You can read Part I here and Part II here

Although I dealt with the special case of Vienna and the 1^{st }Viennese School of Mathematics in the first post of this series, it is now time to turn to the general history of the fifteenth-century university and the teaching of astronomy. Although the first, liberal arts, degree at the medieval university theoretically encompassed the teaching of the quadrivium, i.e. arithmetic, geometry, music and astronomy, in reality the level of teaching was very low and often neglected all together. Geometry was a best the first six books of Euclid and at worst just book one and astronomy was the *Sphaera*of Sacrobosco, a short non-technical introduction.

This all began to change in the fifteenth century. The humanist universities of Northern Italy and of Poland introduced dedicated chairs for mathematics, whose principle purpose was the teaching of astrology to medical students. However, to fully understand astrology and to be able to cast horoscopes from scratch students first had to learn astronomy, which in turn entailed first having to learn arithmetic and geometry, as well as the use of mathematical and astronomical instruments. The level of mathematical tuition on the university increased considerable. The chairs for mathematics that Galileo would occupy at the end of the sixteenth century in Pisa and Padua were two such astrology chairs.

As the first European university, Krakow introduced two such chairs for mathematics and astronomy relatively early in the fifteenth century.

It was here at the end of the century (1491–1495) that Copernicus first learnt his astronomy most probably in the lectures of Albert Brudzewski (c. 1445–c.1497) using Peuerbach’s *Theoricae Novae Planetarum *and Regiomontanus’ *Astronomical Tables*. Brudzewski also wrote an important commentary on Peuerbach’s *Theoricae Novae Planetarum*,*Commentum planetarium in theoricas Georgii Purbachii *(1482).Krakow was well endowed with Regiomontanus’ writings thanks to the Polish astrologer Marcin Bylica (c.1433–1493), who had worked closely with Regiomontanus on the court ofMatthias Corvinus (1443–1490) in Budapest and who when he died bequeathed his books and instruments to the University of Krakow, including the works of Regiomontanus and Peuerbach.

From Krakow Copernicus went on to Northern Italy and its humanist universities. Between 1496 and 1501 he studied canon law in Bologna, Europe’s oldest university.

Here he also met and studied under/worked with the professor for astronomer Domenico Maria Novara da Ferrara (1454–1504), who claimed to be a student of Regiomontanus and it is known that he studied under Luca Pacioli (c. 1447–1517), who was also Leonardo’s mathematics teacher. Although none of Novara da Ferrara writings have survived he is said to have taken a critical attitude to Ptolemaic astronomy and he might be the trigger that started Copernicus on his way. In late 1501 Copernicus moved to the University of Padua, where he studied medicine until 1503, a course that would also have included instruction in astrology and astronomy. In 1503 he took a doctorate in canon law at the University of Ferrara. Sometime in the early sixteenth century, probably around 1510 he wrote an account of his first thoughts on heliocentricity, now known as the *Commentariolus*, which was never published but seems to have circulated fairly widely in manuscript. We will return to this later.

The first German university to install a dedicated chair for mathematics/astronomy was Ingolstadt in the 1470s.

As with the North Italian universities this was principally to teach astrology to medical student. This chair would prove to be an important institution for spreading the study of the mathematical sciences. In 1491/1492 the humanist scholar and poet, Conrad Celtis (1459–1508) was appointed professor of poetics and rhetoric in Ingolstadt. Celtis had a strong interest in cartography as a part of history and travelled to Krakow in 1489 in order to study the mathematical sciences. In Ingolstadt Celtis was able to turn the attention of Andreas Stiborius (1464–1515) and Johannes Stabius (1468–1522) somewhat away from astrology and more towards cartography. In 1497 Celtis received a call from the University of Vienna and taking Stiborius and Stiborius’ star student Georg Tannstetter (1482–1535) with him he decamped to Vienna, where he set up his *Collegium poetarum et mathematicorum*, with Stiborius as professor for mathematics. In 1502 he also fetched Johannes Stabius. From 1502 Tannstetter also began to lecture on mathematics and astronomy in Vienna. Stiborius, Stabius and Tannstetter form the foundations of what is known as the 2^{nd}Viennese School of Mathematics. Tannstetter taught several important students, most notably Peter Apian, who returned to Ingolstadt as professor for mathematics in the 1520, a position in which he was succeeded by his son Philipp. Both of them made major contributions to the developments of astronomy and cartography.

Stabius’ friend and colleague Johannes Werner also studied in Ingolstadt before moving to and settling in Nürnberg. One of the few astronomical writing of Copernicus, apart from *De revolutionibus*, that exist is the so-called *Letter against Werner *in which Copernicus harshly criticised Werner’s *Motion of the Eighth Sphere *an essay on the theory of precession of the equinox.

Another graduate of Ingolstadt was Johannes Stöffler (1452–1531), who having had a successful career as an astronomer, astrologer and globe and instrument maker was appointed the first professor of mathematics at the University of Tübingen.

Amongst his student were Sebastian Münster (1488–1552) the most important cosmographer of the sixteenth century and Philipp Melanchthon (1497–1560), who as a enthusiastic fan of astrology established chairs for mathematics and astronomy at all of the protestant schools and universities that he established starting in Wittenberg, where the first professor for lower mathematic was Jakob Milich (1501–1559) another graduate of the University of Vienna. Milich’s fellow professor for astronomy in Wittenberg Johannes Volmar (?–1536), who started his studies in Krakow. The successors to Milich and Volmar were Georg Joachim Rheticus (1514–1574) and Erasmus Reinhold (1511–1553).

Another Melanchthon appointment was the first professor for mathematics on the Egidien Obere Schule in Nürnberg, (Germany’s first gymnasium), the globe maker Johannes Schöner (1477–1547), who would play a central role in the heliocentricity story. Schöner had learnt his mathematics at the university of Erfurt, one of the few German universities with a reputation for mathematics in the fifteenth century. When Regiomontanus moved from Budapest to Nürnberg he explained his reasons for doing so in a letter to the Rector of Erfurt University, the mathematician Christian Roder, asking him for his active support in his reform programme.

The Catholic universities would have to wait for Christoph Clavius (1538–1612) at the end of the sixteenth century before they received dedicated chairs for astronomy to match the Lutheran Protestant institutions. However, there were exceptions. In Leuven, where he was actually professor for medicine, Gemma Frisius (1508–1555) taught astronomy, astrology, cartography and mathematics. Amongst his long list of influential pupils we find Johannes Stadius (1527–1579), Gerhard Mercator (1512–1594) and John Dee (1527–1609). In France, François I appointed Oronce Fine (1494–1555) Royal lecturer for mathematics at the University of Paris. He was not a very impressive mathematician or astronomer but a highly influential teacher and textbook author. In Portugal, Pedro Nunes (1502–1578) was appointed the first professor of mathematics at the university of Coimbra as well as to the position of Royal Cosmographer.

Over the fifteenth and sixteenth centuries the mathematical sciences, driven mainly by astrology and cartography, established themselves in the European universities, where the professors and lecturers, as we shall see, played a central role in the reform and renewal of astronomy.

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Relating this to the previous post on the series, can you comment on whether, and if so to what extent, print technology was necessary for the reforms of Melanchthon and Clavius? I don’t know how they did things in the 16th century, but in modern universities, it’s pretty typical for every student to get a copy of the textbook. This seems like it would have been prohibitively expensive before print, and it also seems like learning advanced math would be a lot harder if you had to do it from lectures alone or by sharing a single copy of the text among multiple students. But, this is just a guess.

The invention of printing didn’t really improve life for students or at least the ones who were not wealthy. Early books particularly large astronomy of mathematics tomes were prohibitively expensive. Students would have had to do what they had always done in the days of manuscripts, take notes in lectures and makes notes from books in the library. Both Melanchthon and Clavius were extensive textbook writers, so I assume that university libraries would have had copies, perhaps multiple copies, of those textbooks. What print technology did do was make acquiring a faithful, accurate copy of a book easier for those that could afford them and libraries. Previously one had to copy the manuscript for yourself or pay somebody else to do it. This is what Regiomontanus spent about ten years doing for his various patrons, Bessarion, Vitéz, Corvinus. But copies of copies introduces the very real risk of errors. This was something that Regiomontanus was acutely aware of and his aim with his printing office in Nürnberg was to publish printed edition of the major astrological and astronomical works cleaned of their errors through philological analysis. To this end he did not just acquire one manuscript of a given book but several from different sources so that he could compare and analyse the texts and produce the best possible, cleaned up edition.

Thanks for the added context. BTW could the 1497 image of the German Nation at the University of Bologna from the original post be taken as an illustration of the dynamic wherein the professor has the book and the students don’t, or am I misunderstanding the context?

The image from the University of Bologna is of students registering in the German Nation, a sort of Renaissance student support organisation

I guess what I’m asking regarding the image is who’s the figure in the center, what’s the book he’s holding, and what’s he doing with it?

Student Nations had elected officials, who were responsible for the administration. The man in the middle with the book is one such official registering students into the German Nation. Copernicus was registered in the German Nation at Bologna

Student nations still exist in Sweden.

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(Coming in late; forgive me, I only just found this blog)

I wonder what connection there might be between this rise in serious mathematical instruction and the introduction of standardized symbols. It was during the period covered by this essay that the plus, minus, and equals signs were introduced (or came in to wide use). It’s a HECK of a lot easier to do and share calculations when you’re not doing everything as a word problem or couching it in a personal shorthand.