270 
Mr. Wilson's Essay on the 
but another form of the same equation; and gives (jr*— 5* 
-j- 4 = o) as at first. 
7. This view of the algebraic resolution of an equation 
shews, that it does not so much aim at giving us the roots 
themselves, as the basis or common principle of their artificial 
combination in the equation to which it applies; pointing out 
some form of a perfect power, of which they may be conceived 
to be the correspondent natural roots. From which it follows, 
that if the transformation required to be made in the given 
equation be possible, or such as can really be effected, the 
resolution will be real; for every real power has some real 
root: but that if, on the contrary, the power into which the 
equation is conceived to be transformed be merely imaginary, 
the resolution must be so too ; for all the roots of an imaginary 
power are themselves imaginary. It doth not therefore depend 
upon the nature of the roots of the equation themselves , but on 
the form which the equation must assume to become a perfect 
power, to determine, whether the resolution be real or imagi- 
nary : so that the nature of the resolution, and that of the 
roots of an equation may be very different, as we know is fre- 
quently the case ; particularly in the resolution of cubic equa- 
tions by Cardan's rule, where, when the roots are real, the 
resolution is almost always imaginary. This has seemed to 
surprize and perplex some writers very much, who have treated 
it as at best a paradox, if not a contradiction,* but surely 
without cause ; for, as the formula affects only to be an ideal 
representation of the mechanism or structure of a perfect power 
* Vide Playfair on the Arith. of Impossibles, Phil. Trans. 1778, p. 318 ; Dr. 
Hutton on Cubic Equations, ditto, 1780, page 387; and Mr. Baron Maseres, 
Script. Logarithm Vol. II. p. 456. 
