Witch of Agnesi

The witch of Agnesi is a cubic plane curve defined from two diametrically opposite points of a circle. To construct it, take a circle with diameter OM, choose a point A on the circle, intersect the secant line OA with the tangent line at M, and combine that intersection with the foot of the perpendicular from A; the points obtained this way make up the curve. In Cartesian form, with a suitable choice of axes, it is the graph of 1 divided by x squared plus 1, the derivative of the arctangent function, which is also the shape of the Cauchy probability density. The curve is asymptotic to the tangent line to its defining circle at the other defining point, and the area between the curve and that line is four times the area of the circle. Fermat studied it in the seventeenth century, Guido Grandi gave the construction in 1718 and the name versiera, and Maria Gaetana Agnesi included it in her 1748 textbook Instituzioni analitiche. An English translator misread versiera as a word for a witch, which gave the curve its lasting English name.