Although students encountered music early in their training, in none of the medieval definitions of the liberal arts was music presented first amongst the arts of the quadrivium. Instead, as Cassiodorus described, ‘Secular writers maintain that arithmetic is the first mathematical discipline because arithmetic is essential to explain the excellences of music, geometry and astronomy’. The fundamental role of numbers in three of the arts was expressed in a diagram at the end of Boethius’ treatise On Arithmetic. Here geometry, arithmetic, harmony and musical consonance were associated through their use of proportions based on the numbers 12, 9, 8 and 6. A version of this diagram was included in Thierry of Chartres’ Heptateuchon, at the end of Boethius’ text (Chartres, Bibliothèque Municipale MS 498, fol. 113v). As was typical in such diagrams, the numbers were linked by arches reflecting different proportions. However by the twelfth century this display of theoretical unity around numbers may have represented an adequate definition of these arts for many students. In addition to Boethius’ text, only eleven folios of Thierry's volume were given over to arithmetic. These contained Martianus Capella's allegorical account of the art from his Marriage of Merucry and Philology and brief sections of works by Gerbert of Aurillac and Euclid. Moreover Peter Abelard was not ashamed to admit his ignorance of arithmetic, a theme taken up in a twelfth-century story that claimed that he abandoned his study of mathematics because it proved too difficult. Similarly in his allegorical poem on the arts Godfrey of Saint Victor declared that having sipped from arithmetic's stream he quickly abandoned his study of the art. Yet Godfrey's account also described a number of applications for arithmetic. He declared, ‘Sitting on that bank there's a varied delegation’ including computists (computiste), algorists (algoriste) and the one who guards money boxes (fenoris incubator ciste). Arithmetic thus encompassed a range of skills valuable both inside and outside the classroom. Closely related to arithmetic was the art of geometry, which also found wide application. The fragmentary surviving material by Euclid was often associated with this study, but a text attributed to Boethius declared that the art's use was threefold: in construction or architecture, medicine and the training of the soul.