[245] To remind the reader of Isidore’s notation Roman numerals are kept wherever he used them.
[246] The division into even, odd, and numbers sharing the characteristics of even and odd numbers goes back to Nicomachus. It is not a logical division, as the second class contains the third. See Gow, p. 90.
[247] Superflui, diminuti, perfecti.
[248] The examples are found in Du Breul. They do not appear in Arevalo.
[249] Cantor, Vorlesungen über Geschichte der Mathematik, vol. i, p. 521.
[250] The authenticity of the work on geometry that has been handed down under Boethius’ name is questioned. (See Cantor, ibid., pp. 536 et seq.) It contains the complete proof of only three of Euclid’s propositions. It also contains calculations of areas of geometrical figures. See edition of Friedlein (Leipzig, 1867).
[251] Cf. Martianus Capella’s definition: “Geometria vocor quod permeatam crebro admensamque tellurem eiusque figuram, magnitudinem, locum, partes et stadia possim cum suis rationibus explicare neque ulla sit in totius terrae diversitate partitio quam non memoris cursu descriptionis absolvam.” Eyssenhardt, 198, 30.
[252] The whole of Isidore’s De Geometria is here given, with the exception of a few passages that are untranslatable. It is given as a whole to enforce attention to the loss of the traditional content, partial or complete, which was so striking a feature of all the members of the quadrivium in early medieval times.
[253] Hujus ars disciplinae. Ars may be equal to ‘hand-book’ here.
[254] Schmidt, Questiones de musicis scriptoribus Romanis, imprimis de Cassiodoro et Isidoro (Darmstadt, 1899). This dissertation is in part an examination of the question whether the Roman writers associated music with grammar or the mathematical sciences in their enumerations of educational subjects. It contains a useful list of passages bearing on the seven liberal arts.