First Principles of the Differential Calculus. 271 
and dz,-dex are called their differentials, and =A is called 
the differential co-efficient, since we must multiply dz by it to 
obtain doa =e .dv=Adz. The same results are readily obtain- 
ed by writing for 4a in the first member of (3’), dpr+ 4’grx=A4r 
+Bzz, the d relating to the term that involves the first power 
of 4 only, and the 4 to the remaining part of the right mem- 
ber of (3’), .*: we get dpr=Asr=Adz, by using d for 4, in the 
tight 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 dpc =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 
gi» BQ€ aes) . tae _ Aga 
regarding ~7 = Aas expressing the limit of the ratio Zo=. 
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. Tamales amt ES oh Rg 
~ Again, the method of Leibnitz, which consists in rejecting the 
term B4z in comparison with the term Az in (3) when 4z is 
indefinitely small, so that 4¢¢=Adz, or denoting these supposi- 
tions by using d instead of 4, dyr=Adz, has its practical advan- 
> ‘Migess’ ee be ee é 
| brant rc a ee der , 
. Finally, we may consider, (if we please,) 7 ~-=Aas 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(a+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 o(7-+dv)=92+ Adzr+Badz, 
and that the term Adz is the differential of gz, so that we have 
Pe en tee ae 
dgxz= U or a, =A. 
