3 02 Mr. Wilson’s Essay on the 
that of the function, formed by summing the combinations by 
two of the roots in pairs, or summing the roots themselves in 
pairs, or the equation of the halves, or quarters, or doubles, 
trebles, &c. of those functions, is immaterial; no new function 
is employed, no other principle put in action, than what is de- 
rived from the general properties of this degree of quantities 
here explained. 
31. Biquadratics being generally thus reducible to cubics, of 
course, by resolving those cubics, distinguishing what function 
their roots are of the roots of the original biquadratic, they may 
all be found ; and, for practical utility, there is no preference to 
be made of either of the two methods ; for, the first, though a 
real cubic, being formed from products of the roots, it requires 
a quadratic equation to obtain them after the cubic is resolved ; 
whereas the second , though an equation of the sixth power, 
being formed from simple addition of the roots, gives them at 
once. But, as both these cubics necessarily have all their roots 
real, when those of the given biquadratic are so, and the reso- 
lution of cubics is in that case imaginary, it follows, that no 
biquadratic having all its roots real , can admit of a real solution 
by either of these methods. 
32. The formula expressing the actual resolution of a biqua- 
dratic has not been given ; the writers upon algebra going no 
further than to point out the cubics by means of which such a 
resolution may be obtained. To be sure, such a formula would 
be very long, and (till the imperfection in the cubic resolution, 
which must make a large part of it, can be removed, ) embar- 
rassed with radicals, so as to be of little practical use; but it 
would be a valuable accession to the theoretical part of algebra, 
to have the analysis of this degree carried as far as that of the 
