the Lifinitejlmal Calculus. 34$ 



Was the indefinitely fmall quantity MZ, and it's ordinate 



MZ 

 proportional to -~ , that which would anfwer to the ab- 



TP 



fciffe o, would be reprefented by , and not by an arbi- 

 trary quantity. Now this is what dirunguifhcs the quantities 

 which I call evanefcent from thofe which are limply nothing.. 

 Thus, though in general we have o~sXo, = 3X0, 

 = 4X0, &c. yet we cannot treat an evanefcent quantity, 

 fuch as MZ, in the fame manner, and fay MZ = iMZ = 

 3MZ = 4.MZ, &:e. ; for the law of continuity -cannot affign 

 to MZ and MZ, any other ratio than that of equality, nor 

 any other relation than that of identity *. 



43. We have feen that bv introducing into the calculation 

 infinitely fmall quantities, and by neglecting them in com- 

 parifon with finite quantities, the equations became imper- 

 fect, and that the errors which they produced were only 

 compenfated in the required refult. But we have it now in 

 our power to avoid this kind of inconvenience, by means of 

 evanefcent quantities, which being nothing elfe than the 

 ultimate values of the infinitely fmall quantities correfpond- 

 ing to them, may, like any of the other values, be attributed 

 to thefe indefinitely fmall quantities; and which being, in 

 another point of view, abfolute nullities, may be neglected, 

 when they are found connected with any effective quantities, 

 without preventing the calculation from being perfectly ri- 

 gorous. 



44. The Tnfinitefimal Analyfis, then, may be confidered in 

 two different points of view ; by regarding the infinitely fmall 

 quantities, either as real, effective quantities, or as abfolute 

 nullities. In the firlt cafe, the Infinitcfimal Analyfis is 

 nothing elfe than the Calculus of compenfated errors; and in 

 the fecond, it is the art of comparing evanefcent quantities 

 among themfelves, and with others, in order to deduce from 

 thefe companions the proportions and relations, whatever 

 they may be, which fubfift among the quantities propofed. 



* For " identity, '" the author fhould have ufed the word " congruity." 

 They are very r.-r from being fynonymous terms ; though ufed as fuch 

 by fomc muihcmxticians, Sec Euclid's 8th axiom. — W. D. 



Evanefcent 



