Natural Science and the Liberal Arts in Abbo of Fleury's Commentary on the Calculus of Victorius of Aquitaine

NATURAL SCIENCEAND THE LIBERALARTS IN ABBO OF FLEURY'SCOMMENTARY ON THE CALCULUS OF VICTORIUS OF AQUITAINE by G. R. Evans and A. M. Peden At the end of his Quaestionesgrammaticales, Abbo of Fleury quotes Virgil's obser- vation: .'Constat nimirum quia 'numero deus impare gaudet'" (Ecl. 8.75). He says little about the implications of this, and excuseshimself from doing so by explaining that he has adequately covered the matter in a little book on number, measure, and weight which he wrote about the Calculus of Victorius of Aquitaine} Only the pro- logue of this treatise2and some shott extracts on practical calculation3 and on fractions and weights4 are in print. But the work is of considerable general interest, beyond what it tells us of the remarkable range of Abbo's learning; it demonstrates the degree of competence at which it was possible, before the end of the tenth century, to apply the technical methods and terms of one of the liberal arts to another, and in particular, it shows what could be achieved by a dialectical approach to arithmetic. I. THE COMMENTARY AND ITS SOURCES The chronology of Abbo's works is by no meansclear. His Commentary on the Calculusmust have been written beforehis work on grammar,since the latter refers to the Commentary. Cousin dates the Commentary on the Calculus, with the @ 1985 by The Regentsof the University of California 0083-5897/85/010109+ 19Si.oo 'Abbo of Fleury, Quaesttonesgrammaticales 48; ed. A. Guerreau-Jalabert(Leiden 1982)271-273. 2Ibid. 50 (ed. 275). For Victorius's preface,see Appendix below. IN. Bubnov,ed., Gerberti Opera mathematica(Berlin 1899) 199-203, 299. 4W. Christ, "Uber dasArgumentum calculandides Victorius und dessenCommentar," Sitzungsbe- richte der bayenschenAkademie der Wissenschaften,phil.-hist. KI. (Munich 1863)100-152. A. van de Vyver, "Les oeuvresinedites d' Abbon de Fleury," Relluebenedictine 47 (1935) 139-140gives notices of manuscriptsand printed texts. The extant manuscriptsare asfollows: Berlin, DeutscheStaatsbibl. Phil!. 1833(Rose no. 138),from which folio referenceshere aretaken and henceforwardreferred to asF; Vatican Library Reg. Lat. 1281; Bamberg, StaatlicheBibl. H.J. IV, 24; Cusa,Hospitalbibl. 206; Karlsruhe, Lan- desbibl. K., 504; Brussels,Bibl. royale10078-95; Vienna, Nationalbibl. 2269. An edition of the whole text is now being preparedby A. M. Peden. 110 G. R. EVANS AND A. M. PEDEN treatiseson the syllogism,to before A.D. 985, when Abbo was still at Fleury, and before he went to England at the age of about forty-five.s Van de Vyver sawa shift in Abbo's attentions from the scientific studies of the quadrivium to the grammarand logic of the trivium, when Abbo was in his early forties (in the mid-980s),6although Abbo certainlygave instruction on astronomy and computusto the monks of Ramsey,to whom he was sent asteacher in 986-988, and he wrote letters on the Dionysiancycle in 1000and 1004.7The Commentaryon the Calculuswould form a convenientbridge betweenthe two spheresof his in- terests,for his inquiry is by no meanslimited to strictly arithmeticalproblems. Although Victorius's prefaceto the Calculusis very brief, Abbo's Commentary upon it is discursiveand wide-ranging,including citations from classicalliterature. He goesbeyond the simple explanationof words and phrasescharacteristic of many glossesand commentarieson the textbooksof the arts before the eleventhcentury, developinghis points at some length. In this, he wastrue to his own view of the commentator, whose role he discusseswhen he explains Victorius's use of the verb commentor in the phrase "tale argumentum antiqui comment;sunt." Abbo equatescomment; sunt with" invented" (jinxerunt), and explains that commen- tators elucidate truths which are "wrapped up in obscureideas" (uen/atem aliquo modo obscunssententits ;nuolutam) by inventing' 'fictions" which are likenessesof the truth, and theseare called "commentaries."sThese features make the Commen- tary a much richer sourceof information on the rangeand depth of Abbo's learning than his works on the syllogism,9for example, which are distinguished chiefly by their economicaltechnical treatment, and whosewell-defined structureowes much to the availability of Boethius's monographson the categoricaland hypothetical syllogismsas models. Arithmetic, however,was a little-known subjectin Abbo's day, and he realized the need' 'to build a bridge of introduction to arithmetic in the form of an exposition" (sub expositionis tenore ad arithmeticam introduction;s pontem construo),io Accordingly,before he beganhis detailed discussionof the actual text of the Cal- culus, Abbo discussedin a complete tractatusthe question of "number, measure, and weight" to which he refers in his Quaestionesgrammaticales. The sourceswhich Abbo used for his expositionof the Calculuscorrespond to the three principal spheresof his investigation: arithmetic, dialectic, and cosmology. 'P. Cousin,Abbon de Fleury-sur-Loire(Paris 1953)215. 6Vande Vyver 164. 7Ibid. 149-150, 154-155, 164; Cousin 65-73; 84-89. of, fol. 14ra-b. On William of Conches'sdistinction betweencommentum, which expoundsonly the generalmeaning (sententia)of a book, and glasa, which dealswith the detailed analysisof the text, see Glosaesuper Platonem 10, ed. E. )eauneau (1965) 67; )eauneau, "Gloses sur Macrobe: Note sur les manuscrits,"Archives d'histoire doctn.naleet litteraire du moyenage 7 (1960)26-27; idem, "Deuxredac- tionsdes glosesde Guillaume de Conchessur Priscien,"Recherches de theologieancienne et medlcvale27 (1960)223-224. 9Ed. A. van de Vyver, Abbonis FlorillCensisOpera inedita 1 (Bruges1966). 'oF. fol. 7va. ABBO OF FLEURY'S COMMENTARY 111 Fleury was an important center for texts relating to natural science,lland Abbo's early astronomicaland computistical output showsthat he had benefited from the sourcesknown there. Macrobius'scommentary on Cicero's Somnium Scipioniswas to be found at Fleury from the ninth century,and wasused by Abbo in his Compu- tus,I2and later in the Commentary on the Caicu1US.13Calcidius's Commentary on the Ttmaeuswas alsoused by Abbo in his Commentary}4There are fewersurviving manuscriptsof Calcidius'sCommentary from the ninth and tenth centuries than of Macrobius's,but four out of five of thesewere written in northern Franceand so could have beenavailable to Abbo}' For all its excursionsinto philosophyand dialectic, Abbo's Commentarywas closely linked with the scientific sphereof his work, and the Commentarycirculated with his computisticalworks and texts of other authors on astronomyand music}6 In East Berlin, Deutsche StaatsbibliothekMS Phill. 1833(Rose no. 138),a manuscriptprobably put togetherunder the supervision of Abbo himself, the Computusand Commentaryare found togetherwith diagrams and excerptsfrom Macrobius'sCommentary. For basic arithmetic, Abbo used stan- dard texts: Boethius, De anthmetica; Martianus Capella, De nuptits Phtlologiaeet Mercuni' book 7; and Isidore of Seville,Etymologiae; and he alsoquoted two verses from pseudo-priscian,Carmen de pond ere et mensura.Whether or not Abbo, like his contemporaryGerbert, knew the more advancedtreatises of Boethius on Aristo- tle,I7he makesthe fullest use of Boethius'sCommentaries on the Categonaeand De interpretatione, of Cicero's Topicaand De dillisione,and of Marius Victorinus's De definitione. II. CONTEMPLATION AND THE UBERAL ARTS Van de Vyver,who did so much to put Abbo and his world on the medieval map, once attempted to plot the stagesof the scientific developmentof the Middle Ages. He sawthe first stage,up to the end of the Carolingianperiod, asone in which the study of the Bible predominatedover science; the second,the tenth-centuryworld of Abbo and Gerbert, where sciencewas taught alongsideother arts suchas grammar "Van de Vyver (n. 4 above) 145-146, 148-149. For a collection of excerptsfrom Pliny and other sourcesknown at Fleury, seeV. H. King, ..An Investigationof SomeAstronomical Excerpts from Pliny's Natural History Found in Manuscriptsof the Earlier Middle Ages," B.Litt. thesis(Oxfqrd Univ. 1969) 127-128. lIB. C. Barker-Benfield,"The Manuscriptsof Macrobius' Commentaryon the Somnium Sclpionis," D.Phil. thesis(Oxford Univ. 1975) 1.87, 112; A. M. White. .'GlossesComposed before the Twelfth Cen- tury in Manuscriptsof Macrobius'Commentary on the Somnium Scipionis,"D.Phil. thesis(Oxford Univ. 1981) 1.7-9,60-62,105,121,141,167. uSee below after n. 28, and at nn. 47 and 56. 14Seebelow at n. 27. I 5Calcidius, Commentanusin Timaeum,ed.]. H. Waszink (London 1962)cx, cxvi, cxx, cxxvi-cxxvii. '6Van de Vyver(n. 4 above)139-140. 17Ibid. 130-131. 112 G. R. EVANS AND A. M. PEDEN and rhetoric, but not really in conjunction with them; the eleventh century as a period in which newly-discoveredlogic and sciencewere made to work together; and the twelfth century asthe time when pure sciencewas emancipated as an indepen- dent study}8We are now better placed to discernin the late Carolingianworld of Eriugenaand Remigius of Auxerre signs of an increasingfamiliarity with dialectic, and an awarenessof its potential applicationto the other arts and to exegesis.There also seemsto have been a changein the late ninth century in the study of the sci- ences.Whereas the study of arithmetic and natural sciencein the eighth and early ninth centurieswas largely directed either to instruction in practicalelementary calculation and computus19or to the significanceof numbers and cosmologyarising from the study of Scripture,20Eriugena and Remigius approachedthe subject in a more technicalway. Remigius'scomments on book 7 (De arithmetica)of

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