Mr. Horner's new method, (Sc. 309 
it is totally inapplicable. A theorem which should meet this 
deficiency, without sacrificing the great facilitating principle 
of attaching the functional symbols to a. alone, does not 
appear to have engaged the attention of mathematicians, in 
any degree proportionate to the utility of the research. This 
desideratum it has been my object to supply. The train of 
considerations pursued is sufficiently simple ; and as they 
have been regulated by a particular regard to the genius of 
arithmetic, and have been carried to the utmost extent, the re- 
suit seems to possess all the harmony and simplicity that can 
be desired ; and to unite to continuity and perfect accuracy, a 
degree of facility superior even to that of the best popular 
methods. 
Investigation of the Method. 
3. In the general equation 
pX—0 
I assume = r" -{- 
and preserve the binomial and continuous character of the 
operations, by making successively 
x — R -j- z =R-|-r -f z> 
= R' -I- z' = R' + r % «f z" 
= R" -f- z " = &c. 
Where R n represents the whole portion of a? which has already 
been subjected to <p, and z*— the portion still excluded ; 
but of which the part r* is immediately ready for use, and is 
to be transferred from the side of z to that of R, so as to change 
<pR* to pR* 1 without suspending the corrective process. 
4. By Taylor’s theorem, expressed in the more conve- 
nient manner of Arbogast, we have 
qoc = <p ( R + z') = 
<pR + DtpR . z + D 2 <pR . -f D 3 ©R . % 3 •+ 
