| r3 | = | V | M ; |
| T2 | 4π2 |
or, in words, the cube of the distance divided by the square of the periodic time for every planet or satellite of the system under consideration, will be constant and proportional to the mass of the central body.
This is Kepler's third law, with a notable addition. It is stated above for circular motion only, so as to avoid geometrical difficulties, but even so it is very instructive. The reason of the proportion between r3 and T2 is at once manifest; and as soon as the constant V became known, the mass of the central body, the sun in the case of a planet, the earth in the case of the moon, Jupiter in the case of his satellites, was at once determined.
Newton's reasoning at this time might, however, be better displayed perhaps by altering the order of the steps a little, as thus:—
The centrifugal force of a body is proportional to r3/T2, but by Kepler's third law r3/T2 is constant for all the planets, reckoning r from the sun. Hence the centripetal force needed to hold in all the planets will be a single force emanating from the sun and varying inversely with the square of the distance from that body.
Such a force is at once necessary and sufficient. Such a force would explain the motion of the planets.
But then all this proceeds on a wrong assumption—that the planetary motion is circular. Will it hold for elliptic orbits? Will an inverse square law of force keep a body moving in an elliptic orbit about the sun in one focus? This is a far more difficult question. Newton solved it, but I do not believe that even he could have solved it, except that he had at his disposal two mathematical engines of great power—the Cartesian method of treating geometry, and his own method of Fluxions. One can explain the elliptic motion now mathematically, but hardly otherwise; and I must be content to state that the double fact is true—viz., that an inverse square law will move the body in an ellipse or other conic section with the sun in one focus, and that if a body so moves it must be acted on by an inverse square law.
Fig. 59.