It is one of the ironies of the medieval universities that mathematics played almost no role in undergraduate education. It is ironical because the curriculum was nominally based on the seven liberal arts of which the mathematical sciences – arithmetic, geometry, music and astronomy – formed one half, the quadrivium. Although the quadrivium was officially a large part of the curriculum in reality the four mathematical disciplines were paid little attention and hardly taught at all. This only began to change in the fifteenth century with the rise of astro-medicine or iatromathematics, to give it its formal name. With the rise of this astrology-based medicine the humanist universities of Northern Italy and Kraków introduced chairs of mathematics to teach astrology to their students of medicine. This of course entailed first teaching mathematics and then astronomy in order to be able to do astrology and thus mathematics gained a first foothold in the European universities. Ingolstadt became the first German university to introduce a chair for mathematics, also for teaching astrology to medical students, in the 1470s. It became an important centre for seeding new chairs at other universities with its graduates. Stabius and Stiborius going from there to Vienna with Celtis, for example. However there was no systematic introduction of mathematics into the university curriculum as of yet, this would first come as a result of the Reformation and the educational reforms of Philip Melanchthon.

Melanchthon was born Philip Schwartzerdt in Bretten near Karlsruhe on 16 February 1497. A great nephew of Johann Reuchlin a leading humanist scholar Philip changed his name to Melanchthon, a literal Greek translation of his German name, which means black earth, at Reuchlin’s suggestion. Melanchthon was a child prodigy who would grow up to be Germany’s greatest humanist scholar. He studied at Heidelberg University where he was denied his master degree in 1512 on account of his youth. He transferred to Tübingen where he came under the influence of Johannes Stöffler, one of those Ingolstadt graduates, a leading and highly influential mathematician/astrologer.

The cosmograph Sebastian Münster was another of Stöffler’s famous pupils. Stöffler also has a great influence on several of the Nürnberger mathematician-astronomers, especial Johannes Schöner and Georg Hartmann. Under Stöffler’s influence Melanchthon became a passionate supporter of astrology.

On Reuchlin’s recommendation Melanchthon became professor of Greek at Luther’s University of Wittenberg at the age of twenty-one and thus a central figure in the Reformation. One of the major problems faced by the reformers was the fact that the education system was totally in the hands of the Catholic Church, which meant that they had to start from scratch and create their own school and university system; this task was taken on by Melanchthon, who became Luther’s *Preceptor Germania*, Germany’s Schoolmaster.

Because of his own personal passion for astrology Melanchthon introduced mathematics into the curriculum of all the Lutheran schools and universities. He invented a new type of school on a level between the old Church Latin schools and the universities that were devised to prepare their pupils for a university education. The very first of these was the Eigidien Oberschule in Nürnberg, which opened in 1526 with Johannes Schöner, as its first professor for mathematics.

These type of school created by Melanchthon would become the Gymnasium, still today the highest level secondary schools in the German education system.

In Wittenberg he appointed Johannes Volmar (1480-1536) professor for the higher mathematic, music and astronomy, and Jakob Milich (1501- 1559) professor for the lower mathematic, arithmetic and geometry, in 1525. Their most famous students were Erasmus Reinhold, who followed Volmar on the chair for higher mathematics when he died in 1536, and Georg Joachim Rheticus, who followed Milich on the chair for lower mathematics, in the same year when Milich became professor for medicine. Schöner, Reinhold and Rheticus were not the only mathematicians supported by Melanchthon, who played an important role in the dissemination of the heliocentric astronomy. Although following Melanchthon’s lead these Protestant mathematicians treated the heliocentric hypothesis in a purely instrumentalist manner, i.e. it is not true but is mathematically useful, they taught it in their university courses alongside the geocentric astronomy.

As a result of Melanchthon’s passion for astrology the Lutheran Protestant schools and universities of Europe all had departments for the study of mathematics headed by qualified professors. The Catholic schools and universities would have to wait until the end of the sixteenth century before Christoph Clavius did the same for them, although his motivation was not astrology. Sadly Anglican England lagged well behind the continent with Oxford first appointing professors for geometry and astronomy in the 1620s at the bequest of Henry Savile, who had had to go abroad to receive his own mathematical education. Cambridge only followed suit with the establishment of the Lucasian Chair in 1663, whose first occupant was Isaac Barrow followed by that other Isaac, Newton. In 1705 John Arbuthnot could still complain in an essay that there was not one single school in England that taught mathematics.

I’m surprised mathematics had gotten so neglected in its time. I understand it being less favored since, particularly, arithmetic was cumbersome. But geometry and astronomy and music seem like they wouldn’t be harder, or easier, to study in the 13th century than in the 17th.

Thank you for the excellent posts.

While the point about medicine and astrology is important and well made, and Melachthon’s influence is undeniable, I hope I might be allowed to make a few counter-observations on mathematics at medieval universities, especially in England.

Firstly, I would note that the large number of manuscripts of Sacrobosco’s De sphera, various Algorimus (arithmetic textbooks) and similar suggest that maths was being taught at the universities from the early 13th century onwards. Music was the neglected subject of the quadrivium (having more to do with harmonics and ratios) but the others were covered, at least to the basic level of the textbooks mentioned above. They were being taught by regent masters who had the job of lecturing to undergraduates, usually while they were studying for higher degrees themselves. Hence, there was no need for dedicated professors of mathematics, or even the subjects of the trivium, which the regent masters also taught.

Syllabuses for Oxford survive back to 1268 (from memory) and include the quadrivium throughout. For example, the 1431 syllabus requires Arithmetic: Boethius: 1 term; Geometry: Euclid, Alhazen or Witelo: 2 terms; Music: Boethius: 1 term; and Astronomy or Theoricum planetarum or Ptolemy: 1 term. Of course, we should take the authors cited with a hefty pinch of salt as teaching was really from textbooks. No medieval undergraduate was expected to get to grips with Ptolemy in the raw.

By the late fifteenth century, the state of the evidence at Oxford and Cambridge becomes good enough to tell us exactly who was doing the teaching. The university began to pay regent masters £4 a year to lecture, with stipends going to a mathematics lecturer as well as those on philosophy and the trivium. One of the earliest maths lecturers was Robert Collingwood whose unpublished textbook on arithmetic survives in manuscript at Corpus Christi College Oxford. It shows he taught arithmetic, multiplication, roots, and basic algebra. A bit later in the 1520s, Cuthbert Tunstall’s arithmetic was published in London and reprinted several times across the continent in the next few decades. As a maths textbook it is really quite good, turning up frequently on university syllabuses thereafter.

In summary, I think it is a mistake to assume the lack of professor’s chairs meant nothing of maths was being taught. In fact, mathematic education, while basic, was widespread from the thirteenth century onwards.

Rather more on 16th century maths at Oxford and Cambridge was in the PhD thesis here: https://www.repository.cam.ac.uk/handle/1810/218820

Is it possible to get a copy of the full text of your thesis James?

Hi Thony, you can download it free from the Cambridge website linked to above. By the way, I saw the early globes at the Germanic History Museum in Nuremberg today. I was very excited to see that the circumference of the earth is understated on the earlier one from 1492, just like Columbus believed.

“…these Protestant mathematicians treated the heliocentric hypothesis in a purely instrumentalist manner, i.e. it is not true but is mathematically useful…”

Now that’s funny.

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