188 On prime and ultimate Ratios. 



T say, that the measures of such ratios never can attain the 

 limits which we have assigned to them : they may, how- 

 ever, continually approximate towards them ; and when the 

 measures of the ratios differ from those limits by less than 

 any assignable difference, they may be said to be equal. 



This being allowed, it is evident that in making use of 

 those ratios, after having supposed that they have attained 

 to such ultimate states, or limits, we continually approxi- 

 mate towards true results ; and when the results thus ob- 

 tained, differ from the true results by a quantity indefinitely 

 small, they may be said to be indefinitely near the truth, 

 and in practice the indefinitely small error may be neg- 

 lected, as being of no sensible magnitude. 



This is all that Newton meant in his first lemma in the 

 Principia, where he says that Ci quantities, and the ratios 

 of quantities, which in any finite time converge continually 

 to equality, and before the end of that time approach nearer 

 the one to the other than by any given difference, become 

 ultimately equal." 



It was the calling, those results true which are only ap- 

 proximations indefinitely near the truth, that gave the au- 

 thor of the Analyst so much advantage in exposing the 

 errors in the metaphysics of the fluxional calculus ; and it 

 was very inconsiderate in Philalethes (supposed to be Dr. 

 Jurin) to argue that the error occasioned by neglecting a 

 certain very small quantity did not affect the result of any 

 operation : — that did not in the least tend to overthrow the 

 arguments adduced by the author gf the Analyst, since it 

 was the error in principle that he struck at, and not the 

 quantity of error that the making use of a false principle 

 might produce. 



The conclusions obtained by the method of fluxions are 

 not absolutely true, nor did Newton ever consider them as 

 such ; they are approximations, which produce no sensible 

 errors; and had his host of defenders proceeded no further 

 than this, all the arguments that could have been brought 

 forward against this method must have vanished. 



But instead of giving up what was evidently untenable, 

 all the varied arguments which imagination aided by science 

 could suggest, were brought forward in order to get rid of 

 the difficulties which Berkley had pointed out, but without 

 effect; for truth is at all times consistent with itself, and 

 what is once wrong can never be proved to be right. 



Newton was desirous of determining the areas of curvi- 

 linear figures : this was at all times a great desideratum, 

 and hadT exercised the talents of philosophers in all ages. 



The 



