JURIN V. ROBINS AND PEMBERTON 107 



as they diminish, is the true proportion of the 

 fluxions"(p. 438). Here a new difficulty arises : 

 Does the varying ratio reach its limit **actually, 

 perfectly, and absolutely," or does it not ? On the 

 understanding that it does not reach the Hmit, '' ali 

 that has at any time been demonstrated by the 

 ancient method of exhaustions may be most easily 

 and elegantly deduced." Rigour of demonstration 

 does not require ultimate coincidence. *'Coinci- 

 dence of the variable quantity and its limit, could 

 it be always prov'd, would yet bring no addition to 

 the accuracy of these demonstrations " (p. 441). 

 Hence, *'why to the naturai difficulty of these 

 subjects should the obscurity of so strained a con- 

 ception be added ? " Is this view a correct inter- 

 pretation of Newton ? A literal translation of bis 

 Lemma i, Section i, Book I, Principia (see our 

 §§ 4, 6, 8), is : " Ouantities, and the ratios of 

 quantities. that during any finite time constantly 

 approach each other, and before the end of that 

 time approach nearer than any given difference, 

 are ultimately equal. " What is the meaning of 

 *' given difference"? If it be a "difference first 

 assigned" according to which the degree of approach 

 of these quantities may be afterwards regulated ; 

 then . . . ratios, and their limits, tho' they do 

 never actually coincide, will come within the de- 

 scription of this Leinuia ; since the difference being 

 once assign'd, the approach of these quantities may 

 be so accelerated, that in less than any given time 

 the variable quantity, and its limit, shall differ by 



