evans1985 Natural Science and Liberal Arts in Abbo of Fleury's Commentary on the Calculus of Victorius of Aquitaine.pdf

8/19/2019 evans1985 Natural Science and Liberal Arts in Abbo of Fleury's Commentary on the Calculus of Victorius of Aquitai…

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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 he 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, Quaesttones rammaticales 8; ed. A. Guerreau-JalabertLeiden 1982)271-273.

2Ibid. 50 (ed. 275). For Victorius's preface,seeAppendix below.

IN. Bubnov,ed., Gerberti Opera mathematica Berlin 1899) 199-203, 299.

4W. Christ, "Uber dasArgumentum calculandidesVictoriusund dessenCommentar," Sitzungsbe-

richte der bayenschen kademie der Wissenschaften,hil.-hist. KI. (Munich 1863)100-152.A. vande

Vyver, "Les oeuvresneditesd' Abbon de Fleury," Relluebenedictine47 (1935) 139-140givesnoticesof

manuscriptsand printed texts. The extant manuscriptsare as ollows: Berlin, DeutscheStaatsbibl.Phil .

1833 Rose o. 138), rom which folio references ere are aken and henceforwardeferred o asF; Vatican

Library Reg. Lat. 1281; Bamberg, StaatlicheBibl. H.J. IV, 24; Cusa,Hospitalbibl. 206; Karlsruhe,Lan-

desbibl. K., 504; Brussels, ibl. royale10078-95; Vienna, Nationalbibl. 2269. An edition of the whole

text is now being preparedby A. M. Peden.


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G. R. EVANS AND A. M. PEDEN

10

treatiseson the syllogism, o 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 n his early

forties (in the mid-980s),6although Abbo certainlygave nstruction on astronomy

and computus o the monks of Ramsey,o whom he was sent as eacher n 986-988,

and he wrote letters on the Dionysiancycle n 1000and 1004.7 he Commentaryon

the Calculuswould form a convenientbridge between he two spheresof his in-

terests, or his inquiry is by no means imited to strictly arithmeticalproblems.

Although Victorius's preface o the Calculus s very brief, Abbo's Commentary

upon it is discursiveand wide-ranging, ncluding citations from classicaliterature.

He goesbeyond he simple explanationof words and phrases haracteristic f many

glosses nd commentaries n the textbooksof the arts before he eleventhcentury,

developinghis points at some ength. In this, he was rue to his own view of the

commentator, whose role he discusseswhen he explains Victorius's use of the

verb commentor n the phrase tale argumentum antiqui comment;sunt. Abbo

equatescomment; sunt with invented (jinxerunt) and explains that commen-

tators elucidate ruths which are wrapped up in obscure deas (uen/atem aliquo

modo obscuns ententits nuolutam)by inventing' 'fictions which are ikenesses f

the truth, and thesearecalled commentaries. sThese eaturesmake he Commen-

tary a much richersourceof information on the rangeand depth of Abbo's learning

than his works on the syllogism,9 or example, which are distinguished chiefly by

their economical echnical reatment, and whosewell-defined structureowesmuch

to the availability of Boethius's monographson the categoricaland hypothetical

syllogismsasmodels. Arithmetic, however,wasa little-known subject n Abbo's day,

and he realized the need' 'to build a bridge of introduction to arithmetic in the

form of an exposition (sub expositionis enore ad arithmeticam introduction;s

pontem construo),io

Accordingly,before he beganhis detailed discussion f the actual ext of the Cal-

culus, Abbo discussedn a complete tractatus he question of number, measure,

and weight to which he refers n his Quaestiones rammaticales.

The sourceswhich Abbo used or his expositionof the Calculus orrespondo 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's istinction betweencommentum,which expounds nly the

generalmeaning (sententia)of a book, and glasa,which dealswith the detailed analysis f the text, see

Glosae uper Platonem 10, ed. E. )eauneau (1965) 67; )eauneau, Gloses sur Macrobe: Note sur les

manuscrits, Archivesd'histoire doctn.nale t litteraire du moyenage 7 (1960)26-27; idem, Deuxredac-

tionsdes gloses e Guillaume de Conches ur Priscien, Recherchese theologieancienne t medlcvale27

(1960) 23-224.

9Ed. A. van de Vyver, Abbonis FlorillCensis pera nedita 1 (Bruges1966).

'oF. fol. 7va.


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BBO OF FLEURY'S COMMENTARY

Fleury was an important center for texts relating to natural science,lland Abbo's

early astronomicaland computistical output shows hat he had benefited from the

sources nown there. Macrobius'scommentaryon Cicero's Somnium Scipioniswas

to be found at Fleury rom the ninth century,and wasused by Abbo in his Compu-

tus,I2and later in the Commentary on the Caicu1US.13alcidius'sCommentaryon

the Ttmaeuswasalsoused by Abbo in his Commentary}4 here are ewersurviving

manuscriptsof Calcidius'sCommentary rom the ninth and tenth centuries than of

Macrobius's,but four out of five of thesewere written in northern Franceand so

could have beenavailable o Abbo}' For all its excursionsnto philosophyand dia-

lectic, Abbo's Commentarywasclosely inked 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 Roseno. 138),a manuscriptprobably put togetherunder he supervision

of Abbo himself, the Computusand Commentaryare ound togetherwith diagrams

and excerpts rom 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,Carmendepond ere et mensura.Whether or not Abbo, like

his contemporaryGerbert, knew he more advancedreatises f Boethius on Aristo-

tle,I7he makes he fullest use of Boethius'sCommentaries n the Categonae nd De

interpretatione,of Cicero's Topicaand De dillisione,and of MariusVictorinus'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 stages f the scientific developmentof the Middle Ages.

He saw he first stage,up to the end of the Carolingianperiod, asone in which the

study of the Bible predominatedoverscience;he second, he 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 excerpts rom Pliny and other

sources nown at Fleury, seeV. H. King, ..An Investigationof SomeAstronomicalExcerpts rom 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. .'GlossesComposedbefore he Twelfth Cen-

tury in Manuscripts f Macrobius'Commentaryon the Somnium Scipionis, D.Phil. thesis Oxford Univ.

1981) 1.7-9,60-62,105,121,141,167.

uSeebelow after n. 28, and at nn. 47 and 56.

14See elow at n. 27.

I 5Calcidius Commentanus n Timaeum,ed.]. H. Waszink London 1962)cx, cxvi, cxx, cxxvi-cxxvii.

'6Van de Vyver n. 4 above)139-140.

17Ibid. 130-131.


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and rhetoric, but not really in conjunction with them; the eleventh century as a

period in which newly-discoveredogic and scienceweremade o work together; and

the twelfth century as he time when pure sciencewasemancipatedas an ndepen-

dent study}8We are now better placed to discern n the late Carolingianworld of

Eriugenaand Remigius of Auxerre signs of an increasing amiliarity with dialectic,

and an awareness f its potential application o the other arts and to exegesis. here

also seems o have been a change n the late ninth century n the study of the sci-

ences.Whereas he study of arithmetic and natural science n the eighth and early

ninth centurieswas argelydirected either to instruction n practicalelementary al-

culation and computus19 r to the significanceof numbers and cosmology rising

from the study of Scripture,20 riugena and Remigius approached he subject in

a more technicalway. Remigius'scommentson book 7 (De arithmetica)of Martia-

nus Capella's De nupttis Phtlologiaeet Mercuni show hat he had absorbed omeof

Boethius'sDe arithmetica,and he wasable occasionallyo use dialecticalconceptsn

his observations bout the nature of number.21But he wasalso heir to the tradition

of number-symbolismn cosmology,which simply cataloged he powersand mani-

festationsof numbers.z2 bbo takesup these wo threads, he one conservative,he

other more progressive, nd adds o them the fruit of his study of naturalscience.He

still sees is work within a broad Christian ramework, but the scientific contenthas

becomemore important in its own right, not leastbecause f the possibility of treat-

ing it systematically.

The conservative approach equired that the ultimate spiritual aim be kept in

view, he intellect being allowed ree play for the ascentwhich Abbo proposes,rom

the visible through the invisible, to the unchangingTrinity. Here Abbo is working

within an established radition, in which speculative heology s man's love of God

searching or the image of the Trinity in creation, and, most immediately, n man

himself.z3

'sA. van de Vyver, L'evolution scientifique du haut Moyen Age, Archeion 19 (1937) 19-20.

19As emanded by Charlemagnen A.D. 789-MGH Cap. 1 (1835)65-and supplied by, e.g., ps-

Alcuin, Propositionesad acuendosuvenes (PL 101.1145-1160);RhabanusMaurus, De computo (PL

107.669-728), ollowing the tsadition of Bede,De temporum ratione ed. C. W. Jones,CCSL123B 1977]

241-460).

2°E.g., RhabanusMaurus,De universo 9-11, 18 (PL 111.257-330;479-495); cf. M. Rissel, Rezep-

tion antiker und patristischerWissenschaft ei HrabanusMaurus, Lateinische Prache Ind Literatur des

Miltelalters 7 (Bern 1976)41-48, 276-277.

21Remigius f Auxerre, Commentan'usn Martz'anumCapellam,cd. C. Lutz, 2 (1965). e.g., 367.6

(ed. 181.4-16),367.9 (ed. 181.34-182.2).

22E.g.,on the significanceof the number seven:Remigius285.14 ed. 120.12-122.1).This approach

continued to be fruitful; see,e.g., Otloh of St. Emmeran,Dialogus de tribus quaestionibus34-42 (PL

146.103-119);cf. G. R. Evans, Otloh of St. Emmeranand the Seven iberal Arts, Recherches e the-

ologie ancienneet medievale44 (1977)29-54.

23See,or example,Augustine, De libero arbitno 2.3.7; Confess/ones3.11.12; De Trinitate, esp.

bk. 108.11, 10.13, 11.17; De civilate Dei 11.27-28.


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ABBO OF FLEURY'S COMMENTARY 113

Abbo adopts this framework. Since he sees his subject as number, measure, and

weight, the three-fold means by which God has ordered creation (Sap. 11.21), he

establishes from the start the significance of his discussions as an inquiry into the

nature of God, and His various manifestations in the three-fold aspects of man and

nature. Abbo asserts that the ascent to the Trinity is achieved through the love of

wisdom, which is the love of God; the love of wisdom is in some way imitated by the

three-fold power of the soul, which gives the powers of growth, of growth and sense,

and of growth, sense, and reason, respectively, to the three orders of living beings

(plants, animals, and man).24 Abbo does not attempt to establish an exact correspon-

dence between the three powers and the grades of perception required by each cate-

gory in the intellectual ascent. He is simply exploring similar triads suggested t~ him

by his knowledge of the Augustinian and secular traditions.

Abbo also quoted directly (though without acknowledgment) from Claudianus

Mamertus (d. ca. 474), whose De statu antmae was designed to defend Augustine's

teaching on the soul's incorporeality. In this work, Abbo found a discussion of the

nature of number, measure, and weight, and of the way in which they were to be

found in corporeal things and in the soul. This enabled him to draw together the

threads of his introduction to show how number, measure, and weight, as applied to

bodies, are related to the universal presence of the Trinity, from the Creator (the

Trinity itself), to the soul (the image of the Trinity formed inte//t'gibt'liter), to the

body (the vestige of the Trinity, formed uistbiliter). The soul is both one and three,

in its powers of memory, deliberation, and will; the body is also one and three, in its

constitution according to number, measure, and weight.25

Abbo drew not only on patristic thought, but also on the secular representatives of

late antique Platonism. In both traditions, knowledge of science had a metaphysical

dimension. The two traditions share a conception of unity, and in discussing unity,

Abbo felt free to draw not only on theological works, but also on cosmological ones.

At the beginning of his detailed exposition of Victorius's preface, Abbo launches

into a discussion of unity and multitude, so as to set it in the context of the divine

unity from which all things derive. The absolute simplicity of unity represents for

Abbo the notion of the substance of simple absolute being.26 The differentiation of

the multitude which proceeds from one suggests, further, the Platonic theory of

forms which impose individual natures on separate things. Abbo quotes a section of

Calcidius's Commentary on the generation of the World Soul, which proceeds from

divine unity, the source of numbers.27 He then introduces the lambda figure

(representing the World Soul's composition from unity, three odd and three even

UF, ol. 8rb; cf. Macrobius,Commentarii n Somnium Scipionis1.14.10-13,ed.). Willis (Leipzig1970)

57.7-25. This triad is not found in Augustine.

2'F, ol. 9ra-b. quoting ClaudianusMamertus. De statu animae 2.6 (CSEL11 [1885J119.5-25).

26Cf.Boethius. De hebdomadibus2-8; ed. E. Rapisarda Catania 1960)23-31.

27F. ol. 9rb-va, quoting Calcidius 39 (n. 15 above)ed. 88.12-89.2.


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numbers,which form arithmetical,geometrical, nd musical elationships).28e ends

by referring the reader o Calcidiusand Macrobius or further information. By draw-

ing on thesesources f Platonism,Abbo has woven nto his inquiry a philosophical

tradition which assumeshat the universehas an intelligible and rational structure,

an essentiallynumericalorderlinesswhich links cosmology loselywith arithmetic.

The similarity of the Christian and Platonic viewpointsallowed Abbo to preserve

certain freedom of speculation,within traditional limits.

Abbo approaches is taskon the assumptionhat there s nothing in secular uthors

which may not be useful, no branch of secular earning which ought to be avoided

by Christian scholars.He saysat the beginning that the value of studying number,

measure, nd weight ies in what s to be learned rom it about he Creator. t is, in

other words, material for contemplation.This view shapes is approach o mathe-

matics, although he is in a position to make use of more advancedmathematical

techniquesand concepts. n a similarway, Abbo uses raditional mageryand princi-

ples about Sapientiaasa means o approachogic (she esides n a housesupported

by the seven olumns of the liberal arts, hewn o adorn he temple of Solomon;wis-

dom is the contemplationof the divine, the perfectknowledgeof what is unchang-

ing and the complete understanding of truth; it inspiresvirtue and re-createshe

soul n the image of the Creator).29 ut the contemplativespirit of this introduction,

permeatedby the sapientialbooks of the Old Testamentand patristic theology, s

enrichedby Abbo's knowledgeof logic's method. He argues hat logic supports he

work of wisdom,and that rational argumentation rawson philosophy or its subject

matter.30 e mentions natural science physics)asone part of philosophy, inking it

to his proposedsubject(number, measure, nd weight) which is to be investigated

through the four disciplines of the quadrivium.3

On this basis, he methods proper to each art are brought together n Abbo's

approach o arithmetic: he is able to use he tools of dialectic o provide him with a

systemof argumentation,and to take some of his topics from natural science.

But Abbo is well awareof the need o distinguish carefully between he different

disciplines n order o proceedsystematicallyn his investigations. n a passage here

2°F, ol. 9va; cf. Calcidius32 (cd. 82); Macrobius1.6.46 n. 24 abovc)26.22-28. To roc thrcc typcsof

rclationship,Abbo addsastronomy,which is not in Calcidius,and looks ikc a spccious ddition to makc

up roc fourth disciplinc of roc quadrivium.

29F,ol. 7vb; cf. Cassiodorus,nshtuUones ,praef 2, cd. R. A. B. Mynors Oxford 1963)89; Alcuin,

De grllmmllut"1I PL 101.853); n gcncral, M.-Th. d' Alvcmy, "La Sagcssct scsscpt fillcs," in Meillnges

Felix Grllt 1 (Paris 1946) 245-246, 254-257.

3OF,ol. 8rb. Hc dcfincs logic in Ciccro's tcrms (Topit"1I2.6);cf. Bocthius. In top. Cit". (PL 64.1044-

1047).For Abbo, if not for Ciccro, thc divisionsof this dillgens rllUo disserendi thc invcntion of topics

and thc judgmcnt of proofs) wcrc primarily tools of logic. On the dialcctical approach o tcxtbooksof

rhctoric, sccM. Dickcy, "Somc Commentaries n the De inllenuone and Ad Herennium of thc Elevcnth

and Early Twclfth Ccnrurics," Medielllllllnd Renllissant"etudies6 (1968) 1-41.

31F,ol. 8va. For roc usc of natUralscicncc sa sourccof topics or dialcctic, cf. Ps.-Augustine,Det"em

CIItegorille 7-88, (cd. L. Minio-Palucllo, Amtoteles Iliunus 1.1.-5 [1961] 152.14-30),whcrc a discussion

of thc antipodcs s uscd to illustratc t"ontranetatesot"um.


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he discusseshe functioning of memory,he explains hat, if we possess n equal mas-

tery of all the fields of knowledge omnia panter comprehendimus uarum scientia

adepti sumus),we are able o call to mind whichever ody of knowledge s needed n

a given situation. Each,he says, comes fully and completely o mind wheneverhe

occasion emands (utraqueper seplena et integraratiocinanti occumt quotiens-

cumque occasionrepit).32 he application of this principle is seenclearly n Abbo's

handling of the common echnical erms of the disciplineswhich he brings to bear

on eachother. Notae, or punctuation marks, are, he says, Igna vocum; together

with letters hey' 'speak to the literate man, and they may thus be classifiedamong

the signs for words. But the nota or sIgnum s alsocalled he point (punctum).

So as o avoid confusion,we must bear n mind that the' 'point is the beginning or

end of the line in geometry.33When he explains he meaning of anthmetica disci-

plina in Victorius's preface,he warnshis readers hat they must be on their guard

againstmistaking one senseof the term for another. He also notes that thosewho

look carefullywill see hat the term arithmetic is equivocal nomen aequivocum),

and that it is necessaryo throw some ight on it so that the word will not be ambigu-

ous (ne fieret dictio ambigua). This ambiguity, Abbo points out, is called amphi-

bolia by the grammarians.34or, both the art of arithmetic and the artis mulier-the

personified Arithmetica-may be called arithmetic. It is for this reason hat the

word disctplina s added here, o make t plain that it is the art of arithmetic which s

intended.3~ ikewise,Abbo's frequent use of technical erms and distinctionsshows

that he wishes o employ the languageof the arts to the full (into the few lines on

an'thmeticadisciplinaAbbo introduces he terms genus,supponere, pecies, omen,

aequivocum, dictio, ambtgua, grammaticus,amphibolia, denominatio),but he in-

sures hat they are used with meticulous correctness,arefullypointing out any dif-

ferencesof usagewhere the terms are proper to more than one discipline.

III. THE INTERACTION OF THE ARTS

The new rigor of Abbo's approachwasdue not only to the stimulation provided by

new texts, although his was o be an important factor n the increasen speculative

writing about science. The more advanced ogical treatises,and the works on the

abacusand on geometrywhich Gerbert discovered,began o circulate only in the

32F,ol. 19rb; cf. Augustine, Coni 10.11-12.

33F,ol. 15va.b; cf. Calcidius 32 (n. 15 above)ed. 82.1-5, quoted by Abbo, F, fol. 9va.

34Cf.eg., Charisius, nstitutiones grammaticae4 (ed. H. Keil, Grammatici atini 1 [Leipzig 1857]

271.26-32); DonatUs,Ars grammalt'ca .3 (Keil4 [1864]395.20-26).The term is not, apparently,used n

Priscian, nst. Gramm.

35F,ols. 10vb-llra; cf. Boethius, n Categ.AIist. 1 (Pi 64.168)on appellaltoaequivoca:musica s the

sameword as hat used or a musicalwoman, (mulier) musica.Abbo gives sidore's derivation of disci-

plina and mathematica Etymologiae 10.66, 3.1).


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very late tenth and early eleventh century.)36 It was also a case of looking at old texts

in a new way; the sources of arithmetic and natural science on which Abbo drew

were known to the late Carolingian scholars, but the confidence, freshness, and

flexibility with which he used them made them of much wider value in the study of

the artes in general. Of course, this brought its own problems, for the interaction of

the arts became increasingly controversial as the practice of them became more

sophisticated. But the burning issue of the day became the extent to which dialectic

could legitimately be applied to grammar and theology. Abbo attempted compari-

sons of method and subject matter in his own field similar to those which Gilbert of

Poitiers would make, albeit with more sophistication, in the first half of the twelfth

century.37

These comparisons make use of most branches of secular studies. The trivium

makes its appearance at the beginning of the Commentary in the formal introduc-

tion (accessus) hich Abbo provides after his opening words about the circumstances

and intention of his own work.38Abbo's senseof scholarly propriety has been strongly

developed by his studies, and it offends him a little that Victorius breaks the usual

rules of rhetoric by not giving a formal opening (the Calculus begins abruptly with

a statement about the nature of unity).39 Rhetorical correctnesswould dictate certain

procedures to be followed in the exordium of a piece of writing, and an author ought

to be careful to capture the goodwill, attention, and receptiveness of his readers and

listeners.4o Later in the Commentary Abbo, using traditional criteria, analyzes how

Victorius does in fact seek o win his readers' attention, even if he does not do so in a

formal proemium.)41 Then he notes Victorius's intentt o, or purpose, which he re-

lates to the terms of his Tractatus: It is to ensure correct calculation when dealing

with matters of number in the quadrivium, the arles quae numerorum ratione con-

stant, or with any question of measure or weight. The utilitas of the Calculus, even

for the novice, is therefore evident, since it elucidates the fundamental nature of

things: for Omnia creata stint in numero mensura et pondere.42 Abbo is concerned

~6See . van de Vyver, Les erapesdu developpementphilosophique du haut Moyen Age, Revue

beige dephtlologie et d'histoire 8.2 (1929)425-452; C. Thulin, Zur Uberlieferungsgeschichtees Cor-

pus Agrimensorum, Goteborgs ungl. Vetenskaps-Och itterhets.,Handl. 14 (1911)3-68; M. Folkerts,

ed., Boethius Geometrie I (Wiesbaden1970)69-81,95-104; Gerbert, Ep. 8, ed. F. Weigle, MGH

Briefe tier deutschenKaiserzeit 2 (1966) 30-31.

~7TheCommentaries n Boethiusby Gilbert ofPoi hers,ed. N. Haring (Toronto 1966),e.g. 189-190.

~.On accessus,eeBoethius,De differentiis topicis PL 64.1207); Conradof Hirschau,Dialogus super

auctores19,ed. R. B. C. Huygens Leiden 1970)78; E. Quain, The MedievalAccessusad uctores, Tra-

ditio 3 (1945) 215-264; R. W. Hunt, Introductions to the 'Artes' in the Twelfth Century, in Studia

medievalia n honorem R.J. Martin (Bruges 1949)85-112.

~9F,ol. 7va.

4°F, ol. 7va-b; cf. Cicero, De tnventione 1.14.19-15.20.

41F, ol. 14ra; derived from Cicero, except or Victotius's captaho benevolentiae hrough humility:

he attributed the systemof calculation o the ancientsand not to himself (cf. Cicero, De inventione

1.16.22).

41F. ol. 7vb.


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ABBO OF FLEURY'S COMMENTARY

117

to locateVictorius's Calculusand thereby,his own work) in the structureof academic

study; the rhetorical tradition has helped him to do so.

Grammar s not mentioned formally in the preface,and it makes ts appearance

chiefly where it touches on dialectic. For example,Abbo explains that the words

unus and unitas are like magnusand magnitudo, in that one is derived from the

other; the technical erm he uses o describe heir relation s denominatur. Denomi-

natilla were to becomean especially ontentious ssue n the courseof the eleventh

century preciselybecausehey raisedproblems on the borderline betweengrammar

and dialectic. 43Abbo emphasizeshe technical erm here becauset is the proper

one, and one of his major concernswas o give rigor to his subjectand instruction o

his readerby the use of correct erminology.

Already in Abbo' s day medievalscholars ad realized he potential of dialectic as

an intellectual ool. Here was o lie the growing point of the work of several enera-

tions to come. It is not surprising o find some of Abbo's most technicallyadvanced

observations bout arithmetic being drawn from dialectic. But the combination of

gifts which Abbo and his contemporaryGerbert of Aurillac possessed as are. Few

scholarswereable to work with equal facility in dialecticand arithmetic. Evena cen-

tury later, Abelard says hat he heard ectures n which arithmeticalprinciples were

comparedwith the teaching of dialectic about the categoryof quantity, but claims

that he himself has no arithmetical ability and little knowledge of the subject.44

But Abbo wasable o explore he possibilitiesof comparison nd interactionbetween

arithmetic and dialectic in a knowledgeableway. Thesepossibilities all most con-

veniently nto three groups: commonground of sharedconcepts, ommonground of

technical vocabulary,and common ground of method.

IV. SHARED CONCEPTS

In Abbo's Commentary, t was principally the conceptof .'composition in arith-

metic which stimulated Abbo to draw on his knowledgeof other disciplines, n this

casecosmology, or parallels. Abbo writes that things can be composite either by

nature or by will. Those which are composite by nature. and which increase n

a properly egulatedway (si augmentumsui legitimaprogressione apiant),are dis-

solved in exactly he sameway as hey were composed. he samewill be found to

be true of compositesby will, if they are made systematically.45

43F,ol. 9va. On denominatives, eeD. P. Henry, TheLogic oiSI. Anselm Oxford 1967)31-116; on

grammarand dialectic n general,seeR. W. Hunt; Studies on Priscian n the Eleventhand Twelfth Cen-

turies, Mediaeva/and Renaissance ludies 1 (1941-1943)194-231.

Peter Abelard, Dialectica,ed. L. M. de Rijk (Assen 1956)59.1-13. On the study of Boethius'sDe

arilhmehca n the Middle Ages,seeA. M. White, Boethius in the MediaevalQuadrivium, in Boelhius:

His Life, Writings and Influence, ed. M. T. Gibson (Oxford 1981)162-205.

4sF,ol. lIra-va.


10/19

118 G. R. EVANS AND A. M. PEDEN

The first category, hat of the natural composites, uggests o him an illustra-

tion taken from natural science; his demonstrateshe mathematicalprinciple that

a systematicorder governsboth the compositionand the dissolution of composites.

He cites he' 'natural philosophers (phisiologi) on the progress f human ife; they

divided it into periods of sevenyears hebdomads ), ascending o a peak during

the first five hebdomads o the age of thirty-five, then declining, in proportion, dur-

ing the second ive hebdomads o the age of seventy.The same orderly processs

seen,he argues, n the seven-day hasesof the lunar cycle, from crescent, o full

moon, to crescentagain.46 bbo may be borrowing from Macrobiushere, for he

would find there a detailed treatment of the variousways n which the number seven

governsnatUral phenomena.47 bbo, like Macrobius, ooks to the natural world for

a further dimension to add to his initially mathematical nvestigation. Like Macro-

bius too, he goes on to remark on the special qualities of seven, the virgin

number which alone of the numbersone through ten s not a product or a factor of

other numbers through ten.48 n a somewhatawkwardand obscureway,Abbo ties

this numerologydown to a preciseChristiansymbolism.The virginity of the number

seven eveals t to be simple wisdom, and links it thereby to the soul, the seat

of wisdom. This virgin number s diffused through the number ten (seper dena-

n'um diffundit) when the soul, awaiting liberation from the body, observeshe ten

commandments.49

Abbo continues his contemplative considerationof his topic in his discussion f

perfect and imperfect numbers. This providesone of the few nstancesof his use of

a theological subject merely for illustrative purposes, ather than as part of an ana-

logical argument. Such a use was already explicitly present n Abbo's sources:he

simply developsBoethius's characterization, n the De arithmetica, of more-than-

perfect and less-than-perfect numbers (that is, numbers greateror smaller han

the sum of their factors)as he excess nd defect of a quality (which is a vice),con-

trasting them with perfectnumbers (which are as are asvirtuous men).'oThis discus-

sion is not strictly appropriate to Abbo's immediate topic. The relation of a

number's factors o the number tself is certainlyone aspect f its compositeness,ut

the' 'deficiency and excess of imperfectnumbersdo not reallycorrespondo the

growth and decline in the natural cycleswhich Abbo discusses. bbo has simply

included a parallel provided by his source o as o enrichhis treatment of the subject.

He justifies his discussion f natural composites y declaring that compositesmade

%F, ol. 11rb.

47Cf. Macrobius 1.6.67-74 (n. 24 above)cd. 31.6-32.18; 1.6.54-56, cd. 28.11-26.

48Cf. bid. 1.6.11, cd. 20.15-22.

49F,ol. 11rb.

'oF, ol. 11rb-11va; Boethius, De Imthmelica1.19-20, cd. G. Friedlein (Leipzig 1867)39.18-41.25;

cf. Augustine, De cillo Dei 11.30. For the moralization of the concept,cf. MartianusCapella,De nuPltts

7.736 (ed. A. Dick [Leipzig 1925]383.17-19); Remigius 383.17 n. 21 above)ed. 205.13-29. Perfect

numbers are equal to the sum of their factors.


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119


by will aresimilar to those n nature,since hey belong o the art of mathematics,

and' 'all art imitates nature.' I

Abbo's treatment of the division of what is incorporeal is another point at

which the natural world supplied him with illustrations to support his comments.

He explains at some ength the nature of sense-perceptionsopposed o the soul's

perceptionof' 'intelligible objects (citing asexamples bstractions uch as he cir-

clesdividing the sky-the zodiac,ecliptic, parallels,and colures-which are nvisible

but canbe perceivedby the mind).'2 This distinction between he perceptionof cor-

porealand intelligible objectswas crucial o one of the fundamental propositionsof

Abbo's exposition: that number, measure, nd weight are measurements f bodies

and not bodies n themselves.'3 ut Abbo wasmore concernedwith the nature of the

objects of perception han the mechanismby which they are perceived.He quotes

ClaudianusMamertus:what s incorporeal, he soulperceiveshrough ts ownpowers

(per Ie), becauset, too, is incorporeal;what s corporeal, he soulperceiveshrough

the body.'4 But Abbo wants to show that incorporeal hings can be not only per-

ceived but also divided, even hough they cannotsenseor be sensed.First, he uses

dialectic to distinguish exactly what is to be divided: are day and hour sub-

stances r accidents?Using the method of formal definition, Abbo shows hat they

are not bodies but types of quantity which may be predicatedof a subject. To dem-

onstrate hat temporal quantity (time) canbe divided systematically, ven hough it

is incorporeal,Abbo takesan example rom natural science.He uses,with acknowl-

edgment, Macrobius'sdescriptionof the Egyptians' division of the sky nto twelve

signscorresponding o the signs of the zodiac,using the clepsydra water-clock).'6

This is particularly apt for the discussionof abstractdivision, since Macrobius

regardshis account as an answer o the question: Who has ever ound or made

twelve divisions of the sky, since they are in no way apparent o the eye? '1 But

Abbo adaptsMacrobius o his own purposes.Macrobiuswasmostconcernedo show

how the sky wasdivided into twelve zodiacalsections,but Abbo wanted to demon-

strate he division of time into hours (two for eachzodiacalsign), and to showhow

the amount of time called day variedaccording o the time of the year.The mea-

surementof the amount of water flowing through the clepsydraduring daylight at

F. fol. 11va-b: Dico autem 'secundumplacitum' essequae voluntate fiunt uel facta sunt, [fol.

11vb]quae et ipsanon multam a natura discrepant i rationabiliter acteconstant,quoniam omnis ars mi.

tatur naturam, profectaex passionibus nimae, quae credimusnaturales sse. Cf. Boethius, n Top.Cic.

(PL 64.1048) and In AnsI. Perihermenias1.1-2 (ed. Meiser 1877), 36.22-51.19);J. Engels, Origine,

senset survie du terme boeciensecundumplacitum,' , Vivarium 1 (1963)87-114.

'2F, ol. 14rb-vb; cf. ClaudianusMamerrus1.17 n. 25 above)ed. 62.19-64.11. On the invisibility of

thesecircles,seeMacrobius1.15.2 n. 24 above)ed. 61.12-13; 1.15.9, ed. 62.6-9.

F. fol. 8vb; taken from ClaudianusMamerrus2.4, cd. 111.19-113.1

'4F. ols. 8vb-9ra; cf. ClaudianusMamertus2.4, ed. 113.11-114.4.

F, fol. 13va.

'6F, ol. 13vb; cf. Macrobius1.21.9-21 (n. 24 above)ed. 86.24-88.28.

Ibid. 1.21.8, ed. 86.19-21.


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120

different times would revealeven ractional changesn the relative ength of day and

night. So Abbo has shown that day, a quantity of time itself (though variable by

nature), can be measured y hours; hours too, likewisea quantity of time, are divi-

sionswhich canbe perceived hrough their sensiblemanifestations n corporeal ub-

jects, while remaining incorporeal-and-intelligible hemselves.Abbo moves with

ease rom example, o principle, to application,using each o shed ight on the next,

so that the range of his inquiry may not merely enrich he knowledgeof the reader,

but alsoenable him to see he rational and harmoniousstructure of the universeat

its different levels. Moreover,Abbo's use of natural science asenabled him to illu-

minate the nature of conceptual hinking, aswell as o demonstratehe operationof

intelligible principles in the sensibleworld.

IV. TECHNICAL VOCABULARY

Not only in dealing with concepts, but also in his use of technical terms, Abbo finds

it helpful to compare those used in more than one art. He does this most skilfully

where dialectic meets arithmetic. Number, he says, s a species of quantity, which is

counted among the accidents of a body's substance (que computata inter accidentia

sue substantie).58 There is no need to force a comparison here, since number was

discussedby Aristotle in the Categon esn connection with the category of quantity.59

The discussion of unity, which occupies Abbo for some time, raises more profound

questions of common technical terminology. Victorius says hat unity is simple, con-

tains no parts, and cannot be divided. But other things, although they are called

one because hey seem whole and solid, are really composite, and so are unavoid-

ably subject to division. Those other things have existence, and Abbo asserts hat

Everything which exists is one, and whatever is one, it is necessary hat it exists.

To exist and to be one are therefore interchangeable in predication (ad suam

inuicem predicanonem convertibzlis). Those things which are interchangeable in

predication are equal, and therefore, unum and est are equal, for what does not exist

cannot be called one, and what is one cannot be said not to exist.6O here are

major philosophical problems here; Abbo felt it necessary o touch on them in order

to explain Victorius's principle that although unity cannot be divided, yet, mani-

festly, one horse, one day, one hour, can be divided into parts. Abbo attempts to

resolve the paradox of the indivisible and divisible unity with the aid of a discussion

of the dialectical rules of predication. He cannot be said quite to have succeeded,

and indeed he does not pursue the investigation very far. What is important here is

his readiness to use straightforward technical procedures both upon issues of great

complexity, and also upon the simpler problems of elucidating arithmetical theory,

'SF, £01. 9vb.

'9Aristotlc. Cl1legoriae 6.4b.

6OF. 01. 9va; cr. Bocthius, In Porphyrii Isl1gogen 1 (PL 64.83B).


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ABBO OF FLEURY'S COMMENTARY

121

suchas he following: one may be said to have parts not in itself, but in relation

to somethingelse. n itself it is simple, but it may be multiple through an oppositio

re/ationis one of the four kinds of opposita n the Categones).61ne' 'four is two-

fold in relation o eight, and one eight contains wo fours. ,Elsewherewe are

reminded that substancesaveno contraries.62s far asAbbo is concerned,t seems,

Aristotle has aid down awsof thought which are appropriate o everykind of intel-

lectual problem, and which govern mathematicalas well as ogical questions.There

can be no objection o fitting the conceptsAristotle provides o the particular prob-

lemsraisedby mathematics.Equally t canonly be lluminating to makeuseof fami-

liar technical erms of dialectic in discussing hese principles, provided their exact

relevance o the question n hand is made clear.

VI. SHAREDMETHODS

As to methods common o more than one art: here, too, Abbo finds that dialectic

and arithmetic come together naturally at various points. He analyzesVictorius's

opening statement hat unity is the indivisible sourceof all number, with the aid of

what he identifies as a disjunctive conditional syllogism per disjunctionemhuius-

modi conditiona/emcolltgentes yllogismum).The syllogism s as ollows:

Everything which exists s either simple or composite

But unity is truly simple.

Therefore t is in no way composite.

From this it follows that it is indivisible, because t is composed of no parts. Boethius

notes in the De syllogismo hypothetico that every conditional proposition is either

connexa or disiuncta,' '63 and Abbo is simply following here a procedure which

he describes n detail in his own work on hypothetical syllogisms.64 t may be objected

that he has really said nothing further about Victorius's definition; he has done

no more than paraphrase it and cast t in the form of a syllogism. But this was just

the sort of clarificatory operation which Abbo considered it his task to perform

as commentator.

By far the most common methodological borrowing from dialectic in the commen-

tary is Abbo's use of definition, This was a topic to which he had given comparatively

little space in his treatises on the syllogism. He merely notes there that it remains

to deal with the kinds of definition and the topics of argument, which can be more

easily recalled to memory by counting them on the fingers,' '6' The list of kinds

of definition which follows is to be found in the monograph De definitione which

61F,ol. lIra; cf. Aristotle, Categ. 9.llb (17-19).

62F,ol. 13va;cf. Boethius, n Categ. Anst. I (PL 64.195-196).

6~PL 4.837.

64F,ol. lIra; cf. PL 64.835; van de Vyver n. 9 above)64-81.

6'Van de Vyver 50.20-32.


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122

Marius Victoriunus wrote to fill out Cicero's list in the Topica, and it is also in

Isidore.66

Abbo had, then, a sound graspof the technicalprocedures f definition. He was

not content merely o use definitions to make Victorius's terms clear o his readers,

but goes o the trouble of identifying the type of definition which s appropriate n

eachcase.When Victorius says hat "unity is that from which the whole multitude

of numbers proceeds," and that it belongs o the discipline of arithmetic, Abbo

points out that "he first explainswhat unity is, by the secondmode of definition."

Abbo gives both the Greek term for this mode (f:vvol1~attKiI) nd the Latin (notio),

just as Marius Victorinus does,and says hat it explains he thing-which-is-to-be-

defined, by what it "does."67What unity "does" is to give rise to all numbers, or

all numbersproceed rom unity. The eighth mode of definition appears little later.

After Victorius has said that unity is simple, Abbo adds that it is made up of no

parts, 'as if to use he eighth form of definition. ..by denial of the contrary" (per

privationem contrani).68Victorius as defined unity by saying both that it "is sim-

ple" and that it "is not the opposite of simple." The fifteenth form of definition

deals with the rei ratio, or the reason or the thing. Both Marius Victorinus and

Isidore give an example which Abbo makesuse of directly: "Day is the sun above

the earth; night is the sun beneath he earth," and again he identifies this as a

species efinitionis 69

VII. UMITAnONS OF THE COMMENTARY

In his assumption that common laws of thought underlie all the artes, in his careful

clarification of technical terms in context, in his willingness to borrow the methods

of dialectic in particular, to help him in his analysis of arithmetical problems, Abbo

rarely probes far into the deep problems which caused masters such as Gilbert of Poi-

tiers so much difficulty a hundred fifty years ater O His mind was amply furnished

with technical terms; he thoroughly understood the teaching of the textbooks of the

artes he knew; he was well able to expound those books for his pupils. He could go

further; he could adapt their teaching up to a point to make the terms and methods

and concepts of one art serviceable n another. But he did not possess he capacity of

an Anselm of Bec for absorbing technical knowledge to the point at which it served

as an aid to original thought and not an end in itself. Abbo' s transference of meth-

ods and principles is usually sensible and to the point, but it is relatively mechanical.

When he considers the notion of "division," for example, he merely repeats what

66C£.Cicero, ToPica 6.28; Marius Victorinus, De definitt.one PL 64.901-902); Isidore o£ Seville,

Etymologiae2.29.1-16.

67F, 01.10vb; Victorinus, PL 64.902.

68F, 01. Ira; Victorinus, PL 64.904-905; Isidore,Etym. 2.29.9.

69F, 01.13va; Victorinus, PL 64.907; Isidore,Etym. 2.29.16.

7'Haring (n. 37 above)189-190.


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123


Boethiussays n his commentaryon Porphyry:we call something ndivIduum for var-

ious reasons, ecauset hasno parts, ike unity, God, or the soul, or becauset is too

hard to cut, like adamant, or because,f we cut it up, the parts cannotbe called by

the name of the whole}1 Abbo does not pause o consider he philosophical mpli-

cationsof what he hassaid about the relations of parts and wholes.He goesstraight

on to distinguish magnitudesand multitudes, desiring only to introduce the reader

to the nature of the problem in hand and to prepare he ground for his own remarks

on discreteand continuousquantity. His concern s simply to identify and label the

similarities between he notions of "undivided" in dialectic and arithmetic.

Much he samemight be said of his treatment of the idea of ' 'necessity."Victorius

says hat although unity itself is indivisible, there is nothing in the natural world

which cannot n somesense e divided, because verythingbut unity itself s compo-

site, and what s compositemust, necessarily,e divisible. Abbo thinks it helpful to

askwhat Victorius meansby "necessary," or somethingmay be said o be necessary

in three ways.The first and secondare contingent upon the circumstances f the

moment (secundumcondt'tionem emporis contingenter): "It is necessaryor me to

write while I am writing. It is necessaryor me to eatwhile I live." The third is neces-

sary n itself (secundum esimpliciter): "It is necessaryor me to be moral." The first

is necessarys ong as certainconditions obtain. The second s necessaryecause f

somethingelse I must eat n order o live). The third is necessaryy virtue of its very

existence potentia actus simpliciter). Things in the natural world are necessarily

divisible only in a contingent way,Abbo explains,ascircumstancesemand}2Once

again,he entersupon a large and problematicareaof discourse, hows hat he knows

the teachingof the authorities on the matter n hand, but fails to go further than the

immediate demandsof textual interpretation oblige him to do.

As to Abbo's use of natural science: is understandingof it was horough, but he

uses t chiefly for illustration. In considering he four elements earth, water, air,

fire), Abbo first explains Victorius's opening words: "Unity, from which every

multitude of numbersproceeds,"by showing how numbersare perceived n bodies

by sense, nd yet they preserve omethingof their incorporealorigin, unity, in their

harmony.He states hat each ense peratesn conjunctionwith one of the elements;

since there are five senses, e adds aether as the fifth element 3 But Abbo only

noticescertainsimilar ideas without drawing out their full potential importance or

his argument.

His treatment of harmony n the cosmoss also simply illustrative. Prompted by

Plato's descriptionof the harmonious chain ormed by the binding togetherof the

71F,ol. 10ra; cf. Boethius, In Par. Isagogen2, PL 64.97 (Boethius givesunitas and mens or things

which have no parts); cf. also Abelard (n. 44 above)549.4-20.

nF, fol. 14ra. On the Aristotelian and Boethiandiscussion f necessity nd on the work of the tWelfth

century, seeHenry (n. 43 above)172-180.

73F,ols. 9vb-10ra; probably using ClaudianusMarnertus1.6.7 (n. 25 above)ed. 42.7-44.7, 45.1-

46.6; d. Augustine, De magistro 12.39; L. Schrader,Sinne und SinnesverknupfungenHeidelberg 1969),

esp. 181-184.


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124


four elements rom which the universewasmade,74 bbo takesup this motif ofhar-

mony, at first literally. He argues hat although senses re not transferable the eye

cannot hear, nor the ear see), et they can assist achother, for the eyemay perceive

numerical atios which produceaudible harmonies.Then he uses he motif allegori-

cally, when he compares his creation of harmony with the work of God, he who

bound together he elements,who tempers he strings of the organum of the heart

to prevent he dissonance f the senses.75inally, Abbo applies he motif to the sub-

ject itself, number, measure,and weight, which must be equally balancedso as o

produce harmony n creation, he concord of plurality in unity.

Abbo has certainlyexplored he variousparallelssuggestedo him by the idea of

harmony in diversity; but he has not worked through his opening remarks about

how number becomes sensible, in terms of the elements, which are the basic

ingredientsof sensiblematter. It might have beenmore relevanthere o showexactly

how matter is perceivedasquantified body, and at what stage he presence f the

elements n sucha body make their measurement ossible.For the elementswere

thought to be not simply media of sense,asAbbo treats them here, but present

throughout the universeand in everybody, as ts basicmaterial. Abbo's subsequent

investigation of the nature of composites howshe wasperfectly well awareof this

dimension.76 ow can' 'composite propetly be predicatedof something which we

recognizeasone?He takes earth asan example.Earth s composedof more than

one thing, and yet it is one individual element (of the four) and thus apparently

simple. And we read Terra erat t'nvisibtliset t'ncompositaGen. 1.1 [LXX]). One of

the basicproblems here is evidently the different meaningsof the word terra. But

Abbo uses he problem asa point of departure or a discussion f the process f cor.

porealcomposition.He argues hat the earth n primevalchaoswas nvisible and not

ordered (t'ncomposita) ecauset wasconfusedwith the other elements n indistinct

matter; once made visible, it was composite, not because t was constructedout

of other elements but by the acquisition of its own individual qualities. Thus the

diverseparts of the earth (as a composite)are ts underlying substance nd its par-

ticular form, which make t a compositecorporeal eing. Abbo hasmade an attempt

to explain he process f compositionby a method which he defines as igura aethi-

ologica.77 e doesnot explain he actualprocess f creatingvisible body from invisi-

ble and unformed matter, nor doeshe clarify the relationshipbetween he element,

earth, and the visible earth discussedn Genesis which could be considereda

74Plato,Timaeus,31b-32c; cf. Macrobius 1.6.24-33 (n. 24 above)ed. 22.21-24.18; Calcidius 22

(n. 15 above)ed. 72.21-73.4 and 317-318, ed. 313.5-31'4.13.

7'F, ol. 10ra; cf. a similar moralization by Calcidius267, ed. 272:21-273.4.

76F,ols. 12vb-13ra.

77F,ol. 13ra;cf. Isidore,Elym. 2.21.39; Eriugena,Periphyseon2.16-17, ed. I. P. Sheldon-Williams

(Dublin 1972)52-58. J. M. Parent,La doctrine de 111rel1tion l1ns'ecole de Chl1rtresOttowa 1938);T.

Silverstein, Elementl1tum: Its Appearance mong the Twelfth Century Cosmonogists, MediaevlZi tu-

dies 16 (1954) 156-162; R. McKeon, Medicine and Philosophy n the Eleventhand Twelfth Centuries:

The Problemof Elements, The Thomist 24 (1961)211-256.


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125


composite because ompounded of a mixture of elements).78he problems and

refinements of later attempts to harmonize biblical accountswith Platonist cos-

mology were not an issue or Abbo. He simply applies he cosmology f his sources

to the dialectical terms of substanceand accident, which he then applies to the

mathematicalconceptsof Victorius's preface.

In a third passage, e are taken nto the realm of pure natural science.79bbo

seeks o explain the problem of relative weight by considering he four different

qualities which bind together the four elements: coldnessand heat, wetnessand

dryness. Cold makes things dense, and thus heavier-this, Abbo asserts,s one

reasonwhy the furthest planet, Saturn, s the slowest o complete ts circuit (which

is, admittedly, the longest), or it is also he coldest,and thereforeheaviest, lanet.

After considering urther aspects f the naturalpower of certainsubstances, bbo

goeson to discusshe relationship betweenheat and cold, and wetness nd dryness,

and that of all four qualities to weight. Somethingdried by heat becomesighter;

somethingmade wetter by cooling becomes eavier.So, a half-burned torch thrown

into water will surfaceburnt end first. Abbo then considershe application of these

theories o human physiology, n the effectsof the preponderance f humor in the

body, and finally considershe relativewetness nd densityof wine, honey,and oil.

The point which Abbo wishes o make s relativelystraightforward,and he achieves

clarity and vividnessby his use of a wide range of evidence.. e is not attempting to

extend he languageof dialectic or arithmetic in this case;but neither hashe simply

followed the line of the standard Platonic cosmological ources, or he puts his

knowledge o practical use within the limits of his inquiry.

VIII. CONCLUSIONS

These hree examples how he options available o Abbo: he could allow himself to

be led, almost at random, through the various acetsof the topic or problem under

consideration;he could, as n the secondcase,apply his knowledgeof cosmologyo

the dialectical terms in which he proposed o treat his subject; or he could use

natural scienceo elucidate hoseaspects f his subjectwhich were strictly concerned

with natural phenomena.He does all three with confidenceand sureness f touch,

despite he fact that he is exploring severalways of approachinga subject at once,

and does not limit himself to any single discipline.

Abbo moves antalizingly close o several reaswhich were o stimulate speculative

thought in the eleventh and twelfth centuries. But he also displays an admirable

78Calcidius 07 (n. 15 above)ed. 307.20-308.2 egardedvisible elementsas compositebecausehey

consisted f a mixture of all four elements,eachvisible element aking its name rom the elementwhich

waspredominant n it. Cf. J. van Winden, Calcidiuson Matter: His Doctrine and SourcesLeiden 1959)

140-141.

79F,ol. 20rb-vb. This sectionwasedited by Christ (n. 4 above)147-152.


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26

scholarlydiscipline by making everythinghe saysdirectly relevant o the interpreta-

tion of the text beforehim. He wants o label and identify the technicalproblem in

hand, and to include as much learning as will profit, but not confuse,his readers.

The commentary-formwas deal for this purpose: it provided a ground-plan and

criterion of relevance,by keeping to the sequenceof the text, while allowing the

commentator reedom o introduce a body of knowledgewhich he wished o trans-

mit to his audience.The breadth and thoroughness f Abbo's learning made him

better-equipped han most of his contemporariesor this type of work, and he was

able o give his Commentarya richness nd clarity which s rare n suchpieces. t has,

admittedly, little of the air of speculativecuriosity of twelfth-centurywork; Abbo

is concerned o display the detailed correspondence etweenone discipline and

another, ather than to usehis competencen thesedisciplinesasan intellectual tool

for more advanced hinking. But he wasperforming an essential reliminary exer-

cise; it was first necessaryo expand the areas n which the seculararts could be

employed,and to do it with an eye to soundness, ccuracy, nd commonsense.

APPENDIX

The following text of the preface o Victorius of Aquitaine's Calculus s taken from

EastBerlin, StaatsbibliothekMS Phill. 1833 Roseno. 138). ol. 5ra-b.

INCIPIT PRAEFATIO DE RATIONE CALCUli

Unitas ilIa, uncleomnis multitudo numerorumprocedit, quaeproprie ad arithme-

ticam disciplinam pertinet, quia uere simplex est et nulla partium congregatione

subsistit, nullam utique recipit sectionem.De ceterisuero rebus, icet aliquid tale

sit, ut propter integritatem ac soliditatem SUam nitatis meruerit uocabulo nuncu-

pari, tameDquia compositumest, diuisioni necessarioubiacebit.Nihil enim in tota

rerum natura praeter memoratamnumerorum unitatem tam unum inueniri potest

quod non ulla omnino ualeat diuisione distribui. Quod ideo fit quia non simplici-

tate sedcompositionesubsistit.Dicitur enim unus homo,unus equus,unus dies,una

hora, unus nummus et alia huiusmodi innumerabilia, quae icet unitatis sint sortita

uocabulum, ameDpro causae tque rationis necessitate iuiduntur. Ad huius diui-

sionis conpendium ale calculandi argumentum antiqui commenti sunt, ut omnis

diuidendi integritas rationabili per illud possitpartitione secari, iue d corpussiue

res ncorporeasit quod diuidendum proponitur. In hoc argumentounitas assis oca-

tur cuius partes uxta proportionalitatem suam proprius sunt insignitae uocabulis.

Notis etiam ad hoc excogitatisper quas eademuocabula exprimantur, ut per dis-

cretionemnominum et notasnominibus afflXas niuscuiusque articulaenotio faci-

lius aduertatur. Et assisquidem qui per i litteram, sicut in numeris unum scribi

solet, exprimitur, xii partes habet, quarum si unam detraxeris, eliquae undecim

partes iabus dicuntur. IlIa uero quam detraxisti, d est: duodecima,uncia uoca-


19/19

127

ABBO OF FLEURY S COMMENTARY

tur. Si duas sustuleris,decem esiduaedextanset quod sustulisti, d est: duae, sex-

tans appellatur. At si tres dempseris,nouem quae remansetuntdodrans, et tres

demptaequadransuocantur.Quod si quattuor tollere uelis, octo reliquas bissemet

quattuor trientem nominabis.Quinque uero sublatis septem esiduas eptuncem,et

quinque.sublatasquin- [fol. 5rb] cuncemplacuit appellari. Cum uero per medium

fuerit facta diuisio, uttumque dimidium senispartibus constans, emissem ocita-

runt, unciam autem et dimidiam sescunciam,nciaequedimidium semunciam. am

reliquae minuciae quarum congestione imidium unciae conficitur, ut sunt sicilici,

sextulaeet cetera,melius ex ipsius calculi nspectionecognoscuntur. ncipit autem

idem calculusa mille et usque ad quinquaginta progreditur, primo per duplicatio-

nem, deinde per triplicationem, turn per caeterasmultiplicationes ncrementacapi-

ens, tanta numerositate concrescitut usque ad infinitum quantitatis eius summa

perueniat. Scribitur uero ineis a superioriparte in inferiorem descendentibus,upe-

rius milium summas ex multiplicatione uenientes, nferius diuisionum minutias

continentibus (above: scilicet, ineis). A quibus ramen n legendoprincipium est a-

ciendum et sic sursumuersuseundemquousquead milium summam,quae ex lIa

multiplicatione paulatim adcrescitegendoueniatur, ncipiendumquea dimidia sex-

tula per duplicationemusquead n, inde iterum per triplicationem a ~idia sextula

usquead ill, turn a dimidia sextulaper quadruplicationem squead iiii et sic usque

ad finem.

EXPUCIT PRAEFATIO

Fitzwilliam College

Cambridge. England

Saint Hilda s College

Oxford OX4 IDY, England

Documents

evans1985 Natural Science and Liberal Arts in Abbo of Fleury's Commentary on the Calculus of Victorius of Aquitaine.pdf