Maria Gaetana Agnesi’s Analytical Institutions

Chelsea Sprankle

Hood CollegeFrederick, 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 1Table 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 (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

Colson’s Rendition Wrote The Plan of the Lady’s

System of AnalyticksPurpose was to “render it more

easy and useful” for the ladiesDid 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) 1st Derivative: 2nd Derivative:

Leibniz’s differentials 1st Derivative: 2nd Derivative:

or

xx

dx ddx xd 2

Notational Controversy

Myth: Agnesi didn’t mention fluxionsMyth: Colson eliminated Agnesi’s

references to differences *Agnesi used both words*Colson used both wordsTruth: 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 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 d3x 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 .

x

xx x

x

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, 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.