On the USE of NEGATIVE QUANTITIES. 141 



the number which, fubtracted from its fquare, leaves a remain- 

 der equal to a. 



WHAT has been faid is eafily applicable to thofe problems, 

 where more than one quantity is to be determined. 



2. Of Indeterminate Equations. 



IN problems which are folved by indeterminate equations, 

 there is often nothing elfe required, but to determine the dif- 

 ferent magnitudes of one or more of the variable quantities, cor- 

 refponding to any given magnitudes of the remaining variable 

 quantities. There the pofitive roots alone can determine thefe 

 correfponding magnitudes ; fo that if there be no pofitive roots-, 

 or none which come within the limits of the problem, then the 

 problem is impoffible. 



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

 different correfponding magni- 

 tudes of the lines AE (#) and 

 EC (y) ; but alfo, with refpect to 

 each of the values of #, whether it 



lies to the right or left of the given M BAB 



point A ; and with refpecl to each 

 of the correfponding values of y^ 

 whether it is to be taken above or 

 below the line AE. Here we fup- 

 pofe that, whether AE lies to the 

 right or left, and whether the cor- 

 refponding line EC is to be taken 



upwards or downwards, there is nothing, except only thefe op- 

 pofite fituations of AE and BC, to make any difference in the 

 equations, to which the problem would be reduced upon each of 

 thefe fuppofitions. Then, we make any of thefe fuppofitions 

 at pleafure ; as for inftance, that AB lies to the right, and that 

 EC lies above AB ; and on this fuppofition, we exprefs the con- 

 ditions 



