Maria Gaetana Agnesi’s
Analytical Institutions
Chelsea Sprankle
Hood College
Frederick, Maryland
Biographical Information
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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
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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)
Table of Contents
Page 1
Dedication
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Empress Maria Theresa of Austria
Proud to publish during time of
woman ruler
Maria Teresa gave her a gift
Influences on Maria
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Descartes, Newton, Leibniz, Euler
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Belloni, Manara, Casati
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Ramiro Rampinelli
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Reyneau’s Analyse démontrée
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Jacopo Riccati
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Maria Teresa of Austria-role model
Analytical Institutions
(Instituzioni Analitiche)
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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
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John Colson (1680-1760)
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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”
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The Mistake of the Witch
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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
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Newton’s fluxions (English)
1st Derivative:
2nd Derivative:
x
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x
Leibniz’s differentials
1st Derivative:
dx
2nd Derivative:
2
ddx or d x
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 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 x , or the second
fluxion of x, is written thus, x; where it may be
observed, that x and x 2 are not the same, the first
signifying (as said before,) the second fluxion of x, and
the other signifying the square of 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
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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…
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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
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Streets, scholarships, and schools
have been named in her honor
Instituzioni is the first surviving
mathematical work of a woman
Special Thanks!
Thanks to the Summer Research
Institute of Hood College!
References
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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.