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, only 
admits of positive answers. 
The Edinburgh reviewers in their article on M. Bue'e’s “ memoire,” state 
their opinion with respect to the nature of the square roots of negative quan 
tities 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 
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 */ — 1 expresses perpendicularity, the reviewers begin 
with giving his reasoning on that subject, viz. — 1 is a mean proportional 
between + 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-f-1 x — 1 = — 1, so that the line itself would 
be equal to — 1 , and thus — 1 would denote not perpendicularity, or the 
situation in which a line makes the adjacent angles equal, but that 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. 
