V 
(70 
— j -y X isp — s '^ and that =r ^ 4- = -i- 
2'/+ 4*^* Let us fuppofe, 
I. That s is pofitivej then it is manifeft that if ei- 
ther pox qht negative, muft alfo be found negative, 
and confequently that when the vii^^ or ix^*’ Proporti- 
ons (hew any Roots to be imaginary, the xi^h Proporti- 
on muft diicover them at the fame time. But as 
X 
(= q^s =z — X s) may be found negative 
when p and q are both pohtive , it follows that the 
Rule we have deduced from the xi^^ Proportion may ' 
difcover imaginary Roots in an Equation, that do not 
appear by the preceeding Proportions : Thus if you 
examine the Equation — 6 ^ * + 10 — 7 a; 
+ I hy Six Jfaac Newton^s Rule, or by our vii^^ 
Proportion, no imaginary" Roots appear in it from 
cither. But ftnce xB* — 5'AC-i-8D (= ) = 
200 — xio-j- 8 == — X is in this Equation nega- 
tive, it is manifeft that two Roots of the Equation muft' 
be imaginary. ' Let us fuppofe 
X That j is negative, and that from the Signs, of the 
Terms of the Equation, it appears that fome Roots are 
portive and fome negative ; then in Order to fee if the 
Equation has any imaginary Root?, the moft ufeful 
Rule is that we deduced from the fecond Theorem of 
Prop. xii. that if B" does not’ exceed xAC-j- 
4 D fome of the Roots of the Equation muft be ima- 
ginary: For the Excefs ofB^above x AC + 40be- 
, I .1 I 
ing — of* — xq s ^ 2, / -p 4 and s 
4 4 
being 
