242 LIMITS AND FLUXIONS 



they are never, strictly speaking, either prime or 

 ultimate in fact), but those limits to which the 

 ratios • of magnitudes perpetually approach ; which 

 they can never reach, nor pass beyond ; but to 

 which they appear nearer than by any assignable 

 difìference. " . . . " We now proceed to explain this 

 Lemma more particularly than perhaps might seem 

 necessary, had it not been much controverted, mis- 

 represented, and misunderstood. " As one of the 

 conditions of the proposition, Thorp states, is " that 

 quantities and the ratios of quantities must con- 

 tinually tend to equality. The one must never 

 become equal to, nor pass beyond the other : their 

 difference must never either vanish to nothing, or 

 become negative." In this restriction Thorp goes 

 even further than had Robins. The following 

 passage from Thorp's commentary is thoroughly in 

 the spirit of Robins :**... That we may not be led, 

 from the expression ultiniately equal, to suppose, 

 that there is an ultimate state, in which they are 

 actually equal, we are cautioned in the scholium at 

 the end of this Section [of Principia^ Bk. I, Sect. i] 

 in these words, The ultimate ratioSy in which quantities 

 vanishy are not in reality the ratios of ultimate 

 quantities ; but the limits to which the ratios of 

 quantities continually decreasing always approach ; 

 which they never can pass beyond ^ nor arrive at^ unless 

 the quantities are continually and indefinitely dimin- 

 ished, According to Thorp, the inscribed or cir- 

 cumscribed polygon can never arrive at the curve. 

 He quotes from Saunderson's Fluxions. By the 



