the ancients would probably have drawn from the principle of a mutual gravitation directly proportional to the masses and varying inversely as the square of the distance.
For the sake of brevity the mechanical cause may be left out of consideration in the discussion. As these philosophers would have foreseen many difficulties in rigorously testing every consequence to see if it coincided exactly with observation, and would therefore have refrained from embarking upon so serious a task before perceiving that the deductions accorded in gross with the results of experience, I presume they would not seriously have applied geometry and computation to this gravitation without having first determined by simple reasonings what, approximately, would be the effects flowing from it, and seeing that these conjectures accorded roughly with the real constitution of the universe I believe I do no violence to probabilities in presuming that the ancient philosophers would have been acquainted with some such reasonings.
Having fewer matters than we to distract their attention, they were able to make very exact deductions in subjects requiring nothing but meditation. With reference to the acquired knowledge which would be needed in such reasonings, it will be recalled that the theory of conic sections had been discovered and cultivated before the birth of Epicurus, that Archimedes had made great advance in the doctrine of centers of gravity, and that the ancient geometers, and especially the last named, employed approximations with great ingenuity when they were unable to attain to rigorous precision.
Encouraged by these first successes and animated by the grandeur of the enterprise it is highly improbable that these ardent and subtile geometers would have stopped here. They would doubtless have invented for the purpose some means for passing from the ratio of sensible quantities to that of their imperceptible elements, and conversely from elementary quantities to their summation, at least for the simple case required when one wishes to avoid the numerical computation of the small anomalies of the movements of the celestial bodies. 
Certainly they had sufficient patience and sagacity to succeed in finding such a method, since they had had enough of these qualities to discover and advance in considerable degree the admirable doctrine of incommensurables, and of exhaustions, although these were not ordinarily used except in the consideration of the five regular bodies, and were specially derived, it is said, to examine certain very hazardous and even fantastic conjectures of the Pythagoreans and Platonists.
Practically, if one omits from the theory of central forces those curious propositions and generalizations which can only be regarded as its luxuries, as well as the delicate evaluations which are required only for the perfecting of astronomical tables, all the rest may be demonstrated sufficiently for the uses of the physicist by the aid of lemmas less exact and universal than those of the calculus.
This has indeed been pointed out in some degree by several geometers, but it may be realized still further if the reader will undertake by the same or analogous means of simplification to attack other propositions than those already so treated.
But the probability that the ancients would have been able to accomplish such demonstrations is still less necessary to the plan which I have proposed to myself, as stated at the beginning of this essay, than the probability that they would have discovered the simple relations mentioned in the thirteenth paragraph. Consequently the reader may, if he prefers, ignore the last three paragraphs and give attention only to matters which I have expressly engaged to establish.
I declared that the laws of Kepler were necessary consequences of the doctrine that gravitation results from the impulsion of atoms moving in every direction, since Kepler’s laws follow directly from those of Newton. I ought, however, to show, for the benefit of readers less versed in the matter, where it may be found proved that the first-mentioned laws are the natural consequences of the second.
First. That the law of areas proportional to times is a necessary consequence of gravitation, always directed toward a single point, is demonstrated by elementary geometry in the first proposition of Newton’s Principia.
Second. That the law of squares of periodic times proportional to the cubes of the distances, for bodies appearing to describe circles, must necessarily follow from a gravitation inversely proportional to the square of the distance constitutes the second part of the sixth corollary to Proposition IV of the same work, and may be demonstrated by elementary methods also for regular polygons, which represent more nearly than exact circles the orbits traversed by bodies diverted slightly from their paths by intermittent collisions. 
Third. That the ellipticity of an orbit is the necessary consequence of gravitation directed toward its focus, and reciprocally proportional to the square of the distance, is the converse of Proposition XI of the same book. This proposition has been more simply demonstrated as a consequence of the fiftieth of Book III of the conics of Appolonius. I may pause here, since in maintaining that the laws of Kepler are an easy consequence of the system of atoms I have not pretended that their application to complex cases readily follows from the slight knowledge of geometry possessed by the ancients. Nevertheless, I may add– Fourth. That the Proposition XI of the Principia once attained it does not appear to me difficult to establish the fiftieth, which extends our second consequences to ellipses-that is to say, which proves that in ellipses as well, the squares of the periodic times about an attracting body (placed in one of the foci) are proportional to the cubes of the mean distances.
Let us now see how the laws of Galileo may be derived from the hypothesis of the impulsion of the atoms. The blows of corpuscles, moving with a velocity more rapid than light, upon a body which has fallen three or four seconds, would be sensibly of the same strength as the preceding blows had been upon the same body when it had only fallen one or two seconds. Hence the successive accelerations of the body in equal times must be sensibly equal, and the velocity at any instant must be sensibly proportional to the time elapsed since the beginning of the fall. From this it follows necessarily that the spaces traversed since the beginning are sensibly proportional to the squares of the total times, and will be sensibly proportional to the successive odd numbers.
These synthetic demonstrations of laws of falling bodies by the introduction of mechanism whose existence is only surmised, may perhaps be less philosophical than analytic demonstrations which are based entirely upon observed phenomena. Still it must be recalled that in cases where direct observation has been difficult and inexact, error has frequently attended deductions of this latter kind. At all events the former kind of demonstration is much more philosophical than a gratuitous hypothesis, which is, nevertheless, the means of invention employed by Galileo; and its results are quite as well established as are the laws of Galileo since they are proved by exactly the same means, that is by the sensible accord of their consequences with the phenomena. Nothing else than this is claimed by Galileo himself and his principal successors.
But the atomists would have encountered one very serious objection, to which they were necessarily exposed in common with all physicists who undertake an explanation of gravitation. For by having thickness a roof receives not a whit more of hail, or a shield of arrows; whereas, remaining otherwise unchanged, the weight of all bodies is augmented in direct proportion of their thickness.
Conversely when one removes a heavy body from a shop or dwelling, or reduces it to sheets exposed without protection to material influences (the rain, for example) it receives more than when protected or concentrated so as to present a small surface.
But it has never been found by merchants and artisans, who are continually in the habit of weighing, that bodies appear heavier in open air than when under cover, and gold-beaters have never perceived that the weight of the metal augments in proportion to the increase of its surface.
In a word, if the collision of atoms is the cause of heaviness, the weight of bodies ought to be proportional to their surface (or rather to their horizontal projection). How, then, does it happen that the weight is proportional to the mass!
Do the gravitational atoms then act across the thickest and most compact envelopes of all substances as fully as through the air? And  does not the very sensible weight which they impart to these envelopes demonstrate the contrary, that is that all substances arrest the passage of a great number of corpuscles?
To this the Epicureans would have been forced to respond that the atoms doubtless traverse very freely all heavy bodies; as freely, for example, as light passes through diamond and magnetic matter through gold, though one of these bodies is the hardest and the other the heaviest of all known bodies (which shows that they are less porous than most substances).
Thus the number of atoms which are intercepted by the first layers of a heavy body would be absolutely insensible relatively to the number of those which pass through the last layers.
Nevertheless, the relatively small number intercepted would produce a sensible action upon the body, since they have, in virtue of an immense velocity, the force of impact which they would lack by reason of their small mass.