( 5 i0 ) 
Proposition, I. 
Let x* — B x n ~ l C x n ~ 2 — * 1 ) x”** -|~ 
Ex n ~ ♦ — &c. -±e x* + a x % + cx 2 +bx ±_ A— o 
be an /Equation of any Dimenlions all whofe Roots 
are real, let M be any Coefficient of this /Equation, 
Z/, N the adjacent Coefficients, and m the Exponent of 
M. Then the Square of any Coefficient M multi- 
wx® — m 
will al- 
ply’d by the Fraction 
W-f I x» — «? + I 
ways exceed the Rectangle under the adjacent Coeffi- 
cients L x N. Thus in the /Equation x 4 — B x i + 
Cx 2 — T> x + A — o, where n = 4, making M ~C 
and therefore L = B, N = Z), and m =z z, then 
1X4 — 1 
==_ x C 2 or — C a will exceed BxD 
i+ I X4 — x+i 9 
providing all the Roots of the /Equation be real. 
Becaufe (by Lem. 3.) the Roots of the quadratick 
/Equation n x 
n 
x 2 — n — 1 B x -j- C = o, are 
real, therefore (by Lem. 1.) \n— i| 2 xi? 2 mult be 
n — 1 
greater than n x X C and (dividing both by 
n x - — - ^ x B 2 greater than 1 x C. Therefore in 
z J zn 
the /Equation x n — B x n ~~ 1 C x n ~ 2 — E) x n ■~ 3 -f» 
&c. ± A— 0 of the n Degree, all whofe Roots are 
real, the Square of B the Coefficient of the fecond 
Term, 
