Hypatia of Alexandria

How to use this course

Hypatia is the first woman mathematician about whom we have reasonably detailed contemporary evidence. She is also among the most mythologized figures in the history of science. Novels, plays, films and popular histories have made her a martyr of reason, a victim of the burning of the Library of Alexandria, an inventor of instruments, and a young beauty. Very little of this survives contact with the sources.

This course therefore runs on two rules. First, each biographical claim is tied to the source it comes from, and the sources are ranked by date and reliability. Second, the mathematical lessons teach the texts she is documented or reported to have worked on: Ptolemy’s Almagest and Handy Tables, Diophantus’s Arithmetica, Apollonius’s Conics, and her father’s edition of Euclid. No original theorem of hers survives. What survives is the curriculum of an Alexandrian mathematician around 400, and that is what we study.

Timeline

All dates before 415 that concern Hypatia herself are approximate.

Theon’s daughter

Every ancient source agrees that Hypatia was the daughter of Theon of Alexandria. Theon was a mathematician and astronomer who observed two eclipses in Alexandria in 364 and wrote commentaries on Ptolemy and an edition of Euclid. The Suda calls him a member of the Museum, Alexandria’s ancient research institution, though whether the Museum still existed in his day is uncertain. Nothing is known of Hypatia’s mother. Theon addresses a commentary to an Epiphanius as his child, which may mean Hypatia had a brother, but the Greek word can also be used of a student.

When was she born?

No source gives her birth date. Two inferences compete.

• c. 370. Richard Hoche argued in the nineteenth century from Damascius’s description of her as beautiful and from the Suda’s statement that she flourished under the emperor Arcadius (395–408). This became the standard date, and the source of her image as a young woman.
• c. 350–355. John Malalas, writing in the sixth century, calls her an old woman at the time of her death. Her role as a recognised teacher by the 390s, when Synesius came to study with her, also fits an earlier date. Maria Dzielska favoured c. 355.

Robert Penella has argued that neither inference is strong and that the date should be left open. This course follows him: born between about 350 and 370. If the earlier date is right, she was about sixty when she was killed.

Education

No source describes her education in detail. That she learned mathematics from her father is a reasonable inference from their collaboration, not a documented fact. Philostorgius, writing a generation after her death, says she surpassed her father in mathematics. Hesychius calls her a distinguished astronomer.

Teacher and philosopher

By about 400 Hypatia was the best-known teacher of philosophy in Alexandria. Socrates Scholasticus says she succeeded to the school of Plato and Plotinus and drew students from far away. Damascius says she wore the philosopher’s cloak, the tribon, walked through the city, and expounded Plato and Aristotle to anyone who wished to hear. Socrates adds that her self-possession allowed her to appear before magistrates and in the company of men without loss of respect.

What kind of philosophy?

Late Platonism in Alexandria had two strands. One followed Iamblichus and joined philosophy to pagan ritual; it was associated with the Serapeum, destroyed in 391. Hypatia and Theon belonged to the other, which followed Plotinus and emphasised mathematics and contemplation over ritual. Edward Watts and others have argued that this is why Christian bishops, including Theophilus, could tolerate and even value her circle.

Synesius and the students

Our only first-hand witness is Synesius of Cyrene, a landowner from the Libyan Pentapolis who came to Alexandria in 393 with his brother Euoptius. He later became bishop of Ptolemais in 410 and died about 413. Seven of his letters are addressed to Hypatia; her replies are lost. He calls her his teacher, mother, sister and benefactor. The letters show a group of students bound by friendship and by a shared sense of philosophy as a private discipline for the few. Synesius’s letters name other members of the circle, among them Herculianus and Olympius. Alan Cameron and others have noted that all her known students were Christians or became Christians.

Synesius asked her to review his books before publication, to use her influence on behalf of friends in Alexandria, and to build him a hydroscope (Lesson 10). His last letter, written from his deathbed, says he is ill in body and in spirit after the death of his children.

What she wrote

The Suda, drawing on Hesychius, lists three works and says that all are lost:

• a commentary on Diophantus’s Arithmetica;
• an Astronomical Canon;
• a commentary on Apollonius’s Conics.

None survives under her name. The Astronomical Canon was probably an edition of, or commentary on, Ptolemy’s Handy Tables, the tables an astronomer used for daily computation.

The one direct attestation

The only contemporary evidence of her mathematical work is a heading in her father’s commentary on Ptolemy’s Almagest. The commentary on Book III is introduced as Theon’s commentary on an edition revised by the philosopher, his daughter Hypatia. For a long time this was read as Hypatia revising her father’s commentary. Alan Cameron and Alexander Jones read it differently: the text Theon was commenting on was Hypatia’s revised edition of Ptolemy’s Book III itself. On this reading she prepared a working text of the Almagest, and Cameron suggested she may have done so for more books than one. Some historians connect her with an improved tabular method for sexagesimal division in that book.

Possible survivals

Paul Tannery suggested in the nineteenth century that the six Greek books of the Arithmetica survive because they were the ones Hypatia commented on, and that some interpolated material in them may be hers. After the discovery of four further books in Arabic in 1968, Cameron argued that their worked verifications and added steps fit her commentary. Wilbur Knorr disagreed, judging the additions too elementary for her. The question is open.

Hypatia is also often credited with a hand in Theon’s edition of Euclid’s Elements. There is no ancient evidence for this, and it should be treated as a modern guess.

Computing in base sixty

For the general reader. Greek astronomers wrote whole numbers in Greek letters but fractions in base sixty, inherited from Babylon. We still use the system for hours, minutes and seconds. Theon’s Almagest commentary is our best ancient guide to how these computations were done by hand, and Hypatia’s revision of Book III was a working text for exactly such calculations.

Modern notation writes \(67;4,55\) for \(67+\frac{4}{60}+\frac{55}{60^2}\). The semicolon separates the integer part from the fraction.

Theon’s square root of 4500

Ptolemy gives \(\sqrt{4500}\approx67;4,55\) in Book I of the Almagest without showing the work. Theon explains it geometrically: take a square of area 4500, cut off the largest square of whole side, then remove two strips and a small square at each step, as in the expansion \((a+b)^2=a^2+2ab+b^2\).

For \(N=4500\): \(a=67\), remainder \(4500-4489=11\). Next, \(2\cdot67\cdot\frac{b_1}{60}\approx11\) gives \(b_1\approx4.93\), so \(b_1=4\): the gnomon is \(2\cdot67\cdot\frac{4}{60}+\frac{16}{3600}=8;56,16\), leaving \(2;3,44\). Then \(2\cdot67;4\cdot\frac{b_2}{3600}\le2;3,44\) gives \(b_2=55\). Hence \(\sqrt{4500}\approx67;4,55\), too small by less than \(\frac1{3600}\).

Almagest III: the motion of the sun

For the general reader. The seasons are unequal: in the northern hemisphere spring and summer together last about a week longer than autumn and winter. Ptolemy explained this in Book III of the Almagest by putting the sun on a circle whose centre is slightly off the earth. This is the book whose text Hypatia revised.

The year

Ptolemy adopts Hipparchus’s tropical year of \(365\tfrac14-\tfrac1{300}\) days, which is \(365;14,48\) days. From it he derives the sun’s mean daily motion, \(0;59,8,17,13,12,31^\circ\) per day.

Eccentric and epicycle

Proof. In complex notation with the observer at 0, the eccentric model gives \(z(t)=e+Re^{i\omega t}\), with \(e\) a fixed vector toward the apogee. The epicycle model gives \(z(t)=Re^{i\omega t}+e\), the epicycle radius fixed in direction. The two expressions are the same sum taken in the other order.

The equation of centre

Let the sun move uniformly on the eccentric, at angle \(\alpha\) from the apogee as seen from the centre. Seen from the earth, its angle from the apogee is \(\alpha-q\), where

Ptolemy, following Hipparchus, takes \(e/R=2;30/60=\frac1{24}\), derived from the observed lengths of spring (94½ days) and summer (92½ days). He places the apogee at Gemini 5;30°.

Chord tables and interpolation

For the general reader. Ancient astronomers did not have sine tables. They used a table of chords: for each angle, the length of the straight line it cuts off in a circle. An astronomical canon like the one the Suda attributes to Hypatia was largely tables, and using tables meant interpolating between their entries.

Ptolemy takes a circle of radius 60. The chord subtending an angle \(\theta\) at the centre is

His table runs from \(\tfrac12^\circ\) to \(180^\circ\) in steps of \(\tfrac12^\circ\), to two sexagesimal places, with a column of thirtieths of the differences for interpolation.

Proof sketch. Choose \(E\) on \(AC\) with \(\angle ABE=\angle DBC\). Then triangles \(ABE\) and \(DBC\) are similar (equal inscribed angles at \(A\) and \(D\)), giving \(AE\cdot BD=AB\cdot CD\); and triangles \(EBC\), \(ABD\) are similar, giving \(EC\cdot BD=AD\cdot BC\). Add.

Linear interpolation

For \(f\) twice differentiable, the error of linear interpolation between \(x_0\) and \(x_1=x_0+h\) is bounded by \(\frac{h^2}{8}\max|f''|\). For \(f(\theta)=120\sin(\theta/2)\) in degrees, \(|f''|\le120\cdot\big(\frac{\pi}{360}\big)^2\approx0.0091\), so with \(h=\tfrac12^\circ\) the error is at most about \(0.00028\). That is one unit in the second sexagesimal place, the same size as the rounding in the table itself, so interpolation loses almost nothing.

Diophantus and the lost commentary

For the general reader. Diophantus’s Arithmetica is a collection of problems asking for numbers with given properties, solved with a symbol for the unknown. Hypatia wrote a commentary on it, now lost. A commentary of her time would have explained the steps, checked the answers and added variants. This lesson works problems in that style.

The half-sum substitution

Many problems of Book I ask for two numbers with a given sum \(2s\) and some second condition. Diophantus writes the numbers as \(s+x\) and \(s-x\), reducing a system to one equation in one unknown.

Solution. Let the numbers be \(10+x\), \(10-x\). Then \(100-x^2=96\), so \(x^2=4\), \(x=2\), and the numbers are 12 and 8. In general, \(s^2-x^2=P\) has a rational solution exactly when \(s^2-P\) is the square of a rational number: this is the condition he states.

A commentary’s additions

The Greek manuscripts contain worked checks and second solutions that seem to be later additions. Tannery proposed Hypatia as their source. The Arabic books found in 1968 contain systematic verifications of each solution in a style Cameron likened to Theon’s. If he is right, the habit of checking that each answer satisfies the problem, which every reader of the Arithmetica now takes for granted, may owe something to her.

Apollonius and the conics

For the general reader. The Suda says Hypatia wrote a commentary on the Conics of Apollonius, the most advanced geometry text of antiquity. Apollonius described each curve by a single relation between lengths, which he called its symptom. The names parabola, ellipse and hyperbola come from that relation.

In the language of application of areas, the square on the ordinate equals a rectangle applied to \(p\) with width \(x\) exactly (parabolē, “placing beside”), falling short of it (elleipsis), or exceeding it (hyperbolē).

Derivation for the parabola. Cut a circular cone by a plane parallel to a generator. Take a horizontal circular section through a point of the curve with ordinate \(y\). The ordinate is a half-chord of that circle perpendicular to a diameter, so \(y^2\) equals the product of the two segments of that diameter. One segment is proportional to \(x\) by similar triangles; the other is constant along the curve because the cutting plane is parallel to a generator. Hence \(y^2=px\) for a fixed \(p\).

Theon’s Euclid

For the general reader. For more than a thousand years, almost everyone who read Euclid read Theon’s edition, made in Alexandria in Hypatia’s lifetime. Theon smoothed the wording, filled small gaps and added a few results. A different, older version came to light only in the nineteenth century, and comparing the two shows what an Alexandrian teacher thought a student needed.

Nearly all Greek manuscripts of the Elements describe themselves as “from the edition of Theon”. In the early nineteenth century François Peyrard found that the Vatican manuscript Vat. gr. 190 lacks this label and preserves an older text. In his Almagest commentary, Theon says that he had proved in his edition of the Elements that sectors in equal circles are to one another as their angles. That addition is in the Theonine manuscripts and missing from Vat. gr. 190, which confirms the identification.

Proof (Eudoxus’s method). Let the arcs be \(a\) and \(b\), with sectors \(S_a,S_b\). For any whole numbers \(m,n\), lay off \(m\) copies of arc \(a\) and \(n\) copies of arc \(b\) (wrapping around if needed). Equal arcs in equal circles give congruent sectors, so the sector on \(m\) copies of \(a\) is \(mS_a\), and similarly \(nS_b\). Larger arcs give larger sectors. Thus \(ma>nb\iff mS_a>nS_b\), and similarly for \(=\) and \(<\). By Euclid’s Definition V.5, \(a:b=S_a:S_b\).

Astrolabe and hydroscope

For the general reader. Popular accounts credit Hypatia with inventing the astrolabe and the hydrometer. The sources say less. Synesius describes a silver astrolabe he designed with her guidance, and he asked her in a letter to have a hydroscope made for him. Both instruments were older than she was. The mathematics behind them is still worth learning.

The astrolabe

In his essay On the Gift, addressed to Paeonius, Synesius describes a silver plane astrolabe he had made, and says he designed it with the help of his teacher. The theory may go back to Hipparchus and is set out in Ptolemy’s Planisphaerium; Theon wrote a treatise on the instrument, now lost. A plane astrolabe maps the celestial sphere onto the plane of the equator by projection from the south celestial pole.

The distance formula is a single inscribed-angle argument: the angle at the south pole between the axis and the line to the point is \(p/2\). Circles of declination \(\delta\) have \(p=90^\circ-\delta\), so they project to concentric circles of radius \(R\tan(45^\circ-\delta/2)\). The horizon and the circles of equal altitude are not centred on the pole, and the hero image shows them for the latitude of Alexandria.

The hydroscope

In Letter 15 Synesius asks Hypatia to have a hydroscope made for him. He describes it in detail: a cylindrical tube the size of a flute, with a line of notches, weighted with a cone at the bottom, which settles at a depth that shows the weight of the water. The level of detail suggests he did not expect her to know the device. It is the earliest known description of a hydrometer.

Alexandria, 412–415

Hypatia’s school had prospered under Bishop Theophilus, who cultivated the city’s elite. In October 412 Theophilus died and his nephew Cyril succeeded him after a contested election. Cyril was more aggressive. Socrates Scholasticus describes a quick escalation: Cyril’s conflict with the Jewish community, the expulsion of Jews from the city after a night-time massacre of Christians, and a breach with Orestes, the imperial prefect of Egypt, who resented the bishop’s growing power. Some five hundred monks came down from Nitria to support Cyril; one of them, Ammonius, struck Orestes on the head with a stone and was tortured to death.

The account of Socrates Scholasticus

Socrates, the earliest source, writes as a Christian who condemns the killing. He says Hypatia often met Orestes, and that a rumour spread among Christians that she prevented the prefect’s reconciliation with the bishop. A group led by a church reader named Peter waylaid her as she returned home, pulled her from her carriage and took her to the Caesareum, a former imperial temple then used as a church. They stripped her and killed her with roof tiles or potsherds, tore apart her body and burned the remains at a place called Cinaron. Socrates dates the killing to March, in Lent, in the fourth year of Cyril’s episcopate, under the consulships of Honorius (tenth) and Theodosius (sixth): that is, 415. He says the deed brought great disgrace on Cyril and on the church of Alexandria.

Other accounts

• Damascius, a pagan writing a century later, makes Cyril personally responsible. He says Cyril, seeing crowds at her door, was consumed by envy and plotted her death, and that he escaped punishment by bribery. Damascius was hostile to Christians and also critical of Hypatia, whom he ranks below his own teacher Isidore.
• John of Nikiu, a Coptic bishop writing in the late seventh century, approves of the killing. He calls her a pagan devoted to magic, astrolabes and musical instruments who had bewitched the prefect.

Historians still disagree about Cyril’s personal role. Christopher Haas argued that the killers were lay members of Cyril’s faction, not the parabalani. The imperial edict of 416, which cut the number of parabalani and placed them under the prefect, suggests that the court in Constantinople held the bishop’s followers responsible.

Legend and memory

Hypatia’s afterlife has been far longer than her life, and most of it was written for causes she never knew. The legend took shape in stages.

• Late antiquity. Damascius made her a symbol of pagan learning crushed by a jealous bishop. He also recorded an anecdote in which she discouraged a lovesick student by showing him a menstrual cloth to show him the body was nothing to love. That story, and his claim that she remained a virgin, shaped later portraits.
• Byzantine and medieval. Elements of her story may have fed into the legend of Saint Catherine of Alexandria, a learned virgin martyred by a tyrant, whose cult is first attested in the eighth century. The connection is suggestive, not proven.
• Enlightenment. John Toland’s Hypatia (1720) used her death against clerical power; Thomas Lewis replied defending Cyril. Voltaire presented her as a victim of Christian fanaticism. Edward Gibbon’s Decline and Fall (chapter 47, 1788) told her death as an indictment of Cyril, and his version became standard.
• Nineteenth century. Charles Kingsley’s novel Hypatia, or New Foes with an Old Face (1853) made her a young, beautiful heroine and popularized her as the last of the Hellenes. Paintings and plays followed. An illustration of 1908 for Elbert Hubbard’s invented biography became her most reproduced image.
• Twentieth century. She was linked, wrongly, to the destruction of the Library of Alexandria, which no longer existed as an institution in her time; Carl Sagan’s Cosmos (1980) helped spread the link. Feminist scholarship adopted her as a founding figure for women in science.

Recovering the historical Hypatia

Modern scholarship has moved in the other direction. Maria Dzielska’s Hypatia of Alexandria (1995) separated the documented figure from the legend. Alan Cameron, Edward Watts and Michael Deakin placed her in the Alexandrian schools and in the history of mathematical texts. The picture that emerges is of a senior, respected teacher in a Plotinian tradition, with Christian students and powerful friends, whose mathematical work was editorial and pedagogical: the careful preparation and explanation of the most advanced texts of her time. That is a real achievement. It is how Greek mathematics survived.

Honours

The asteroid 238 Hypatia (1884) and a lunar crater bear her name, as does Hypatia, a journal of feminist philosophy founded in 1986. Alejandro Amenábar’s film Agora (2009) retold her story for a wide audience, with many of the legendary elements.

Connections

People linked to Hypatia of Alexandria across this collection. A name without a link has no page here yet. See Hypatia of Alexandria in the network of all 337 lives →

Students

• Synesius of Cyrene 393-413

Influenced by

• Apollonius of Perga
• Claudius Ptolemy
• Diophantus of Alexandria
• Euclid
• Pappus of Alexandria
• Plotinus

Rivals

• Cyril of Alexandria

Friends

• Orestes

Family

• Theon of Alexandria

Also mentioned in this life

• Archimedes
• Aristotle
• Richard Dedekind
• Eudoxus of Cnidus
• Hipparchus
• Plato

Fields

• astronomy 50 lives
• mathematics 15 lives

On CoursesHub

• Diophantus & Late Antiquity · History of Mathematics
• Astrophysics for Beginners

Sources and further reading

Ancient sources in translation

1. Socrates Scholasticus, Ecclesiastical History 7.13–15. Chapter on Hypatia, Internet Medieval Sourcebook.
2. Synesius of Cyrene, Letters to Hypatia, trans. A. FitzGerald (1926): Letter 10, Letter 16, Letter 81, Letter 154, at Livius.org.
3. Damascius, The Philosophical History, ed. and trans. P. Athanassiadi, Apamea, 1999.
4. John of Nikiu, Chronicle 84, trans. R. H. Charles, 1916.

Modern studies

1. M. Dzielska, Hypatia of Alexandria, Harvard University Press, 1995.
2. E. J. Watts, Hypatia: The Life and Legend of an Ancient Philosopher, Oxford University Press, 2017.
3. A. Cameron, Isidore of Miletus and Hypatia: on the editing of mathematical texts, Greek, Roman and Byzantine Studies 31 (1990); reprinted in Wandering Poets and Other Essays, Oxford, 2016.
4. M. A. B. Deakin, Hypatia of Alexandria: Mathematician and Martyr, Prometheus, 2007.
5. C. Haas, Alexandria in Late Antiquity, Johns Hopkins University Press, 1997.
6. R. J. Penella, When was Hypatia born?, Historia 33 (1984).
7. J. J. O’Connor and E. F. Robertson, Hypatia of Alexandria and Theon of Alexandria, MacTutor.
8. Hypatia, Wikipedia (with extensive references).

Mathematics

1. G. J. Toomer, Ptolemy’s Almagest, Duckworth, 1984; Princeton, 1998.
2. O. Pedersen, A Survey of the Almagest, rev. ed., Springer, 2011.
3. T. L. Heath, The Thirteen Books of Euclid’s Elements, vol. 1, Cambridge, 1908 (introduction on Theon’s edition).
4. J. Sesiano, Books IV to VII of Diophantus’ Arithmetica in the Arabic Translation Attributed to Qusṭā ibn Lūqā, Springer, 1982.
5. W. R. Knorr, Textual Studies in Ancient and Medieval Geometry, Birkhäuser, 1989.
6. J. D. North, The astrolabe, Scientific American 230 (1974).