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whole theory of logarithms was founded on a basis as purely geo- 

 metrical 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 investi- 

 gation, 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 con- 

 ventional in this representation, I was riot 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. 



The properties of the Logocyclic curve, a curve which hitherto 

 has escaped discussion, if not discovery, are many of them as re- 

 markable as they are simple, and I will now proceed to mention 

 some of the most obvious. 



The equations of the Logocyclic curve. 

 Let r and be the polar coordinates of the curve, then its equation 



is 



r=a(sec0Hrtan0) (1) 



Since sec = -, tan = -, r = 



we get for the equation of the curve in rectangular coordinates, 



Let the focus F of the parabola, whose vertical focal distance FO is 

 0, be taken as the origin, the axis and parameter of the parabola 

 being the axes of x and y ; let the vertical tangent OT and the 

 directrix of the parabola DS be drawn, then the following proper- 

 ties of this curve may easily be shown : 



T2 



