ibe Irifimtefimal V-alail-vs, aj? 



The reafoa of this procedure is fimple. Stippofe we Lave 

 occafion to inveftigate the relations which fubfift between Se- 

 veral propofed quantities. If it be difficult to find directly 

 equations to exprefs thefe relations, we naturally recur ty 

 fome intermediate quantities, which may ferve as terms Qjf 

 coaiparifon. Bv this means we obtain, if not the very equa*- 

 tions fought, at Icaft other equations, in which the propofed 

 quantities are blended with auxiliary ones ; and there can be. 

 no queftion that thefe hut ought to be eliminated. But if, 

 additionally, the values of thefe auxiliary quantities be arbU 

 trary, aud may be fuppofed as fmall as we pleafe, without 

 affecting the propofed quantities, it is eafv to fee, that, if iu 

 the equations expreiEng the relations fought, arbitrary quan- 

 tities be mixed with propofed ones, each of thefe equations 

 may be decornpofed into two, one containing afiigned, and 

 the other arbitrary quantities. It is nearly in this manner, 

 that an equation containing real and imaginary quantities 

 may be decornpofed into two equations, the one confining of 

 real, and the other of imaginary quantities. Now, as we. 

 only want the equation which exifts between the propofed 

 quantities, it is evident that, in thofe equations where they 

 are mixed with arbitrary ones, we may fafely neglect the 

 quantities which embarrafs our calculation, when the refund 

 ing errors can only affect the equation between the arbitrary 

 quantities which it contains. Now this is precifely what 

 takes place in the Infinitefimal Calculus, where we confidaf 

 infinitely fmall quantities as nullities when compared with 

 finite ones. 



In order to render this explanation dill clearer, let us re- 

 fume our former example. In article q, we found 

 " m MZ , MZ 2 y + RZ 



TP+ T'T=j> "~v > aml 157 = T~^~r w> * ■ 



•s HZ 7iZ la — 2.v — MA 



Thefe two equations are both perfectly exact, whatever be the 



values ©fMZ and RZ. Deducing, then, from. the firft equation, 



MZ 

 the value of -_=-, and fubftituting it in the fecond, I get 



TP -f T'T _ _ ~y + RZ 



y 2fl.-» %X — .'1/Z 



And this equation is accurate, as it ought to be, wlmtever 



«iiit£ftte 



