the Infnltejimal Calculus. 229 



to be able to diftinnuifh the cafes in which this accuracy 

 takes place, and thus to convert this method of approxima- 

 tion into a calculus perfectly exact and rigorous; and fuch is 

 the object of the infinitefimal analyfis. 



8. Let us firft fee, then, how it came to pafs that, in the equa- 

 tion TP = ■ > . ( -~ y ~. — -sir, found in article -nl, though we 

 %Ll _ 2x - MO ° ° 



neglefted MO and NO, the juftnefs of the rcfult was not af- 

 fected ; or rather, how the refult became exact from the fup- 

 preffion of thofe quantities, and why it was not exact before 

 that fuppreffion. 



Now, the rcafon of what happened in the folution of the 

 above problem will eafily appear, on remarking that the hy- 

 pothefis on which that folution is grounded, is falfej it being 

 abfolutely impoffible that a circle can be a true polygon, what- 

 ever be the number of its fides ; fo that there muft refult from 

 that hypothefis an error of fome kind in the equation TP — 



y (zy + NO ) Bui ^ ^^ rfp = _y_ being ncvertbe- 

 la — ix — MO a — x ° 



lefs exact, as was proved by the comparifon of the two tri- 

 angles CPU, MPT, we found that MO and NO might be 

 neglected in the firft equation. Indeed thefe quantities ought 

 to be neglected, in order to correct the calculus, and to de- 

 ft roy the error which had arifen from the falfe hypothefis firft 

 affumed. The rejection of quantities of this nature is not 

 barely allowable, but is abfolutelv neceffary in fuch cafes, 

 as being the only manner of expreffing accurately the condi- 

 tions of the problem. 



The Accuracy of fuch Rtfults is owing' to a Compcnjattort 

 of Errors. 



9. The accurate refult TP = — hasbecnobtained then, ' 



a — x 



only in confequenee of a compenfatiqn of errors ; and this 

 compenfation may be rendered (till more evident, by treating 

 the example above given in a manner fomewhat different, 

 namely, by considering the circle as (what it really is) a truo 

 Curve, and not a polygon, 

 for this purpofe, from the point R (G'y. 1.) taken arbitra- 

 8 rily 



