the Infimtejimal Calculus. 233 



Will fugged an anfwer to this queftion. " In fact," might 

 the inventor continue, " let me fuppofe for a moment that 

 the defired competition has taken place, and let me inquire 

 by what mark it ought to be indicated, in the conchifion of 

 the calculation. Now it muft naturally happen, that when 

 the quantities which occaiion the errors difappear, the errors 

 themfelves will difappear at the fame time. For thofe quan- 

 tities (fuch as MZ and RZ) having, by the fuppofition, ar- 

 bitrary values, ought no longer to exift in formula, or rc- 

 fulls, which are not arbitrary, and which depend, not on the 

 will of the calculator, but 011 the nature of the things whole 

 relation, expreffed bv thofe remits, he propofed to inveftigate. 

 The mark, then, which announces that the defired compen- 

 fation has taken place, is the abfence of the arbitrary quan- 

 tities which produced the errors ; and therefore I have nothing 

 more to do, in order to effect that compenfation, than to eli- 

 minate thofe arbitrary quantities." 



12. With a view to fix thole ideas the more firmly in the 

 mind, and to give to the principles thence derived the ne- 

 ceffary degree of prccifion and generality, I fhall remark, that 

 the quantities which we have occafion to confider in the fub- 

 ject before us, may be diftinguiihed into two claffes. The 

 lirll: clafs confiits of quantities which are either given, or de- 

 termined by the conditions of the problem, fuch as MC, 

 MP, PT, MT. The fecond clafs is compofed of quantities, 

 fuch as RS, RT', ST', which depend on the arbitrary pofition 

 of the point R, and which, as the point R approaches to the 

 point M, do respectively approach their correfponding quan- 

 tities in the firft clafs. Thus MP, for example, is the limit 

 of RS, that is, the fixed term to which it continually ap- 

 proaches, or, if you will, its laft or ultimate value. In 

 like manner, MT is the limit or ultimate value of RT', and 

 PT that of iST / . For the fame reafon, it is clear that the 

 limits or ultimate values of MZ, RZ, MR, T'T, are, every one 

 of them, o. In fine, it is alfo evident that the ultimate ratio 



DO 



oficS to MP, (that is, the uhimate value of ,) is the ratio 



MP 



of equality, and fuch is the ratio of RT' to MT, of ST' to PT, 



or, in a word, fuch is that of anv other quantity to its limit. 



Vol. VIII. II h " j*. To 



