Medieval Mathematics

detail Euclid text 888AD

detail from MS d’Orville 301. (courtesy Clay Museum)

“The Renaissance Mathematicus” is one of my favourite blogs, but occasionally just like everyone else,Thony Christie gets a little too caught up with theoretical constructs and then comes a cropper because the general idea sounds fine, but the facts beg to differ.

Even on the rare occasions when this happens on Thony’s blog (and they are very rare), some of the corrective comments just make it better

One such was added recently (here) by James Hannam, who included a link to his  doctoral thesis, written in 2008. (see details at end). Hannam has also published a book  about the true state of scientific studies in medieval England, one which has received enthusiastic reviews from the popular and from the more staid sort of journal – not very easy to achieve.  (see here)

His thesis  opens with a quotation from late sixteenth century England that instantly recalled a letter written in seventeenth century Prague, a lament along the same lines. But the English text runs:

“Arethmetike, musicke,  geometrie and astronomie, and with them all skill in perspectives, are now smallie regarded.” – William Harrison in his Description of England (1577)

The point in relation to MS Beinecke 408 is that its containing numerous folios generally agreed to relate to astronomy means that to understand it, one would need a fair level in arithmetic and geometry, even supposing the work is not about astrology. (which is not to suggest that the text in MS Beinecke 408 is arithmetic, but you never know).

So late as the end of the sixteenth century, still, a very small number of texts were studied in relation to astronomy and mathematics – which is handy for us, isn’t it? . Hannam says:

The de Sphaera of Proclus or of Sacrobosco; the works of Boethius on mathematics; and otherwise the writings of Duns Scotus are pretty much it –  even in Oxford, and even in the late sixteenth century.

Among the interesting points brought up in Hannam’s thesis is a distaste for astrology in France and elsewhere  – as a rule.

Thus he notes that,

“In 1535, Michael Servetus was unable to persuade even the physicians of Paris that astrology was essential to their profession. The Cologne Theology Faculty ordered the local astrologers Hartungus (fl. 1488) and Johann Lichtenberger (fl. 1492) to desist from their practices, as it considered them altogether too ignorant to study such dangerous things. The latter was even to be investigated by a local inquisitor.” (Hannam p.29)

Hannam’s thesis is chiefly concerned with England during the  sixteenth century and later, but his first two chapters refer to the period before the 1530’s.

In his comment to the Renaissance Mathematicus (which I hope it’s alright to excerpt here),  Hannam said among other things 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 … but the others …. were being taught…

… the 1431 syllabus requires Arithmetic: Boethius… Geometry: Euclid, Alhazen or Witelo… Music: Boethius…and Astronomy or Theoricum planetarum or Ptolemy:….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 university began to pay regent masters …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 … mathematic education, while basic, was widespread from the thirteenth century onwards.”


which reminds me of another hypothesis-defying incident – the curious case of the mathematician Leon:

Leon the mathematician

Just for the record –  and as I’ve said before – nothing in MS Beinecke 408 appears to me to relate to astrology; Lynn Thorndike saw nothing reminiscent of any work of magic, science or pseudoscience; Adam McLean saw nothing in the imagery related to the western or the Arabic alchemical traditions.

Though if you have a theory you really like, and which can’t co-exist with these opinions, I daresay you won’t be much bothered by them.


1. James Hannam, “Teaching Natural Philosophy and Mathematics at Oxford and Cambridge 1500-1570” which can be accessed and downloaded as a pdf (here).


2. In case you haven’t heard of it, too, there is a marvellous  old two-volume history of mathematics that is still well worth having by you. David Eugene Smith, A History of Mathematics (2 vols.)  It was reprinted in a Dover edition, and I recently re-acquired a copy myself.  (I see it is now available on the internet archive (vol. 1  and vol. 2)

2b… along with what seems to be the first volume (only) of Florian Cajori’s A History of Mathematical Notations (vol. 1).

3. For another fascinating mathematics-centred blog try Johnathan Crabtree’s (here).


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