( IC9 ) 
the Coefficient of the third Term is negative,* and if 
the Coefficient of the third Term is not affcded with 
a negative Sign, it is a Proof that two of the Roots 
are Impoffible. ' 
PROP. 111. 
t 
In any Cubic -Equation, all whofe Roots are real, 
the Square of the third Term mu(f be greater than the 
triple Produd of the fecond and fourth Terms. 
In the fame Cubic. Equation, whofe Roots are a, 
the Square of the third Term is ab-\-nc-\-bc^ the Pro-' 
dudf of the fecond and fourth Terms is a^hc~\~ah^c 
-{•ahc^^ as is plain from the Infpedlion of the 
quation ,* and it is obvious that h c^a c^a h 
is the Sum of the Produces of any two of the 
Terms 4 ^ c ; and therefore by Carol, i Prop. 
2. the Square of the Sum of thefe Terms, that is, 
a b a c -\- b~c' mufl be greater than 34^ 3 4^'r + 
3 ac^ h. So that a b ac b~'c mud be greater than 
3 /I b^ c^^T^b X ,* that is, the’ Square of the 
third Term mufl be greater than the triple Produdl of 
the fecond and fourth Terms. 
Cor. I. It follows from the Demondration , that 
is always greater than be % ;r+Tqr^. 
Cor. 2. If the Square of the third Term is found 
to be lefs than the triple Produd of the fecond and 
fourth Terms, then the Roots of the ^^quation cannot 
be all real Quantities ; and this concludes with the fe- 
cond Part of Sir Ifaac Newton's Rule for finding when 
the Roots of a Cubic i^quation are 
impoffible: For this Cafe gives — ■ to 
be placed under the third Term, and ~ -f 
P 
con- 
