LATER BOOKS AND ARTICLES 251 



equality of inequality of those limits from some 

 relation of the variable quantities, which are never 

 supposed absolutely to reach their limits, it cer- 

 tainly requires something more than a definition to 

 shew this. ... It is not my intention to detain 

 the reader, with answering the objections of the 

 Arialyst and his foUowers, because it has been 

 already done by others in a satisfactory manner. 

 . . . Notwithstanding the assertions of some 

 modem writers, the method of ultimate ratios is 

 extremely perspicuous, strictly logicai, and more 

 concise than any other of modem invention ; . . . 

 it neither involves the strange notion, that a 

 straight line may be a part of a curve, and a piane 

 superficies a part of a concave or convex one ; nor 

 the unintelligible idea of adding and subtracting 

 indivisibles, or inconceivably small magnitudes. 

 Whatever magnitudes are compared, according to 

 this method, they are always supposed to be 

 finite." 



T. Newton begins with the following two defini- 

 tions (p. i) : 



'*If a variable quantity, either increasing or de- 

 creasing, approaches to a fixed quantity, the differ- 

 ence between them being continually diminished, so 

 as at length to become less than any assignable 

 quantity ; the fixed quantity is called the Limit of 

 the variable quantity." 



'' If the ratio of two variable quantities continu- 

 ally approaches to a fixed ratio, so as to come 

 nearer to it than by any assignable difference ; the 



