Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/310



And becaue there is given the ratio of the ditance of either body from that common centre to the ditance between the two bodies, there is given of coure the ratio of any power of one ditance to the ame power of the other ditance; and alo the ratio of any quantity derived in any manner from one of the ditances compounded any how with given quantities, to another quantity, derived in like manner from the other ditance, and as many given quantities having that given ratio of the ditances to the firt. Therefore if the force with which one body is attracted by another be directly or inverely as the ditance of the bodies from each other, or as any power of that ditance; or latly as any quantity derived after any manner from that ditance compounded with given quantities; then will the ame force with which the ame body is attracted to the common centre of gravity, be in like manner directly or inverely as the ditance of the attracted body from the common centre, or as any power of that ditance, or latly as a quantity derived in like fort from that ditance compounded with analogous given quantities. That is, the law of attracting force will be the ame with repect to both ditances. Q. E. D.