136 On theVSE of NEGATIVE 



" gative roots had never been admitted into algebra, or were 

 ' again difcarded from it * " yet ftill he has carried us to the 

 length only of cubic equations. The truth is, that the whole 

 bufinefs of algebra might be carried on without the confidera- 

 tion of the negative roots. The difference between fuch a fyftem 

 and the prefent, is precifely this j that wherever a problem re- 

 quired us to confider any of the quantities, as exifting in oppo- 

 fite fituations ; wherever, for inftance, a line or a point was to be 

 confidered, as fituated firft on the right hand, and then on the 

 left y it would be neceflary, to find and to refolve a feparate equa- 

 tion for each of thefe cafes. Thus, in the analyfis of any par- 

 ticular curve, it would be neceflary to have a feparate equation 

 for each of the four angles of the co-ordinates ; except, indeed, 

 the axes were fo chofen, as to make us certain that there were 

 fome of thefe angles, in which no part of the curve was to be 

 found. Since, therefore, the ufe of negative quantities frees us 

 from this inconvenience,, which, in many cafes, particularly in 

 the analyfis of curves, would be exceedingly perplexing ; and 

 fince it evidently affords fo great elegance and univerfality to 

 algebraical folutions ; to find our author gravely declaring that 

 he can fee no advantage in it, is perfectly aftonifliing : As it is 

 to be lamented, that he did not exert his induftry and ingenui- 

 ty, rather to confirm than to deflroy ; rather to demonftrate, how 

 far we might rely on the method of negative quantities, than 

 to overturn at once fo great a part of the labours of the modern 

 algebraifls. 



WHAT follows is an attempt to explain this fubjecl, without 

 confidering the negative fign in any other light, than as the fign 

 of fubtra&ion ; and without propofing any alteration in the 

 received fyftem of algebra. 



* Diflert. on the Neg. Sign, p. 34. 



I. Of 



