248 A Short History of Astronomy [CH. x. 



expressed in such a way as to be capable of being inter- 

 preted in terms of the original problem, whereas in the 

 analytical treatment the problem is first expressed by 

 means of algebraical symbols ; these symbols are manipulated 

 according to certain purely formal rules, no regard being 

 paid to the interpretation of the intermediate steps, and 

 the final algebraical result, if it can be obtained, yields on 

 interpretation the solution of the original problem. The 

 geometrical solution of a problem, if it can be obtained, 

 is frequently shorter, clearer, and more elegant ; but, on 

 the other hand, each special problem has to be considered 

 separately, whereas the analytical solution can be con- 

 . ducted to a great extent according to fixed rules applicable 

 in a larger number of cases. In Newton's time modern 

 analysis was only just coming into being, some of the most 

 important parts of it being in fact the creation of Leibniz 

 and himself, and although he sometimes used analysis to 

 solve an astronomical problem, it was his practice to translate 

 the result into geometrical language before publication ; in 

 doing so he was probably influenced to a large extent by 

 a personal preference for the elegance of geometrical proofs, 

 partly also by an unwillingness to increase the numerous 

 difficulties contained in the Prindpia, by using mathematical 

 methods which were comparatively unfamiliar. But though 

 in the hands of a master like Newton geometrical methods 

 were capable of producing astonishing results, the lesser 

 men who followed him were scarcely ever capable of using 

 his methods to obtain results beyond those which he 

 himself had reached. Excessive reverence for Newton and 

 all his ways, combined with the estrangement which long 

 subsisted between British and foreign mathematicians, as 

 the result of the fluxional controversy (chapter ix., 191), 

 prevented the former from using the analytical methods 

 which were being rapidly perfected by Leibniz's pupils and 

 other Continental mathematicians. Our mathematicians 

 remained, therefore, almost isolated during the whole of the 

 1 8th century, and with the exception of some admirable 

 work by Colin Madaurin (1698-1746), which carried 

 Newton's theory of the figure of the earth a stage further, 

 nothing of importance was done in our country for nearly 

 a century after Newton's death to develop the theory of 



