IV. On the logocyclic curve, and the geometrical origin of logarithms

In a paper read before the Mathematical Section of the British Association during its meeting at Cheltenham in 1856, and which was printed among the reports for that year, I developed at some length the geometrical origin of logarithms, and showed that a trigonometry exists as well for the parabola as for the circle, and that every formula in the latter may be translated into another which shall indicate some property of parabolic arcs analogous to that from which it has been derived. I showed, moreover, that the hole theory of logarithms was founded on a basis as purely geometrical as the trigonometry of the circle, and I pointed out how the imaginary expressions in the latter—such as DeMoivre’s theorem—have real counterparts in the trigonometry of the parabola. As the principle of duality is of the widest application in geometrical investigation, and as every property of circumscribed space has its dual, it would be strange if the dual of circular trigonometry had no existence. In the paper.to which I have referred, I showed how, by the help of certain arbitrary lines drawn about the parabola, numbers and their logarithms might be exhibited. But as there was something conventional in this representation, I was not quite satisfied with the construction. I suspected that the geometrical theory of logarithms was just as little conventional as the trigonometry of the circle. With this view I have again lately considered the whole subject, and by the help of a new curve, which I have called the Logocyclic Curve , from the similarity of many of its properties to those of the circle, and from its use in representing numbers and their logarithms , I have succeeded in exhibiting the whole theory of logarithms in a geometrical form as complete as it is comprehensive, and simple as it is beautiful.