In this great work Newton propounds the principle that “every particle of matter in the universe is attracted by, or gravitates to, every other particle of matter with a force inversely proportional to the squares of their distances.”  From the second law of Kepler, namely, the proportionality of the areas to the times of their description, Newton inferred that the force which keeps a planet 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 quantities of matter.

Having established this universal law, Newton was able not only to determine the weight which the same body would have at the surface of the sun and the planets, but even to calculate the quantity of matter in the sun and in all the planets that had satellites, and also to determine their density or specific gravity.