172 Sir Is a a c N e w t o n’s Book II.

any bent line whatever, whole concavity fhould be everywhere turned towards the center of the force. It was far-ther remarked, that the llrength of the centripetal force,in each place, was to be collected from the nature of theline, wherein the body moved 1 *. Now fince each planetmoves in an ellipfis, and the fun is placed in one focus;Sir Isaac Newton deduces from hence, that the llrengthof this power is reciprocally in the duplicate proportion of thedillance from the fun. This is deduced from the properties,which the geometers have difcovered in the ellipfis. The pro-cels of the realoning is not proper to be enlarged upon here ■but I fhall endeavour to explain what is meant by the recipro-cal duplicate proportion. Each of the term s reciprocal pro-portion, and duplicate proportion, has been already defined b .Their fenfe when thus united is as follows. Suppofe the planetmoved in the orbit ABC (in fig. 93. ) about the fun in S.,Then, when it is laid, that the centripetal power, which adds onthe planet in A, bears to the power adding on it in B a propor-tion, which is the reciprocal of the duplicate proportion of thedillance S A to the dillance SB; it is meant that the powerin A bears to the power in B the duplicate of the proportionof the dillance S B to the dillance S A. The reciprocal du-plicate proportion may be explained alfo by numbers as fol-lows. Suppofe feveral dillances to bear to each other propor-tions exprefled by the numbers 1, 1, 3, 4, y ; that is, let thefecond dillance be double the firft, the third be three times?the fourth four times, and the* fifth five times as great as the

* Book I. Ch. j> § 23. b Ibid. Ch. i. § 30, 17..

firft-

*