274 
Mr. Wilson’s Essay on the 
endeavours to improve these general methods, had, instead of 
involving themselves in a labyrinth of substitution and process, 
upon the chance of some means of simplification presenting 
itself, considered beforehand the probability of success, the 
imperfection of Cardan’s rule would never have appeared a 
paradox, nor the interruption of all further progress by it have 
given room for surprise. They must have seen, that no equation 
beyond a quadratic can admit of a real extinction of its inter- 
mediate terms. In the general equation ( x” — px n ~ l -f- qx n ~* 
— rx n ~ a -|- sx n ~ * .&c. = o), ( p ) being the sum of the roots, 
and (q) the sum of their combinations in pairs, by Sir I. New- 
ton’s theorem for finding the sums of the powers of the roots, 
( p 1 — 2 q) will be the sum of their squares; and therefore, if 
both (p) and (q) vanish, the sum of the squares of the roots 
must vanish also ; which can never happen with real quantities. 
Besides this, in attempting to destroy many intermediate terms 
at once, we know by experience, the equations that become in- 
cidentally necessary to be solved, rise to a much higher dimen- 
sion than the given equation ; so that our labour, in this respect, 
defeats itself. 
10. Nor will these difficulties be avoided, if we abandon the 
idea of a general resolution, and attempt to work out the roots 
separately : although the number of coefficients is always suf- 
ficient to afford a distinct equation to each root, and therefore, 
by the common principles of indeterminate equations, will 
clearly determine them all ; and would also find them, if the 
equations afforded by the coefficients were all of the same de- 
gree ; but they rise successively, and, from the drawing them 
together, in order to expunge the several unknown quantities, 
the index of the reducing equation increases so as to defeat the 
