the Injiniteji/nal CahaJus. 339; 



*ti infinitely fmall quantity; and, reciprocally, the fecond is 

 find to be infinite or infinitely great, relatively to the firfh 



29. Two quantities are faid to differ infinitely little, or to- 

 be infinitely little different, from one another, when the ratio 

 Of the one to the other differs from unity only by an in- 

 finitely fmall quantity; fo that their ultimate ratio is a ratio 

 of equality, and fuch evidently are RS and MP. 



30. The Infinitejimal Calculus may be defined to be ths 

 art of difcovering, by the help of quantities called Infinitefi- 

 mals, the proportions or relations, whatever thev may be, 

 which fubfiff. between the different parts of any fyftem of 

 quantities propofed. 



Thcfe infinitefimal quantities being nothing more than 

 auxiliary quantities, introduced into the calculation only to 

 facilitate the expreflion of the conditions of any propofed 

 problem, it is evident that they muft neceflarily be eliminated 

 out of the calculation, in order to obtain the defired rcfult., 

 namely, the relations fought. Thus we may in fome re- 

 fpe&s affirm, That every Infinitefimal Calculation is an un- 

 finijhed calculation ; for in facl, as foon as the auxiliary 

 quantities; which are not effential, come to be eliminated, 

 the calculation ceafes to be infinitefimal, and in every thing 

 refembles an ordinary algebraic refnlt *. 



To complete the explanation of the principal terms which 

 relate to the theory of Infinites in general, it remains for me 

 to (fate what I underftand by an imperfect equation. 



31. I call every equation An imperfetl equation, whofe 

 two fides are unequal quantities, the difference however be- 

 tween them being infinitely fmall ; or, which comes to the 

 tame thing, Every equation is imperfect, the two fides of 

 which, though unequal, have for their ultimate ratio,, the 

 ratio of equality. 



MZ 

 Thus, for example, the falfc equations, TP =y— — . 



RZ 



and — ~ — — — - (fee article 9.) are what I call imperfecl 



• Every one knows, indeed, that a calculation, in which Infinitefimal 

 quantities have any place, is not fuppofed to be fmifhed, and that vtz. 

 never think the rcfult accurate till all the infinittfirrjal quantities are en- 

 tirely eliminated. 



equations-; 



