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.

**PROPOSITION 76. THEOREM
36**

*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.

**[End of Book 1. Section 1. Proposition 75-76]**