394 -Dr. hutton on Cubic Equations 
Thus then we eafily perceive in all cafes the nature of 
the roots as to real and imaginary ; namely, partly from the 
lign of p, and partly from the relation of p to q: for the 
equation has always two imaginary roots when p is pofi- 
tive ; it has alfo two imaginary roots when p is negative, 
and lefs than 1^1 ; in the other cafe the roots are all 
real, namely, when p is negative and either equal to 
• or greater than ±ql . 
12. Moreover, when p is = o, the equation has two 
imaginary roots ; for this cannot happen but by — e 1 be- 
coming + <?% in the value of p, and = to |r 2 ; and then 
p - - ir z + e 1 = - + |f 2 = o, and q — r . ±r 2 + e z = 
r • \r % + |r 2 = r . r 2 = r 3 , and confequently the above 
becomes barely x z — r 3 , which therefore, belides one real 
general equation root x - r, has alfo two imaginary roots. 
13. Hence alfo it again appears, that the greateft root 
is always of the fame affection, in refpedt of poiitive and 
negative, w T ith q on the right-hand fide of the equation, 
they being either both pofitive or both negative toge- 
ther; and the other two roots of the contrary fign. For 
if r be the greateft root, then is \r greater than £, and 
ir 2 greater than <?% and ^r 2 - e 1 always pofitive, and con- 
fequently the product r .~r L - e z , or <7, will have the fame 
fign with r. But if r be one of the lefs roots, the con- 
trary 
