The Witch of Agnesi: History, Geometric Construction, and the Runge Effect

The witch of Agnesi is one of the most beautiful curves in mathematics — and it owes its ominous name to a translator’s mistake.

The original term, coined by Guido Grandi, was versiera — meaning “turning curve.” But in Italian, that word sits uncomfortably close to avversiera — “witch.” And so, in English textbooks of the 17th and 18th centuries, the witch of Agnesi was born.

Behind the linguistic confusion, however, lies genuine mathematical poetry.

Geometric Construction of the Witch of Agnesi Curve

Imagine a circle and a tangent line touching it at one end of a diameter. Take an arbitrary point on that tangent line and draw a line through it to the opposite end of the diameter — this line intersects the circle at some point.

Through that point on the circle, draw a line parallel to the tangent. Then drop a perpendicular from the original point on the tangent down to this parallel line.

The intersection of that perpendicular and the parallel line gives you a single point on the witch of Agnesi curve. For all such points, a simple geometric relationship holds: BM : BC = OA : OB.

As the original point slides along the tangent, its projection traces a smooth, characteristic arc — the witch itself: a symmetric, gentle, bell-shaped curve arising from a surprisingly simple geometric mechanism.

From Fermat to Agnesi: A Brief History of the Curve

In the 17th century, Pierre de Fermat investigated the area beneath this strange new curve. A few decades later, Grandi gave it a clear geometric construction.

Then came 1748. Maria Gaetana Agnesi — one of the first women mathematicians in Europe — gathered the scattered results, derived the algebraic equation, described the inflection points, and permanently established the curve in the history of mathematical analysis.

The Witch of Agnesi and the Runge Effect

Why does this curve matter today?

It has become a classic example of how a smooth, well-behaved curve can be unexpectedly tied to the Runge effect — the phenomenon where approximating a function with polynomials on a uniformly spaced grid leads not to greater accuracy, but to an explosive growth in error near the boundaries.

Curves like the witch of Agnesi are precisely what forced mathematicians to search for better approximation methods.

A Curve Worth Remembering

So the witch of Agnesi has nothing to do with magic. It is about how a simple geometric game gives rise to an entire chapter of modern numerical mathematics.

Which mathematical curve do you think deserves more attention?