Achard of St. Victor (d. 1171) and the Eclipse of the Arithmetic Model of the Trinity

8/16/2019 Achard of St. Victor (d. 1171) and the Eclipse of the Arithmetic Model of the Trinity

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ACHARD OF ST. VICTOR (D. 1171) AND THE ECLIPSEOF THE ARITHMETIC MODEL OF THE TRINITY

By DAVID ALBERTSON

In the first book of De doctrina christiana, Augustine of Hippo famouslyteaches that only the Trinity is to be enjoyed; all other things and evenpeople are to be used toward this singular end. The brevity of Augus-tine’s passing remarks on the Trinity gives no hint that he will laterdevote many pages to the topic. He writes:

These three have the same eternal nature, the same unchangeableness,the same majesty, the same power. In the Father there is unity, in theSon equality, and in the Holy Spirit a harmony of unity and equality.And the three are all one because of the Father, all equal because of theSon, and all in harmony because of the Holy Spirit.1

These few lines comprise the lengthiest discussion of the Trinity in Dedoctrina christiana. We can now trace Augustine’s triad of unitas, aequali-tas, and concordia to a saying of the neo-Pythagorean Moderatus of Gades

(Cádiz) (fl. ca. 50 CE), as reported by Porphyry.2

Marius Victorinus hadalready adopted a portion of the same passage on Moderatus (along withothers from Porphyry) when formulating his own Pythagorean analogyof the Trinity.3 But Augustine, immediately after introducing his triadin De doctrina christiana, repudiates the high-minded philosophical anal-

1 “Eadem tribus aeternitas, eadem incommutabilitas, eadem maiestas, eadempotestas. In patre unitas, in filio aequalitas, in spiritu sancto unitatis aequalitatisqueconcordia, et tria haec unum omnia propter patrem, aequalia omnia propter filium,conexa omnia propter spiritum sanctum” (Augustine, De doctrina christiana, 1.12

[V.5], ed. R. P. H. Green [Oxford, 1995], 16–17).2 Moderatus had suggested that Pythagorean philosophers adverted to numbers asa pedagogical device, since the primal forms of things, being invisible and difficultto conceive, are best conveyed by definite numbers. The number three, he states,represents the perfection of things, while the number one denotes “unity [henotês],equality [isotêtos] or sameness, and the cause of harmony [sympnoia] and sympathy”(Porphyry, Vita Pythagorae 49, ed. August Nauck, Porphryrii philosophi Platoniciopuscula selecta [Leipzig, 1886; Hildesheim, 1963], 44:8–12; trans. in K. S. Guthrie,The Pythagorean Sourcebook [Grand Rapids, 1987], 133).

3 “Deus est monos, monadem ex se gignens, in se unum reflectens ardorem. . . .Sic quidem etiam in multis: unaquaeque unitas proprium habet numerum quia super

diversum ab aliis reflectitur” (Françoise Hudry, Le Livre des XXIV Philosophes:résurgence d’un texte du IVe siècle [Paris, 2009], 150). For details on Victorinus’ssources, including Vita Pythagorae 50–51, see ibid., 24–29.


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ogy, calling it an example of the painful failure of language in the face ofGod’s ineffability.4 Indeed, the bishop of Hippo disowned the idea alto-

gether, never repeating it again. There is no trace of the triad in Augus-tine’s earlier works, despite his avowedly Pythagorean views of numberin De ordine, De musica, and De libero arbitrio, and despite the arithmo-logical interests of his several commentaries on Genesis.5 Nor does Augus-tine so much as mention the triad in the fifteen books of De trinitate thathe wrote two decades later.

Despite such inauspicious origins, Augustine’s triad of unitas, aequali-tas, and concordia in hindsight now belongs on any short list of classicalanalogies of the Trinity. Medieval readers of Augustine found it a fruit-ful analogy of Trinitarian relations and considered its meaning alongside

other notable triads: Hilary of Poitiers’s aeternitas, species, usus, muchdiscussed by Augustine and then by early scholastics; Augustine’s ownmore noteworthy memoria, notitia, amor from De trinitate; variationson the Plotinian triad of One, Mind, and World Soul, such as survivein Macrobius; later Neoplatonic triads conveyed from Ps.-Dionysius’sDivine Names via John Scotus Eriugena’s translations (mansio, proces-sio, reditus; or less commonly, esse, vivere, intelligere); and finally, thecontroversial potentia, sapientia, benignitas from Hugh of St. Victor andPeter Abelard.6 Peter Lombard carried Augustine’s triad of unity, equal-

ity, and harmony into the Sentences, and Aquinas found a home for it inthe Summa.7 But where these authorities cited the analogy as a confir-mation of standard accounts of intradivine relations, its most noteworthy

4 Augustine, De doctrina christiana, 1.13–14 (VI.6), ed. Green, 16–19.5 See Christoph Horn, “Augustins Philosophie der Zahlen,” Revue des Études

Augustiniennes 40 (1994): 389–415.6 The triad has long been viewed as an invention of Peter Abelard, but new

research by Dominique Poirel has raised the possibility of Hugh of St. Victor’s pri-

ority (Livre de la nature et débat trinitaire au XIIe siècle: Le De tribus diebus deHugues de Saint-Victor [Turnhout, 2002], 345–420). See further, however, ConstantJ. Mews, “The World as Text: The Bible and the Book of Nature in Twelfth-CenturyTheology,” in Scripture and Pluralism: Reading the Bible in the Religiously PluralWorlds of the Middle Ages and Renaissance, ed. Thomas J. Heffernan and Thomas E.Burman (Leiden, 2005), 95–122; cf. comments by Boyd Taylor Coolman and HughFeiss in Trinity and Creation, ed. Boyd Taylor Coolman and Dale M. Coulter, Victo-rine Texts in Translation 1 (Turnhout, 2010), 28–35 and 52–58, respectively. On theprehistory of the triad in ancient Greek philosophy, see further P. L. Reynolds, “TheEssence, Power and Presence of God: Fragments of the History of an Idea, FromNeopythagoreanism to Peter Abelard,” in From Athens to Chartres: Neoplatonism and

Medieval Thought: Studies in Honour of Édouard Jeauneau, ed. Haijo Jan Westra(Leiden, 1992), 351–80.7 On Peter Lombard, see below; cf. Summa theologiae I, q. 38, art. 8.


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defenders, Thierry of Chartres and Nicholas of Cusa, relished the deepermathematical mystery that the three words seemed to conceal.

In his boldly naturalistic hexaemeral commentary from the 1130s,Thierry of Chartres (d. 1157) breathed new life into the Augustinian triadby illuminating its connections with contemporary medieval science.8

Thierry sought to interpret the six days of creation secundum physicam,and likewise the Creator, so to speak, secundum quadrivium. Unitas andaequalitas were the grounding principles, respectively, of arithmetic andharmonics (or music), the two pillars of the mathematical disciplines inBoethius’s account.9 Thierry went on to pursue this arithmetical recon-struction of Augustine’s triad throughout his commentaries on Boethi-us’s De trinitate in the 1140s and 1150s.10 He adjusted Augustine’s third

term from concordia to conexio, a minor amendment but one that drewattention to the implicit arithmetical link between unity and equality.Numerical oneness, explained Thierry, can be multiplied by itself andremain oneness. This perfect self-equality of unity is an analogy of theSon’s generation by the Father. But the oneness that results after theself-generation stems both from unity and from unity’s generated equal-ity: it is their connection.11 Hence the intrinsic unity, equality, and con-

8 Thierry of Chartres, Tractatus de sex dierum operibus 30–47, in Commentaries on

Boethius by Thierry of Chartres and His School, ed. Nikolaus M. Häring (Toronto,1971), 568–75.

9 Boethius, Institutio arithmetica 1.1.4, ed. Jean-Yves Guillaumin, Institution Arith-métique (Paris, 2002), 7. Without being able to explore this further, one shouldnote that in the late eleventh century a new office for the feast of the Trinitywas instituted at Cluny that included this antiphon: “In patre manet aeternitas infilio aequalitas in spiritu sancto aeternitatis aequalitatisque connexio.” The officewas compiled by Stephen of Liège, who drew from Alcuin’s prayers and treatises,which in turn borrowed especially from Marius Victorinus’s theology of divine unity.See Hugh Feiss, “The Office for the Feast of the Trinity at Cluny in the LateEleventh Century,” Liturgy O. C. S. O. 17.3 (1983): 39–66; cf. Josef Andreas Jungmann,

“Marius Victorinus in der karolingischen Gebetsliteratur und im römischen Drei-faltigkeitsoffizium,” in Kyriakon: Festschrift Johannes Quasten, ed. Patrick Granfieldand Josef A. Jungmann (Münster, 1970), 691–97 and Pierre Hadot, “Marius Victori-nus et Alcuin,” Archives d’histoire doctrinale et littéraire du moyen âge 29 (1954): 5–19,at 7. I owe thanks to an anonymous reviewer for illuminating this connection.

10 See Thierry of Chartres, Commentum super Boethii librum de Trinitate 2.30–38,in Commentaries, ed. Häring, 77–80; idem, Lectiones in Boethii librum de Trinitate5.16–19, in Commentaries, ed. Häring, 218–19; idem, Glosa super Boethii librum deTrinitate 5.17–29, in Commentaries, ed. Häring, 296–99.

11 “Unitas ergo ex se per semel equalitatem gignit. Unitas enim semel unitas est.Gignit ergo unitas equalitatem unitatis ita tamen ut res eadem sit unitas et unitatis

equalitas. Unitas ergo in eo quod gignit Pater est; in eo quod gignitur Filius est.Unum igitur Pater est et Filius. . . . Amor autem hic et conexio nec gignitur necgignit sed ab unitate et ab unitatis equalitate procedit: non ab uno scilicet illorum


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nection of number make a compelling analogy for the threefold unity ofGod. Thierry’s triad is often called “Pythagorean” because of its evidentbasis in the mathematical arts of the quadrivium, which Augustine’susage lacked.12

Three centuries later, Nicholas of Cusa, “searching for jewels in thedecaying body of scholastic thought,”13 rediscovered the analogy. He firstcelebrated Thierry’s arithmetic Trinity in a sermon of 1440 and thenexplored it at length in his magnum opus of the same year, De doctaignorantia.14 He continued to experiment with the triad in ten differentworks over the next twenty years.15 In his classic study of Cusanus’s the-ology, Rudolf Haubst contended that such triadic images of the Trinityin Nicholas’s writings — chief among them Thierry’s — are the key to

understanding the whole, “like a Sphinx in which all the secrets of hisphilosophy are shrouded.”16

sed ab utroque. Nec enim amor uel conexio unius tantum est. Hic amor igitur etconexio ab unitate et ab unitatis equalite procedens Spiritus sanctus est ut quoniamunitas Pater est, equalitas essendi Filius, a Patre et Filio procedat Spiritus sanctus”(Thierry, Commentum 30, 38, in Commentaries, ed. Häring, 78, 80).

12 See M.-D. Chenu, “Une définition pythagoricienne de la vérité au moyen âge,”Archives d’histoire doctrinale et littéraire du moyen âge 28 (1961): 7–13; Édouard Jeau-neau, “Mathématique et Trinité chez Thierry de Chartres,” in Die Metaphysik imMittelalter, ed. Paul Wilpert (Berlin, 1963), 289–95; Klaus Riesenhuber, “Arithmeticand the Metaphysics of Unity in Thierry of Chartres: On the Philosophy of Natureand Theology in the Twelfth Century,” in Nature in Medieval Thought — SomeApproaches East and West, ed. Chumaru Koyama (Leiden, 2000), 43–73; BernardMcGinn, “Does the Trinity Add Up? Transcendental Mathematics and TrinitarianSpeculation in the Twelfth and Thirteenth Centuries,” in Praise No Less Than Char-ity: Studies in Honor of M. Chrysogonus Waddell, ed. Rozanne Elder (Kalamazoo,2002), 237–64.

13 R. W. Southern, Scholastic Humanism and the Unification of Europe, vol. 2: TheHeroic Age (Oxford, 2001), 88.

14 Nicolaus Cusanus, Dies Sanctificatus (Sermo 22), §22, in Nicolai de Cusa opera

omnia, vol. 16.4, ed. Rudolf Haubst and Martin Bodewig (Hamburg, 1984), 346. Cf.De docta ignorantia 1.7–10, §§18–29, in Nicolai de Cusa opera omnia, vol. 1, ed. ErnstHoffmann and Raymond Klibansky (Leipzig, 1932), 14–21.

15 Cusanus’s version of the arithmetic Trinity has been well explained elsewhere.See Giovanni Santinello, “Mittelalterliche Quellen der ästhetischen Weltanschauungdes Nikolaus von Kues,” in Die Metaphysik im Mittelalter, ed. Wilpert, 679–85; Ber-nard McGinn, “Unitrinum Seu Triunum: Nicholas of Cusa’s Trinitarian Mysticism,” inMystics: Presence and Aporia, ed. Michael Kessler and Christian Sheppard (Chicago,2003), 90–117; Jan Bernd Elpert, “Unitas-Aequalitas-Nexus: Eine textkommentie-rende Lektüre zu De venatione sapientiae (Kap. XXI–XXVI),” in Nikolaus von Kues:De venatione sapientiae, ed. Walter Andreas Euler, Mitteilungen und Forschungs-

beiträge der Cusanus-Gesellschaft 32 (Trier, 2010), 127–82.16 Rudolf Haubst, Das Bild des Einen und Dreieinen Gottes in der Welt nach Niko-laus von Kues (Trier, 1952), 1.


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I mention Cusanus only because Thierry’s triad is so often approachedthrough the lens of its reappearance in Cusan theology.17 In fact, it wasthis concept that first linked the two authors in the minds of historians ofmedieval philosophy.18 And it is true that until its reprise in the works ofthe German cardinal, Thierry’s arithmetic Trinity did not fare especiallywell, either in the twelfth century or in the centuries that followed. Butthis common comparison with Cusanus can also occlude a better under-standing of the fate of Thierry’s doctrine in the decades after its initialdebut and before its eclipse around 1160, when both political conditionsand theological distinctions shunted attention away from the Bretonmaster’s radical rereading of Augustine’s triad and toward the conserva-tive gloss favored by Peter Lombard and the tradition after him.

These circumstances convey at least three disadvantages. Students ofThierry are left rightly wondering why the famed teacher failed to winmore support for his ingenious rereading of Augustine. Likewise, stu-dents of Cusanus, lacking evidence of any other significant engagementswith Thierry’s ideas before 1440, may assume that the German cardinalsimply transcribed the triad into his texts, much like Thierry’s studentsauditing his Parisian lectures. This is, of course, false: Cusanus changesthe triad as it suits him and experiments with new applications withinthe altered parameters of late medieval theology.19 Finally, this ahistori-

cal comparison has encouraged some to classify Thierry’s efforts vaguelyas “number speculation,” rather than as participation in a serious, pro-tracted debate over the role of the arts in theological language or as aprovocative rereading of a major patristic authority.20

17 See, e.g., Werner Beierwaltes, “Einheit und Gleichheit: Eine Fragestellung imPlatonismus von Chartres und ihre Rezeption durch Nicolaus Cusanus,” in idem,Denken des Einen: Studien zur neuplatonischen Philosophie und ihrer Wirkungsge-schichte (Frankfurt am Main, 1985), 368–84. Cf. Christian Trottmann’s comparison ofAlan of Lille and Nicholas of Cusa in “Unitas, aequalitas, conexio: Alain de Lille dans

la tradition des analogies trinitaires arithmétiques,” in Alain de Lille, Le DocteurUniversel: philosophie, théologie et littérature au XIIIe siècle, ed. Jean-Luc Solère,Anca Vasiliu, and Alain Galonnier (Turnhout, 2005), 401–27.

18 Pierre Duhem, “Thierry de Chartres et Nicolas de Cues,” Revue des sciences phi-losophiques et théologiques 3 (1909): 525–31; cf. Chenu, “Une définition Pythagorici-enne.”

19 See McGinn’s appendix in “Unitrinum Seu Triunum,” 105–9. Nicholas invents,for example, triads of absolute equality, equality of equality, and their nexus (Deaequalitate, 1459); the unity of love, the equality of love, the connection of love(Cribratio Alkorani, 1461); and possibility, equality, and their union (Compendiumtheologiae, 1464).

20

E.g., Michel Lemoine, “Le Nombre dans l’École de Chartres,” PRIS-MA 8(1993): 65–78. This may contribute to Lemoine’s misperception that although Thier-ry’s mathematical “speculation” would return again with Meister Eckhart and Nich-


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The missing context that would dispel such misapprehensions cannotbe found by turning yet again to Thierry or Nicholas, but by surveyingthe territory between them. To redress this situation, what we require isa richer account of the earliest reception of Thierry’s arithmetic modelof the Trinity. Such an account should register what his students foundimmediately stimulating about the doctrine and how they deployed itbefore the decline of Thierry’s bold interpretation. In a valuable article,Édouard Jeauneau surveyed the legacy of the “school of Chartres,” buthis sketch needs to be fleshed out with reference to sources he did notpossess.21 Moreover, Jeauneau inclines toward the third mistake notedabove. In light of the triad’s popularity with Cusanus and his readers,he remarks, “Pythagorean speculations on the Trinity, dear to Thierry

of Chartres, have had a long history.”22 In fact, a closer inspection ofthe evidence shows that the fortunes of the doctrine were rather morerestricted before its accidental revival in the fifteenth century. BernardMcGinn has recently filled in some of the details of Jeauneau’s narrative,including, crucially, the tension between two hermeneutics of Trinitar-ian theology that structured the discursive options in the mid-twelfthcentury. But McGinn devotes his attention mostly to Meister Eckhart’shenology and then to Cusanus.23 Marcia Colish situates Thierry’s Trini-tarian theology within contemporary debates over theological language

in the wake of Peter Abelard and Gilbert of Poitiers, but without con-sidering the hermeneutical divide noted by McGinn.24 Moreover, none ofthese studies consider the new details conveyed by several newly editedtexts from the decade between 1150 and 1160, including sources fromThierry’s earliest circle of students, from Alan of Lille, and most impor-tantly, as we will see, from Achard of St. Victor.

In what follows, I explore these and other texts in order to build up apicture of the reception of Thierry of Chartres’s arithmetic Trinity before1160. All of them adopt (or are at least sympathetic towards) Thierry’snew reading of Augustine, despite their awareness of the mainstream

olas of Cusa, his theology was already in the twelfth century enjoying great successand broad diffusion (ibid., 73–74).

21 Édouard Jeauneau, “Note sur l’École de Chartres,” in idem, “Lectio philosopho-rum”: Recherches sur l’Ecole de Chartres (Amsterdam, 1973), 5–36. Jeauneau lists thefollowing sources: two shorter glosses from Thierry’s circle, Clarembald of Arras, Deseptem septenis, Helinand of Froidmont’s Christmas sermon, Alan of Lille’s Regulae,and Achard of St. Victor’s De unitate, as well as a collection of others who cite theAugustinian triad without giving it Thierry’s meaning.

22 Ibid., 11.23

McGinn proposes that Meister Eckhart may have mediated Thierry’s arithmeticTrinity to Cusanus. See McGinn, “Does the Trinity Add Up?” (n. 12 above), 258–64.24 Marcia L. Colish, Peter Lombard, vol. 1 (Leiden, 1994), 101–5.


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Porretan or Lombardian alternative, that is, the scholastic unreadingof Thierry’s arithmetic Trinity back toward Augustine’s simpler triadicanalogy. Given the paucity of evidence of Thierry’s legacy between thetwelfth and fifteenth centuries, even this minor contribution can point ustoward a more capacious understanding of Thierry’s Wirkungsgeschichtebefore the Cusan restoration. It can also illuminate why defenders ofthe arithmetic Trinity found the conceptual language of mathematics togrant, as one of them put it, a “refuge” to theology in a stormy period ofintellectual life marked by profound institutional change.

Two Readings of Augustine’s Triad

Having moved from Chartres to Paris around 1124, Thierry was wellregarded by contemporaries from the 1120s on as an erudite educationalreformer and a brilliant if sometimes caustic teacher.25 Thierry seems tohave first explored the arithmetical triad in his Tractatus on Genesis inthe 1130s. He then revisited it in his Commentum on Boethius’s De trini-tate in the late 1130s or early 1140s; his subsequent Lectiones and Glosain the next decade turn to other Boethian topics.26 Thierry’s Commentumgained him some fame, as enthusiastic citations by some of his studentstestify. His renown reached its high point after he became chancellor of

the Chartres cathedral school in 1141, when Gilbert left for Paris. By1155 he had retired to a Cistercian house, where he died two years later.27

These were not the best years to be proposing a new triad as an anal-ogy of the Trinity.28 Thierry had been present at Soissons in 1121 when

25 Life and Works of Clarembald of Arras, ed. Nikolaus M. Häring (Toronto, 1965),4. On what we know of Thierry’s life and his standing in the schools, see J. O. Ward,“The Date of the Commentary on Cicero’s ‘De Inventione’ by Thierry of Chartres(ca. 1095–1160?) and the Cornifician Attack on the Liberal Arts,” Viator 3 (1972):219–73.

26

Commentaries, ed. Häring (n. 8 above), 47. Häring first dated Commentum to1135, but subsequently revised it to 1148. Mews argues for the “early 1120s” (Con-stant J. Mews, “In Search of a Name and Its Significance: A Twelfth-Century Anec-dote about Thierry and Peter Abaelard,” Traditio 44 [1988]: 171–200, at 192). Mewsconcurs (as do I) with Häring’s sequence of Tractatus, Commentum, Lectiones, andGlosa, pace Enzo Maccagnolo (see Maccagnolo, Rerum universitas: Saggio sulla filoso-fia di Teodorico di Chartres [Florence, 1976], 211–15) and Peter Dronke (“Thierry ofChartres,” in A History of Twelfth-Century Western Philosophy, ed. idem [Cambridge,1988], 358–85, at 360).

27 Karen M. Fredborg, Latin Rhetorical Commentaries by Thierry of Chartres(Toronto, 1988), 8–9; Life and Works of Clarembald, ed. Häring, 23–27.

28

On this tumultuous period in which the cathedral schools developed a new pro-gram for theological education, see Stephen C. Ferruolo, The Origins of the Univer-sity: The Schools of Paris and Their Critics, 1100–1215 (Stanford, 1985); Heinrich


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Alberic of Rheims failed to secure Abelard’s condemnation. (Abelardrecords that Thierry betrayed his sympathies during the proceedingswith an ironic remark mocking the papal legate.) But by 1141 Bernard ofClairvaux had successfully sanctioned Abelard at the Council of Sens forhis interpretation of the triad of power, wisdom, and goodness.29 Gilbertof Poitiers was nearly censured at Rheims in 1148, among other thingsfor his contention — in counterpoint to Abelard — that divine unitywas so supreme that Trinitarian persons could differ only by number. 30

By 1160, however, the whole generation that had fought the battles overnew school curricula and their methodological consequences had passedfrom William of St. Thierry (d. 1148) and Bernard of Clairvaux (d. 1153)to Gilbert of Poitiers (d. 1154), William of Conches (d. 1154), and Thierry

himself (d. 1157). Not surprisingly, the decade of the 1160s witnessed aflurry of consolidating activity among cadres of former students. Somedefended Gilbert’s legacy, some worked to abate the suspicions of Ber-nard and William about employing state-of-the-art dialectical analysis intheology, and some sought to rebalance the weight of competing authori-ties, in the pattern of the Sentences of Peter Lombard (d. 1160).31

Historians of early scholasticism have unearthed a network of mastersaround Paris influenced by Gilbert’s methods in similar ways during thisdecade. A partial roster would include Alan of Lille, Simon of Tournai,

Fichtenau, Heretics and Scholars in the High Middle Ages, 1000–1200, trans. DeniseA. Kaiser (University Park, PA, 1998); and C. Stephen Jaeger, The Envy of Angels:Cathedral Schools and Social Ideals in Medieval Europe, 950–1200 (Philadelphia,1994). Two fine surveys are Constant J. Mews, “Philosophy and Theology 1100–1150:The Search for Harmony,” in Le XIIe siècle: Mutations et renouveau en France dansle première moitié du XIIe siècle, ed. François Gasparri (Paris, 1994), 159–203 andPeter Gemeinhardt, “Logic, Tradition, and Ecumenics: Developments of Latin Trini-tarian Theology between c. 1075 and c. 1160,” in Trinitarian Theology in the Medi-eval West, ed. Pekka Kärkkainen (Helsinki, 2007), 10–68.

29 Constant J. Mews, “The Council of Sens (1141): Abelard, Bernard, and the Fear

of Social Upheaval,” Speculum 77 (2002): 342–82. On Abelard’s possible connectionsto Thierry and his similar interests in Plato and the Timaeus, see further D. E.Luscombe, The School of Peter Abelard (Cambridge, 1970), 57–58 and Tullio Gregory,“Abélard et Platon,” Studi Medievali, ser. 3a, 13 (1972): 539–62.

30 For an overview of Gilbert’s academic career, see Theresa Gross-Diaz, ThePsalms Commentary of Gilbert of Poitiers (Leiden, 1996), 1–24. A good introductionto his doctrine of God is Michael E. Williams, The Teaching of Gilbert Porreta on theTrinity, Analecta Gregoriana 56 (Rome, 1951); cf. Lauge Olaf Nielsen, Theology andPhilosophy in the Twelfth Century (Leiden, 1982), 142–63.

31 Marcia Colish argues that Peter Lombard’s engagement with Gilbert’s viewsin the wake of Rheims was both more substantive and more positive than is often

assumed; see Colish, “Gilbert, the Early Porretans, and Peter Lombard: Semanticsand Theology,” in Gilbert de Poitiers et ses contemporains, ed. Jean Jolivet and Alainde Libera (Naples, 1987), 229–50.


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and Peter of Poitiers, as well as others at one remove like Richard of St.Victor or Gandulph of Bologna, in writings ranging between 1155 and1175. Cognizant of Peter Lombard’s looming monument — its final edi-tion was taught in lectures in 1158, but had been underway since themid-1140s — this group was, at the same time, not yet confined by itsarchitecture.32 But, like the Lombard, they analyzed the Augustiniantriad alongside Hilary’s famous and Abelard’s notorious alternatives. Outof a desire to avoid Abelard’s errors, but also to recover a sanitized ver-sion of his triad, all six men invoked the same distinction, one that theysaw as implied in the passage from De doctrina christiana cited above,and that would survive well into the thirteenth century.33

The distinction they proposed is between “proper” triadic analogies

and “appropriated” or “attributed” analogies.34 Names assigned to theTrinity fall into two categories. Some are “proper” names, whose nomi-nal differences articulate real differences among the three divine persons;others are “appropriated” or “attributed” names, whose differences area matter of custom or language only, since strictly speaking they eachname the same undifferentiated divine essence. Hence, “paternity, fili-ation, and spiration” is a clear instance of a proper Trinitarian name,but the triad of “power, wisdom, and goodness” is only an appropriatedname. Abelard’s chief error, from this perspective, was to misconstrue

32 For this dating, see Colish, Peter Lombard, 23–25. On the theologies of unitasand forma in the Porretani and Chartrians, cf. Stephan Otto, Die Funktion des Bild-begriffes in der Theologie des 12. Jahrhunderts, Beiträge zur Geschichte der Philoso-phie und Theologie des Mittelalters (= BGPhThM) 40 (Münster, 1963), 176–99, 224– 50. Otto notes (ibid., 185) that Gilbert never used the mathematical triad.

33 See, for example, Albertus Magnus, Commentarii in I Sententiarum, dist. 31,art. 1–2, in Opera omnia, vol. 27, ed. S. C. A. Borgnet (Paris, 1893), 99–101; ThomasAquinas, Commentum in Quatuor Libros Sententiarum, vol. 1, dist. 31, q. 1, art. 2, inOpera omnia, vol. 6 (Parma, 1856, repr. New York, 1948), 250.

34

On the history of the theory of appropriated Trinitarian names, see LudwigOtt, Untersuchungen zur theologischen Briefliteratur der Frühscholastik, BGPhThM34 (Münster, 1937), 254–66 and 581–94 and Ludwig Hödl, Von der Wirklichkeit undWirksamkeit des dreieinen Gottes nach der appropriativen Trinitätstheologie des 12.Jahrhunderts, Mitteilungen des Grabmann-Instituts der Universität München(Munich, 1965), 5–14, and in more doctrinal perspective, 28–59. Hödl argues that thedivisions of the twelfth century are ultimately the fruit of two responses to Arianismafter Nicaea: “Der Unterschied zwischen der Trinitätstheologie Augustins und desHilarius ist der Unterschied der abendländischen und morgenländischen Theologie,der Unterschied einer an der Proprienspekulation orientierten Trinitätslehre und derappropriativen Trinitätsbetrachtung” (ibid., 50). This situation is then repeated in

the twelfth century and exacerbated by the divide between Peter Lombard’s almostexclusive use of Augustine and Gilbert of Poitiers’s adoption of Boethius and Hilaryof Poitiers (cf. ibid., 26, 35, 52).


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Hugh’s appropriated triad as if it were a proper one.35 All six of theauthors cited above, not to mention their thirteenth-century successors,read Augustine’s triad of unitas, aequalitas, and concordia as an “appro-priated” analogy. As Ludwig Ott notes, if Abelard’s triad created theproblem of appropriative Trinitarian theology, it was Augustine’s triadthat put early scholastic theologians through their paces in exercisingtheir new appropriation theory, as we will examine below.36

Thierry of Chartres, in marked contrast to his contemporaries, quiteclearly viewed Augustine’s triad as a proper name for the Trinity.37 Inthe Tractatus on Genesis where he introduces his doctrine of the arith-metical Trinity, Thierry states that of course “unity” and “equality” des-ignate the same divine substance. But this does not of itself entail that

the arithmetic triad is merely an appropriated name. For “the divinephilosophers assigned the term ‘person’ . . . for the purpose of designat-ing certain properties”; and if to beget is the “property of unity,” to bebegotten is the “property of equality.”38 Therefore, he reasons, the essen-tial properties to which the personal names refer are perfectly expressed

35 As Théodore de Régnon remarked in 1892, “le système d’Abailard est la théo-rie des appropriations, mais renversée” (Ott, Untersuchungen, 256 n. 48). Abelard’sdeliberations begin when he asks whether diversity in the Trinity is real, nominal, or

somehow both; none of the answers is immediately satisfactory. “Aut enim, inquiunt,haec diuersitas personarum in solis uocabulis consistit, non in re, ut uidelicet uoca-bula tantum diuersa sint et nulla sint in Deo rei diuersitas, aut in re sola et non inuocabulis; aut simul et in re et in uocabulis” (Abelard, Theologia christiana 3.90, ed.E. M. Buytaert, Petri Abaelardi Opera Theologica, CCM 12 [Turnhout, 1969], 230).Peter of Poitiers begins his account of appropriative names with an allusion to thispassage: “Fit autem personarum distinctio bipartito: tum appropriatione nominum etrerum, tum appropriatione nominum, sed non rerum” (Sententiae Petri Pictaviensis,Lib. 1, cap. 22, ed. Philip S. Moore and Marthe Dulong, vol. 1 [Notre Dame, 1943],183, lines 5–7).

36 “Dieser Satz bereitete dem theologischen Denken der Frühscholastik, das sich

noch nicht allenthalben zur vollen Klarheit über den Unterschied zwischen Proprie-täten und Appropriationen durchgerungen hatte, erhebliche Schwierigkeiten” (Ott,Untersuchungen, 569).

37 McGinn, “Does the Trinity Add Up?” (n. 12 above), 256 n. 53. McGinn citesLectiones 5.16, but in my opinion there are other passages that provide stronger evi-dence.

38 “Quamuis autem unitas et eius equalitas sint una penitus substantia tamenquoniam nichil se ipsum gignere potest et alia proprietas est genitorem esse queproprietas est unitatis: alia uero proprietas est genitum esse que proprietas estequalitatis idcirco ad designandum has proprietates que sunt unitatis et equalitatiseterna identitate diuini philosophi uocabulum persone apposuerunt ita ut ipsa eterna

substania dicatur persona genitoris secundum hoc quod ipsa est unitas: personauero geniti secundum hoc quod ipsa est equalitas” (Thierry, Tractatus 41, in Com-mentaries, ed. Häring [n. 8 above], 572).


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by the arithmetic terms unitas and aequalitas. Hence the triad of unitas,aequalitas, and conexio is not an arbitrary mathematical symbol (let alonean instance of “number speculation”), but in fact a sturdy conceptualbasis for grasping just what a Trinity of persons might mean. Beneaththe traditional metaphor of generation lies the more solid foundation ofnumber.

Thierry seems to have left aside this concern in Commentum from the1130s, but in Lectiones and Glosa of the next decade, he found it nec-essary to confront the issue, which must have been gaining publicitythroughout the 1140s, given the affairs at Sens and Rheims. In Com-mentum, Thierry had discussed the arithmetic Trinity in an excursus onBoethius’s account of how divine form relates to the different disciplines

of theology, mathematics, and physics (De trinitate 2). But in Lectionesand Glosa, he relocates his comments on the arithmetic Trinity to a pas-sage in which Boethius, too, grappled with predicating Trinitarian rela-tions (De trinitate 5). And even while Thierry simply recycles past formu-lations of the arithmetic Trinity from Tractatus and Commentum in theselines, he also adds a new emphasis: the validity of the names unitas andaequalitas arises from the fact, he says, that they express divine “prop-erties.” God is named “Son” because of the “property” of equality that“Son” designates.39 The members of the triad are not “discrete things”

but rather distinct “properties,” he now maintains, and for this reason,one cannot simply identify conexio and aequalitas, as one might if theywere appropriated names.40 Thierry’s certainty about his position evenemboldens him to venture another triad. The divine Trinity of unitas,aequalitas, and conexio, he writes, is the eternal foundation of the quad-rivium itself: numerus (i.e., arithmetic), proportio (harmonics), and pro-portionalitas (geometry).41

Thus we find two interpretive options for approaching Augustine’striad, represented by Peter Lombard but mainly the Porretani on theone hand, and Thierry of Chartres and some sympathetic students on

39 “Equalitas uero diuine substantie ascribitur per hoc quod est Filius quia inUerbo i.e. in Filio genito a Patre cuncta creauit. . . . Et per talem proprietatemhoc uocabulum filius refertur ad deum” (Thierry, Lectiones 7.7, in Commentaries, ed.Häring, 225).

40 “Istud amborum relatiuum est ad proprietates has quas dixi equalitatem et uni-tatem: non ad res discretas. Non enim est nisi sola unitas: trina tamen in propri-etate. Conexio enim unitas est. . . . Tamen non concedimus quod conexio equalitassit: propter personales proprietates” (ibid.; cf. ibid., 7.6, in Commentaries, ed. Häring,225; idem, Glosa 5.22–29, in Commentaries, ed. Häring, 297–98).

41

Ibid. On the distinction of proportio and proportionalitas and their significancefor the broader study of the quadrivium, see Boethius, Institutio arithmetica 2.40.1– 3, ed. Guillaumin (n. 9 above), 140.


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the other. I will call these, for short, the weak (or nominalist) versusthe strong (or realist) readings.42 In the weak reading, the triad does notdefine the different persons, but only the one divine substance; it doesnot add any new information about intratrinitarian difference. Neverthe-less, one can still parse out the triad provisionally, associating a giventerm with a given divine person, since it may seem more fitting to callthe Spirit “connection” than the Father. But in truth, the Father is asmuch “connection” as the Son or Spirit. But in Thierry’s strong readingAugustine’s triad actually defines the proper names of Father, Son, andSpirit in their differentiation. This is plausible because of the universal-ity, eternity, and stability of the arithmetical names unitas, aequalitas,and conexio (in Thierry’s amended version). Hence the arithmetic Trin-

ity in the strong reading actually does deliver new knowledge aboutGod; “equality” specifies what “filiation” essentially means. On this viewAugustine’s triad provides a supplementary conceptual basis to help fur-ther determine the fundamental metaphors of generation and procession.

Strong Readings by Thierry’s Students

All of the twelfth-century authors whom we can most easily iden-tify as Thierry’s students embraced his strong reading.43 Häring has

identified two short anonymous works on the Trinity that echo several ofThierry’s doctrines, including the arithmetic Trinity, which he designatesas Tractatus de Trinitate and Commentarius Victorinus.44 Häring dates

42 In this way I hope to distinguish this limited case of reading Augustine’s triadfrom the larger debate on the logic of universals between “nominalists” and “real-ists.” For a survey of old and new scholarship on twelfth-century nominalism, seethe special issue of Vivarium 30.1 (1992) edited by William J. Courtenay.

43 The Cistercian Helinand of Froidmont (1162–1237), another important wit-ness of Thierry’s views, is too late to be considered here. Helinand’s Sermon 2 (PL

212:486A–498C) seems to draw on Thierry’s Commentum or Tractatus — his mostwell-circulated texts, particularly among Cistercian monasteries — in its references(sometimes verbatim) to the Son as aequalitas and veritas and to God as formaessendi. At the same time, some passages of the sermon resemble the Johannine exe-gesis of De septem septenis; see, e.g., PL 212:491C on John 14:6. For background onHelinand’s sermons, see Beverly M. Kienzle, “Hélinand de Froidmont et la prédica-tion cistercienne dans le Midi (1145–1229),” in La prédication en Pays d’Oc (XIIe– début XVe siècle), Cahiers de Fanjeaux 32 (Toulouse, 1997), 37–67.

44 Both texts follow the sole exemplar of Thierry’s Lectiones in MS Paris BN Lat14489, fols. 62r–66r (Tractatus de Trinitate) and fols. 67r–95v (Commentarius Victori-nus, formerly called In titulo and attributed to Ps.-Bede). On Commentarius Victori-

nus, see the general account in Ermenegildo Bertola, “Il ‘De Trinitate’ dello PseudoBeda,” Rivista di Filosofia Neoscolastica 48 (1956): 316–33. Nikolaus M. Häring (“AShort Treatise on the Trinity from the School of Chartres,” Mediaeval Studies 18


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the former to the early 1140s and the latter to the early 1150s. Thereare signs that these are derivative works by close students rather thandrafts for another Boethian commentary by Thierry himself, and ifso, they may represent the first moment of the reception history ofThierry’s theology. The treatises closely resemble each other and assumea familiarity with Thierry’s other Boethian commentaries. Both treatisesadd corroborating details to clarify Thierry’s ideas, but stand at oneremove from the originality of the master’s undisputed works, spellingout what remains implicit in his lectures, and at times weighing theirtheological consequences.

Both student treatises are enthusiastic about Thierry’s arithmeticTrinity. They display a palpable eagerness both to apply the quadrivium

within theology and to articulate the sources that authorize this strat-egem in greater detail than one finds in Thierry’s works themselves. InLectiones and Glosa, Thierry simply cites Augustine as the source of theterms.45 The students go further, describing how, armed with this “ver-bal formula,” Augustine could confront the ineffability of the Trinitydescribed in De doctrina christiana and, in response, “take refuge in math-ematical learning.”46 The turn to mathematics, according to Thierry’sstudents, is the path Augustine charted forward in response to the “inef-

[1956]: 125–34) notes several reasons for dating the two works after Thierry’s Glosa,whether they were written by an aging Thierry or by one of his students: “Espe-cially the manner of handling the ‘mathematical’ explanation of the Trinity, basedon the Augustinian dictum cited above, offers impressive evidence to the effect thatboth works belong to the school of Thierry of Chartres” (ibid., 128). Häring wouldlater argue, however, that Commentarius Victorinus could well have been written byThierry himself, noting “very striking points of contact” with the anonymous Trac-tatus de Trinitate (Commentaries on Boethius, 40–45); cf. Ott, Untersuchungen, 571.

45 Thierry, Lectiones 7.5, in Commentaries, ed. Häring, 224–25; Glosa 5.17, in Com-

mentaries, ed. Häring, 296–97.46 “Procedat igitur Augustinus in medium qui trium personarum distinctionem subhac forma uerborum diligens ueritatis speculator assignat: In Patre inquit unitas inFilio equalitas in Spiritu sancto unitatis equalitatis conexio. Sancte Trinitatis statumnon de facie ad faciem intuens ad mathematicam ut ex forma uerborum datur intel-ligi disciplinam confugit ut saltem sic aliquam distinctionis personarum insinuaretnoticiam. Arimetici namque unitatem primum omnium constituunt numerorum prin-cipium” (Tractatus de Trinitate 12, in Commentaries, ed. Häring, 306). “Ad hanc enimpro modulo capacitatis nostre declarandam dicit Augustinus: In Patre unitas in Filioequalitas in Spiritu sancto unitatis equalitatisque conexio uel concordia. Sed sicutex formula uerborum haberi potest uolens Augustinus quoquo modo insinuare quod

ineffabile erat et incomprehensibile confugit ad mathematicam. Arithmetici unitatemprincipium numerorum constituunt” (Commentarius Victorinus 81, in Commentaries,ed. Häring, 498).


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fability and incomprehensibility” of God’s names.47 In their view, Augus-tine does not despair of his Trinitarian formula in the passage but rathercelebrates the foothold won by the “arithmeticians”; namely, that unity,as the source of number, is not a number itself.48 Compared to Thierry,the student treatises also display a more literal appeal to Boethius’squadrivial terminology. Tractatus de Trinitate conspicuously claims theauthority of arithmetica and arithmetici for doctrines that Thierry sim-ply asserts in his text in accordance with arismethica ratione.49 LikewiseCommentarius Victorinus informs readers that Thierry’s notion of “unity”stems from what “Boethius says in the Musica.”50

The two authors are equally committed to connecting Thierry’s math-ematical trinity to past authorities in Trinitarian theology. They fre-

quently cite Augustine and Hilary of Poitiers by name on the Trinity,as well as Ps.-Dionysius on the “theology of negation,” but they shyaway from the Hermetic Asclepius.51 Thierry, on the other hand, namesHermes as often as he does Augustine. Both student treatises cite fromJohn’s Gospel frequently. They also feel a special burden to refute theo-logical heresies. Tractatus de Trinitate, for instance, finds that Thierry’smathematical triad cures two mistakes: that the generation of the Son isa second God (rather than the perfect “equality” of unity), or that theSpirit is generated (rather than proceeding as a “connection”).52

Commentarius Victorinus wishes to revive the theology of aequalitasthat Thierry aired in Commentum.53 The author is also fascinated by oneof Thierry’s more unusual suggestions in Commentum, when Thierry callsthe divine Son the eternal square. Thierry refers here to a prophetic frag-ment known as the Spanish Sibyl that circulated shortly before the Sec-

47 This suggestion by the two student treatises — that mathematical symbols can

work hand in hand with negative theology — is a striking anticipation of Nicholasof Cusa’s entire theological project.48 Tractatus de Trinitate 12, in Commentaries, ed. Häring, 306; Commentarius Vic-

torinus 81, in Commentaries, ed. Häring, 498.49 Tractatus de Trinitate 12, 13, 17, 18, in Commentaries, ed. Häring, 306–7. See,

for example, Thierry, Commentum 2.34 and 4.4, in Commentaries, ed. Häring, 78,96; Tractatus 30, in Commentaries, ed. Häring, 568; but cf. Thierry, Lectiones 3.5, inCommentaries, ed. Häring, 178.

50 Commentarius Victorinus 87, in Commentaries, ed. Häring, 499.51 Tractatus de Trinitate 26, 28, in Commentaries, ed. Häring, 309–10; citations of

both authors in Commentarius Victorinus are numerous.52

Tractatus de Trinitate 13–18, in Commentaries, ed. Häring, 306–7.53 Thierry, Commentum 2.31–36, 2.46–49, in Commentaries, ed. Häring, 78–79,82–84; Commentarius Victorinus 86–88, in Commentaries, ed. Häring, 499.


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ond Crusade.54 Thierry cites a fragment of the oracle that states: “Whenyou reach the side of the eternal sitting square and the side of the eter-nal standing squares.”55 After adducing supplementary rationes fromBoethius’s Institutio arithmetica, Thierry interprets the Sibyl to show thatthis eternal square is in truth the divine Son. We know “by arithmeticreason,” he explains, that two taken twice makes a square; but unitymultiplied by itself is the “first square.” This “squaring” is an instance ofgeneration, and the first and eternal generation is that of the divine Son,the equality of unity.56 Thierry draws his conclusion:

And because the first squaring is the generation of the Son, also the Sonis the first square. But such squaring is a figure. Therefore rightly is theSon named the figure of the substance of the Father (Heb. 1:3). . . . The

square was thus well attributed to the Son since this figure is judged asmore perfect than the others on account of the equality of its sides.57

In Thierry’s gloss, when the apostle calls the Son the perfect figura ofthe Father (Heb. 1:3), he is literally referring to an arithmetical conceptor geometrical shape.58 The fact that Commentarius Victorinus focuses onthis particular doctrine reveals his enthusiasm for Thierry’s agenda to

54 The texts from MSS Munich (clm) 5254 and 9516 are transcribed as two inde-

pendent versions in Wilhelm von Giesebrecht, Geschichte der deutschen Kaiserzeit, vol.4 (Leipzig, 1877), 502–6. The original sense of the Sibyl concerns German noblestraveling first to Constantinople, where the Greek emperor sits eternally and thenobility stand eternally, and thence toward Jerusalem (Giesebrecht, Geschichte, 502).On the Sibyls generally in medieval literature, see Peter Dronke, “Hermes and theSibyls: Continuations and Creations,” in idem, Intellectuals and Poets in MedievalEurope (Rome, 1992), 219–44.

55 “Cum perueneris ad costam Tetragoni sedentis eterni et ad costam tetragono-rum stantium eternorum” (Thierry, Commentum 2.34, in Commentaries, ed. Häring,79). Thierry cites only a fragment of the Sibyl but must have used the version fromMS Munich (clm) 5254, viz. from Otto of Freising’s Gesta Friderici Imperatoris, which

continues “et ad multiplicationem beati numeri per actualem primum cubum” (Gie-sebrecht, Geschichte, 505).56 Thierry, Commentum 2.34, in Commentaries, ed. Häring, 78–79.57 “Et quoniam tetragonatura prima generatio Filii est, et Filius tetragonus pri-

mus est. Tetragonatio uero figura est. Merito ergo Filium figuram substantie Patrisappellat. . . . Bene autem tetragonus Filio attribuitur quoniam figura hec perfectiorceteris propter laterum equalitatem iudicatur” (Thierry, Commentum II.34, in Com-mentaries, ed. Häring, 79). Cf. Thierry, Tractatus 41, in Commentaries, ed. Häring,572: “Est igitur ipsa unitatis equalitas eiusdem unitatis quasi quedam figura etsplendor. Figura quidem quia est modus secundum quem ipsa unitas operatur inrebus. Splendor uero quia est id per quod omnia discernuntur a se inuicem. Fine

enim modoque proprio cuncta inuicem a se discreta sunt.”58 Commentarius Victorinus 95, in Commentaries, ed. Häring, 501; cf. Thierry,Commentum 2.33–34, in Commentaries, ed. Häring, 78–79.


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approach theology through the language of the quadrivium. These twoearly receptions, therefore, champion Thierry’s strong reading of Augus-tine.

A second example to consider is Clarembald of Arras (d. ca. 1187),who proudly counted himself as a student of both Thierry and Hugh ofSt. Victor. 59 Clarembald of Arras studied under Hugh before the Victo-rine’s death in 1141 and under Thierry between 1136 and 1146 in Paris.By 1152, Clarembald was the provost of the cathedral school at Arras,but by 1156, he was promoted to the archdeaconate, a transfer from aca-demic to church affairs. Before his death in 1187, however, it seems thatClarembald enjoyed two leaves from his administrative work to lectureat the famed school of Laon. During his first stay (1157–59) he wrote

two commentaries on Boethius, and during his second (1165–68) he wrotea commentary on Genesis. All were intended as tributes to Thierry’s sim-ilar works. Clarembald discusses Thierry’s arithmetic Trinity in his com-mentary on Boethius’s De trinitate.

Clarembald states his objective in a prefatory letter to his friend Odoof Ourscamp, the noted Cistercian abbot and student of Peter Lombard. 60

Clarembald wrote his commentary in order to provide a clearer guideto Boethian theology than Gilbert of Poitier’s notoriously abstruse one;in particular, he wanted to expose the falsehood of Gilbert’s contention

that persons of the Trinity differ according to number.61

To this end, heexplains his plan to “imitate the lectures” of his teachers Thierry andHugh.62 But Clarembald believed that Thierry’s Trinitarian doctrine was,even more, an antidote to the Abelardian error as well.63 Sifting throughThierry’s different works as a guide, Clarembald opts for Commentumover Lectiones for presenting the arithmetic Trinity and simply repeatsThierry’s formulae verbatim. Given his project of correcting both Abelardand Gilbert with Thierry’s theology, there is no question that Clarembalddesired to reinforce his master’s strong reading of the Augustinian triad.

59 On Clarembald’s life, see Life and Works of Clarembald, ed. Häring (n. 25above), 4–23 and John R. Fortin, Clarembald of Arras as a Boethian Commentator(Kirksville, MO, 1995).

60 Ibid., 17.61 See Clarembald, Tractatus super librum Boetii De Trinitate 1.24–28, in Life and

Works, ed. Häring, 95–97; 1.52–53, in Life and Works, ed. Häring, 105; 2.48–50, inLife and Works, ed. Häring, 126–27; and 3.36–39, in Life and Works, ed. Häring,145–46.

62 Clarembald, Epistola ad Odonem 2–3, 7–8, in Life and Works, ed. Häring, 63–65.63

Häring discusses Clarembald’s critique of Abelard and Gilbert of Poitiers in Lifeand Works, 38–45; cf. Ott, Untersuchungen (n. 34 above), 264; Fortin, Clarembald ofArras, 44–48.


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Clarembald’s own theological interests seem to echo the two studenttreatises considered above. First, he accentuated Thierry’s attention toaequalitas as the Son.64 To Thierry’s account Clarembald adds that theSon’s equality is also the power that effects any scriptural figuration,by rendering the two signs semantically equivalent. This is why spiritualexegesis of the Old Testament requires attention to exact numerical ana-logues in order to decipher the Christological code of a story. “You muststrive to distinguish singular elements,” writes Clarembald, “that is, simi-lar details and ceremonial accompaniments known in the case of Abra-ham and Isaac, so far as possible, also in the figures themselves, underthe same number and the same quantity. Otherwise you will not verywell represent what you wish through figures.”65 Second, much like the

student treatises, Clarembald is fascinated by the passage in Commentumwhere Thierry names the Son the “primal square.” He reproduces it inhis own text and tries to clarify a few of his master’s terms. 66

A third testimony to Thierry’s arithmetic Trinity is found in the anon-ymous treatise De septem septenis. The author indirectly claims Thierryas a master and, like Clarembald, evinces signs of Victorine influenceas well. While the exact date and author of this Hermetic fragmentremain unknown,67 the author attributes several of Thierry’s ideas toan unnamed magister and repeats phrases found in the Commentarius

Victorinus from the 1150s.68

While it is conceivable that the text waswritten after 1160, the author’s implicit reference to Thierry assumes a

64 Of the nine paragraphs on the arithmetic Trinity in Clarembald’s Tractatus, sixconcern aequalitas. During this discussion of the Son as equality, Clarembald citesverbatim from every paragraph of Thierry’s text on the arithmetic Trinity in Com-mentum: cf. Thierry, Commentum 2.30–38, in Commentaries, ed. Häring, 77–80; cf.Tractatus 2.34–40, in Commentaries, ed. Häring, 120–23.

65 “Necesse est ut singula membra i.e. consimilia et officialia quae in Abrahamet Ysaac fuisse cognovisti in figuris ipsis sub eodem numero et eadem si fieri potest

insignire studeas. Alioquin non bene quod volueras per figuras representabis” (Cla-rembald, Tractatus 2.36, in Life and Works, ed. Häring, 121).66 Ibid., 2.38, in Life and Works, ed. Häring, 122.67 PL 199:945D–964D. The sole known manuscript is in the British Museum, Lon-

don, MS Harley 3969, fols. 206v–215v. The final page seems to be missing, sincethe author’s conclusion is broken off in midsentence. The treatise was formerlyattributed to John of Salisbury, but Peter Dronke rightly refers to “the anonymoustwelfth-century author of the De septem septenis, whose precise date and milieu arestill uncertain” (Fabula: Explorations into the Uses of Myth in Medieval Platonism[Leiden, 1974], 35; cf. Carl Schaarschmidt, Johannes Saresbariensis nach Leben undStudien, Schriften und Philosophie [Leipzig, 1862], 278–81; Hans Daniels, Die Wis-

senschaftslehre des Johannes von Salisbury [Kaldenkirchen, 1932], 91–94).68 “Haec, magistrum nostrum sequentes, pro viribus succincte diximus” (Septem,PL 199:960A).


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degree of fame established at Paris since the 1130s, making a date in the1150s or early 1160s likely. Unlike the two student treatises, however,

which retained Thierry’s orientation to Boethius’s De trinitate, the authorof Septem combines Thierry’s concepts with a stew of other influences,appropriating them as it suits his objective. Amidst the controversy overthe new arts curriculum in the mid-twelfth century, and its place vis-à-vis traditional monastic education, Septem seeks to delimit the roleplayed by the liberal arts within a life of contemplation.

In the seventh of the eponymous septets, the author revives the the-ological critique of the liberal arts found in the Hermetic Asclepius.69

Hermes Mercurius taught that even the learned, despite their studies,can remain ignorant of the ultimate principles grounding their disci-

plines.70 Accordingly, the author of Septem compiles seven fundamental“principles of things, or primordial causes.” The first four can be tracedto ideas from Thierry of Chartres, and the first of these is the arithmeticTrinity. The author collates the prologue to John with fragments attrib-uted to Heraclitus, Hermes, Boethius, and the Sibyl, all borrowed from aPs.-Augustinian apologetic sermon. All have perceived and hinted, some-times in cryptic utterances, that God’s Son is co-eternal with God.71 Toclarify and correct such prophecies, Septem adverts to Thierry’s doctrine:

Again [Parmenides] says: “God is unity: from unity is born the equalityof unity. But the connection proceeds from unity and the equality ofunity.” Whence, therefore, Augustine says: “To all those who perceiverightly, it is clear why from sacred scripture the doctors assign unityto the Father, equality to the Son, connection to the Holy Spirit. Andalthough from unity is born equality, connection proceeds from both; yet

69 See Paolo Lucentini, “L’Asclepius Hermetico nel Secolo XII,” in From Athensto Chartres: Neoplatonism and Christian Thought, Studies in Honour of Édouard

Jeauneau, ed. Haijo Jan Westra (Leiden, 1992), 397–420. Lucentini notes that whileAsclepius had been known to Christian thought since Lactantius and Augustine, itsinfluence was greatest during the twelfth century, including such anonymous Her-metic texts as Liber de VI rerum principiis, and is discussed in works by Abelard,Hermann of Carinthia, Bernardus Silvestris, Alan of Lille, and Thierry of Chartres(in Tractatus and Commentum). Lucentini mentions De septem septenis only briefly asa “fragment” by John of Salisbury, noting that John mentions “Hermes Trismegis-tus” once by name in Policraticus. See Policraticus 2.28, ed. Clement C. J. Webb,vol. 1 (Oxford, 1909), 163.

70 Septem, PL 199:960BC, 962D; cf. Asclepius 8, in Hermetica, ed. and trans. Wal-ter Scott, vol. 1 (London, 1968), 301–3.

71

The citations from Hermes, the Sibyl, the Gospel of John, and Ps.-Augustine inSeptem, PL 199:960D–961B stem from Quodvultdeus, Adversus quinque haereses, PL42:1102–3 (“Ex Hermete et Sibylla adversus Paganos”).


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they are one and the same.” This is that unity of three, as Pythagorastaught, which alone deserves to be adored.72

For the author of Septem, the arithmetic Trinity was bequeathed fromPythagoras to Parmenides to Augustine, and thence in turn passed tohis master Thierry. Septem’s repetition of Thierry’s arithmetic Trin-ity is obviously unconcerned with the theory of appropriative names,but clearly endorses a strong reading. This sacred triad, in his opinion,already worshiped by the ancients, is the mystical foundation of thequadrivial sciences of number, and thus legitimates their existence withinthe spiritual life of contemplation.

To summarize: all four of these minor examples of Thierry’s earliest

students championed the master’s strong reading of Augustine withoutquestion. All found the arithmetic Trinity fruitful for further theologi-cal uses of their own devising, including warding off heresies, and theyshared in common a special attention to the second person of aequali-tas. Several of them focused in particular on the image of God’s Sonas an eternal square. But unlike Thierry himself, none of them reflectsany awareness of the alternative weak reading of Augustine’s triad inaccord with the theory of appropriated Trinitarian names. This makesthe evidence of two further students of Thierry of Chartres all the morevaluable. The writings of Alan of Lille and Achard of St. Victor bothexpressly weigh the Breton master’s strong reading in light of the weakreading. Their deliberations in response to the controversy greatly enrich

72 “Parmenides quoque dicit: Deus est cui esse quidlibet quod est esse omne idquod est. Item idem: Deus est unitas: ab unitate gignitur unitatis aequalitas. Con-nexio vero ab unitate et unitatis aequalitate procedit. Hinc igitur Augustinus: Omnirecte intuenti perspicuum est, quare a sanctae Scripturae doctoribus Patri assignaturunitas, Filio aequalitas, Spiritui sancto connexio; et licet ab unitate gignitur aequali-tas, ab utroque connexio procedat: unum tamen et idem sunt. Haec est illa triumunitas: quam solam adorandam esse docuit Pythagoras. . . . Opinor ideo cum quiillam veram unitatem considerare desiderat, mathematica consideratione praeter-missa, necesse est ad intelligentiae simplicitatem animus sese erigat” (Septem, PL199:961B–C). It is difficult to determine the precise relationship between this passageand known texts of Thierry’s circle. One good conjecture for a mediating source isthe Commentarius Victorinus, which includes the first sentence quoted by Parmenidesbut does not attribute the triad to him: “Et secundum theologicam affirmationisdata est illa descriptio de deo a Parmenide philosopho quam utinam dedisset aliquissanctorum: deus inquit est cui quodlibet esse quod est est esse omne id quod est”(Commentarius Victorinus 99, in Commentaries, ed. Häring, 502). Septem’s summary

of Thierry’s arithmetic Trinity resembles the account at Commentarius Victorinus83–85, in Commentaries, ed. Häring, 498–99, in the pages preceding the Parmenidespassage; but cf. also Thierry, Commentum 2.38, in Commentaries, ed. Häring, 80.


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our understanding of the reception of Thierry’s arithmetic Trinity in themiddle decades of the twelfth century.

The Ambivalence of Alan of Lille

Paradoxically, the writings of Alan of Lille preserve perhaps the clear-est defense of the rationale for the appropriation theory, but at the sametime perpetuate one of the most renowned traditions of the arithme-tic Trinity. Alan represents, therefore, an ambivalent testimony to thedebates over Augustine’s triad in these decades. His complex, evolvingviews were no doubt shaped by virtue of his education. By the 1160s, itwas especially students of Gilbert of Poitiers such as Alan who tended

to advocate the weak reading.73 But the loyal Porretaner seems to havebeen impressed by Thierry as well, having studied in Paris during thelate 1140s and early 1150s, at the height of the Breton’s fame.74 As wewill see, this leads to a somewhat conflicted approach to Augustine’striad across Alan’s works; it is quite conceivable that this tension wasfelt by other contemporaries as well.

Alan and the Weak Reading

Alan’s first theological work was the Summa “Quoniam homines,” writ-

ten between 1155 and 1165 and probably right around 1160.75 As Glo-rieux points out, Alan’s Summa is neither a commentary on Lombard’sSentences, nor a treatise against heresies (as he would later write), nora scriptural commentary, nor yet one of Alan’s later poetic works, butrather a comprehensive systematic theology.76 The second half of the

73 P. Glorieux, “La Somme ‘Quoniam Homines’ d’Alain de Lille,” Archivesd’histoire doctrinale et littéraire du moyen âge 20 (1953): 116.

74 For a general comparison, see Michel Lemoine, “Alain de Lille et l’école de

Chartres,” in Alain de Lille, ed. Solère, Vasiliu, and Galonnier (n. 17 above), 47–58.For further evidence of Alan’s Chartrian influences and his mediation of Thierry’sideas beyond France, see Lucy Pick, Conflict and Coexistence: Archbishop Rodrigoand the Muslims and Jews of Medieval Spain (Ann Arbor, 2004), 88–90 and 122–25.

75 On Alan’s biography generally, see G. R. Evans, Alan of Lille: The Frontiers ofTheology in the Later Twelfth Century (Cambridge, 1983); Françoise Hudry, Règles dethéologie (Paris, 1995), 7–47; eadem, “Mais qui était donc Alain de Lille?” in Alainde Lille, ed. Solère, Vasiliu, and Galonnier, 107–24.

76 Glorieux, “La Somme,” 114. For an overview of the contents and the method,respectively, of the Summa “Quoniam Homines” (hereafter SQH), see P. Glorieux,“L’auteur de la Somme ‘Quoniam homines,’” Recherches de théologie ancienne et

médiévale 17 (1950): 29–45 and Alain de Libera, “Logique et théologie dans la Summa‘Quoniam Homines’ d’Alain de Lille,” in Gilbert de Poitiers, ed. Jolivet and de Libera(n. 31 above), 437–69.


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Summa’s first book is devoted to Trinitarian theology, and there Alandistinguishes three categories of divine names: “essential names” (suchas Deus, deitas, essentia, substantia; but also implied “co-essentials” likeprincipium or origo); “personal names” designating “personal properties”(Father, Son, Spirit, but also paternitas, filiatio, processio); and finally,“appropriated” names.77 Whereas personal names are directly assignedto the person on account of the personality itself (nomine et re), Alanexplains, appropriated names are only indirectly assigned on account ofthe name (nomine et non re), that is, because the name designates thedivine unity and therefore the persons in common.78

Alan approaches the arithmetic Trinity from two different directions inthese passages of his Summa. First, he brings up Augustine’s triad as the

point of departure for his entire section on Trinitarian theology, treatingit as a case study in divine names generally.79 Alluding to Romans 1:20,Alan describes how “the philosophers” name God through their percep-tion of the natural world: “Seeing that unity is the beginning and originof all numbers, [the philosophers] have likewise conjectured that in thecreation of things there is one Creator from which all alterity (i.e., every-thing changeable) proceeds, as if from the original and supreme unity.”80

This insight leads the philosophers to reflect on how unity generates onlyitself, and that this “mind” or “wisdom” sprung from divine unity must

be equal to unity and indeed connected by love. Such philosophers thusappear to have discovered Augustine’s triad from natural reason alone,and indeed they not only speak of “God and his mind and the world soul”

77 The second part begins at SQH 1.31, ed. Glorieux, “La Somme,” 167.78 Alan distinguishes the first two kinds of names summarily at SQH 1.55 (ed.

Glorieux, “La Somme,” 198–99), but the third at 1.80 (ed. Glorieux, “La Somme,”226): “Pertractatis hiis que de nominibus personalibus dicenda erant que personis

appropriantur nomine et re; agendum est de illis que appropriantur nomine et nonre, ut hoc nomen potentia, sapientia, bonitas.”79 Christian Trottmann (“Unitas, aequalitas, conexio: Alain de Lille dans la tradi-

tion des analogies trinitaires arithmétiques,” in Alain de Lille, ed. Solère, Vasiliu,and Galonnier, 401–27) considers this first instance of the triad in SQH 1.31 as wellas the Regula, but not the second instance in SQH 1.114 nor De fide catholica.

80 “Unde videntes unitatem esse principium et origo omnium numerorum, simileconiectaverunt in creatione rerum ut unum esset creator a quo, tamquam a princi-pali et suprema unitate procederet omnis alteritas, id est omne mutabile” (SQH 1.31,ed. Glorieux, “La Somme,” 167). See further Andreas Niederberger, “Naturphiloso-phische Prinzipienlehre und Theologie in der Summa ‘Quoniam Homines’ des Alain

von Lille,” in Metaphysics in the Twelfth Century: On the Relationship among Phi-losophy, Science and Theology, ed. Matthias Lutz-Bachmann, Alexander Fidora, andAndreas Niederberger (Turnhout, 2004), 185–99.


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but even attempt to identify the Holy Spirit and the anima mundi!81

(Here Alan alludes to Abelard or William of Conches, if not Thierry him-self.82) But in fact, Alan continues, they “have apparently discovered acertain trace of the Trinity, but only as if in a dream, not distinctly, ina catholic way.”83 For although the philosophers believe they are nam-ing the Trinity through triads like power, wisdom, and goodness, in fact“they have no such knowledge of the three persons that would allowthem to distinguish such notions.”84 Alan points to Hermes Trismegistus,the Sibyl, the Liber XXIV philosophorum, and Macrobius as examples ofmurky philosophical conjectures about the second divine person — allsources dear to Thierry of Chartres.

So for Alan, in the first place, the strong reading of the arithmetic

Trinity is not a fruitful path forward for catholic theology. Instead, it isthe paradigmatic instance of authentic, but ultimately inadequate, antic-ipations of the revealed Word among philosophers, much as Augustinehad said of the Platonists. Because these nascent intuitions of the Trin-ity are real but incomplete, Alan’s Summa must properly “distinguish”Trinitarian naming in detail, as he then proceeds to do. But it is difficultto decide how much this prologue, which coordinates the mathematicalTrinity, number, and alterity in a way so reminiscent of Thierry, tellsus about Alan’s views of the Breton master. It is important to note that

the foolish “philosophers” in Alan’s account start off well enough withThierry’s reasoning and only become culpable when they make Abelard’smistake. Alan certainly never targets Thierry for explicit critique as hedoes Abelard; indeed Alan’s examples of pagan wisdom especially recallThierry’s lectures. And Alan’s implication that the strong reading of thetriad finds its roots in late antique Platonic sources is remarkably astuteand belies a degree of familiarity with Thierry’s doctrine.

81

“Tamen multa dixerunt de Deo et mente eius et anima mundi, que ad trespersonas referri possunt. Et ideo dicuntur habuisse noticiam [sic] de Trinitate. . . .Nonne et plura dixerunt de anima mundi que possunt ad Spiritum Sanctum referri?”(SQH 1.31, ed. Glorieux, “La Somme,” 168).

82 See further Bernard McGinn, “The Role of the Anima Mundi as Mediatorbetween the Divine and Created Realms in the Twelfth Century,” in Death, Ecstasy,and Other Worldly Journeys, ed. John J. Collins and Michael Fishbane (Albany,1995), 289–319.

83 “Et ita videntur invenisse quedam vestigia Trinitatis; sed quasi per sompnium;nec ita distincte ut catholici” (SQH 1.31, ed. Glorieux, “La Somme,” 168).

84 “Sed quasi quedam in divinitate considerabant quorum nominibus persone

solent distingui, ut potentia, sapientia, benignitas. . . . Sed non habuerunt notitiamde tribus personis ut scirent eas distinguere suis notionibus” (ibid., 1.31, ed. Glo-rieux, “La Somme,” 168).


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Alan’s second meditation on the arithmetic Trinity comes when, pre-cisely in order to solve the problem his prologue raises, he applies thetheory of appropriated names to the triads of Abelard, Augustine, andHilary of Poitiers. In the cases of Abelard and Hilary, Alan could not beclearer. On Abelard: “For although there is one power of the three per-sons, one wisdom and one goodness, nevertheless the name of power, butnot the thing named, is appropriated to the Father; likewise the nameof wisdom is appropriated to the Son and the name of goodness to theHoly Spirit.”85 Alan lists the familiar scriptural citations that associatepower with the Father. On Hilary: Some names are appropriated as asso-ciated terms but not as strict significations (appropriantur voce et nonsignificatione). For example, the Father is called “eternity” because he

is the beginning without beginning, and the term “image” is more “spe-cially appropriated” (specialiter appropriatur) to the Son than the Spiritbecause of the Son’s “likeness” (the Augustinian similitudo).86 In bothcases Alan uses “appropriation” repeatedly and confidently.

In between the triads of Abelard and Hilary, Alan addresses the onefavored by Thierry. It must be more than a coincidence that here Alanprefers the milder potius dicitur to the formal term appropriatur.87 Hewrites: “unity, indeed, is rather said to be [potius dicitur] in the Fatherthan in the Son, for just as unity depends on nothing for existence, but

every number depends on unity, so the Father is from nothing and allthings are from the Father.”88 Here Alan’s reasoning follows from thenatural theology of the “philosophers” in his prologue, not from scrip-tural images or theological tradition, as it did with Abelard and Hilary.Equality is “rather said” of the Son not only because of the priority ofthe Son’s similitudo, as in Hilary’s triad, but also to underscore that theSon, though “other” than the Father, is not “lesser.”89 The Spirit is sim-ply “said to be the community or connection” of unity and equality. 90

85

“Quamvis enim una sit potentia trium personarum, una sapientia, una bonitas,tamen nomen potentie appropriatur Patri et non res nominis; similiter nomen sapi-entie Filio, nomen bonitatis Spiritui Sancto” (SQH 1.80, ed. Glorieux, “La Somme,”226).

86 Ibid., 1.122, ed. Glorieux, “La Somme,” 255.87 On the broad range of possible terms for appropriation, see Ott, Untersuchungen

(n. 34 above), 580.88 “Unitas ideo potius dicitur esse in Patre quam in Filio quia sicut unitas a nullo

est et omnis numerus ab unitate, sic Pater a nullo et omnia a Patre” (SQH 1.114,ed. Glorieux, “La Somme,” 248).

89 “In Filio autem dicitur equalitas non alteritas esse; quia si diceretur alteritas esse

in Filio videretur esse minor Patre” (ibid., 1.114, ed. Glorieux, “La Somme,” 248).90 “Spiritus autem Sanctus ideo communitas dicitur unitatis et equalitatis siveconnexio” (ibid.).


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Thus after explaining the necessity for the property-appropriation dis-tinction in his prologue, particularly in light of the triad of unity, equal-

ity and connection, Alan pulls his punch, so to speak, when it comes tothe arithmetic Trinity favored by Thierry. Abelard’s potentia led Alaninto a dissertation on divine omnipotence, and Hilary’s aeternitas into adiscussion of time and space. Only the case of Augustine’s triad inspireshim to plunge further into issues of Trinitarian predication. Alan weighsthe tensions between Augustine’s and Boethius’s views on number in theTrinity — the subject of Thierry’s own Boethian commentaries — andeven sketches an account of what he calls “theological number” in con-trast to merely “logical number.” Alluding to the Boethian distinctionbetween two kinds of number, Alan contends that the number of the

Trinity is not a quantitative number, but “a number sui generis, namelytheological number.”91 What could such a “theological number” be, if notprecisely the triple unitas defined in the arithmetic Trinity? Ultimately,Alan appears uncertain about applying appropriation theory to the arith-metic Trinity as forcefully as he had originally planned. He seems wellaware that his theological preceptors — Augustine, Boethius, and John ofDamascus — put the mystery of number at the center of their Trinitar-ian meditations.92 Here is a reader of Thierry’s works uncertain whetherto walk one mile with the controversial master or two.

Alan and the Strong Reading

In this light it is unsurprising to find Alan still preoccupied with Thier-ry’s arithmetic Trinity in two other works, preserving the ambivalenceon display in the Summa.93 In his better-known Regulae theologiae, writ-

91 “Nec nos numerum theologicum quantitatem dicimus, sed potius pluralitatempersonarum quam faciunt distinctiones que attenduntur secundum paternitatem,filiationem, spirationem. . . . Non concedimus ergo quod ibi predicatur numerus sed

numerus sui generis, scilicet numerus theologicus. Nec inde sequitur quod quanti-tas predicatur, quid de numero theologico non potest inferri numerus logicus, id estnumerus qualis apud logicum consideratur” (ibid., 1.115, ed. Glorieux, “La Somme,”250). Cf. Boethius, De sancta trinitate 3, in Boethius: De Consolatione Philosophiae,Opuscula Theologica, ed. Claudio Moreschini (Munich, 2005), 171:132–34: “Numerusenim duplex est, unus quidem quo numeramus, alter vero qui in rebus numerabilibusconstat.”

92 John of Damascus’s De fide orthodoxa was translated around 1150 and cited fre-quently by Peter Lombard. See M.-D. Chenu, La théologie au douzième siècle (Paris,1957), 283–84. Although he does not discuss the case of Alan in particular, see fur-ther Nikolaus M. Häring, “The Porretans and the Greek Fathers,” Mediaeval Studies

24 (1962): 181–209.93 Glorieux rightly notes that the Summa (for Glorieux, as yet anonymous) closelyresembles passages in Alan’s Regulae and Contra haereticos; but he assumes that the


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ten several years after the Summa, Alan constructs his entire axiomaticsystem on the foundation of the mathematical triad, echoing some of his

own formulations from the Summa.

94

Häring has emphasized that Alan’srules must have been compiled over time in collaboration with Gilbert’sother students, such as Peter of Poitiers, and possibly with the supportof Gilbert himself. Like Thierry, Gilbert had lectured on Boethius’s Dehebdomadibus, where the axiomatic ideal is first modeled in Christian the-ology.95 Nevertheless, they belong to a later period after Alan has leftParis for the Languedoc.96

Since the quadrivium, trivium, and physics all begin with axiomaticrules, Alan reasons, so too should theology. The first rule is that God isnot only one (unus) but unity (unitas or monas): “unity begets unity from

itself; from itself it brings forth equality.”97 The second rule states thatGod’s unity is sui generis, the unique transnumeric source of all numbers(unitas singularitatis).98 In the third rule, Alan once again — though nowat greater length than in the Summa — decodes the cryptic first lemmaof the Liber XXIV philosophorum as a figure of the mathematical Trini-ty.99 Finally, Alan reveals the fourth rule — the arithmetic Trinity itself

— as the culmination of the first three: “in the Father is unity, in theSon equality, in the Holy Spirit the connection of unity and equality.”100

Does Alan’s fourth rule represent a weak or a strong reading of Thier-

ry’s triad? On the one hand, Alan again presents it from the perspectiveof appropriative theology (“unity is specially said to be in the Father,and equality in the Son”) and repeats his reasoning from the Summa.101

arithmetic Trinity has the same meaning in each text regardless of context; see Glo-rieux, “L’auteur” (n. 76 above), 33–34, 37–38.

94 The work is also known as Regulae caelestis iuris or De maximis theologicis. Com-paring these two texts on the mathematical triad encourages one to suppose thatAlan wrote the Summa first and then adapted portions of it within his ongoing proj-ect of the Regulae.

95 Nikolaus Häring, “Magister Alanus de Insulis: Regulae Caelestis Iuris,” Archivesd’histoire doctrinale et littéraire du moyen âge 48 (1981): 97–226, at 99, 118.

96 Hudry dates the Summa to 1155–67, the Regulae to 1192–94, and De fide catho-lica (see below) to 1190–1200; see Règles de théologie (n. 75 above), 85–89.

97 “Vnitas de se gignit unitatem, de se profert equalitatem” (Regulae 1.5, ed.Häring, “Magister Alanus,” 125).

98 Ibid., 2.3, ed. Häring, “Magister Alanus,” 126.99 Ibid., 3.1–4, ed. Häring, “Magister Alanus,” 127–28. Cf. SQH 1.31, ed. Glorieux,

“La Somme,” 168.100 “In Patre unitas, in Filio equalitas, in Spiritu Sancto unitatis equalitatisque

connexio” (Regulae 4 [regula], ed. Häring, “Magister Alanus,” 128).101 “In Patre specialiter dicitur esse unitas, in Filio equalitas” (ibid., 4.2, ed.Häring “Magister Alanus,” 128). Cf. SQH 1.114, ed. Glorieux, “La Somme,” 248.


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On the other hand, Alan again avoids the formal term “appropriated,”and more importantly, the context of the Summa is quite differentfrom that of the Regulae. In the Summa, Alan sought to outline all ofChristian theology as part of a coherent whole and to weigh competingauthorities, just like Peter Lombard. But in the Regulae Alan is up tosomething quite original among his twelfth-century peers.102 FollowingBoethius’s example in De hebdomadibus, Alan seeks “common concep-tions of the mind,” that is, self-evident truths that anyone will acceptimmediately upon hearing, because they are known in themselves. Theseare the highest truths of reason, he writes in his prologue, because theyare the most general truths, commonly owned by every rational mind. 103

To that end, Alan first tries to establish that the Trinity is the primary

instance of such universal “common conceptions” when grasped in termsof the mathematical triad. But for this to work, Alan must considerunity, equality, and connection as linguistic equivalents of the Father,Son, and Spirit and not simply provisional qualities associated with thedifferent persons, such as power, eternity, beauty, or goodness. If themathematical triad were anything less, then Alan would not have provedthat the divine Trinity was a self-evident common conception, but onlythat certain other divine attributes like eternity or goodness were. Inorder for the triad to function as Alan’s rationalist project in the Regu-

lae requires, the three terms need to be strictly nomina personales, notnomina appropriativae.Given this peculiar context of the arithmetic Trinity’s reappearance

in the Regulae, Alan should be understood as a de facto proponent ofthe strong reading. Much the same can be said about his later apolo-

102 Mechthild Dreyer (More mathematicorum: Rezeption und Transformation derantiken Gestalten wissenschaftlichen Wissens im 12. Jahrhundert, BGPhThM 47 [Mün-ster, 1996], 106–61) notes that the most important context for Alan’s axiomaticmethod in the Regulae are the commentaries on Boethius’s De hebdomadibus by Gil-

bert of Poitiers, Thierry of Chartres, and Clarembald of Arras. See further G. R.Evans, “Boethian and Euclidean Axiomatic Method in the Theology of the LaterTwelfth Century,” Archives internationale d’histoire des sciences 30 (1980): 36–52;Charles H. Lohr, “The Pseudo-Aristotelian Liber de causis and Latin Theories ofScience in the Twelfth and Thirteenth Centuries,” in Pseudo-Aristotle in the MiddleAges: The Theology and Other Texts, ed. Jill Kraye, Charles B. Schmitt, and W. F.Ryan (London, 1986), 53–62; Françoise Hudry, “Métaphysique et Théologie dans lesRegulae Theologiae d’Alain de Lille †1202),” in Metaphysics in the Twelfth Century,ed. Lutz-Bachmann, Fidora, and Niederberger (n. 80 above), 201–15.

103 “Communis animi conceptio est enuntiatio quam quisque intelligens probatauditam. Hec omnes maximas, cuiuscumque sint facultatis, sua generalitate con-

plectitur. . . . Vnde indemonstrabilis, per se nota et maxima nuncupatur” (Regulae,Prologus, 10, ed. Häring, “Magister Alanus,” 123). Cf. Boethius, Quomodo substantiae[De hebdomadibus] 1, ed. Moreschini, Opuscula (n. 91 above), 187:17–18.


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getic work De fide catholica contra haereticos sui temporis.104 In the thirdbook Contra Iudaeos, Alan defends the Christian belief in the Trinity.He introduces the distinction between essence and persons and then listsscriptural texts that support his views. The only non-scriptural authoritycited is the Hermetic Asclepius, the same source Alan had used in bothSumma and Regulae to introduce the arithmetic Trinity.105 But now inaddition Alan promises “to prove the same with reasons.”106 His rationalproof in De fide catholica is none other than the triad of unity, equality,and connection, and here there is not a trace of the weak appropriativereading. Quite to the contrary: in order to purify divine “sonship” and“generation” from the whiff of polytheism, Alan adverts to the clarity —the apologetic “refuge,” we might say — of mathematics:

For just as all divisible plurality proceeds from all indivisible unity, sodoes everything variable proceed from the invariable Creator, since “he,remaining still, gives motion to all” [Boethius]. And just as an image ofCreator and creature arises in unity and number, so too a likeness of theTrinity: for in the property of unity there arises a trace of the Trinity,since, as arithmetic teaches, unity generates itself. Between generatedunity and generating unity one discovers an equality. But in what exist-ing thing could we possibly encounter this, unless in God? God gener-ates God and by generating nothing other than God from God indeedbrings forth one who is the same God as the one generating. And thereis the perfect equality, or meeting-point or connection, of generating andgenerated, which is called the Holy Spirit, in whom the Father and Sonmeet. Thus the philosopher said: “Monad generates monad, and in itselfreflects its passion.” Therefore you will not discover in any existing thingwhat is said of unity; it is only discovered in divine uni

Documents

Achard of St. Victor (d. 1171) and the Eclipse of the Arithmetic Model of the Trinity