The latter definition involves the conceptions of a rectilinear
directrix, and a varying ratio in the cases of the different curves,
this ratio being unity for the parabola, less for the ellipse, and
greater for the hyperbola. The former involves the conception of a
circular directrix with a ratio equal to unity in all cases; and the
two definitions become identical in the construction of the parabola,
which is in fact the only curve of which a clear idea is given by
either of them. That of Boscovich has been given a prominence far in
excess of its merits, being made the foundation for the discussion of
these important curves, and this in a textbook whose preface contains
the following true and emphatic statement, viz.:
"The abstract nature of a ratio, and the fact that it is a
compound concept, peculiarly unfit it for elementary
purposes."
The definition herein set forth has not been given in any treatise on
the subject, so far as we have been able to ascertain. And it is
presented with the distinctly expressed hope that it never will be,
except as a mere matter of abstract interest.
Of this it may, like the other, possess a little, but both have the
great disadvantage that, except in relation to the parabola, the idea
which they convey to the mind of the curves to which they relate, if
indeed they convey any at all, is most obscure and indirect; and of
practical utility neither one can claim a particle.
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