For the attraction of every particle is inversely as the square of its distance from the centre of the attracting sphere (by Prop. 74), and is therefore the same as if that whole attracting force issued from one single corpuscle placed in the centre of this sphere. But this attraction is as great as on the other hand the attraction of the same corpuscle would be, if that were itself attracted by the several particles of the attracted sphere with the same force with which they are attracted by it. But that attraction of the corpuscle would be (by Prop. 74) inversely proportional to the square of its distance from the centre of the sphere; therefore the attraction of the sphere, equal thereto, is also in the same ratio.—Q.E.D.

COR. I. The attractions of spheres towards other homogeneous spheres are as the attracting spheres applied to the squares of the distances of their centres from the centres of those which they attract.

COR. II. The case is the same when the attracted sphere does also attract. For the several points of the one attract the several points of the other with the same force with which they themselves are attracted by the others again; and therefore since in all attractions (by Law III) the attracted and attracting point are both equally acted on, the force will be doubled by their mutual attractions, the proportions remaining.

COR. III. Those several truths demonstrated above concerning the motion of bodies about the focus of the conic sections will take place when an attracting spheres is placed in the focus, and the bodies move without the sphere.

COR. IV. Those things which were demonstrated before of the motion of bodies about the centre of the conic sections take place when the motions are performed within the sphere.

If spheres be however dissimilar (as to density of matter and attractive force) in the same ratio onward from the center to the circumference; but everywhere similar at every given distance from the center, on all sides round about; and the attractive force of every point decreases as the square of the distance of the body attracted: I say that the whole force with which one of these spheres attracts the other will be inversely proportional to the square of the distance of the centres.

Imagine several concentric similar spheres AB, CD, EF, &c., the innermost of which, added to the outermost may compose a matter more dense towards the centre, or subtracted from them may leave the same more lax and rare. Then, by Prop. 75, these spheres will attract other similar concentric spheres GH, IK, LM, &c., each the other, with forces inversely proportional to the square of the distance SP. And, by addition or subtraction, the sum of all those forces, or the excess of any of them above the others; that is, the entire force with which the whole sphere AB (composed of any concentric spheres or of their differences) will attract the whole sphere GH (composed of any concentric spheres or their differences) in the same ratio. Let the number of the concentric spheres be increased in infinitum, so that the density of the matter together with the attractive force may, in the progress from the circumference to the centre, increase or decrease according to any given law; and by the addition of matter not attractive, let the deficient density be supplied, that so the spheres may acquire any form desired; and the force with which one of these attracts the other will be still, by the former reasoning, in the same inverse ratio of the square of the distance. Q.E.D.

COR. I. Hence if many spheres of this kind, similar in all respects, attract each other, the accelerative attractions of each to each, at any equal distances of the centres, will be as the attracting spheres.

COR. II. And at any unequal distances, as the attracting spheres divided by the squares of the distances between the centres.

COR. III. The motive attractions, or the weights of the spheres toward one another, will be at equal distances of the centers conjointly as the attracting and attracted spheres; that is, as the products arising from multiplying the spheres into each other.

COR. IV. And at unequal distances directly as those products and inversely as the squares of the distances between the centres.

COR. V. These proportions hold true also when the attraction arises from the attractive power of both spheres exerted upon each other. For the attraction is only doubled by the conjunction of the forces, the proportions remaining as before.

COR. VI. If spheres of this kind revolve about others at rest, each about each, and the distances between the centres of the quiescent and revolving bodies are proportional to the diameters of the quiescent bodies, the periodic times will be equal.

COR. VII. And, again, if the periodic times are equal, the distances will be proportional to the diameters.

COR. VIII. All those truths above demonstrated, relating to the motions of bodies about the foci of conic sections, will take place when an attracting sphere, of any form and condition like that above described, is placed in the focus.

COR. IX. And also when the revolving bodies are also attracting spheres of any condition like that above described.