Agnesi Publishes Analytical Institutions

Date 1748

Maria Agnesi published one of the first introductory textbooks for beginning students in the new field of calculus. Her text, by defining the terms in which the new discipline was taught, shaped the understanding of a generation of mathematicians. One of the curves discussed in her text is still associated with her name.

Locale Milan (now in Italy)

Key Figures

• Maria Gaetana Agnesi (1718-1799), Italian mathematician and philanthropist
• Leonhard Euler (1707-1783), Swiss mathematician and physicist
• Laura Bassi (1711-1778), Italian physicist

Summary of Event

Maria Gaetana Agnesi’s Instituzioni analitiche ad uso della gioventù italiana (1748; Analytical Institutions, 1801) provided students with a guide to the basic elements of calculus as it had been developed by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late seventeenth century. The book was translated into several other European languages, bearing witness to its importance in the transmission of calculus throughout Europe. Agnesi herself was admired by the mathematical community, which was impressed by the mastery achieved by a woman in what was at the time thought of as a male domain.

Agnesi was born into a wealthy family that valued education. Her father was a professor of mathematics at the University of Bologna and provided every educational opportunity for his children. In particular, Maria was tutored by university professors after she displayed a remarkable talent for both languages and mathematics at an early age. She is said to have been fluent in Latin, Greek, and Hebrew by the age of nine, and her father arranged for her to give discourses on scholarly subjects at home while she was still a child.

It is not clear that Maria Agnesi enjoyed being thrust into the spotlight, and in addition to her scholarly attainments, she also displayed ill health throughout her life. This may have been a way of refusing to be made the center of attention. By 1738, she expressed an intention to enter a convent, but this plan was frustrated by her father. She was quite accomplished in mathematics, and one reason for her taking up that pursuit may have been her recognition that it was less likely that people would be invited into the house to hear her talk about the newly created subject of calculus than to hear about literary and philosophical subjects. Her work in mathematics was guided by Father Remiro Rampinelli, a member of the Benedictine order and professor of mathematics at Pavia. She acknowledged his guidance throughout her published work.

In her years as a teenager, Agnesi was already solving problems of some difficulty, but she was following well-trodden paths, so her work did not make many ripples in the mathematical community. The publication in 1748 of the two-volume Analytical Institutions attracted attention well beyond anything that she had accomplished earlier. The work was aimed at a younger readership than most earlier discussions of calculus, so it began from an elementary level. In addition, to reduce the level of difficulty for students in Italy, Agnesi wrote her text in Italian rather than Latin, which was the scholarly international language of the time. She did not write in the native dialect of Milan, which was at the time still under the control of the Holy Roman Empire. Instead, she wrote Italian prose suited for a broad readership throughout the Italian states.

Analytical Institutions proceeds from the basis of algebra to the limits of the calculus that was known at the time. Agnesi was careful in her discussions to make sure that her target audience would be able to understand the process for solving problems in calculus. In fact, her presentation has been criticized by later mathematicians for concentrating too much on examples, at the expense of explicating the theory behind the practical applications she teaches. It was not until the next century, in fact, that any exposition of the field took a primarily theoretical approach. The method that Agnesi used had the merit of ensuring that students would be able to do mathematics and not just read about it.

The single mathematical object to which Agnesi’s name is attached is a curve defined by an equation of the third degree. It is commonly known as the “witch” of Agnesi, but this is the result of a mistranslation of the Italian word for “turn.” There is a certain irony in such a term being attached to the name of someone as pious as Agnesi. She was responsible neither for the invention of the curve nor for the name attached to it, but her name has been firmly associated with the curve by later generations.

Agnesi’s book, as a introductory textbook rather than an analysis of the field, did little to advance the knowledge or level of discussion of the foundations of the calculus. She provided interesting discussions of arbitrarily small quantities that could be added to finite quantities without markedly changing their value. Her primary concern in dealing with the calculus, however, was simply to enable students to handle problems involving maxima and minima, and for such calculations the theoretical foundations of the discipline were not crucial. If her readers were able to appreciate how to solve differential equations, the most advanced topic she discussed, the treatise would have achieved its purpose.

The wealth of her family enabled Agnesi not only to have the book printed but also to send copies to eminent people outside the sphere of mathematics. For example, Austrian archduchess Maria Theresa received a copy and rewarded the author (who had dedicated the book to her) with a diamond ring. Even more to her taste, however, may have been the recognition she received from Pope Benedict XIV, who presented her with a gold medal and arranged for her appointment to a chair in mathematics at the University of Bologna. Within scientific circles, Agnesi received similar acclaim. The French Académie des Sciences (Academy of Sciences) arranged for a translation of the second, more advanced volume, and one of Newton’s successors himself translated the book into English.

It is not clear that Agnesi ever took up the chair provided by the pope. She had moved in the direction of religious activity, and her later years were entirely devoted to philanthropy in the setting of the Church. When asked to comment on the work of a younger mathematician, she avowed that she had given up on mathematics herself. With the publication of her Analytical Institutions, she may have been saying farewell to the field she hoped to shape.

Significance

It is easy to attribute to the Analytical Institutions a strictly mathematical importance it did not possess. The originality of exposition does not compare with that of Leonhard Euler, whose introduction to analysis came out the same year as Agnesi’s book. While Euler’s approach to the foundations of the calculus may not have been entirely satisfactory, he rethought the issues in a way that Agnesi did not.

Similarly, Agnesi’s position as a woman in the European intellectual world may not have been quite so distinctive as historians of mathematics suggest. While she may have been the first woman to be offered a chair in mathematics at a European university, Laura Bassi had been teaching physics at the University of Bologna since 1732. Bassi encouraged Agnesi to take up the position there but failed to convince her younger colleague.

On the other hand, there is no doubt that the publication of the Analytical Institutions made a difference in the perception of the ability of women to do mathematics at the highest level. Women were an important part of the intellectual life of Europe in the eighteenth century, but there may have been skepticism about their interest in something so abstract as mathematics. Agnesi demonstrated that a woman could master the full extent of the mathematics of her time and give back to her students what she had learned. It is not a coincidence that the Analytical Institutions is the earliest mathematical treatise by a woman that survives.

Bibliography

Burton, David M. The History of Mathematics: An Introduction. 5th ed. New York: McGraw-Hill, 2003. Careful distillation of the prevalent scholarship in the history of mathematics.

Kennedy, Hubert. “Maria Gaetana Agnesi.” In Women of Mathematics, edited by Louise Grinstein and Paul Campbell. New York: Greenwood Press, 1982. Provides cultural context rather than mathematical details.

Kramer, Edna E. “Maria Gaetana Agnesi.” In Dictionary of Scientific Biography, edited by Charles C. Gillispie. Vol. 1. New York: Charles Scribner’s Sons, 1970. Kramer, a historian of mathematics, pays tribute to Agnesi with some enthusiasm but also technical details.

Osen, Lynn M. Women in Mathematics. Cambridge, Mass.: MIT Press, 1974. Tribute to Agnesi’s “vital and inspiring” memory in the fight for women in science.

Stigler, Stephen M. Statistics on the Table. Cambridge, Mass.: Harvard University Press, 1999. Examination of the sequence of events that led to the term “witch” being applied to the curve associated with Agnesi.

Truesdell, Clifford. “Maria Gaetana Agnesi.” Archive for History of the Exact Sciences 40 (1989): 113-142. Demythologizing study of the historical impact of Agnesi’s treatise.