The movie Agora mentioned in the previous post got me interested in the actual Hypatia, so I picked up “Hypatia of Alexandria, Mathematician and Martyr” by Michael B. Deakin (2007).

Portrait by Elbert Hubbard, 1908

It discusses the little that is actually known about her, and goes into some depth on what her mathematical contributions were.  The author is an Honorary Research Member of the School of Mathematical Science at Monash University, which is near Melbourne Australia.   He taught there from 1973 until his retirement in 2003, and got his doctorate from the University of Chicago in 1966.

According to Deakin, Hypatia was the leading mathematician in the world in 400 CE.  This, unfortunately, was because she was the leading mathematician in Alexandria.  Nothing was happening at that time in Athens, the other major center of learning in the Roman world, and China and India were in down phases.    So it may seem a limited honor, but it’s a rare one.  One could say that Marie Curie was the world’s leading chemist in 1910, but no other female mathematicians since Hypatia’s time could really claim to be the best in the world.

Even so, Hypatia herself cannot be called one of the world’s great mathematicians.    We have no work directly by her.  Instead, she is credited with having edited Ptolemy’s Alamgest and Apollonius’ Conics.  The most direct source is the inscription of her father Theon’s commentary on Ptolemy:

Theon of Alexandria’s commentary on the third [book] of the Mathematical Syntaxis of Ptolemy, the edition having been prepared by the philosopher, my daughter Hypatia.

The one piece of math that we can most directly credit to her comes from this third Book, where there is a discussion of a method of doing long division.  This may sound obvious, but it’s not, especially when done in base-60.  The Greeks did not use the absurd Roman numeral system for real calculations – they used a positional notation where each pair of characters represented a value from to 1 to 60.  The first position was the 60s digit, the second the 3600 digit, and so on.  Her method is substantially faster and more accurate than earlier schemes, but is still not what we do today.

Well, that’s a fairly important advance, but it doesn’t exactly lend itself to big movie climaxes.  Here’s where the writers of the movie about Hypatia, Agora, showed real brilliance.

Massive spoiler alert! Seriously – the plot point below is a slow reveal through the movie, and is one of its pleasures to a technically-minded viewer.

The writers, Alejandro Amenábar and  Mateo Gil, used her expertise in Ptolemy’s planetary system and in conic sections to propose that she discovered the heliocentric model with elliptical planetary orbits.  This would have been 1200 years before Kepler!    The Greeks knew that planetary motion could be explained by having the sun at the center of the solar system, but Ptolemy just didn’t find that necessary.  His scheme of an epicycle on a circular orbit that was not quite centered on the Earth worked quite well.  Well enough, anyway, to match the limited astronomical data of the time.

In the movie, Hypatia finds that system inelegant.  That too is a nice interpretation.  To a Neo-platonist like Hypatia, mathematics was of spiritual, not material interest.   Mathematics was one method of abstraction, of stripping away the dross of the world to achieve the purity of Idea.   As neo-platonists ascend higher and higher in the world of abstraction, they approach the One, the dazzling unity of all space, time, form, and flow.

Her philosophy gave her a serenity that was much remarked upon in her life, and a striking part of Rachel Weisz’s portrayal of her.   Even as her city is in upheaval, even as the Library is being ransacked, even as her beloved father dies, she still keeps poised and calm.  It’s not that she’s unaware or indifferent, it’s that she remains steady.

Anyway, Hypatia finds these epicycles and equants distasteful, and looks for a better solution.   A heliocentric model gets rid of the epicycles, but if the earth is moving around the sun, why don’t we feel it?  She finds the answer to that by re-creating one of Galileo’s thought experiments, where moving bodies on something that is itself moving, such as a smoothly sailing ship, behave just the same in spite of the overall motion.  Velocity and position are relative to a frame of reference, not absolute.

But what about the change in size of the sun?  The sun looks larger in winter (in the Northern hemisphere) than in the summer.   If the earth’s orbit is a circle it should be the same size all the time.   The writers give her an epiphany – all circles are just special cases of ellipses.   A circle is just an ellipse whose two foci happen to coincide.   It’s a horizontal section through a cone instead of one at a tilt.  With elliptical orbits, the sun and the earth are not always at the same distance, nor are the moon and earth, or the planets and sun.

Kepler’s actual discovery of this in 1609 is one of the highlights of the Scientific Revolution.   It finally gave an explanation of something – the positions of the planets and Moon – that people had observed for millenia.  It kicked off an explosion of scientific development that has now been going for 400 years.   If Hypatia had discovered this in 415 CE, it really would have been a fantastic achievement, one that would have rendered insignificant the turmoil of her life and the tragedy of her death.

Of course, it wouldn’t really have been possible.  Ptolemy himself, a far greater figure than Hypatia, couldn’t have proven this theory.   The Greeks didn’t have the mathematical tools to do it.  They didn’t have algebra, or the concept of zero, or the basic arithmetic for it.  Even long division, as mentioned above, was an advanced skill.   Nor did they have the experimental data for it.  It takes large instruments and long periods of observation to really measure positions in the heavens.   You don’t need telescopes – Tycho Brahe did without – but you need an effort that we don’t see in the classical world.

But this is a quibble.  It’s rare to see any kind of math in the movies, so math that is central to a movie’s character and fits in with her actual life is extraordinary.  Usually math in the movies drives people mad, as in Pi or A Beautiful Mind, but here it kept a woman sane in the most difficult of times.