IV. On the USE of NEGATIVE Q^J A N T I T I E S in the SOLU- 

 TION O/'PROBLEMS ^ALGEBRAIC EQJJATIONS. 

 % WILLIAM GREENFIELD, M. A. F. R. S. EDIN. Mini- 

 Jier of St Andrew's Church, and ProfeJJor of Rhetoric in the' 

 Univerftty, of EDINBURGH. 



[Read by the Author > April 12. 1784.] 



BY the introduction of letters into algebra, to denote all the 

 quantities, both known and unknown, involved in an 

 equation, this very important advantage was .gained, that the 

 final equation exhibited both a general rule for the folution of 

 all fimilar problems, and alfo the limitations within which fuch 

 problems were poflible. 



THIS, however, could not be underftood univerfally, if the 

 figns -f- and were not ufed in that extenfive fenfe in which 

 they are now taken. For there are innumerable problems, which 

 require us to confider fome of the quantities, as capable of ex- 

 ifting in two oppofite fituations. Thus the diftance of a flar 

 from the horizon, may include both its elevation at one time, 

 and its depreffion at another. Hence, in the general invefliga- 

 tion of this diftance, two different cafes arife, which may feem 

 to require two different equations. 



MATHEMATICIANS, however, came naturally to confider 

 fuch oppofite fituations as analogous to addition and fubtrac- 

 tion. And, upon this ground, they made the equation to 

 which the problem was reduced in the one cafe, ferve alfb for 

 the other, in the following manner : If it was one of the un- 

 known quantities which had changed its fituation, they took a 

 negative root ; if it was one of the known quantities, they 



R 2 changed 



