9 
term, and whereof the roots are x 1 and x 2 has both its roots 
expressed by the formula 
2 ^ 4x x x 2 > 
now we know that this formula may be cleared from 
radicality. There is a way, and it seems to me a unique 
and perfectly definite way, of doing so. Since the quad- 
ratic is imperfect 
x 1 -\-x 2 — 0 ^ 
and since we may introduce a zero (additively) under the 
radical sign, we may put the radical formula under the form 
JO — 4x x x 2 — J(cc i +# 2 ) 2 — 4x x x 2 = J(x 1 —x 2 ) 2 =±(x 1 — x 2 ), 
and then 0±.(x x — x i )=x x <\-x. 1 ±.{x x — x 2 )=2x x , or 2x 2 , 
and thus - J—4x x x 2 may be made to yield the roots. More- 
over, there seems no other way of obtaining the roots, and 
the result and process are each unique. We may always 
supply zero functions whether simple as a = 0, or complex 
as a? — 6 = 0, and the conditions which they are to satisfy 
are such as to enable us to extract the roots of the radical 
expressions in a^, x 2 , &c., for Hargreave (see p. 121) does 
not dispute Abels principle, but rather regards his theorem 
as a truism (ibid). Again, I cannot concur with Hargreave 
in supposing that there is an impassable gulf between the 
trinomial quartic or quintic, and the perfect forms (see, for 
instance, pp. 90, 100-1). Surely from 
Z n +AZ+B=0 
we can, by assuming 
C— « 0 Z” — 1 a 1 Z ,i ~ 2 -f- . . • 
pass to the n- ic 
&c.=0 
whose n co-efficients will involve n arbitraries a 0 , a h . . a,^. 
and, therefore, be arbitrary, and the equation consequently 
general. With regard to Hargreave’s theory of the cubic, 
the formulse at p. 24 seem to show that, interesting though 
his theory be, it ultimately coincides with Lagrange’s, for 
