TEXT-BOOKS, 1736-1741 



77 



tions ; and, on the contrary, the Fluxion for the 

 Increment " (pp. 5, 6). Accordingly, he deduces the 

 rules of operation by the use of increments, and in 

 the result substitutes the fluxion for the increment. 



In finding the fluxion of xy he lets x' and y' be 

 the increments, then the " increase in its nascent 

 state" is such that x'y' ''bears no assignable Ratio 

 to either x'y or xy (for as x'y' : x'y : :y' : y and y' by 

 Supposition is infinitely less than y,'' and can be 

 " expunged or rejected. " 



160. The third edition (1767) was commented upon 

 by J. Stubbs, Fellow of Queen's College, Oxford, 

 as follows : *' I received your valuable present, and 

 was much surprised to find 

 it so prodigiously improved. 

 Indeed, it so much resembles 

 a New Work, when compared 

 with the First Edition, that I 

 almost wish you had made 

 no mention of its being the 

 Third ; but left the two former 

 to be forgotten." 



The fluxion of xy is now 

 deduced thus : "The fluxion 

 of the curvilinear space AEI is less than the fluxion of 

 the rectangle (of Constant altitude) AH before EH 

 reaches BC, and greater after EH passes BC ; hence 

 at BC the two fluxions are alike and equal to yi: 

 Similarly, it follows that the rectangle AG (of Con- 

 stant base) has the same fluxion xy at DB as has 



the curvilinear space AFI. Hence the rectangle 



12 



E C 



FlG. IO. 



