I io8 ) 
• - 
He deHres you may write above the middle Terms 
of the Equation the Fradions 3-, 3’ as 
in the Margin ; and placing the 6ign 
+ under the f:rft and lalt Term, -f. _ % -f 
he multiplies the Square ‘of the fe- 
cond Term by the Fradion i that is above it ,* and if 
the Produd is greater than the Produd of the adjacent 
7 ^erms> he places ~f- under the fecond Term ; but if 
that Produd is Ids, he places — under the fecond 
Term, and fa>s, there are as many impodible Roots 
as changes in the S^gns. Now by this Propohtion, if 
X' is not greater than 3 or greater than 
y A* the Roots cannot be all real. 'J he fame vS'uppofi- 
tion makes two Changes in the .Signs, whatever Sign 
you place under the third Term, fince the Signs under 
the firfi; and laif are both -f.; and therefore this Propo- 
fition demonftrates the firlt Part of Sir '/Jaac Newton's 
Rule, as far as it relates to Cubic iTquations. 
* 
Cor. 3. If the fecond Term is wanting in a Cubic 
’Equation, and the third is Pofitive, two of the Roots 
oi theiTquation muft be impojfible : for the Square of 
the fecond Term ft qua! to nothing in this Caiej will 
be Ids than the triple Produd o\ the adjacent Terms. 
But this will better appear from confiden'ng that, when 
the fecond Term vaniflies in an ^^quation, the Pofitive 
and Negative Roots are equal, and W'hen added together, 
deflroy each other. Suppofe the Roots to be 4-^ and 
— — c ;■ then in this Cafe 4-^ -|- c, and the Co- 
efticient of the third Term will be --ah — ac-\-hc 
— — ^2 — 2, be — d-j-^c = — — be — and con- 
fcquentiy Ne.g^ive. Or, if you fuppofe two Roots 
Poiitive and offe^tgatlve, let them be — a -f-^> 
then the CdeRicient of the third Term w.ll be Uill 
— h — he — d. Therefore when the Roots are real, 
the 
