Resolution of Algebraic Equations. 269 
lutions are commonly deemed simple equations : they may in 
this view be defined to be, the simple equations that the original 
quadratic, cubic, or other higher given equation, contained in 
power , since they express the nature and form of a quantity 
which , by involution or reverting the operation, re-produces it ; 
as the root of any power, being reinvolved, returns to the power 
from which it was extracted. This fixed and visible connection be- 
tween the equation and the general formula for its roots, throws 
a beauty and elegance into the method of pure algebraic reso- 
lution , which none of the others, such as the method of divisors, 
and all the contrivances for approximation, can pretend to. 
For, when by any of those methods we have obtained one or 
more separate roots, the relation to the original equation is no 
longer perceivable ; but here the chain is perfect. The equa- 
tion leads to the resolution : the resolution embraces at once 
all the correspondent roots ; and, when reinvolved, proves the 
operation, by reproducing the original equation. Thus, for 
example, if ( x 2 — 5a: -p 4 = 0), and it be perceived, or found 
by any conjectural method, that unity is one of the roots of 
that equation, there is no discernible connection between the 
simple equation expressing (a; = 1 ) and the original equation ; 
no transformation of one will produce the other. This latter 
equation (a: = 1), though truly expressing a numeral root of 
the former, is no more a resolution of it than of the equations 
(af — 6 x -}- 5 = 0), (a: 2 — jx 6 — 0), or any other of 
the infinite number of equations of which unity is a root; 
whereas, the algebraic resolution of (a: 2 — gx -f- 4 = 0) viz. 
(g = 5 ^ — ■ — ), which equally expresses (1), and (4) the 
other root, needs only to be cleared of its radical, to shew itself 
mdccxcix. N n 
