Resolution of Algebraic Equations. 299 
in one of the steps taken, when it is, in truth, of much deeper 
origin than any particular method, being the necessary conse- 
quence of the constitution of the cube power. 
27. The result of these observations upon cubic equations 
shews, that directly they are not resolvable, i. e. they cannot, 
like quadratics, be always brought to a mere extraction of their 
correspondent root : that, however, by means of the peculiari- 
ties inseparable from the number of three quantities , a relation 
is discoverable, which inevitably gives equal roots to the equa- 
tion of the cubes of a particular function of them ; but that, 
that function involves sometimes a quadratic surd which was 
not in the roots themselves, but arose from the form necessary 
to be given them ; that the equal relation not taking place in 
any case, till the cube of that function, and, in some cases, not 
being rational , till the square of that cube, the equation is not 
lowered in degree, by the operation, but rather increased. 
28. Let [n = 4), and the equation become the general bi- 
quadratic (x 4 — px 3 -{- qx z — rx -{• s = o), the number of 
differences are twelve ; we cannot, therefore, hope to obtain a 
direct simple resolution. But, in Art. 14, two peculiarities be- 
longing to sets of four quantities were pointed out, from which 
it is easy to obtain a reduction of the equation to a cubic form. 
The first peculiarity there mentioned, was shewn to subsist 
among the sums of the combinations of the roots in pairs. 
If ( a , b, c, d,) be supposed the roots of the given equation, 
and their combinations by two ( ab , ac, ad, be, bd, cd,) be 
summed in pairs, though the number of quantities so formed 
are no fewer than 30, yet there is an evident distinction ob- 
servable amongst them; for, in some, (the first six,) no letter 
occurs twice. If, therefore, instead of simply requiring the 
