Resolution of Algebraic Equations. 271 
answering to the given affected equation, it may be expected to 
be clear or complicated, real or imaginary, not as the roots 
themselves are simple and real, but as the principle of their 
union, of which only it is truly the index , is near or remote : it 
merely shews the central point of their combination, which, 
like the centre of gravity, suspension, or any other power, may 
not actually exist in any of the bodies whose motions it governs, 
but in some imaginary point without, and remote from them 
■all. Had the nature of the algebraic resolution of an equation 
been considered in this light, and the forms to which they are 
proposed to be reduced, been compared with the original forms 
of the roots in the given equation, no surprize or appearance of 
paradox could have arisen in the matter; but it must have been 
clearly perceivable, what cases would admit of real } and what 
only of imaginary resolutions, as will be shewn hereafter. 
I have dwelt the longer upon the nature of the algebraic reso- 
lution of an equation, because it is a very curious subject, about 
which many errors and inconsistencies have been fallen into, 
though hardly any direct examination of it is to be found in 
any of our books. It is the sole method of obtaining a com- 
plete general answer to any problem. It makes algebra con- 
sistent with itself, and sufficient to solve its own difficulties, 
without foreign aid, (from series or other branches;) and, in 
all cases where any general ulterior use is to be made of the 
resolution of an equation, is the only method that avails at all. 
8. In order to obtain this general resolution, the common 
methods have been, (without considering the nature of the 
roots,) to attempt some universal reduction in the forms of 
equations; as. 
N n 2 
