Case 3. Imagine another ſphere compoſed of innumerable corpuſcles P; and becauſe the force with which every corpuſcle is attracted is as the diſtance of the corpuſcle from the centre of the firſt ſphere, and as the ſame ſphere conjunctly, and is therefore the ſame as if it all proceeded from a ſingle corpuſcle ſituate in the centre of the ſphere; the entire force with which all the corpuſcles in the ſecond ſphere are attracted, that is, with which that whole ſphere is attracted, will be the ſame as if that ſphere were attracted by a force iſſuing from a ſingle corpuſcle in the centre of the firſt ſphere; and is therefore proportional to the diſtance between the centres of the ſpheres. Q. E. D.
Case 4. Let the ſpheres attract each other mutually, and the force will be doubled. but the proportion will remain. Q. E. D.
Case 5. Let the corpuſcle be placed within the ſphere AEBF; (Fig. 3.) and becauſe the force of the plane ef upon the corpuſcle is as the ſolid contained under that plane and the diſtance pg; and the contrary force of the plane EF as the ſolid contained under that plane and the diſtance pG; the force compounded of both will be as the difference of the ſolids, that is as the ſum of the equal planes drawn into half the difference of the diſtance that is, as that ſum drawn into PS, the diſtance of the corpuſcle from the centre of the ſphere. And by a like reaſoning, the attraction of all the planes EF, ef throughout the whole ſphere, that is, the attraction of the whole ſphere, is conjunctly as the ſum of all the planes, or as the whole ſpheres and as pS, the diſtance of the corpuſcle from the centre of the ſphere. Q. E. D.
