318 Mr. Horner's new method of solving numerical 
popular methods of approximation ; a circumstance in favour 
of the latter, which is overlooked by many algebraists, both 
in employing those methods, and in comparing them with 
processes pretending to superior accuracy. The radical fea- 
ture which distinguishes them from ours is this : they forego 
the influence of all the derivees, excepting the first and 
perhaps the second ; ours provides for the effectual action 
of all. 
20. Concerning these derivees little need be said, as their 
nature and properties are well known. It is sufficient to state 
that they may be contemplated either as differential coeffi- 
cients, as the limiting equations of Newton, or as the nume- 
rical coefficients of the transformed equation in R + z. This 
last elementary view will suffice for determining them, in 
most of the cases to which the popular solutions are adequate ; 
viz. in finite equations where R, an unambiguous limiting 
value of x , is readily to be conjectured. When perplexity 
arises in consequence of some roots being imaginary, or dif- 
fering by small quantities, the second notation must be called 
in aid. The first, in general, when <px is irrational or trans- 
cendental. 
si. The fact just stated, namely, that our theorem con- 
tains within itself the requisite conditions for investigating the 
limits, or presumptive impossibility, of the roots, demonstrates 
its sufficiency for effecting the developement of the real 
roots, independently of any previous knowledge of R. For this 
purpose, we might assume R = o ; r, r, &c. = i or . i &c. and 
adopt, as most suitable to these conditions, the algorithm of 
Theorem II, until we had arrived at R*, an unambiguous 
limiting value of x. Rut since these initiatory researches 
