Maria Agnesi: Calculus Pioneer and Charity Leader

The “Witch of Agnesi” is one of the most misleading labels in the history of mathematics. Welcome to Math! Science! History! I’m Gabrielle Birchak, and to close out Women’s History Month, I’m telling the story of Maria Gaetana Agnesi, the woman behind that infamous nickname. There was no witchcraft, no scandal, and no dark legend. Instead, there was a brilliant mind that helped make advanced mathematics teachable, then spent decades caring for the sick and unfortunate, giving away her fortune and choosing a quiet ending.
Milan and its intellect on display
Milan in the early eighteenth century was a city where status could be curated like a gallery, and wealthy families used culture, language, and learning as proof that they belonged among the elite. Smithsonian Magazine notes that Maria Gaetana Agnesi grew up as the eldest child of Pietro Agnesi, a wealthy Milanese silk merchant who hosted salons in the family palazzo as part of a deliberate climb in social standing.[1]

Those salons were not just parties with exquisite appetizers and intellectual elites. They were staged performances of refinement where guests watched young prodigies display skill in music, languages, and argument, the way other households displayed art. Pietro Agnesi showcased Maria Gaetana and her sister, Maria Teresa, to visitors at these gatherings, turning intellect into a kind of social currency.[2]
Agnesi’s abilities made her an astonishing centerpiece for that world, and they showed up early. A detailed biographical profile hosted by Oxford Mathematics reports that she became famous as “oracolo settelingue,” a “miracle in seven languages,” reflecting the breadth of languages she learned as a child.
This is the part of her story that often gets told like a wholesome montage. Still, the salon setting mattered because it explained a tension that followed her for years. Historian Paula Findlen argues that Agnesi was shaped by her father’s ambition and by the social machinery that turned a learned daughter into a symbol of family advancement, even as she became a serious thinker in her own right.[3]
If you zoom out, the salon was a doorway and a cage at the same time. It gave Agnesi access to tutors, books, and intellectual networks. Still, it also placed her under a spotlight she did not fully control, where her learning could be admired. At the same time, her choices were still constrained by expectations about women’s roles. The Vatican-related outlet L’Osservatore Romano even emphasizes that Agnesi did not enjoy salon life in the way her father did, highlighting her distance from the social ambitions wrapped around her public image. She did not want a life of social ambition nor one of academia. She had begged her father to spare her a life of marriage and a life of fitting in with the societal norms. Instead, she wanted to live in a convent and serve the impoverished. Her father, however, refused to let her live the life she chose. In the span of history and across the world, women have had that luxury for only a little over a century; some have not at all. And even today, we are expected to exist under the constraints of family and societal expectations.[4]
So, Agnesi was not simply known as a brilliant young lady. Still, rather, as Findlen notes, “she was brilliant inside a system that treated brilliance as a family asset.” Findlen’s historiography frames Agnesi as one of the best-known learned women of her generation, but also as someone whose legacy is easy to misunderstand if you ignore the social forces that put her on stage.[5]
That said, how does someone raised in a culture of learned performance decide to produce real scholarship, on her own terms, in a world that praises women’s intellect while still policing their lives? Much to her dislike, Agnesi was living inside a public wrapper of salon performances as an intellect under duress. Still, despite that, she took on a private seriousness that set her up to become a major mathematical genius. So, even though she was under constraint, she was still able to use her extensive knowledge for the benefit of mathematics.[6]
By the time she was thirty years old, she was no longer just performing cleverness for a room. She was doing something far more lasting. She wrote mathematics in a way that made it teachable. That shift moved her from being admired to being unignorable. In 1748, she wrote a two-volume work on algebra and calculus, titled Instituzioni analitiche ad uso della gioventù italiana (Analytical Institutions for the Use of Italian Youth), which was central to her reputation as a female mathematician.[7]

Agnesi’s Mathematical Legacy
Agnesi’s real mathematical legacy is not a single curve. Her legacy is a teaching project built at full scale that organizes a chaotic frontier of math in order to feel learnable to a human mind. In 1748, she published Instituzioni analitiche ad uso della gioventù italiana. This two-volume work in English is titled Analytical institutions for use by Italian youth. It is remarkably comprehensive and systematic, covering algebra and “analysis,” including integral and differential calculus, which was new at that time.[8]
The Mathematical Association of America’s historical note makes the point bluntly. This book was written as a teaching text. It was praised for “synthesizing and clarifying” the work of others in calculus, meaning Agnesi is acting less like a lone inventor and more like a brilliant translator between experts and students.[9]
Agnesi’s title, “for the youth of Italy,” is not decorative because she is telling you her goal right on the label, which is that the book is a comprehensive text of algebra and calculus written in the vernacular and meant to be usable.[10]
This is also where her mentors and collaborators matter, because the book did not emerge from a vacuum. In Historian Clara Silvia Roero’s article, she documents an intense correspondence (1745–1752) between Agnesi, her teacher Ramiro Rampinelli, and the Riccati family, including Jacopo Riccati and his sons, centered on the writing, composition, and printing of the Instituzioni.[11]
Agnesi’s primary teacher, Rampinelli, was considered an outstanding mathematician and teacher. Amid the expectations she endured, she considered Rampinelli one of her greatest mentors. In the preface of her book, she pays tribute to Rampinelli’s help. She describes how he provided her with mathematical clarity. He was her greatest supporter and encouraged her to write her teaching guide on differential calculus. Agnesi writes, “I should have become altogether entangled in the great labyrinth of insuperable difficulty… to him I owe all advances that my small talent has sufficed to make.”[12]
Jacopo Riccati enters the story not as a footnote but as part of the technical and editorial ecosystem around the book, and Professor Roero’s recent analytical study treats the Riccati family as active participants in the scientific dialogue shaping the text as it was being prepared for publication. Agnesi also makes a very strategic dedication, dedicating the Instituzioni to Empress Maria Theresa, and multiple collection notes discuss this dedication as both a patronage move and a statement about women operating in a male-dominated intellectual world.[13] Even the way modern libraries summarize it makes the point that this dedication is not random, noting that she dedicated the book to Maria Theresa and frames the dedication in the context of Enlightenment-era reforms and cultural politics.[14]
As for her contribution to the “witch’s curve,” it was not one famous curve. Her concept took a sprawling set of ideas, organized them, and wrote them in a way that made analysis feel teachable, stepwise, and coherent to students.
Now, for my math nerds, this is the moment we talk about what “analysis” meant in practice, because a huge amount of eighteenth-century analyses is the science of curves.[15]
When we study curves in this era, we are asking questions that sound simple but turn out to be powerful: what is the curve’s equation, where does it peak, where does it change concavity, what tangent line kisses it at a given point, and what area sits beneath it. In modern language, we are learning to treat a curve as the graph of a function, and then using derivatives to measure local behavior and integrals to measure accumulated behavior, which is why a “textbook of analysis” is also a practical handbook for understanding change.
Here is one anchor, before I get into formulas. A curve is not just a drawing, because in analysis, a curve is a rule that pairs an input with an output, and the graph is the visible trace of that rule.
Once you have that rule, calculus gives you two superpowers: the derivative tells you how steep the curve is at a point, and the integral tells you how much total “stuff” piles up under the curve across an interval. For Agnessi, she was not providing groundbreaking information. It had been studied earlier by Fermat and Guido Grandi. And this matters because she was not claiming it from nothing; she was doing what no other mathematician had done before: teaching it.
In a clean, modern form, one common scaling of the Agnesi curve is
y=\frac{a^3}{x^2+a^2}which you can view as a tall central hump that falls off toward zero as |x| grows.
Now the derivative is where the curve starts talking.
Differentiate and you get
\frac{dy}{dx}=-\frac{2a^3x}{{(x^2+a^2})^2} which immediately tells you the slope is zero at x=0, positive for x<0, and negative for x>0, so the curve rises to a single peak and then falls.
That one line is the calculus version of “this curve has one summit,” and it is exactly the kind of thing a student can learn to do again and again with different families of curves.
You can push it one step further and ask about concavity, because concavity is the difference between a hill that rounds gently and a hill that suddenly “tightens” as you move away from the center.
If you compute the second derivative, you can locate inflection behavior. The Wikipedia summary of the curve’s properties points to the existence of inflection points in standard scalings, which is the qualitative geometry students are training their intuition to see.
Then there is the integral, which is where the curve becomes a measuring instrument.
The total area under
y=\frac{a^3}{x^2+a^2}across all real x is
\int_{-\infty}^{\infty}\frac{a^3}{x^2+a^2}\ dx=\pi a^2which is a gorgeous result because the answer is not some messy expression, it is a clean constant times a2.
This is also one reason the curve appears later in probability and physics contexts: the same functional shape is tied to the arctangent derivative and the Cauchy distribution in common normalizations.

So, back to her writing, Agnesi’s deeper contribution here is not that she found a derivative nobody could find; instead, she helped make the mathematics legible, repeatable, and teachable to students who were not already living inside the mathematicians’ private correspondence networks or attending the salon gatherings.[16]
And that returns us to why the Rampinelli and Riccati connections matter, because the archival record shows her book, Analytical Institutions for the Use of Italian Youth, being shaped through teaching, correspondence, and revision, not merely through solitary inspiration.[17]
Agnesi took the mathematics of curves and turned it into a curriculum, which is one of the most beneficial forms of influence a mathematician can have.
The Witches Curve
When Agnesi wrote about curves, she wrote in the language of her century’s cutting edge, because in the 1700s, “analysis” was largely the art of turning geometry into calculation and calculation back into geometry. For a listener, a curve can sound like a doodle, but for eighteenth-century mathematicians, a curve is a disciplined object: it has an equation, it has a shape, and that shape contains measurable facts such as maxima, tangents, inflection points, and areas.
That is why curves mattered so much, because once you can express a curve algebraically, you can differentiate it to understand local behavior, and you can integrate it to understand total accumulation, which is the toolkit that ends up powering physics, astronomy, and engineering.
The curve that later gets tied to her name is often presented as a kind of celebrity curve. Still, it is better understood as a teaching example inside a much larger project of making analysis coherent for students.
Intuitively, the “Witch of Agnesi” curve has a single smooth peak and then falls away toward the x‑axis on both sides, which makes it perfect for showing students how to locate maxima, study concavity, and think about asymptotes without needing a complicated function. If you want a clean mental picture, imagine a hill that rises to one rounded summit and then slowly flattens out into long, low tails, rather than dropping off like a cliff.
Now, the nickname is the part people remember, and it is the least important part of her life. The Mathematical Association of America notes that Agnesi called the curve “la versiera,” and that the English translator John Colson rendered it as “witch,” and the name stuck in the anglophone world.[18] [19]
Wikipedia’s etymology section gives the core mechanism: “versiera” is tied to older Latin and trigonometric terminology, but it also sits close enough to words used for “the adversary,” which is why Colson’s translation drifted into “witch,” creating a label that sounds like folklore when it is really linguistic static.[20]
So for this podcast, I’m using the nickname only as a signpost for listeners who have heard it before. Then I am going to put it back on the shelf where it belongs, because Agnesi’s story is not a spooky anecdote; it is a story about teaching, reputation, and what gets remembered. Her work makes her impossible to ignore, and then comes Bologna.
Bologna: Her appointment and prestige
You see, in 1750, Agnesi was appointed by Pope Benedict XIV to the chair of mathematics and natural philosophy at the University of Bologna, and yet she never actually took up the post in the everyday sense we imagine when we hear “professor,” or in her case, “professoressa.”
It is interesting because she never traveled to Bologna to accept and serve in the role, even though the appointment itself was real and public.[21] This appointment by the pope was part of her broader public reputation, and it emphasized the appointment while also noting she never filled the post, which is exactly the tension you want to explore rather than smooth over.
The deeper story here is about patronage and prestige, because Pope Benedict XIV was also known for supporting learned culture and bolstering Bologna’s intellectual standing. By elevating a famous scholar, he was creating a public signal about what kind of institution Bologna intended to be.
That interpretation is strengthened by the archival-based scholarship summarized in a review by Massimo Mazzotti, which explains that she was proclaimed an honorary lecturerby Bologna’s senate on October 5, 1750. This phrasing points toward honorific recognition more than a modern employment contract with mandatory weekly lectures.[22]
In other words, the appointment can be read as Bologna and the papacy saying, “This scholarship counts, this author matters, and our intellectual life is not provincial,” even if Agnesi’s actual daily life remained anchored in Milan by family obligations, health, and the direction her values were pulling her. And what is so very cool about this honorary appointment she didn’t fill is that, after this, her life didn’t shrink; it expanded. It broadened her life view, giving her the momentum to pivot.
Agnesi’s second life
History has a ruthless compression algorithm, because it keeps the catchy label and discards the slow labor that actually shaped lives. The curve nickname survived because it is sticky. Still, the real achievement is that Agnesi wrote a teaching masterpiece that organized calculus-era analysis for learners, and then spent decades directing care for people society preferred not to see.
If you only know Agnesi from a curve nickname, you miss the fact that she spent decades building something as demanding as any book: a life organized around care for people who had very little protection. She had a major pivot after her father’s death. She devoted herself to theology and charitable work, and that was the start of what would become the dominant chapter of her life.
This is where the story stops being “a brilliant woman who did math” and becomes “a brilliant woman who chose a different kind of rigor.” Agnesi chose altruism. Because running care for the sick and the poor is an administrative, emotional, and logistical grind that does not reward you with applause.[23]
In 1771, she took on a formal leadership role at Milan’s Pio Albergo Trivulzio, which is a hospital and retirement home for the underprivileged, and was invited to preside there as Priora for the women’s ward, which in English means she was the director for the women’s ward, in an institution serving the most destitute patients.[24]
Multiple modern historical summaries converge on the same stark outcome: she gave away substantial resources, lived simply, and died in 1799, which makes her life less of an “eccentric genius” and more of a human with a sustained selfless commitment to the unfortunate.”
In her later years, Agnesi’s work shifted from the abstract clarity of mathematics to the daily, grinding clarity of care. She took on real responsibility for women who were sick and poor, not as a symbolic patroness but as someone involved in the administration and human reality of suffering. In other words, she moved from teaching people how to understand curves on paper to helping people survive the harsh curves of life, where resources run out, and institutions decide who is worth attention.
What makes that turn so striking is that it was not a brief detour or a sentimental ending. Accounts of Agnesi’s final decades consistently describe a person who gave away her resources, lived a quiet life, and accepted the material consequences of that choice, dying in 1799 with little left for herself. That is not the arc of an “eccentric genius” who lost interest in her talent. It is the arc of someone who treated intellect as a form of responsibility, then followed that responsibility into the hardest kind of work. This kind does not come with applause.
When she passed away, she was buried in a mass grave for the poor, alongside about fifteen others. Some accounts describe them as women from the very institution where Agnesi had spent years caring for the sick.
To me, that detail changes the emotional temperature of her whole life story, because it turns her ending into a decision about belonging. She had been offered honors, titles, and the kind of reputation that usually demands a monument, and she seems to have stepped away from all of it in the most final, physical way possible. If she chose to be buried without a marker, in common ground, then she was refusing the idea that her life needed to end as a public symbol. She was placing herself, literally, among the women she had lived alongside, the women she had served, and the women whose suffering she did not treat as a separate category from her own humanity. That is not a rejection of her intellect. It is the completion of it: the same mind that made knowledge more accessible also made dignity more accessible, and in the end, she asked for no special treatment, only proximity to those she held dear.
[1] Lamb, Evelyn. “The 18th-Century Lady Mathematician Who Loved Calculus and God.” Smithsonian Magazine. Accessed February 11, 2026. https://www.smithsonianmag.com/science-nature/18th-century-lady-mathematician-who-changed-how-calculus-was-taught-180969078/.
[2] Lamb, Evelyn. “The 18th-Century Lady Mathematician Who Loved Calculus and God.” Smithsonian Magazine. Accessed February 11, 2026. https://www.smithsonianmag.com/science-nature/18th-century-lady-mathematician-who-changed-how-calculus-was-taught-180969078/.
[3] Findlen, Paula. “Calculations of Faith: Mathematics, Philosophy, and Sanctity in 18th-Century Italy (New Work on Maria Gaetana Agnesi).” Historia Mathematica 38, no. 2 (2011): 248–91. https://doi.org/10.1016/j.hm.2010.05.003.
[4] Eduati, Laura. “What We Owe to Maria Gaetana — L’Osservatore Romano.” Journal. L’Osservatore Romano, Vatican, June 1, 2024. https://www.osservatoreromano.va/en/news/2024–06/dcm-006/what-we-owe-to-maria-gaetana.html.
[5] Findlen, Paula. “Calculations of Faith: Mathematics, Philosophy, and Sanctity in 18th-Century Italy (New Work on Maria Gaetana Agnesi).” Historia Mathematica 38, no. 2 (2011): 248–91. https://doi.org/10.1016/j.hm.2010.05.003.
[6] Lamb, Evelyn. “The 18th-Century Lady Mathematician Who Loved Calculus and God.” Smithsonian Magazine. Accessed February 11, 2026. https://www.smithsonianmag.com/science-nature/18th-century-lady-mathematician-who-changed-how-calculus-was-taught-180969078/.
[7] Findlen, Paula. “Calculations of Faith: Mathematics, Philosophy, and Sanctity in 18th-Century Italy (New Work on Maria Gaetana Agnesi).” Historia Mathematica 38, no. 2 (2011): 248–91. https://doi.org/10.1016/j.hm.2010.05.003.
[8] “Instituzioni Analitiche Ad Uso Della Gioventù Italiana | Work by Agnesi | Britannica.” Accessed February 11, 2026. https://www.britannica.com/topic/Instituzioni-analitiche-ad-uso-della-gioventu-italiana.
[9] Huffman, Cynthia J. “Mathematical Treasure: Maria Agnesi’s Analytical Institutions in Italian and English | Mathematical Association of America.” Convergence, Mathematical Association of America, January 2017. https://old.maa.org/press/periodicals/convergence/mathematical-treasure-maria-agnesi-s-analytical-institutions-in-italian-and-english?utm_source=chatgpt.com.
[10] “Instituzioni Analitiche Ad Uso Della Gioventu’ Italiana · Duke University Library Exhibits.” Accessed April 8, 2026. https://exhibits.library.duke.edu/items/show/4035?utm_source=chatgpt.com.
[11] Roero, Clara Silvia. “M.G. Agnesi, R. Rampinelli and the Riccati Family: A Cultural Fellowship Formed for an Important Scientific Purpose, the Instituzioni Analitiche.” Historia Mathematica 42, no. 3 (2015): 296–314. https://doi.org/10.1016/j.hm.2014.12.001.
[12] “This Month in Physics History.” Accessed April 8, 2026. https://www.aps.org/archives/publications/apsnews/201006/physicshistory.cfm.
[13] Bishop, Amy. Maria Gaetana Agnesi. July 7, 2021. https://iastate.pressbooks.pub/cardinaltales1/chapter/rare-book-highlights-maria-gaetana-agnesi/.
[14] Bishop, Amy. Maria Gaetana Agnesi. July 7, 2021. https://iastate.pressbooks.pub/cardinaltales1/chapter/rare-book-highlights-maria-gaetana-agnesi/.
[15] Huffman, Cynthia J. “Mathematical Treasure: Maria Agnesi’s Analytical Institutions in Italian and English | Mathematical Association of America.” Convergence, Mathematical Association of America, January 2017. https://old.maa.org/press/periodicals/convergence/mathematical-treasure-maria-agnesi-s-analytical-institutions-in-italian-and-english?utm_source=chatgpt.com.
[16] Huffman, Cynthia J. “Mathematical Treasure: Maria Agnesi’s Analytical Institutions in Italian and English | Mathematical Association of America.” Convergence, Mathematical Association of America, January 2017. https://old.maa.org/press/periodicals/convergence/mathematical-treasure-maria-agnesi-s-analytical-institutions-in-italian-and-english?utm_source=chatgpt.com.
[17] Roero, Clara Silvia. “M.G. Agnesi, R. Rampinelli and the Riccati Family: A Cultural Fellowship Formed for an Important Scientific Purpose, the Instituzioni Analitiche.” Historia Mathematica 42, no. 3 (2015): 296–314. https://doi.org/10.1016/j.hm.2014.12.001.
[18] Physics Today. “Maria Gaetana Agnesi.” May 16, 2018. https://doi.org/10.1063/PT.6.6.20180516a.
[19] Huffman, Cynthia J. “Mathematical Treasure: Maria Agnesi’s Analytical Institutions in Italian and English | Mathematical Association of America.” Convergence, Mathematical Association of America, January 2017. https://old.maa.org/press/periodicals/convergence/mathematical-treasure-maria-agnesi-s-analytical-institutions-in-italian-and-english?utm_source=chatgpt.com.
[20] Osen, Lynn M. Women in Mathematics. With Internet Archive. Cambridge, Mass., MIT Press, 1974. http://archive.org/details/womeninmathemati00osen.
[21] Gunderman, Richard, and David Gunderman. “Maria Agnesi, the Greatest Female Mathematician You’ve Never Heard Of.” Scientific American. Accessed April 8, 2026. https://www.scientificamerican.com/article/maria-agnesi-the-greatest-female-mathematician-youve-never-heard-of/.
[22] Mazzotti, Massimo. The World of Maria Gaetana Agnesi, Mathematician of God. Edited by Ronald Calinger. Vol. 1. Milano, 1836. Reprint, The Johns Hopkins University Press, 2007.
[23] Eduati, Laura. “What We Owe to Maria Gaetana — L’Osservatore Romano.” Journal. L’Osservatore Romano, Vatican, June 1, 2024. https://www.osservatoreromano.va/en/news/2024–06/dcm-006/what-we-owe-to-maria-gaetana.html.
[24] Eduati, Laura. “What We Owe to Maria Gaetana — L’Osservatore Romano.” Journal. L’Osservatore Romano, Vatican, June 1, 2024. https://www.osservatoreromano.va/en/news/2024–06/dcm-006/what-we-owe-to-maria-gaetana.html.
