and the force of a particle exerted at the distance PF conjunctly; that is, by the supposition, as DN × Dd, or as the evanescent area DNnd. Therefore the forces of all the laminæ exerted upon the corpuscle P are as all the areas DNnd, that is, the whole force of the sphere will be as the whole area ANB. Q.E.D. COR. 1. Hence if the centripetal force tending to the several particles remain always the same at all distances, and DN be made as , 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 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 will be as the area ANB. COR. 3. If 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 , 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; and DN be made as ; the force with which a corpuscle is attracted by the whole sphere will be as the area ANB. PROPOSITION LXXXI. PROBLEM XLI. The things remaining as above, it is required to 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, [Pg 229]bisect PI in L; and (by Prop. XII, Book II, Elem.) PE2 is equal to PS2 + SE2 + 2PSD. But because the triangles SPH, SHI are alike, SE2 or SH2 is equal to the rectangle PSI, Therefore PE2 is equal to the rectangle contained under PS and PS + SI + 2SD; that is, under PS and 2LS + 2SD; that is, under PS and 2LD. Moreover DE2 is equal to SE2 - SD2, or SE2 - LS2 + 2SLD - LD2, that is, 2SLD - LD2 - ALB. For LS2 - SE2 or LS2 - SA2 (by Prop. VI, Book II, Elem.) is equal to the rectangle ALB. Therefore if instead of DE2 we write 2SLD - LD2 - ALB, the quantity , which (by Cor. 4 of the foregoing Prop.) is as the length of the ordinate DN, will now resolve itself into three parts ; where if instead of V we write the inverse ratio of the centripetal force, and instead of PE the mean proportional between PS and 2LD, those three parts will become ordinates to so many curve lines, whose areas are discovered by the common methods. Q.E.D.