Gillian R. Evans

This paper considers the nature of some of the evidetw for the study of the subjects of tke quad- riviun in the eleventh and twelfth centuries, and the sources of contemporary interest in the math-

ematical arts. The survival in the eleventh and twelfth centuries of manuscripts of the quadrivium texts of such late-classical writers as Boethius and Martianus Capella is considered, and their math- ematical ideas are analysed in relation particularly to the thought of William of Conches, Peter

Abelard and others. An examination of mathematical notions employed in the spect$c context of textbook

comme:rtary on Aristotles Categories is followed by discussion of the method of proving the unity

of the Trinip by means of analogy with the ProBerties of points, lines and surfaces.

In recent years, it has hecome increasingly

clear to what extent the study of grammar

and of dialecti: underlies the thinking of the

speculative philosophers of the eleventh (and twelfth centuries (Henry 1967); at a rrlore

workaday level, the commentators and gloss-

ators of the standard textbooks show some-

thing of how these works were studied, in

greater or lesser depth, in the schools (Hunt 1941, 1950; de Rijk 1956, 1967). The study

of the quad-ivium has not yet received an

equal attention. All the subjects of the quadrivium, arith-

mztic, geometry, music .and astronomy, v/yerc

alike treated as mathematical subjects;; a

considerable variety of mathematically de.

rived notions and issues testifies to their

importance in the work of the schools of the

eleventh and twelfth centuries. That is not

to suggest that mathematics was at any time

the centre of attention, a real growing-point,

in the way that dialec:tic appears to have been in Abelards day. Rut the evident

interest of his contemporaries in dialectic has

perhaps tcndcd to obscu rc the very significant,

if more diffuse, contriilution of mathema-

tics.

The influence of each field of study mani- fests itself at several levels, from simple borrowing of method and technical termin-

ology, to the inspiration it gave to a whole

complex of what I propose to call think.ing

schemes. It is not only in the methods of

solution which it suggests, but also in the

nature of the very problems which presem

themselves for solution that the influence of the study ofan individual art makes itsclffelt. The widespread discussions of certain issues

may, to some extent, be explained by the

fact that those who delivered outstanding sets oflectures had their admirers and imitators -

even plagiarizers. There is remarkable uni-

Journal of Medieval History 1 (1975), 151-164. 0 North-Holland Publishing Company 151

formity o,f subject-matter and treatment

amon: the logical treatises published by

L.M. de Rijk in his Logica modernorum, and in the Cbmwentaries o.n Roethius of the school of Thicrry of Char .j, published by N.M.

Hgring; sse\reral pieces, which are not, it

seems, Thierrys own finished work, appear to be Iecture:s or even lecture notes, abridge-

me:ats of Thierrys lectures, made perhaps with his permission and under his supervision,

perhaps without. Fhe schools bred sparkers-

c,ff of controversy, from Berengar to Roscelin, Aheiard and Gilbert of Foitiers; in this way, issues were kept in the air. Even so, the clecp sources of at least the small, detailed preoccu-

pations of these thinkers 1a.y in the texts on which ihe). lectured zr.lA Those on which they had cut tiheir own teerh as academics. A

newly popular text bred new issues of contro- versy. This is notably true of the Trinitarian

debates of the twelfth century, where Boethius theological treatises raised fresh problems in addition to those considered i>y Augustine and his successors in writing on the

Trinity. It is in the textbooks that we find traces of rnathemattcal influence which find the ir way i,nto the: commentaries and heyonll; m works appa.rently on quite different sub$ects, mathematical discussion occurs.

In Aristotles Catqyries, various numerical principles, to be discussed more fully later:, are raised in the sections (6-7) which deal with quantity and relatives; the notions of figure and shape occur in the discussion

ofqiuality (8). These are worked over in detail.

by Boethius and by the twelfth-century writers af the textbook treatises which arncurit to little more than commentaries (in that the) adopt the Aristotelian subject- matter and order of treatment). In Boethius theological treatises, it number of math-

ematical issues are shown to be relevant to

the solution of the problem of the Trinity. Again, the twelfth-century commentators en-

large upo.n these aspects. Mathematical

notions arc dealt with b) patristic writers, too. In the De ciuitate Dei (1.30; 18.23) Augustine, for example, discusses the signifi-

cance of the cube of three. Gregory writes of

the number of Jobs cattle after his return to prosperity; in this part of the Moralia in Job (MPL 76: 773), hc describes the arithmetical properties of perfect numbers, and mentions

linear, plane and solid numbers. The study of the quadrivium, however,

depended on the availability of the textbooks

of the arts themselves. Broadly, students had at their disposal three means of acquiring a workin? kcowlcdge of a chosen subject: they could read the textbooks, or the summaries

presented by the encyclopaedists, or they might hear the commentary cf a master. 1Jp to a point, it seenls likely that the survival of a large number of copies of a given text

means that that text M.as widely studied. Thr more often a gloss or commentary is found wirh the text, the more probable it seems that

it was popular in th(s schools. On such an assumption, it is clear that the study of the subjects ofthe quadrivium did not match thirt of grammar and dialectic, nor even, perhaps,

that of the two Uceronian Rhetorics. L. Thorndike and P. Kibre list only a few copies of Boet hius rlrihhetica which appear to b,elong to the eleventh century (1963: cols. 178, 907, 1019, 1491). A gloss and commen-

tary to this work is to be found in a manu- script of St Augustines Canterbury, now Corpus Christi Collegcb, Cambridge MS. 352; the hand is for the most part contemporary

with that of the mc.in bodv of the text,

which would suf;gcst that the commentary is not the product of ecti\rc tc aching in thr

school at Canterbury in the ttllth or eleventh

I 52

i i&3.

Figuw I. Iblathematical diagram. 2ambridge Corpus Christi College MS. 352 1: 1.

centm y, but perhaps rather of earlier origin. Figure 1 shows the mathematical diagram

with t kich the war;: is pre!Bccd. An intcr-

linear gloss consists largely of expansions of

abbreviations, synonyms or cxplanatior,s fbl

;lifficult words. In the margins, longer exp.a- uations or summaries of pwtions 01 I

simple; the glossator seeks to be strai&ht-

fi)rwardly helpful, providing place-finders

in the margins, and symbols to link gloss with text. There are no instances of com- Farisons with the views OF other glossators,

no speculative explorations, none of the apparatus of the later schoolroom (unless WC count a hoc pertinet ad Platonen on f. 2va). There were sckolia to Boethius Aritkmetica fi-om an early date. Rubnov prints sckolia of ?Iol:ker, Gerbert of Aurillac, Abbo of Fleury ancl others (Bubnov 1899:31-5, 297-9, all to Eaok 2.1, and 147-50, a uersus 3n the work as :, whole. None appears to be related to OUI glossj. Despite these evidences of a period- ically active interest in the work, the study of

arithmetic does not appear to have caught on in the way that the study of dialectic did in the course of the eleventh and twelfth centuries, .Abelard himself remarks, in citing the De aritkmetica, that he wishes he knew more about the study of arithmetic (de Rijk 1956 :59). Nevertheless, he remembers his masters te;aching* on the subject ofpointe and lines, so we may take it that the teaching of at least such elements of arithmetic as were

fimnd to he relevant to the works of the schools was practised in his student days at tjle end of the eleventh century. Thierry of Chartres cites the Prologue to the Aritkmeti.a ~II his rlefinition of number in the gloss on Eoethius L)e Trinitate (H&ring 1971:267) arrd in the commentary on the same work (Hgring

197 1:70). The text as a whole may not have been frequently studied, but the fact that t:lese references are explicitly to Boethius work, and not to some encyclopaedists

summary, suggests that copies were available, and that the source-text was being consulted in the original.

The demand for Boethius De musica may have been of a rather different kind, although

Gu;do Aret .nus firmly distinguishes musica

from the pn.ctical art of cantus:

Musicorrunn et cantorum~ magna est distantia Isti dicunt, ilili sciunt., quit

Figure 2. Linear, plane and solid numbers.

Mathematics in astronomy manifests itself

in the twelfth century in the form of dis- cussionsaboutcosmology.C~illiamofCorlches

Gha super Platoneu draws on Boethius L)e Trinitute, but in Jcauneaus view (l965:27) it is Boethius De auiikmetica and Ue musica

which provide the mosi important influence, especially in that part of the commentary

which is concerned wrth the anima mundi. Chalcidius, of course, provides an inter-

mediary source between Plato and the

medieval commentator, and Williams cos-

mology draws heavily on Macrobius (Jcauneau 1965:28), but the more obviously

mathematical elements in his thinking are

dcrivrd from BoetHius. In Chapter 74 (Jeauneau 1965:154) we meet the notion

of the perfection of number: nothing, after

God, is so perfect as number. Unity is the

beginning of number; after unity come the

linear numbers, then the plane and solid numbers: see Figures 2 and 3. So much is

27

familiar in clementar;* Nichomachran- Boethian number theo~;;. IJnity, William

goes on to say, signifies the irldivisible ess:nce

of the soul; the linear r?umt#ers show, its

capacity for movement lineally : potelltia movendi corpus in longum (Jeauneau 1965 : 155))

lvhile the plane numbers show its power to rno1.e laterally: in latum, and the solid numbers demonstrate its power 0 rno1.c in

depth: in @sum. Evenness and oddness are introduced: numbers which ;re even l:par)

can be divided into equal parts; the impares are indissolubilia. Proportion and harn1oc.y and ratio, with their musical and arithmetic:11

ramifications, are discussed at length -n

succeeding chapters. Among Williams sto :k c)f ideas, the primary notions of mathematics,

as they werrP set out by Kichomachus aild

rendered by Boethius, evidently come nat- urally and conveniently to mind when hc

discusses the nature of the soul. So much for the textbooks and their

3. 3 6 10

1

cl a

a A a a 4

a a

El a d

a A a a a aa 9

a .s. a L --I a a a a a a -_.._-A ITipre 3, Triangii!ur and sq-uare numbers.

commentaries. The encyclopaedists were

read, perhaps by less serious or advanced stlrdents, but no rjoubt much more widely. X?e subordinate place of the subjects of the citradrivium is r!:flected in their treatment. TYhere I&lore, for example, gives a whole book of tire Dymologi~ to the study of

grammar, a ser:ond to rhetoric and dialectic, the four subjects of the quadrivium share Book, amongst them. Cassiodorus gives them

even sbortrr shrift. The essentials of arith- metic, geometry, music, astronomy, are sum- rnarised in turn. While there is little doubt

tha,: tie thinker who required a brie; explanaition of some elementary principle in

crder to furnish himself with an argument

would often Gnd what he wanted here - s-Vera\ ideas to be found in Abelard 01 tVilliam of St Thierry, for example, are in t le encyclopaedias : the indivisibility of the @ncium or monad, the notions of equality

and inequality, of odd and even - it is much

more probable that such ideas would suggest

themselves to him tnore forcefully if he himself had a first-hand acquaintance with

the source-texts; and certain notions, es- pecially the import;mt ona? that a point imposed directly on another results in only a

single p&n, are to be found only in Boethius (Jrithme~ica 2.4, ed. Friedlein 1867).

In a sense, Martianus Capellas De au/&s may be regarded as an encyclopaedic work,

depending as it does on Euclid as well as Nichomachus for its arithmetical resources.

In the DP geamebia (Book 6) Martianus Capella indulges himself in a geographical tour, ending with the briefest of remarks on geometry itself; the De arithmetica (Book 7) is

a far more systematic exposition of the familiar basic principles found, with indiT:id- ual variations, in all these writers on arith-

metic and the arithmetical geor retry of linear, plane and solid numbers. Bo3k 8 and

9 deal with astronomy and mu ic. The

156

numerous surviving manuscript: (Dick 1969 :

xxxv, ff.). show that this work, pompously

affected as it appears to mcdern tastes, often

obscure and unnecessarily difficult, possessed

considerable attractions for medieval readers;

perhaps, viewed simply as an encyclopaedi.a,

it should be considered an important source

of mathematical knowledge.

number is necessarily plural. One is the

source and origin of m.mter, but not in

itself a number:

The writers of all these primary and

secondary works on the theory of the subjects of the quadrivium shared certain common

opinions about what it was important for the

student of mathematics to know. Their

preoccupations are reflected in the USC

scholars made of their material in the eleventh and twelfth ccncuries. First came a definition

of number, in which it is made clear that

of equality and inequality is dependent upon that of plT_u-ality. The .author of the Victorine

We meet a variety of types of jbaritai

Gloss explains that, just as unity is the

in Abelard; paritas secundum praedicationem,

beginning of number. so equality is the be-

ginning of mt2ltiplicita.s; equality presupposes

secundum comitationem secundtim quantitatem, se-

the existence of two equals; in more-than-two

lies the possibility of larger and smaller, and thus of inequalitas: sicut enim u?zita.s

principium est numeri ita . . . equal&s principium est multiplic5tatis (Hgring 197 1 :499). For Boethius, the notion

mathematics, again we find nuInerous traces. Abelard tells us that he has heard in his youth many solutions drawn from arithmetic :

m vitas ab arithmeticis solutiones audierim (de Rij k 1956:59). The author of the Victorme C!.oss

says that when Augustine wished to der:ron-

st rate so.mething ineffable, he turneo to mathematics : confugit ad mathematicum (Ha ling 1!~71:4!#). The Chartrean Lectiones on the De TrimWe mention mathematicae rafiones (Haritg 1971:154) and the difference be- tween the demr,nstrations of mathem:itics and th: proofs of*dialectic is mentioned in a twelfth-century logical treatise : silloglsmus alks demonstrutiws - vel philosophicus sive mathematicus sive apodicticus - a&us dialecticus, alks temptativus, alius sqpksticlls (de Rijk IQ67 280). These means of formal demon- -G _ stration based on mathematics come into their own in the handling of that complex of theological prloblems which commends itself particularly in the eleventh and twelfth centuries to solution by mathematical means:

those of the Trinity and the two Natures of Christ. Boethius points out in the Contra Eztychn that in his day, in all the debates of the c:pposed factions of here&s, the subject- matter of debate is concerned with persons and natures (Stewart and orhers 1973:76); in the debates over the paradox of the Triruty, .hree in one presen.ts a numerical

impossibiity, and for the contemporaries ofThierr:J of Chartres the paradoxes of three- in-one and two-in-one demanded arithmetical aids to solution.

In the remainder of this paper, two specific

ways in which mathematics was wed will be considered: in textbook commentary on the DC qilantitate passage in the Categories, an4 in the pun&urn stiperpuncto met x~d of proving the unity. of the Trinity, whit h is employed by

a variety of wri?ers of he eleventh and

twelfth centuries, and which helps to suggest that the study of the subjects of the quad- rivium provided a more than mechanical aid

to proble:m-solving. It was of (course Bot:t.hius commentary on

the Ca,tegork which gave definition to the mathematics! aspects of Aristotles remarks

on quantity (Cutegorkr, Rook 6) and which encouraged later writers to follow him in their interpretations. Arist :ule makes a series of points: quantity is either discrete or

continuous. Among discrete quantities,

Aristotlle names number and speech; of continuous quantities, he lists line,

surface, :;olid, timf, place. Some

quantities abut onto, I r have positions

rekctivc to, others. TW 1 ves make ten, but there is no common 1i.n-G~ at which the two fives mus- join. So numb :r is discrete; simi-

larly, spel*ch, in the sense of the spoken word, consists of a series of sepai.ate syllables which

never merge into one ano:hcr. Continuous quantity I~, however, to be

found in ths iine, the pane and the solid, which are bounded: r?speztively, by a terminal point, line or phne. There is a point

at which a line may be sanl to snd : just as the present, past and future are linked in con- tinuous time, or here and there, in space,

so the parts of a line arte continuous, up to the point where the line en&. The parts ofa line,

or a plane or a solid have positions which can be ascertained, andl whoseorder is fixed. But the elements of a number occupy no necessary relative positi~ons; the parts of

time - moments or instants - do not endure,

and so their relative positions cannot bc dctermlned.

Some lquitntities arc relative: great or

small. &ml.: are absolute, suc11 a? two cubits. Quantities have no contraries. Pr. is

peculiar to contraries that we comp,tre and

contrast them on grounds of equality and

inequality: of all qcalities we can predicate

equal or unequal. This is the material with

which &?stotle f%nijhes Boethius and his

successors, and they find ip it meat for a

great deal of mathematical discussion.

Boethius, incidentally, provided Latin equiv-

alents for Aristotles terms, and helped to

make them technical terms in their own

right.

In his Categories commentary, Boethius,

closely followed by Garlandus Compotista,

early introduces the idea of number; he

appears to be associating, if not equating,

quantity with number. He first points out

that everything which exists falls into the province of number : in numerum cadunt (MPL

64:201). Everything is either one or many

(compare Abelard: vel unum vel m&a, de Rijk 1956:56). But not everything possesses qual-

ity, so, in Boethius view, the treatment of

number!quantity properly comes first. He

goes on to explain that every body (CO$JUS)

possesses three dimensions, and these three

dimensions are quantities both in number

itnd in spatial continuity: sed tres dimensiones et numero et continuatione sjatii quantitates sunt (MPL 64~202). Thus from the first, Boethius

enlarges the mathematical dimension of what

Aristotle has to say and sets out to emphasise it. Quantity is either discrete (disgregatum) or

continuous (continuum) (MPL 64: 203).

Boethius attempts to clarify Aristotles mean- ing here. As an instance of the discreteness

of the syllables in speech (oratio) he gives the word CZi-ce-ro. If the syllables are jumbled,

the result is nonsense, but the fact that they

can bc jumbled at all indicates that they

remain separate and separable elements in

the col:.lposition of the word. Their parrs are

not joined by an.11 common term: disgregatum est cuius partes null0 communi termino conjunguntur

(MPL 64:2(13). This problem of the common

t :rm or boL.ndary is the crux of the matter.

iarts which ihare a common boundary cannlot

be separated without the boundary being

torn away from one of the pair; they are

envisaged by Boethius and his successors as

being like Siamese twins. I:: is in this sense

that certain quantities are said to be con-

tinuous. Garlandus Compo&a, again faith-

ful to Boetlius, explains that continuous

quantity is that in which the parts join at a

common boundary, while discreta quantitas has no common point of union with any other

quantity (de Rijk 195,1:22). But, says Boethius, again in the commcn-

tax-y on the Categories., supposing a line, for example, is divided in two. Each part of the

line will now end in a point. We must

therefore assume that before the line was

divided, those two points lay side by side within the line, because the point itself is

indivisible; the line cannot have been c:.lt

through a point (M.FL 64:204). Yet if tlic

whole quantity which is a lint can tillus bc

divided into self-conllained parts, the lir,e cannot bc a continuous quantity in the ser,~

already defined. It is a property of con-

tiriuous quantity that if one part is movetl,

the rest must follow; if I throw a stick into thz air, the whole stick traveis as one (:MPL

64:204-5). In discrctc quantity this is not so; I can move one away from ten and

letve nine. Must we then argue that tt-.,c

pints do not make up or constitute the line,

but merely provide it with ends? in this way,

we -,an see how a line may function as a

conr inuous quantity. B,,ethius has put his finger on a weakness

in P ristotles argument at which his successors

rep~~atedly worry, employing all their skills

of ;.rithmetical geometry. IJnlike Aristotle,

the! have to reconcile the Djichomachean

150 . .

view that a linz is necessarily made up of a

Per~ieS of points, with the problem which

Bo&ius has isolated for them. Garlandus

says that according to the dialectici.ans

(set.un&n dialecticr r) thr line IS made up of

three points [or zn ode, number of points?]

and the middle point is the common boundary

(de Rijk 1954:22). Abelard reminds us that

the line cannot be divided through a point (per punctum) because the point is indivisible, hul: remarks that it seems that the knife

cannot be taken between two points, either, ,:ince they are contiguous and there is no ;pace between into which the point of the knife might be inserted (de Rij is 1956:60). lhis I iew brings home the fact that thcse

:hi-nkers envisaged what Aristotle describes n an immensely literal and concrete fashion

and the problems they perceive are cor- :-es?ondingly concrete -- aimost practical. .%belard acknowledges that the notion of the iwo-point or three-point line - the line composed of only two or three points - is not

easily conceived or perceived by the human mind (de Rijk 1956:58). But it raises special :rnd important problems. If the linea bipunctalis is tli\.ided, it ceasrs to be a line at all, but :-esoh,es itsel into two points; it is therefore ;t simple line, whereas the trij!wnctaris is composite, because even when divided, it contains a residual two-point line. The quadri- jhctalis is the shortest line which can be divided into two equal parts (de Rijk 1956:72). The same details are discussed by

r.he author of the Gfosa Worina, so we may I:ake it that the special problems to do with l.wo- and three-point lines were under dis- cussion in the schools (H&ring 1971:544).

The author of the Victorine Gloss uses the material i:? his efforts to show that, in the Trinity, the &sons arc not united in the way

l.hat three points are joined to make a

three -point line. 11 would seem that textbook

corn~,lentary had, in this instance, generated at least on; mathematical notion which was

of direct relevance :o the speculative theo-

logian. The commentators add to Aristotles classi-

fication the division of quantities into simple

and composite - a categorization which is not

quite synonymous with the division into discrete and conrinous quantity. Abelard lists

five simple, seven composite quanti . . s: the

poirrts 01 units of geometry, arithmetic, time, spee:ch, place, punctum, unitas, instans, elementem (= uox- ir1hiuidua or single sound), simplex locus, a.nd their compounds. Iinea, superj&s corpus. ternpus, l0ch.r corn htus, rbratio, numews (de Rijk 1%6:56). TIP complicate matters, a linea bipuwctalis must be regarded as a simple line, beca.use it cannot be divided

and remain a line; so - as a line - it is indivisible. Such refinements of Boethian exposition all h;id their usefulness: the simpleness and inclivisibility of the God-

head was a prime datum of theology. Certain quantities, says Aristotle, have

prcjperties of place and position. Boethius experiments with the results of positioning a yoint ori a line, a line on a surface, a surface on a solid. Ihere are, hc says, three points to be considered:

Ergo haec trix consideranda sunt, ut ad se invicrm positionem pertes habere videantur, id ~xt locum ir, quo partes ipsae sun1 pwilae, ut park- i!lae mm peteant, ut sit partium continentia atqucm. ~inuat I , (hfPL 64 :2(h6),

the relative and ah:;olul:e place or lllisition of t:le parts, the mc:~.u~s by which tbcy may

be envisaged as Farts of somethi!lg else,

without. perishing ;:z individuals, and their discrszteness or coninuity. (We might add the

question : in what place is place itself?

Locus autem in loco. esse z$se non poterit.) Garlandux, again, bases his remarks closely

upon those of Boethius. Abelard pursues his

own line of thought. In a line or surface or

solid, the parcs occupy unalterable position.;

in relation to one another, side by side; r.hr,

parts of time - instants - do not endure, and

so they can have no position; the parts oi

number do not lie side by side in a con-

tinuous sequence (de Rijk 1956:73-4). Hc

agrees with Boethius that, although the

positions of the parts of time cannot be

established, their order can (MPL 64:207;

de Rijk 1956:74) but later, under qualities,

hc takes the problem up again, in order to

point out that although we understand the

notion of position in relation, for example, to

a iine or a surface, it is something quite

different to suggest that line, surface, body,

are themselves positions; they are not pos-

itions, but quantities: net tamen sunt positiones, sed quL atitates (de Rijk 1956 : 100). It is in such ways that these mathematical riddles re-

peate:!ly reappear. If they are nut central preoccupations, they are, at any rate, evi-

dently niggling problems at the back of

Abelards mind. It would perhaps be tedious to describe

here the way in which .4beI;rrd ant1 his

fellows handle the problems raised by the

fact that quantities have no contraries, and

that it is peculiar to qua] tities that we

compare them on grounds c,i equalit)l and

inequality. Abelard, for examDle, gives rather

less space to these issues thal he does tr, the

point-and-tine questions. This Bould suggest

that it is in arithmetical areas ti here Aristotle,

and Boethius rendering of t!le Nichomachean

Arithneh, most obviously Tlret rhat con-

temporary interests were chil:fly aroused. This \,iew appears to be borne out by the

number of occasions on WE ;rh points and

lines are introduced into discussions about thr

unity of the Trinity. One multiplied by

one is one. One plus one is two.

If God begets God, there is still only one God,

says the author of the iictorine Commentary-

on the De Trinitate; the begetting of the Son

by the Father is analogous with the multipli-

cation of one by one (Haring 1971 :498).

Anselm makes use of a similar idea, in his

punctum analogy in the De incarnatione verbi:

ideo panctum cum punc:to sine intervallo non est nisi unum punctum: situ aeterniras cum aeicrnitatr non est nisi una aeternitas. Ergo c:uoniam deus aeterniias est, non sunt plures dii: quia Ned est extra deum net drug in drum addit numerum dw:l

CYe might compare the further use of this

analogy in the De processione, where hnselm shows, by comparing point on poinr, line on

line, that if God begets God, H,* is not

increased in quantity; nor does He become plural : si in se rt$icentur, net quantitatem augent net pluralitafem admittunt (Schmitt 1940:218). What is crucial here is a notion derived from arithmetical geometry, that of

the intervallum or dimension. Bocthius explains in the De arithmetica (2.4) that if one point is iml)osed on anotlu?r without going into

another dimension, the result must be still

a single point, line or surface. In the ,work of the Greek arithmeticians to whom he is

indebted, it IS a cardinal doctrine that in

order to depart from unity, some sort of otherness must be introduced. The same

idea is to be found in the Lectiones on thtb

De Trinitate: From unity proceeds all other-

ness ; ab unitate descent it omnis alteritas i,HSrin;

1971 : 155). If an additional point or line is to

be placed outside thle first, it must bc

placed in a higher dimension, Abelard speaks

of distance, spacing, dispositio (clc Ri.ik 1956:6?). Thtk author of the Victorine Gloss

161

c:xplains that, in order to form a line, the

points must bc set side-by-side and outside

one another: iunck ad invicem sed sunt extra :I invicem (Waring 197 1:545). In Gilbert of :Poitiers commentary on the Contra FU&he?Z

of Boethius, he remarks on Boethius dis-

cussion of the proposition that the two :\Tatures lay in Christ like two bodies side by

side (Stewart and others 1973:92). Gilbert suggests an analogy with two stones, one

bla.ck, one white, neither of which bestows ;tnything of its colour on the other as a result of their proximity. He, too, bases his thinking on the assumption that the pun&urn is a discrete unit which can only lie outside

another poilrt, contiguous but not continuous Tnrith it (Laier, he considers the implications of commixtio; coaiunctin and compositio in con- nection with the two Natures (Haring ! 966:290-2). It seem:;, then, a presupposition of ali these thinkers that since a point possesses no dimension, if another point is to

be set beside it that must be done in another dimension, and so a line will be formed; since a line possesses only one dimension, if another line is set. beside it that must be done

in another dimension, and so a surface is

formed. There is, of course, no otherness in God,

and so, as the author of the Victorine Gloss

t mphasises, the Trinity may not be compared tvith a tripunctakis linea in which the three lersons stand side by side. ibelard makes a

rather different use of the principle, in his discussion of the problem Aristotle leaves unresolved in the Categories: ma; a line be said to be composed of points, or merely terminated by points? He shows that if the points are to be envisaged as making LIP the

line,, they must be considered to be di.$osita, set out along thcb line side by side, each point outside its neighbour in the dimension of

lengthl. He cites Boethius remarks directly, e\ven quoting him (de Rijk 1956:59); later,

he extends his rcma.rks to show that, as in space, pomts without intervals add up to no

line, so in time, moments without intervals

(simultaneous moments) involve no passage

of time, while moments divided by intervals

add up to measurable time (de Rijk 1956:61) ; compare Anselms use of the notion in con- nection with eternity (Schmitt 1940:33-4).

Given so concrete a picture of the nature

ofnumber:;, where the unit is a discrete whole, used as a building-bllock in order to make up number-based geometrical patterns in ever- higher dimensions, it would have been difricult for any medil:val thinker to disengage

himself from the assumption that he will find both unit and dimension in the composition of every plural entity, and unity without dimension in every kind ofsingleness. Abelard

puts it plainly: non so/urn punctorum pluraLik2tem exigit, verum etiam ctrtam ewum dispositionem lon,yitudini.s continuatiorwm (de Rij k 1956 :62). The existence of a :ine demands not onl) plurality of points, but also their disposition

along the dimensiGn .jf length. Thus to add is to move a counter (the

unit) so many stages along a (linear) dimen- sion. It would not bl: clu;te true to say that the differen:-e envisa,qed i3y Anselm and his

successors between multiplying one by one and adding one to one is a difference in kind.

To multiply five by two, for example, is merely to add two to two five times. But it is perhaps #worth noting that certain multipli-

cation enuercises had ii special interest for the

student of Nichomachean arithmetic. If the two first linear numbers (2,3) are squared, they produce the iin,t two plane, or square numbers (4,9) (see F igure 3 above). if they are multiplied by t!hemselves again, the

result is tire first two solid or cubic numbers

162

(8,27). So much may seem obvious, but the been thoroughly impregna ted with those process of multiplication in these special elements in the quadrivium subjects which instances appears to raise the numbers into were of philosophical .snd theological - even progressively higher dimensions. The exercise cosmological - relevance. Perhaps th,:rc is might Se eupressed in terms of successive more behind, additions, but it is multiplication which gives it that symmetry which so pleased the medieval mind. It was especially pleasing to those trained in arithmetical qeometry Notes who were accustomed, in a way that we are not, to think of numbers as linear, plane and solid numbers. But one multiplied by one is still one. The operation does not give rise to a number in a higher dimension. One is a special case. One, however, is not a number at all, because numbers arc always plural, and it possesses no dimension

(0 x 0 = 0). It becomes a number only when it becomes plural; it is the origin and source of number only in that sense. Thus, to multiply one by one does not involve the generation of plurality, because there is in that activity nothing of the process of separation, spacing, distinction, other ne;s, whicl is essential to plt,rality. It would seem perfectly proper that the Divine Unity should display such singlllar characteristics even at the level of simple arithmetic, and a number of thinkers took full advantage of their knowledge of arithmetical geometry to argue this notion in defence of the doctrine of the Trinity.

It would be premature as yet to do more than hint at the full richness of the contri- bution of the study of the quadrivium to the thought of the eleventh and twelfth centuries, as it would be to attempt to assess the extent to which the textboeoks were actually studied. But in the perhaps limited areas where arithmetical notions had been traditionally rmploycd in discussion and commentary, a number of minds show themselves to have

I L.M. de Rijk suggests in a footnote (de Rijk 1956:59) that the Magister would probably have been WilliamofChampeaux. I should like to thank l.he Mas- ter and Fellows of Corpus Christi College, Cambridge for permission to consult, cite and reproduce their manuscript of Boethius Ai$hm&a. 2 Avranches, Bibliothi*que Municipale MS. 236, src Nortier 1971:204. 3 According LO Thornciike (1913:544), one of the treatises most frequently encountered in early mediae- val manuscripts. 4 And so a point put togei ht*r with a point, but without an intervening dimeniian, makes only one point, just as eternity added to etlernity makes only a single eternity. Therefore, since God is eternity, there are not several gods. For God is not external to God, nor does God in God confer the property of number on God (Schmitt 1940:34).

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