40 8 Dr. hutton on Cubic Equations 
equal to \p with its fign changed. But this may always 
take place when p is pofitive ; for then - \p will be ne- 
gative, and two numbers, the one pofitive and the other 
negative, may always be taken fuch that their product 
ihall be equal to any negative number whatever, and yet 
their Turn be equal to a given quantity x ; and this is done 
by taking the pofitive one as much greater than x as the 
other is negative ; for thus it is evident the pofitive and 
negative numbers may be increafed without end : where- 
fore there is no impoffibility in the fuppofition when p is 
pofitive ; and therefore then the formula ought to exhi- 
bit only real quantities, that is, in all the cafes after the 
1 6th in the table of forms, as we have before found. 
But the fame thing cannot always happen when p is ne- 
gative, or - jp - zy is pofitive : for that zy may be pofi- 
tive, the figns of the two factors z and y mull be alike, 
either both + or both -, that is, both + when the fign 
of o' is +, or both - when that is - : but it is well known, 
that the greateft product which can be made of the two 
parts into which a conftarit quantity a; may be divided, is 
when the parts are equal to each other, or each equal bx, 
and therefore the greateft product is equal to i aJJ- or | x z : 
wherefore if ‘£-x* be equal to or greater than - jp, the 
condition which fuppofes that zy is = — jp, is poffible, 
and the formula ought to exprefs the root by real quan- 
tities 
