Who was Maria Gaetana Agnesi to calculus teachers?

Most names in calculus arrive attached to a theorem. Maria Gaetana Agnesi (1718–1799) arrives attached to a textbook — and that is precisely why she is worth a lesson. She is the subject of our Who's Who entry; this is the longer version, and it is the textbook, not the curve, that earns the space.

The book, and the people who made it possible

Agnesi published Instituzioni analitiche ad uso della gioventù italiana — roughly, Analytical Methods for the Use of Italian Youth — in two volumes, the first in 1748 and the second in 1749. It was written in Italian rather than Latin, so that students could actually read it, and it was printed privately in her father’s house, where she supervised the printing herself.

The verdict that made its reputation came from a committee of the Académie des Sciences in Paris, and it is worth reading as a description of good teaching material:

“It took much skill and sagacity to reduce, as the author has done, to almost uniform methods these discoveries scattered among the works of modern mathematicians and often presented by methods very different from each other. Order, clarity and precision reign in all parts of this work. … We regard it as the most complete and best made treatise.”

One honesty note, because it is the sort of thing a sharp student will find if we leave it out: the book contains essentially no new mathematics. MacTutor describes it as an “exposition by examples rather than by theory” — the value is in the selection, the order and the clarity, not in an original result. That is a defensible thing to be remembered for, and students should hear it said plainly.

Agnesi was also careful about credit, which is the part of the story with the most practical use in class. She had no formal instruction in mathematics until the monk Ramiro Rampinelli, a mathematician who had taught at Rome and Bologna, became a regular visitor to her father’s house. In the preface she wrote that without his “secure guidance and wise direction” she “should have become altogether tangled in the great labyrinth of insuperable difficulty”. When Rampinelli connected her with the mathematician Jacopo Riccati, Riccati read the draft and offered his own earlier work on integration for inclusion — and Agnesi printed it with proper acknowledgement.

What a teacher can borrow from it

Her stated aim is in the preface, and it reads like a style guide written yesterday: to present the material “endowed with proper clarity and simplicity”, proceeding “with that natural order which provides, perhaps, the best instruction and the greatest light”. Clarity, simplicity, natural order. Three hundred years of textbooks have not improved on the brief, and teachers who hand students a worked-example-first sequence are following the same instinct.

The second thing to borrow is the acknowledgement paragraph. A textbook, a course, a department — none of it is really a solo act. Agnesi says so in print, names the people who helped, and thanks the monk who taught her. It is a five-minute exercise to set against a class: who actually helped you learn this, and did you say so?

The curve — and the translator’s slip that named it

The curve known in English as the witch of Agnesi is a cubic. Agnesi discussed it in the book; MacTutor gives it in Cartesian form as

y (x² + a²) = a³, or parametrically x = a t, y = a / (1 + t²).

The shape is easy to describe and easy to plot: it peaks at y = a when t = 0, flattens towards the line y = 0, which is an asymptote, and carries points of inflection at y = 3a/4. Guido Grandi gave a construction for it in 1703 and named it in 1718; Fermat had looked at it earlier still.

The classical construction makes a good board exercise, because it is a locus a student can build by hand. Draw a circle centred at (0, a/2) through the origin. From the origin draw any straight line; it cuts the circle at a point L and the horizontal line y = a at a point M. Then the point P whose x-coordinate comes from M and whose y-coordinate comes from L traces the curve as the line sweeps round.

The name is a mistranslation, and a memorable one. Grandi’s Latin name was versoria — a rope that turns a sail — and its Italian form was versiera. Agnesi, correctly, called the curve la versiera. John Colson translated her book into English (finished before his death in 1760, published in 1801) and read la versiera as l’aversiera: the witch, or the she-devil. The English name stuck, on a curve that nothing about is diabolical. MacTutor notes plainly that there has been much argument over the reason for the name; what is not in doubt is where the English word came from.

So the curve is a ready-made exercise with a maximum, an asymptote, two inflection points and a story attached — and the story is a useful lesson in itself: the names in mathematics are history, not mathematics. A student who has heard this one will never again assume a theorem’s name explains the theorem.

The chair she never took

The honour is often reported as “she held a chair at Bologna”. The real version is odder and better. Pope Benedict XIV wrote to her, then appointed her an honorary reader at the University of Bologna, and she was invited by the president of the Academy of Bologna and three of its professors to accept the chair of mathematics. A papal rescript in October 1750 confirmed the appointment. MacTutor’s account is exact about what happened next: her name stayed on the rolls of the university for forty-five years, and she never went to Bologna.

The second half of the life

Agnesi’s father died in 1752, and she turned from mathematics to charity: she set up a hospice for old and infirm women, gave away her money, and died on 9 January 1799 in the poorhouse she herself had directed. The textbook is a fraction of the person, and it is worth saying to a class that she chose the rest of it.

One more correction that belongs in any honest telling. The claim that her father was a professor of mathematics at Bologna is simply wrong — MacTutor states it clearly, and notes that the error has been copied forward from one reference work to the next. It is a compact demonstration of why we date and link our sources: a wrong fact, repeated often enough, starts to read like background.

What we are not claiming

Three limitations, so the page can be trusted rather than just enjoyed. We are not claiming the book contains original mathematics — it does not, and MacTutor says so. We are not claiming she occupied the Bologna chair, only that it was offered and confirmed. And we are not claiming the witch’s-name story is the only version in circulation; MacTutor is careful to say the naming has been argued about, and we follow its account of the mistranslation rather than asserting a settled history. Everything above comes from the two MacTutor pages linked below, read on the date shown.

• Sources (both checked 16 October 2026): MacTutor History of Mathematics — Maria Gaetana Agnesi (the two volumes and their dates; the Paris Académie committee’s report; “exposition by examples rather than by theory”; Rampinelli and the preface; Riccati’s contribution; the Bologna chair and the forty-five years on the rolls; the 1752 turn to charity and her death in 1799; Grandi’s versoria and Colson’s mistranslation; the incorrect claim about her father); Witch of Agnesi (the Cartesian and parametric forms, the peak at y = a, the asymptote y = 0, the inflection points at y = 3a/4, the circle construction, and its earlier study by Fermat and Grandi). MacTutor’s reference list points on to A. Cupillari, “Maria Gaetana Agnesi’s Other Curves (more than just the Witch)”, Mathematics Magazine 87(1), 2014, for readers who want to go further. See also our Who's Who.