ults to
which they lead, and omit the details of the reasoning. Newton first
took the law which asserted that the planet moved over equal areas in
equal times, and he showed by unimpeachable logic that this at once gave
the direction in which the force acted on the planet. He showed that the
imaginary rope by which the planet is controlled must be invariably
directed towards the sun. In other words, the force exerted on each
planet was at all times pointed from the planet towards the sun.

It still remained to explain the intensity of the force, and to show how
the intensity of that force varied when the planet was at different
points of its path. Kepler's first law enables this question to be
answered. If the planet's path be elliptic, and if the force be always
directed towards the sun at one focus of that ellipse, then mathematical
analysis obliges us to say that the intensity of the force must vary
inversely as the square of the distance from the planet to the sun.

The movements of the planets, in conformity with Kepler's laws, would
thus be accounted for even in their minutest details, if we admit that
an attractive power draws the planet towards the sun, and that the
intensity of this attraction varies inversely as the square of the
distance. Can we hesitate to say that such an attraction does exist? We
have seen how the earth attracts a falling body; we have seen how the
earth's attraction extends to the moon, and explains the revolution of
the moon around the earth. We have now learned that the movement of the
planets round the sun can also be explained as a consequence of this law
of attraction. But the evidence in support of the law of universal
gravitation is, in truth, much stronger than any we have yet presented.
We shall have occasion to dwell on this matter further on. We shall show
not only how the sun attracts the planets, but how the planets attract
each other; and we shall find how this mutual attraction of the planets
has led to remarkable discoveries, which have elevated the law of
gravitation beyond the possibility of doubt.

Admitting the existence of this law, we can show that the planets must
revolve around the sun in elliptic paths with the sun in the common
focus. We can show that they must sweep over equal areas in equal times.
We can prove that the squares of the periodic times must be proportional
to the cubes of their mean distances. Still further, we can show how the
mysterious movemen