On the USE of NEGATIVE QUANTITIES. 137 



I. Of the Negative Roots of Equations. 



I . Of Determinate Equations. 



IN the folution of problems by means of equations, the 

 analyft fixes upon one or more quantities, by the determination 

 of which all that is required may be known or performed. 

 We fhall at prefent fuppofe that there is only one quantity to be 

 determined. The problem {hews the conditions which are re- 

 quired of this quantity ', and thefe conditions, as far as they 

 can be fo ^xprefled, are reduced to .an equation of the common 

 form, 



L . . . a + bx+cx 2 + Wf. rr o. 



Here, according to the common method in the general notation 

 of equations, the fign + denotes, at pleafure, either the addition 

 or fubtradUon of the terms to which it is prefixed. 



IN many cafes, nothing elfe is required, but to determine 

 the magnitude of the quantity fought. There the pofitive roots 

 alone can determine the magnitude j fo that if the equation 

 has no pofitive roots, or none which come within the limits of 

 the problem, then the problem is impomble. 



BUT let the problem require us to determine, not only the 

 different magnitudes of the line AB (#), 



but alfo, with refpec"l to each of thefe B & _ ^ 

 magnitudes, whether it lies on the right 



or left of the given point A. Here we fuppofe the problem to 

 be fuch, that whether we reduce it to an equation, upon the 

 fuppofition that AB lies to the right, or upon the fuppofition 

 that it lies to the left, there is no circumftance, except only the 

 oppofite fituation of AB y to make any difference in thefe equa- 

 tions. 



IN this cafe, we make either fuppofition at pleafure ; as, for 

 ihftance, that it lies to the right ; and, on this fuppofition, 



S we 



