Maria Gaetana Agnesi’s Analytical Institutions Chelsea Sprankle Hood College Frederick, Maryland

Biographical Information  Born in Milan on May 16, 1718  Parents: Pietro & Anna Agnesi  Fluent in many languages by the time she was an adolescent  Discussed abstract mathematical and philosophical topics with guests at her father’s home

Biographical Information  Published Propositiones Philosophicae (191 theses on philosophy and natural science) in 1738  Wanted to enter a convent at age 21  Took over household duties  Studied theology and mathematics

Analytical Institutions (Instituzioni Analitiche)

Page 1 Table of Contents

Dedication  Empress Maria Theresa of Austria  Proud to publish during time of woman ruler  Maria Teresa gave her a gift

Influences on Maria  Descartes, Newton, Leibniz, Euler  Belloni, Manara, Casati  Ramiro Rampinelli  Reyneau’s Analyse démontrée  Jacopo Riccati  Maria Teresa of Austria-role model

Analytical Institutions (Instituzioni Analitiche)  Two-volume work (4 books)  Tomo I Libro Primo – Dell’ Analisi delle Quantità finite  Tomo II Libro Secondo – Dell Calcolo Differenziale Libro Terzo – Del Calcolo Integrale Libro Quarto – Del Metodo Inverso delle Tangenti

English Translation  John Colson ( ) Lucasian professor at Cambridge Published Fluxions in English in 1736 Learned Italian late in life Died in 1760 before it was published  Edited by Reverend John Hellins  Published in 1801

Colson’s Rendition  Wrote The Plan of the Lady’s System of Analyticks Purpose was to “render it more easy and useful” for the ladies Did not get past the first book  Responsible for the “witch”

The Mistake of the Witch  Original Italian version: a versiera – versed sine curve  Derived from Latin vertere – to turn  Colson’s version: avversiera – witch “…the equation of the curve to be described, which is vulgarly called the Witch.”

Notational Controversy  Newton’s fluxions (English) 1 st Derivative:2 nd Derivative:  Leibniz’s differentials 1 st Derivative:2 nd Derivative: or

Notational Controversy Myth: Agnesi didn’t mention fluxions Myth: Colson eliminated Agnesi’s references to differences *Agnesi used both words *Colson used both words Truth: Colson did change Agnesi’s differential notation to fluxional notation

5. In quella quisa che le differenze prime non-ânno proporzione assegnabile alle quantità finite, così le differenze seconde, o flussioni del secondo ordine non ânno proporzione assegnabile alle differenze prime, e sono di esse infinitamente minori per mondo, che due quantità infinitesima del primo ordine, masono assumersi per equali. Lo stesso si dica delle differenze terze rispetto alle seconde, e così di mano in mano. Le differenze seconde si sogliono marcare condoppia d, le terze con trè d ec. La differenza adunque di dx, cioè la differenza seconda di x si scriverà ddx, o pure d 2 x, e dx 2, perchè il primo significa, come ô deto, la differenza seconda di x, ed il secondo significa il quadrato di dx ; la differenza terza sarà dddx, o pure d 3 x ec. Così ddy sarà la differenza di dy, cioè la differenza seconda di y ec.

5.After the same manner that first differences or fluxions have no assignable proportion to finite quantities; so differences or fluxions of the second order have no assignable proportion to first differences, and are infinitely less than they: so that two infinitely little quantities of the first order, which differ from each other only by a quantity of the second order, may be assumed as equal to each other. The same is to be understood of third differences or fluxions in respect of the second; and so on to higher orders. Second fluxions are used to be represented by two points over the letter, third fluxions by three points, and so on. So that the fluxion of, or the second fluxion of x, is written thus, ; where it may be observed, that and 2 are not the same, the first signifying (as said before,) the second fluxion of x, and the other signifying the square of.

Problem I Let there be a certain sum of shillings, which is to be distributed among some poor people; the number of which shillings is such, that if 3 were given to each, there would be 8 wanting for that purpose; and if 2 were given, there would be an overplus of 3 shillings. It is required to know, what was the number of poor people, and how many shillings there were in all.

Solution Let us suppose the number of poor people to be x; then because the number of shillings was such, that, giving to each 3, there would be 8 wanting; the number of shillings was therefore 3x – 8. But, giving them 2 shillings a-piece, there would be an overplus of 3; therefore again the number of shillings was 2x + 3. Now, making the two values equal, we shall have the equation 3x – 8 = 2x + 3, and therefore x = 11 was the number of poor. And because 3x – 8, or 2x + 3, was the number of the shillings, if we substitute 11 instead of x, the number of shillings will be 25.

Comparison: Agnesi & Euler  Introductio in Analysin Infinitorum and Analytical Institutions published in 1748  Both thought it was important to know English notation and Leibniz notation  Began their texts with basic definitions and explanations of concepts  Used many examples

After 1748…  Appointed as honorary reader at University of Bologna by Pope Benedict XIV  Later asked to accept the chair of mathematics  Devoted the rest of her life to charity  Cared for poor older women  Died January 9, 1799

Recognition  Streets, scholarships, and schools have been named in her honor  Instituzioni is the first surviving mathematical work of a woman

Thanks to the Summer Research Institute of Hood College! Special Thanks!

References  Agnesi, Maria. Analytical Institutions (English translation). John Colson. London: Taylor and Wilks,  Agnesi, Maria. Instituzioni Analitiche ad uso Della Gioventu Italiana. Milan,  Dictionary of Scientific Biography. “Agnesi, Maria Gaetana”  Findlen, Paula. "Translating the New Science: Women and the Circulation of Knowledge in Enlightenment Italy." Configurations 3.2(1995) June  Gray, Shirley. Agnesi. 1 Jan California State University. 22 Jul  Mazzotti, Massimo. "Maria Gaetana Agnesi: Mathematics and the Making of a Catholic Enlightenment." Isis 92(2001):  Mount Holyoke College Library web page..  Mulcrone, T. F. “The Names of the Curve of Agnesi.” The American Mathematical Monthly 64(1957):  Archimedes/Newton/Agnesi/Euler: A Sampler of Four Great Mathematicians. Ohio State University,  Truesdell, Clifford. "Maria Gaetana Agnesi." Archive for History of Exact Science 40(1989):