Maria Gaetana Agnesi’s Analytical Institutions

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

2. 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

3. 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

4. Analytical Institutions (Instituzioni Analitiche)

5. Table of Contents Page 1

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

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

8. 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

9. English Translation • John Colson (1680-1760) • 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

10. 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”

11. 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.”

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

13. 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

14. 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 differenzeterze rispetto alle seconde, e così di mano in mano. Le differenze seconde si sogliono marcare condoppia d, le terze con trè dec. La differenza adunque di dx, cioè la differenza seconda di x si scriverà ddx, o pure d2x, e dx2, 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 d3xec. Così ddy sarà la differenza di dy, cioè la differenza seconda di y ec.

15. 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 2are not the same, the first signifying (as said before,) the second fluxion ofx, and the other signifying the square of .

16. 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.

17. 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.

18. 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

19. 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

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

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

22. References • Agnesi, Maria. Analytical Institutions (English translation). John Colson. London: Taylor and Wilks, 1801. • Agnesi, Maria. Instituzioni Analitiche ad uso Della Gioventu Italiana. Milan, 1748. • Dictionary of Scientific Biography. “Agnesi, Maria Gaetana”. 75-77 • Findlen, Paula. "Translating the New Science: Women and the Circulation of Knowledge in Enlightenment Italy." Configurations 3.2(1995) 167-206. 27 June 2007 http://muse.jhu.edu/journals/configurations/v003/3.2findlen.html>. • Gray, Shirley. Agnesi. 1 Jan. 2001. California State University. 22 Jul 2007 <http://instructional1.calstatela.edu/sgray/Agnesi/>. • Mazzotti, Massimo. "Maria Gaetana Agnesi: Mathematics and the Making of a Catholic Enlightenment." Isis 92(2001): 657-683. • Mount Holyoke College Library web page. <http://www.mtholyoke.edu/lits/library/arch/col/rare/rarebooks/agnesi/>. • Mulcrone, T. F. “The Names of the Curve of Agnesi.” The American Mathematical Monthly 64(1957): 359-361. • Archimedes/Newton/Agnesi/Euler: A Sampler of Four Great Mathematicians. Ohio State University, 1990. • Truesdell, Clifford. "Maria Gaetana Agnesi." Archive for History of Exact Science 40(1989): 113-142.