**Maria Gaetana Agnesi**(1718-1799) and hermost famous work was

*Instituzioni analitiche ad uso della gioventù italiana*(

*Analytical institutions for italian young people*), in which she studied an old famous curve callde

*versiera*, in anglosaxon worlds

*the witch of Agnesi*. The name derives from a mistranslation by

**John Colson**who intend the original italian therm

*averisera*, that means

*versed sine curve*, like

*avversiera*, that in english became

*witch*or

*wife of devil*.

The curve was studied for the first time by

**Pierre de Fermat**in 1630, and after by

**Guido Grandi**in 1703 and in 1748 by Maria in her treatise. The curve was defined by:

(commons)

Starting with a fixed circle, a point $O$ on the circle is chosen. For any other point $A$ on the circle, the secant line $OA$ is drawn. The point $M$ is diametrically opposite $O$. The line $OA$ intersects the tangent of $M$ at the point $N$. The line parallel to $OM$ through $N$, and the line perpendicular to $OM$ through $A$ intersect at $P$. As the point $A$ is varied, the path of $P$ is the witch.Its cartesian equation is \[y = \frac{8a^3}{x^2 + 4a^2}\] In 1918

**Frederick H. Hodge**proofed that the witch is generated from the following curve \[a^2 ( x^2 - 2 (a+k) (2a - y))^2 = k^2x^2 (2ay - y^2)\] when $k$ goes to $\infty$.

**Links and bibliography**:

Witch of Agnesi (Wikipedia)

**Weisstein, Eric W**.

*Witch of Agnesi*. From MathWorld--A Wolfram Web Resource.

Hodge, F. (1918). Discussions: Relating to Generalizations of the Witch and the Cissoid The American Mathematical Monthly, 25 (5) DOI: 10.2307/2972650

Witch of Agnesi

Maria Gaetana Agnesi, biography on MacTutor

**Applets**:

The living witch of Agnesi

Witch of Agnesi on Wolfram Demonstrations Project

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