First Principles of the Differential Calculus. 271 
dga 
and dx, dg are called their differentials, and sa = A is called 
the differential co-efficient, since we must multiply dz by it to 
obtain dyno .dx=Adz. "The same results are readily obtain- 
ed by writing for 4z in the first member of (3’), dpr+4’grx=A4z 
+Bzz, the d relating to the term that involves the first power 
of Jz only, and the 4’ to the remaining part of the right mem- 
ber of (3’), .°. we get dpr=Asxr=Adz, by using d for 4, in the 
right member of the equation ; this process shows the propriety 
of calling the method of obtaining the expression der=Adz, 
(together with its various applications, ) the differential calculus, 
since Adz is only a part of the entire difference Ah+ Bh, obtain- 
ed by putting dpyy=Ah=Adz. We consider the method which 
we have given (deduced from considering (3’) or (3) as an iden- 
tical equation) for obtaining (4’) or (4”), as being the true founda- 
tion of the differential calculus. 
These remarks however are to be understood as referring to the 
principles of the science ; for in practice the common method of 
. der ; lon . Ape 
regarding eee A as expressing the limit of the ratio an 
A-+B when 4c is diminished in infinitum, is generally more sim- 
ple and expeditious than any known method, and is therefore by 
no means to be abandoned. 
Again, the method of Leibnitz, which consists in rejecting the 
term B4z in comparison with the term Aaz in (3’) when 47 is 
indefinitely small, so that 4gr=Azz, or denoting these supposi- 
tions by using d instead of 4, dyxr=Adz, has its practical advan- 
tages. 
d 
Finally, we may consider, (if we please, ) =A as denoting 
the operation that must be performed on ¢gz in order to obtain A, 
the co-efficient of the first power of h (only) in the expansion 
of g(z+h); for it is only this co-efficient that is obtained by the 
several methods that we have noticed ; and we may observe that 
if we change h into dz, we shall get 9(7-+-dr)=9xr+Adz-+ Bdz, 
and that the term Adz is the differential of gr, so that we have 
dpx 
der = Adz, or G- =A. 
