542 Cartiot on the Theory of 



be neglected, in the courfe of the calculation, without alter- 

 ing that firft term. 



The Infinitefimal Analyfis, therefore, differs from the 

 method of indeterminates only in this, that in the former, 

 quantities which, were they allowed to remain, would, in the 

 end, always deftroy one another, are treated as nothing, or 

 rather are underjlood throughout the calculation ; while, in 

 the Method of Indeterminates, we wait till the end. of the 

 calculation, and then cancel the arbitrary quantities which 

 ought to be eliminated. This laft method may therefore very 

 eafily be made to fupply the ufe of the Infinitefimal Calculus, 

 without the help of imperfect equations, and without com- 

 mitting any error in the courfe of the calculation. 



38. There is yet another method of coming at the refults 

 of the Infinitefimal Analyfis, without overpaffing the bounds 

 of ordinary algebra; and that is, by the Method of Limits, or 

 Ultimate Ratios. For though this analyfis be founded entirely 

 on the properties of limits and ultimate ratios, it differs never- 

 thelefs from what is properly called the method of limits, in 

 this, that in the latter, the quantities which we call Infini- 

 tefimal, do not enter feparately into the calculation, nor even 

 their ratios, but only the ultimate values of thefe ratios, 

 which being finite quantities, do not fo properly confiitute 

 this method a particular calculus, as a fimple application of 

 ordinary algebra. 



The bufmefs before us, then, is by barely introducing into 

 ordinary algebra, not Infinitefimal quantities themfehcs, 

 but the ultimate ratios of thefe quantities, to fupply the 

 means which the Infinitefimal Analyfis furnifhes, for dif- 

 covering any properties, ratios and relations whatfoever, of 

 the magnitudes which conftitute any propofed fyfiem ; and 

 this is that which is properly called the Method of Limits. 



To explain the procedure, and give fome idea of the fpirit, 

 of this method, we fliall again refume the example before 

 treated of. 



Explanation of the Method of Limits, properly fo called. 

 It is evident, from what was delivered in article. 9, that, 



MZ TP 



though - - be not equal to '-—^ , vet the firft of thefe 

 ti/t Mr 



3 quantities 



