344 Carnat on the Theory of 



The Origin of the Name, Infinitely Small Quantities. 



40. From what has been faid (in article 2.) on the origin 

 which the Infinitefimal Analyfis might have had, it appears, 

 that the quantities called Infinitely Small, received that name 

 from it's being at firft believed that it was neceffary, for the 

 fuccefs of the calculations in which they were employed, to 

 attribute to thofe arbitrary quantities values, which were really 

 lefs than any which could be recognized by the fenfes, or 

 conceived by the imagination. But a better digefted theory 

 has made it appear, that fuch a fuppofition is unnecefl'ary ; 

 fmce the fuccefs of this calculus proceeds not from the atte- 

 nuation of thofe arbitrary quantities, but folely from the 

 compenfation of errors which they occafion in the procefs. 



We have feen indeed, in illuftrating the example fo often 

 adduced, that the procedure and the remits of the calculation 

 were precifely the fame, whatever value we attributed to the 

 infinitely fmall quantities MZ and RZ, and that confequently 

 the character of this kind of quantities confifts not in their 

 real minutenefs, but rather in their being abfolutely indeter- 

 minate, that is, in their property of remaining arbitrary 

 throughout the calculation, and fo independant on the pro- 

 pofed quantities, that we can always take them as fmall as 

 we pleafe, without changing, in any refpecl:, the conditions 

 of the problem. 



Infinitefimal quantities, as was obferved in article 24, are 

 by no meuns chimerical beings, but (imply variable quan- 

 tities, characterized by the nature of their limit, which is o 



for infinitely fmall quantities, and — for thofe which are 



infinitely great. To thefe indeterminate quantities, as well 

 as to all other indefinite quantities, may be fucceflively attri- 

 buted, divers arbitrary values, and among thofe values ought 

 to be included the ultimate values of all, that is, o, for 



quantities infinitely fmall, and — for thofe which are infi- 

 nitely great. 



D\/lin6lionr 



