From the first law of Kepler, namely, the proportionality of the areas to the times of their description, Newton inferred that the force which retained the planet in its orbit was always directed to the sun. From the second, namely, that every planet moves in an ellipse with the sun as one of foci, he drew the more general inference that the force by which the planet moves round that focus varies inversely as the square of its distance therefrom. He demonstrated that a planet acted upon by such a force could not move in any other curve than a conic section; and he showed when the moving body would describe a circular, an elliptical, a parabolic, or hyperbolic orbit. He demonstrated, too, that this force or attracting, gravitating power resided in even the least particle; but that in spherical masses it operates as if confined to their centres, so that one sphere or body will act upon another sphere or body with a force directly proportional to the quantity of matter and inversely as the square of the distance between their centres, and that their velocities of mutual approach will be in the inverse ratio of their quantities of matter. Thus he outlined the universal law.
The System of the World
It was the ancient opinion of not a few (writes Newton in Book III.) in the earliest ages of philosophy that the fixed stars stood immovable in the highest parts of the world; that under the fixed stars the planets were carried about the sun; that the earth, as one of the planets, described an annual course about the sun, while, by a diurnal motion, it was in the meantime revolved about its own axis; and that the sun, as the common fire which served to warm the whole, was fixed in the centre of the universe. It was from the Egyptians that the Greeks derived their first, as well as their soundest notions of philosophy. It is not to be denied that Anaxagoras, Democritus and others would have it that the earth possessed the centre of the world, but it was agreed on both sides that the motions of the celestial bodies were performed in spaces altogether free and void of resistance. The whim of solid orbs was[1] of later date, introduced by Endoxus, Calippus and Aristotle, when the ancient philosophy began to decline.
As it was the unavoidable consequence of the hypothesis of solid orbs while it prevailed that the comets must be thrust down below the moon, so no sooner had the late observations of astronomers restored the comets to their ancient places in the higher heavens than these celestial spaces were at once cleared of the encumbrance of solid orbs, which by these observations were broken to pieces and discarded for ever.
Whence it was that the planets came to be retained within any certain bounds in these free spaces, and to be drawn off from the rectilinear courses, which, left to themselves, they should have pursued, into regular revolutions in curvilinear orbits, are questions which we do not know how the ancients explained; and probably it was to give some sort of satisfaction to this difficulty that solid orbs were introduced.
The later philosophers pretend to account for it either by the action of certain vortices, as Kepler and Descartes, or by some other principle of impulse or attraction, for it is most certain that these effects must proceed from the action of some force or other. This we will call by the general name of a centripetal force, as it is a force which is directed to some centre; and, as it regards more particularly a body in that centre, we call it circum-solar, circum-terrestrial, circum-jovial.
Centre-Seeking Forces
That by means of centripetal forces the planets may be retained in certain orbits we may easily understand if we consider the motions of projectiles, for a stone projected is by the pressure of its own weight forced out of the rectilinear path, which, by the projection alone, it should have pursued, and made to describe a curve line in the air; and through that crooked way is at last brought down to the ground, and the greater the velocity is with which it is projected the further it goes before it falls to earth. We can, therefore, suppose the velocity to be so increased that it would describe an arc of 1, 2, 5, 10, 100, 1,000 miles before it arrived at the earth, till, at last, exceeding the limits of the earth, it should pass quite by it without touching it.
And because the celestial motions are scarcely retarded by the little or no resistance of the spaces in which they are performed, to keep up the parity of cases, let us suppose either that there is no air about the earth or, at least, that it is endowed with little or no power of resisting.