SEC. XII.] OF NATURAL PHILOSOPHY. 2'3\. 2SI3 4 SI3 ~oj-p And these by subducting the last from the first, become -oT~r Therefore the entire force with ,7hich the corpuscle P is attracted towards the centre of the sphere is as-^, that is, reciprocally as PS3 X PJ Q.E.I. By the same method one may determine the attraction of a corpuscle situate within the sphere, but more expeditiously by the following Theorem. PROPOSITION LXXXIL THEOREM XLI. In a sphere described about the centre S with the interval SA, if there be taken SI, SA, SP continually proportional ; ! sat/, that the attraction, of a corpuscle within the sphere in any place I is to its attraction without the sphere in the place P in a ratio compounded of the subduplicate ratio of IS, PS, the distances from the centre, and the subduplicate ratio of tJie centripetal forces tending to the centre in those places P and I. As if the centripetal forces of the particles of the sphere be reciprocally ;is the distances of the corpuscle at tracted by them ; the force with which the corpuscle situate in I is attracted by the entire sphere will be to the force with which it is attracted in P in a ratio compounded of the subdu plicate ratio of the distance SI to the distance SP, and the subduplicate ratio of the centripetal force in the place I arising from any particle in the centre to the centripetal force in the place P arising from the same particle in the centre ; that is, in the subduplicate ratio of the distances SI, SP to each other reciprocally. These two subduplicate ratios compose the ratio of equality, and therefore the attractions in I and P produced by the whole sphere are equal. By the like calculation, if the forces of the particles of the sphere are reciprocally in a duplicate ratio of the distances, it will be found that the attraction in I is to the attraction in P as the distance SP to the semi -diameter SA of the sphere. If those forces are reciprocally in a triplicate ratio of the distances, the attractions in I and P will be to each other as SP2 to SA3 ; if in a quadruplicate ratio, as SP3 to SA3. There fore since the attraction in P was found in this last case to be reciprocally as PS 3 X PI, the attraction in I will be reciprocally as S A 3 X PI, that is, because S A 3 is given reciprocally as PI. And the progression is the same in injinitnm. The demonstration of this Theorem is as follows : The things remaining as above constructed, and a corpuscle being in anj