SEC. Xll.] OP NATURAL PHILOSOPHY. 225 were attracted by a force issuing from a single corpuscle in the centre of the first sphere ; and is therefore proportional to the distance between the centres of the spheres. Q,.E.D. CASE 4. Let the spheres attract each other mutually, and the force will be doubled, but the proportion will remain. Q..E.D. CASE 5. Let the corpuscle p be placed within ^- ^\E the sphere AEBF ; and because the force of the plane ef upon the corpuscle is as the solid contain ed under that plane and the distance jog' ; and the contrary force of the plane EF as the solid con tained under that plane and the distance joG ; the ^ force compounded of both will be as the difference ** of the solids, that is, as the sum of the equal planes drawn into half the difference of the distances ; that is, as that sum drawn into joS, the distance of the corpuscle from the centre of the sphere. And, by a like reasoning, the attraction of all the planes EF, ef, throughout the whole sphere, that is, the attraction of the whole sphere, is conjunctly as the sum of all the planes, or as the whole sphere, and as joS, the distance of the corpuscle from the centre of the sphere. Q.E.D. CASE 6. And if there be composed a new sphere out of innumerable cor puscles such as jo, situate within the first sphere AEBF, it may be proved, as before, that the attraction, whether single of one sphere towards the other, or mutual of both towards each other, will be as the distance joS of the centres. Q, E.D. PROPOSITION LXXVIII. THEOREM XXXVIII. If spheres it* the progress from the centre to the circumference be hoivMtv dissimilar a->id unequable, but similar on every side round about af all given distances from the centre ; and the attractive force of evsrt/ point be as the distance of the attracted body ; I say, that the entire force with which two spheres of this kind attract each other mutitallij is proportional to the distance between the centres of the spheres. This is demonstrated from the foregoing Proposition, in the same man ner as Proposition LXXVI was demonstrated from Proposition LXXY. COR. Those things that were above demonstrated in Prop. X and LXJV, of the motion of bodies round the centres of conic sections, take place when all the attractions are made by the force of sphaerical bodies of the condi tion above described, and the attracted bodies are spheres of the same kind. SCHOLIUM. i have now explained the two principal cases of attractions; to wit, when the centripetal forces decrease in a duplicate ratio of the distances •r increase in a simple ratio of the distances, causing the bodies in botli 15