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First Woman to Make a Substantial Contribution to the Development of Mathematics
350 – 415 A.D.
Hypatia stands as one of the most remarkable women of antiquity, and she was famous in an unusual line, that of philosophy. Her father, Theon, was at the head of the Platonic school at Alexandria and was noted for his philosophic attainments, but his fame and name are preserved to us more on his daughter’s account than on his own. She was his devoted pupil and his very life passed over into hers. She made astonishing progress in all branches of learning and soon surpassed her father and all other philosophers in their special pursuits.
She succeeded her father as head of the Alexandrine school. Pupils came from all parts of the Roman empire and eagerly listened to the beautiful and learned woman.
She was considered an oracle of wisdom, and magistrates consulted her on many important cases. Men gathered about her in great numbers. Probably no woman was ever more praised and petted than she. In the midst of it all she maintained a modest reserve. Her mind was too thoroughly trained to lose its perfect poise through vanity.
Orestes was governor of Alexandria and Cyril was bishop. Orestes frequently consulted Hypatia as did other leading men and naturally admired her. The bishop disliked Orestes and was bitterly intolerant of Hypatia’s philosophy. He is credited with having incited the mob to an attack on the governor. Feeling became so intense on the part of the bishop’s followers when it was rumored that Hypatia had prevented a reconciliation between the two men, that some conspirators, headed by one Peter, waylaid the noble woman, dragged her from her carriage into a church, stripped her naked, killed her with broken tiles, tore her body in pieces and then burned the remains to ashes. Thus was Hypatia a martyr to philosophy, suffering at the hands of a mob, martyred in 415 A.D. Cyril was the intolerant and bigoted instigator. When he became bishop, one of his first acts was to lead a mob and drive out the Jews from Alexandria, though for centuries they had enjoyed many privileges.
Reference: Woman: Her Position, Influence and Achievement Throughout the Civilized World. Designed and Arranged by William C. King. Published in 1900 by The King-Richardson Co. Copyright 1903 The King-Richardson Co.
Mathematician and Philosopher
370 – 416 B.C.
Hypatia of Alexandria, a mathematician and philosopher, one of the most eminent women teachers of antiquity, and one of the ablest of the later Greeks who preached the pagan philosophy. Her father, Theon, was a distinguished scholar of the tie, but the daughter, Suidas says, surpassed her father in ability. In spite of her remarkable position, few passages relating to her survive, but they uniformly ascribe her an exceptional distinction for culture and influence no less than beauty and virtue. Her great eloquence, combined with her intellectual gifts, attracted to her classroom a large number of pupils. Among these was Synesius, afterwards bishop of Ptolemais, several of whose letters to her, full of chivalrous admiration and reverence, are still extant.
She wrote a number of important books, but they have not survived.
The story of this eminent woman forms the basis of the well-known historical romance Hypatia, by Charles Kingsley, in which the author thus describes her person and her philosophy: “In the light arm-chair, reading a manuscript which lay on the table, sat a woman, of some five and twenty years, dressed in a simple snow-white Ionic robe, entirely without ornament, except the two narrow purple stripes down the front, which marked her rank as a Roman citizen, the gold-embroidered shoes upon her feet, the gold-embroidered shoes upon her feet, and the gold net, which looped back, from her forehead to her neck, hair the color and gloss of which were hardly distinguishable from that of the metal itself, such as Athene herself might have envied for tint, and mass, and ripple, while her features, arms and hands were of the severest and grandest type of old Greek beauty.
“She has lifted her eyes off her manuscript, and is talking to herself. Listen!
“‘Yes. The statues are broken. The libraries are plundered. The alcoves are silent. The oracles are dumb. And yet – who says a that the old faith of heroes and sages is dead? The beautiful can never die. If the gods have deserted their oracles, they have not deserted the souls who aspire to them. If they have ceased to guide nations, they have not ceased to speak to their own elect. If they have not ceased to speak to their own elect. If they have cast off the vulgar, they have not cast off Hypatia.
“Ay. To believe in the old creeds, while everyone else is dropping away from them. To believe in spite of disappointments. To hope against hope. To show oneself superior by the herd, by seeing boundless depths of living glory in myths which have become dark and dead to them. To struggle to the last against the new superstitions of a rotting age, for the faith of my forefathers, for the old gods, the old heroes, the old sages who gauged the mysteries of heaven and earth – and perhaps to conquer – at least to have my reward! To be welcomed into the celestial ranks of the heroic – to rise to immortal gods, to the ineffable powers, onward, upward ever, through ages and through eternities, till I find my home at last, and vanish in the glories of the Nameless and the Absolute One!'”
Reference: Famous Women An Outline of Feminine Achievement Through the Ages With Life Stories of Five Hundred Noted Women By Joseph Adelman. Copyright, 1926 by Ellis M. Lonow Company.
Hypatia: The Life and Legend of an Ancient Philosopher. Women in antiquity
This monograph, dedicated to reconstructing the life and career of the Alexandrian mathematician and philosopher Hypatia, is part of the Women in Antiquity series. The study has a strong historical focus, so that little is said about Hypatia’s philosophical views, apart from identifying Hypatia as a Plotinian Platonist, that is, one who did not engage in theurgical practices popular among contemporary Platonists. The choice of a historical focus might seem surprising as the evidence for her life is very sparse, but Watts presents a detailed picture of Hypatia’s career by means of innovative use of a large variety of texts. The book is comprised of introduction, ten chapters and concluding remarks.
The introductory chapter, “A Lenten Murder”, starts with the most notorious event in Hypatia’s life, her murder at the hands of the Alexandrian mob. The murder made Hypatia a symbol, but what her life symbolised, as Watts shows, has varied from century to century (p. 4). The starting point for a historically accurate picture, therefore, must be the study of Hypatia’s immediate surroundings. By discussing the historical, social and political circumstances of the city of Alexandria, Watts introduces one of the main themes of the book, that Hypatia’s life and career cannot be separated from the context in which she lived and worked.
A discussion of Alexandria’s history in late antiquity can be found in the first chapter, entitled “Alexandria”. The topics most pertinent to Hypatia’s life are the Serapeum library (which was central to cultural and philosophical life in Alexandria), the Alexandrian population (its labour market, density and class divisions) and the religious dynamics of fourth-century Alexandria. In regard to the last, Watts argues that religious divisions were much less prominent than often portrayed in the scholarship (pp. 17- 19). The societal structure and divisions imposed by collegia (associations of individuals working on the same crafts) and by class were significantly more palpable to an Alexandrian.
The topic of the second chapter is Hypatia’s childhood and education, although the discussion involves a much wider array of subjects. While Hypatia’s father, Theon, was a noted mathematician, Watts argues that her mother probably came from an intellectual family. The argument is based on quite an extensive discussion of the education of women in Alexandria and the Roman Empire. Watts’ study of the education of elites more generally reveals a significant point about women and philosophy in late antiquity. The education of the male youth was geared towards gaining prestigious posts in imperial administration (p. 24). As these posts were not open to women, women from wealthy families were able to study whatever they wanted (sometimes to a very high level, see p. 25), including less prestigious subjects such as philosophy.
Hypatia’s own education was fairly standard for a philosopher. She studied with her father Theon, and her studies culminated in a large work, comparable to a doctoral dissertation (p. 29), which was written in collaboration with Theon. This early work might have been an edition of Ptolemy’s Handy Tables or, more likely, an edition of Ptolemy’s Almagest books 3-13 (pp. 30-1). This chapter also contains a discussion of the relationship between mathematics and philosophy in Alexandria. Watts argues that, as far as the contemporaries of Hypatia were concerned, the issue at stake was not whether one ought to choose mathematics over philosophy, but which subject ought to be privileged. In Alexandria, where mathematicians held the stage for a long time, Hypatia’s choice to privilege philosophy represented a new intellectual trend.
The following chapter, “The School of Hypatia”, discusses the curriculum that Hypatia introduced in the school she inherited from her father. The curriculum that Proclus followed in Alexandria about a decade after Hypatia’s death serves as a basis for Watts’ suggestion that the studies most likely started with mathematics and ascended to philosophy. This chapter also contains the most explicit discussion of Hypatia’s philosophical views. Hypatia was a Plotinian Platonist, which is best understood when contrasted with another branch of Platonism that emerged during Hypatia’s lifetime, namely Iamblichean Platonism, notable for the practice of theurgy (pp. 43-5). Theurgy was much more cutting-edge philosophy than Hypatia’s Plotinian Platonism, but Hypatia’ teachings suited the needs of her native city. This is partly due to the fact that most of Hypatia’s students were Christians, including a Libyan aristocrat Synesius, who is an important source for this book. While theurgy would have been problematic for Christian students, Hypatia’s approach was not only non-problematic but also not inconsistent with their theology, as the analysis of Synesius’ hymns shows (pp. 47-9).
Most of the chapter “Middle Age” is dedicated to locating Hypatia’s school within broader philosophical trends in approximately the 390s. The most significant development in philosophy during this period was the emergence of theurgy, which revitalized the Athenian philosophical scene. Synesius’ letter shows that the Alexandrian school, led by Hypatia, and the Athenian school, led by Plutarch, were thought of as rivals. Iamblichean Platonism showed up in Alexandria too, although it was not long-lived since the head of this school, Olympus, and his followers were involved in the infamous standoff in the Serapeum. The chapter provides a detailed explanation of how Theophilus’ anti-pagan agenda, which involved not only restrictions in religious practices but also public mockery, led to the riot and the standoff. The conflict ended with the emperor Theodosius’ granting amnesty to the Serapeum fighters, but they subsequently fled Alexandria, thus ending the theurgist school in Alexandria. Watts argues that Hypatia’s teaching, meanwhile, proved to be especially fitting for her time and place, serving as a guide for her students in how to organise their lives and the city better (p. 62).
The fifth chapter, “A Philosophical Mother and her Children”, starts with the question of whether there is enough evidence to suggests that Hypatia drew a public salary, i.e. was a public teacher. Watts argues that it is unlikely and that Hypatia probably simply made herself more accessible than most other teachers of her time. Apart from publicly accessible lectures, Hypatia, like most philosophers, had a close-knit inner circle to which access was restricted. The relationship between Hypatia and her close disciples is the main topic of the chapter. Synesius’ letters serve as the main evidence here as well (pp. 67-72). His letters also discuss the notion of philosophical love, which might have been the way in which the members of the inner circle conceptualised their relationships. The chapter also includes the discussion of the anecdotes about Hypatia’s celibacy.
Following the discussion of Hypatia’s school, the chapter The Public Intellectual deals with the social status of Hypatia and the role she had in Alexandrian society. As in the previous chapters, the topic is extensively contextualised by discussing not only the role of philosophers as state advisors but also evidence of ‘false’ philosophers, that is, people who used their advisory role for personal gain in late antiquity. Synesius’ letters again prove to be an invaluable source for Hypatia’s role in the public sphere. The letters in which Synesius asks for favours on behalf of his acquaintances show that she had some significant influence in Alexandrian social circles and that the members of her inner circle expected certain kinds of assistance from their teacher in the public sphere. The letters also provide some insight into how the members of Hypatia’s inner circle approached the seeming conflict between their philosophical pursuits and social duties.
“Hypatia’s Sisters” is dedicated to the discussion of four other contemporary women whose careers are comparable to Hypatia’s, namely Pandrosion (a mathematician), Sosipatra (a philosopher), Asclepigenia (a philosopher) and the wife of Maximus of Ephesus (a philosopher). This is an especially interesting chapter because these women’s careers shed some light on the professional life of Hypatia, despite the fact that, overall, less is known of these philosophers. The comparisons with this evidence help to clarify the challenges that Hypatia faced as a woman philosopher and show to what extent Hypatia’s circumstances were unusual (pp. 103-6).
Chapter eight, “Murder in the Street”, contains a historical exposition of the early career of Cyril, the bishop who succeeded Theophilus. Cyril’s rise to power and his subsequent conflict with Orestes, the prefect, formed the background for the murder of Hypatia. The chapter also includes the discussion of the murder, its aftermath and the effect it had on Alexandria (pp. 117-20). Watts notably argues that it is unlikely that the mob set out to kill Hypatia, and that the murder was a circumstantial, rather than premeditated, event. The murder of Hypatia had an impact reaching far beyond Alexandria. The following chapter, “The Memory of Hypatia”, discusses the reception of Hypatia in ecclesiastical histories and chronicles, Damascius’ Life of Isidore, Suda and the Egyptian historical tradition. Watts discusses why, in antiquity, Hypatia’s life was often seen as a turning point in history and shows how various authors’ commitments determined whether the turn was seen as a positive or a negative change.
The final chapter, “A Modern Symbol”, discusses the reception of Hypatia in the modern period. The texts include John Toland’s essay dedicated to Hypatia in Tetradymus, a response to Toland by Thomas Lewis with his The History of Hypatia, a Most Impudent School-Mistress of Alexandria, Voltaire’s Dictionnaire Philosophique, Diodata Saluzzo Roero’s poem Ipazia Ovvero Delle Filosofie, and others. There is also discussion of the reception of Hypatia in late-twentieth-works, from Ursule Molinaro’s (1989) composition in poetic prose to Alejandro Amenábar’s film Agora. Although the murder of Hypatia is often treated as signifying the end of an enlightened period, Watts argues that such treatments fail to take into account the history of the centuries following Hypatia’s life and thus mistreat Hypatia’s legacy.
The main arguments of the book are summarised in the epilogue, “Reconsidering A Legend”. The first half of this section recaps the historical circumstances of Alexandria during Hypatia’s lifetime, and it shows how substantially the life of the city shaped the life of the philosopher. The rest of the chapter addresses the extent to which Hypatia’s death marked a cultural turning point.
Arguably, the central claim of this book is that Hypatia was a product of late-antique Alexandria and that the study of her life cannot be separated from the study of the social, political and, to a certain extent, religious circumstances of Alexandria. The monograph is a study not only of Hypatia but also of her world. It is, therefore, somewhat surprising that Watts emphasises Hypatia’s personal qualities as the reasons for her achievements (pp. 104, 155). One might argue that this detailed and contextualised study shows that, apart from being talented, Hypatia was extremely fortunate in her circumstances. No matter how talented women philosophers were, the cultural and economic circumstances were working against them, and thus one cannot underestimate the advantage that Hypatia had in being Theon’s daughter and inheriting her own school.
This monograph is undoubtedly an important addition to the scholarship on Hypatia, not least because there has been no monograph-length study for more than two decades. Arguably one of the most significant contributions of this study is the argument that the murder of Hypatia was not, as is often argued in the existing scholarship, a pre-meditated attack but rather a circumstantial event. Apart from Hypatia’s life, this book is also notable for painting a detailed picture of late-antique Alexandria and for showing how much information about an ancient figure can be teased out of indirect evidence about historical circumstances, parallel cases and similar. The writing style is extremely accessible. Quite a few comparisons to modern history and popular culture—although almost exclusively American—make the book very approachable to a wide range of audiences, not only the specialists.
Hypatia – Teacher & Pioneer
For every era of history, be it within mathematics or other domains, there are catalyzing characters who push their domains to new heights. At times, this takes place through discoveries made in some field by the individual but, in some cases, this comes through their ability to share their love and knowledge of the field with others.
Hypatia of Alexandria was no ordinary individual. She was raised in a time when it was virtually unheard of to have a woman be a teacher, let alone a student, of mathematics, science, astronomy, or philosophy. She stood above all, however, pioneering these subjects for future generations of women and girls to come. Hypatia was, and still remains, one of the most renowned female mathematicians of all time. However, religious fanaticism and anti-intellectualism would lead to an angry mob attacking her and ending her life in an incredibly brutal way. Her death is often cited as a marker for the end of the classical world.Hypatia of Alexandria. Public domain. Fresco from Pompeii believed to be a rendering of Hypatia.
Hypatia was born the daughter of Theon of Alexandria, who was both a well-regarded mathematician and accomplished astronomer in his time. The exact date of her birth remains a mystery with conflicting dates ranging from 350-370 AD. There is a reasonable estimate that it was in 355 AD, as one of her most renowned students, Synesius of Cyrene, is believed to have been born in 370 AD and would have almost surely been substantially younger than his teacher, Hypatia.
Theon of Alexandria was also known to have been the last professor at the University of Alexandria and wrote commentaries on some of the greatest scientific works of Ancient Greece, including Euclid’s Elements and Ptolemy’s Almagest. Hypatia is widely believed to have contributed to these commentaries, however, and her own reputation as a mathematician would far surpass her father’s.
Theon of Alexandria’s, or possibly Hypatia’s, edits on Euclid’s Elements shown in an ancient manuscript. Public Domain.
Having worked under her father’s guidance, Hypatia remarkably became the head of the Platonist School at Alexandria around 400 AD where she would lecture on mathematics and philosophy. She was particularly well versed in Neoplatonism, which was a strand of Platonic philosophy that sought to bridge the gaps between philosophy, religion, and even literature. This played an important role in Hypatia’s mathematical development. The Neoplatonic drive to find connections between the “Ideal” (thought) and the “Form” (reality) would in many ways resemble the chasm between the infinite and the finite concepts that Hypatia would grapple with throughout her studies and lecturing. She was renowned for being a particularly skilled speaker, and people would come from all over to hear her lectures.
Within mathematics, Hypatia is best known for her work on Apollonius’s treatise of conic sections, as well as her commentaries on Diophantus’s Arithmetica, and Ptolemy’s astronomical works. Unfortunately, all of her work is lost except for its titles and references to it after the mass exodus of the scholars of the city which marked the decline of Alexandria as the major hub for learning and intellectualism in the world.
- Cover of the Greek 1621 edition translated by Claude Gaspard Bachet de Méziriac. Public Domain.
- 9th century Arabic translation of Apollonius’s “Conics” © Bodleian Libraries/Wikimedia Commons CC BY 4.0
- Ptolemaic system of planetary paths. Public Domain.
Hypatia paid a heavy price for her pursuit of knowledge, as it is believed that the Bishop of Alexandria spread awful rumours about her, which led to her being attacked by a Christian mob, stripped, stabbed to death with broken pottery, and dragged through the streets. The manner of her death and its symbolism as the end of an age of intellectual enlightenment in lieu of religious extremism continues to permeate historical anecdotes. Hypatia was a martyr for knowledge at her time, but her story has empowered many women, and men, to continue to pursue mathematics, philosophy, and science.
Hypatia - History
Women and Minorities in Mathematics
Incorporating Their Mathematical Achievements Into School Classrooms
Hypatia, the First Known Woman Mathematician
Appalachian State University, Boone, North Carolina
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 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 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.
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.
Deakin, M. (1996). Mathematician and martyr: A biography of Hypatia of Alexandria. Victoria, Australia: Department of Mathematics, Monash
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/
Greenwald, S. (2001b). Classroom Worksheet on Hypatia’s Possible Work on Archimedes Dimension of the Circle. [On-line]. Available: http://www.mathsci.appstate.edu/
Heath, T. (1921). A history of Greek mathematics. Oxford: Clarendon.
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Knorr, W. (1989). Textual Studies in Ancient and Medieval Geometry . Boston, MA: Birkhauser.
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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.
The Ancient History of Sexism Begins with Hypatia’s Murder
Hypatia was born in Alexandria, Egypt, in 355. She was murdered there by a Christian mob. Her story is eloquently told in the 2009 film, Agora.
Hypatia invented the plane astrolabe, the graduated brass hydrometer, and the hydroscope. Alejandro Almenábar captures her story and her time in Agora.
It was not unusual then for women to teach science, mathematics, astronomy, and philosophy. Hypatia sought to revive the glory of Greece, when Socrates and Plato were followed by the greatest group of intellectuals the world has ever seen. She was the daughter of Theon, who taught mathematics at the Museum of Alexandria, the center of Greek intellectual and cultural life and home to the great library of Alexandria. Theon, a dogmatic liberal, set out to make his daughter the perfect human being.
Only 100 years before Hypatia’s birth, the ruler of the Roman Empire, Constantine, embraced Christianity and from that moment everyone in the empire became a Christian by his edict. But they remained Pagans by character, despite his order that made every Pagan temple a Christian church and every Pagan priest a Christian preacher.
Hypatia, a Pagan, was First to Realize Planets Orbit the Sun in Ellipses, Not Circles
She was 5’9” tall and weighed 135 pounds when she was 20 and easily walked 10 miles without fatigue, rowed, drove her own chariot, rode bareback, and climbed mountains. She was said to have had “a body of rarest grace.” Rachel Weisz, who plays her in the film, apparently bears a close resemblance.
As director of the Library’s Neo-Platonist school of philosophy, Hypatia studied conic sections to understand the motions of the planets. She appears to have been the first to realize, long before Kepler, that the sun is the focus, not the center, of the universe, and that planets therefore orbit the sun in ellipses, not circles.
Simple Faith-Based Acceptance vs. Scientific Investigation
When she was just a girl, Theon taught Hypathia that to know but one religion is to know just one superstition whereas to know one philosophy is to know no absolute truth. Religions are accepted passively in faith, but science demands constant doubt to motivate the investigation necessary to discover new knowledge.
Hypatia wrote, “Neo-Platonism is a progressive philosophy, and does not expect to state final conditions to men whose minds are finite. Life is an enfoldment, and the further we travel the more truth we can comprehend. To understand the things that are at our door is the best preparation for understanding those that lie beyond.”
As Elbert Hubbart wrote about Hypatia in his 1908 book, Great Teachers: “A man living in a certain environment, with a certain outlook, describes the things he sees and out of these, plus what he imagines, is shaped his philosophy of life. If he is repressed, suppressed, frightened, he will not see very much, and what he does see will be out of focus.”
Organized Religion’s Suppression of Women Begins with the Murder of Hypatia
Alexandria was ruled by a Roman Prefect, or Governor, named Orestes, a Pagan like Hypatia. Rome exercised great religious tolerance. As a Pagan, Orestes was an adversary of the new Christian bishop, Cyril, and he vigorously objected to Cyril’s expulsion of the Jews from the city. For this opposition, he was murdered by Christian monks.
Cyril next began to plot against his other major Pagan opponent in Alexandria, Hypatia. As a woman who represented heretical teachings, including experimental science and pagan religion, she made an easy target.
He preached that Christ had no female apostles, or teachers. Therefore, female teachers had no place in Christianity. This sermon incited a mob led by fanatical Christian monks to attack Hypatia as she drove her chariot through Alexandria. They dragged her from her chariot and, according to accounts from that time, stripped her, killed her, chopped and cut away the flesh from her bones, scattered her body parts through the streets, and burned what was left in the library of Caesareum.
The Dark Ages Begin
Hypatia’s students fled to Athens. The Neo-Platonism school she headed continued in Alexandria until the Arabs invaded in 642. When they burned the library of Alexandria, using it as fuel for their baths, the works of Hypatia were destroyed. Her writings are only known today through the works of others who quoted her along with a few letters written to her by contemporaries.
Cyril, the fanatic Christian who incited her destruction, was made a saint. With the supreme reign of Christian and then Muslim religious dogmatism came the suppression of women.
Adler, A. ed. Lexicographi Graeci. Vol. I. Suidae Lexicon A-W.Index. Pars 4: P-Y. Leipzig: B.G. Teubner, 1935, 644–646. The entry in Suidae Lexicon, vol. IV: 644–646 is the main source on Hypatia. Adler draws in part from Damascius (hence a portion of the entry is edited as fragments 102–105 of Damascius, Vita Isidori (see following citation).
Damascius. Vita Isidori. Edidit Adnotationibusque Instruxit C. Zintzen. Hildesheim: Georg Olms Verlag, 1967, fragments. 102–105. A part of the S uidas entry on Hypatia comes from this work and is here re-edited in the form of fragments.
Cameron, Alan. “Isidore of Miletus and Hypatia: On the Editing of Mathematical Texts.” Greek, Roman and Byzantine Studies 31 (1990): 103–127. This article contains the analysis of the inscription to Theon’s Commentary, Book III and the proposal that Hypatia edited the Almagest.
Cameron, Alan, and Jacqueline Long. Barbarians and Politics at the Court of Arcadius. Berkeley: University of California Press, 1993, 39–62. The authors present a detailed discussion of the extant evidence, still over-optimistic in its assessment of the sources.
Charles, R.H. (trans.), The Chronicle of John, Bishop of Nikiu. Oxford: Oxford University Press, 1916, 100–102. A further source on Hypatia. The text is currently available only as an English translation of an Ethiopian version of a (lost) Arabic translation.
Dzielska, Maria. Hypatia of Alexandria. Cambridge, MA: Harvard University Press, 1996. Presents an extensive discussion of the extant evidence and contains a very rich bibliography.
Jones, Alexander. “Uses and Users of Astronomical Commentaries in Antiquity.” In Commentaries – Kommentare, edited by Glenn W. Most. Aporemata: Kritische Studien zur Philologiegeschichte, Band 4.Göttingen: Vandenhoeck und Ruprecht, 1999, 149–172. Alan Cameron’s conclusions are criticized in this article.
Knorr, Wilbur R. Textual Studies in Ancient and Medieval Geometry. Boston: Birkhäuser, 1989, 753–84. Hypatia’s contributions to several extant commentaries and edited works are here singled out.
Rome, Adolphe. Commentaires de Pappus et de Théon d’Alexandrie sur l’Almageste. Tome III: “Théon d’Alexandrie,” Commentaire sur les livres 3 et 4 de l’Almageste. Texte établi et annoté par A. Rome. Città del Vaticano: Biblioteca Apostolica Vaticana, 1943: cxvi-cxxi. The introduction contains the first stylistic analysis of Theon’s Commentary.
Saffrey, Henri D. “Hypatie d’Alexandrie.” In Dictionnaire des Philosophes Antiques, edited by Richard Goulet. Paris: CNRS Editions, 2000. A first orientation, with bibliography, on Hypatia’s possible philosophical affiliations.
Sesiano, Jacques, trans. Books IV to VII of Diophantus’ Arithmetica. Attributed to Qustâ ibn Lûqâ. New York: Springer-Verlag, 1982, 68–75. Hypatia’s role in the tradition of Diophantus’ Arithmetica is discussed here.
Socrates. Historia Ecclesiastica. In Patrologiae Cursus Completus, Accurante J.-P. Migne. Patrologiae Graecae Tomus 67. Paris: 1864, VII.13–15. A well-balanced exposition of Hypatia’s life by a her contemporary.
Synésios de Cyrène, Tome II: Correspondance: Lettres I-LXIII Tome III: Correspondance: Lettres LXIV–CLVI. Texte établi par A. Garzya, traduit par D. Roques. Paris: Les Belles Lettres, 2000.
Tannery, Paul. “L’article de Suidas sur Hypatia.” Annales de la Faculté des Lettres de Bordeaux, 2 (1880): 197–201, reprinted in Mémoires Scientifiques, tome I (1912), no. 7: 74–79. This first critical discussion of the Suidas entry was valuable into the early 2000s.
Implications of Witch Hunts: Blaming Social Outcasts, the Poor, and Women
The parallels between victims of these trials indicate that witchcraft accusations disproportionally fixate on people viewed as socially disposable, such as women of color, poor women, single mothers, widows, older women, and others who provided no &ldquoproductive value&rdquo within society. Most of the time, it took only a single witness to successfully condemn a witch to death, revealing just how skewed the whole system was in favor of well-respected Christian men and their families.
Actually, the American colonies were the perfect breeding ground for witch hunts. Life in colonial times was quite difficult, and, just like they had during the Middle Ages in Europe, many settlers blamed Satan &ndash and, by proxy, anyone who deviated from Puritanical norms &ndash for their woes.
Of course, they didn&rsquot forget to blame American Indians. Some people believe that the colonists&rsquo heightened fear of Satanic forces was due to military insecurities. Afraid that they would not be able to defeat the native inhabitants, desperate to control the unknown, and oblivious to their own hypocrisy (as most of America still is today with regards to indigenous peoples&rsquo rights), the European invaders immediately began equating Native Americans with Satanic beliefs, seemingly oblivious to the fact that it was they who were creating the violence in the first place.
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