Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/377

to each other is as $\scriptstyle PS\times IE\times IE^{n}$ to $\scriptstyle IS\times PE\times PE^{n}$. Becauſe SI, SE, SP are in continued proportion, the triangles SPE, SEI are alike; and thence IE is to PE as IS to SE or SA. For the ratio of IE to PE write the ratio of IS to SA; and the ratio of the ordinates becomes that of $\scriptstyle PS\times IE^{n}$ to $\scriptstyle SA\times IE^{n}$. But the ratio of PS to SA is ſubduplicate of that of the diſtances PS, SI; and the ratio of $\scriptstyle IE^{n}$ to $\scriptstyle PE^{n}$ (becauſe IE is to PE as IS to SA) is ſubduplicate of that of the forces at the diſtances PS, IS. Therefore the ordinates, and conſequently the areas which the ordinates deſcribe, and the attractions proportional to them, are in a ratio compounded of thoſe ſubduplicate ratio's. Q. E. D.

Proposition LXXXIII. Problem XLII.

To find the force with which a corpuſcle placed in the centre of ſphere is attracted towards any ſegment of that ſphere whatſover.

Plate 23, Figure 5

Let P (Pl. 23. Fig. 5.) be a body in the centre of that ſphere, and RBSD a ſegment thereof contained under the plane RDS and the ſphærical ſuperficies RBS. Let DB be cut in F by a ſphærical ſuperficies EFG deſcribed from the centre P, and let the ſegment be divided into the parts BREFGS, FEDG. Let us ſuppoſe that ſegment to be not a purely mathematical, but a phyſical ſuperficies, having ſome, but a perfectly inconſiderable thickneſs. Let that thickneſs be called O