316 Mr. Horner’s new method of solving numerical 
<pR"=?.R' + 0 A' 
A'=(A+ ,B) + C B' 
B'=(B + ,C+ 2 C)-f 0 C' 
C' = (C+,D+ 2 D + 3 D) + 0 D' 
• • * * • • • 
V' = ( V + ,u + 2 u + 3 u + „_ 2 U ) + 0 U' 
U' = (U + n— l . r) + / . . . [Ill] 
the subordinate derivations being understood. 
17. The Theorems hitherto give only the synthesis of <px, 
when x = R + r -|- r' &c. is known. To adapt them to 
the inverse or analytical process, we have only to subtract 
each side of the first equation from the value of <px ; then 
assuming <px — cpR* = A*, we have 
a' = A — q A 
A = a + B 
&c. as in Theorem I. 
a"= A 7 - — 0 A' 
A’= (A + jB) + 0 B' 
& c. as in Theorem II. or III. 
The successive invention of R, r, r', &c. will be explained 
among the numerical details. In the mean time, let it be 
observed that these results equally apply to the popular for- 
mula q>x = constant, as to <px — 0. 
18. I shall close this investigation, by exhibiting the whole ' 
chain of derivation in a tabular form. The calculator will 
then perceive, that the algebraic composition of the addends 
no longer requires his attention. He is at liberty to regard 
the characters by which they are represented, in the light of 
mere corresponding symbols, whose origin is fully explained 
