238 Varkbt on the Theory of- 



&nd confequcntly we may confide r A' — I' as the dlffereuce 



between an auxiliary quantity A •+■ (A — Y) and its limit 

 A y and this difference (by article 19.) is an infinitely final! 

 quantity. It may therefore be affirmed, in general, that An 

 irijinitcly finall quantity is the difference between two auxiliary 

 quantities which have the fame limit. 



26. Two quantities cannot have the fame third quantity 

 for their limit, unlcfs the ultimate ratio between thofe two 

 quantities themfelv.es be a ratio of equality. For, fince, by 



X 



the fuppofition, the limit, or ultimate value of — is 1, and 



that of— is alfo 1, it is evident that the limit, or ulti- 

 mate value or — ■%—. is hkewne unitv. JNow — p— =s 



(-!r) ' (4-) 



Y X 



— — : and therefore the limit, or ultimate value of ~ -- 

 T 2 



is 1 ; that is, the ultimate ratio of X to Y is a ratio of 



equality. It may therefore be affirmed, in general, That 



An infinitely f mall quantity is the ratio which the difference of 



two quantities, ultimately in the ratio of equality, has for 



each of thefe two quantities. 



27. In fine, it may evidently be farther affirmed, That An 

 infinitely fmall quantity is an unqfjighed quantity, to which 

 is atfrft attributed any arbitrary value whatever, and which 

 is afterwards fupfofed to decreafe i/fenfib/y till it arrive at O, 

 or zero. Thus, in general, when it is faid, Let Z, for ex- 

 ample, be an infinitely fmall quantity, it is precifely tanta- 

 mount to faying, Let Z be any arbitrary quantity whatever, 

 (and consequently auxiliary, for afligned quantities cannot 

 be arbitrary) and let us afterwards fuppofe that quantity (Z) 

 to decreafe continually till it come to o, or zero *. 



38. A quantity is faid to be infinitely fmall relatively to 

 another quantity, when the ratio of the firft to the fecond is 



• : In the 25th, 26th and 27th articles, the author gives us different 

 ideas of an infinitely fipall quantity ; but which, if duly confidered, will 

 he found to amount nearly to the fame thia^;. W. D. 



. . an 



