24 LIMITS AND FLUXIONS 



''In the same time that the quantity x, by 

 flowing, becomes x-\-o^ the quantity x"^ will 

 become x-^-o^, that is, by the method of infinite 

 series's, x''-^nox''-^-\-{n^ — n)l2 oox''-^-\-etc. And 

 the augments 6> and nox*'-^-{-{n'^ — n)l2 oox''-^-{-etc. 

 are to one another as i and nx''-'^-j-(n^ — n)l2 ox^-'^ 

 + etc. Now let these augments vanish, and their 

 ultimate ratio will be i to nx""-^. 



42. " By like ways of reasoning, the fluxions of 

 lines, whether right or curve in ali cases, as likewise 

 the fluxions of superficies's angles and other quan- 

 tities, may be collected by the method of prime and 

 ultimate ratios. Now to institute an analysis after 

 this manner in finite quantities and investigate the 

 prime or ulti^nate ratios of these finite quantities 

 when in their nascent or evanescent state, is con- 

 sonant to the geometry of the ancients : and I was 

 willing to show that, in the Method of Fluxions, 

 there is no necessity of introducing figures infinitely 

 small into geometry. Yet the analysis may be 

 performed in any kind of figures, whether finite or 

 infinitely small, which are imagin'd similar to the 

 evanescent figures ; as likewise in these figures, 

 which, by the Method of Indivisibles, used to be 

 reckoned as infinitely small, provided you proceed 

 with due caution." 



43. In the Quadrature of Curves proper, under 

 '* Proposition I" the proof of the rule for finding 

 the fluxion of expressions like x^ — Xf^ -^ a'^z — l?^ = o 

 contains the following passages which indicate the 

 use made of the symbol " (? " and of the term 



