As the moon’s revolution round the earth, in an orbit nearly circular, is caused by the combination of the earth’s attraction, and an original projectile impulse, so also the singular phenomena of the planets’ revolution round the sun in orbits nearly circular, must be considered an effect of the same class, as well as the revolution of the satellites of those planets which are attended by such bodies. Although the orbits in which the comets move deviate very much from circles, yet this does not hinder the application of the same principle to them, their deviation from circles not depending on the sun’s attraction, but only on the direction and force of the original impulse which put them in motion. (113.) We therefore conclude that gravitation is the principle which, as it were, animates the universe. All the great changes and revolutions of the bodies which compose our system, can be traced to or derived from this principle. It still remains to show how that remarkable law, by which this force is declared to increase or decrease in the same proportion as the square of the distance from the attracting body is decreased or increased, may be verified and established. It has been shown, that the curvilinear path of a projectile depends on, and can be derived, by mathematical reasoning, from the consideration of the intensity of the earth’s attraction, and the force of the original impulse, or the velocity of projection. In the same manner, by a reverse process, when we know the curve in which a projectile moves, we can infer the amount of the attracting force which gives the curvature to its path. In this way, from our knowledge of the curvature of the moon’s orbit, and the velocity with which she moves, the intensity of the attraction which the earth exerts upon her can be exactly ascertained. Upon comparing this with the force of gravitation at the earth’s surface, it is found that the latter is as many times greater than the former, as the square of the moon’s distance is greater than the square of the distance of a body on the surface of the earth from its centre. (114.) If this were the only fact which could be brought to establish the law of gravitation, it might be thought to be an accidental relation, not necessarily characterising the attraction of gravitation. Upon examining the orbits and velocities of the several planets, the same result is, however, obtained. It is found that the forces with which they are severally attracted by the sun are great, in exactly the same proportion as the squares of the several numbers expressing their distances are small. The mutual gravitation of bodies on the surface of the earth towards each other is lost in the predominating force exerted by the earth upon all of them. Nevertheless, in some cases, this effect has not only been observed, but actually measured.