Introductions to Boethius's “Arithmetica” of the Tenth to the Fourteenth Century

Isidore , Etymologiae, ed. Lindsay W. M. (Oxford, 1911), I. xvii. 21–28.

Halm C. , Rhetores Latini minores (Leipzig, 1863), and Keil H. , Grammatici Latini (8 vols, Leipzig, 1855–80).

On the study of grammar, see Hunt R. W. , “Studies on Priscian in the eleventh and twelfth centuries”, Mediaeval and Renaissance studies, i (1941–43), 194–231; ii (1950), 1–56. On dialectic, see Boehner P. , Mediaeval logic (Manchester, 1952); Henry D. P. , Mediaeval logic and metaphysics (London, 1972). On rhetoric in general, see Murphy J. J. , Rhetoric in the Middle Ages (Berkeley, California, 1974); and on the relation between rhetoric and dialectic see Dickey M. , “Some commentaries on the De inventione and the Rhetorica ad Herennium of the eleventh and twelfth centuries”, Mediaeval and Renaissance studies, vi (1968), 1–41. Haskins C. , Studies in the history of mediaeval science (Cambridge, Mass., and London, 1927) is still useful on quadrivium subjects. On Nicomachus, see Heath T. L. , A history of Greek mathematics (2 vols, Oxford, 1921) where Boethius's debt to this Pythagorean mathematician is explained. A more recent general history is that of Taton R. (ed.), A history of science. I: Ancient and mediaeval science (Paris, 1957). I have looked at some aspects of the revival of interest in quadrivium studies in “The influence of quadrivium studies in the eleventh and twelfth century schools”, in Journal of mediaeval history, i (1975), 151–64.

Friedlein G. , Boetii de institutione arithmetica; De institutions musica (Leipzig, 1867), 4, 1.6.

Corpus Christi College, Cambridge, MS. 352 f.2v.

Friedlein , op. cit. (ref. 4), 3, 15.

The text has doctrinabilis, possibly in error. On doctrina and its derivatives, see Marrou H. I. , “Doctrina et disciplina dans la langue des Pères de l'Eglise”, Bulletin du cange, ix (1934), 5–25.

A notable exception is the ‘Victorine’ gloss in C.L.M. 4643 f.100–107v, which borrows heavily from the early-twelfth century Parisian master, Hugh of St Victor.

This appears to be a thirteenth or fourteenth century manuscript, containing various mathematical items. On appellatio, cf. Boehner , op. cit. (ref. 3). 12.

Friedlein , op. cit. (ref. 4), p. 4. 1. 24 - p. 5, 1. 4.

f. 5.

Ed. Migne J. P. , Patrologia Latina (222 vols, Paris, 1844–64), vol. lxiv, 875–92.

Ibid., 891–910, printed among Boethius's works. On Victorinus's claim to authorship, see Hadot P. , Marius Victorinus: Recherches sur la vie et ses oeuvres (Paris, 1971).

Generale nomen is itself a technical term of grammar, meaning a noun which is the name “of many things”. Isidore gives animal as an example, in Etymologiae , I. vii. 5.

C.L.M. 18764 f.3v.

Patrologia Latina (ref. 12), vol. lxiv, 891–910, and 1098, in the course of Boethius's comments on Cicero's Topics, V, 25–8, have similar instances.

Friedlein , op. cit. (ref. 4), p. 8, lines 4–7.

Patrologia Latina, (ref. 12), vol. lxiv, 245–55.

Anselm St Abelard Peter borrow the example in the eleventh and twelfth centuries respectively. On the question of the circulation of a complete Latin text of the Categories, apart from the Lemmata provided by Boethius, see Minio-Paluello L. , “The genuine text of Boethius's translation of Aristotle's Categories”, Mediaeval and Renaissance studies, i (1941–3), 151–77.

Compare Patrologia Latina (ref. 12), vol. lxiv, 220.

Ibid., 216–19.

In Peter Abelard's Dialectics for instance the Praedicamenta or Categories come very early on. See Petrus Abaelardus Dialectica, ed. de Rijk L. M. (Assen, 1956). The point should perhaps be made here that once such works as Abelard's were in circulation, it was possible to study elementary dialectic from them, and not directly from Boethius. Garlandus is the author of a Dialectica which would serve this purpose, and there are other examples. See Garlandus Dialectica, ed. de Rijk L. M. (Assen, 1959); Logica modernorum, ed. de Rijk L. M. (2 vols, Assen, 1967).

Friedlein , op. cit. (ref. 4), 10, 15.

Patrologia Latina (ref. 12), vol. lxiv, 87–116.

Cassiodorus , Institutiones, ed. Mynors R. A. B. (Oxford, 1937), 112.

Patrologia Latina (ref. 12), vol. lxiv, 1039–1218 contains Boethius's commentaries on Cicero's Topics and his De differentiis topicis. Abelard devotes the third treatise of his Dialectica to ‘topics’: op. cit. (ref. 22), 253–468.

Murphy J. J. gives full details of the editions of some of the early material in his Bibliography of mediaeval rhetoric (Toronto, 1971).

Friedlein , op. cit. (ref. 4), 3, 6, 5, 14.

Munich C.L.M. 4643 f.100v, a thirteenth century manuscript.

Munich C.L.M. 13021 f.31v, a twelfth century manuscript.

Accessus ad auctores, ed. Huygens R. B. C. (Leiden, 1970) gives details of earlier work in this field in notes and bibliography.

This blurring of boundaries is not confined to the arithmetical accessus. Elements of the accessus are to be found in the first lines of some of the Commentaries on Boethius of the School of Thierry of Chartres, ed. Häring N. M. (Toronto, 1971).

The probable date of the manuscript makes it clear that this introduction and commentary must have been written at least before the end of the eleventh century. Unfortunately L. M. de Rijk's illuminating study, “On the curriculum of the arts of the trivium at St Call from c. 850-c. 1000”, Vivarium, i (1963), 35–86, is not greatly concerned with quadrivium studies there, and N. Bubnov gives few details of commentary material of the Carolingian and post-Carolingian period in his treatment of scholia to Boethius's Arithmetica in Gerberti Opera mathematica (Berlin, 1899). He does, however, note the scholium of Notger which occurs in this manuscript ff.78v–79r, ibid., p. 297.

Huygens , op. cit. (ref. 31), 1.

Quain E. A. , “The mediaeval Accessus ad auctores”, Traditio, iii (1945), 215–64.

Thorndike L. Kibre P. , A catalogue of incipits of mediaeval scientific writings in Latin (New York and London, 1963), col. 682, identifies it as a summary of Boethius's Arithmetica, but it does not follow the text of the Arithmetica in this manuscript, which ends on f.26, but the Liber ysagogorum Alchorismi translated by (?) Adelard of Bath and edited in Abhandlungen zur Geschichte der Mathematik, ii (1887), 1–27, which occupies ff.27–31v. It precedes the Astronomia of (?) William of Conches (cf. Thorndike and Kibre, 443 and 656). It is not at all clear to which of these the author of our ‘introduction’ addresses himself, since his description fits neither work very well. It certainly does not refer to Boethius.

f. 133. This differentiation of causae echoes the dialectical emphasis of the schools of the thirteenth and fourteenth centuries, to which this manuscript belongs.

Patrologia Latina (ref. 12), vol. clx, 217–18. There is some dispute over the relation between the accessus and works of this kind. See Quain , op. cit. (ref. 35), 217–18.

Patrologia Latina (ref. 12), vol. clxxii, 224–5.

Huygens , op. cit. (ref. 31), p. 47, lines 2–5, and p. 105, lines 1054–6.

As he is in the Bodleian Library, MS. Laud. Misc. 644 f. 133.

This is a thirteenth century manuscript.

Friedlein , op. cit. (ref. 4), p. 4, line 17 - p. 5, line 4.

Isidore , Etymologiae, III. i. 1.

Ibid., III. i. 2. Cf. Cassiodorus , Institutiones, p. 140, and Huygens , op. cit. (ref. 31), 129, lines 1086–90.

Ibid., p. 129, line 1809 where geometry is said to have come from the Egyptians.

Friedlein , op. cit. (ref. 4), p. 3, line 1.

Huygens , op. cit. (ref. 31), p. 48, line 48.

Ibid., p. 48. line 48, and p. 106, line 1095.

Ibid., p. 48, line 46, and p. 106, line 1094.

f. 2.

f. 132v, cf. Thorndike Kibre , op. cit. (ref. 36), col. 1567. The piece ends f. 139.

f. 127v. The reference is to the opening words of Book I. i. of the Arithmetica.

Ed. Curtze M. (Copenhagen, 1897).

Op. cit. (ref. 33).

Ralph of Laon , Liber de abaco, ed. Nagl A. , Abhandlungen zur Geschichte der Mathematik, v (1890), 96–133, p. 96.

Adelard of Bath, Regule abaci, ed. Boncompagni B. , Bulletino di bibliografia e di storia delle scienze matematiche e fisiche, xiv (1881), 1–134, p. 96.

Drawn from Hugh of St Victor, Didascalicon , VI. 14, Patrologia Latina (ref. 12), vol. clxxvi, 810.

Munich , C.L.M. 18764 f. 5v.