The Renaissance saw not only the introduction of new branches of mathematics, as I have outlined in the last three episodes in this series, but also over time major changes in the teaching of mathematics both inside and outside of the universities.

The undergraduate or arts faculty of the medieval university was nominally based on the so-called seven liberal arts, a concept that supposedly went back to the Pythagoreans. This consisted of the trivium – grammar, logic, and rhetoric – and the quadrivium – arithmetic, geometry, music, and astronomy – whereby the quadrivium was the mathematical disciplines. However, one needs to take a closer look at what the quadrivium actually entailed. The arithmetic was very low level, as was the music, actually in terms of mathematics the theory of proportions. Astronomy was almost entirely non-technical being based on John of Sacrobosco’s (c. 1195–c. 1256) *Tractatus de Sphera* (c. 1230). Because Sacrobosco’s *Sphera* was very basic it was complemented with a *Theorica planetarum*, by an unknown medieval author, which dealt with elementary planetary theory and a basic introduction to the cosmos. Only geometry had a serious mathematical core, being based on the first six books of *The Elements of Euclid*.

I said above, nominally, because in reality on most universities the quadrivium only had a niche existence. Maths lectures were often only offered on holidays, when normal lectures were not held. Also, the mathematical disciplines were not examination subjects. If a student didn’t have the necessary course credit for a mathematical discipline, they could often acquire it simply by paying the requisite tuition fees. Put another way, the mathematical disciplines were not taken particularly seriously in the early phase of the European universities. There were some exceptions to this, but they were that, exceptions.

Through out much of the Middle Ages there were no chairs for mathematics and so no professors. Very occasionally a special professor for mathematics would be appointed such as the chair created by Francois I at the Collège Royal in the 16^{th} century for Oronce Fine (1494–1555) initially there were only chairs and professorships for the higher faculties, theology, law, and medicine. On the arts faculty the disciplines were taught by the postgraduate masters. The MA was a teaching licence. If somebody was particularly talented in a given discipline, they would be appointed to teach it, but otherwise the masters were appointed each year by drawing lots. To get the lot for mathematics was the equivalent of getting the short straw. This changed during the Renaissance, and we will return to when and why below but before we do, we need to first look at mathematics outside of the university.

During the medieval period preceding the Renaissance, trades people who had to do calculations used an abacus or counting board and almost certainly a master taught his apprentice, often his own son, how to use one. This first began to change during the so-called commercial revolution during which long distance trade increased significantly, banks were founded for the first time, double entry bookkeeping was introduced, and both the decimal place value number system and algebra were introduced to aid business and traded calculations. As I said earlier this led to the creation of the so-called abbacus, or in English reckoning schools with their abbacus or reckoning books.

The reckoning schools and books not only taught the new arithmetic and algebra but also elementary geometry and catered not only for the apprentice tradesmen but also for apprentice artists, engineers, and builder-architects. It is fairly certain, for example, that Albrecht Dürer, who would later go on to write an important maths textbook for apprentice artists, acquired his first knowledge of mathematics in a reckoning school. This was a fairly radical development in the formal teaching of mathematics at an elementary level, as the Latin schools, which prepared youths for a university education, taught no mathematics at all.

The first major change in the mathematic curriculum on the European universities was driven by astrology, or more precisely by astrological medicine or iatromathematics, as it was then called. As part of the humanist Renaissance, astro-medicine became the dominant form of medicine followed on the Renaissance universities; a development we will deal with later. In the early fifteenth century, in order to facilitate this change in the medical curriculum the humanist universities of Northern Italy and also the University of Cracow introduced chairs and professorships for mathematics, whose principal function was to teach astrology to medical students. Before they could practice astro-medicine the students had to learn how to cast a horoscope, which meant first acquiring the necessary mathematical and astronomical skills to do so. This was still the principal function of professors of mathematics in the early seventeenth century and Galileo, would have been expected to teach such courses both at Pisa and Padua.

As with other aspects of the humanist Renaissance this practice spread to northwards to the rest of Europe. The first chair for mathematics at a German university was established at the University of Ingolstadt, also to teach medical student astrology. Here interestingly, Conrad Celtis, know in Germany as the Arch Humanist, when he was appointed to teach poetics subverted the professors of mathematics slightly to include mathematical cartography in their remit. He took two of those professors, Johannes Stabius and Andreas Stiborius, when he moved to Vienna and set up his *Collegium poetarum et mathematicorum*, that is a college for poetry and mathematics, this helped to advance the study and practice of mathematical cartography on the university.

Astrology also played a central role in the next major development in the status and teaching of mathematics on school and universities. Philipp Melanchthon (1497–1560) was a child prodigy. Having completed his master’s at the University of Heidelberg in 1512 but denied his degree because of his age, he transferred to the University of Tübingen, where he became enamoured with astrology under the influence of Johannes Stöffler (1452–1531), the recently appointed first professor of mathematics, a product of the mathematics department at Ingolstadt.

Melanchthon was appointed professor of Greek at Wittenberg in 1518, aged just twenty-one. Here he became Luther’s strongest supporter and was responsible for setting up the Lutheran Protestant education system during the early years of the reformation. Because of his passion for astrology, he established chairs for mathematics in all Protestant schools and university. Several of Melanchthon’s professors played important rolls in the emergence of the heliocentric astronomy.

The Lutheran Protestants thus adopted a full mathematical curriculum early in the sixteenth century, the Catholic education system had to wait until the end of the century for the same development. Founded in 1540, the Society of Jesus (the Jesuits) in their early years set up an education system to supply Catholics with the necessary arguments to combat the arguments of the Protestants. Initially this strongly Thomist education system did not include mathematics. Christoph Clavius (1538–1612), who joined the Jesuits in 1555, was a passionate mathematician, although it is not exactly clear where he acquired his mathematical education or from whom. By 1561 he was enrolled in the Collegio Romano, where he began teaching mathematics in 1563 and was appointed professor of mathematics in 1567. Clavius created an extensive and comprehensive mathematical curriculum that he wanted included in the Jesuit educational programme. Initially, this was rejected by conservative elements in the Society, but Clavius fought his corner and by the end of the century he had succeeded in making mathematics a central element in Jesuit education. He personally taught the first generation of teachers and wrote excellent modern textbooks for all the mathematical disciplines, including the new algebra. By 1626 there were 444 Jesuit colleges and 56 seminaries in Europe all of which taught mathematics in a modern form at a high level. Many leading Catholic mathematicians of the seventeenth century such as Descartes, Gassendi, and Cassini were products of this Jesuit education network.

By the beginning of the seventeenth century mathematics had become an established high-level subject in both Protestant and Catholic educational institutions throughout the European mainland, the one exception which lagged well behind the rest of Europe was England.

Well aware that the mathematical education in England was abysmal, a group of influential figures created a public lectureship for mathematics in London at the end of the seventeenth century. These lectures intended for soldiers, artisans and sailors were held from 1588 to 1592 by Thomas Hood (1556–1620), who also published books on practical mathematics in the same period. Other English practical mathematicians such as Robert Recorde, Leonard and Thomas Digges, Thomas Harriot and John Dee also gave private tuition and published books aimed at those such as cartographers and navigators, who needed mathematics.

In 1597, Gresham College was set up in London using money bequeathed by Sir Thomas Gresham (c. 1519–1597) to provide public lectures in both Latin and English in seven subjects, including geometry and astronomy. The professorships in these two mathematical disciplines have been occupied by many notable mathematical scholars over the centuries.

The two English universities, Oxford, and Cambridge, still lagged behind their continental colleagues, as far as the mathematical sciences were concerned. The first chairs at Oxford University for astronomy and geometry were the result of a private initiative. Henry Savile (1549–1622), an Oxford scholar, like many others in this period, travelled on the continent in order to acquire a mathematical education, primarily at the North German Universities, where several prominent Scottish mathematicians also acquired their mathematical education.

In 1619, he founded and endowed the Savilian Chairs for Astronomy and Geometry at Oxford. Many leading English mathematical scholars occupied these chairs throughout the seventeenth century, several of whom had previously been Gresham professors.

Cambridge University held out until 1663, when Henry Lucas founded and endowed the Lucasian Chair for Mathematics, with Isaac Barrow (1630–1677) as its first incumbent, and Isaac Newton (1642–1626) as his successor. Despite this, John Arbuthnot (1667–1735) could write in an essay from 1705 that there was not a single grammar school in England where mathematics was taught.

In the High Middle Ages the mathematical disciplines were treated as niche subjects on the medieval university. Throughout the Renaissance period this changed and with it the status and importance of mathematics. This change was also driven by the need for mathematics in the practical disciplines of cartography, navigation, surveying, astrology, and the emerging new astronomy; we will deal with these developments in future episodes. However, by the end of the Renaissance, mathematics had gained the high academic status that it still enjoys today.

Excellent post, thank you.

It’s sad to see how far behind the level of mathematical knowledge was the scholastic teaching in universities. If building had only been possible with university-educated architects, we’d have no Notre Dame de Paris, no ships, none of the marvellous fourteenth-century maps. Seamen wouldn’t have been able to calculate astronomical positions and geometries without instruments (as many did). And traders’ manuals show the level of mental arithmetic was very good among the sort of traders who dealt in overseas ports – they memorised the fluctuating rates of exchange, the varieties of different systems of weights and measures, and obviously did the conversions in their heads in the middle of a hectic bazaar environment, while simultaneously calculating tax owed in that city for each particular good, and space remaining in the hull… I wonder if the universities were so behind hand because they had previously considered such knowledge the preserve of menials.

I agree. Neither were universities at the forefront of natural philosophy during the early modern period. Perhaps they were too deeply embedded in the intellectual and social positions they had striven so long and hard to establish.

I wonder if calculation skills weren’t assumed because they were taught in places like cathedral schools, for instance. Certainly, advanced calculation skills were necessary to cast horoscopes, something physicians did regularly. Don’t forget that music as a university discipline was quite advanced in the medieval university (I only know the situation in England and France in any detail). Michael McVaugh’s work on pharmaceutical compounding in the medieval period demonstrates an exquisite knowledge of calculation, and of course who can forget the importance of knowing the proper day on which Easter fell? Anyway, food for thought.

Faye, it’s interesting that you mention horoscopes because when this came up in another context, one contributor pointed out that the majority of physicians didn’t need to do much calculation at all. They had set tables for the position of the moon etc., and the need for any sophisticated knowledge of astronomical calculation was much reduced. Sorry I can’t cite sources about this. Maybe Thony is in a better position. And of course it all depends on when, and where, we’re talking about.