.( 74 ) 
joined with the fame Signs. Now it is obvious that 
if thefe Quantities be feparately found pofitive, any 
fuch Aggregates of them muft be pofitive ; but thefe 
Aggregates may be pofitive, and yet fome of the Quan- 
•tities a'y h\ c\ e^, themfelves may be found negative : 
From which it follows, that if the -Roots of the 
Equation are all alfeded with the fame Sign, and no 
impoflible Roots appear by Propofition xi*'^, none will, 
appear by the preceeding Propofitions- but that fome' 
imaginary Roots may be difcovered by Propofition 
xi*^,when none appear in the Equation examined by 
the Propofitions that proceed the xi^*’. If fome of 
the Roots of the Equation are pofitive, and fome ne- 
gative (which always eafily appears by confidering the 
Signs of the Terms of the Equation) then the xipi^ 
Propofition will be in many Cafes more apt to difco- 
ver imaginary Roots in an Equation than thofe that 
proceed it. 
The Rule that flows from the firfl: Theorem of. the 
xi^^^ Propofition, obtains when, the Roots of the Equa- 
tion are affeded with different Signs, as well as when 
they all have the fame Sign, and it is this. Multiply 
the Number of the Terms in an Equation thatpreceeds 
any Term, asEA? by the 'Number of Terms that 
follow it in the fame Equation, and call the Produd^/. 
Suppofe that -|— D, — — C, — |— B, — A, -F I are the Co- 
efficients preceeding the T*erm Ea; ”■“^and that -f- F, 
G, -F H, — I, +.K are the Coefficients that follow 
" T . ■ 
it ; then if E* does not exceed + i x D F 
2 ' 
— + 4 X CG xBH — m +. I X A I 
4 -^ + ^ ^be Equation muft have fome imagina- 
• ry Roots 5 where the Coefficients + i, ;;; 4- 4, m 4- 9, 
iXc. 
