( MI ) 
be negative and be lefs than ^ r% it appears that 
two Roots of the Equation mufl be impoITible, from 
this Corollary, and from Cor. 3. Prop. z. taken toge- 
ther. 
PROF. IV. 
In a Biquadratic i^^quation, all whofe Roots arc real 
Quantities, \ of the Square of the fecond Term is al- 
ways greater than the Produdf of the firft and third 
Terms ; and i of the Square of the fourth Term is al- 
ways greater than the Produdt of the third and fifth 
Terms. 
I. Let the ^Equation bex^ — p x" — r x-^s=o; 
and fince the Roots are fuppofed to be all real, let 
them be reprefented by a, h, c, </, then p=:a-\-h-\^c-\-d^ 
and q^=-a l-^a c-\-a d-\-b d^c d. But it is plain from 
Lemma 3, that 3 -j- 3 4. 3 c' 4. 3 is greater than 
z a I z a c-\-z a d^ zlc-\-z h d-\^ zed; and con- 
fequently by adding 6 ah 6 ac ^ 6 a d-\- 6 hc-\- 6hd 
6 c d to both, we fliall find that 3 x a'-{-b-^ c4-/ 
mufl: be greater than 8 ^ ^ 4 ta 8 ^ ^/-f 8 h r-j- 8 
h d-\~ Sc d; that is, 3 greater than 8 q 5 and there- 
fore ^ x^ greater than q x^. 
z. Since r^ahc-\-ahd-\-a c d h c d^ and 
ah c d; and fince ^ j is equal to c d 4- r' ^ ^ 4 
d^ h c h'- c^ a d^ k d^ ac-\>c" d^ a h, which are the 
Produces can be made of any two of the Quantities 
ah Cy ahd, ac d, h c d, whofe Sum is r multiplied by 
one another ^ it follows, that 3 r* is always greater than 
S q s : So that I of either, the Square of the fecond 
Term, or of the Square of the fourth Term, mufl al- 
P z, ways 
