Witch of Agnesi a lasting contribution from the first surviving mathematical work written by a woman, The

Witch of Agnesi a lasting contribution from the first surviving mathematical work written by a woman, The

Gray, S I B

A commemorative on the 200th anniversary of her death

For mathematicians, there is one and only one Witch, the Witch of Agnesi. This famous curve is so named only in English world and is the result of a mistake. In other European languages, the curve is known as the Locus of Agnesi. Why is there a difference in translations from the original Italian? Maria Gaetana Agnesi (1718-1799) wrote a highly successful calculus textbook, Instituzioni analitiche ad uso della gioventu italiana (1748). With clear explanations and good organization, the book was widely studied and admired in France and Germany, and was translated into several languages. However, in an English translation of 1801, the Cambridge Lucasian Professor of Mathematics, John Colson, made a mistake of confusing the cubic averisera, or versed sine curve (from the Latin vertere, “to turn”), with the Italian aversiera, meaning “witch” or “wife of the devil.”

One might argue that this colorful mistake saved Agnesi’s work from anonymity and gave her curve a permanent position in future calculus texts. From Newton to Stephen Hawking, the title “Lucasian Professor” has been bestowed only upon the most distinguished of mathematicians. Colson’s mistake, a small error on the part of a recognized scholar, amounted to giving prominence and longevity to a work that might otherwise have been ignored. Today we recognize Instituzioni analitiche as the first surviving mathematical work written by a woman. January of her was the 200th anniversary of her death.’

In later times, both Newton and Liebniz studied the cycloid, the locus of a point on a circle as the circle rolls along a straight line (Figure 6). Their work was continued by the next generation of mathematicians, L’Hopital, Johann Bernoulli, and Jakob Bernoulli, which led to the solutions of the brachistochrone problem (from the Greek meaning “shortest time”) and the tautochrone problem (from the Greek meaning “equal time”). These provided the genesis of the calculus of variations [5,12].

Johann Bernoulli’s death occurred the year Agnesi published Instituzioni analitiche (1748), the same year that Euler’s Introductio in Analysin Infinitorum was published. Agnesi’s classic work was considered by her contemporaries to be the best comprehensive textbook on the calculus since l’Hopital’s Analyse des infiniment petits (1696). A reviewer described it as the “most complete and best made” calculus, with the only other candidate being Maclaurin’s Treatise of Fluxions, (1742), a work “so dense, torve, and ugsome as scarcely to have been read through by anyone but its author and would plunge any beginner into the slough of despond.”2

Also, note that thirty years later, Lagrange would publish his monumental four-volume Mecanique analytique (1788). There is a remark in its preface that there were no diagrams in its pages. This may or may not have been a boast. The technology for printing was still in its infancy. As is typical of all 18th century mathematics books, Agnesi’s work has “plates” of carefully numbered illustrations attached at the back of the book. These oversized pages were to be opened outward enabling the reader to view the figures while also reading the text. See Figure 7.

Let us briefly note the writing, teaching, and organizational style of the Instituzioni. In her first volume (see Figure 8), Agnesi develops with increasing order of difficulty, finite quantities, analytic geometry, construction of loci, tangents, and the elementary theory of maxima and minima. Near the end we find her famous “Witch” (Figure 10). Her detail is impressive. She notes AQ is “l’asintoto della curva”, i.e., the asymptote; and at “lo punto C”, there is a maximum. The curve for “la x negative” is “ramo simile,” or symmetric for the negative branch. Significantly, she concludes the introductory volume with her now famous curve, an exquisitely clear example of both a maximum and an asymptote, before she goes on to introduce differential calculus in the second volume.

Much of Maria Agnesi’s life span (1718-1799) coincided with that of Euler (1707-1783). Unlike Euler, Agnesi devoted very few years of her life to mathematics. Born in Milan on May 16, she enjoyed a childhood of wealth and privilege fostered within an intellectual environment nourished by her father. Don Pietro Agnesi Mariami occupied the chair of professor of mathematics at the University of Bologna, one of the most venerable European centers of learning. He hosted many visiting scholars. The French would use the word “salon” to describe the atmosphere within the Agnesi home. Being both the first born and extremely precocious, she was able to capitalize on the environment in her family. Before the age of nine, she is said to have mastered Latin, Greek, Hebrew and some modern European languages. Significantly, her first treatise, composed at the age of nine and published at the age of eleven, was an essay in Latin advocating higher education for women. At the age of 20 she wrote Propositiones philosophicae, a collection of essays on philosophy and natural science that her father soon had published.

A visitor to the Agnesi home during this time would write that he had “witnessed something more stupendous than the cathedral of Milan” [7, 21]. Maria and her younger sister, Maria Teresa, entertained their parents’ guests with original discourses and musical compositions. A letter describes Agnesi and a party at her home: “I was brought into a large and fine room, where I found about thirty people from all the countries of Europe, ringed in a circle. . . Mlle Agnesi is a girl of 18 to 20 years of age, neither ugly nor pretty, with a very simple and very sweet manner.”3

But abrupt change came to the Agnesi family. The mother died, leaving Maria, the eldest, and her father as caretakers of twenty other brothers and sisters. She had already gained invaluable experience as a teacher, having assumed the role of mathematics tutor for her younger brothers. Thus she began to write her monumental Instituzioni analitiche. A project that started at the age of 20 as an amusement with mathematics evolved into a serious effort to instruct her younger brothers and ended as a highly respected scholarly treatise. She never married, spending the crucial decade of her 20s engaged in her writing and caring for her immediate family.

Her work was recognized in several ways. She was elected a member of the Bologna Academy of Sciences, an extremely rare honor for a woman. In 1749, the year following publication of Instituzioni analitiche, Pope Benedict XIV appointed her an honorary lecturer at the University of Bologna, a position that she may or may not have accepted. Since her father died three years later, some writers have suggested the appointment may have been a compassionate gesture to support a family in stress. It is known that thereafter she abandoned mathematics and devoted her life to extensive charitable work and to religious studies. She died in the January before her 81st birthday in May, 1799 [4, 9, 10, 13, 15, 20]. She had spent her final years as a Blue Nun at the Luogo Pio, a hospice known throughout Italy. On the centennial of her death, in 1899, streets were named for her in Milan, Monza, and Masciago [11].4 One engraved stone was placed on the Villa Montevecchia and another on the facade of Palazzo de via della Signora:



Maria Gaetana Agnesi most erudite in mathematics glory of Italy and of her century even more gifted in caring for the impoverished elderly in this albergo a hunble servant of charity died in the year 1799

Acknowledgments. The authors thank Dr. Ronald Brashear, Curator, History of Science, Huntington Library, San Marino, CA, Dr. Angel J. Di Bilio, Senior Research Fellow, Beckman Institute, California Institute of Technology, Pasadena, CA, and Prof. John Webb, Murdoch University, Perth, Western Australia.

1The French translation of Newton’s Principia by Emilie de Breteuil, Marquise du Chatelet, was published posthumously in 1759. See [14] for a history of the names and various translations of the names of the curve.

2″Torve” means “stern, wild, or fierce.” “Ugsome” is derived from “ugh.”

3For the sketch of the portrait that is frequently reproduced on the Internet, see [4]. For photographs of a portrait and a bust of Agnesi that were executed by a contemporary artist, see [21]. The original works are housed in the Ambrosian Library of Milan. The famous eleventh edition of the Encyclopedia Britannica, and others, note that her younger sister, Maria Teresa, was a “well-known Italian pianist and composer” of cantatas, concertos and five operas. Her portrait hangs in the La Scala Opera House museum.

4See Table 27: D1-D2 of http://www.CitylightsNews.com/ztmit27.htm to view the street in Milan named for Agnesi on the centennial of her death in 1899.

5Additional references and information on web sites: http://www.scottlan.edu/lriddle/women/agnesi.htm http://www.astro.virginia.edu/~ ewww6n/math/WitchofAgnesi.html http://www-groups.dcs.st-and.ac.uk/~history/Curves/Witch.html http://alephO.clarku.edu/~djoyce/mathhist/agnesi.html http://scidiv.bcc.ctc.edu/Math/Agnesi.html

6To evaluate the qualuty of information on web pages: http://ernie.bgsu.edu/~ vrickey/math311 /web-quality.html


1. Maria Gaetana Agnesi, Instituzioni analiticbe ad uso della gioventu italiana, (Foundations of Analysis for the use of Italian youth), Milan, 1748. See Figure 8.

2. Maria Gaetana Agnesi, Analytical Institutions, translated by John Colson, Taylor and Wilks, 1801.

3. Howard Anton, Calculus with Analytic Geometry, 4th ed., John Wiley, 1992; 823-824.

4. M. Jacques Boyer, “Sketch of Maria Agnesi,” Appleton’s Popular Science Monthly, 53 (July 1898) 289, 403-409.

5. David M. Burton, Burton’s History of Mathematics, 3rd ed., Wm. C. Brown, 1995; p. 391.

6. Robert Decker and Dale Varberg, Calculus Preliminary Edition, Prentice Hall, 1996; p. 635.

7. Encyclopedia Britannica, 11th edition, 1(1910), p. 378.

8. Ross L. Finney, George B. Thomas, Franklin Demana and Bert K. Waits, Calculus: Graphical, Numerical and Algebraic, Addison-Wesley,1994; p. 799.

9. S. I. B. Gray and Robert Mena, Amusements in the History of Mathematics, PRIMUS 7:4, (1997) 317-328.

10. S. I. B. Gray, A Priceless Collection, The Mathematical Intelligencer, 20:2 (1998) 41-46.

11. Emilio Guicciardo, Il Pio Albergo Trivulzio nella storia e nell’attualita (1771-1961), Consiglio Orfanotori e Luogo Pio Trivulzio, 1961; 59-60.

12. Victor J. Katz, A History of Mathematics: An Introduction, Harper Collins, 1993; 511-512.

13. Edna E. Kramer, Maria Agnesi, Biographical Dictionary of Mathematicians, Scribner’s 1992; vol. 1, 21-23.

14. T. F. Mulcrone, The names of the curves of Agnesi, American Mathematical Monthly, 64 (1957) 359-361.

15. Lynn M. Osen, Women in Mathematics, MIT Press, 1995; 33-48.

16. Murray Protter and Philip E. Protter, Calculus with Analytic Geometry, 4th ed., James and Bartlett, 1986; p. 495.

17. George F. Simmons, Calculus with Analytic Geometry, 2nd ed., McGraw-Hill, 1996; p. 591.

18. James Stewart, Single Variable Calculus Early Transcendentals, 3rd ed., Brooks/Cole, 1995; p. 533.

19. James Stewart, Calculus: Concepts and Contexts, Single Variable Preview Edition, Brooks/Cole, 1997; p. 54.

20. Sister Mary Thomas a Kempis, “The walking polyglot, Scripta Mathematica, 6(1940) 211-217.

21. C. Truesdell, Maria Gaetana Agnesi, Archive for History of Exact Science, 40(1989) 113-142. Correction and addition 43(1991) 385-386.

S. I. B. Gray (sgray@calstatela.edu) received her Ph.D. at the University of Southern California. At USC she had the pleasure of taking classes with B. A. Troesch who had studied with Heinz Hopf and Wolfgang Pauli at the ETH in Zurich. She learned that the European and American historical interpretations of mathematical works are not as carefully aligned as one might think.

Tagul Malakyan was born and educated in Gyumri, Armenia (Europe) where she worked in advanced mathematical research applied to science. She immigrated to the United States in 1990. While taking a History of Mathematics course with Dr. Gray at California State University, Los Angeles, she was amazed that a curve she had seen in Russian as a student was called a “Witch ” in English.

Copyright Mathematical Association Of America Sep 1999

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