NEWTON 35 



fluxions cannot be given on the basis of infini- 

 tesimals or that infinitely small quantities are 

 impossible ; for he says/ ''the analysis may be 

 performed in any kind of figures whether finite or 

 infinitely small, which are imagined similar to the 

 evanescent figures." 



In fact, not even in 1704 did Newton succeed in 

 completely banishing from his doctrine of fluxions 

 the infinitely little. If what he used in 1704 is 

 not the infinitely little, it is so closely related thereto, 

 that it cannot be called either a finite magnitude or 

 an absolute zero. 



In 1704, fluxions are "in the fif-st ratio of the 

 nascent augments, " or ' ' in the ultimate ratio of the 

 evanescent parts. " ^ Unless the fully developed 

 theory of limits is read into these phrases, they 

 will involve either infinitely little parts or other 

 quantities no less mysterious. At any rate, the 

 history of fluxions shows that these expressions 

 did not meet the demands for clearness and freedom 

 from mysticism. Newton himself knew full well 

 the logicai difiiculty involved in the words "prime 

 and ultimate ratios " ; for in 1687 he said,^ "it is 

 objected, that there is no ultimate proportion of 

 evanescent quantities ; because the proportion, 

 before the quantities have vanished, is not ultimate ; 

 and, when they have vanished, is none." How does 

 Newton meet this, his own unanswerable argument ? 

 He does so simply by stating the difificulty in another 



1 See our §§ 33, 42. '^ See our §§ 29, 30, 33, 36, i%, 39, 42. 

 ' See our §§ u, 14. 



