The ?sub-Euclidean? geometry of the earlier middle ages, up to the mid-twelfth century

Abstract

The problem of the transmission of Euclid has tended to stand at the centre of investigations into the geometry of the eleventh and twelfth centuries and rightly so. As M. Clagett puts it, "the question of the translation into Latin of Euclid's Elements in the Middle Ages can be resolved into two principal sub-questions"1 He refers to the survival of older translations from the Greek and to the making of fresh translations from the Arabic in the twelfth century. 2 A third question, which lies outside the scope of his paper, is that of the independent survival of a residue of "suo-Euclidean" geometry side by side with the Euclid material. That is not to suggest that any serious alternative to the Euclidean geometry was offered, but merely that a good deal of very simple geometrical knowledge was clearly being transmitted by writers who had, in many cases, not read Euclid for themselves. Important though the twelfth century translations of Euclid were to be for the future, it was these simpler versions which exerted a perhaps greater immediate influence, if only because they seem to have been more widely read, and read by scholars not primarily interested in mathematical topics. A number of late classical and early mediaeval sources contain some account of geometry: Macrobius' Commentary on the Dream of Scipio,3 Martianus Capella's De Nuptiis Philologiae et Mercurii,4 Cassiodorus in the Institutions5 and Isidore in the Etymologiae,6 Boethius in the Arithmetica1 and the commentary on the Categories. 8 Chalcidius' commentary on Plato's Timaeus provides