In a sphere described about the centre S with the interval SA, if there be taken SI, SA, SP continually proportional; I say, 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 the 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 as the distances of the corpuscle attracted 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 subduplicate 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 SA2; if in a quadruplicate ratio, as SP3 to SA3. Therefore since the attraction in P was found in this last case to be reciprocally as PS3 × PI, the attraction in I will be reciprocally as SA3 × PI, that is, because SA3 is given reciprocally as PI. And the progression is the same in infinitum. The demonstration of this Theorem is as follows: The things remaining as above constructed, and a corpuscle being in any[Pg 232] place P, the ordinate DN was found to be as . Therefore if be drawn, that ordinate for any other place of the corpuscle, as I, will become (mutatis mutandis) as . Suppose the centripetal forces flowing from any point of the sphere, as E, to be to each other at the distances and as to (where the number n denotes the index of the powers of PE and IE), and those ordinates will become as and whose ratio to each other is as