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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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G. R. EVANS AND A. M. PEDEN
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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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G. R. EVANS AND A. M. PEDEN
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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BBO OF FLEURY'S COMMENTARY
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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G. R. EVANS AND A. M. PEDEN
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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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.
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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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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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G. R. EVANS AND A. M. PEDEN
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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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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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G. R. EVANS AND A. M. PEDEN
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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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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G. R. EVANS AND A. M. PEDEN
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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BBO OF FLEURY'S COMMENTARY
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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G. R. EVANS AND A. M. PEDEN
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-
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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