If we could rely implicitly upon this number we might make it determine for us the distance of the sun through the law of gravitation as follows: It was suggested in [§ 38] that Newton proved Kepler's three laws to be imperfect corollaries from the law of gravitation, requiring a little amendment to make them strictly correct, and below we give in the form of an equation Kepler's statement of the Third Law together with Newton's amendment of it. In these equations—
T = Periodic time of any planet;
a = One half the major axis of its orbit;
m = Its mass;
M = The mass of the sun;
k = The gravitation constant corresponding to the particular set of units in which T, a, m, and M are expressed.
(Kepler) a3/T2 = h; (Newton) a3/T2 = k (M + m).
Kepler's idea was: For every planet which moves around the sun, a3 divided by T2 always gives the same quotient, h; and he did not concern himself with the significance of this quotient further than to note that if the particular a and T which belong to any planet—e. g., the earth—be taken as the units of length and time, then the quotient will be 1. Newton, on the other hand, attached a meaning to the quotient, and showed that it is equal to the product obtained by multiplying the sum of the two masses, planet and sun, by a number which is always the same when we are dealing with the action of gravitation, whether it be between the sun and planet, or between moon and earth, or between the earth and a roast of beef in the butcher's scales, provided only that we use always the same units with which to measure times, distances, and masses.
Numerically, Newton's correction to Kepler's Third Law does not amount to much in the motion of the planets. Jupiter, which shows the greatest effect, makes the circuit of his orbit in 4,333 days instead of 4,335, which it would require if Kepler's law were strictly true. But in another respect the change is of the utmost importance, since it enables us to extend Kepler's law, which relates solely to the sun and its planets, to other attracting bodies, such as the earth, moon, and stars. Thus for the moon's motion around the earth we write—
(240,000)3/(27.32)2 = k (1 + 1/81),