( 75 ) 
6cc. are found by adding to m the S(]uares of the 
Numbers i,x,3,4, 6cc. which (hew the Diftances of 
the Coefficients to which they are prefixed, from the 
Coefficient E. The fecond Theorem of the xii^^ Pro- 
pofition (hew?, that if E* does not exceed 
% 
— nJ — ~ IX X CG m — x BH ^ 240 
X A I + — 600 X K, the Equation mufthavefome 
Roots imaginary. ' 
For an Example, If the four Roots of the Biquadra- 
tick Equation x* — Ax^ + B X — C X — j— D ^ — o 
are real Quantities, it will follow equally from the 
vt*’, vihh, ix^h, and Propo(ition?,that — A* muft be 
3 ^ 
greater than B, and that — C* mud exceed B D. The 
o 
r 
vii^^'further (hews that — B* mud exceed AC D* 
IX ’ 
the ix^^demondrates that B* mud exceed A C ; but 
9 
• our Rule deduced from Prop xi. (hews that x B* mud 
exceed 5* A C — 8 D,the excefs being a', and the 
X 
Rule deduced from the fecond Theorem of the xih*' 
Propofition (hews that B* mud always exceed x A G 
•+ 4 D, the Excefs being It appears from fe- 
veral preceeding Propodtions, that if the Roots of the 
Equation have all the fame Sign, then A C mud ex- 
ceed 1(5 D : Let the ExceiTes 5*B*-^ ixAC -f ixD 
= 4B — 9AC = y, AC-^ — 16 0 = ^* and 
it is plain that a' ( = 4B’ — 10 AG + 16 D) == ^ 
— s 
