( 77 ) 
being negative, it is manifeft, that if ^ or / be nega* 
tive — a'' muft be negative^ and that — may bene- 
4 4 ^ 
gative when q and p are both pofitive ; that is, This 
Rule muft always ditcover fome Roots to be imaginary 
when the vii^^ orix*^ Propolitions difcover any impofli- 
ble Roots in an Equation ; and will very often difcover 
fuch Roots in an Equation when thefe Propofitions dif- 
cover none. For Example, if you examine the Equa- 
tion x"" + ^ X = o, you will 
difcover no imaginary Roots in it by the vii^^' or ix'^ 
Propofitions^ and though AC — i6D( = j) be ne^ 
gative, it does not follow, that the Equation has any 
impoflible Roots, becaufe it appears from the Signs of 
the Terms, ?nat the Equation has Roots alFeded with 
diifefent Signs. But fince B* — xAC — 4D(=: 
36 -f- 10 — 48= — i)is negative, it appears 
from our Rule, that the Equation muft have fome ima- 
ginary Roots. 
I might {hew in the next Place, how the Rules de- 
duced from the and xii*^ Propofitions may be ex- 
tended fo as to difcover when more than two Roots of 
an Equation are imaginary^ and in general to determine 
the Number of imaginary Roots in any Equation ; but 
. as it would require a long Difeuflion, and {omQ Lemma-- 
ta to demonftrate this ftridly, I ihall only obferve that 
i thefe xi^^ and xii*^ Propofitions will be found to be 
iftillthe inoft ufeful of all thofe we have given for that 
IPurpofe. To give one Example of this ; If we are to ex- 
amine the Equation x"’ — ^ a x^ 6 x"" — ^ a h^x 
-j- • — — 
o by Sir IJ'a-ac Ne^ton^s Rule, it is found 
to have four impoftible Roots when a is greater than h ; 
for though the Square- of the fecond Term multipli- 
ed 
