Hypatia is estimated to have been born in Alexandria around 370 AD [4]. Interestingly, there is more known about her death than her life [5]. She is the first woman mathematician who we have evidence of [4]. Her commentaries on older works helped them survive and be better understood for years after.
Her story begins with her father, Theon of Alexandria, who was a famous mathematician and philosopher. He was the director of the library of Alexandria during its peak and raised his daughter, Hypatia, in an environment filled with education and thought [4]. They had a strong bond and shared passions [6]. Though her father desired for her to become as well-rounded as possible, he focused mostly on mathematics and astronomy [4]. Hypatia studied astronomy, astrology, religion, and mathematics, and kept a strict physical routine to maintain a healthy body [6]. She took the various teachings of her father to a new level by studying philosophy and in about 400 AD she became head of the Platonist school at Alexandria [4]. She was viewed highly for her way of teaching and respectable morals [9].
As a teacher, she focused primarily on mathematics and the philosophy of Neoplatonism, based on the ideas of its founders, Plotinus and Iamblichus. This particular ancient Greco-Roman philosophy reflected upon centuries of intellectual culture and merged the scientific and moral theories of Aristotle, Plato, and the Stoics with literature, myth, and religious practice. The Neoplatonists were very idealistic and laid their foundation on the meaning of what is now phrased as “mind over matter.” The stark newness they brought to the realm of philosophy, and perhaps the reason they are separated from the Platonists by the prefix “neo” was their investigation of the beginning of the universe [2]. Hypatia taught these ideas with more scientific background than the former Neoplatonists [4].
Many students traveled from around the world to learn from her and some wrote correspondences from afar [6]. One of her most famous pupils was Synesius of Cyrene, who went on to become the Bishop of Ptolemy. His letters provide greater insight into the life Hypatia lived. He gives her credit for creating two important astronomical devices, the astrolabe and the planesphere. He also refers to her creation of instruments to distill water, measure the level of water, and determine specific gravity of certain liquids [9]. Synesius shows great reverence for Hypatia in these letters, addressing her as “the Philosopher” [10]. Hypatia was indeed well respected in a male-dominated field [1].
Her works are not completely known, but there is evidence that she wrote commentaries and treatises on several important mathematical works [4]. Though many were lost, some remain and it is through these editions that ancient Greek mathematical thought was preserved and more easily studied [1]. The first of these is her commentary on Ptolemy’s Book III of the Almagest, written with her father Theon. In it’s introduction reads “the edition having been prepared by the philosopher, my daughter Hypatia” [1]. The Almagest was the top resource for astronomical study in the world from its writing until 16th century Copernicus. Another important commentary which she used to teach her students was that on Diophantus’ Arithmetica. Below, [10] explores the general structure of a problem found in Arithmetica in Book I:6.
(a) General problem
Example: To divide a given number into two numbers such that a given fraction of the first number exceeds a given fraction of the second number by a given number.
(b) Necessary condition
Example: It is necessary, that the later given number be smaller than the number obtained when the greater given fraction is taken of the first given number.
(c) Specific values for the given number(s)
Example: Let it be required to divide 100 into two numbers such that the difference of one fourth of the first number and one sixth of the second number be 20 units.
(d) Detailed solution in paragraph form followed by careful check that the found quantities indeed satisfy the given conditions
Example: Let one sixth of the second number be 1 arithmos, then this number will be 6 arithmoi. From then on, one forth of the first number will be 1 7 Not all problems have necessary condition. Some necessary conditions prevent having negative solutions since Diophantus avoided these as being absurd numbers, such as the condition in I:6; others are genuinely necessary for the proper solution of the equation, such as the condition in V:11. 4 arithmos plus 20 units; so the second number will be 4 arithmoi plus 80 units. Moreover, we want the numbers added together to produce 100 units. Now, these two numbers added together form 10 arithmoi plus 80 units, which equal to 100 units. Subtract like from like: it remains that 10 arithmoi are equal to 20 units, and the arithmos becomes 2 units. Returning to our conditions, it was proposed that one sixth of the second number be 1 arithmos, which is 2 units, then the second number will be 12 units. On the other hand, since one fourth of the first number is 1 arithmos plus 20 units, which is equal to 22 units, then the first number will be 88 units. From then on, it is established that one fourth of the first number exceeds one sixth of the second number by 20 units and that the sum of the required numbers is the given number.
This work on algebraic methods veers from the typical focus of Greek mathematics on topics relating to geometry and number theory. For this reason, it demonstrates Hypatia’s versatility and rank in the field [9]. Her comments likely brought this material to a level where more students could grasp it.
Apollonius also studied in Alexandria and developed the important work, Conics, which Hypatia also wrote and taught about. This was again heavy material that she not only was able to understand but well enough to teach it. The concepts presented in this work influenced Ptolemy’s study of planetary orbits as well as the development of analytical geometry by Descartes and Fermat [5]. It introduced the ideas of hyperbolas, parabolas, and ellipses [6]. The potential of the effect Hypatia’s comments and teachings on this work had in future mathematics is astounding. Hypatia also may have written on Archimedes’ Measurement of the Circle. Arabic manuscripts found in the 20th century have the careful explanation that matches the style of Hypatia’s teachings [3]. Proposition 3 in Measurement of the Circle that gives a bound for pi, the ratio of the circumference to the diameter of a circle.
It could be argued that mathematical achievements are just as important as their accessibility to others. Hypatia took these works and probably many more important mathematical ideas and presented them in ways that were more easy to digest. She created activities and scenarios that demonstrated the material [6]. It is difficult to know for certain the impact she had on future mathematical achievements. However, one can speculate that if it were not for her efforts the timeline of mathematical history probably would have taken a large hit. For one example, imagine if Diophantus’ ideas had not been so effectively translated and studied leading up to the time of Fermat. Would he have made the same strides that he did in number theory? How would it have affected the topic of calculus? She is linked directly by [6] to the later work of Descartes, Newton, and Leibniz. Anyone who has studied the connections and building nature of mathematical history can understand the amazing role of this woman despite any solid proof. One could even argue that every mathematician who read one of her editions of mathematical works was impacted in some way by her orate skill.
Unfortunately, the to the religious climate of Alexandria during Hypatia’s life was not ideal. There was a divide between the Christians, led by Bishop Cyril, and the Neoplatonists, led by the prefect Orestes. Cyril was conservative, dogmatic, and power-hungry. He organized a political reprisal against Orestes and created a riot. Hypatia, an easy target due to her fame, outspokenness, and ties to Orestes, was accused of being a heretic [3]. She was taken from her chariot, brutally tortured, and burned. Her story becomes one of a scientific martyr. At a time when most women were occupied with domestic duties, Hypatia is remembered as one of the last great thinkers and teachers of ancient Alexandria [1].
References
[1] B. Bibel, “Meet Hypatia, the ancient mathematician who helped preserve seminal texts,” July 2018, https://massivesci.com/articles/hypatia-math-science-heroes/.
[2] C. Brown, Christianity & western thought, Apollos, 1 (1990) 34.
[3] J. Lienhard, Hypatia’s mathematics, The Engines of Our Ingenuity, 215 (1997), https://uh.edu/engines/epi215.htm.
[4] J. O’Connor & E. Robertson, “Hypatia of Alexandria,” April 1999, https://mathshistory.st-andrews.ac.uk/Biographies/Hypatia/.
[5] L. Osen, Women in Mathematics, Cambridge, MA, MIT Press (1974) 21—32.
[6] L. Riddle, “Hypatia,” January 2017, https://www.agnesscott.edu/lriddle/women/hypatia.htm.
[7] M. Betrand, “Archimedes and Pi,” May 2014, http://nonagon.org/ExLibris/archimedes-pi.
[8] M. Mercer, “Hypatia,” http://www.math.wichita.edu/history/Women/hypatia.html.
[9] S. Greenwald & E. Mendez, “Hypatia, the First Known Woman Mathematician,” https://mathsci2.appstate.edu/~sjg/ncctm/activities/hypatia/hypatia.htm.
[10] T. Davis, “Forty Two Problems of First Degree from Diophantus’ Arithmetica,” December 2010, http://www.math.wichita.edu/~pparker/classes/Davis_Tinka_FL2010.pdf.
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