# Definition:Witch of Agnesi

## Definition

Let $OAM$ be a circle of radius $a$ whose center is at $\tuple {0, a}$.

Let $M$ be the point such that $OM$ is a diameter of $OAM$.

Let $OA$ be extended to cut the tangent to the circle through $M$ at $N$.

Generate $NP$ perpendicular to $MN$ and $AP$ parallel to $MN$.

As $A$ moves around the circle $OAM$, the point $P$ traces the curve known as the **witch of Agnesi**.

### Properties

This page or section has statements made on it that ought to be extracted and proved in a Theorem page.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed.To discuss this page in more detail, feel free to use the talk page. |

Various properties of the **witch of Agnesi** are as follows.

- It is defined for all $x$.
- $0 < y \le 2 a$.
- $y$ reaches its maximum at $x = 0$.
- The curvature $K$ of the curve is such that $0 \le K \le \dfrac 1 a$, and it achieves that maximum at $x = 0$.

This article is complete as far as it goes, but it could do with expansion.In particular: And there's more.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Also known as

The **witch of Agnesi** is also known as the **versiera**.

## Also see

- Results about
**Witch of Agnesi**can be found**here**.

## Source of Name

This entry was named for Maria Gaëtana Agnesi.

## Historical Note

The **witch of Agnesi** was studied in detail by Maria Gaëtana Agnesi in the $18$th century, but does not actually originate from her.

It had previously been written about by others, for example Pierre de Fermat.

## Linguistic Note

The **witch of Agnesi** was originally named the **versorio** by Luigi Guido Grandi, from the Italian **vertere** (**to turn**: the term comes from the rope used to turn a sail).

Maria Gaëtana Agnesi confused the word with **versiera**, from **avversiera**, which means **witch** or **she-devil** (from the same root as the word **adversary**, an archaic soubriquet for **Satan**).

I has been suggested that the initial misnaming may have been mischievous.

When referred to in other languages, the term **witch** is not seen, and the less colorful term **curve of Agnesi** is usually used instead.

Note the name **Agnesi** is Italian: its pronunciation is something like ** an-ye-zi**, and never in the apparently obvious way

**ag-**.

*nee*-zee

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 11$: Special Plane Curves: Witch of Agnesi: $11.23$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**witch of Agnesi**or**versiera** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**witch of Agnesi (versiera)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**witch of Agnesi (versiera)**