Quadratrix of Hippias

$\begin{figure}\begin{center}\BoxedEPSF{hippias_quadratix.epsf scaled 800}\end{center}\end{figure}$

The quadratrix was discovered by Hippias of Elias in 430 BC, and later studied by Dinostratus in 350 BC (MacTutor Archive). It can be used for Angle Trisection or, more generally, division of an Angle into any integral number of equal parts, and Circle Squaring. In Polar Coordinates,

$\begin{displaymath} \pi \rho=2r\theta\csc\theta, \end{displaymath}$

$\begin{displaymath} r = {\rho \pi\sin\theta\over \theta}, \end{displaymath}$

which is proportional to the Cochleoid.

References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 195 and 198, 1972.

MacTutor History of Mathematics Archive. ``Quadratrix of Hippias.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Quadratrix.html.