Maria Gaetana Agnesi

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Agnesi, Maria Gaetana

(b. Milan, Italy, 16 May 1718; d. Milan, 9 January 1799)


Maria Gaetana Agnesi, the first woman in the Western world who can accurately be called a mathematician, was the eldest child of Pietro Agnesi and Anna Fortunato Brivio. Her father, a wealthy Milanese who was professor of mathematics at the University of Bologna, encouraged his daughter’s interest in scientific matters by securing a series of distinguished professors as her tutors and by establishing in his home a cultural salon where she could present theses on a variety of subjects and then defend them in academic disputations with leading scholars. Agnesi invited both local celebrities and foreign noblemen to his soirees. During the intermissions between Maria Gaetana’s defenses, her sister, Maria Teresa, a composer and noted harpsichordist, entertained the guests by playing her own compositions.

In all her discourses at these gatherings, Maria Gaetana demonstrated her genius as a linguist. At age live she spoke French fluently. At age nine, she translated into Latin, recited from memory, and released for publication a lengthy speech advocating higher education for women, By age eleven, she was thoroughly familiar with Greek, German, Spanish, and Hebrew. The disputations were conducted in Latin, but during the subsequent discussions a foreigner would usually address Maria in his native tongue and would be answered in that language. The topics on which she presented theses covered a wide range—logic, ontology, mechanics, hydromechanics, elasticity, celestial mechanics and universal gravitation, chemistry, botany, zoology, and mineralogy, among others. Some 190 of the theses she defended appear in the Propositiones philosophicae (1738), her second published work.

Although the 1738 compilation does not contain any of Agnesi’s purely mathematical ideas, various other documents indicate her early interest in mathematics and her original approach to that subject. At fourteen she was solving difficult problems in analytic geometry and ballistics. Her correspondence with some of her former tutors indicates that, as early as age seventeen, she was beginning to shape her critical commentary on the Traité analytique des sections coniques of Guillaume de L’Hospital, a leading mathematician of the Newtonian era. The manuscript material that she prepared, although judged excellent by all the professors who examined it, was never published.

In 1738, after the publication of the Propositiones philosophicae, Agnesi indicated that the constant public display of her talents at her father’s gatherings was becoming distasteful to her, and she expressed a strong desire to enter a convent. Persuaded by her father not to take that step, she nevertheless withdrew from all social life and devoted herself completely to the study of mathematics. In the advanced phases of the subject she was guided by Father Ramiro Rampinelli, a member of the Olivetan order of the Benedictines, who later became professor of mathematics at the University of Pavia. A decade of concentrated thought bore fruit in 1748 with the publication of her Istituzioni analitiche ad use della gioventu italiana, which she dedicated to Empress Maria Theresa of Austria. This book won immediate acclaim in academic circles all over Europe and brought recognition as a mathematician to Agnesi.

The Islituzioni analitiche consisted of two huge quarto volumes containing more than a thousand pages. Its author’s objective was to give a complete, integrated, comprehensible treatment of algebra and analysis, with emphasis on concepts that were new (or relatively so) in the mid–eighteenth century. In this connection one must realize that Newton was still alive when Agnesi was born, so that the development of the differential and integral calculus was in progress during her lifetime. With the gioventu (youth) in mind, she wrote in Italian rather than in Latin and covered the range from elementary algebra to the classical theory of equations, to coordinate geometry, and then on to differential calculus, integral calculus, infinite series (to the extent that these were known in her day), and finally to the solution of elementary differential equations. She treated finite processes in the first volume and infinitesimal analysis in the second.

In the introduction to the Istituzioni analitiche, Agnesi—modest as she was, with too great a tendency to give credit to others had to admit that some of the methods, material, and generalizations were entirely original with her. Since there were many genuinely new things in her masterpiece, it is strange that her name is most frequently associated with one small discovery which she shared with others: the formulation of the versiera, the cubic curve whose equation is x’v = a 2 (a–r) and which, by a process of literal translation from colloquial Italian, has come to be known as the “witch of Agnesi.” She was apparently unaware (and so were historians until recently) that Fermat had given the equation of the curve in 1665 and that Guido Grandi had used the name versiera for it in 1703.

Agnesi’s definition of the curve may be stated as follows: If C is a circle of diameter a with center at (O, 1/2a), and if the variable line OA through the origin O intersects the line y = a at point A and the circle at point B, then the versiera is the locus of point P, which is the intersection of lines through A and B parallel to the Y axis and X axis, respectively. The curve, generated as the line OA turns (Latin vertere, hence the name versiera), is bell-shaped with the X axis as asymptote. There are interesting special properties and some applications in modern physics, but these do not completely explain why mathematicians are so intrigued by the curve. They have formulated a pseudo versiera by means of a change in the scale of ordinates (a similarity transformation). Even Giuseppe Peano, one of the most formidable figures in modern axiomatics and mathematical logic, could not resist the temptation to create the “visiera of Agnesi,” as he called it a curve generated in a fashion resembling that for the versiera.

The tributes to the excellence of Agnesi’s treatise were not numerous that its is impossible to list them all but those related to translations of the work will be noted. The French translation (of the second volume only) was authorized by the French Academy of Sciences. In 1749 an academy committee recorded its opinion: “This work is characterized by its careful organization, its clarity, and its precision. There is no other book, in any language, which would enable a reader to penetrate as deeply, or as rapidly, into the fundamental concepts of analysis. We consider this treatise the most complete and best written work of its kind.”

An English translation of the Istituzioni analitiche was mede by John Colon, Lucasian professor of mathematics at Cambridge, and was published in 1801 at the expense of the baron de Maséres. In introducing the translation, John Hellins, its editor, wrote: “He [Colson] found her [Agnesi’s] work to be so excellent that he was at the pains of learning the Italian language at an advanced age for the sole purpose of translating her book into English, that the British Youth might have the benefit of it as well as the Youth of Italy.”

The recognition of greatest significance to Agnesi was provided in two letters from Pope Benedict XIV. The first, dated June 1749, a congratulatory note on the occasion of the publication of her book, was accompanied by a gold medal and a gold wreath adorned with precious stones. In his second letter, dated September 1750, the pope appointed her to the chari of mathematics and natural philosophy at Bologna.

But Agnesi, always retiring, never actually taught at the University of Bologna. She accepted her position as an honorary one from 1750 to 1752, when her father was ill. After his death in 1752 she gradually withdrew from all scientific activity. By 1762 she was so far removed from the world of mathematics that she declined a request of the University of Turin to act as referee for the young Lagrange’s papers on the calculus of variations.

The years after 1752 were devoted to religious studies and social work. Agnesi made great material sacrifices to help the poor of her parish. She had always mothered her numerous younger brothers (there were twenty-one children from Pietro Agnesi“s three marriages), and after her father’s death she took his place in directing their education. In 1771 Agnesi became directress of the Pio Albergo Trivulzio, a Milanese home for the aged ill and indigent, a position she held until her death.


I. Original Works. Agnesi’s main works are Propositiones philosophicae (Milan, 1738) and Analytical Institutions, an English translation of the Istituzioni analitiche by the Rev. J. Colson (London, 1801).

II. Secnodary Literature. Further information about Agnesi and her wok may be found L. Anzoletti, Maria Gaetana Agnesi (Milan, 1990); A.F. Frisi, Elogio storicos di Dominia Maria Gaetana Agnesi milanese (Milan, 1799); and A. Masotti, “Maria Gaetana Agnesi,” in Rendiconti del seminrio matematico e fisico di Milano, 14 (1940), 1–39.

Edna E. Kramer

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(b. Milan, Italy, 16 May 1718; d. Milan, 9 January 1799)

mathematics. For the original article on Agnesi see DSB, vol. 1.

Agnesi was the first woman ever to publish a book of mathematics in her own name. The book, titled Instituzioni analitiche ad uso della gioventù italiana, appeared in Milan in 1748. In its two volumes Agnesi presented the principles and methods of algebra, Cartesian geometry, and calculus. Hers was among the first textbooks to offer such a complete introductory survey—and certainly the most accessible to beginners. The only other text published by Agnesi in her life was Propositiones philosophicae(1738), a collection of theses that, when she was twenty years old, signed the completion of her philosophical studies. This kind of publication was routinely expected from college students at the end of their course. Being a woman, Agnesi was prevented from accessing college; however, she carried out her training privately, under the supervision of eminent tutors. Agnesi also authored some writings of religious character, including a mystical treatise that remained unpublished. From 1752 onward she devoted her life to teaching young girls and to the assistance of the old and sick of her city, first at the Ospedale Maggiore and then, from 1771, at the Pio Albergo Trivulzio, a charitable institution she directed for almost thirty years.

Maria Gaetana Agnesi was born into a prominent Milanese family, which had accumulated remarkable wealth with the trade of wool and silk textiles. Pietro, her father, abandoned the trading activities of his ancestors and, by the time of Gaetana’s birth, was engaged in a difficult attempt to bring his family into the city’s aristocracy. Contrary to what was stated in the original entry, Pietro was not a university lecturer, nor had he any significant connection to the academic world. Gaetana’s exceptional education was entrusted entirely to private tutors, mainly local ecclesiastics, and should be understood within Pietro’s overall strategy of social enhancement. In the periodical gatherings at Palazzo Agnesi—or conversazioni, as these soirées were called—Gaetana performed in front of prestigious guests from all over Europe. Mainly she engaged in disputations with university lecturers over topics in natural philosophy and mathematics. On these occasions she defended the positions of the “moderns”— Isaac Newton, the Dutch experimental philosophers, and Italian natural philosophers such as Antonio Vallisneri— against the criticisms of scholastic and Cartesian philosophers. Together with her sister Maria Teresa, who was to become a renowned harpsichord player and composer, Gaetana provided Pietro with remarkable social visibility through the 1730s, just as he was pursuing a royal title and a coat of arms for his family.

Agnesi’s precocious talent for the study of languages, philosophy, and mathematics was always coupled with an intense religious piety. In fact her religiosity was nurtured by those same tutors who were supervising her philosophical and mathematical studies. These learned ecclesiastics were actively involved in an attempt to reform the theological, liturgical, and social features of contemporary Catholicism. Essentially they supported an antibaroque devotion, characterized by a more sober and rational set of practices. In opposition to contemporary Jesuit teaching, these reformers deemed necessary the introduction of the experimental sciences and modern mathematics in universities and religious institutions of higher education.

Agnesi herself would contribute importantly to this reformist current with her textbook, the first to be expressly designed to help young students to learn algebra and calculus. The distinguishing features of the book depend largely on its didactic purpose. Thus, for example, unlike the authors of other contemporary treatises and

articles on the techniques of calculus, Agnesi gave priority to the systematization of this diverse literature and to the clarification of the concepts she used, which was often achieved by relating them to the student’s most familiar intuitions. The Instituzioni were also shaped by Agnesi’s belief that the study of mathematics had particular relevance in the context of a truly Christian upbringing. The truths of geometry were exemplary in their certainty and assumed metaphysical relevance in the framework of a Malebranchian theory of knowledge of the sort she embraced. Agnesi was very clear about the priority of geometrical evidence over algebraic manipulation of symbols, hence her choice to present the most recent developments in calculus in purely geometrical terms and her emphasis on techniques such as the geometrical construction of equations. Throughout the book she keeps analytical formalism detached from mechanical and empirical considerations in order to preserve the simplicity, rigor, and evidence she held to be distinctive of the classical tradition in geometry. Unique among the texts on calculus that appeared in those years, Agnesi’s does not contain a single example of the application of its techniques to the solution of problems in experimental physics and rational mechanics. Analytic methods are presented as significant for the solution of geometrical problems and the study of curves interesting exclusively for their metric properties, such as the versiera, the curve most commonly associated with Agnesi’s name.

Following the publication of her book Agnesi reached the apex of her fame, and in 1750 she was offered a lectureship in mathematics at the University of Bologna through the personal intervention of the pontiff, Benedict XIV. Agnesi, however, never went to Bologna, convinced that her contribution to the pedagogy of mathematics was now concluded. Rather, she began devoting most of her time to teaching young girls from poor backgrounds to read, write, and count and to deepen her knowledge of religious literature. Agnesi also began volunteering at the great public hospital of Milan and soon opened the gates of her family palace to sick and infirm women who could not be assisted at the hospital. In 1752 the sudden death of her father freed her from her last commitments to the conversazione at the family palazzo. In 1771 Agnesi was nominated director of the female section of the Pio Albergo Trivulzio, a major charitable institution for the care of the sick and old in Milan. She ran it through the troubled years of the end of the century with her usual passion and determination. After bequeathing her inheritance to the city’s charitable institutions and to her relatives, Agnesi spent the last part of her life in poverty. She died in 1799 and was buried in a common grave off the city walls.


Findlen, Paula. “Maria Gaetana Agnesi.” In The Contest for Knowledge: Debates over Women’s Learning in Eighteenth-Century Italy, edited and translated by Rebecca Messbarger and Paula Findlen. Chicago: Chicago University Press, 2005.

Mazzotti, Massimo. “Maria Gaetana Agnesi: Mathematics and the Making of the Catholic Enlightenment.” Isis 92 (2001): 657–683.

Truesdell, Clifford. “Maria Gaetana Agnesi.” Archive for History of Exact Sciences40 (1989): 113–142.

Massimo Mazzotti

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Maria Agnesi

One of the great figures of Italian science, Maria Gaëtana Agnesi (1718-1799) was born and died in the city of Milan. Her principal work, Analytical Institutions, introduces the reader to algebra and analysis, providing elucidations of integral and differential calculus. Among the prominent features of Agnesi's work is her discussion of a curve, subsequently named the "Witch of Agnesi."

In early childhood, Agnesi demonstrated extraordinary intellectual abilities, learning several languages, including Greek, Latin, and Hebrew. Her father, who taught mathematics at the University of Bologna, hired a university professor to tutor her in mathematics. While still a child, Agnesi took part in learned discussions with noted intellectuals who visited her parents' home. Her knowledge encompassed various fields of science, and to any foreign visitor, she spoke fluently in his language.

Her brilliance as a multilingual and erudite conversationalist was matched by her fluency as a writer. When she was 17 years old, Agnesi wrote a memoir about the Marquis de l'Hospital's 1687 article on conic sections. Her Propositiones Philosophicae, a book of essays published in 1738, examines a variety of scientific topics, including philosophy, logic, and physics. Among the subjects discussed is Isaac Newton's theory of universal gravitation.

Following her mother's death, Agnesi wished to enter a convent, but her father decided that she should supervise the education of her numerous younger siblings. As an educator, Agnesi recognized the educational needs of young people, and eloquently advocated the education of women.

Witch of Agnesi

Agnesi's principal work, Instituzione analitiche ad uso della gioventu' italiana (1748), known in English as her Analytical Institutions, is a veritable compendium of mathematics, written for the edification of Italian youth. The work introduces the reader to algebra and analysis, providing elucidations of integral and differential calculus. Praised for its lucid style, Agnesi's book was translated into English by John Colson, Lucasian Professor of Mathematics at Cambridge University. Colson, who learned Italian for the express purpose of translating Agnesi's book, had already translated Newton's Principia mathematica into English. Among the prominent features of Agnesi's work is her discussion of a curve, subsequently named the "Witch of Agnesi," due in part to an unfortunate confusion of terms. (The Italian word versiera, derived from the Latin vertere, meaning "to turn," became associated with avversiera, which in Italian means "devil's wife," or "witch.") Studied previously by Pierre de Fermat and by Guido Grandi, the "Witch of Agnesi" is a cubic curve represented by the Cartesian equation y (x2 + a2) = a3, where "a" represents a parameter, or constant. For "a" = 2, as an example, the maximum value of y will be 2. As y tends toward 0, x will tend, asymptotically, toward ± ∞.

Received Papal Recognition

In 1750, Pope Benedict XIV named Agnesi professor of mathematics and natural philosophy at the University of Bologna. As David M. Burton explained, it is not quite clear whether she accepted the appointment. Considering the fact that her father was gravely ill by 1750, there is speculation that she would have found the appointment difficult to accept. At any rate, after her father's death in 1752, Agnesi apparently lost all interest in scientific work, devoting herself to a religious life. She directed charitable projects, taking charge of a home for the poor and infirm in 1771, a task to which she devoted the rest of her life.

Further Reading

Alic, Margaret, Hypatia's Heritage: A History of Women in Science from Antiquity through the Nineteenth Century, Beacon Press, 1986.

Burton, David M., Burton's History of Mathematics: An Introduction, Wm. C. Brown, 1995.

Dictionary of Scientific Biography. edited by Charles Coulston Gillispie, Charles Scribner's Sons, 1970.

Olsen, Lynn M., Women in Mathematics, MIT Press, 1974. □

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Agnesi, Maria Gaëtana

Italian Mathematician and Philosopher 17181799

Maria Gaëtana Agnesi was born in Milan, Italy. By the time she was 5 years old, she could speak both Italian and French. Her father was a professor of mathematics in Bologna, and Agnesi enjoyed a childhood of wealth and privilege. Her father provided her with tutors, and she participated in evening seminars, speaking to the guests in languages as varied as Latin, Greek, Hebrew, and Spanish.

In her teens, Agnesi mastered mathematics. She became a prolific writer and an advocate for the education of women. After her mother died, Agnesi managed the household of eight children, and educated her brothers. Her father remarried and after her stepmother died, Maria became housekeeper for twenty siblings.

Agnesi continued studying mathematics, mostly at night, and often to the point of exhaustion. In 1748 her mathematical compendium , Instituzioni analitiche ad uso della gioventù italiana (Analytical Institutions), derived from teaching materials she wrote for her brothers, was published in two volumes. In this work, Agnesi had not only written clearly about algebra , precalculus mathematics, differential calculus , and integral calculus , but she had also added her conclusions and her own methods. Analytical Institutions remains the first surviving mathematical work written by a woman.

So successful was Agnesi's textbook that it became the standard text on the subject for the next 100 years. Her book was studied and admired not only in Agnesi's native Italy but also in France and Germany, and was translated into a number of other languages.

In his 1801 English translation of Agnesi's work, John Colson, the Cambridge (England) Lucasian Professor of Mathematics, made the mistake of confusing the Italian word for a versed sine curve, aversiera, with another Italian word for witch or wife of the devil, avversiere. Although 200 years have passed, the curve Colson misnamed the "Witch of Agnesi" still bears that name in many calculus texts as an illustration of an "even" function with a "maximum" value. For a = 2, for example, the maximum value of y will be 2. The curve illustrates many basic concepts in calculus.


Agnesi was recognized during her lifetime with election to the Bologna Academy of Science. The Pope at the time was interested in mathematics and in 1750 made certain that Agnesi was invited to be an honorary lecturer in mathematics at the University of Bologna. However, Agnesi turned down the appointment and instead adopted a religious life, spending much of her time working among the elderly poor and sick women of Milan.

Although a hospice Agnesi founded has become famous throughout Italy, and Italian streets, a school, and scholarships have been named in her honor, Agnesi is perhaps best known today for her contributions to mathematics.*

*On the centennial of Maria Gaëtana Agnesi's death in 1899, the Italian cities of Milan, Monza, and Masciago chose to remember her by naming streets after the noted mathematician and humanitarian.

see also Calculus.

Shirley B. Gray


Cooney, Miriam P., ed. Celebrating Women in Mathematics and Science. Reston, VA: National Council of Teachers of Mathematics, 1996.

Olsen, Lynn M. Women in Mathematics. Cambridge, MA: MIT Press, 1992.

Internet Resources

Gray, Shirley B. Maria Gaetana Agnesi. <>.