# The Dictionary of Mathematical Eponymy: The Witch of Agnesi

So, we got to Z in the Dictionary of Mathematical Eponymy. What now? Well, in the course of my research, I found that there were more than 26 people with things named after them. Shocking! In particular, despite my efforts, the original list was rather heavy on dead white men. I could make excuses for that, but instead I’m going to try to redress the balance. I’m not going through the alphabet again, but I am going to start with A.

### What is the Witch of Agnesi?

The Witch of Agnesi is a pleasing-looking curve constructed from a circle.

Take a circle and pick a point $M$ on its circumference. Let $M$ be the right-angle of a triangle with the circle’s diameter $OM$ as a leg, and $N$ (generally not on the circle) as the third point. The hypotenuse $ON$ cuts the circle at $A$.

Construct the perpendicular to $MN$ through $N$, and the parallel to $MN$ through $A$. These lines meet at point $P$.

The Witch of Agnesi is the set of all possible such points $P$.

If $M$ is at $(0,2a)$, then the equation of the curve is $y = \frac{8a^3}{x^2 + 4a^2}$.

### Why is it a witch?

It doesn’t look anything like a witch, does it? Apparently, the word *versiera*, which is Italian for ‘sail’ or ‘sheet’, was mistranslated: *avversiera* can mean ‘she-devil’ or ‘witch’. It’s not clear whether it was meant as a joke, or as an error.

### Why is it important?

The Witch of Agnesi has several applications. By picking $a$ appropriately, the curve is the probability density function of a Cauchy distribution – interestingly, despite its symmetry, it has no well-defined mean.

Because the area under the curve is equal to four times that of the original circle (i.e., $4\pi a^2$), the witch can be used to estimate the value of $\pi$. Indeed, this is how Leibniz derived the formula $\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \dots$. With $a = \frac{1}{2}$, the equation of the curve is $y=\frac{1}{1+x^2}$, which can be expanded binomially and integrated term by term.

It also looks like a hill, and is used in topography as a simple model.

### Who was Maria Gaetana Agnesi?

Maria Gaetana Agnesi (1718-1799) was one of twenty-one (twenty-one!) children, and a precocious linguist – she could speak seven languages by the time she was 11. Her father made a deal with her that she could do all the charitable work she wanted and stay out of society, so long as she kept up her mathematical research and educated her siblings.

She wrote a famous textbook (Analytical Institutions for the Use of Italian Youth), considered to be an excellent introduction at the time to the works of Euler.

She was appointed a professor at the University of Bologna in 1750 (although she never served); she fell ill the following year, and after the death of her father in 1752 devoted herself to charitable work.