$$1 Car not on the Infinitejimal Calculus. 



it nothing, it may be treated as fuch, and yet no error will 

 exift in the refult; becaufe that error, if there were any, 

 would be arbitrary, like the quantity which occasioned it. 

 Now it is evident, that no fuch error can exift, except among 

 quantities, one of which at leaft is arbitrary. When, there- 

 fore, we arrive at a refult containing no arbitrary quantity, 

 and which expreffes any relation whatever between quantities 

 given, and thofe determined by the conditions of the problem, 

 we may reft affured that that refult is accurate ; and that, 

 confequently, the errors necefiarily committed in expreffing 

 thefe conditions, muft have been compcnfated and have difap- 

 peared, bv the neceffary and infallible effects of the operation. 



48. Other mathematicians, apparently embarrafied by the 

 objection juft difcufTed, have {imply confined themfelves to 

 prove, that the Method of Limits, the procefles of which are 

 rlgoroufiy accurate, in all refpects, muft neceflarily lead to 

 the fame refults as the Infinitefimal Calculus. But, while it 

 is agreed that the principle of that method is very luminous, 

 it cannot be diffembled, that the difficulty is thus only elud- 

 ed, not removed ; that the Method of Limits leads to the fame 

 refults as the Infinitefimal Calculus, only by a difficult and 

 circuitous way; and, in line, that that method, far from 

 being the fame with the Infinitefimal Calculus, is, on the 

 contrary, only the art of difpenfing with this calculus, and of 

 fupplying it by ordinary Algebra. It appears to me, that 

 they would fucceed, in a more fimple manner, by the Method 

 of lndcterminates. But why adopt one of thefe methods to 

 the exclufion of the reft, when they can afford us their mutual 

 affiftanee? Let us then employ them all — the Infinitefimal 

 Calculus, properly fo called, the Method of Limits, and the 

 Method of lndcterminates, as circumftances may require, and 

 let us neglect none of the means which can conduct us to 

 truth, or Amplify our refearches. 



It remains for me to fhow, by fome examples, the applica- 

 tion of the general principles, which I have explained. This 

 I fhall do, by giving my reader an idea of the Differential 

 and Integral Calculi, which, properly fpeaking, are the Infi- 

 fiitefimal Analyfis itfelf reduced to practice. 

 £To be continue (f.j 



XT. Ap- 



