On the USE of NEGATIVE 



IN like manner, let it be required to determine, not only the 

 value of a certain fum of money, but alfo, whether it is part of 

 a certain perfon's Jlock, or whether it is part of his debts. 

 Then, if we reduce the problem to an equation, upon the fup-~ 

 pofition that it is part of his flock ; we are to aflame X a fup- 

 pofed additional quantity of ftock. 



THERE are cafes, therefore, where the quantity foxight is to 

 be confidered in two different fituations, which may be repre- 

 fented by addition and fxtbtraction from another aflignable 

 quantity of the fame kind ; and where there is nothing in the 

 problem, except only this oppofite fituation, which can produce 

 any difference in the equations to which it is reducible upon 

 each fuppofition. From the preceding obfervations it will ap- 

 pear what may be underftood by the negative roots, and how 

 mathematicians are juftified in the conclusions which they draw 

 from them in fuch cafes. 



THE negative roots are fometimes alfo ufeful in the folutioni 

 of problems relating to abflrael quantities. For the equation 



L . . . a-\-bx-\-cx* -f- &c. n o 

 has the fame roots with the equation: 



L' . . . a bx+cx* <b'c. o, 



except only that the negative roots of the one are the pofitive 

 roots of the other. Wherever, therefore, any problem pro- 

 ducing the equation L, is fo connected with any other problem. 

 prodticing the equation L', that they may be confidered as dif- 

 ferent cafes of the fame problem, or that the confideration of 

 the one fuggefls the other ; there it is evident, that the negative 

 roots will be ufeful, by affording, from either of thefe equa- 

 tions, the folution of both problems. Thus, the equation 

 ** + * = a, gives us, not only the number, which, added to its 

 fquare, makes a fum equal to a j but alfo, by the negative root, 



the 



