Women and Minorities in Mathematics

Incorporating Their Mathematical Achievements Into School Classrooms

Hypatia, the First Known Woman Mathematician

Sarah J. Greenwald,

Appalachian State University, Boone, North Carolina

and

Edith Prentice Mendez,

Sonoma State University, Rohnert Park, California

Hypatia is the first woman mathematician about whom we have either biographical knowledge or knowledge of her mathematics. Hypatia developed commentaries on older works, probably including those by Ptolemy, Diophantus, and Apollonius, in order to make them easier to understand. Hypatia was probably the first woman to have a profound impact on the survival of early thought in mathematics.

Since Hypatia lived so long ago, it is hard to know exactly what she worked on, although we do have some specific historical evidence of her mathematics (Deakin, 1996, pp. 79-81; Fitzgerald, 1926, p. 90) and an account of her horrible death. Other fictional accounts of her life have added to the confusion about her. We do know that original scholarship was not Hypatia’s focus. Together with her father Theon, she helped preserve some of the treasures of ancient Greek mathematics and astronomy. While she cannot compare in originality with the mathematicians that she wrote commentaries on, her reputation as a teacher and scholar is secure, and as research and analysis of ancient texts continues, we may indeed learn more about her mathematical contributions.

We will examine what we do and do not know about her life, the mathematics that she might have worked on, and ways to incorporate these ideas into the mathematics classroom.

**Hypatia’s
Life**

Hypatia’s birthdate is unknown. During the time that she lived, in the 4th century of the common era, Alexandria was the center of learning for Western civilization. According to a 6th century report by Damascius (Deakin, 1996) Hypatia was born and educated in Alexandria. She went beyond the mathematics and astronomy of her father's expertise, learning philosophy. She then taught philosophy, and presumably the prerequisite mathematics, to students who came from distant places. She was held in high esteem for her teaching, her virtue, and her civic-mindedness. There is no evidence that she traveled outside of Alexandria.

Much of what is published about Hypatia's life is fiction written in the 19th and 20th centuries. Hubbard (1908) wrote an entertaining, but fictional account of Hypatia’s educational training and life, and even invented a quote that he attributed to Hypatia. While this fictional account is highly romantic and may encourage student interest in Hypatia, there is no evidence supporting most of the tale. In her book, Osen (1974) used Hubbard as one of her primary sources on Hypatia. Unfortunately, this fictional account has been spread as truth in other publications that depend on Osen. (e.g., Johnson, 1999; Smith, 1996).

This example of mistaking fiction for fact and the spread of poor scholarship can be a great starting point for a discussion on the importance of using numerous reputable references, trying to get as close to original sources as possible, and the fact that books as well as web pages are not necessarily correct. Students will be interested in the idea that a fictional account in one book can propagate as truth, spreading to many sources.

In Hubbard’s book, he includes a fictionalized sketch of Hypatia. We have no historical basis for Hypatia’s appearance. There are no statues nor sketches of her that have survived, as far as we know. In fact, she may have resembled Egyptian women of the time instead of the woman represented as Hypatia in Figure 1. Before showing students this fictionalized sketch of Hypatia, you could ask them to imagine what Hypatia might have looked like. The fact that this fictionalized picture has been stated to be a real picture of Hypatia would also be a good beginning of a discussion of racial issues.

__Figure 1__.
Fictionalized Sketch of Hypatia (Hubbard, 1908).

Hypatia's death in 415 is authenticated by an ancient, nearly contemporary, account of the church historian Socrates Scholasticus (Valesius, 1680; Deakin, 1996, pp. 82-84). Hypatia was an associate of Orestes, the Roman political leader of Alexandria and a rival of the Christian bishop Cyril for control of the city. Although Orestes and some of her students were Christians, Hypatia never converted to that religion. A Christian mob was incited to lynch and kill her. The mob dragged her through the streets and scraped the flesh from her bones with oyster shells before burning her body.

Smith (1996, pp. 45-46) and Lumpkin & Strong (1995, pp. 145-146) contain worksheets that engage students with questions related to Hypatia’s life and death.

** **

**Hypatia’s
Mathematics**

Hypatia’s reputation as the leading mathematician and philosopher of her time is authenticated in ancient writings. We have neither evidence of mathematical advances made by Hypatia, nor writings that are assuredly hers. Yet there is evidence of her commentaries on the work of others that have helped to make these older works clearer for students and to preserve them through the centuries. Let's look at the ancient evidence about Hypatia that is available, some conjectures that can be made about her mathematics, and ways to incorporate these into the classroom.

**Evidence of Hypatia’s
Commentaries and their Role in the Preservation of Mathematics**

The one contemporaneous citation
of Hypatia's mathematical work is in the introduction to Theon's commentary on
Ptolemy's Book III of the *Almagest*.
Theon describes this as "the edition having been prepared by the
philosopher, my daughter Hypatia" (Rome, 1931-1943, p. 807). The other
direct ancient report of Hypatia's mathematics comes from Hesychius in the 6th
century: "She wrote a commentary on Diophantus, the Canon of Astronomy,
and a commentary on the Conics of Apollonius" (Deakin, 1996). Since this
list does not include the commentary on Ptolemy, it is obviously not
exhaustive. Wilbur Knorr uses this fact and a close comparison between
Hypatia's edition of the Ptolemy commentary and others by Theon to conjecture
that Hypatia may well have edited and made commentaries on other ancient texts,
including those of Archimedes (Knorr, 1989). Socrates Scholasticus’s
report of Hypatia’s death also speaks of Hypatia's high achievements in
science and philosophy, surpassing all the other philosophers of her time. We
also hear of Hypatia's teaching and philosophy from one of her students,
Synesius of Cyrene. His extant letters do not mention her mathematics, but he
describes Hypatia as "the most holy revered philosopher" and
addresses his letters to her "to the Philosopher" (Fitzgerald, 1926).

Students will be interested in the fact that detective work is needed to guess what Hypatia worked on and that historians are still debating and researching the possibilities today. They will also be interested in the role commentaries played in preserving ancient texts. These texts were written on fragile papyrus and would have disintegrated under the best of circumstances. Centuries of unrest, wars, and lack of interest in scholarship provided a poor climate for preservation. Copies of copies of copies found their way to surviving centers of culture such as Constantinople and Baghdad. Arab scholars translated ancient Greek works, wrote their own commentaries, and produced original mathematics in the Greek tradition as well. Some ancient works are known today only because of their Arabic copies, others have a Greek tradition with later commentaries as well, or in Latin translation of the Greek. Commentaries not only provided copies of ancient texts, but assistance for students who had only the text, not a teacher, from which to learn. As Western civilization fragmented and ancient schools were disbanded, those few who had access to learning were often solitary scholars. Thus Hypatia's expertise as a teacher may well have influenced untold generations.

**Greek Number System**

** **

Hypatia must have used the standard Greek numbering system, which was based on the Greek alphabet, with some archaic letters included (Heath, 1921, p. 32). Each decade had a different symbol [I for tens and K for 20s, for example], but this was not a positional base-ten system. There was no use of zero, except in the base-60 fractions used in astronomy. Neither was there a subtractive principle such as the Roman's use of IV to indicate V - I. Roman influence helped solidify the standardization of the order of writing Greek numerals with the higher value on the left. In earlier times, 25 might have been written as either KE or EK.

__Figure 2__.
Greek Number System

**Hypatia’s Work on
Ptolemy’s Almagest**

We know that Hypatia worked on
Ptolemy's Book III of the *Almagest*.
Hence Hypatia worked in astronomy, a field that relied heavily on careful
calculation and on the geometry required to describe Ptolemy's geocentric
universe. The *Almagest* remained
the leading resource for astronomical study in the West and in Arabic regions
from the time of its writing in Alexandria in the second century of the common
era until the time of Copernicus in the 16^{th} century.

In Book III of the *Almagest*, there is a sexigesimal (base 60) computation of the
orbit of the earth around the sun. Students can look at the Greek version
(Knorr, 1989, pp. 802-804) and then work through a version translated into
English (Knorr, 1989, pp. 780-781). It is interesting to note that in Book I,
a similar computation was done very differently. Instead of the precise answer
found in Book III, an approximate solution was found in Book I (see Greenwald,
2001a). In addition, stylized differences in the writing were also found.
Hypatia might have worked on this section, but it also might have been Theon or
someone else, and we will never know for certain.

**Hypatia’s work with
Diophantus**

Historians have debated the
extent of Hypatia's work with Diophantus. Diophantus probably lived in the
third century of the common era in Alexandria. He is best known for his work *Arithmetica*, part of which has survived in Greek, and part only
in Arabic translation. Unlike earlier mathematicians in the Greek tradition who
focused on geometry and number theory, Diophantus wrote on algebra. He made
innovations in introducing symbols to a field that had been one of verbal
algorithms since early Babylonian times. He introduced problems with many
solutions in indeterminate analysis. Hypatia's ability to teach and write
commentary on these works would be an indication of her versatility as a
mathematician. Although the subject matter is different from Hypatia’s known
work related to astronomy and the mathematics associated with it, Diophantus
was an Alexandrian mathematician, and Hypatia, as the leading mathematician of
her time, must have known of his work.

There are a number of activity
sheets related to Hypatia’s possible commentaries on Diophantus’ *Arithmetica*. Perl (1978, p. 27) engages students with the
number of ways to make change for a dollar using nickels, dimes and quarters.
Johnson (1999, pp. 41-42) asks students to find multiple solutions to an
algebraic statement. Lumpkin & Strong (1995, pp. 144-146) discuss number
patterns. Waithe (1987, pp. 176-183) contains a translation of sections from
Diophantus’ *Arithmetica*
that Hypatia might have worked on. While this is not an activity sheet,
students can engage the material by translating the problems into modern
algebraic notation. They can then solve them and present their work to the
rest of the class.

**Hypatia’s work on
Apollonius’ Conics**

Apollonius of Perga lived in the
third century before the common era and studied in Alexandria. His work on
conic sections is massive and difficult. The names of the curves parabola,
ellipse, and hyperbola are his. Apollonius' work not only influenced Ptolemy
in his studies of planetary orbits, but Descartes and Fermat in the 17th
century in their development of analytical geometry. Any help that Hypatia
gave to the elucidation and preservation of the works of Apollonius may have
had far reaching consequences. The use of the *Conics* in astronomy and Apollonius' connection to
Alexandria are arguments for Hypatia's involvement with commentaries on the *Conics*.

Students can investigate conic sections through a worksheet designed to engage them with both geometric and algebraic formulations of parabolas, hyperbolas and ellipses (see Perl, 1978, pp. 13-26). This worksheet encourages visualization through hands on activities such as cutting and taping sections of a cone and the exploration of each conic section as a path of points satisfying algebraic conditions. The worksheet concludes with real-life applications of conic sections.

**Hypatia and Archimedes’ Dimension
of the Circle**

Hypatia may have written a
commentary on Archimedes’ *Dimension of the Circle*. Archimedes, the greatest mathematician of ancient
times, was killed by Roman soldiers in 212 before the common era. He lived in
Syracuse, Sicily, but was in correspondence with mathematicians at Alexandria.
One of his works, the *Method* was
lost in the middle ages and only rediscovered in 1906. Another, the *Dimension
of the Circle* is found in both Greek and
Arabic copies. The Arabic manuscripts contain further clarification and careful
explanation, such as might be written by a master teacher. Since Hypatia was
known to be a commentator and an excellent teacher, it is certainly possible
that she was one of the scholars who helped preserve this work.

Wilbur Knorr, a math historian,
identified a certain style of writing that he attributes to Hypatia. He
learned new languages so that he could analyze different versions of
Archimedes’ *Dimension of the Circle*
in Hebrew, Arabic, Latin and Greek. Although there is no historical evidence
of the existence of commentaries developed by Hypatia on Archimedes’
work, Knorr suggests that Hypatia's influence can be found there. As research
and analysis of ancient texts continues, we may learn more about
Hypatia’s mathematical contributions.

This diagram shows the inscribed polygons needed in the proof of Proposition 1: Every circle is equal to a right-angled triangle, one of whose sides containing the right angle is equal to the circumference of the circle, and the other such side equals the radius of the circle.

__Figure 3__.
Arabic Diagram for Archimedes *Dimension of the Circle*.

The activity sheet found at the
end of this column engages students with constructions related to
Archimedes’ *Dimension of the Circle*,
while a worksheet aimed at a higher level (see Greenwald, 2001b) details the
proof of Proposition 1. Students can also explore these activities in a
dynamic software package.

**References**

Deakin, M. (1996). *Mathematician
and martyr: A biography of Hypatia of Alexandria.* Victoria, Australia: Department of Mathematics, Monash

University, Clayton.

Fitzgerald, A. (1926), *The
Letters of Synesius of Cyrene.* London:
Oxford University Press.

Greenwald, S. (2001a). *Hypatia’s
work on Ptolemy’s Almagest. *[On-line].
Available: http://www.mathsci.appstate.edu/~sjg/womeninmath/book3.html

Greenwald, S. (2001b). *Classroom
Worksheet on Hypatia’s Possible Work on Archimedes Dimension of the
Circle.* [On-line]. Available:
http://www.mathsci.appstate.edu/~sjg/wmm/hypatia/Hypatiafinalsheet.html

Heath, T. (1921). *A history
of Greek mathematics.* Oxford: Clarendon.

Hubbard, E. (1908). *Little
journeys to the homes of the great teachers.*
East Aurora, NY: the Roycrofters. Available:
http://www.polyamory.org/~howard/Hypatia/Hubbard_1928.html

Johnson, A. (1999). *Famous
problems and their mathematicians. *Englewood,
CO: Teacher Ideas Press.

Knorr, W. (1989). *Textual
Studies in Ancient and Medieval Geometry*.
Boston, MA: Birkhauser.

Lumpkin, B. & Strong, D.
(1995). *Multicultural science and math connections: Middle school projects
and activities.* Portland, ME: J. Weston
Walch.

Osen, L. (1974). *Women in
mathematics.* Cambridge, MA: MIT Press.

Perl, T. (1978). *Math equals:
Biographies of women mathematicians + related activities. *Menlo Park, CA: Addison-Wesley Publishing Company.

Rome, A. (1931-1943). *Commentaires
de Pappus et de Theon d'Alexandrie sur*

*l'Almageste*. Vatican City: Biblioteca Vaticana.

Smith, S. (1996). *Agnesi to
Zeno: Over 100 vignettes from the history of math. *Berkeley, CA: Key Curriculum Press.

Valesius. (1680). *Ecclesiastical
History of Socrates Scholasticus* Cambridge,
England: John Hayes, Microtext 015 4 981:8.

Waithe, M.E. (1987). *A
history of women philosophers: Ancient women philosophers 600 B.C. - 500 A.D.* place, Dordrecht, Netherlands: Martinus Nijhoff.

**Activity Sheet: Hypatia and ****Archimedes’ Dimension of the Circle**

Hypatia is the first woman mathematician about whom we have either biographical knowledge or knowledge of her mathematics. Hypatia developed commentaries on older works, probably including those by Ptolemy, Diophantus, and Apollonius, in order to make them easier to understand. Hypatia was probably the first woman to have a profound impact on the survival of early thought in mathematics

Since Hypatia lived so long ago, it is hard to know exactly what she worked on, although we do have some specific historical evidence of her mathematics and an account of her horrible death. We know that original scholarship was not Hypatia’s focus. Together with her father Theon, she helped preserve some of the treasures of ancient Greek mathematics and astronomy. While she cannot compare in originality with the mathematicians that she wrote commentaries on, her reputation as a teacher and scholar is secure.

Hypatia may have written a
commentary on Archimedes’ *Dimension of the Circle*. Wilbur Knorr, a math historian, identified a
certain style of writing that he attributes to Hypatia. He learned new
languages so that he could analyze different versions of Archimedes’ *Dimension
of the Circle* in Hebrew, Arabic, Latin and
Greek. Although there is no direct evidence of the existence of commentaries
developed by Hypatia on Archimedes’ work, Knorr suggests that Hypatia's
influence can be found there. As research and analysis of ancient texts
continues, we may indeed learn more about Hypatia’s mathematical
contributions. We will explore mathematical ideas from Archimedes’ *Dimension
of the Circle* and in this way we will see
some mathematics that Hypatia might have worked on.

Archimedes worked to establish a very good estimate of the value of the ratio of circumference to diameter that we today call “π”, and he proved the following theorem: The area of any circle is equal to the area of a right-angled triangle in which one of the sides about the right angle is equal to the radius and the other to the circumference of the circle.

1. Archimedes’ theorem states that for any circle, one-half the perimeter times the radius is equal to the area. Using formulas for the area and perimeter (circumference) of a circle, in terms of the radius, show that this statement is true.

Since Archimedes and the mathematicians who later wrote commentaries on his work, such as possibly Hypatia, were not working with the formulas that we use today, they were interested in proving this statement. Archimedes proved the theorem by inscribing and circumscribing polygons about a circle. Here are some of the constructions related to his clever proof.

2. Construct and find the area of the square inscribed in a circle of diameter 6 inches as shown below. Archimedes knew the Pythagorean Theorem and several useful facts about circles and squares that you already know. State any facts that you use.

3. If we bisect the arcs formed by the inscribed square, then we will obtain four new points. We can connect these points and the corners of the square with straight lines in order to obtain the octagon in the picture below. Find the area of the inscribed regular octagon. Give your answer both in exact radical notation and approximated to 4 decimal places.

4. Find the area of the square circumscribed about the same circle we started with in question 2.

5. What bounds have you now found for π? You should approximate your results to four decimal places.

6. Calculate the area of the your circle with diameter 6 inches, using 3.1416 as the approximation of π. Compare this value to the approximations in questions 2-4.