The "witch of Agnesi" is a curve studied by Maria Agnesi in 1748 in her book Instituzioni analitiche ad uso della gioventù italiana (the first
surviving mathematical work written by a woman). The curve is also known as cubique
d'Agnesi or agnésienne, and had been studied earlier by Fermat and Guido Grandi
in 1703.

The name "witch" derives from a mistranslation of the term averisera ("versed sine curve," from the Latin vertere, "to turn")
in the original work as avversiera ("witch" or "wife of the
devil") in an 1801 translation of the work by Cambridge Lucasian Professor of
Mathematics John Colson (Gray).

The curve is obtained by drawing a line from the origin through the circle
of radius
and center ,
then picking the point with the coordinate of the intersection with the circle and the coordinate of the intersection of the
extension of line
with the line .

In parametric form,

(1)

(2)

for ,
or

(3)

(4)

for .

The Cartesian equation can be obtained by eliminating in the parametric equations, giving