Having further proved the, for that day, wonderful proposition that,
with the law of inverse squares, the attraction by the separate
particles of a sphere of uniform density (or one composed of
concentric spherical shells, each of uniform density) acts as if the
whole mass were collected at the centre, he was able to express the
meaning of Kepler's laws in propositions which have been summarised as
follows:--

The law of universal gravitation.--_Every particle of matter in the
universe attracts every other particle with a force varying inversely
as the square of the distance between them, and directly as the
product of the masses of the two particles_.[2]

But Newton did not commit himself to the law until he had answered
that question about the apple; and the above proposition now enabled
him to deal with the Moon and the apple. Gravity makes a stone fall
16.1 feet in a second. The moon is 60 times farther from the earth's
centre than the stone, so it ought to be drawn out of a straight
course through 16.1 feet in a minute. Newton found the distance
through which she is actually drawn as a fraction of the earth's
diameter. But when he first examined this matter he proceeded to use
a wrong diameter for the earth, and he found a serious discrepancy.
This, for a time, seemed to condemn his theory, and regretfully he
laid that part of his work aside. Fortunately, before Newton wrote the
_Principia_ the French astronomer Picard made a new and correct
measure of an arc of the meridian, from which he obtained an accurate
value of the earth's diameter. Newton applied this value, and found,
to his great joy, that when the distance of the moon is 60 times the
radius of the earth she is attracted out of the straight course 16.1