loo LIMITS AND FLUXIONS 



prime and ultimate ratios ; that Robins carefully 

 distinguished both these methods from the method 

 of indivisibles and also from each other. After 

 an historical excursion viewing the works of the 

 ancients, of Cavalieri and Wallis, the introduction 

 by Newton of the concept of motion is taken up. 

 " If the proportion between the celerìty of increase 

 of two magnitudes produced together is in ali parts 

 known, " then '' the relation between the magnitudes 

 themselves must from thence be discoverable." This 

 is the basis for fluxions. The "method of prime 

 and ultimate ratios proceeds entirely upon the 

 consideration of the increments produced." By it 

 Newton avoids "the length of the ancient demon- 

 strations by exhaustions, " on which, according to 

 Robins, the method of fluctions rests. "Newton 

 did not mean, that any point of time was assign- 

 able, wherein these varying magnitudes would 

 become actually equal, or the ratios really the 

 same ; but only that no difìference whatever could 

 be named, which they should not pass." Newton's 

 term momentum is used simply for greater brevity, 

 hence need not be considered. Newton's descrip- 

 tion is capable of an interpretation too much 

 resembling the language of indivisibles — in fact, he 

 sometimes did use indivisibles at first ; Robins has 

 freed the doctrine from this imputation in a manner 

 that "shall agree to the general sense of his 

 [Newton's] description. " 



