258 LIMITS AND FLUXIONS 



221. The reviewer states that foreign mathe- 

 maticians have written treatises in which motion is 

 entirely excluded, "and in some of these treatises, 

 the principles of the doctrine in question have been 

 laid down with a considerable degree of evidence 

 and exactness." The Residuai Analysis of Landen 

 rests on *'a process purely algebraical : but the 

 want of simplicity . . . is a very great objection to 

 it." The reviewer is of the opinion that Euler and 

 D'Alembert give " the most clear and precise notions 

 of the principles on which the differential calculus 

 is established. " He refers to Euler's Institutiones 

 calculi differentialis, 1755. D'Alembert, says the re- 

 viewer, *' observes that the method is really founded 

 on that of prime and ultimate ratios, or of limits, 

 which latter method is only an algebraical transla- 

 tion of the former ; that, in fact, there are no such 

 things as inftnitely small quantities ; and that, when 

 such quantities are mentioned, it is by the adoption 

 of a concise mode of speech for the purpose of 

 simplifying and abridging the reasoning ; — that the 

 true object of consideration is the limit of the ratio 

 of the finite differences of quantities." 



The reviewer continues : ''The explanations given 

 by Euler and D'Alembert, beyond ali doubt, deserve 

 much consideration, yet their method of consider- 

 ing the doctrine of fluxions is not completely satis- 

 factory, but is objectionable on two grounds : first, 

 that we have no clear and precise notion of the 

 ratio of quantities, when those quantities are in 

 their vanishing state, or cease to be quantities ; 



