404 Dr. hutton on Cubic Equations 
that which fuppofes 3 zy to be = - p ; and in confe- 
quence of this fuppofition, the equation becomes barely 
z 3 + y 3 = q. Now from the fquare of this equation let 
four times the cube of zy - - \p be fubtrafted, and 
there will remain z 6 - zz 3 y 3 + y 6 = q~ + pj p 3 , the fquare 
root of which is z 3 - y 3 — ^ q 1 + p- p 3 ; this lafl being 
added to, and fubtracled from, the equation z 3 + y 3 = q> 
lZ T =q+ s f q 1 + pjp 3 = q + 2 y/t? + 
2 f = q~ \/q z + pjp 3 - q - 2 , 
hence dividing by 2, and extracting the cube roots, we 
r« »•&-»+ V!S> 1 1!* — 
we have 
have 
7 
A 
-s/^a- 
p' x 1 or x 
+ x 1 or x - ■ 
~ 2 
three values of z andjy; for every quantity has three dif- 
ferent forms of the cube root, and the cube root of 1, 
is not only 1, but alfo - or - 1 ~ v "~ 3 . Hence then 
the three values of z +y or x, or the three roots of the 
v4<7 + "JV?* + IF ' 
2 
V — 3 
the 
equation a? 3 + px = q, are V^ + 
I + \/ 2 I — V' — 2 
x 2 or x — 
x — 
2 
V — 3 
or x - 
a ■ A-V-A 
, + — 3 
p' x i or 
x 1 or 
, where the figns of \/— 3 
muftbe oppolite in the values of z and y, that is, when 
it is 
^—3 
in the one,, it muft be 
1 y—3 
2 
in the other, 
otherwife 
