The Epicurean Propositions

Section 11

The Epicureans had the following propositions:

1. The atoms which pass to one side of any central body contribute nothing to the gravity which it exercises towards 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.

• Their gravitational effect is inversely proportional to the square of the distance of the attracted body from the central body.

2. The gravitational atoms are directed towards:

• the centers of the greater bodies and
• each of their particles as well

This is because they move indiscriminately in all directions in space.

The atoms, moreover, act effectively in those directions in which their antagonists are intercepted – in all directions in which there are matter particles.

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 towards 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.

Section 12

The proposed that gravitation is mutual or reciprocal.

• Gravity is subject to the ancient law of mechanics, which states that action and reaction are equal.

This proposition therefore forms a gradation between those which I have established by the first method and those which I shall establish by the second.

Method 1

One body is pushed toward another by the atoms which the second body has deprived of direct antagonists.

• The latter body is pushed towards the former by these same antagonists.
• The 2 bodies are necessarily pushed towards each other with equal force, whatever be the inequality of their masses or the differences in their forms.

Method 2

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.

The impetus or momentum of the first body is the summation of the impetus of its separate particles.

• Then it is proportional to the total mass of the first body.

Thus the impetus of the first body is proportional to the product of the masses of the two bodies.

Likewise, the impetus of the second body is also proportional to this product.

Therefore the usual bodies are urged together with equal forces.

Section 13

The ancients thus have the principle of a mutual gravitation directly proportional to the masses and varying inversely as the square of the distance.

The mechanical cause may be left out of consideration in the discussion.

These philosophers would have foreseen many difficulties in rigorously testing every consequence to see if it coincided exactly with observation.

• They would have refrained from embarking on so serious a task before their deductions accorded with the results of experience.

This is why they did not apply geometry and computation to this gravitation. They would have to first determine it by simple reasonings. Then approximate the effects flowing from it.

Having fewer matters than we to distract their attention, they were able to make very exact deductions in subjects requiring only meditation.*

The theory of conic sections would be needed in such reasonings. This had been discovered and cultivated before the birth of Epicurus.

Archimedes had made great advances in the doctrine of centers of gravity.

The ancient geometers, especially Archimedes, employed approximations with great ingenuity when they were unable to attain precision.

Section 14

Encouraged by these first successes and animated by the grandeur of the enterprise, these ardent and subtile geometers[13] would have invented a way to pass:

• from the ratio of sensible quantities to that of their imperceptible elements
• conversely from elementary quantities to their summation

This is 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. [149]

Section 15

The bare the theory of central forces can give the physicist, by the aid of lemmas, less exact and universal astronomical tables than those of the calculus.