And becauſe there is given the ratio of the diſtance of either body from that common centre to the diſtance between the two bodies, there is given of courſe the ratio of any power of one diſtance to the ſame power of the other diſtance; and alſo the ratio of any quantity derived in any manner from one of the diſtances compounded any how with given quantities, to another quantity, derived in like manner from the other diſtance, and as many given quantities having that given ratio of the diſtances to the firſt. Therefore if the force with which one body is attracted by another be directly or inverſely as the diſtance of the bodies from each other, or as any power of that diſtance; or laſtly as any quantity derived after any manner from that diſtance compounded with given quantities; then will the ſame force with which the ſame body is attracted to the common centre of gravity, be in like manner directly or inverſely as the diſtance of the attracted body from the common centre, or as any power of that diſtance, or laſtly as a quantity derived in like fort from that diſtance compounded with analogous given quantities. That is, the law of attracting force will be the ſame with reſpect to both diſtances. Q. E. D.
