OF NEGATIVE QUANTITIES. 253 



are good ; he evidently proceeds on the principle, that whenever in the alge- 

 braic solution of any question, we arrive at imaginary quantities as answers, 

 we must consider that the question might have been expressed in more general 

 terms, and that the imaginary quantities are answers to the question in this 

 extended sense. This appears to me to be the true principle, and is analogous;)! 

 to our usual method of reasoning, when we arrive at a negative answer in 

 resolving a question, which, from the manner in which it is expressed, oniric 

 admits of positive answers. ja oiU-iytiii kioiA 



The Edinburgh reviewers in their article on M. Bue'e's "m^moire," state 

 their opinion with respect to the nature of the square roots of negative quan 

 titles in these words : 



"The essential character of imaginary expressions is to denote impossibility; 

 and nothing can deprive them of this signification, nothing like a geometrical 

 construction can be applied to them ; they are indications of the impossibility 

 of any such construction, or of any thing that can be exhibited to the senses." 



As I have already answered this objection, it will not be necessary for me.c 

 to make any further remarks on this point. 



In considering the evidence adduced by M. Bue'e in support of his funda- 

 mental proposition, that aJ — I expresses perpendicularity, the reviewers begin 

 with giving his reasoning on that subject, viz. ^ — I is a mean proportional 

 between -f- 1 and — 1, and therefore a perpendicular; and they observe with 

 respect to his arguments, that " any imaginable conclusion might have been 

 obtained in the same manner, the third line for example, needed not have been 

 placed at right angles to the other two, but making an angle, suppose of 120° 

 with one, and of 60° with the other ; it would still be a mean proportional 

 between them, and its square would be therefore, according to the above me- 

 thod of reasoning equal to -|- 1 x — I = — 1, so that the line itself would 

 be equal to >>y — 1, and thus ^/ — 1 would denote not perpendicularity, or the 

 situation in which a line makes the adjacent angles equal, but tliat in which 

 it makes one of these angles double of the other ; the one of these arguments 

 is just as good as the other, and neither of them of course is of any value." 



The above objection derives its force from the want of a definition of pro- 

 portion in M. Bue'e's " memoire," as is evident from what has already been 

 proved in this paper. 



