C 308 3 
XXI. A new method of solving numerical equations of all 
orders , by continuous approximation * By W. G. Horner, Esq. 
Communicated by Davies Gilbert, Esq. F. R. S. 
Read July 1, 1819. 
i.The process which it is the object of this Essay to esta- 
blish, being nothing else than the leading theorem in the 
Calculus of Derivations, presented under a new aspect, may 
be regarded as a universal instrument of calculation, extend- 
ing to the composition as well as analysis of functions of every 
kind. But it comes into most useful application in the nume- 
rical solution of equations. 
2. Arbogast’s developement of 
<p ( oc -j- fox -J- yx* Jx 3 -j- ex* -J- . . • . ) 
(See Calc, des Der. § 33) supposes all the coefficients within 
the parenthesis to be known previously to the operation of q>. 
To the important cases in which the discovery of y, J, &c. 
depends on the previous developement of the partial functions 
<p (a -f- / 3 x), (a + + <yx 2 ), &c. 
* The only object proposed by the author in offering this Essay to the acceptance 
of the Royal Society, for admission into the Philosophical Transactions, is to secure 
beyond the hazard of controversy, an Englishman’s property in a useful discovery. 
Useful he may certainly be allowed to call it, though the produce of a purely mathe- 
matical speculation ; for of all the investigations of pure mathematics, the subject of 
approximation is that which comes most directly, and most frequently into contact 
with the practical wants of the calculator. 
How far the manner in which he has been so fortunate as to contemplate it has 
conduced, by the result, to satisfy those wants, it is not for him to determine ; but 
his belief is, that both Arithmetic and Algebra have received some degree of improve- 
ment, and a more intimate union. The abruptness of transition has been broken 
down into a gentle and uniform acclivity. 
