Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/362

276

Imagine another phere compoed of innumerable corpucles P; and becaue the force with which every corpucle is attracted is as the ditance of the corpucle from the centre of the firt phere, and as the ame phere conjunctly, and is therefore the ame as if it all proceeded from a ingle corpucle ituate in the centre of the phere; the entire force with which all the corpucles in the econd phere are attracted, that is, with which that whole phere is attracted, will be the ame as if that phere were attracted by a force iuing from a ingle corpucle in the centre of the firt phere; and is therefore proportional to the ditance between the centres of the pheres. Q. E. D.

Let the pheres attract each other mutually, and the force will be doubled. but the proportion will remain. Q. E. D.



Let the corpucle be placed within the phere AEBF; (Fig. 3.) and becaue the force of the plane ef upon the corpucle is as the olid contained under that plane and the ditance pg; and the contrary force of the plane EF as the olid contained under that plane and the ditance pG; the force compounded of both will be as the difference of the olids, that is as the um of the equal planes drawn into half the difference of the ditance that is, as that um drawn into PS, the ditance of the corpucle from the centre of the phere. And by a like reaoning, the attraction of all the planes EF, ef throughout the whole phere, that is, the attraction of the whole phere, is conjunctly as the um of all the planes, or as the whole pheres and as pS, the ditance of the corpucle from the centre of the phere. Q. E. D.