172 Sir Isaac Newton’s Book II.

any bent line whatever, whose concavity should be everywhere turned towards the center of the force. It was far-ther remarked, that the strength of the centripetal force,in each place, was to be collected from the nature of theline, wherein the body moved*. Now st nee each planetmoves in an ellipsts, and the fun is placed in one focus;Sir I s a a c Newton deduces from hence, that the strengthof this power is reciprocally in the duplicate proportion of thedistance from the fun. This is deduced from the properties,which the geometers have discovered in the ellipsis. The pro-cess of the reasoning is not proper to be enlarged upon here;but I shall 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 fense when thus united is as follows. Suppose the planetmoved in the orbit ABC (in fig. 93. ) about the fun in S.Then, when it is said, that the centripetal power, which acts onthe planet in A, bears to the power acting on it in B a propor-tion, which is the reciprocal of the duplicate proportion of thedistance SA to the distance SB; it is meant that the powerin A bears to the power in B the duplicate of the proportionof the distance SB to the distance S A. The reciprocal du-plicate proportion may be explained also by numbers as fol-lows. Suppose several distances to bear to each other propor-tions expressed by the numbers 1, 2, g, 4, f ; that is, let thesecond distance be double the first, the third be three times,the fourth four times; and the fifth five times as great as the

* Book I. Ch. 3. § 19. *> Ibid. Ch, § 30, 17.

first.