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

Rh to each other is as $$\scriptstyle PS \times IE \times IE^n$$ to $$\scriptstyle IS \times PE \times PE^n$$. Becaue 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 ubduplicate of that of the ditances PS, SI; and the ratio of $$\scriptstyle IE^n$$ to $$\scriptstyle PE^n$$ (becaue IE is to PE as IS to SA) is ubduplicate of that of the forces at the ditances PS, IS. Therefore the ordinates, and conequently the areas which the ordinates decribe, and the attractions proportional to them, are in a ratio compounded of thoe ubduplicate ratio's. Q. E. D.

To find the force with which a corpucle placed in the centre of phere is attracted towards any egment of that phere whatover.

Let P (Pl. 23. Fig. 5.) be a body in the centre of that phere, and RBSD a egment thereof contained under the plane RDS and the phærical uperficies RBS. Let DB be cut in F by a phærical uperficies EFG decribed from the centre P, and let the egment be divided into the parts BREFGS, FEDG. Let us uppoe that egment to be not a purely mathematical, but a phyical uperficies, having ome, but a perfectly inconiderable thicknes. Let that thicknes be called O