288 SIB ISAAC NEWTON. 



That he must have tried the process of integrating by parts 

 in attempting to generalize the inverse problem of central 

 forces before he had recourse to the geometrical approximation 

 which he has given, and also when he sought the means of 

 ascertaining the comet's path (which he has termed by far the 

 most difficult of problems), is eminently probable, when we 

 consider how naturally that method flows from the ordinary 

 process for differentiating compound quantities by supposing 

 each variable in succession constant ; in short, differentiating 

 by parts. As to the calculus of variations having substan- 

 tially been known to him no doubt can be entertained. 



Again, in estimating the ellipticity of the earth, he pro- 

 ceeded upon the assumption of a proposition of which he gave 

 no demonstration (any more than he had done of the isoperi- 

 metrical problem) that the ratio of the centrifugal force to 

 gravitation determines the ellipticity. Half a century later, 

 that which no one before knew to be true, which many 

 probably considered to be erroneous, was examined by one of 

 his most distinguished followers, Maclaurin, and demonstrated 

 most satisfactorily. 



Newton had not failed to perceive the necessary effects of 

 gravitation in producing other phenomena beside the regular 

 motion of the planets and their satellites, in their course 

 round their several centres of attraction. One of these phe- 

 nomena, wholly unsuspected before the discovery of the 

 general law, is the alternate movement to and fro of the earth's 

 axis, in consequence of the solar (and also of the lunar) at- 

 traction combined with the earth's motion. This Libration, 

 or Nutation, distinctly announced by him as the result of the 

 theory, was not found by actual observation to exist till sixty 

 years and upwards had elapsed, when Bradley proved the 

 fact.* 



* The Nutation, and by name, is given in Prin. Lib. III. prop. 21, the 

 demonstration being referred to as in Lib. I. prop. 66, cor. 20. Clainuit, 

 ' Princ. de Du Chatelet,' torn. II. p. 72, 73, refers to the same proposition. 

 F. Walmsley, ' Phil. Trans.' 1746, lias an excellent paper on Precession 

 and Nutation, treated Geometrically. It is stated in Montucla, IV. 216, 



