( ^5 ) ' 
of the Parts of E by each other, that is, than E'x E. 
but X E' X E, = E* ^ E' E' = (by the firft Theo- 
rem in the laft Coroliaa-y) 2DF — xCG + xBH 
— 2 I -f- ^ K!, and therefore fince x E* 
» 'll 
muft always exceed E'E, , it follows that —Hi- E* 
2 I 
muft always be greater thanDF — CG-)-BH 
A 1 -f K when the Roots of the Equation are real 
(Quantities. 
S c H o L. In following my Method this was the firft 
general Propofltion prefented itfelf. For having firft ob- 
ferved that if /expreftes the Number of any Quantities 
the Square of their Sura multiplied by I muft 
a ways exceed the Sum of theProdufts made bv multi- 
plying any two of them by each other-, and that the Ex- 
cels was the Sum of the Squares of the Differences of 
the auantmes divided by i / , it was eafy to fee in the 
Equation -v ’ — A w Ba? _ Cx ’ -j_ 
D A- ” * 8 cc. = o. . Since B is the Sum of the Pro- 
dufts of any two of the Parts of A, that if / expreffes 
theNumber of the Roots of the Equation, — LL,^ x* 
2 / 
muft -always exceed B; and this is one Part of 
•the 5th Propofition. In the next Place, I comnared 
the Sum of the Produfts of any two Parts of B with 
AQ and found that it was not equal to AC but to 
A C — D from which I inferred, that if / expreffes 
^ ^ the 
