et in its orbit is always
directed to the sun. From the first law of Kepler, that every planet
moves in an ellipse with the sun in one of its foci, he drew the still
more general inference that the force by which the planet moves round
that focus varies inversely as the square of its distance from the
focus. From the third law of Kepler, which connects the distances and
periods of the planets by a general rule, Newton deduced the equality of
gravity in them all towards the sun, modified only by their different
distances from its centre; and in the case of terrestrial bodies, he
succeeded in verifying the equality of action by numerous and accurate
experiments.

By taking a more general view of the subject, Newton showed that a conic
section was the only curve in which a body could move when acted upon by
a force varying inversely as the square of the distance; and he
established the conditions depending on the velocity and the primitive
position of the body which were requisite to make it describe a
circular, an elliptical, a parabolic, or a hyperbolic orbit.

It still remained to show whether the force resided in the centre of
planets or in their individual particles; and Newton demonstrated that
if a spherical body acts upon a distant body with a force varying as the
distance of this body from the centre of the sphere, the same effect
will be produced as if each of its particles acted upon the distant body
according to the same law.

Hence it follows that the spheres, whether they are of uniform density,
or consist of concentric layers of varying densities, will act upon each
other in the same manner as if their force resided in their centres
alone. But as the bodies of the solar system are nearly spherical, they
will all act upon one another and upon bodies placed on their surface,
as if they were so many centres of attraction; and therefore we obtain
the law of gravity, that one sphere will act upon another sphere with a
force directly proportional to the quantity of matter, and inversely as
the square of the distance between the centres of the spheres. From the
equality of action and reaction, to which no exception can be found,
Newton concluded that the sun gravitates to the planets and the planets
to their satellites, and the earth itself to the stone which falls upon
its surface, and consequently that the two mutually gravitating bodies
approach one another with velocities inversely proportional to their
quan