no LIMITS AND FLUXIONS 



not necessary ; only, Robins's method is not that of 

 Newton. To establish this last point, Philalethes 

 quotes Newton's lemma in Latin, then prints Robins's 

 and his own translation of it. In case of variation, 

 the upper line is Robins's translation, the lower is 

 Jurin's : — 



Quantities ^ and \ j \ ratio's of quantities, that 



, . ^ -. .- . .7 {approach eacÌLOthevA 



ciurins!; any finite time constantlyX , ,. J- 



^ -"-^ -^ \ tend to equahty , J 



and before the end of that time approach nearer 



i than any given difference, are ultimately equal. \ 



\ to one another than to have any given difference^ do \ 



at last become equal. ) 



It is not clear to Jurin what Robins means by 

 **are ultimately equal," nor can Jurin conceive 

 "how quantities, which do never become actually 

 equal, . . . can come within the description of a 

 Lemma, which Lemma expressly affirms, that they 

 become equal. " Fiunt ultimo cequales means * * become 

 at last equal." Jurin quotes different places in the 

 Principia which support his point. He denies that 

 Newton proceeds, in the case of inscribed and 

 circumscribed rectangles, by continuai divisions of 

 the base of the figure, and gives references in 

 support of his contention. Of interest are the 

 foUowing admissions made by Jurin (p. 87): ''This 

 equality therefore we are obliged to acknowledge, 

 although we should not be able, by stretch of 

 imagination, to pursue these figures, and, as it 

 were, to keep them in sight ali the way, till the 



