The third is found by 
fubftituting in mf! E'E' _ yio B'H' = a>" the Va- 
B'H'; and by a like Subftitution 
thele Iheorems may be continued. 
^ General COROLLARY. 
From thefe Propofitions a great Variety oif Rules 
may be deduced for 'difcovering when an Equation has 
imaginary Roots. The Foundation of Sir Ifaac New- 
tons Rule is demonftrated in the ninth Propofition, and 
Its Corollary. The feventh Propofition {hews that if 
~~l' not exceed DF — CG+BH A I 
'+ K, fome of the Roots of the Equation muft be iina. 
ginary ; and fometiines this Rule will difcover impof- 
hble Roots in an Equation, that do not appear by Sir 
f’fvat’ Intro V\ 1*1. the only two Rules 
that, have been Imherto publilhed. But the Rules that 
arife from the Theorems in the eleventh and twelfth 
Propofitions, are preferable to both ; becaufe any imaei- 
nary Roots that can bedifeoveredbythe vii'i'or ix'’' al- 
ways appear from the xi‘i> and xii'h Propofitions • and 
impolfible Roots will often be difeovered by the’ xi* 
and xii'h Propofitions in an Equation, that do not an. 
pear in that Equation when examined by the vii* and 
The Advantage which the Rules 
deduced from the xi«i> Propofition, have above thofe de- 
duced from the preceeding Propofitions, wiU be mani- 
feft by confidering that in the xi'» Propofition we have 
the Values of the auantities a', h>,c', d',e', feparately • 
S VaTues nf " Propofitions, we have Jly 
the Values of certain Aggregates of thefe Quantitira 
L i. joined • 
