( «°7 ) 
PROP. II, 
In any Cubic i^quation, all whofe Roots are real, 
the Square of the fecond Term is always greater than 
the triple Produd of the firfl: and third. 
If the Cubic Equation has all its Roots real,' they - 
may be reprefented with their Signs by c, and the 
i^quation will be expreffed thus : 
— ay^J^a hy — a h c=o 
• — ^ 
— ^ cy 
But by Lemma is always greater than 
a h-\^ac-{-h c ; and confequently adding 2 a h-\-z ac^zlc 
to both fides, a h-\-^ a r-f-2 Ic 
will be greater than 3 ^^+3 he; and therefore 
Tp+Z xjy^ muft be greater than nc-\-'ih c xjS 
that IS, the Square ot the’ fecond Term muft be greater 
than the triple Produd of the firft and third Terms. 
Cor. I. In general, it appears from the Demonftra- 
tlon, that the Square of the Sum of three real Quan- 
tities, a-\-b-\-c' is always greater than the triple Sum 
of all the Produds, that can be made by multiplying 
any two of them into one another. 
Cor. 2. It follows from the Propofition, that when 
the Square of the fecond Term is not greater than the 
triple Produd of the firft and third Terms, the Roots 
of thc i^quation cannot be all real ; but two of them 
muft be impoffible : And this plainly coincides with 
one Part of Sir Ifaac Newtons Rule for difeovering 
when the Roots of Cubic /Equations are impoftible. 
He 
