Category Archives: History of Mathematics

Everything you wanted to know about Simon Marius and were too afraid to ask – now in English

Regular readers of this blog should by now be well aware of the fact that I belong to the Simon Marius Society a small group of scholars mostly from the area around Nürnberg, who dedicate some of their time and energy to re-establishing the reputation of the Franconian mathematicus Simon Marius (1573–1625), who infamously discovered the four largest moons of Jupiter literally one day later than Galileo Galilei and got accused of plagiarism for his troubles. Galileo may have discovered them first but Marius won, in the long term, the battle to name them.

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Frontispiece of Mundus Iovialis Source:Wikimedia Commons

In 2014 the Simon Marius Society organised many activities to celebrate the four-hundredth anniversary of the publication of his opus magnum, Mundus Jovialis (The World of Jupiter). Amongst other things was an international conference held in Nürnberg, which covered all aspects of Marius’ life and work. The papers from this conference were published in German in 2016: Simon Marius und seine Forschung (Acta Historica Astronomiae), (AVA, Leipzig).

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Now after much effort and some delays the expanded translation, now includes the full English text of Mundus Jovialis, has become available in English: Simon Marius and his Research, Springer, New York, 2019.

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The ebook is already available and the hardback version will become available on 19 August. I apologize for the horrendous price but the problem of pricing by academic publishers is sadly well known. Having copyedited the entire volume, which means I have read the entire contents very carefully I can assure you that there is lots of good stuff to read not only about Simon Marius but also about astronomy, astrology, mathematics, court life in the seventeenth century and other topics of historical interest. If you can’t afford a copy yourself try to persuade you institutional library to buy one! If your university library buys a copy from Springer then students can order, through the library, a somewhat cheaper black and white copy of the book.

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Filed under History of Astrology, History of Astronomy, History of Mathematics, Renaissance Science

History of science on the Internet – the gift that keeps giving

Dear readers it is time once again for the Hist–Sci Hulk to flex his muscles and give a couple of Internet authors a good kicking for the crime of spreading history of science nonsense. First up we have a website called StarTeach Astronomy Education, who describe themselves as follows:

 

StarTeach Astronomy Education is a multimedia web-based astronomy education program designed especially for K-12 students and their teachers. The primary goal of the program is to aid in the classroom instruction of astronomy by providing a multimedia supplement that is available to schools with internet capability. In addition, it can also be used as research material for individual students.

The StarTeach Astronomy Education Program was created by Leslie Welser when she was a graduate student in the University of Nevada, Reno Physics Department. StarTeach has been recently updated to include topics for more advanced readers.

The section to which my attention was inadvertently directed was under the drop down menu Ancient Cultures with the title Arab and Islamic Astronomy.

Alone this title causes problem. Only yesterday I had an Internet exchange with Peter Adamson, the excellent creator of the History of Philosophy Without Any Gaps podcast and author of many books and articles based on that podcast, how exactly to describe the philosophy and science produced in the Middle Ages by the inhabitants of those areas of the world dominated by Islamic culture. The answer is certainly not Arab and Islamic, which would seem to suggest two separate areas of production or if only one that the astronomy was only produced by Islamic Arabs. The latter is definitively not the case as we have Arabs, Jews, Persians, Syriac Christians and various others. The subject is complex enough to warrant its own blog post, which I might or might not write sometime in the future.

The article opens with the following:

During the period when Western civilization was experiencing the dark ages [my emphasis], between 700-1200 A.D., an Islamic empire stretched from Central Asia to southern Europe.

The sound you can hear echoing around the world, is the sound of thousands of medievalist weeping and rending their clothes at the use of the term ‘the dark ages’. The preferred term is Early Middle ages. I should point out that the Arabic-Islamic dominance of a substantial part of the Eurasian landmass extended several centuries past 1200 CE and one of the things that Peter Adamson and I discussed is the fact that this empire actually split up fairly early into competing caliphates. Leaving out some minor quibbles we now come to this:

Another important religious use for astronomy was for the determination of latitude and longitude. Using the stars, particularly the pole star, as guides, several tables were compiled which calculated the latitude and longitude of important cities in the Islamic world.

Whereas you can determine latitude using altitude of the sun or the pole star and a bit of relatively simple trigonometry, you cannot determine longitude in this way. In fact the only way available in the Middle Ages to determine longitude astronomically is by simultaneous observation of eclipses, lunar or solar, and then comparing the timings. Next up with have:

Aside from religious uses, astronomy was used as a tool for navigation. The astrolabe, an instrument which calculated the positions of certain stars in order to determine direction, was invented by the Greeks and adopted and perfected by the Arabs (see picture below).

To quote David King, one of the world’s leading authorities on both the astrolabe and Islamic astronomy, the astrolabe was never used for navigation! It was actually the next sentences that finally provoked me into writing this blog post:

The sextant was developed by the Arabs to be a more sophisticated version of the astrolabe. This piece of technology ultimately became the cornerstone of navigation for European exploration.

When I first read this wonderful bullshit-double-whammy, I seriously started banging my head on my desk and praying for quick release from my pain. We are here confronted by a common problem in the history of science, the use of the same name for two or more completely different instruments. Our, by the way anonymous, I wouldn’t want to attach my name to this crap either, author apparently has no idea that the astronomical sextant and the navigational sextant are two completely different beast that apart from a scale representing one sixth of a circle, hence the name, have very little in common.

The astronomical sextant, to which the author intended to refer here, is anything but more sophisticated than the astrolabe and is in fact much, much simpler. It is a simple scale engraved on a segment, one sixth of a circle, with sights mounted on it used to determine the position of stars. I refer the reader to the Wikipedia article for more details. The navigational sextant, also not derived from the astrolabe, is a much more complex beast invented in the seventeenth century, the story of which you can read here. A change of topic:

Science was considered the ultimate scholarly pursuit in the Islamic world, and it was strongly supported by the nobility. Most scientists worked in the courts of regional leaders, and were financially rewarded for their achievements. In 830, the Khalifah, al-Ma’muun, founded Bayt-al-Hikman, the ‘House of Wisdom’, as a central gathering place for scholars to translate texts from Greek and Persian into Arabic. These texts formed the basis of Islamic scientific knowledge.

I think Qur’anic scholars might well dispute the claim that “Science was considered the ultimate scholarly pursuit in the Islamic world.” Another topic to which I might one day devote a whole blog post is the fact that the House of Wisdom, as it is usually presented, such as here, is a myth. For details see Dimitri Gutas, Greek Thought, Arabic Culture: The Graeco-Arabic Translation Movement in Baghdad and Early ‘Abbāsid Society (2nd–4th/8th–10thcenturies).

I could go on and on but I will just consider one last gem:

One of the greatest Islamic astronomers was al-Khwarizmi (Abu Ja’far Muhammad ibn Musa Al-Khwarizmi), who lived in the 9th century and was the inventor of algebra[my emphasis].

The ancient Babylonians and the Indians would be amazed to discover that al-Khwarizmi invented algebra. His book, Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala, introduced algebra into medieval Europe and part of its title gave the discipline its modern name but algebra was being practice literally millennia before al-Khwarizmi was born.

Interestingly this dog’s dinner of an article, that is intended to educate school kids, isn’t based on academic literature on the subject but offers as its sources links to five other Internet articles, three of which no longer exist and one of which is about the navigational sextant and as such irrelevant.

My second candidate for a Hist_Sci Hulk kicking is from Quillette a self styled platform for free thought and is a review by Jared Marcel Pollen (whoever he is) of Ben Shapiro’s The Right Side of History. An extremely tedious review of an extremely tedious book by an extremely tedious person that wouldn’t normally occupy my thought stream for longer than thirty seconds were it not for the following totally fucked up piece of history of astronomy.

This also allows Shapiro to skirt the obvious hostility the church showed toward intellectual inquiry for centuries. Shapiro writes as if the church had never banned a book or burned a heretic:

Contrary to popular opinion, new discoveries weren’t invariably seen as heretical or dangerous to the dominion of the Church; in fact, the Church often supported scientific investigation.

The Church was indeed the only place where any kind of inquiry could be conducted at the time. It was the only place where people were literate and enjoyed steady funding and access to instruments. But the Church only encouraged inquiry to the extent that it could reinforce and expand its own doctrine, which is a bit like the state telling you that you are free to make whatever art you like, just as long as you don’t criticize the regime.

What we have here is the standard popular presentation of the medieval Catholic Church as some sort of all-powerful totalitarian state. The reality was actually somewhat different. The Church was not able to exercise the sort of total intellectual control that Pollen is here claiming and surprisingly diverse positions were possible and existed in reality .

Also Pollens claim that, “The Church was indeed the only place where any kind of inquiry could be conducted at the time. It was the only place where people were literate and enjoyed steady funding and access to instruments” is of course total rubbish. Just to take the example of the three leading writers on astronomy in the reception phase of Copernican heliocentricity–Tycho Brahe, Johannes Kepler and Galileo Galilei–all three of them were court mathematicians outside of the various churches. There are plenty of other examples of important and influential scholars from this period, who found employment outside of the churches.

The first major scientific challenge to the Church was heliocentrism. But Shapiro claims this was hardly an issue: “Nicolas Copernicus studied in parochial school and served the Church of Warmia as medical advisor; his publication of De revolutionibus… in March 1542, included a letter to Pope Paul III.”

Copernicus was in fact a canon of Frombork cathedral and as such part of the government of the prince-bishopric of Warmia. His role was much wider than that of cathedral physician. De revolutionibus was published in 1543 not 1542 but did in fact contain a dedicatory letter to Pope Paul II.

In fact, Copernicus had finished his treatise years earlier (there are records indicating that the manuscript had been completed as early as the 1530s), but he withheld it, aware that its publication could be life-threatening, and circulated only a few anonymous copies to his close friends.

The bulk of De revolutionibus was, as far as we can tell, finished by about 1530, however the reasons for Copernicus not publishing at that time are complex and contrary to popular opinion had very little to do with any fear of persecution by the Church and its publication would certainly not have been life-threatening; that claim is complete rubbish based on hindsight and fail interpretations of the cases of Giordano Bruno and Galileo Galilei. Despite censoring Copernicus for his relationship with his housekeeper, Danticus, the Prince-Bishop of Warmia, supported Copernicus work and even invited Gemma Frisius, who Danticus supported, to come to Frombork to work with Copernicus. Tiedmann Giese, Bishop of Kulm, an influential cleric and Copernicus’ best friend, had been urging him to publish for years. Lastly Nicholas Schönberg, Cardinal of Capua, wrote a letter to Copernicus urging him to make his work public. This letter was included in the front material of De revolutionibus. As can be seen Copernicus had solid support for his work within the Church hierarchy. After its publication there was no initial opposition to De revolutionibus from the side of the Church. Copernicus is considered to have held back with publication because he couldn’t provide the proof of the heliocentric hypothesis that he wished to. There were no anonymous copies of De revolutionibus circulated to close friends! I assume our author is confusing De revolutionibus with the Commentariolus, Copernicus unpublished manuscript, which first propagated his heliocentric hypothesis from around 1510.

The book was only published in its entirety on the eve of Copernicus’s death, and the letter to the pope, which was also anonymous, was not written by Copernicus, but by Andreas Osiander, a Lutheran preacher who had been given the job of overseeing the book’s publication. It was an attempt to soften the blow, and states, inter alia, that the author’s findings are only meant to aid the computation of the heavens, and do not even need to be considered true in order for the calculations to be useful.

The implication that Copernicus only gave De revolutionibus free for publication when he was dying is once again total rubbish. Rheticus took the finished manuscript away from Frombork in September 1541 more than eighteen months before Copernicus’ demise. The next sentence is mindboggling for anybody who knows anything about De revolutionibus and its publication history. Copernicus dedicated De revolutionibus to His Holiness Pope Paul III. The anonymous text added by Andreas Osiander during publication was the infamous ad lectorem a totally different text altogether.

The Church would continue to uphold the geocentric model for at least another 150 years, and wouldn’t get around to officially pardoning Galileo until 1992. However, Shapiro claims the persecution of Galileo was merely a PR move by the Church; an attempt to crack down on dissent in response to Protestant accusations of leniency and hypocrisy. The trial of Galileo also saw dozens of astronomical works, including De Revolutionibus, placed on the Church’s “List of Prohibited Books”

Not the geocentric but the geo-heliocentric model of Tycho Brahe, was upheld not only by the Church but also by a very large number astronomers for many decades because the heliocentric model proved extremely difficult to prove. I can’t really comment on Shapiro’s claim, not having read his book and not intending to do so, but whatever the Church’s reasons for the persecution Galileo, and they are very complex, they have absolutely nothing to do with Protestant accusations of leniency and hypocrisy. Galileo is said to have obtained permission from Urban to write his Dialogo, because he claimed that the Protestants were mocking the Catholic Church because of its ignorant stance in the astronomy/cosmology debate. It would appear that Shapiro got his fact confused to put it diplomatically.

“Dozens of astronomical works” is a crass exaggeration and those books that were placed on the Index were placed there in 1616, following Galileo’s first run in with the Church and not in 1633 following his trial. Interestingly, and I get weary of repeating this, De revolutionibus was only placed on the Index until corrected, which surprisingly was completed by 1620; the corrected version with only those parts corrected which claimed heliocentricity to be a fact rather than a hypothesis, was then given free by the Church.

What we have here in total is a collection of half remember facts and myths stirred up together with an added sauce of nonsense and then spewed onto the page without any consideration for truth, facts or accuracy. Kind of sums up Quillette in my opinion.

 

 

 

 

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Filed under History of Astronomy, History of Mathematics, Renaissance Science

An important 13th-century book on optics

The thirteenth-century Silesian friar and mathematician Witelo is one of those shadowy figures in the history of science, whose influence was great but about whom we know very little.

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Page from a manuscript of Perspectiva with a miniature of the author Source: Wikimedia Commons

His biography can only be pieced together from scattered comments and references. In his Perspectiva he refers to “our homeland, namely Poland” and mentions Vratizlavia (Wroclaw) and nearby Borek and Liegnitz suggesting that he was born in the area. He also refers to himself as “the son of Thuringians and Poles,” which suggests his father was descended for the Germans of Thuringia who colonized Silesia in the twelfth and thirteenth centuries and his mother was of Polish descent.

A reference to a period spent in Paris and a nighttime brawl that took place in 1253 suggests that he received his undergraduate education there and was probably born in the early 1230s. Another reference indicates that he was a student of canon law in Padua in the 1260s. His Tractatus de primaria causa penitentie et de natura demonum, written in Padua refers to him as “Witelo student of canon law.” In late 1268 or early 1269 he appears in Viterbo, the site of the papal palace. Here he met William of Moerbeke  (c. 1220–c. 1286), papal confessor and translator of philosophical and scientific works from Greek into Latin. Witelo dedicated his Perspectiva to William, which suggest a close relationship. This amounts to the sum total of knowledge about Witelo’s biography.

In the printed editions of the Perspectiva he is referred to as Vitellio or Vitello but on the manuscript copies as Witelo, which is a diminutive form of Wito or Wido a common name in thirteenth century Thuringia, so this is probably his correct name. Family names were uncommon in thirteenth-century Poland, and there is no evidence to suggest that Witelo had one.[1]

Witelo’s principle work, his Perspectiva, was not started before 1270, as he uses William of Moerbeke’ translation of Hero of Alexandria’s Catoptrica, which was only completed on 31stDecember 1269. Witelo is one of three twelfth century authors, along with Roger Bacon (c. 1219–c. 1292) and John Peckham (c. 1230–1292), who popularised and disseminated the optical theories of  Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham, known in Latin as Alhazen or Alhacen. Al-Haytham’s Kitāb al-Manāzir (Book of Optics) was the most important Islamic texts on optics and one of the most important in the whole history of optics. It was translated into Latin by an unknown translator in the late twelfth or early thirteenth century with the title De aspectibus. Bacon was the first European author to include De aspectibus in his various writings on optics and Witelo and Peckham followed his lead. Although it is clear that Witelo used Ptolemy’s Optica, Hero’s Catoptrica and the anonymous De speculis comburentibus in composing his Perspectiva, and that he was aware of Euclid’s Optica, the Pseudo-Euclid Catoptrica and other prominent works on optics, it is very obvious that his major debt is to al-Haytham’s De aspectibus, although he never mentions him by name.

The Perspectiva is a monumental work that runs to nearly five hundred pages in the printed editions. It is divided into ten books:

Book I: Provides the geometric tools necessary to carry out geometrical optics and was actually used as a geometry textbook in the medieval universities.

Book II: Covers the nature of radiation, the propagation of light and colour, and the problem of pinhole images.

Book III: Covers the physiology, psychology, and geometry of monocular and binocular vision by means of rectilinear radiation.

Book IV: Deals with twenty visible intentions other than light and colour, including size, shape, remoteness, corporeity, roughness darkness and beauty. It also deals with errors of perception.

Book V: Considers vision by reflected rays: in plane mirrors

Book VI: in convex spherical mirrors

Book VII: in convex cylindrical and conical mirrors

Book VIII: in concave spherical mirrors

Book IX: in concave cylindrical, conical, and paraboloidal mirrors

Book X: Covers vision by rays refracted at plane or spherical surfaces; it also includes a discussion of the rainbow and other meteorological phenomena.

Witelo’s Perspectiva became a standard textbook for the study of optics and, as already mentioned above, geometry in the European medieval universities; it was used and quoted extensively in university regulations right down to the seventeenth century. The first printed edition of this important optics textbook was edited by Georg Tannstetter (1482–1535) and Peter Apian (1495–1552) and printed and published by Johannes Petreius (c. 1497–1550) in Nürnberg in 1535 under the title Vitellionis Mathematici doctissimi Peri optikēs, id est de natura, ratione & proiectione radiorum visus, luminum, colorum atque formarum, quam vulgo perspectivam vocant.

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Georg Tannstetter Portrait ca. 1515, by Bernhard Strigel (1460 – 1528) Source: Wikimedia Commons

Georg Tannstetter born in Rain am Lech in Bavaria had studied at the University of Ingolstadt under Andreas Stiborius (c. 1464–1515) and when Stiborius followed Conrad Celtis (1459–1508) to Vienna in 1497 to become professor for mathematics on the newly established Collegium poetarum et mathematicorum Tannstetter accompanied him. In 1502 he in turn began to lecture on mathematics in Vienna, the start of an illustrious career.

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Conrad Celtis: Gedächtnisbild von Hans Burgkmair dem Älteren, 1507 Source: Wikimedia Commons

Peter Apian, possibly his most famous pupil, was born, Peter Bienewitz, in Leisnig. He entered the University of Vienna in 1519 graduating B.A. in 1521. He then moved first to Regensburg and then to Landshut where he began his publishing career with his Cosmographicus liber in 1524.

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Apianus on a 16th-century engraving by Theodor de Bry Source: Wikimedia Commons

Following several failed attempts to acquire the position, Apian was appointed printer to the University in Ingolstadt in 1527, as well as lecturer for mathematics, positions he would hold until his death in 1552, when he was succeeded by his son Philipp (1531–1589), who had begun to take over his teaching duties before his death.

Apian’s Ingolstadt printing office continued to produce a steady stream of academic publications, so it comes as somewhat of a surprise that he chose to farm out the printing and publication of his own Instrumentum primi mobilis (1534) and the Tannstetter/Apian edited Witelo Perspectiva (1535) to Johannes Petreius in Nürnberg. Although both books were large and complex it should have been well within Apian’s technical capabilities to print and publish them in his own printing office; in 1540 he printed and published what is almost certainly the most complex science book issued in the sixteenth century, his Astronomicon Caesareum. The problem may have been a financial one, as he consistently had problems getting the university to supply funds to cover the advance cost of printing the books that he published.

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Source: Wikimedia Commons

Johannes Petreius, actually Hans Peter, was born in the Lower Franconian village of Langendorf near Hammelburg. He studied at the university in Basel graduating MA in 1517. Here he also learnt the printing trade in the printing office of his uncle Adam Petri (1445–1527). In 1523 he moved to Nürnberg where he set up his own printing business. By the early 1530s, when Apian approached him, he was one of the leading German printer publishers with a good reputation for publishing mathematical works, although his most famous publication Copernicus’ De revolutionibus orbium coelestium still lay in the future. In fact his publishing catalogue viewed as a whole makes him certainly the most important printer publisher of mathematical books in Germany and probably in the whole of Europe in the first half of the sixteenth century. As was his style he produced handsome volumes of both Apian’s Instrumentum and Witelo’s Perspectiva.

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Apian’s Instrumentum Title Page Source: Sothebys

Although he died in 1550 the Petreius printing office would issue an unchanged second edition of the Witelo in 1551, which was obviously in preparation before his death. After his death his business ceased as he had no successor and his catalogue passed to his cousin Heinric Petri (1508–1579) in Basel.

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Vitellionis Mathematici doctissimi Peri optikēs… title page Source: Christie’s

The Witelo volume would come to play a role in the eventual publication of Copernicus’ magnum opus by Petreius. When Georg Joachim Rheticus (1514-1574) set out in 1539 to seek out Copernicus in Frombork he took with him the Witelo tome as one of six specially-bound-as-a-set books, four of which had been printed and published by Petreius, as a gift for the Ermländer astronomer. The Petreius books were almost certainly meant to demonstrate to Copernicus what Petreius would do with his book if he allowed him to print it. The mission was a success and in 1542 Rheticus returned to Nürnberg with Copernicus’ precious manuscript for Petreius to print and publish in 1543.

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Copernicus De revolutionibus title page Source: Wikimedia Commons

There was a third printed edition of Witelo’s Perspectiva printed and published from a different manuscript by Friedrich Risner (1533–1580) together with al-Haytham’s De aspectibus in a single volume in Basel in 1527 under the title, Opticae thesaurus: Alhazeni Arabis libri septem, nuncprimum editi; Eiusdem liber De Crepusculis et nubium ascensionibus, Item Vitellonis Thuringopoloni libri X.

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Friedrich Risner edition Opticae Thesaurus (Basel, 1572) Title Page Source

This is the edition that Johannes Kepler (1571–1630) referenced in his Astronomiae pars optica. Ad Vitellionem Paralipomena (The Optical Part of Astronomy: Additions to Witelo) published in Prague in 1604, the most important book on optics since al-Haytham’s.

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Astronomiae pars optica. Ad Vitellionem Paralipomena  Source: University of Reading

Witelo remains an obscure thirteenth century scholar but his optics magnum opus cast a shadow down more than four hundred years of European history of optics. [2]

[1]All of the biographical information, and mush else in this article, is taken from David C. Lindberg, Witelo in Complete Dictionary of Scientific Biography, Charles Scribner’s Sons, 2008. Online at Encyclopedia.com

[2]For more on Witelo’s influence on the history of optics see David C. Lindberg, Theories of Vision from al-Kindi to Kepler, University of Chicago Press, Chicago and London, 1976, ppb. 1981.

On Peter Apian as a printer Peter Apian: Astronomie, Kosmographie and Mathematik am Beginn der Neuzeit mit Ausstellungskatalog, ed. Karl Röttel, Polygon-Verlag, Buxheim, Eichstätt, 1995 and Karl Schottenloher, Die Landshunter Buchdrucker des 16. Jahrhundert. Mit einem Anhang: Die Apianusdruckerei in Ingolstadt, Veröffentlichungen der Gutenberg-Gesellschaft XXXI, Mainz, 1930

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Filed under Early Scientific Publishing, History of Astronomy, History of Mathematics, History of Optics

The emergence of modern astronomy – a complex mosaic: Part II

You can read Part I here

Before we progress we need to take stock and deal with a couple of points that came up in a comment to Part I. This series is about the factors that led to the emergence of heliocentricity in Europe in the Early Modern Period. It doesn’t deal with any of the factors from earlier periods and other cultures that also explicitly and implicitly flowed into European astronomy. If one were to include all of those, it would be a total history of western astronomy that doesn’t even start in the West but in Babylon in about 2000 BCE. That is not what I intend to write and I won’t be doing so.

The other appears to contradict what I said above. At my starting point circa 1400 CE people became aware of a need to increase their usage of mathematical astronomy for a number of reasons that I sketched in Part I. Ptolemaic mathematical astronomy had been available in Europe in two Latin translations, the first from Greek the second from Arabic, since the twelfth century. However, medieval Europeans in general lacked the mathematical knowledge and to some extent the motivation to engage with this highly technical work. The much simpler available astronomical tables, mostly from Islamic sources, fulfilled their needs at that time. It was only really at the beginning of the fifteenth century that a need was seen to engage more fully with real mathematical astronomy. Having said that, at the beginning the users were not truly aware of the fact that the models and tables that they had inherited from the Greeks and from Islamic culture were inaccurate and in some cases defective. Initially they continued to use this material in their own endeavours, only gradually becoming aware of its deficiencies and the need to reform. As in all phases of the history of science these changes do not take place overnight but usually take decades and sometimes even centuries. Science is essential conservative and has a strong tendency to resist change, preferring to stick to tradition. In our case it would take about 150 years from the translation of Ptolemaeus’ Geographiainto Latin, my starting point, and the start of a full-scale reform programme for astronomy. Although, as we will see, such a programme was launched much earlier but collapsed following the early death of its initiator.

Going into some detail on points from the first post. I listed Peuerbach’s Theoricarum novarum planetarum(New Planetary Theory), published by Regiomontanus in Nürnberg in 1472, as an important development in astronomy in the fifteenth century, which it was. For centuries it was thought that this was a totally original work from Peuerbach, however, the Arabic manuscript of a cosmology from Ptolemaeus was discovered in the 1960s and it became clear that Peuerbach had merely modernised Ptolemaeus’ work for which he must have had a manuscript that then went missing. Many of the improvements in Peuerbach’s and Regiomontanus’ epitome of Ptolemaeus’ Almagest also came from the work of Islamic astronomers, which they mostly credit. Another work from the 1st Viennese School was Regiomontanus’ De Triangulis omnimodis Libri Quinque (On Triangles), written in 1464 but first edited by Johannes Schöner and published by Johannes Petreius in Nürnberg in1533.

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Title page of a later edition of Regiomontanus’ On Triangle

This was the first comprehensive textbook on trigonometry, the mathematics of astronomy, published in Europe. However, the Persian scholar Abū al-Wafā Būzhjānī (940–988) had already published a similar work in Arabic in the tenth century, which of course raises the question to what extent Regiomontanus borrowed from or plagiarised Abū al-Wafā.

These are just three examples but they should clearly illustrate that in the fifteenth and even in the early sixteenth centuries European astronomers still lagged well behind their Greek and Islamic predecessors and needed to play catch up and they needed to catch up with those predecessors before they could supersede them.

After ten years of travelling through Italy and Hungary, Regiomontanus moved from Budapest to Nürnberg in order to undertake a major reform of astronomy.

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City of Nürnberg Nuremberg Chronicles Workshop of Michael Wohlgemut Printed by Aton Koberger and published in Nürnberg in 1493

He argued that astrological prognostications were inaccurate because the astronomical data on which they were based was also inaccurate, which it indeed was. He had an ambitious two part programme; firstly to print and publish critical editions of the astronomical and astrological literature, the manuscripts of which he had collected on his travels, and secondly to undertake a new substantial programme of accurate astronomical observations. He tells us that he had chosen Nürnberg because it made the best scientific instruments and because as a major trading centre it had an extensive communications network. The latter was necessary because he was aware that he could not complete this ambitious programme alone but would need to cooperate with other astronomers.

Arriving in Nürnberg, he began to cooperate with a resident trading agent, Bernhard Walther, the two of them setting up the world’s first printing press for scientific literature. The first publication was Peuerbach’s Theoricae novae planetarum (New Planetary Theory)

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followed by an ambitious catalogue of planned future publications from the astrological and astronomical literature. Unfortunately they only managed another seven publications before Regiomontanus was summoned to Rome by the Pope to work on a calendar reform in 1475, a journey from which he never returned dying under unknown circumstances, sometime in 1476. The planned observation programme never really got of the ground although Walther continued making observations, a few of which were eventually used by Copernicus in his De revolutionibus.

Regiomontanus did succeed in printing and publishing his Ephemerides in 1474, a set of planetary tables, which clearly exceeded in accuracy all previous planetary tables that had been available and went on to become a scientific bestseller.

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However he didn’t succeed in printing and publishing the Epytoma in almagesti Ptolemei; this task was left to another important early publisher of scientific texts, Erhard Ratdolt (1447–1528, who completed the task in Venice twenty years after Regiomontanus’ death. Ratdolt also published Regiomontanus’ astrological calendars an important source for medical astrology.

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Calendarius by Regiomontanus, printed by Erhard Ratdolt, Venice 1478, title page with printers’ names Source: Wikimedia Commons

The first printed edition of Ptolemaeus’ Geographia with maps was published in Bologna in 1477; it was followed by several other editions in the fifteenth century including the first one north of the Alps in Ulm in 1482.

The re-invention of moveable type printing by Guttenberg in about 1450 was already having a marked effect on the revival and reform of mathematical astronomy in Early Modern Europe.

 

 

 

 

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The emergence of modern astronomy – a complex mosaic: Part I

I have recently been involved in more that one exchange on the subject as to what tipped the scales in favour of heliocentricity against geocentricity in the Early Modern Period. People have a tendency to want to pin it down to one crucial discovery, observation or publication but in reality it was a very gradual process that took place over a period of at least three hundred and fifty years and involved a very large number of people. In what follows I intend to sketch that process listing some, but probably not all, of the people involved. My list might appear to include people, who at first might not appear to have contributed to the emergence of modern astronomy if one just considers heliocentricity. However, all of those who raised the profile of astronomy and emphasised its utility in the Early Modern Period raised the demand for better and more accurate astronomical data and improved models to produce it. The inclusion of all these factors doesn’t produce some sort of linear progress but more a complex mosaic of many elements some small, some simple, some large and some spectacular but it is not just the spectacular elements that tells the story but a sum of all the elements. So I have cast my nets very wide.

The first question that occurs is where to start. One could go back all the way to Aristarchus of Samos (c.310–c.230 BCE) but although he and his heliocentric theories were revived in the Early Modern Period, it was largely with hindsight and he played no real role in the emergence of heliocentricity in that time. However, we should definitely give a nod to Martianus Capella (fl.c. 410–420), whose cosmos model with Mercury and Venus orbiting the Sun in an otherwise geocentric model was very widespread and very popular in the Middle Ages and who was quoted positively by Copernicus.

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The Capellan system Source: Manuscript Florenz, Biblioteca Medicea Laurenziana, San Marco 190, fol. 102r (11th century) via Wikimedia Commons

Another nod goes to Jean Buridan (c.1300–c.1358/61), Nicole Oresme (c.1320-1325–1382), Pierre d’Ailly (1351–1420) and Nicholas of Cusa (1401–1464) all of whom were well-known medieval scholars, who discussed the model of geocentrism with diurnal rotation, a model that was an important step towards the acceptance of heliocentricity.

I start with a figure, who most would probably not have on the radar in this context, Jacopo d’Angelo (c.1360–1411). He produced the first Latin translation of Ptolemaeus’ Geōgraphikḕ Hyphḗgēsis(Geographiaor Cosmographia) in Florence in 1406.

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Manuscript: d’Angelo’s translation of Ptolemy’s Geography Source: Scan from Nancy Library (Hosted at Wikicommons, early 15th century).

This introduced a new concept of cartography into Europe based on a longitude and latitude grid, the determination of which requires accurate astronomical data. Mathematical, astronomy based cartography was one of the major forces driving the reform or renewal of astronomy in the Early Modern Period. Another major force was astrology, in particular astro-medicine or as it was known iatromathematics, which was in this period the mainstream school medicine in Europe. Several of the astronomy reformers, most notably Regiomontanus and Tycho, explicitly stated that a reform of astronomy was necessary in order to improve astrological prognostications. A third major driving force was navigation. The Early Modern Period includes the so call great age of discovery, which like mathematical cartography was astronomy based. Slightly more nebulous and indirect were new forms of warfare, another driving force for better cartography as well as the collapse of the feudal system leading to new forms of land owner ship, which required better surveying methods, also mathematical, astronomy based. As I pointed out in an earlier post the people working in these diverse fields were very often one and the same person the Renaissance mathematicus, who was an astrologer, astronomer, cartographer, surveyor or even physician.

Our next significant figure is Paolo dal Pozzo Toscanelli (1397–1482), like Jacopo d’Angelo from Florence, a physician, astrologer, astronomer, mathematician and cosmographer.

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Paolo dal Pozzo Toscanelli. Detail taken from the 19th century honorary monument to Columbus, Vespucci and Toscanelli dal Pozzo in the Basilica di Santa Croce in Florence (Italy). Source: Wikimedia Commons

Most famous for his so-called Columbus world map, which confirmed Columbus’ erroneous theory of the size of the globe. In our context Toscanelli is more important for his observation of comets. He was the first astronomer in the Early Modern Period to treat comets as astronomical, supralunar objects and try to record and measure their trajectories. This was contrary to the ruling opinion of the time inherited from Aristotle that comets were sublunar, meteorological phenomena. Toscanelli did not publish his observations but he was an active member of a circle of mathematically inclined scholars that included Nicholas of Cusa, Giovanni Bianchini (1410 – c.1469), Leone Battista Alberti (1404 – 1472),Fillipo Brunelleschi (1377 – 1446) and most importantly a young Georg Peuerbach (1423–1461) with whom he probably discussed his ideas.

Here it is perhaps important to note that the mathematical practitioners in the Early Modern Period did not live and work in isolation but were extensively networked, often far beyond regional or national boundaries. They communicated extensively with each other, sometimes in person, but most often by letter. They read each other’s works, both published and unpublished, quoted and plagiarised each other. The spread of mathematical knowledge in this period was widespread and often surprisingly rapid.

We now turn from Northern Italy to Vienna and its university. Founded in 1365, in 1384 it came under the influence of Heinrich von Langenstein (1325–1397), a leading scholar expelled from the Sorbonne in Paris, who introduced the study of astronomy to the university, not necessarily normal at the time.

langenstein_heinrich_von_1325-1397_in_rationale_divinorum_officiorum_des_wilhelmus_durandus_codex_2765_oenb_1385-1406_106.i.1840_0-2

Probably Heinrich von Langenstein (1325-1397), Book illumination im Rationale divinorum officiorum des Wilhelmus Durandus, circa 1395 Source: Archiv der Universität Wien, Bildarchiv Signatur: 106.I.1840 1395

Heinrich was followed by Johannes von Gmunden (c.1380–1442) who firmly established the study of astronomy and is regarded as the founder of the 1stViennese School of Mathematics.

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Johannes von Gmunden Calendar Nürnberg 1496 Source: Wikimedia Commons

Georg Peuerbach the next member of the school continued the tradition of astronomical studies established by Heinrich and Gmunden together with his most famous student Johannes Regiomontanus.

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Johannes Regiomontanus Source: Wikimedia Commons

It can’t be a coincidence that Peuerbach and Regiomontanus extended Toscanneli’s work on comets, with Regiomontanus even writing a pamphlet on the determination of parallax of a moving comet, which was only publish posthumously in the sixteenth century. The two Viennese astronomers also designed and constructed improved astronomical instruments, modernised the trigonometry necessary for astronomical calculations and most importantly with Peuerbach’s Theoricarum novarum planetarum(New Planetary Theory),

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Georg von Peuerbach, Theoricae novae planetarum, Edition Paris 1515 Source: Wikimedia Commons

first published by Regiomontanus in Nürnberg in 1472, and their joint Epytoma in almagesti Ptolemei, a modernised, shortened improved edition of Ptolemaeus’ Syntaxis Mathematiké

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Epytoma in almagesti Ptolemei: Source

first published by Ratdolt in Venice in 1496, produced the standard astronomy textbooks for the period right up into the seventeenth century.

The work on the Viennese School very much laid the foundations for the evolution of the modern astronomy and was one of the processes anchoring the ‘modern’ study of astronomy an the European universities, How the journey continues will be told in Part II of this series.

 

 

 

 

 

 

 

 

 

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You shouldn’t believe everything you read

One of the things that I have been reading recently is a very interesting paper by John N. Crossley, the Anglo-Australian logician and historian of mathematics, about the reception and adoption of the Hindu-Arabic numbers in medieval Europe.[1]Here I came across this wonderful footnote:[2]

[…]

It is interesting to note that Richard Lemay in his entry “Arabic Numerals,” in Joseph Reese Strayer, ed., Dictionary of the Middle Ages(New York, 1982–89) 1:382–98, at 398 reports that in the University of Padua in the mid-fifteenth century, prices of books should be marked “non per cifras sed per literas claras.” He gives a reference to George Gibson Neill Wright, The Writing of Arabic Numerals(London, 1952), 126. Neill Wright in turn gives a reference to a footnote of Susan Cunnigton, The Story of Arithmetic: A Short History of Its Origin and Development(London, 1904), 42, n. 2. She refers to Rouse Ball’s Short History of Mathematics, in fact this work is: Walter William Rouse Ball, A Short Account of the History of Mathematics, 3rded. (London, 1901), and there one finds on p. 192: “…in 1348 the authorities of the university of Padua directed that a list should be kept of books for sale with the prices marked ‘non per cifras sed per literas claras’ [not by cyphers but by clear letters].” I am yet to find an exact reference for this prohibition. (There is none in Rouse Ball.) Chrisomalis Numerical Notations, p. 124, cites J. Lennart Berggren, “Medieval Arithmetic: Arabic Texts and European Motivations,” in Word, Image, Number: Communication in the Middle Ages, ed. John J. Contreni and Santa Casciani (Florence, 2002), 351–65, at 361, who does not give a reference.

Here we have Crossley the historian following a trail of quotes, references and footnotes; his hunt doesn’t so much terminate in a dead-end as fizzle out in the void, leaving the reader unsure whether the university of Padua really did insist on its book prices being written in Roman numerals rather than Hindu-Arabic ones or not. What we have here is a succession of authors writing up something from a secondary, tertiary, quaternary source with out bothering to check if the claim it makes is actually true or correct by looking for and going back to the original source, which in this case would have been difficult as the trail peters out by Rouse Ball, who doesn’t give a source at all.

This habit of writing up without checking original sources is unfortunately not confined to this wonderful example investigated by John Crossley but is seemingly a widespread bad habit under historians and others who write historical texts.

I have often commented that I served my apprenticeship as a historian of science in a DFG[3]financed research project on Case Studies into a Social History of Formal Logic under the direction of Professor Christian Thiel. Christian Thiel was inspired to launch this research project by a similar story to the one described by Crossley above.

Christian Thiel’s doctoral thesis was Sinn und Bedeutung in der Logik Gottlob Freges(Sense and Reference in Gottlob Frege’s Logic); a work that lifted him into the elite circle of Frege experts and led him to devote his academic life largely to the study of logic and its history. One of those who corresponded with Frege, and thus attracted Thiel interest, was the German meta-logician Leopold Löwenheim, known to students of logic and meta-logic through the Löwenheim-Skolem theorem or paradox. (Don’t ask!) Being a thorough German scholar, one might even say being pedantic, Thiel wished to know Löwenheim’s dates of birth and death. His date of birth was no problem but his date of death turned out to be less simple. In an encyclopaedia article Thiel came across a reference to c.1940; the assumption being that Löwenheim, being a quarter Jewish and as a result having been dismissed from his position as a school teacher in 1933, had somehow perished during the holocaust. In another encyclopaedia article obviously copied from the first the ‘circa 1940’ had become a ‘died 1940’.

Thiel, being the man he is, was not satisfied with this uncertainty and invested a lot of effort in trying to get more precise details of the cause and date of Löwenheim’s death. The Red Cross information service set up after the Second World War in Germany to help trace people who had died or gone missing during the war proved to be a dead end with no information on Löwenheim. Thiel, however, kept on digging and was very surprised when he finally discovered that Löwenheim had not perished in the holocaust after all but had survived the war and had even gone back to teaching in Berlin in the 1950s, where he died 5. May 1957 almost eighty years old. Thiel then did the same as Crossley, tracing back who had written up from whom and was able to show that Löwenheim’s death had already been assumed to have fallen during WWII, as he was still alive and kicking in Berlin in the early 1950s!

This episode convinced Thiel to set up his research project Case Studies into a Social History of Formal Logic in order, in the first instance to provide solid, verified biographical information on all of the logicians listed in Church’s bibliography of logic volume of the Journal of Symbolic Logic, which we then proceeded to do; a lot of very hard work in the pre-Internet age. Our project, however, was not confined to this biographical work, we also undertook other research into the history of formal logic.

As I said above this habit of writing ‘facts’ up from non-primary sources is unfortunately very widespread in #histSTM, particularly in popular books, which of course sell much better and are much more widely read than academic volumes, although academics are themselves not immune to this bad habit. This is, of course, the primary reason for the continued propagation of the myths of science that notoriously bring out the HISTSCI_HULK in yours truly. For example I’ve lost count of the number of times I’ve read that Galileo’s telescopic discoveries proved the truth of Copernicus’ heliocentric hypothesis. People are basically to lazy to do the legwork and check their claims and facts and are much too prepared to follow the maxim: if X said it and it’s in print, then it must be true!

[1]John N. Crossley, Old-fashioned versus newfangled: Reading and writing numbers, 1200–1500, Studies in medieval and Renaissance History, Vol. 10, 2013, pp.79–109

[2]Crossley p. 92 n. 42

[3]DFG = Deutsche Forschungsgemeinschaft = German Research Foundation

 

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Hypatia – What do we really know?

The fourth century Alexandrian mathematician and philosopher Hypatia has become a feminist icon. She is probably the second most well known woman in #histSTM after Marie Curie. Unfortunately, down the centuries she has been presented more as a legend or a myth intended to fulfil the teller’s purposes rather than a real human being. As Alan Cameron puts it in his excellent essay, Hypatia: Life, Death, and Works:[1]

A pagan in the Christian city of Alexandria, she is one of those figures whose tragic death inspired a legend which could take almost any form because so few facts are known. As a pagan martyr, she has always been a stick to beat Christians with, a symbol in the continuing struggle between science and revealed religion. The memorable account in Gibbon begins wickedly “On a fatal day in the holy season of lent.” As a woman she can be seen as a feminist as well as a pagan martyr. Her name has been a feminist symbol down the centuries more recently a potent name in lesbian and gay circles. As an Egyptian, she has also been claimed as a black woman martyr. There is an asteroid named after her, a crater on the moon, and a journal of feminist studies. As early as 1886, the women of Wichita Kansas, familiar from the movies of our youth as a lawless western cattle town, formed a literary society called the Hypatia Club. Lake Hypatia in Alabama is a retreat for freethinkers and atheists. Rather less in tune with her scholarly activity, there is Hypatia Capital, a merchant bank whose strategy focuses on the top female executives in the Fortune 1000.

A few minutes’ googling will produce countless eulogies of Hypatia as a uniquely gifted philosopher, mathematician and scientist, the second female scientist after Marie Curie, the only woman in antiquity appointed to a university chair, a theorist who anticipated Copernicus with the heliocentric hypothesis. The 2009 movie Agora goes even further in this direction. A millennium before Kepler, Hypatia discovered that earth and its sister planets not only go round the sun but do so in ellipses, not circles. She remained unmarried, and could therefore be seen as a model of pagan virginity. Alternatively, since the monks are said to have killed her because of her influence on the prefect of Egypt, she could be seen as a slut. It is fascinating to observe how down the centuries she served as a lay figure for the prejudices of successive generations.

So what do we know about the real Hypatia? The answer is almost nothing. We know that she was the daughter of Theon (c.335–c.405) an Alexandrian mathematician and philosopher, most well known for his edition of The Elements of Euclid. We don’t know her birth date with estimates ranging from 350 to 370 CE. Absolutely nothing is known about her mother to whom no references whatsoever exist. It is assumed that she was educated by her father but once again, whilst highly plausible, no real evidence exists for this assumption. If we take a brief looked at the available sources for her biography the reason for all of this uncertainty becomes very clear.

The only source we have from somebody who actually knew Hypatia is Synesius of Cyrene (c.373–probably 413), who was one of her Christian students around 393 CE. In 410 CE he was appointed Bishop of Ptolemais. There was an edition of his letters, which contains seven letters to Hypatia and some to others that mention her. Unfortunately his letters tell us nothing about her death as he predeceased her. His last letter to her was written from his deathbed in 413 CE. Two of his letters, however, request her assistance for acquaintances in civil matters, which indicates that she exercised influence with the civil authorities.

Our second major source is Socrates of Constantinople (c.380–died after 439) a Christian church historian, who was a contemporary but who did not know her personally. He mention her and her death in his Historia Ecclesiastica:

There was a woman at Alexandria named Hypatia, daughter of the philosopher Theon, who made such attainments in literature and science, as to far surpass all the philosophers of her own time. Having succeeded to the school of Plato and Plotinus, she explained the principles of philosophy to her auditors, many of whom came from a distance to receive her instructions. On account of the self-possession and ease of manner which she had acquired in consequence of the cultivation of her mind, she not infrequently appeared in public in the presence of the magistrates. Neither did she feel abashed in going to an assembly of men. For all men on account of her extraordinary dignity and virtue admired her the more.

The third principle source is Damascius (c.458–after 538) a pagan philosopher, who studied in Alexandria but then moved to Athens where he succeeded his teacher Isidore of Alexandria (c.450–c.520) as head of the School of Athens. He mentions Hypatia in his Life of Isidore, which has in fact been lost but which survives as a fragment that has been reconstructed.

We also have the somewhat bizarre account of the Egyptian Coptic Bishop John of Nikiû (fl. 680–690):

And in those days there appeared in Alexandria a female philosopher, a pagan named Hypatia, and she was devoted at all times to magic, astrolabes and instruments of music, and she beguiled many people through her Satanic wiles. And the governor of the city honoured her exceedingly; for she had beguiled him through her magic. And he ceased attending church as had been his custom… And he not only did this, but he drew many believers to her, and he himself received the unbelievers at his house.

It is often claimed that she was head of The Neo-Platonic School of philosophy in Alexandria. This is simply false. There was no The Neo-Platonic School in Alexandria. She inherited the leadership of her father’s school, one of the prominent schools of mathematics and philosophy in Alexandria. She however taught a form of Neo-Platonic philosophy based mainly on Plotonius, whereas the predominant Neo-Platonic philosophy in Alexandria at the time was that of Iamblichus.

If we turn to her work we immediately have problems. There are no known texts that can be directly attributed to her. The Suda, a tenth-century Byzantine encyclopaedia of the ancient Mediterranean world list three mathematical works for her, which it states have all been lost. The Suda credits her with commentaries on the Conic Sections of the third-century BCE Apollonius of Perga, the “Astronomical Table” and the Arithemica of the second- and third-century CE Diophantus of Alexandria.

Alan Cameron, however, argues convincingly that she in fact edited the surviving text of Ptolemaeus’ Handy Tables, (the second item on the Suda list) normally attributed to her father Theon as well as a large part of the text of the Almagest her father used for his commentary.  Only six of the thirteen books of Apollonius’ Conic Sections exist in Greek; historians argue that the additional four books that exist in Arabic are from Hypatia, a plausible assumption.

All of this means that she produced no original mathematics but like her father only edited texts and wrote commentaries. In the history of mathematics Theon is general dismissed as a minor figure, who is only important for preserving texts by major figures. If one is honest one has to pass the same judgement on his daughter.

Although the sources acknowledge Hypatia as an important and respected teacher of moral philosophy there are no known philosophical texts that can be attributed to her and no sources that mention any texts from her that might have been lost.

Of course the most well known episode concerning Hypatia is her brutal murder during Lent in 414 CE. There are various accounts of this event and the further from her death they are the more exaggerated and gruesome they become. A rational analysis of the reports allows the following plausible reconstruction of what took place.

An aggressive mob descended on Hypatia’s residence probably with the intention of intimidating rather than harming her. Unfortunately, they met her on the open street and things got out of hand. She was hauled from her carriage and dragged through to the streets to the Caesareum church on the Alexandrian waterfront. Here she was stripped and her body torn apart using roof tiles. Her remains were then taken to a place called Cinaron and burnt.

Viewed from a modern standpoint this bizarre sequence requires some historical comments. Apparently raging mobs and pitched battles between opposing mobs were a common feature on the streets of fourth-century Alexandria. Her murder also followed an established script for the symbolic purification of the city, which dates back to the third-century. There was even a case of a pagan statue of Separis being subjected to the same fate. There is actually academic literature on the use of street tiles in street warfare[2]. What is more puzzling is the motive for the attack.

The exact composition of the mob is not known beyond the fact that it was Christian. There is of course the possibility that she was attacked simply because she was a woman. However, she was not the only woman philosopher in Alexandria and she enjoyed a good reputation as a virtuous woman. It is also possible that she was attacked because she was a pagan. Once again there are some contradictory facts to this thesis. All of her known students were Christians and she had enjoyed good relations with Theophilus the Patriarch of Alexandria (384–412), who was responsible for establishing the Christian dominance in Alexandria. Theophilus was a mentor of Synesius. Also the Neoplatonic philosophy that she taught was not in conflict with the current Christian doctrine, as opposed to the Iamblichan Neoplatonism. The most probably motive was Hypatia’s perceived influence on Orestes (fl. 415) the Roman Prefect of Egypt who was involved in a major conflict with Cyril of Alexandria (c.376–444), Theophilis’ nephew and successor as Patriarch of Alexandria. This would make Hypatia collateral damage in modern American military jargon. In the end it was probably a combination of all three factors that led to Hypatia’s gruesome demise.

Hypatia’s murder has been exploited over the centuries by those wishing to bash the Catholic Church but also by those wishing to defend Cyril, who characterise her as an evil woman. Hypatia was an interesting fourth-century philosopher and mathematician, who deserves to acknowledged and remembered for herself and not for the images projected on her and her fate down the centuries.

[1]Alan Cameron, Hypatia: Life, Death, and Works, in Wandering Poets and Other Essays on Late Greek Literature and Philosophy, OUP, 2016 pp. 185–203 Quote pp. 185–186

[2]You can read all of this in much more detail in Edward J. Watts’ biography of Hypatia, Hypatia: The Life and Legend of an Ancient Philosopher, OUP, 2017, which I recommend with some reservations.

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