272 Mr. Wilson’s Essay on the 
1st. The destroying their intermediate terms, and converting 
them into pure powers. Or, 
2dly. The discovering some constant complement which will 
always raise them to the nearest perfect power. In both which 
cases, the resolution will afterwards be nothing more than 
simple extraction of the proper root. Or, 
3dly. The assuming some convenient formula with indeter- 
minate coefficients; and, by assigning their values properly, 
adapting it to every case. 
It would be going to too great a length, to give distinct ex- 
amples here, of the application of these methods. Numerous 
instances of each of them are given in the common books of 
algebra, which usually treat them as separate and distinct from 
each other; but the fact is, they are all in truth the same. 
Whoever tries them separately, will find, however variously 
they seem to set out, they lead precisely to the same conclu- 
sions, and fail precisely in the same points. A quadratic, whe- 
ther resolved by completing the square, or by expunging the 
second term : a cubic, whether resolved by Cardan’s rule, or 
by completing the cube, or by assuming a resolution, as sug- 
gested in Dr. Waring’s Meditationes Algebraic re, (p. 179, 180.) 
present the same formula of resolution, and the same limita- 
tions and irreducible cases. And the reason is easily found. To 
complete the requisite power, (according to the index of the 
equation,) or to destroy the intermediate terms, occasions an 
alteration in just the same number of terms ; it is only the parti- 
cular relation they are required to bear to each other that is 
varied. In the one case, they are all to be equal, (or equal to 
nothing;) in the other, to correspond respectively with the 
