406 Br, hutton on Cubic Equations 
always the greateft root as it has been commonly 
thought to be. 
46. The firft root r - s + d - b + v b l + a 1 
■ — . 
+ \f b — V0* + (2 3 , although it be always a real quantity, 
yet often affumes an imaginary form when particular 
numbers are fubftituted inftead of the letters a and b, or 
p and q. And this it is evident will happen whenever a is 
. — .. 
negative and a> greater than b 1 , or ~p s greater than \q ; 
for then v 7 q + 4 s becomes vV — a 1 = sl\(f - j/>* the 
fquare root of a negative quantity, which is imaginary. 
And this will evidently happen whenever the equation. 
has three real roots, but at no time clfe, that is in all the 
firft 1 3 cafes of the foregoing table, wherein |/>T is 
greater than fqf and p negative; the 4th and 1 3th only 
excepted, when j/^’ is = and therefore b' 
a 3 = o, 
and two of the roots become equal, but with contrary 
iigns. This root can never affume an imaginary form 
when a or p is politive, nor yet when /* is negative and 
| ~q greater than |/>f ; for in both thefe cafes the quantity 
■s/^ ± « 3 is real, or the fquare root of a pofitive quantity; 
And thefe take plate after the firft 13 cafes of the table 
Of “forms, that is, J in all the cafes which have only one 
real foot. So that this rule of cardan’s always gives the 
Y 
ft O’ j £}.*'■* j 4% «>'1 V .wii 
root 
