in Mathematical Reasoning. 349 
introduced by the algebraic operations that were necessary to 
arrive at the final equation: these ought to be pointed out, and 
the step at which they were introduced should be noticed, and 
also whether they can admit of any translation into the language 
of the problem considered. Imaginary roots are very frequently 
introduced; they sometimes imply impossibility or contra- 
diction amongst the data; their origin ought to be carefully 
traced, and such a course will frequently make us acquainted 
with the maxima and minima which belong to the question. 
It is still more necessary to attend to all the real roots whether 
positive or negative, and to explain the various circumstances in 
the solution to which they refer. 
It is by no means uncommon with algebraical authors, when 
they have led their readers through a process which terminates in 
an equation, to select that root which gives the answer they 
require, without explaining the signification of the other roots 
that are equally comprised in it; and this incomplete mode of 
solution, which is censurable from revealing only a part of the 
truth, has in some instances caused the most interesting circum- 
stances attending a question to be entirely overlooked. A singular 
example of this occurs in several authors who have sought analy- 
tically the side of a heptagon inscribed in a circle, or the radius 
of a circle which would circumscribe a regular heptagon whose 
side is given. In neither of these questions can the equation to 
which we are led, be reduced below the third degree, and the 
three roots of the cubic are always real: the largest of the 
positive roots gives the answer to the latter of these questions 
for the common heptagon of Euclid: but no reason is stated 
why this root should be considered as the true answer to the 
question in preference to either of the others. In the Analytical 
Institutions of M. Agnesi*, where the first problem is solved, no 
* Vol. 1. p. 168. English Translation. 
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