the Infin'itefimal Calculus. 343 



'quantities differs fo much the lefs from the fecond, as RS 



MZ 

 "approaches nearer to MP; or, in other words, that -^ — 



TP 



±=p is an imperfecT: equation ; but that (putting L for the 



_ MZ TP . rA 



limit, or the ultimate value,) L . -^ = -jjp IS a P erlect > 



or rigoroufly exa&, equation. 



In like manner, L . - D7 - = -£— is proved to be a per- 

 feci:, or rigorouflv exact, equation. Equating then thefe two 

 values of L . -^-. there arifes, as before, 



w = sb? or (MP bcins = ■>' TP = ^ ■ 



Thus, this new calculus contains neither the infinitely fmall 



MZ 

 quantities MZ and RZ, nor even their ratio ^ ; but only 



, r MZ 

 the limit or ultimate value of that ratio, namely, L, . -^ , 



which is a finite quantity. 



39. If this method could be always as eafily put in practice 

 as the ordinary Infinitefimal Analyfis, it might even appear 

 the mod eligible of the two : for it would have the advantage 

 of conduaing us to the fame refults, by a path which is 

 always dired and luminous; whereas the other conducts us 

 to the truth, only after having made us traverie, fo to fpeak, 

 the regions of error. 



But it muft be owned that the Method of Limits is attended 

 with a confiderable difficulty, which has no place in the ordi- 

 nary Infinitesimal Calculus. In the former, the infinitely 

 fmall quantities cannot, as in the latter, be feparaled from 

 each other ; and thefe quantities being always conne&ed two 

 and two, afford no opportunity of introducing into the com- 

 binations, the properties of each in particular, or of fubjea- 

 ing the equations into which they enter, to thofe transform- 

 ations which may affift in their elimination. This difficulty 

 is much lefs felt in the operations themfelves, than in the 

 preparatory and fupplemental propositions and reafonmgs. 



Yyi The 



