228 THE MATHEMATICAL PRINCIPLES [BOOK I. distance PE or PF, conjunctly. But (by the last Lemma) Dd is to F/ as PE to PS, and therefore F/ . is equal to PE F/ is equal to Dd X ; and DE2 X DE2 X PS PET~; and therefore the force of the laDE2 X PS mina EF/e is as Do? X PT?~ and the force of a particle exerted at the distance PF conjunctly ; that is, by the supposition, as DN X D(/7 or as the evanescent area DNwrf. Therefore the forces of all the lamina) exerted upon the corpuscle P are as all the areas DN//G?, that is, the whole force of the sphere will be as the whole area ANB. Q.E.D. COR. 1. Hence if the certripetal force tending to the several particles •p)F2 vx po remain always the same at all distances, and DN be made as ™ ; Jr Jli the whole force with which the corpuscle is attracted by the sphere is as the area ANB. COR. 2. If the centripetal force of the particles be reciprocally as the DE2 X PS distance of the corpuscle attracted by it, and DN be made as -— ^^ , the force with which the corpuscle P is attracted by the whole sphere wil] be as the area ANB. Cor. 3. Jf the centripetal force of the particles be reciprocally as the cube of the distance of the corpuscle attracted by it, and DN be made as T)F2 y PS — --- — — . the force with which the corpuscle is attracted by the whole sphere will be as the area ANB. COR. 4. And universally if the centripetal force tending to the several particles of the sphere be supposed to be reciprocally as the quantity V ; DE2 X PS and D5& be made as „„ — ^- ; the force with which a corpuscle is atJr Jtj X ' tracted by the whole sphere will be as the area ANB. PROPOSITION LXXXI. PROBLEM XLI. T/Le things remaining as above, it is required lo measure the area ANB. From the point P let there be drawn the right line PH touching the sphere in H ; and to the axis PAB, letting fall the perpendicular HI, bisect PI in L; and (by Prop. XII, Book II, El em.) PE2 is equal tf