[ 2 +7 ] 
The following obfervations relative to the appli- 
cation of the abovementioned rulers may be ufeful : 
1. The roots of equation are of three forts, af- 
firmative, negative , and impofible ; which laid are 
fometimes called imaginary. 
2. Every equation contains as many roots as it has 
dimentions. 
3. The impoflible roots go by pairs. Thus, if an 
equation has an impoflible root of this form a-\-b 
‘'/—i, it has another impoflible one of this, viz. 
a — b ^ — 1, which may be called its fellow: from 
hence it follows, that every equation, if it has any 
impoflible roots, has either two, four, or fix , &cc. that is, 
an even number of them : and every time the curve 
defcribed by the rulers abovementioned approaches 
the' bafe, and leaves it again without eroding it, 
it implies one impoflible root and its fellow; fo that 
if it approaches the bafe in this manner three times, 
it implies, that the equation has fix impoflible roots. 
And this is all the rulers can do with regard to this 
fort of roots ; they cannot fhew what thofe are, but 
only how many there are. I fhall diredt to a method 
of finding what they are in the 8th article below. 
Inafmuch then, as the impoflible roots go by pairs, 
and the number of roots in any equation is equal to 
the number of its dimenfions, it follows, 
4. That every equation of an odd number of 
dimenfions, muff contain at lead one real root. 
5. Every equation whofe fir ft and Lift terms 
(when brought to one fide) have contrary figns, will 
have at leaft alfo one real root : and therefore, when 
this is the cafe, and the number of its dimenfions is 
1 alfo. 
