Resolution of Algebraic Equations. 281 
properties, though without any order or connection, and confined 
merely to particular ranks or numbers of quantities, being ge- 
neral to all possible or imaginable quantities of those classes , 
afford methods general, as to those degrees, but without pro- 
ducing any result really general to equations at large. 
15. Having shewn that an indeterminate general equation 
cannot be resolved by any of the methods whose principle is 
yet known, because they are all grounded upon the assumption 
of some particularity, either inherent in the roots, or universally 
communicable to them, which, so far from being general, is 
seldom found, and absolutely incompatible with many sorts of 
roots; that the difficulty is in all cases the same, — the intrusion 
of superfluous roots and higher radicals ; that a relation of any 
kind (when known) obviates that difficulty, as far as it extends; 
and that some orders of quantities have generally a constant and 
necessary relation, more or less remote, I proceed to examine, 
more minutely, the application of these observations to the se- 
veral degrees of equations to which they materially apply. 
CHAP. II. 
Of the Resolution or Reduction of Equations of particular 
Degrees. 
16. In examining those degrees of equations which submit 
to be resolved, I shall observe the same order as I did before; 
i. e. first consider the power of obtaining a general formula, or 
complete resolution ; and, if that is not attainable directly, in- 
quire by what general means the roots can be separately inves- 
