BOETHIUS NOSTER: Thierry of Chartres's ARITHMETICA

BOETHIUS NOSTER: Thierry of Chartres’s Arithmetica Commentary as A missing source of Nicholas of Cusa’s De docta ignorantia

David Albertson

Abstract

Nicholas of Cusa is known to have used several sources stemming from the twelfth-century master Thierry of Chartres in the composition of his treatise De docta ignorantia (1440). To date these works have included Thierry’s famous hexaemeral commentary, his three commentaries on Boethius’s De trinitate, and later anonymous works by Thierry’s readers. Given Irene Caiazzo’s recent discovery of Thierry’s commentary on Boethius’s De arithmetica, we can now evaluate the possibility that this work was another Cusan source before 1440. Numerous parallels in Books I and II of De docta ignorantia suggest that Cus- anus made use of Thierry’s Arithmetica commentary while adapting four Boethian concepts to theological ends: precise equality, primordial number, the quadrivial arts, and folding. This finding raises new questions about the status of the controversial Fundamentum naturae manuscript.

After over a century of scholarship on Nicholas of Cusa’s De docta ignorantia, one might expect its fundamental mystery to have been solved. How did a busy lawyer with modest philosophical training compose such an original volume, far from the usual ways of the medieval schools, as Nicholas himself boasted? As Hans Gerhard Sen- ger has observed, “the fact that a 40-year-old suddenly came up with such a well thought-out and substantive philosophy is surprising.”1

1. H. G. Senger, Die Philosophie des Nikolaus von Kues vor dem Jahre 1440: Unter- suchungen zur Entwicklung einer Philosophie in der Frühzeit des Nikolaus (1430-1440), Münster 1971, p. 7. See also H. G. Senger, “‚in mari me ex Graecia redeunte, credo superno dono‘ – Vom Wissensfrust zur gelehrten Unwissenheit. Wie platzte 1437/1438

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Martin Honecker has shown that the two years preceding the work’s completion in February 1440 were exceptionally busy for Cusanus. Only eight of those twenty-four months would have permitted him sufficient leisure to write.2 In order to explain the novelty of De docta ignorantia – unless we credit Cusanus’s claim of divine illumination – we must consider other sources that might have assisted its author.3 The ideas of Ps.-Dionysius, Meister Eckhart, and Ramon Llull all inform the agenda of De docta ignorantia, but their own words appear infrequently throughout the text. Senger has proposed that perhaps Nicholas’s early interest in calendrical calculations and astronomical instruments might explain his sudden turn to arithmetic and geometry in De docta ignorantia.4 Alongside these influences, one should also examine the complex body of source materials stemming from Thierry of Chartres (d. 1157), a well-regarded master in twelfth-century Paris. Thierry’s creative re- reading of the Boethian tradition led him to what we might call a Trinitarian theology of the quadrivium and therefore, if accidentally, to the first substantive Christian engagement with Neopythagoreanism. Unlike his other sources, Nicholas regularly cited these texts connected with Thierry of Chartres word for word. To date there are three categories of such Chartrian sources. (1) As Pierre Duhem and Raymond

der Knoten?,” in: A. Speer – Ph. Steinkrüger (eds.), Knotenpunkt Byzanz. Wissensformen und kulturelle Wechselbeziehungen, Berlin 2012, pp. 481-495. 2. See M. Honecker, “Die Entstehungszeit der ‘Docta ignorantia’ des Nikolaus von Cues,” in: Historisches Jahrbuch 60 (1940), pp. 124-141; and R. klibansky, “Zur Geschichte der Überlieferung der Docta ignorantia des Nikolaus von Kues,” in: Nikolaus von Kues: Die belehrte Unwissenheit, Buch III, ed. P. Wilpert – H. G. Senger, Hamburg 1977, pp. 216-219. 3. Nicholas of Cusa, “Epistola auctoris,” De docta ignorantia, III, 263, in: Nikolaus von Kues. Philosophisch-theologische Werke, Band 1: De docta ignorantia, ed. P. Wilpert – H. G. Senger, Hamburg 1999, pp. 98-100: “Accipe nunc, pater metuende, quae iam dudum attingere variis doctrinarum viis concupivi, sed prius non potui, quousque in mari me ex Graecia redeunte, credo superno dono a patre luminum a quo omne datum opti- mum, ad hoc ductus sum, ut incomprehensibilia incomprehensibiliter amplecterer in docta ignorantia per transcensum veritatum incorruptibilium humaniter scibilium.” Cf. H. L. Bond, “Nicholas of Cusa from Constantinople to ‘Learned Ignorance’: The Historical Matrix for the Formation of De docta ignorantia,” in: G. christianson – T. M. Izbicki (eds.), Nicholas of Cusa on Christ and the Church, Leiden 1996, pp. 135- 163; M. O’Rourke Boyle, “Cusanus at Sea: The Topicality of Illuminative Discourse,” in: Journal of Religion 71 (1991), pp. 180-191; and H. G. Senger, “‚in mari me ex Graecia redeunte, credo superno dono‘.” 4. See H. G. Senger, Die Philosophie des Nikolaus von Kues, pp. 106-129, 153-154.

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Klibansky first discovered, several passages from Thierry’s own commentaries are peppered throughout De docta ignorantia as well as later texts like Idiota de mente (1450). Nikolaus Häring’s editions of Thier- ry’s hexaemeral commentary, Tractatus de sex dierum operibus, as well as his three successive commentaries on Boethius’s De trinitate, have allowed for more precise identifications, especially in the first book of De docta ignorantia.5 (2) Nicholas also borrowed from anonymous works by Thierry’s twelfth-century readers, above all, De septem septenis,6 a Hermetic tract long misattributed to John of Salisbury, and the student notes edited as Commentarius Victorinus.7 (3) Maarten Hoenen has presented compelling evidence that Cusanus drew upon a philosophical treatise recently found in the library of a fifteenth-century Dominican in Eichstätt. Hoenen argues that Cusanus inserted long portions of this anonymous work, Fundamentum naturae quod physicos ignorasse videtur, into Book II of De docta ignorantia.8 Thierry’s ideas are prominent in the treatise, but only because the author intends to refute them, a fact that Nicholas may have overlooked.9 That is to say: Nicholas seems to have jumbled together Thierry’s own works with those of his reception history. This medley of sources is fragmentary, sometimes confused, and occasionally contradictory, but the family resemblance is hard to miss. Even when stretched or upended by his medieval readers, Thierry’s signature doctrines remain unmistakeable. Nicholas must have noticed their similar parentage, and appreciated their collective force, for in Apologia doctae ignorantiae

5. See Commentaries on Boethius by Thierry of Chartres and His School, ed. N. M. Häring, Toronto 1971. 6. See De septem septenis, PL 199, cols. 945-964; cf. MS London, British Library, Harley 3969, fols. 206v-215v. On the state of the text, see C. Németh, “Fabricating Philosophical Authority in the Twelfth Century: The Liber Egerimion and the De septem septenis,” in: S. Kangas – M. Korpiola – T. Ainonen (eds.), Authorities in the Middle Ages: Influence, Legitimacy and Power in Medieval Society, Berlin 2013, pp. 75-76. 7. MS Paris, Bibliothèque nationale de France, Lat. 14489, fols. 67r-95v; cf. Com- mentarius Victorinus, ed. N. M. Häring, pp. 479-528. 8. See M. J. F. M. Hoenen, “‘Ista prius inaudita.’ Eine neuentdeckte Vorlage der De docta ignorantia und ihre Bedeutung für die frühe Philosophie des Nikolaus von Kues,” in: Medioevo: Rivista di storia della filosofia medievale 21 (1995), pp. 375-476. For the text of Fundamentum naturae quod videtur physicos ignorasse, MS Eichstätt Cod. st 687, fols. 4r-10r, see ibid., pp. 447-476. 9. See D. Albertson, “A Late Medieval Reaction to Thierry of Chartres’s (d. 1157) Philosophy: The Anti-Platonist Argument of the Anonymous Fundamentum naturae,” in: Vivarium 50 (2012), pp. 53-84.

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(1449) he celebrated their common progenitor, an unnamed “Boethian commentator,” as “easily the most brilliant man of all those whom I have read” – and this after acknowledging his debts to Eckhart, Ps.- Dionysius, and Proclus without such hyperbolic praise.10 Cusanus borrowed frequently from Thierry’s treasury: the schema of complicatio and explicatio; the mathematical model of the Trinity as unitas, aequalitas, and conexio; the dialectic of unitas and alteritas; the system of four modes of being spanning from absolute necessity to absolute possibility; and a theology of the divine Mind as the unique equality of beings. Each of these doctrines was once attributed to Cusanus, but in fact first belonged to the Parisian master. On the basis of Häring’s editions we can now understand the dynamic nature of Thierry’s influence. Not only can we detect diachronic change within Thierry’s own theology from the 1130s to 1150s; we can also measure Nicholas’s improving grasp of those ideas over his own writing career from the 1440s to 1460s. As Cusanus acquired increasing access to the full range of Char- trian sources, he had better purchase on how they could be harmonized and applied in his own century.11 A full appraisal of Nicholas of Cusa’s debts to Thierry of Chartres has been an obvious desideratum since the outset of the Heidelberg edition. For several reasons, that task has been deferred for decades. One recent complication has been the reluctance of some scholars to accept Hoenen’s controversial claims regarding the anonymous Fun- damentum naturae treatise. According to Hoenen, Nicholas layered sections of this treatise into Book II of De docta ignorantia and added his own expositions. Hoenen contends that some of Cusanus’s most famous concepts – God as absolute maximum, finitude as contractio – originated from this anonymous source. I remain persuaded by Hoenen’s theory and have presented additional evidence in his favor.12 Since even those scholars who resist Hoenen’s conclusions

10. Nicholas of Cusa, Apologia doctae ignorantiae, 35, in: Nicolai de Cusa Opera omnia iussu et auctoritate Academiae Litterarum Heidelbergensis, Vol. II, ed. R. Klibansky, Leipzig 1932, p. 24: “Unde ait commentator Boethii De Trinitate, vir facile omnium, quos legerim, ingenio clarissimus.” 11. See D. Albertson, Mathematical Theologies: Nicholas of Cusa and the Legacy of Thierry of Chartres, New York 2014. 12. See D. Albertson, “A Learned Thief? Nicholas of Cusa and the Anonymous Fundamentum Naturae: Reassessing the Vorlage Theory,” in: Recherches de Théologie et Philosophie médiévales 77 (2010), pp. 351-390.

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remark that his argumentation is sound, there may be other reasons that account for their hesitation. Nicholas of Cusa is still depicted as a pivotal figure in the emergence of a modern European worldview, even if without the Cartesian or Neo-Kantian zeal of the last century. His achievements are undeni- able: Cusanus embraced an infinite cosmos amenable to mathematical order, posited a subject-oriented epistemology in which knowers participate in constructing the known, and valued interreligious dia- logue on the grounds of a principled perspectivalism. What raw materials enabled this Renaissance cardinal to sound at times like a seventeenth-century philosopher? The search for Nicholas’s motivating sources can become freighted with philosophical, if not political, significance. Scholars frequently contrast earlier theological works (e.g., De docta ignorantia or his sermons) with later philosophical works (e.g., De coniecturis or De mente), and then search for a cor- responding shift in sources that propelled him from faith to reason. Perhaps Nicholas cast aside medieval authorities in favor of something more enlightened and classical. Perhaps, for instance, he was so liberated by his encounter with Proclian Neoplatonism that he could leave Rhineland mysticism behind. Given these expectations, the sug- gestion that Cusanus borrowed extensively from Thierry of Chartres, a twelfth-century Neopythagorean, that he employed Thierry’s ideas in both early and late works, and that he conspicuously praised him above more respectable philosophers – this fact has not always been readily acknowledged or welcomed.13 There are two opposed tendencies in studies of Cusanus’s sources. On the one hand, scholars have credited the slightest hint of influence by the great Neoplatonist philosophers that would accord with the image of Cusanus as a proto-modern thinker. For all its virtues,

13. Another complicating factor is that many accounts of Pythagoreanism in the Mid- dle Ages or Renaissance fail to treat the Neopythagorean contributions of Thierry and Nicholas adequately. See e.g. the entries by A. Hicks, “Pythagoras and Pythagoreanism in Late Antiquity and the Middle Ages,” in: C. A. Huffman (ed.), A History of Pythago- reanism, Cambridge 2014, pp. 416-434, largely on Boethius; and in the same volume, M. J. B. Allen, “Pythagoras in the Early Renaissance,” pp. 435-453, largely on Marsilio Ficino. Neither mentions Thierry or Nicholas; both are influenced by C. l. Joost- Gaugier, Measuring Heaven: Pythagoras and His Influence on Thought and Art in Antiquity and the Middle Ages, Ithaca 2006; and ead., Pythagoras and Renaissance Europe: Finding Heaven, Cambridge 2009.

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Kurt Flasch’s magisterial study of Cusan development is the epitome of this whiggish view. Through extraordinary contact with Greek philosophy, Cusanus rose above the limitations of medieval mysticism, passing from theology to philosophy, from darkness to light.14 Jean- Michel Counet organized his book on Cusanus’s mathematical theology around a passage in Eckhart which, he concedes, Cusanus may not even have read, rather than those in Thierry that we know he studied closely.15 Thomas McTighe, who understood Thierry’s importance better than most, still sought to validate the Cusan adop- tion of folding by tracing the doctrine past Thierry, past Boethius, all the way back to Plotinus.16 Such sources can even be named specu- latively, in anticipation of textual validation to come. Josef Koch hypothesized that Nicholas’s turn from the negative theology of De docta ignorantia to the mathematized philosophy of De coniecturis was enabled by a new, unknown Neoplatonist source. With little evidence in hand, Koch named Proclus and Eckhart as the most likely candidates.17 Thierry’s Neopythagoreanism was out of the question. On the other hand, when it comes to discoveries that might prove Nicholas’s habitual reliance on Chartrian texts, we are urged to proceed with scrupulous caution. In 1909 Duhem exposed Cusanus’s use of Thierry’s Genesis commentary, calling it “plagiarism.”18 Edmond Van- steenberghe responded with a plea to suspend judgment and wait for a common source to emerge that might exonerate Nicholas from such direct dependence.19 In 1995 Hoenen revealed Cusanus’s redoubled

14. See K. Flasch, Nikolaus von Kues: Geschichte einer Entwicklung, Frankfurt am Main 1998. 15. See J.-M. Counet, Mathématiques et dialectique chez Nicolas de Cuse, Paris 2000. 16. See T. p. mcTighe, “A Neglected Feature of Neoplatonic Metaphysics,” in: P. J. Casarella – G. P. Schner (eds.), Christian Spirituality and the Culture of Modernity: The Thought of Louis Dupré, Grand Rapids 1998, pp. 27-49. 17. See J. Koch, Die Ars coniecturalis des Nikolaus von Kues, Köln 1956, p. 35; id., “Nikolaus von Kues und Meister Eckhart. Randbemerkungen zu zwei in der Schrift De coniecturis gegebenen Problemen,” in: R. Haubst (ed.), Das Cusanus-Jubiläum in Bernkas- tel-Kues vom 8. bis 12. August 1964. MFCG 4, Mainz 1964, p. 165. For a balanced analysis of Proclus’s influence after 1450, see S. Gersh, “Nicholas of Cusa,” in: id. (ed.), Interpreting Proclus: From Antiquity to the Renaissance, Cambridge 2014, pp. 318-349. 18. See P. Duhem, “Thierry de Chartres et Nicolas de Cues,” in: Revue des sciences philosophiques et théologiques 3 (1909), pp. 525-531. 19. See E. Vansteenberghe, Le Cardinal Nicolas de Cues (1401-1464): L’action, la pensée, Paris 1920, p. 411.

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debt to Thierry via the Fundamentum naturae treatise, and again the response was caution and delay. Senger joined Heinrich Pauli in con- ceding Hoenen’s argumentation, yet resolving to suspend judgment indefinitely, until such time as a new manuscript discovery could confirm their expectation of a common source.20 In his preface to the critical edition of De docta ignorantia in 1932, Klibansky dismissed such anxieties over Chartrian influence as an “error.” By documenting the origins of Cusan thought, he wrote, one does not thereby demean the cardinal’s real achievements.21

1. A New Work by Thierry of Chartres Happily we can now add another Chartrian source to the list of direct influences on Nicholas of Cusa. Irene Caiazzo has unearthed a medieval gloss on Boethius’s De arithmetica and has successfully attributed it to Thierry of Chartres.22 Over two books of commentary, Thierry scrutinizes the Boethian arithmetical and geometrical concepts most amenable to philosophical interpretation, weighing their ethical and theological significance. Among other themes Thierry explores the dialectics of unity, the eternity of divine Equality, and the “unfolding” of points into lines and surfaces. Caiazzo’s remarkable discovery fits like a missing piece into the puzzle of Chartrian documents that we currently possess. As I will show, there is considerable textual evidence that Nicholas used Thierry’s Arithmetica commentary in the construction of De docta ignorantia.

20. See H. Pauli, “Neues aus der Cusanusforschung,” in: Aktuelle Mitgliederinformation der deutschen Cusanus-Gesellschaft 1 (1996), pp. 4-5; and H. G. Senger, “De docta ignorantia – Eine Provokation?” in: id., Ludus Sapientiae: Studien zum Werk und zu Wirkungsge- schichte des Nikolaus von Kues, Leiden 2002, pp. 43-62. Cf. the note by H. G. Senger, in: De docta ignorantia, ed. P. Wilpert – H. G. Senger, Hamburg 2002, p. 124, n. 67. 21. De docta ignorantia. Nicolai de Cusa Opera omnia iussu et auctoritate Academiae Litterarum Heidelbergensis, Vol. I, eds. E. hoffmann – R. klibansky, Leipzig 1932, p. xii: “Sunt, qui putent philosophiam Cusanam vilioris venire, si origines eius demonst- rantur; huiusmodi errorem extorquere nullum operae pretium est.” 22. See Thierry of Chartres, The Commentary on the De arithmetica of Boethius, ed. I. caiazzo, Toronto 2015. Cf. the partial transcription of selected passages in I. Caiazzo, “Il rinvenimento del commento di Teodorico di Chartres al De arithmetica di Boezio,” in: P. Arfé – I. Caiazzo – A. Sannino (eds.), Adorare caelestia, gubernare terrena: Atti del colloquio internazionale in onore di Paolo Lucentini, Turnhout 2011, pp. 183-203.

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As is the case with all of Thierry of Chartres’s works that are undoubtedly Cusan sources, some caveats are necessary. None of Thier- ry’s commentaries have been located in Nicholas’s library, and it remains unclear when and where he accessed the manuscripts. Hence we have no marginalia, as in the cases of Proclus, Eriugena, Llull, or Eckhart. Moreover, as with many sources, it always remains possible that a prior, common text mediated the concepts in question to Thierry and Nicholas alike. At the same time, these methodological considera- tions should not obscure our recognition that this represents the fourth major discovery of source materials stemming from Thierry of Chartres that later reappeared in Cusan works. The more visible Thierry’s influence becomes, the less one needs to await further textual discoveries – perhaps in the hopes of qualification or mitigation – before confirming Thierry’s stature as a preeminent Cusan source. It comes as no surprise to students of Thierry of Chartres that he wrote a commentary on De arithmetica.23 His exceptionally broad teaching facility was already beginning to be unusual by the end of his career. As chancellor of the cathedral school at Chartres, before his return to Paris, Thierry drafted a master plan for liberal arts education, the Heptateucon, combining the classics of the trivium and the quadrivium.24 Thierry’s earliest known commentaries are on classical works of rhetoric; like the young Augustine and the young Boethius, perhaps the young Thierry had ambitions to lecture his way through the seven arts, including Boethius’s De arithmetica.25 We already have ample evidence that Thierry referenced the latter in his theological commentaries on Genesis and on Boethius’s De trinitate. For example, Thierry leans heavily on De arithmetica when he discusses geometrical equality as a divine name, or when he alludes to the Pythagorean ontology of the

23. See I. Caiazzo, “Introduction,” in: Commentary on the De arithmetica of Boethius, pp. 25-33. 24. See É. Jeauneau, “Le ‘Prologus in Eptatheucon’ de Thierry de Chartres,” in: Mediaeval Studies 16 (1954), pp. 171-175; and G. R. Evans, “The Uncompleted Hep- tateuch of Thierry of Chartres,” in: History of Universities 3 (1983), pp. 1-13. Cf. I. Caiazzo, “Introduction,” pp. 12-17. For context see the biographical essay by 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,” in: Viator 3 (1972), pp. 219-273. 25. See The Latin Rhetorical Commentaries by Thierry of Chartres, ed. K. M. Fred- borg, Toronto 1988.

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unitary (odd) and the binary (even).26 Thierry’s students casually indicate the authority of the arithmetici, suggesting that Boethius’s textbook was commonly cited in the master’s lectures.27 So even before Caiazzo’s discovery, it was clear that Thierry had meditated upon Boethian arithmetic at a profound level, that his reading of the text was highly original, and that he could apply it through fully-formed doctrines in other settings and registers. Given this likelihood that Thierry penned such a work, scholars have long searched for it among anonymous Arithmetica commentaries redolent of Chartres.28 Caiazzo has now convincingly identified Thierry’s commentary within a twelfth-century Stuttgart manuscript first described by Arno Borst and Menso Folkerts.29 The codex originated from the Benedictine abbey of Comburg (Schwäbisch Hall), and both its script and binding date to the twelfth century. Thierry’s commentary takes up the first third of the volume. The other folios are filled with quadrivial pursuits, namely astronomical works on the use of astrolabes and calendrical calculations (computus) – vindicat- ing Senger’s prediction about Nicholas’s early interests.30 Caiazzo shows how two other Arithmetica commentaries are connected to Thierry’s gloss. Thierry drew upon a prominent tenth-century commentary in his work (also edited by Caiazzo), just as the well-known

26. See e.g. Thierry of Chartres, Tractatus de sex dierum operibus, 30-39, ed. N. M. Häring, pp. 568-571; and id., Commentum super Boethii librum de Trinitate, II, 28-36, ed. N. M. Häring, pp. 77-79. (Hereafter I abbreviate Thierry’s works to the first word of their title in Häring’s edition.) 27. See Thierry of Chartres, Lectiones in Boethii librum de Trinitate, III, 5, ed. N. M. Häring, p. 178; and id., Glosa super Boethii librum de Trinitate, I, 38, ed. N. M. Häring, p. 267. Cf. the anonymous works, Commentarius Victorinus, 81-88, ed. N. M. Häring, pp. 498-499; and Tractatus De Trinitate, 12-18, ed. N. M. Häring, pp. 306-307. 28. A fine introduction to the genre is G. R. Evans, “A Commentary on Boethius’s Arithmetica of the Twelfth or Thirteenth Century,” in: Annals of Science 35 (1978), pp. 131-141. 29. MS Stuttgart, Württembergische Landesbibliothek, Cod. math. 4° 33, fols. 11ra- 34ra; cf. I. Caiazzo, “Introduction,” pp. 23-25. Cf. A. Borst, “Rithmimachie und Musiktheorie,” in: M. Bernhard – A. Borst – D. illmer – A. riethmüller – K.-J. Sachs – F. Zaminer (eds.), Geschichte der Musiktheorie, Vol. 3: Rezeption des antiken Fachs im Mittelalter, Darmstadt 1990, p. 256; and M. Folkerts, “Die Rithmachia des Werinher von Tegernsee,” in: M. Folkerts – J. P. Hogendijk (eds.), Vestigia mathemat- ica: Studies in Medieval and Early Modern Mathematics in Honor of H.L.L. Busard, Amster- dam 1993, pp. 121-122. 30. See I. Caiazzo, “Introduction,” pp. 81-85.

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Bern Arithmetica discovered by Klibansky made use of Thierry’s commentary in turn.31 Caiazzo has also located Thierry’s Arithmetica commentary among his other established works. Since his writings often differ subtly in their philosophical formulae, close analysis of his terms can illuminate his intellectual development over time and the sequence of his works. By these methods, Caiazzo concludes that Thierry’s Arithmetica can be placed between the Tractatus on Genesis, typically dated to the early 1130s, and the Commentum on Boethius’s De trinitate, dated between 1130 and 1140.32 As Caiazzo writes, the Arithmetica seems to fall “prior to the grand speculations developed in the commentaries on the Opuscula sacra, and is probably contemporaneous with (or slightly later than) the Tractatus, with which it shares interest in natural philosophy and the liberal arts.”33 I have also attempted a ‘genetic’ account of Thierry’s theological project based on lexicographical analysis, although at the time I could only use Caiazzo’s partial transcriptions of the Stuttgart commentary.34 Caiazzo’s completed edition sheds light on Thierry’s development and confirms many of my conclusions. I would only add two points. First, it is important to emphasize that Thierry’s Arithmetica commentary is an unusually theological enterprise. It is not simply a work of philosophy, but is in fact a Trinitarian theology of the divine Word, crafted out of the rare elements of Boethian number theory. One must reckon fully with the strangeness of this project, even when compared with, say, contemporary Cistercian arithmologies or Augus- tine’s De musica. Precisely as theology, the commentary places Thierry outside the Augustinian mainstream, from which eccentric position

31. On the former, see I. Caiazzo, “Un commento altomedievale al De arithmetica di Boezio,” in: Archivum Latinitatis Medii Aevi 58 (2000), pp. 113-150; and ead., “Intro- duction,” pp. 70-80. On the latter, see K. M. Fredborg (ed.), Latin Rhetorical Com- mentaries, p. 3. A transcription of the Bern commentary (MS Bern Bürgerbibliothek, Cod. 633, fols. 19ra-27ra) can be found in V. rodrigues, “Creatio numerorum: Nature et rationalité chez Thierry de Chartres,” Ph.D. Diss., École Pratique des Hautes Études, Paris 2006; cf. ead., “Pluralité et particularisme ontologique chez Thierry de Chartres,” in: I. Rosier-Catach (ed.), Arts du langage et théologie aux confins des XIe-XIIe siècles: Textes, maîtres, débats, Turnhout 2011, pp. 509-536. 32. See I. Caiazzo, “Introduction,” pp. 18-22; cf. D. Albertson, Mathematical The- ologies, pp. 330-331. 33. I. Caiazzo, “Introduction,” p. 40. 34. See D. Albertson, Mathematical Theologies, pp. 18-19, 119-139.

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the Parisian master ventured an epochal re-introduction of Neopy- thagoreanism into Latin Christian thought.35 By the time of Tractatus Thierry is already convinced that not only Boethius’s opuscula sacra, but indeed his quadrivial translations, are valuable authorities for Christian doctrinal theology. Second, for these reasons, it seems to me that the Arithmetica commentary clearly antedates and informs Tractatus, and not the other way around. Within a few pages at the end of Tractatus, Thierry accom- plishes two highly original and difficult tasks.36 He integrates the mathematical and theological aspects of Boethius into one uniform discourse, a Latin Neopythagorean theology of divine Number; and he reconciles the latter with the regnant Augustinian theology of the divine Word. Given the substantial overlap between Tractatus and the Arith- metica commentary, is it more reasonable to suppose that Thierry completed both of these tasks within Tractatus? Or that he took the first step while glossing Boethius, and then tested its durability within the Augustinian parameters of scriptural exegesis? The latter seems more plausible. The parade of innovations in Tractatus is almost unintelligi- ble unless Thierry had introduced some of them previously, and a theological commentary on De arithmetica fits the bill. This contested sequence might seem trifling, but it matters a great deal. At stake is whether the Parisian master merely happened upon Neopythagoreanism in the course of hexaemeral exegesis – even Augustine veered momentarily in this direction37 – or whether, as Gillian Evans has written, “making no distinction between Boethius’s original works and his commentaries, and his translations, he looked for unity of thought in his author and believed he had found it.”38 The sheer existence of the Arithmetica commentary seems to suggest that the daring plan, to which Thierry only alludes in Tractatus, was indeed deliberate and fully pre-formed prior to that work. Yet this entails that the incorporation of Neopythagoreanism into Christian theology was, from the outset, central to Thierry’s entire intellectual

35. See ibid., pp. 93-118. 36. See Thierry of Chartres, Tractatus, 30-47, ed. N. M. Häring, pp. 568-575. 37. See Augustine of Hippo, De Genesi ad litteram, IV, 1-7, in: CSEL 28, ed. J. Zycha, Vienna 1894, pp. 93-103. 38. G. r. evans, “Thierry of Chartres and the Unity of Boethius’ Thought,” in: Studia Patristica 17 (1983), p. 440.

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enterprise – and therefore, being neither accident nor whim, that it must have been grounded in his contextual response to the intellectual conditions of the twelfth century. Yet it contradicts many of our assumptions to imagine that such a radical mathematization of nature, of thought, and of God arose organically out of the twelfth century – and not, say, the new world of the fifteenth or seventeenth.39 Thierry’s immediate medieval reception proved dismal. The sole heir to his theological legacy, and his peculiarly modern project, turned out to be Nicholas of Cusa. While Caiazzo has well situated the Arith- metica commentary within Thierry’s oeuvre, she has not fully traced its re-emergence as a possible source for Cusanus.40 In what follows I show that the Arithmetica commentary made an indispensable contribution to De docta ignorantia itself, inspiring some of Nicholas’s most creative ideas. Then I briefly suggest how Caiazzo’s discovery sheds light on some outstanding questions about Cusanus’s Chartrian sources, including the enigmatic treatise, Fundamentum naturae. In this way Thierry’s Arithmetica commentary begins to fill out our emerging portrait of Nicholas as an ingenious steward, connoisseur, and editor of his collected Chartrian sources. One cannot but credit the cardinal’s claim in 1449 that, in the end, this unnamed Boethian commentator stood head and shoulders above his other great influences.

2. Thierry’s Arithmetica in De docta ignorantia, Book I There is significant textual evidence that Nicholas of Cusa used Thierry’s Arithmetica commentary in composing De docta ignorantia alongside other confirmed Chartrian traditions. Thierry’s commentary not only contributed key ideas and verbal formulae to the 1440 treatise, but might have inspired Cusanus to imagine the theological possibilities of arithmetic and geometry in the first place. One can easily imagine that his encounter with that commentary, more than other Chartrian texts, led Cusanus to imitate Thierry’s gesture of

39. See A. Speer, Die entdeckte Natur. Untersuchungen zu Begründungsversuchen einer “scientia naturalis” im 12. Jahrhundert, Leiden 1995, pp. 293-294; but cf. N. Germann, De temporum ratione. Quadrivium und Gotteserkenntnis am Beispiel Abbos von Fleury und Hermanns von Reichenau, Leiden 2006, pp. 245-255 and esp. pp. 287-309. 40. Caiazzo cites De docta ignorantia I, 8, and II, 3: see I. Caiazzo, “Introduction,” pp. 57, 67.

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reviving Neopythagoreanism within Latin Christianity. When we sur- vey De docta ignorantia, we first find traces of the Arithmetica commentary in the major themes championed by Nicholas throughout Book I. This initial evidence, while only suggestive, is considerably strengthened once we consider Book II. There Nicholas conspicuously relies upon two doctrines that appear to be unique to Thierry’s Arithmetica commentary.

2.1. Equality and precise measure Book I of De docta ignorantia unfolds in four parts: on God as the maximum (I, 1-6); on the mathematical Trinity (I, 7-10); on geometrical figures as divine symbols (I, 11-23); and on naming God through negation (I, 24-26). The careful reader will note that Cusanus does not begin De docta ignorantia with his famous ideas of maximum absolutum, contractio, coincidentia oppositorum, and indeed docta ignorantia. These all arrive shortly in the second chapter (I, 2), perhaps drawn, per Hoenen, from Fundamentum naturae. But before any of these (I, 1), Nicholas foregrounds a constellation of philosophical terms associated with Boethian arithmetic: proportio, numerus, medium, unitas, and alteritas. According to Cusanus, mathematicians teach that certainty comes only by mediation of proportional means. Proportion joins unity and alterity, permitting reason to comprehend difference, and every proportion is expressible through number. To grasp the dig- nity and universality of number was the genius of Pythagoras, he concludes.41 This opening chapter of De docta ignorantia could serve as an accurate digest of Boethius’s De arithmetica. Of course, due to that work’s diffusion throughout medieval learning, this alone does not yet indicate the presence of Thierry’s commentary. In chapters I, 3 and I, 4, Nicholas already offers a tentative synthesis of chapters I, 1 and I, 2. The problem is how to harmonize the vocabulary of absolute and contracted with that of number and proportion. Cusanus finds his solution in introducing another term from Boethian arithmetic, a term that also captivated Thierry of Chartres: aequalitas, and particularly, the precision of equality (aequalitas

41. See Nicholas of Cusa, De docta ignorantia, I, 1, 3, ed. P. Wilpert – H. G. Sen- ger, pp. 6-8; cf. ibid., I, 11, 32, pp. 42-44.

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praecisa) as an exact measure, without a single degree of deviation either more or less (excedens et excessum). Since the world of beings exists on a continuum of greater and lesser, inevitably we can only approximate truth by degrees, Nicholas explains. We cannot arrive at the divine maximum (or its coincident minimum) gradually, any more than we can attain circularity by adding more sides to a polygon. Therefore the intellect relates to truth as to an unattainable, perfect equality. In other words, it experiences equality negatively, or as Nich- olas prefers, incomprehensibiliter (I, 3). This is the fundamental ignorantia that fully understands (docta) its predicament. God is the impossible coincidence of maximum and minimum; or, stated otherwise, God is the impossible occasion of pure Equality itself, unthink- able in our world of infinitesimally continuous quantity. Nicholas clarifies that while his argument holds in arithmetic with respect to finite quantitas, he intends it in the trans-quantitative, divine instance even more so (I, 4).42 In the next two chapters, Nicholas assigns two further divine names: in the register of number, God is the One (I, 5); in the register of being, God is Absolute Necessity (I, 6). His treatment of the former manifests further echoes of Boethius’s De arithmetica. While the maximum is beyond number, number remains indispensable for thinking discretio, ordo, proportio, harmonia, and pluralitas. Cusanus envisions all numbers arranged along a continuous scale running up (ascensus or additio) and down (descensus or subtractio). Pure divine Equality, the coincidence of maximum and minimum, does not exist along these relative degrees, but is absolute from them all, just as the One (unitas) is not a number, but the fount of all number.43 These two chapters play an important structural role in Books I and II. While chapter I, 5 prefaces Nicholas’s discussion of Thierry’s mathematical Trinity to follow immediately in chapters 1, 7-10, likewise chapter I, 6 foreshadows his discussion of Thierry’s four modes of being in chapters II, 7-10.44 It is important to listen carefully to Cusanus’s overture in chapters I, 1-6. To begin with, their sequence reveals that the fundamental

42. See ibid., I, 3-4, 9-11, pp. 14-16. 43. See ibid., I, 5, 13-14, pp. 20-22. 44. See D. Albertson, Mathematical Theologies, pp. 175-190.

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theme of De docta ignorantia is not what it is often said to be. It is not docta ignorantia as an epistemological category: this is a slogan for a more profound intuition. Nor is it the coincidence of maximum and minimum: this is only one set of terms, which Nicholas repeat- edly tries to combine with another set, namely the lexicon of numerus, proportio, and harmonia. Rather, the fundamental insight of De docta ignorantia, Nicholas clearly states, is the notion of God as perfect aequalitas, namely, an ineffably precise Equality neither greater nor lesser than itself. “The root of learned ignorance,” he writes, is “the inconceivable precision of truth.”45 Nicholas will soon (I, 7) link this insight to Thierry of Chartres’s triad of unitas, aequalitas, and conexio, where equality names the divine Word, a doctrine common to nearly every Chartrian source. But this convenient Trinitarian corollary is not the reason why Nicholas privileges equality from the beginning, nor does it add anything to determine the concept. His definition comes instead from the field of Boethian arithmetic. In chapters I, 1-6 of De docta ignorantia, Nicholas defines aequalitas in three ways that specifically recall Thierry’s Arithmetica commentary. Indeed his account of equality parallels Thierry’s commentary more closely than either Boethius’s De arithmetica itself or Thierry’s other theological commentaries that espouse the mathematical Trinity. The three features of aequalitas in Book I to consider are the precision of equality, inequality as alterity, and the equality of “beingness” (entitas). (1) Aequalitas praecisa. Cusanus defines aequalitas as perfect precision that neither overwhelms nor underwhelms its exact measure. Tracking the cardinal’s preferred terms allows us to compare which sources may have led him to determine aequalitas as neither more nor less than itself. At first Nicholas favors excedens and excessum to denote degrees of difference above and below pure equality.46 Then he shifts

45. Nicholas of Cusa, De docta ignorantia, I, 2, 8, ed. P. Wilpert – H. G. Senger, p. 12: “[…] radicem doctae ignorantiae in inapprehensibili veritatis praecisione […] .” Cf. ibid., I, 26, 89, p. 112. All translations of De docta ignorantia are my own, but I have consulted and benefited from those of H. L. Bond and J. Hopkins. 46. See ibid. I, 3, 9, p. 12; I, 4, 11-12, pp. 16-18; I, 5, 13-14, pp. 20-22; I, 6, 15, p. 24; and II, 1, 91-95, p. 8. Thierry occasionally uses the same verb (excedere) to denote transgression of the mean (modus). See Thierry of Chartres, Commentum super Arith- meticam Boethii, I, 32 (1200, 1218), ed. I. Caiazzo, pp. 145-146. To avoid confusion with Thierry’s Commentum on Boethius’s De trinitate, I abbreviate this commentary as “Super Arithmeticam.”

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to other vocabulary: maius (or plus or magis) and minus, and occasionally ultra and infra.47 To take one example: “The equality of being is that which in a given thing is neither more nor less, neither beyond it, nor below. Were it greater than the thing, it would be monstrous; were it less, it would not exist.”48 One must “subtract” the excess of a greater part in order to reach precise equality.49 There are several Chartrian traditions that might have inspired Cusanus’s unusually literal definition of divine Equality in terms of arithmetic. The Fundamentum naturae treatise frequently describes the contracted cosmos as a continuum of magis and minus in which the absolute maximum cannot be found.50 Yet the treatise’s author – despite evidently knowing Thierry of Chartres’s Lectiones or Glosa quite well – never mentions the theology of divine Equality at all.51 Instead of Fundmentum naturae, we must look to the Tractatus on Genesis, the Commentum on De trinitate, and the Arithmetica commentary, texts where Thierry discusses equality extensively. Did Cus- anus use any of these sources to define equality, and can we detect any sequential development among their conceptions? If one compares Thierry’s doctrine of precise equality across these three texts, two differences become apparent. First, in his Arithmetica commentary Thierry expounds the meaning of equality slowly, cir- cling back to earlier statements and elaborating others at greater length. His goal in this commentary is simply to develop a theological account of pure equality. By contrast, in Tractatus and Commen- tum, Thierry seems to assume the doctrine as a complete, recognizable whole, which can be quickly referenced in the course of other tasks. In Tractatus, precise equality helps Thierry explain how the Creator’s

47. See Nicholas of Cusa, De docta ignorantia, I, 7, 19, p. 28; I, 8, 22, pp. 30-32; I, 18, 53, pp. 70-72; I, 24, 80, p. 102; II, 1, 96, p. 20; and III, 1, 186-188, pp. 6-10. 48. Ibid., I, 8, 22, ed. P. wilpert – H. G. Senger, pp. 30-32: “Aequalitas vero essendi est, quod in re neque plus neque minus est, nihil ultra, nihil infra. Si enim in re magis est, monstruosum est. Si minus est, nec est.” Cf. Boethius, Institutio arithmetica, I, 19, 8-9, in: J.-Y. Guillaumin (ed.), Institution Arithmétique, Paris 2002, pp. 40-42. 49. See Nicholas of Cusa, De docta ignorantia, I, 7, 19, ed. P. Wilpert – H. G. Sen- ger, p. 28. 50. See e.g. Fundamentum naturae, fol. 5r, ed. M. Hoenen, p. 454 = Nicholas of Cusa, De docta ignorantia, II, 8, 136, ed. P. Wilpert – H. G. Senger, p. 60. 51. See D. Albertson, “A Late Medieval Reaction to Thierry of Chartres’s (d. 1157) Philosophy,” pp. 81-82.

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Wisdom or Word sustains the ideas (notiones) of creatures in their number, measure, and weight (Wisdom 11:21).52 In Commentum, Thierry repeats the same exegesis of Wisdom 11 within a Trinitarian excursus that explains Boethius’s ontology of form. He uses precise equality again to define his “trinity of perpetuals” that reflects the divine Trinity throughout the cosmos.53 Second, while Boethius’s De arithmetica could have easily inspired Thierry’s doctrine of precise equality, it is unclear how the texts of Genesis or De trinitate would have led Thierry to this topic in the first place. For these reasons, let us posit the priority of Thierry’s Arithmetica commentary for the purpose of analyzing the sources of De docta ignorantia. Here we must consider three features of precise equality in Thierry’s Arithmetica commentary and judge which of them, if any, is reflected in De docta ignorantia. First, Thierry defines equality as the “mean” of the Good, the eternal divine law, from which evil deviates “beyond or beneath” (citra vel ultra).54 Thierry uses similar language in Tractatus, but not in Commentum.55 Nicholas does not take up this thread in De docta ignorantia. Second, Thierry associates the precision of equality with the perduring integrity of things: their “equality of being” (aequalitas essendi) excludes every excess and deficiency, everything “above or below” the thing itself (ultra et infra, plus vel minus), thus securing its identity.56 Nicholas treats the same theme in De docta ignorantia, but Thierry repeats this feature of equality in both Tractatus and

52. See Thierry of Chartres, Tractatus, 42-44, ed. N. M. Häring, pp. 572-574. 53. See Thierry of Chartres, Commentum, II, 35-36, ed. N. M. Häring, p. 79; and ibid., II, 40-41, p. 81. The seed of the trinitas perpetuorum doctrine, however, already appears in Thierry of Chartres, Super Arithmeticam, I, 32 (1334-1338), ed. I. Caiazzo, p. 150. 54. Thierry of Chartres, Super Arithmeticam, I, 32 (1172-1176), ed. I. Caiazzo, p. 144: “Cognito igitur quomodo inaequalitas ab aequalitate procedat, cognoscetur nimirum quomodo bonum sit origo et principium mali. Modus autem citra vel ultra quem non est bonum, a quibusdam lex divina vocatur, ab aliis praefinitio aeterna, a non- nullis vero providentia sive mens divina.” 55. See Thierry of Chartres, Tractatus, 42 and 44, ed. N. M. Häring, pp. 572- 574. Thierry also relies on Augustine’s sense of modus: see J. J. O’Donnell (ed.), Augus- tine: Confessions, Vol. 2, Oxford 1992, pp. 46-51. 56. Thierry of Chartres, Super Arithmeticam, I, 32 (1228-1232), ed. I. Caiazzo, p. 146: “Eadem vero unitas entitatis est aequalitas. Haec enim est essendi forma, perfectio et integritas, ultra quam et infra nihil est quod ad esse rei pertineat, quo scilicet nihil plus de rei esse est, vel minus. Quae videlicet aequalitas rerumque perfectio omnem excludit superfluitatem, omnemque perficit et implet indigentiam.”

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Commentum, so these could also be Nicholas’s source.57 Third, Thierry contends in his Arithmetica commentary that one makes something perfect by “subtracting or adding” to it until one “returns it to equality […] neither more, nor less, neither above nor below” what it is.58 This lengthy argument from subtraction is missing from both Tractatus and Commentum. However, Cusanus repeats the same argument from subtraction in Book I of De docta ignorantia.59 These comparisons suggest that it is possible, but not yet certain, that Thierry’s Arithmetica commentary was a Cusan source in 1440. (2) Inaequalitas as alteritas. Nicholas famously repeats Thierry’s Trinity of unity, equality, and connection in De docta ignorantia, and in nearly every work thereafter.60 It is well known that Thierry derived his formula by giving a mathematical interpretation to a common Augustinian triad. But in fact Thierry has two different methods for proving the eternal union of “unity” and “equality,” both of which derive from Boethius’s De arithmetica, one geometrical and one arithmetical. The geometrical method shows that if every inequality returns to equality, such absolutely prior equality must be eternal, therefore singular, and therefore one with eternal unity.61 The arithmetical method shows that counting unity only generates unity again,

57. See Nicholas of Cusa, De docta ignorantia, I, 7, 19, ed. P. Wilpert – H. G. Sen- ger, p. 28; I, 8, 22, pp. 30-32; and I, 24, 80, p. 102. Cf. Thierry of Chartres, Trac- tatus, 41, 43, and 45-46, ed. N. M. Häring, pp. 572-575; and id., Commentum, II, 31 and 35, ed. N. M. Häring, pp. 78-79. 58. Thierry of Chartres, Super Arithmeticam, I, 32 (1234-1236, 1264-1268), ed. I. Caiazzo, pp. 146-148: “Quamdiu enim demendum aliquid est vel addendum rei, nec est adhuc res ipsa perfecta. Postquam vero nihil demendum est vel addendum, res quoque integre perfecta est, id est formata. […] Quamvis enim ab aequalitate illa descen- dat, illa tamen aequalitas ad aequalitatem quandam illud redigit, quoniam illud, ut praedictum est, ad bonum exitum ducit, sicut arithmetica ratio omnem inaequalitatem redigit ad aequalitatem. Patet itaque quod bonitas definita est, quoniam in ea nihil plus, nihil minus, nihil ultra, nihil infra est.” Cf. however the brief reference in id., Commentum, II, 36, ed. N. M. Häring, p. 79 (cited below). 59. Nicholas of Cusa, De docta ignorantia, I, 7, 19, ed. P. Wilpert – H. G. Sen- ger, p. 28: “Nam inaequale inter maius et minus est. Si igitur demas quod maius est aequale erit. Si vero minus fuerit, deme a reliquo quod maius est, et aequale fiet. Et hoc etiam facere poteris, quousque ad simplicia demendo veneris. Patet itaque, quod omnis inaequalitas demendo ad aequalitatem redigitur.” Cf. ibid., I, 5, 13, p. 20. 60. See B. McGinn, “Unitrinum seu Triunum: Nicholas of Cusa’s Trinitarian Mysti- cism,” in: M. Kessler – C. Sheppard (eds.), Mystics: Presence and Aporia, Chicago 2003, pp. 90-117. 61. See Boethius, Institutio arithmetica, I, 32, 1-2, ed. J.-Y. Guillaumin, pp. 66-67.

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a unity equal to itself and thus one with equality.62 In his Arithmetica commentary, Thierry moves back and forth between both methods.63 In Tractatus, Thierry leads with the geometrical and then turns to the arithmetical; but in Commentum, Thierry reverses the order, suppress- ing the geometrical method.64 In the later Lectiones and Glosa, Thierry references only the arithmetical method, and De septem septenis and Commentarius Victorinus follow suit.65 Among Chartrian texts using the geometrical method, we can make a further distinction. Across many of his writings Thierry discusses unitas and alteritas in general, deriving alterity from unity.66 But rarely does he state the corollary regarding equality: that inequality’s deviation from perfect equality is itself a manifestation of alteritas, such that unity is to equality as alterity is to inequality. Thierry states this analogy in only two texts: the Arithmetica commentary and Commentum; however, the latter is quite brief compared to the former.67 In his Arithmetica commentary, by contrast, his argument has several steps. Alterity, ever mutable, descends from the stability of unity. While unitas corresponds to odd numbers, form, and square figures (equilaterals), alteritas represents the even, matter, and irregular polygons (figures of unequal sides).68 Since the most equal figure is the square, divine form is rightly conceived as square-like. Divinitas expresses itself as the square’s power to restrain the flux and disorder

62. See ibid., I, 20, 8, p. 45; and ibid., II, 4, 5, p. 90. 63. For the first method, see Thierry of Chartres, Super Arithmeticam, I, 32 (1220- 1226), ed. I. caiazzo, p. 146. For the second method, see ibid., I, 32 (1170-1176), p. 144; I, 32 (1202-1215), pp. 145-146; and II, 31 (814-835), pp. 185-186. 64. See Thierry of Chartres, Tractatus, 37-40, ed. N. M. Häring, pp. 570-572; and id., Commentum, II, 30-36, ed. N. M. Häring, pp. 77-79. 65. See Thierry of Chartres, Lectiones, VII, 6, ed. N. M. Häring, p. 225; id., Glosa, V, 17-18, ed. N. m. häring, pp. 296-297; Commentarius Victorinus, 81-85, ed. N. M. Häring, pp. 498-499; and De septem septenis, PL 199, col. 961BC. 66. See e.g. Thierry of Chartres, Super Arithmeticam, I, 2 (572-574), ed. I. Caiazzo, p. 123; and ibid., I, 32 (1334-1338), p. 150. Cf. id., Tractatus, 30, ed. N. M. Häring, p. 568; and id., Glosa, V, 18-19, ed. N. M. Häring, p. 297. 67. Thierry of Chartres, Commentum, II, 36, ed. N. M. Häring, p. 79: “In his enim quedam est inequalitas: quare magis et minus. Sed inter hec est equalitas. Immo ab equalitate descendit hec inequalitas. Si enim inequalitati aliquid dematur minus est. Si superadditur aliquid magis est. Merito ergo dictum est quod ab unitate per semel gignitur unitatis equalitas quam per binarii uel alterius alteritatis multiplicationem gigni non posse monstratrum est.” 68. See Thierry of Chartres, Super Arithmeticam, II, 27 (666-696), ed. I. Caiazzo, pp. 180-181.

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of alteritas, and therefore to lead inequalities back to equality. The unequal sides of irregular polygons express the mutable alterity of matter.69 There is only one perfectly equilateral square. But by increasing or decreasing internal angles, one can generate an infinite range of deviations, as the square shifts into every possible degree of rhombus or trapezoid.70 Thierry writes: “Since unity and odd numbers, along with squares, participate in equality and simplicity, whatever among them departs from equality into inequality must neces- sarily advance by greater and lesser degrees.”71 Boethius’s principle holds that equality always precedes inequalities, as unity precedes alterity, and as simplicity precedes comparison. Now let us examine how Cusanus introduces Thierry’s mathematical Trinity in De docta ignorantia. Which method does he use? It turns out that when Nicholas first explains the triad, he does not use the arithmetical method that prevails in most Chartrian texts (including two of Nicholas’s confirmed sources, De septem septenis and Com- mentarius Victorinus). Instead, Cusanus conspicuously adopts Thier- ry’s geometrical method. He elaborates and even refines what we find in Thierry’s Arithmetica commentary regarding inequality and alterity. According to Cusanus, equality is always prior to inequality, and inequalities arise from exceeding equality by greater or lesser, by subtracting or adding. Inequality is coextensive with alterity, writes Nich- olas, since the comparison of inequals denotes their binary differentiation. Just as alterity cannot be eternal, neither can inequality; but both unity and equality are eternal; therefore they must be one.72 Cusanus thinks so highly of this geometrical method that he applies the same reasoning to conexio, something Thierry never ventured.73 Not until the end of the next chapter does he finally mention Thier- ry’s arithmetical method of self-multiplying unity.74 Once again, this evidence is not dispositive; theoretically, Nicholas could have used

69. See ibid., II, 31 (809-839), pp. 185-186. 70. See ibid., II, 34 (1022-1025), p. 193. 71. Ibid., II, 34 (1025-1027), p. 193: “Cum enim unitas et impares cum tetragonis participent aequalitate et simplicitate, quicquid ab eorum aequalitate discedit in inaequalitatem, oportet ut in maius vel in minus prodeat.” 72. See Nicholas of Cusa, De docta ignorantia, I, 7, 19, ed. P. Wilpert – H. G. Sen- ger, p. 28. 73. See ibid., I, 7, 20-21, pp. 28-30. 74. See ibid., I, 8, 23, p. 32.

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Commentum for this passage. But given the substance of his argument, it seems probable that his source was the more extensive treatment in Thierry’s Arithmetica.75 (3) Aequalitas entitatis. As is well known, Nicholas borrows from Thierry the notion that God is forma essendi or aequalitas essendi, namely that divine Equality has an ontological function to preserve individuation.76 On a few rare occasions, however, Nicholas uses the strange term aequalitas entitatis, the “equality of beingness,” to mean the same thing.77 Is this a Cusan innovation? Thierry uses aequalitas essendi throughout most of his theological works.78 Thierry also has the peculiar habit of naming perfect being entitas.79 Before Caiazzo’s

75. Another point of comparison regarding alteritas: both Thierry and Nicholas teach that as soon as reason attempts to think the One, that very intellectual motion produces a potential opposition, which precludes the absoluteness of divine unity beyond contraries. Cf. Nicholas of Cusa, De docta ignorantia, I, 24, 76, ed. P. Wilpert – H. G. Senger, p. 98; and Thierry of Chartres, Super Arithmeticam, II, 31 (822-825, 839-841), ed. I. Caiazzo, p. 186. 76. On aequalitas and forma essendi in Cusanus, see H. Schwaetzer, Aequalitas: Erkenntistheoretische und soziale Implikationen eines christologischen Begriffs bei Nikolaus von Kues, Münster 1978; T. Leinkauf, “Die Bestimmung des Einzelseienden durch die Beg- riffe contractio, singularitas und aequalitas bei Nicolaus Cusanus,” in: Archiv für Begriffs- geschichte 37 (1994), pp. 180-211; and C. Rusconi, “Cusanus und Thierry von Chartres: Die Einteilung der Spekulativen Wissenschaften und der Begriff forma essendi in ‘De possest’ und in Kommentar ‘Librum hunc’,” in: H. Schwaetzer – K. Zeyer (eds.), Das europäische Erbe im Denken des Nikolaus von Kues: Geistesgeschichte als Geistesgegenwart, Münster 2008, pp. 285-302. On the interplay between mathematical and theological senses of aequalitas before Cusanus, see N. Germann, De temporum ratione, pp. 255-278. 77. Nicholas of Cusa, De docta ignorantia, I, 8, 22, ed. P. Wilpert – H. G. Senger, pp. 30-32: “Forma enim essendi est, quare et entitas. Aequalitas vero unitatis quasi aequalitas entitatis, id est aequalitas essendi sive exsistendi. Aequalitas vero essendi est, quod in re neque plus neque minus est, nihil ultra, nihil infra.” Ibid., I, 24, 80, p. 102 (my emphasis): “ […] clarum est filium nominari filium ex eo, quod est unitatis sive entitatis aut essendi aequalitas. […] Ex hoc enim filius, quod est aequalitas essendi res, ultra quam vel infra res esse non possent. Ita videlicet quod est filius ex eo, quod est aequalitas entitatis rerum […] .” 78. In Tractatus, Thierry prefers aequalitas existentiae rerum and does not mention entitas: see Thierry of Chartres, Tractatus, 45-46, ed. N. M. Häring, pp. 574-575. By the time of Commentum and Lectiones, Thierry uses entitas regularly, but still prefers aequalitas essendi or aequalitas existendi: see id., Commentum, II, 37-48 passim, ed. N. M. Häring, pp. 80-83; and id., Lectiones, II, 38, ed. N. M. Häring, p. 167. 79. See e.g. Thierry of Chartres, Commentum, II, 20-22, ed. N. m. häring, pp. 74-75; id., Lectiones, II, 35, ed. N. M. Häring, p. 166; and ibid., II, 48, p. 170. See further among the unattributed works from Thierry’s circle: Fragmentum Admuntense: De Hebdomadibus, 8, ed. N. m. häring, pp. 120-121; Abbreviatio Monacensis: Contra Eutychen, I, 26-37, ed. N. M. Häring, pp. 445-446; and Commentarius Victorinus, 223, ed. N. M. Häring, p. 526.

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discovery, when we only had Thierry’s commentaries on Genesis and De trinitate, it could appear that Cusanus engineered the composite formula aequalitas entitatis. But now we can confirm that in fact Thierry uses aequalitas entitatis in his Arithmetica commentary, and only there – where, however, it dominates his discussion of equality.80 Since Nicholas foregrounds this term both in De docta ignorantia and in subsequent works, it seems more likely that he discovered the term in Thierry’s Arithmetica than that he reinvented it for himself.81

2.2. Number as divine exemplar So far we have seen that Nicholas first conceived Book I in terms of numerus, proportio, and harmonia, terms derived from Boethian arithmetic. He then connected this “Pythagorean enquiry” (his words) to his doctrine of God as maximum.82 But to do so Nicholas had to rely upon another term from Boethian arithmetic, the concept of aequalitas. As we have seen, Cusanus’s definition of aequalitas matches Thierry’s Arithmetica commentary more closely than other Chartrian commentaries. Beyond this evidence, there is a third instance of Boethian arithmetic in Book I to consider. For Boethius, number operates as an eternal exemplar in the divine Mind, guiding the world’s creation; and such number must be divided into multitudo (discontinuous integers) and magnitudo (continuous quantity). Together multitude and magnitude structure the four mathematical sciences of the quadrivium, which collectively are not simply human

80. Thierry of Chartres, Super Arithmeticam, I, 32 (1236-1241, 1257-1260), ed. I. Caiazzo, pp. 147-148 (my emphasis): “Haec est ergo illa essendi forma et entitatis aequalitas, a qua omnes rerum formae descendunt […] ultra quam videlicet nihil est vel infra […]. Est ergo haec illa entitatis aequalitas quae vere et proprie dicitur unitatis ipsius figura. […] Illa ergo quam praediximus bonitas seu forma essendi, seu unitatis figura, vel entitatis aequalitas, regula quaedam est, quam quicquid transgreditur malum est, quoniam quicquid plus vel minus, ultra vel infra est, vitium est.” 81. See e.g. Nicholas of Cusa, De coniecturis, II, 17, 179, in: Nicolai de Cusa Opera omnia iussu et auctoritate Academiae Litterarum Heidelbergensis, Vol. III, eds. J. Koch – K. Bormann, Hamburg 1972, p. 179; id., Idiota de sapientia, I, 22, in: Nicolai de Cusa Opera omnia iussu et auctoritate Academiae Litterarum Heidelbergensis, Vol. V, eds. R. Steiger – L. Baur, Hamburg 1983, p. 46; and id., Idiota de mente, XI, 132, ed. R. Steiger, p. 185. 82. Nicholas of Cusa, De docta ignorantia, I, 9, 26, ed. P. Wilpert – H. G. Sen- ger, p. 36.

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arts but the fundamental matrix of cosmic order from the Creator.83 This Platonist doctrine of number as exemplar – number as quadrivial order grounded in the divine Mind – originates with Nicoma- chus of Gerasa and is fully retained by Boethius in De arithmetica.84 Now, the way that Thierry and Nicholas handle this doctrine (let us call it the exemplar doctrine) is quite illuminating for our pur- poses. Thierry only rarely cites the doctrine straight from De arithmetica in his major works. His most prominent reference, for example, is a casual paraphrase in Glosa.85 This is surprising, because several of Thierry’s most distinctive innovations assume that primeval number abides within God. None of the other Chartrian sources that we know Nicholas used in 1440 – De septem septenis and Commen- tarius Victorinus – cite the exemplar doctrine. It turns out that Thierry only refers to the exemplar doctrine directly within the pages of his Arithmetica commentary. This means that as far as Chartrian sources go, when Nicholas clearly states this idea, he could be using Boethi- us’s De arithmetica itself, which is certainly possible, or he could be inspired by Thierry’s Arithmetica commentary.

83. Boethius, Institutio arithmetica, I, 1, 8, ed. J.-Y. Guillaumin, pp. 8-9: “Haec [arithmetica] enim cunctis prior est, non modo quod hanc ille huius mundanae molis conditor deus primam suae habuit ratiocinationis exemplar et ad hanc cuncta constituit quaecumque fabricante ratione per numeros adsignati ordinis inuenere concordiam […].” Ibid., I, 2, 1, p. 11: “Omnia quaecumque a primaeua rerum natura constructa sunt numerorum uidentur ratione formata. Hoc enim fuit principale in animo conditoris exemplar.” 84. The immediate source of Boethius’s doctrine is Nicomachus of Gerasa, Intro- ductio arithmetica, I, 4, 2, ed. R. G. Hoche, Leipzig 1866, p. 9; and ibid., I, 6, 1, p. 12. Seneca, however, similarly writes: “Haec exemplaria rerum omnium deus intra se habet numerosque universorum, quae agenda sunt, et modos mente conplexus est: plenus his figuris est, quas Plato ideas appellat, inmortales, inmutabiles, infatigabiles.” (See Seneca, Ad Lucilium, 65, 7, in: Epistulae Morales, ed. R. M. Gummere, Cambridge 1917, p. 448.) On the divine ideas as mathematicals in Neoplatonism, see H. A. Wolfson, “Extradeical and Intradeical Interpretations of Platonic Ideas,” in: Journal of the History of Ideas 22 (1961), pp. 3-32; H. J. Krämer, Der Ursprung der Geistmetaphysik. Untersuchungen zur Geschichte des Platonismus zwischen Platon und Plotin, Amsterdam 1964, pp. 23-26; and L. m. napolitano Valditara, Le idee, i numeri, l’ordine: La dottrina della mathesis universalis dall’Accademia antica al neoplatonismo, Naples 1988, pp. 426-429. 85. Thierry of Chartres, Glosa, I, 38, ed. N. M. Häring, p. 267: “Quod non uidetur eo quod numerus prior est omni accidente. Numerus enim, ut habet Arithmet- ice prologus, principale exemplar extitit in mente conditoris. Quod si ex accidentibus profluit numerus in substantias ergo preiacent numero accidentia. Quare exemplar conditoris precessisse uidentur. Quod falsum est.” Cf. id., Lectiones, I, 51, ed. N. M. Häring, pp. 149-150.

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Cusanus first broaches the topic of number in God’s mind amidst his apologia for the application of mathematics in theology (I, 11). All the wisest philosophers have held that divine wisdom is only attained by those well trained in mathematics. Pythagoras was the first to teach this, Nich- olas writes, followed by the Platonici and the Christians sages, Augustine and Boethius. He states that for both men, “the number of things to be created was the ‘principal exemplar in the mind of the Creator’.”86 Augus- tine had recourse to mathematics to explain the immortality of the soul, and “this way seemed to please our Boethius, since he frequently asserted that every true teaching could be comprehended in multitude and magnitude.”87 Here Cusanus juxtaposes both aspects of the exemplar doctrine. Number abides eternally in the Creator’s wisdom; and number leads to wisdom only through the categories of multitude and magnitude. Elsewhere in De docta ignorantia Nicholas repeats each aspect again.88 In his Arithmetica commentary, Thierry was evidently attracted to the same passage. He first defines some of Boethius’s terminology for his students, and reminds them that by numerus Boethius intends arithmetic, not the entire quadrivium. Then Thierry glosses the theological sense of the passage as follows: God is said to have created things according to the exemplar of number, since all things have being from difference [discretio], which comes from numbers. And note that God is said to possess arithmetic (that is, number) as the exemplar of his reasoning (that is, his disposition), because he ordered ‘all things’ according to number (that is, according to the difference of things as foreseen in his mind). ‘All things,’ I say, which are harmonized through number, that is, which are arranged by means of proportion, which is to say,

86. Nicholas of Cusa, De docta ignorantia, I, 11, 32, ed. P. Wilpert – H. G. Sen- ger, p. 42: “Quem Platonici et nostri etiam primi in tantum secuti sunt, ut Augustinus noster et post ipsum Boethius affirmarent indubie numerum creandarum rerum ‘in animo conditoris principale exemplar’ fuisse.” 87. Ibid., I, 11, 32, p. 44: “Ista via Boethio nostro adeo placere visa est, ut constanter assereret omnem veritatis doctrinam in multitudine et magnitudine comprehendi.” Regarding Augustine, Nicholas’s words strongly suggest that he is using Commentarius Victorinus: cf. Commentarius Victorinus, 81, ed. N. M. Häring, p. 498. 88. Nicholas of Cusa, De docta ignorantia, II, 12, 172, ed. P. wilpert – H. G. Senger, p. 104: “[…] quod eorum [particularorum] nullus est numerus nisi apud eum [Deum], qui omnia in numero creavit.” Ibid., II, 3, 108-109, pp. 24-26: “Sicut igitur ex nostra mente per hoc, quod circa unum commune multa singulariter intelligimus, numerus exoritur, ita rerum pluralitas ex divina mente, in qua sunt plura sine pluralitate quia in unitate complicante. […] Quis rogo intelligeret, quomodo ex divina mente rerum sit pluralitas, postquam intelligere dei sit esse eius, qui est unitas infinita?”

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all things which are the order arranged in the divine Mind. For it is from these [numbers] that there proceeds the order of all things.89 God’s mind possesses arithmetic as the matrix of created order. Thier- ry’s reference to discretio signals that he is indicating Boethius’s tech- nical definition of number as multitudo.90 As Caiazzo has noted, this observation is no doubt the origin (and antecedent) of Thierry’s famous passing remark in Tractatus that “the creation of number is the creation of all things.”91 Without the context of his Arithmetica commentary, Thierry’s irenic embrace of high Neopythagoreanism in his Genesis commentary is startling; but as an allusion to his prior Boethius lectures, it becomes entirely comprehensible. Thierry often discusses plural forms in the divine Mind. Like many Christian Platonists, he reasons that since God is the form of forms, God’s mind eternally contains divine ideas, or the formal exemplars of all things.92 In Thierry’s later commentaries on De trinitate, he even connects mens divina with folding and with the four modes.93 But in none of these cases does Thierry link the divine Mind specifically to number or arithmetic. Even in Tractatus, despite his many allusions to De arithmetica, Thierry never quotes the Boethian exemplar doctrine. Granted, on one occasion, Thierry does compare the procession of forms from the divine Mind to the procession of numbers from eternal unity.94 It is conceivable that this passage influenced

89. Thierry of Chartres, Super Arithmeticam, I, 1 (343-350), ed. I. Caiazzo, p. 114: “Ad exemplar numeri dicitur deus res creasse, quia omnia ex discretione, quae est ex numeris, habent esse. Et nota quod deus dicitur habuisse arithmeticam, id est numerum, exemplar suae ratiocinationis, id est dispositionis, quia secundum numerum, id est secundum rerum discretionem in mente provisa, cuncta disposuit. ‘Cuncta’ dico quae concordant per numerum, id est iuxta proportionem assignantur, id est quae sunt assignatus ordo in mente divina. Ex his enim omnis rerum ordo procedit.” Cf. ibid., I, 1 (186-192), pp. 108-109. 90. See Boethius, Institutio arithmetica, I, 1, 3, ed. J.-Y. Guillaumin, p. 7; and ibid., I, 2, 3, p. 11. 91. Thierry of Chartres, Tractatus, 36, ed. N. M. Häring, p. 570: “Unitas igitur est omnipotens in creatione numerorum. Sed creatio numerorum rerum est creatio.” 92. On the divine ideas, see Thierry of Chartres, Lectiones, II, 43-44, ed. N. M. Häring, pp. 168-169; and id., Glosa, II, 32-36, ed. N. M. Häring, pp. 275-276. 93. On folding, see Thierry of Chartres, Commentum, II, 49, ed. N. M. Häring, p. 84; on the second mode, see id., Lectiones, II, 66, ed. N. M. Häring, p. 176. Both are combined in Abbreviatio Monacensis: De Hebdomadibus, 25-42, ed. N. M. Häring, pp. 409-412. 94. Thierry of Chartres, Tractatus, 43, ed. N. M. Häring, p. 573: “Sicut igitur ipsa equalitas unitatis rerum notiones et intra se continet et ex se generat ita etiam illa eadem formas omnium rerum et intra se continet et ex se producit. Et sicut ipsa unitas

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Cusanus instead of the text cited above from Thierry’s Arithmetica. Yet on closer inspection, these lines in Tractatus neither reference exemplarity, nor conceive arithmetic as multitude and magnitude. Only in Thierry’s Arithmetica commentary is the Nicomachean doctrine – arithmetical categories as divine exemplars – reproduced in full, just as we find it in De docta ignorantia. To be sure, Cusanus would have been independently familiar with the Boethian notion of number as divine exemplar. In principle, his two discussions of number and the divine Mind could have stemmed from Boethius’s De arithmetica alone. On the other hand, Nicholas had good reason to echo Thierry’s Arithmetica commentary in De docta ignorantia. His agenda in this passage is to illustrate the orthodoxy of applying mathematics within theology – a project not evident in De arithmetica itself, but the very raison d’être of Thierry’s Arithmetica commentary. Here Nicholas envisages Boethius not simply as another textual source, but as Boethius noster, an authoritative fellow contributor to the Christian Platonist tradition that Nicholas consciously shares with him, with Augustine, and presumably with his favored Chartrian sources. Moreover, we know that in the same chapter Nicholas has already deployed another Chartrian source (Commentarius Victorinus) when he mentioned Augustine’s recourse to mathematics. Given this context, does it seem more likely that Nicholas combined Commentarius Victorinus with his own readings in Boethius’s De arithmetica? Or that having uncovered an extraordinary theological commentary on De arithmetica, he wove together its insights with other comparable texts of Chartrian heritage? I find the latter more plausible. On this view, the shared tradition assumed in Cusanus’s appella- tion Boethius noster might even denote his collected Chartrian sources, grasped as a unified set of commentaries on Boethius’s De trinitate and Boethius’s De arithmetica. The possessive pronoun of the first- person plural not only aligns Nicholas with fellow medieval readers of Boethius, but depicts them as partners cooperating to steward the same Boethian legacy. It also suggests that the appeal of these anonymous Chartrian confrères is their fundamental insight, evinced by the

omnes numeros ex se procreat ita ipsa unitatis equalitas omnes proportiones et inaequalitates omnium rerum ex se producit. Et in ipsam eadem omnia resoluuntur.”

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mere existence of the Arithmetica commentary, that since the opuscula sacra and the quadrivial translations flow from the same pen of one author, their Boethius, those works demand to be read univocally, as Christian doctrines transformed by Boethian arithmetic into an inte- grated mathematical theology.

3. Thierry’s Arithmetica in De docta ignorantia, Book II When Hoenen exposed Cusanus’s reliance on Fundamentum naturae, he showed that circumstantial hints in Book I of De docta ignorantia were confirmed by unequivocal evidence in Book II.95 The same pattern holds for Thierry’s Arithmetica commentary. Thus far I have listed several reasons to suspect that Nicholas may have modeled his own mathematizing theology in Book I after Thierry’s example. But now, based on two crucial passages in Book II, it appears even more likely that Cusanus referenced the Arithmetica commentary directly in his 1440 work. The first passage concerns the quadrivium (II, 1); the second concerns Nicholas’s most important doctrinal innovation of Book II, his account of enfolding and unfolding (II, 3).

3.1. Inequality and the quadrivium Cusanus argues in Book I that the experience of mathematical measurement – namely its constant failure – is a trace of divine transcendence. God can only be known negatively, just as pure equality can only be experienced negatively, through infinite approximation to perfect precision. This comparison, however, is a real analogy, since God is the Triune source of number as unity, equality, and connection, and since the exemplar of creation is eternal arithmetic in the divine Mind. Hence as Cusanus transitions from Book I on the divine maximum to Book II on the cosmic maximum, he hits upon an elegant theme: the four arts of the quadrivium as different ways to experience mathematical measurement. The “rule” of learned ignorance teaches that only

95. After comparing general themes in Book I and Fundamentum naturae, Hoenen exposes textual evidence that Cusanus used the treatise in Book II (“‘Ista prius inaudita’,” pp. 401-402, 409-421). On this basis he argues that heretofore ambiguous passages in Book I could be signs of influence as well (ibid., pp. 421-422).

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God possesses precise equality, because God is beyond every opposition. By contrast, every human attempt at measurement fails to surpass approximate comparisons of greater or lesser. Even the tools of mathematical precision cannot escape the conditions of alterity. According to Cusanus, nothing teaches this ontological and theological lesson better than the failures of the four quadrivial arts. Nicholas observes rather ingeniously that the quadrivium generates only imperfect inequalities and never actually reaches equality itself; and yet, nonetheless, that absent equality precedes and grounds every inequality thus produced. Strangely Nicholas proceeds in the opposite order from Boethius’s explanation in De arithmetica, moving in reverse from astronomy to geometry, and then from music to arithmetic, devoting a paragraph in exposition of each science. He shows that no matter how carefully one calculates planetary motions, figural dimensions, proportional harmonies, or numerical sums, the result is always imperfection, inequality, and therefore negation.96 Following these four paragraphs, Nicholas adds two more on the mathematical problem of infinitely divisible continuous quantities, and cites the example of an “infinite line.”97 He concludes the chapter (II, 1) with a nod to the common scholastic distinction between privative infinity (in mathematics) and negative infinity (reserved for God alone).98 At the close of Book II (II, 13), Nicholas returns to the quadrivium. He lauds the skill of the Creator, who used arithmetic, geometry, music, and astronomy (now returned to their proper order) to tame the unruly elements through proportion and harmony.99 Before Caiazzo’s discovery, it was easy to appreciate Nicholas’s meditations on the quadrivium as little more than a clever bookend- ing of Book II. After all, we know that the cardinal entertained a lifelong fascination with technologies of instrumentation and geometrical proofs. His musings on the quadrivial arts and the creation seem to evoke patristic and medieval hexaemeral commentaries.100

96. Nicholas of Cusa, De docta ignorantia, II, 1, 91-94, ed. P. Wilpert – H. G. Sen- ger, pp. 4-8. 97. Ibid., II, 1, 95-96, pp. 8-10. 98. Ibid., II, 1, 97, p. 12. 99. Ibid., II, 13, 175-180, pp. 108-114. 100. See M. Führer, “The Evolution of the Quadrivial Modes of Theology in Nich- olas of Cusa’s Analysis of the Soul,” in: American Benedictine Review 36 (1985), pp. 325- 342; and W. Schulze, Zahl, Proportion, Analogie. Eine Untersuchung zur Metaphysik und Wissenschaftshaltung des Nikolaus von Kues, Münster 1978.

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One might even suspect that Nicholas is indebted to Thierry of Char- tres, who briefly alludes to the quadrivium in Tractatus: “There are four kinds of reasons which lead humankind to the knowledge of the Creator: namely, the proofs of arithmetic, music, geometry, and astronomy. Such tools can be used briefly in our present theology, so that the craft of the Creator in things becomes evident and so that what we have set out above can be rationally demonstrated.”101 After this lapidary decree Thierry presents his mathematized model of the Trinity, with frequent recourse to the principles of Boethian arithmetic. But with Thierry’s Arithmetica commentary in hand, we can now identify a far more probable source for De docta ignorantia II, 1 (and by extension II, 13 as well). I have shown that Nicholas borrowed aspects of his theology of equality from Thierry’s Arithmetica commentary. As a commentator, naturally, Thierry makes gradual progress through the text of De arithmetica, so that his conceptualization of equality arises within a particular exegetical context. To see how Nicholas came to use Thierry’s Arithmetica in II, 1, we have to appreciate this context. Deep into the first book of De arithmetica, Thierry encounters the foundational Boethian maxim that “every inaequalitas proceeds from aequalitas.”102 He immediately advises his students that this is a difficult but essential doctrine, and that it must be approached from multiple disciplinary perspectives. According to Thierry, there are theological, mathematical, and physical interpretations of the maxim, or in a later formulation, the natural (physics), the rational (mathematics), and the moral (ethics).103

101. Thierry of Chartres, Tractatus, 30, ed. N. M. Häring, p. 568: “Adsint igitur quatuor genera rationum que ducunt hominem ad cognitionem creatoris: scilicet arith- metice probationes et musice et geometrice et astronomice. Quibus instrumentis in hac theologia breviter utendum est ut et artificium creatoris in rebus appareat et quod propo- suimus rationabiliter ostendatur.” Cf. K. Takashima, “Nicolaus Cusanus und der Einfluss der Schule von Chartres,” in: K. Yamaki (ed.), Nicholas of Cusa: A Medieval Thinker for the Modern Age, Surrey 2002, pp. 97-105. 102. Boethius, Institutio arithmetica, I, 32, 2, ed. J.-Y. Guillaumin, p. 67: “Hoc autem erit perspicuum, si intellegamus, omnes inaequalitatis species ab aequalitatis creui- sse primordiis, ut ipsa quodammodo aequalitas matris et radicis obtinens uim ipsa omnes inaequalitatis species ordinesque profundat.” Cf. Thierry of Chartres, Super Arith- meticam, I, 32 (1161-1425), ed. I. Caiazzo, pp. 143-154. 103. See Thierry of Chartres, Super Arithmeticam, I, 32 (1164-1166), ed. I. Caiazzo, p. 144; and ibid., I, 32 (1390-1391), p. 152.

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(1) The theological or ethical interpretation has several compo- nents, but the most prominent theme is the nature of divine goodness (bonitas divina). Good is always prior to evil, just as equality precedes inequality; and just as each instance of inequality is a failure of equality, evil is nothing but a perversion (transgressio) of good. God preserves goodness like an eternal mean (modus) that neither exceeds beyond nor recedes from perfect equality, as divine law or divine providence.104 Thierry clarifies that while divine goodness is infinite and incomprehensible, the goodness that we can participate and know is finite (and hence amenable to number, or mathematizable). Boethius’s principle applies only in the latter case, since the perfect exactitude of goodness entails its termination, completion, and hence finitude, whereas evil is limitless in the range of its possible deforma- tions of the Good.105 Thierry shows that the maxim of equality grounds the Trinity as unity, equality, and connection, and allows us to understand the divine Son as the form of being of each creature, neither more than less than what it is.106 Equality radiates divine beauty; indeed, nothing more beautiful than equality can be thought.107 Thierry even contemplates a kind of Neopythagorean emanationism: inequalities (matter, evil, numbers) proceed from equality (God, goodness, unity), but ultimately revert back to divine perfection. If the first book of De arithmetica concerns the procession, Thierry proposes, the second book concerns the return.108 (2) The physical interpretation of equality concerns number and causation. Created things cannot exist without numbers, in their

104. Ibid., I, 32 (1170-1176), p. 144; cf. ibid., I, 32 (1243-1268), pp. 147-148. 105. Ibid., I, 32 (1268-1287), p. 148. Cf. Boethius, Institutio arithmetica, I, 1, 6, ed. J.-Y. Guillaumin, p. 8. 106. Thierry of Chartres, Super Arithmeticam, I, 32 (1220-1242), ed. I. Caiazzo, pp. 146-147. 107. See ibid., I, 32 (1310-1317), ed. I. Caiazzo, p. 149; cf. id., Lectiones, II, 50, ed. N. M. Häring, p. 174. Achard of St. Victor, likely a student of Thierry, makes similar arguments in his remarkable treatise De unitate dei et pluralitate creaturarum, I, 5-6, in: L’Unité de Dieu et la pluralité de créatures, ed. E. Martineau, Saint-Lambert des Bois 1987, pp. 72-74; and ibid., I, 48, p. 130. See further D. Albertson, “The Beauty of the Trinity: Achard of St. Victor as a Forgotten Precursor of Nicholas of Cusa,” in: W. A. Euler (ed.), Akten des Forschungskolloquiums in Freising vom 8. bis 11. November 2012, MFCG 34, Trier 2016, pp. 3-20. 108. See Thierry of Chartres, Super Arithmeticam, I, 32 (1211-1215), ed. I. Caiazzo, p. 146; and esp. ibid., II, 1 (1-14), p. 155.

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plurality and diversity. Yet they subsist in the integrity of their natures through the monad, the principle of unity, and therefore through the priority of equality.109 As the supreme efficient cause, God is unchang- ing identity, but as perfect equality, God is the ground of inequality and change in the world. The final cause of the world is God’s goodness, and the formal cause is God’s wisdom, and both abide in divine Equality. The material causes of the world (the elements of matter, the humors of the body), originally equal, are now mixed in discord- ant proportions. Natural philosophy is nothing more than knowledge of inequalities and equalities.110 (3) When Thierry finally arrives at the “rational” or mathematical sense of Boethius’s equality maxim, his interpretation entirely concerns the four arts of the quadrivium. Physics perceives the world as full of inequalities, and theology observes their common destination in divine Equality. By contrast, mathematics handles both inequalities and equalities. According to Thierry, astronomy, geometry, music, and arithmetic (in that order) all demonstrate that mathematical measurements reduce to inequalities and equalities. Astronomy divines future events by calcu- lating the equality or inequality of planets’ courses to their proper paths. Geometry compares greater and lesser shapes by measuring their inequalities: as the leg and base of a right triangle increase, so too the length of the hypotenuse increases. When the geometer reaches equality, she attains the goal, the “certain reason of measuring” (certa ratio mensurandi). In the quantitative harmonic science of Boethius, music measures equal motions as equal sounds, and unequal motions as unequal sounds. On arithmetic, Thierry has nothing to add beyond a terse reference to the “prologue.”111 He probably has in mind Boethius’s discussion

109. See ibid., I, 32 (1181-1204), p. 145. 110. See ibid., I, 32 (1391-1412), pp. 152-153. Cf. id., Tractatus, 2-3, ed. N. M. Häring, pp. 555-556. 111. Thierry of Chartres, Super Arithmeticam, I, 32 (1412-1424), ed. I. Caiazzo, pp. 153-154 (my emphasis): “[…] quia in astronomia, cum scilicet divinamus secundum motum astrorum quando aequaliter in casis suis dicuntur morari planetae, aliquotiens inaequaliter vel aequaliter a casis suis recedunt; et secundum hoc quarumdam rerum futurarum proventus aequales et inaequales praenoscuntur. In geometria quoque secundum quod maior vel minor triangulus ab eadem ypothenusa dependet, consideratur catheti et basis unius trianguli ad cathetum et basim alterius trianguli; et secundum hanc aequalitatem constat certa ratio mensurandi. Si vero fuerint inaequales, erit in mensurando fallacia. Musica etiam ex aequalitate motuum aequalitates sonorum, ex inaequalitate vero

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of multitude and magnitude as fundamental categories of numerical measurement.112 Nicholas’s interpretation of the quadrivium in chapter II, 1 matches Thierry’s interpretation here quite closely. In the first place, they share the same reverse sequence of the four arts. In medieval Boethian traditions, the order of the quadrivium was quite relevant to its philosophical meaning, and variations were not undertaken lightly. In De arithmetica, Boethius always begins with arithmetic and ends with astronomy. Thierry’s Arithmetica commentary usually does the same, even if he inverts music and geometry.113 Nicholas follows the stand- ard Boethian sequence in chapter II, 13. But in the two passages in question, Thierry and Nicholas both deviate from the established sequence. Like Thierry, Nicholas’s treatment of arithmetic is shorter and perfunctory, much as we would expect if the cardinal depended on the Arithmetica commentary for guidance. Beyond the sequence of four arts, both authors use the quadrivium to discuss the same theme: mathematical inequalities and equalities as a sign of God’s presence in the cosmos. It is already rare for any author to discover a theological insight deep in the thickets of Boethius’s De arithmetica; it is even rarer for one to go out of his way to list the four arts of the quadrivium, yet in reverse order, as supporting evidence. That Thierry does so in his Arithmetica commentary is a testament to his extraordinary creativity. But to suppose that Cusanus just happened to per- form the same feat strains credulity. It is true that Nicholas’s treatment of each quadrivial art is longer and far more descriptive than Thierry’s. He is not copying exact words, but borrowing an idea that he sharpens and embellishes.114

motuum sonorum inaequalitates et consonantias metitur. Quomodo autem arithmetica consideret aequalitatem et inaequalitatem, post prologum demonstrabit evidenter.” 112. See Boethius, Institutio arithmetica, I, 1, 3-4, ed. J.-Y. Guillaumin, pp. 6-7. Cf. e.g. Thierry of Chartres, Super Arithmeticam, I, 1 (80-86), ed. I. Caiazzo, pp. 104- 105: “Haec autem tractat quadruvium […] quia quaecumque comprehenduntur sive sint qualitates sive quantitates sive quaelibet alia huiusmodi praedicamentorum, aut in magnitudine aut in multitudine comprehenduntur; id est aut discernendo ea ab aliis quod refertur ad multitudinem, id est ad numerum, aut iterum comprehenduntur in se, id est in magnitudine earum. Magnitudo autem rei est integritas ipsius, ita ut ne quid ultra nec citra sit.” Cf. also ibid., I, 1 (186-194), pp. 108-109. 113. Cf. e.g. ibid., I, 1 (222-236), pp. 109-110; and ibid., Praefatio (85-87), p. 98. 114. Readers should consult both texts to compare each paragraph in question. For reasons of length I will compare only the first art (astronomy) here in full. Thierry of

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But this pattern could well be counted as evidence in favor of Cusan dependence. We already know several other occasions when Cusanus referenced Chartrian concepts and then added his own exposition in accord with his particular interests. He takes this approach, for example, when handling Thierry’s mathematical Trinity, passages in De septem septenis, or triads from Commentarius Victorinus – not to mention the possibility that Cusanus interspersed his own expositions between whole paragraphs from Fundamentum naturae.115 Further- more, in the case of the passage in question, Nicholas’s exposition adds something new to Thierry’s ideas. In his physical and theological interpretations of the Boethian maxim, Thierry viewed inequalities as negative traces of God’s perfection. Then in his mathematical interpretation, he linked the inequalities of measure to the four quadrivial arts. Thierry does not, however, state that the inequalities measured by the quadrivium themselves point to God. In Book II of De docta ignorantia, Cusanus does just that. He simply applies the lessons of Thierry’s physical and theological interpretations to the mathematical interpretation. It is a brilliant editorial stroke that develops Nicholas’s source beyond its own limits.

3.2. Folding and the quadrivium Further evidence that Cusanus used Thierry’s Arithmetica commentary arrives when Nicholas presents his most famous doctrinal innovation in Book II, the reciprocal concepts of “enfolding” (complicatio) and “unfolding” (explicatio). In Book I, Cusanus rarely mentions

Chartres, Super Arithmeticam, I, 32 (1413-1417), ed. I. Caiazzo, p. 153: “[…] quia in astronomia, cum scilicet divinamus secundum motum astrorum quando aequaliter in casis suis dicuntur morari planetae, aliquotiens inaequaliter vel aequaliter a casis suis recedunt; et secundum hoc quarumdam rerum futurarum proventus aequales et inaequales praenoscuntur.” Nicholas of Cusa, De docta ignorantia, II, 1, 91, ed. P. Wilpert – H. G. Sen- ger, p. 4: “Haec quidem etsi ad infinita tibi deserviant, tamen si ad astronomiam trans- fers, apprehendis calculatoriam artem praecisione carere, quoniam per solis motum omnium aliorum planetarum motum mensurari posse praesupponit. Caeli etiam disposi- tio, quoad qualemcumque locum sive quoad ortus et occasus signorum sive poli elevatio- nem ac quae circa hoc sunt, praecise scibilis non est. Et cum nulla duo loca in tempore et situ praecise concordent, manifestum est iudicia astrorum longe in sua particularitate a praecisione esse.” 115. See M. Hoenen, “‘Ista prius inaudita’,” pp. 418-420; cf. D. Albertson, Math- ematical Theologies, pp. 186-190.

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folding. There its sense is restricted to limited cases: geometrical figures enfolding others, fated events enfolded in providence, divine names unfolding one God.116 In Book II, after the prologue on the quadrivium just discussed (II, 1), he marvels at the unbridgeable dis- tance between Creator and creature (II, 2), a theme from Fundamen- tum naturae, which he will cite shortly (II, 7-10).117 Cusanus then introduces enfolding and unfolding (II, 3) specifically as an answer to the unresolvable antinomies of Creator and creature. None can understand how God’s eternity descends into the relativity of time, or God’s unity into plurality; there is no determinate “proportion” between finite forms and the infinite Form of being.118 Nevertheless, Cusanus explains, we can affirm a relation of Creator to creature that is indissoluble, intimate, and dynamic, if we advert to the figure of “folding.” He defines folding for the first time as follows: Infinite unity is therefore the enfolding of all things. For indeed “unity” means that it unifies all things. But unity is maximal not only as the enfolding of number, but as the enfolding of all things. And just as in number unfolding unity, there is nothing but unity, so too in all things that exist, there is nothing but the maximum.119 Nicholas’s spare lines in fact conceal a definite sequence of thought. Unity enfolds number, and enfolds it exhaustively, without reserve: there is nothing in any number not originally present in oneness. Divine unity is homologous to arithmetical unity, but where numerical unity enfolds only numbers, divine unity enfolds all things. This is what makes it the maximum. That is, Nicholas takes for granted a prior understanding of arithmetical enfolding and unfolding, but wishes to apply it a fortiori to the case of God as absolute maximum.

116. See e.g. Nicholas of Cusa, De docta ignorantia, I, 20, 60-61, ed. P. Wilpert – H. G. Senger, p. 80; ibid., I, 22, 68-69, pp. 88-92; I, 24, 75-77, pp. 96-98; and I, 25, 84, p. 106. 117. Fundamentum naturae, fol. 8r, ed. M. Hoenen, p. 468 = Nicholas of Cusa, De docta ignorantia, II, 9, 150, ed. P. Wilpert – H. G. Senger, p. 76: “Nec cadit eo modo medium inter absolutum et contractum, ut illi Platonici imaginati sunt, qui ani- mam mundi mentem putarunt post deum et ante contractionem mundi.” 118. See Nicholas of Cusa, De docta ignorantia, II, 3, 100-104, ed. P. Wilpert – H. G. Senger, pp. 16-20. 119. Ibid., II, 3, 105, p. 22: “Unitas igitur infinita est omnium complicatio. Hoc quidem dicit unitas, quae unit omnia. Non tantum ut unitas numeri complicatio est, est maxima, sed quia omnium. Et sicut in numero explicante unitatem non reperitur nisi unitas, ita in omnibus quae sunt non nisi maximum reperitur.”

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To illustrate his meaning, Nicholas then proceeds to list several examples of reciprocal folding. Yet as he does so, he hardly mentions the divine maximum, as one would expect, given that this was the thrust of his initial definition. Instead, his list of examples depicts folding primarily in mathematical terms. His first illustration of folding is the geometrical point. As unity enfolds number, the point enfolds quantity. However one divides a line, surface, or solid, one finds only the single, universal point as its “limit, perfection, and totality.” Line is the “first unfolding” of the point.120 Nicholas’s next example is motion and time. Rest is the enfolding of motion, and motion the unfolding of rest. The present moment enfolds time, and time unfolds the present.121 Likewise equality enfolds inequality; identity enfolds diversity, and simplicity enfolds division.122 Cusanus’s next instance of folding is number itself: “number is the unfolding of unity.”123 He suggests that this example holds the key to profound theological mysteries, even the riddle of the One and the Many. “Who, I ask, can understand how there could be a plurality of things in the divine Mind, given that God’s understanding is God’s being, which is infinite unity?”124 If we think number as a multiplica- tion of ones, then it would seem that God is “multiplied” into a plurality of things. But if we think number in terms of folding, we can conceive plurality and unity as co-determined by each other: Therefore just as number arises from our mind through the fact that we understand many things singularly through a common one, so too plurality arises from the divine Mind, in which there are many things without plurality

120. Ibid., II, 3, 105, p. 22: “Ipsa quidem unitas punctus dicitur in respectu quantitatis ipsam unitatem explicantis, quando nihil in quantitate reperitur nisi punctus. Sicut undique in linea est punctus, ubicumque ipsam diviseris, ita in superficie et corpore. Nec est plus quam unus punctus, qui non aliud quam ipsa unitas infinita, quoniam ipsa est punctus, qui est terminus, perfectio et totalitas lineae et quantitatis, ipsam complicans. Cuius prima explicatio linea est, in qua non reperitur nisi punctus.” 121. On the folding of rest and motion, cf. Fundamentum naturae, fol. 9r, ed. M. Hoenen, p. 472 = Nicholas of Cusa, De docta ignorantia, II, 10, 155, ed. P. Wil pert – H. G. Senger, p. 84. On the folding of the present and time, cf. Thierry of Chartres, Commentum, IV, 42-43, ed. N. M. Häring, p. 107. 122. Nicholas of Cusa, De docta ignorantia, II, 3, 106, ed. P. Wilpert – H. G. Sen- ger, p. 24. 123. Ibid., II, 3, 108, p. 24: “Numerus est explicatio unitatis.” 124. Ibid., II, 3, 109, p. 26: “Quis rogo intelligeret, quomodo ex divina mente rerum sit pluralitas, postquam intelligere dei sit esse eius, qui est unitas infinita?”

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since they are in enfolding unity. For through the fact that things are not able to participate in an equal way the very equality of being, God in eternity has understood one thing in this way, another in another, from which plurality arises, which in God is unity. But plurality or number does not possess any being other than what it is from unity. Therefore unity, without which number would not be number, exists in plurality. And indeed what it means to unfold unity is precisely for all things to be in plurality.125 In this surprising discussion of number in the divine Mind, Cusanus puts to work several conceptual tools that he has borrowed from Chartrian theology.126 He suggests that due to the uniqueness of divine aequalitas, God also uniquely possesses a version of plurality that does not introduce difference. He contends that the reciprocity of folding effectively harmonizes unity and plurality as co-given, that this is already achieved in God’s eternity, and therefore that creaturely plurality in no way disturbs divine unity. Note that Nicholas’s argument only holds up if enfolding and unfolding are constitutively mathematical. He seems to assume that the case of numerical folding is not one among others, but the prime instance and indeed the very model of what enfolding and unfolding mean. Let us consider the sources Nicholas had at his disposal for the remarkable passage just cited. Folding is not discussed in De septem septenis or Commentarius Victorinus. It appears infrequently in Funda- mentum naturae, and within the author’s polemic against Thierry, only complicatio is sanctioned, while explicatio is a sign of falsehood.127

125. Ibid., II, 3, 108, p. 24: “Sicut igitur ex nostra mente per hoc, quod circa unum commune multa singulariter intelligimus, numerus exoritur, ita rerum pluralitas ex divina mente, in qua sunt plura sine pluralitate quia in unitate complicante. Per hoc enim, quod res non possunt ipsam aequalitatem essendi aequaliter participare, deus in aeternitate unam sic, aliam sic intellexit, ex quo pluralitas, quae in ipso est unitas, exorta est. Non habet autem pluralitas sive numerus aliud esse quam ut est ab unitate. Unitas igitur, sine qua numerus non esset numerus, est in pluralitate. Et hoc quidem est unitatem explicare, omnia scilicet in pluralitate esse.” 126. Note that this passage stands in tension with the exclusion of plura distincta exemplaria and the rejection of explicatio in Nicholas of Cusa, De docta ignorantia, II, 9, 148-150, ed. P. wilpert – H. G. Senger, pp. 74-78 = Fundamentum naturae, fols. 7v-8r, ed. M. Hoenen, pp. 466-468. Cusanus revisits the idea a few years later, although he changes mens divina to deus mens infinita. See Nicholas of Cusa, De coniecturis, I, 2, 7, eds. J. Koch – K. Bormann, p. 12; cf. Thierry of Chartres, Lectiones, II, 66, ed. N. M. Häring, p. 176. 127. See Fundamentum naturae, fols. 7v-9r passim, ed. M. Hoenen, pp. 466-472; cf. D. Albertson, “A Late Medieval Reaction to Thierry of Chartres’s (d. 1157) Phi- losophy,” pp. 80-81.

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Thierry never mentions folding in his Tractatus. In Commentum he experiments with the term once or twice, but it never congeals into a coherent doctrine. Only in his later Lectiones and Glosa on De trinitate does Thierry systematically pair complicatio and explicatio as a reciprocal couplet.128 He applies it first to define the relationship of God and world, and then as scaffolding for his system of “four modes of being” in Lectiones, in which folding specifically clarifies the relationship between theology and mathematics. Theology’s object is the first mode of being (necessitas absoluta); when this is unfolded, it generates the second mode of being (necessitas complexionis), the object of mathematics.129 Beyond this meta-disciplinary statement in Lectiones, Thierry never applies enfolding and unfolding to discrete examples from the quadrivium, such as number, quantity, points, lines, time, or motion, in any of his commentaries on Genesis or De trinitate. By contrast, as we have just seen, Cusanus does just that, applying the model of folding to various quadrivial topics. Number unfolds unity, lines unfold points, motion unfolds rest, and time unfolds the present. Yet despite his many examples, Nicholas underscores that these are not different species of folding scattered across the categories of quality, quantity, and so forth. Ultimately there is only one universal instance, the God who enfolds all things, and the totality of things unfolding God. According to Cusanus, the unity of folding follows the logic of perfection: “For just as unity precedes alterity, so the point, which is perfection, precedes magnitude. The perfect precedes everything imperfect, as rest precedes motion, identity diversity, equality inequality – and likewise all those things convertible with unity, which is eternity itself.”130 Moreover, as we have seen, Cusanus postulates two further ideas that underpin his argument: that folding has a numerical basis, and that numerical plurality originates in the divine Mind. Since these are all absent from Tractatus and the three

128. See Thierry of Chartres, Commentum, II, 49, ed. N. M. Häring, p. 84; and ibid., IV, 42-43, p. 107. Cf. id., Lectiones, II, 4-13, ed. N. M. Häring, pp. 155-158. On the evolution of Thierry’s doctrine of folding, see D. Albertson, Mathematical Theologies, 121-131. 129. See Thierry of Chartres, Lectiones, II, 14-29, ed. N. M. Häring, pp. 158-164. 130. Nicholas of Cusa, De docta ignorantia, II, 3, 107, ed. P. Wilpert – H. G. Sen- ger, p. 24: “Sicuti enim unitas alteritatem praecedit, ita et punctus, qui est perfectio, magnitudinem. Perfectum enim omne imperfectum antecedit, ita quies motum, identitas diversitatem, aequalitas inaequalitatem, et ita de reliquis, quae cum unitate convertuntur, quae est ipsa aeternitas.”

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commentaries on De trinitate, it would seem that Nicholas has independently drawn connections between Thierry’s account of folding in Lectiones and Glosa, on the one hand, and Boethian arithmetic, on the other. By this reading, his application of folding to new fields of thought would seem to be a paradigmatic instance of Cusan innovation and originality, indeed a pillar of Cusan philosophy.131 This conclusion was justified in the past, given our knowledge of Chartrian sources at the time. But now we can clarify that this chapter (II, 3) is yet another occasion where Cusanus follows Thierry quite closely – if not word for word, then thought for thought. Caiazzo has demonstrated that Thierry’s Arithmetica commentary is the true origin of his doctrine of folding, grounded in the quadrivium, and that its later deployment in Lectiones and Glosa presumes this original mathematized model.132 But we should hasten to add that this passage in the Arith- metica commentary is also the point of departure for Nicholas of Cusa’s own creative, lifelong engagement with complicatio and explicatio. In the second book of his Arithmetica commentary, Thierry turns to Boethius’s account of geometrical construction. Boethius explains an ancient paradox: although atomistic points are without quantity even when they are added together, the point is nevertheless the source and principle of linearity, the first appearance of quantity in space. As Boethius states: “Therefore from this principle (i.e., from unity), arises the original length, which unfolds [explicat] from the principle of binary number into all numbers themselves, since line is the first extension.”133 The sudden transition from point to line is the leap from number to magnitude, from monad to limitless dyad, from indivisibility to extension, from the invisible to the visible.

131. See inter alia T. P. McTighe, “The Meaning of the Couple, ‘Complicatio-Expli- catio,’ in the Philosophy of Nicholas of Cusa,” in: Proceedings of the American Catholic Philosophical Association 32 (1958), pp. 206-214; C. Ricatti, “Processio” et “explicatio”: La Doctrine de la création chez Jean Scot et Nicolas de Cues, Naples 1983; M. De Gandil- lac, “Explicatio-Complicatio chez Nicolas de Cues,” in: G. Piaia (ed.), Concordia discors. Studi su Niccolò Cusano e l’umanesimo europeo offerti a Giovanni Santinello, Padua 1993, pp. 77-106; and A. Moritz, Explizite Komplikationen: Der radikale Holismus des Nikolaus von Kues, Münster 2006. 132. See I. Caiazzo, “Introduction,” pp. 64-67; cf. ead., “Il rinvenimento del commento di Teodorico di Chartres al De arithmetica di Boezio,” pp. 201-203. 133. Boethius, Institutio arithmetica, II, 4, 6, ed. J.-Y. Guillaumin, p. 90: “Ex hoc igitur principio, id est ex unitate, prima omnium longitudo succrescit quae a binarii numeri principio in cunctos sese numeros explicat, quoniam primum interuallum linea est.”

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To explain this passage to his students, Thierry decides to focus on the verb explicare. Whatever philosophical mysteries are revealed in the transition from point to line, he wagers, will be found in that term: But unity takes the place of the point, and rightly so. Because just as unity is the enfolding [complicatio] of number, and number is the unfolding [explicatio] of unity, so also the point is the enfolding of every magnitude, and magnitude is the unfolding of the point. […] Therefore I say that the point is the limit and perfection of a given thing; it is the very totality of that thing. It delimits the entire thing wholly and perfectly, and there is properly speaking only [one] limit of one single thing. But when we say “limits,” we misspeak, just as when we say “unities.” Hence the extension of its line unfolds this totality, which we name limit; for line is the unfurling [evolutio] of the point. […] However, where is there one perfection of a thing, there is not one totality that includes its magnitude; the totality is rather the enfolding of magnitude. To unfold this totality is nothing other than to divide the whole into parts.134 This is a seminal text for the future development of Thierry’s thought. In the first place we should observe that here Thierry originally grounds complicatio and explicatio on the infrastructure of Boethian arithmetic. The point enfolds the line, and unity enfolds number. Only years later in Lectiones will Thierry apply folding to divine transcendence: God is the unity that enfolds the plurality of the cosmos, and the diversity of the cosmos is the total unfolding of God.135 Second, Thierry’s initial treatment of folding here in his Arithmetica commentary is distinctly non-theological. Unlike his treatment of aequalitas in Boethius, Thierry theorizes folding exclusively as a way to analyze the foundations of the quadrivium, not to raise any speculative consequences. Thierry restricts his attention to Boethius’s difficult claims about geometrical points, as if this task were challenging enough.

134. Thierry of Chartres, Super Arithmeticam, II, 4 (230-232, 236-241, 266-269), ed. I. Caiazzo, pp. 163-165: “Unitas vero locum obtinet puncti, et merito, quia sicut unitas est complicatio numeri, et numerus explicatio unitatis, ita quoque punctum est complicatio omnis magnitudinis, et magnitudo est explicatio puncti. […] Dico igitur quod punctus est terminus rei ipsius et perfectio. Haec est ipsa rei totalitas, quae rem totam integre terminat et perfecte, nec est nisi unius rei proprie terminus. Cum enim terminos dicimus, abutimur, quemadmodum cum unitates dicimus. Hanc ergo totalitatem, quam terminum nominamus, explicat ipsius lineae extensio; linea namque est puncti evolutio. […] Nec tamen est ubi una rei perfectio, una totalitas quae ipsam magnitudinem includit; ipsa enim est magnitudinis complicatio. Explicare vero totalitatem hoc nihil aliud est quam totum per partes dividere.” 135. See Thierry of Chartres, Lectiones, II, 4-6, ed. N. M. Häring, pp. 155-156.

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Thierry must explain how the dimensionless point is ontologically prior to the fullness of space, in the same way that unity precedes all numbers. That is, in what sense can an infinitesimal point define and delimit the contours of a solid object extended in space? To answer this question, Thierry decides to redefine the nature of spatial delineation. Geometrical circumscription through lines is, in fact, a derivative mode of ontological delimitation, according to Thierry. When we trace a line around a planar or solid figure, we measure in the dimension of quantitative magnitude, but only in a penultimate sense. For Thierry contends that there are two inextricable aspects of magnitude: unfolded magnitude and enfolded magnitude. The enfolding of the point, however, precedes the unfolding of the line. So if we want to establish the true limit (terminus) or integral perfection (perfectio) of a given whole (totalitas), what we require is not the complex delineation of a corporeal surface, but those lines as they are devolved into their original unity, namely, a single point. This is what Thierry means when he states that the point is the limit, perfection, and totality of every given being, regardless of spatial magnitude. Clearly Thierry’s vision of points “enfolding” lines in geometry is modeled after unity “enfolding” numbers in arithmetic, as he states. But this train of thought leads well beyond a difficult passage in Boethius. By theorizing a common mechanism for establishing the respective principles of multitude and magnitude, Thierry has done nothing less than to unify the bifurcated conceptual foundation of the ancient quadrivium. In one stroke he has formulated a univocal model, reciprocal folding, which generates both quadrivial dimensions homologously: multitude as unfolded unity, and magnitude as unfolded point, producing arithmetic and music, on the one hand, and geometry and astronomy, on the other. These two orders, which remain parallel but disjunct in Nicomachus and Boethius, are now united by the dynamic auto-differentiation of folding, a distantiation that nevertheless preserves a non-contradiction between the original unity (qua enfolded) and the resulting plurality (qua unfolded).136

136. See Nicomachus of Gerasa, Introductio arithmetica, I, 2, 4-5, ed. R. G. Hoche, pp. 4-5; and ibid., I, 3, 1-2, pp. 5-6. See Boethius, Institutio arithmetica, I, 1, 1-12, ed. J.-Y. Guillaumin, pp. 6-11. Cf. L. M. Napolitano Valditara, Le idee, i numeri, l’ordine, pp. 413-434.

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The next robust account of folding in Thierry’s extant works does not arrive until Lectiones and Glosa, more than a decade later. From the perspective of the Arithmetica commentary, we can now register two facts about the reappearance of complicatio and explicatio in Thierry’s later works. First, Thierry is not theorizing folding for the first time in the pages of Lectiones, but deliberately re-tooling this lemma of Boethian geometry for a secondary application, namely, the first principles of theological method in Boethius’s De trinitate. Second, therefore, folding retains its inalienable mathematical basis, even in its later reincarna- tion in Lectiones, and even if Thierry’s insights into the foundations of the quadrivium are not argued on the page but silently assumed without notice to the reader. That is to say: the structure of folding that harmonizes physics, mathematics, and theology in Lectiones is not a neutral device, but an already indelibly mathematized structure. In the same manner, thanks to Caiazzo’s discovery, we can also specify which instance of reciprocal folding Nicholas of Cusa appro- priated as he composed De docta ignorantia. Cusanus did not adopt the derivative sense of folding from Lectiones, as it might have appeared to modern readers in the past. Rather, it now appears that he used the original, mathematical definition of complicatio and explicatio from the Arithmetica commentary itself. We can be relatively certain of this on account of five distinct elements in De docta ignorantia II, 3. All of them indicate Cusanus’s direct dependence on Thierry’s Arithmetica in particular, rather than other candidates from his dossier of Chartrian sources. (1) In the first place, note Nicholas’s immediate turn to the quadrivium in chapter II, 3 as the proper domain for explaining how the absolute maximum enfolds all things. So rapid is his pivot to arithmetic and geometry that Nicholas already wants to supersede this mathematical model for folding – a model he presumes to be valid and intelligible to his readers – in the name of a putatively more comprehensive one. Cusanus states that unity is not merely “the enfolding of number,” but in his view is “the enfolding of all things.”137 That is,

137. Nicholas of Cusa, De docta ignorantia, II, 3, 105, ed. P. Wilpert – H. G. Sen- ger, p. 22 (my emphasis): “Non tantum ut unitas numeri complicatio est, est maxima, sed quia omnium. Et sicut in numero explicante unitatem non reperitur nisi unitas, ita in omnibus quae sunt non nisi maximum reperitur.”

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number provides a model for complicatio, but remains penultimate. Several pages later, he adds the analogue for explicatio. While “number is the unfolding of unity,” it is more important to affirm that divine unity unfolds into all created things, not only number.138 Nicholas’s intention is clearly to underscore the ontological supremacy of the maximum. But along the way he has already assumed a very specific premise, which, however, he has broken into two pieces for the sake of ease as he draws an analogy from the case of number to the case of divine maximum. These two ideas are that unity enfolds number, and that number unfolds unity. Surprisingly, Nicholas never states this simple formula, uniting the two halves of complicatio and explicatio, in De docta ignorantia. Yet it appears very precisely in Thierry’s Arithmetica commentary: “Unity is the enfolding of number, and number is the unfolding of unity.”139 Furthermore, this arithmetical interpretation of folding never occurs in Thierry’s other extant works. When Thierry discusses number in terms of Boethian arithmetic in Lectiones, he never mentions folding.140 To be sure, unitas and pluralitas are folded together, as are the first mode of being (the domain of theology) and the second mode (the domain of mathematics).141 But these are distinct from the idea that folding as such occurs as numerical difference, namely the reciprocal folding of unitas and numerus. Since Cusanus’s account of folding takes this idea for granted, an idea that only appears as a whole within Thierry’s Arithmetica commentary, it is probable that Nicholas had access to that text. (2) As discussed above, after this initial argument Nicholas lists several examples of folding. His first example is the geometrical point: “Unity is called ‘point’ in respect of quantity unfolding unity itself,

138. Ibid., II, 3, 108, pp. 24-26 (my emphasis): “Et ut in numeris intentionem declaremus: Numerus est explicatio unitatis […] Unitas igitur, sine qua numerus non esset numerus, est in pluralitate. Et hoc quidem est unitatem explicare, omnia scilicet in pluralitate esse.” Cf. his discussion of unity unfolding into the decad at ibid., II, 6, 123, pp. 42-44. 139. Thierry of Chartres, Super Arithmeticam, II, 4, 230-231, ed. I. caiazzo, p. 163: “[…] unitas est complicatio numeri, et numerus explicatio unitatis […] .” 140. See e.g. Thierry of Chartres, Lectiones, III, 4-23, ed. N. M. Häring, pp. 178- 184. 141. See respectively ibid., II, 4-5, pp. 155-156; and ibid., II, 11-18, pp. 158-160. The same holds for Commentum and Glosa.

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as nothing is found in unity except the point. […] The first unfolding of the point is the line, in which is nothing but the point.”142 This follows Thierry’s Arithmetica commentary very closely. Thierry first posits the fold of unity and number, and then the fold of point and line. Once again, Thierry’s account provides what seems the missing premise of Cusanus’s exposition.143 Outside of the Arithmetica commentary, I have found only one other similar discussion of geometrical points among Chartrian sources, in an anonymous work loosely associated with Thierry’s school.144 (3) In his account of the enfolding point, Nicholas calls the point “the limit, perfection, and totality of line and quantity.”145 These are the same three titles given by Thierry to the point in his Arithmetica commentary, immediately after introducing complicatio and explicatio for the

142. Nicholas of Cusa, De docta ignorantia, II, 3, 105, ed. P. Wilpert – H. G. Sen- ger, p. 22: “Ipsa quidem unitas punctus dicitur in respectu quantitatis ipsam unitatem explicantis, quando nihil in quantitate reperitur nisi punctus. […] Cuius prima explicatio linea est, in qua non reperitur nisi punctus.” 143. Thierry of Chartres, Super Arithmeticam, II, 4 (230-232, 239-241), ed. I. caiazzo, pp. 163-164: “Unitas vero locum obtinet puncti, et merito, quia sicut unitas est complicatio numeri, et numerus explicatio unitatis, ita quoque punctum est complicatio omnis magnitudinis, et magnitudo est explicatio puncti. […] Hanc ergo totalitatem [sc. punctum] […] explicat ipsius lineae extensio; linea namque est puncti evolutio.” 144. See Glosa Victorina, III, 1-7, ed. N. M. Häring, pp. 544-546. Cf. Boethius, Institutio Arithmetica, II, 4, 4-5, ed. J.-Y. Guillaumin, pp. 89-90; and Thierry of Char- tres, Super Arithmeticam, II, 4 (210-300), ed. I. Caiazzo, pp. 163-167. Häring does not attribute Glosa Victorina De Trinitate (as opposed to Glosa super Boethii librum De Trin- itate) to Thierry of Chartres. This anonymous, twelfth-century work follows immediately after Lectiones and Commentarius Victorinus in MS Paris, BnF Lat. 14489, fols. 95v-109v (see N. M. Häring, “Introduction,” in: Commentaries on Boethius by Thierry of Chartres and His School, p. 45). Häring observes that the treatise fails to repeat any of Thierry’s signature doctrines, despite being written by the same scribe as Lectiones and Commen- tarius Victorinus, and despite glossing Boethius’s De trinitate like them. Nevertheless, in light of Thierry’s Arithmetica commentary, it is no longer the case, in Häring’s words, that the treatise’s “annotations […] do not reveal any association with Thierry’s ideology” (ibid., p. 45). Both Glosa Victorina and Super Arithmeticam find theological significance in the addition of dimensionless points; but where Thierry’s commentary extends this to reciprocal folding and divine Equality, Glosa Victorina draws a Trinitarian conclusion that the sum of three points is one point. While Caiazzo discusses Commentarius Victorinus and MS , BnF Lat. 14489 (see “Introduction,” pp. 60-61), Glosa Victorina represents an impressive further testimony to the authenticity of Super Arithmeticam. 145. Nicholas of Cusa, De docta ignorantia, II, 3, 105, ed. P. Wilpert – H. G. Sen- ger, p. 22: “Nec est plus quam unus punctus, qui non aliud quam ipsa unitas infinita, quoniam ipsa est punctus, qui est terminus, perfectio et totalitas lineae et quantitatis, ipsam complicans.”

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first time.146 In other extant works Thierry occasionally uses the three terms, but never together or in the context of the geometrical point.147 (4) While discussing the point, Nicholas offers an observation about the divisibility of lines: “Just as the point is everywhere in a line no matter where you divide the line, so too does the point exist in a surface or body.”148 It is for this reason, he concludes, that the point is limit, perfection, and totality. Now Nicholas’s remark may seem a commonplace of geometry, but it was widely discussed among twelfth-century philosophers, particularly the paradox of indivisible points forming divisible lines.149 In the course of his discussion of folding in the Arithmetica commentary, Thierry makes similar statements. The point is “indivisible” in its simplicity and perfection, and unfolding a point into a given geometrical object amounts to “divid- ing the whole into parts.” Even if the point is only visible quantita- tively in lines and surfaces, its perfection lies in its enfolded simplicity.150 I am not aware of any discussions of divisible lines in Thierry’s other commentaries on Genesis or De trinitate.

146. Thierry of Chartres, Super Arithmeticam, II, 4 (236-238), ed. I. Caiazzo, p. 164: “Dico igitur quod punctus est terminus rei ipsius et perfectio. Haec est ipsa rei totalitas, quae rem totam integre terminat et perfecte, nec est nisi unius rei proprie terminus.” 147. In Tractatus and Commentum, the divine Word “terminates” creatures by cir- cumscribing and defining their being. See Thierry of Chartres, Tractatus, 45, ed. N. M. Häring, p. 574; and id., Commentum, II, 31, ed. N. M. Häring, p. 78. In Commentum, the divine aequalitas essendi conveys integritas et perfectio to the material beings it forms: see id., Commentum, II, 40-46, ed. N. M. Häring, pp. 81-83. In Lec- tiones, unitas is the universal terminus of all things: see id., Lectiones, VII, 7, ed. N. M. Häring, p. 225. In none of these instances, however, does Thierry connect terminus with perfectio or totalitas, let alone all three. 148. Nicholas of Cusa, De docta ignorantia, II, 3, 105, ed. P. Wilpert – H. G. Sen- ger, p. 22: “Sicut undique in linea est punctus, ubicumque ipsam diviseris, ita in superficie et corpore.” 149. Caiazzo cites several examples of this tradition: the definitions of punctus in the first Latin translations of Euclid’s Elements by Adelard of Bath and Robert of Chester; discussions of Boethian points in commentaries on the Aristotelian categories by William of Champeaux and Peter Abelard; and commentaries on Macrobius in William of Conches and the Cologne gloss. See notes by I. Caiazzo at Thierry of Chartres, Super Arithmeticam, pp. 163-168. Thierry himself not only addresses the discussion in Boethius but also adduces that of Macrobius. See Thierry of Chartres, Super Arithmeticam, II, 4 (276-300), ed. I. Caiazzo, pp. 165-167. Cf. Macrobius, Commentarii in Somnium Scipionis, I, 5, 11, ed. J. Willis, Leipzig 1963, pp. 16-17. 150. Thierry of Chartres, Super Arithmeticam, II, 4 (260-264, 268-273), ed. I. Caiazzo, pp. 164-165: “Quare et perfectio simplex quiddam est, ideoque punctum simplex esse dicitur et indivisibile. Illud quoque sciendum quod, cum dicimus duos

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(5) After the point, Nicholas lists several further examples of folding, apparently of his own invention. Yet he insists that despite his examples, there is only “one enfolding,” which is universal and eternal, and therefore God. His reasoning relies on a common Aristote- lian maxim regarding the ontological priority of what is perfect or complete. “Just as unity precedes alterity,” he writes, “so the point, which is perfection, precedes magnitude. For the perfect is prior to everything imperfect, just as rest to motion, identity to diversity, equality to inequality, and so with everything interchangeable with unity, which is eternity itself.”151 Where does Nicholas find this principle, and why would he invoke it to defend reciprocal folding?152 Boethius does not cite it in De arithmetica, nor does Thierry in his later works, so far as I can tell. A plausible conjecture is that Nicholas read it in Thierry’s Arithmetica commentary. Immediately following his initial definition of complicatio and explicatio, Thierry writes: “Perfection is prior by nature to imperfection, because imperfection descends from it, nor is anything the cause of perfection. […] It should also be known that, since perfection is prior to every imperfection, perfection in fact is simple. […] But the point is the first perfection, while the line is imperfection – not because it is not the perfection of the surface, but in comparison to the point, which is the first perfection, as just stated.”153 It is also possible that Cusanus

terminos lineae esse, abutimur dicendo terminos, cum unus sit terminus, quia unum punctum solum est, id est una sola totalitas, licet multa sint tota. […] Explicare vero totalitatem hoc nihil aliud est quam totum per partes dividere. In unaquaque vero partium est eadem totalitas, quae et in ipso toto, quamvis totum non remaneat idem. Illud quoque sciendum quod, quamvis punctum sit complicatio omnis magnitudinis, tamen soli [de]terminatur ad superficiem, superficies ad lineam, linea vero ad punctum.” At l. 272, one should correct the edition and conjecture (or read) “solidum terminatur” instead of “soli determinatur” (Thierry usually employs the verbum simplex in this context). 151. Nicholas of Cusa, De docta ignorantia, II, 3, 107, ed. P. Wilpert – H. G. Sen- ger, p. 24: “Sicuti enim unitas alteritatem praecedit, ita et punctus, qui est perfectio, magnitudinem. Perfectum enim omne imperfectum antecedit, ita quies motum, identitas diversitatem, aequalitas inaequalitatem, et ita de reliquis, quae cum unitate convertuntur, quae est ipsa aeternitas.” 152. See Aristotle, De caelo, I, 3, 269a19-21; and id., Metaphysics, V, 16, 1021b12- 1022a3. 153. Thierry of Chartres, Super Arithmeticam, II, 4 (247-248, 256-258, 273-275), ed. I. Caiazzo, pp. 164-165: “Perfectio enim prior natura est imperfectione, quia imperfectione ab ipsa descendit, nec aliquid causa perfectionis est. […] Illud quoque sciendum quod, quia perfectio prior est omni imperfectione, ideo ipsa perfectio quiddam simplex

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found the doctrine in Fundamentum naturae, but given the context and the evidence for the Arithmetica commentary’s influence elsewhere, this seems less likely.154 These two passages in Book II of De docta ignorantia – on inequality and the arts of the quadrivium (II, 1), and on folding and the foundations of the quadrivium (II, 3) – are compelling evidence that Cusanus knew Thierry’s Arithmetica commentary before 1440 and used it in the composition of his great work. But as I have sketched above, we can also wager that this work by Thierry probably preceded his other four commentaries on Genesis and De trinitate. This two- fold reordering of the Chartrian legacy, now enabled by Caiazzo’s discovery, exposes, in turn, two noteworthy reversals of our former perceptions of Thierry’s ideas and influence. First, we used to judge De docta ignorantia II, 1 as an echo of Thierry’s remark in his hexaemeral commentary that the quadrivium provides four probationes to lead reason back to her Creator. But now we can affirm, on the contrary, that the idea Cusanus repeats in II, 1 – repeats from the Arithmetica commentary – most likely antedates those famous lines from Thierry’s Tractatus. Without our knowing it, Nicholas had always enjoyed access to an earlier, and consequential, stage of the Parisian master’s theological development. Similarly, we used to read Lectiones as the first instance of Thierry’s doctrine of folding; in this light, the Nicholas’s revival of folding in De docta ignorantia II, 3 seemed a derivative recycling of those concepts. But now we can see that all along, Nicholas’s account in II, 3 secretly preserved Thierry’s original formulation of complicatio and explicatio dating back to the 1130s, as we find it in the Arithmetica commentary. The esteemed passages in Lectiones are now revealed to

est. […] Sed punctum est prima perfectio, linea vero est imperfectio, non quod non sit perfectio superficiei, sed imperfectio dicitur ad comparationem puncti, quod est prima perfectio, ut praedictum est.” 154. The doctrine appears in the late medieval florilegium Auctoritates Aristotelis, a book Cusanus possessed in his library: “Perfectum naturaliter prius est imperfecto.” Auc- toritates Aristotelis, 3 (11), in: Les Auctoritates Aristotelis. Un florilège médiéval. Étude his- torique et édition critique, ed. J. Hamesse, Louvain 1974, p. 160. According to Hoenen, Fundamentum naturae references the same doctrine at Fundamentum naturae, fol. 9v, ed. M. Hoenen, p. 474: “Et ordine naturae perfectum prius est imperfecto.” This is not the only curious intersection between Fundamentum naturae and Super Arithmeticam, but their relationship requires a separate study.

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be the secondary re-purposing of folding (by Thierry himself) that De docta ignorantia was once deemed to be. Such ironic reversals of priority allow us to revisit an unsolved mystery regarding the status of Fundamentum naturae within Cusan development.

4. Thierry’s Arithmetica after De docta ignorantia We know that Nicholas combined multiple Chartrian sources together as he wrote, interweaving texts by Thierry with those of his medieval readers on the basis of definite family resemblances. Even today, readers of Thierry’s Arithmetica quickly recognize multiple points of contact with his other extant works. The Arithmetica commentary shares the mathematical Trinity and theology of equality with Tractatus and the three De trinitate commentaries. It shares reciprocal folding with Fundamentum naturae, Lectiones, and Glosa, and it shares the Spanish Sibyl prophecy with Commentum and Com- mentarius Victorinus. It shares the triad of perpetuals with Commen- tum and De septem septenis and the four Trinitarian causes of creation with Tractatus. This is the same evidence that Caiazzo has marshalled to establish Thierry’s authorship of the Stuttgart Arithmetica. But since we know that Cusanus had access to Commentum, De septem septenis, and Commentarius Victorinus, at the least, it is only reasonable to suppose that he too observed similar connections to the Arith- metica commentary and thus surmised a common parentage. The presence of Thierry’s Arithmetica commentary clarifies how and why Cusanus handled his other Chartrian sources as he did. It accounts for his conspicuous and consistent emphasis on doctrines from Boethian arithmetic that otherwise appear rarely in Thierry’s established commentaries: arithmetic as divine exemplar; number as essential to folding; the categories of multitude and magnitude; unity and the tenfold decad; and a lifelong habit of metaphysical specula- tion on geometrical figures. The list of amendments Cusanus makes to Thierry’s theology in the De trinitate commentaries is, at the same time, a decent résumé of Thierry’s Arithmetica commentary. Surely this is no coincidence. Having identified a key source of De docta ignorantia, we can briefly consider whether Cusanus continued to consult Thierry’s Arithmetica even after 1440. We know that Cusanus returned to other Chartrian

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sources, including De septem septenis and Commentum, in different ways as his interests changed in the 1450s and 1460s.155 One can guess that as he began composing actual mathematical proofs in 1445, a commentary on De arithmetica would have only grown more valuable to the cardinal. This is not the place to analyze how or why Cusanus returned to Thierry’s Arithmetica commentary in later works, but a few examples will suffice to establish, at least, that he did. After completing De docta ignorantia in February 1440, Nicholas revisits Chartrian themes in two Christmas sermons later that year. In the first he declares that all knowledge comes from multitude and magnitude, and that equality always precedes inequality. The principle of multitude is unity, and the principle of magnitude is the three- fold dimensions of space, a vestigium Trinitatis.156 In the second he examines how multitude, inequality, and division descend from unity, equality, and connection.157 A few years later in De coniecturis (1440- 1445), Cusanus repeats the triad but adds that the intrinsic mathematical structure of the human mind descends immediately from the divine Mind.158 In this new treatise he equates rationality with numerability even more explicitly than he had in De docta ignorantia.159 Following Boethius he defines number as the composition of even and odd, and reiterates once more that number is the first exemplar in the mind of the Creator.160 He amplifies the Neopythagorean

155. See D. Albertson, Mathematical Theologies, pp. 232-235. 156. See Nicholas of Cusa, Sermo XXII, 19-20, in: Nicolai de Cusa Opera omnia iussu et auctoritate Academiae Litterarum Heidelbergensis, Vol. XVI/4. Sermones I (1430- 1441). Fasciculus 4: Sermones XXII-XXVI, eds. R. haubst – M. Bodewig, Hamburg 1984, pp. 344-345. 157. See Nicholas of Cusa, Sermo XXIII, 15-17, eds. R. Haubst – M. Bodewig, pp. 367-368. Cf. id., De docta ignorantia, I, 10, 28, ed. P. Wilpert – H. G. Senger, p. 38; and Commentarius Victorinus, 87-88, ed. N. M. Häring, p. 499. On this triad, see Boethius, Institutio arithmetica, II, 40, 1-3, ed. J.-Y. Guillaumin, p. 140; and id., Insti- tutio musica, II, 7, ed. G. Friedlein, Leipzig 1867, p. 232. On divisio, cf. Thierry of Chartres, Super Arithmeticam, I, 4 (601-636), ed. I. Caiazzo, pp. 124-126. 158. See Nicholas of Cusa, De coniecturis, I, 1, 6, eds. J. Koch – K. Bormann, pp. 9-10; and ibid., I, 8, 35, p. 40. 159. See Nicholas of Cusa, De coniecturis, I, 2, 7, eds. J. Koch – K. Bormann, pp. 11-12; cf. id., De docta ignorantia, II, 3, 108, ed. P. wilpert – H. G. Senger, pp. 24-26. 160. On even and odd, see Nicholas of Cusa, De coniecturis, I, 2, 8, eds. J. Koch – K. Bormann, pp. 12-14; cf. ibid., II, 1, 75-76, pp. 74-75. On number as divine exemplar, see ibid., I, 2, 9, p. 14; and ibid., I, 5, 17, p. 21.

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doctrine of the decad and compares the relative powers of the hexad and heptad.161 Finally, Nicholas engages several geometrical issues from De arithmetica: the relation of point, line, and surface; maxima and minima as problems of divisibility; and the construction of harmonies and proportions as ratios of multitudes and magnitudes.162 If De docta ignorantia and De coniecturis contain significant moments of Neopythagoreanism, the treatise Idiota de mente (1450) makes even deeper investments in Boethian arithmetic, to the point of reorienting Cusan epistemology entirely around number. Nicholas repeats Boethi- us’s admonition that all wisdom flows from certain multitudes and magnitudes. As never before in previous works, he now takes the time to explain how these two categories construct the fourfold quadrivial system.163 As in De coniecturis, Cusanus defines number as even and odd and praises harmonies and proportions.164 But now even more radically, he speculates that both divine and human minds share number as their principal exemplar, and hence both unfold number from unity in parallel, Creator and creature mathematizing in synchrony.165 These topical correspondences in De mente not only match elements of Thierry’s Arithmetica; they match those elements already echoed by Cusanus in De docta ignorantia and De coniecturis. Beyond these general similarities, one specific passage suggests that Thierry’s Arithmetica may have directly influenced De mente. In chapter IX, Cusanus makes six discrete statements that all find parallels within Thierry’s Arithmetica commentary, indeed within passages that we know had already impressed Cusanus in De docta ignorantia.

161. On the decad, see ibid., I, 3, 10, pp. 15-16; ibid., I, 7, 29, p. 36; ibid., I, 12, 61, p. 61; and ibid., I, 13, 67, pp. 66-67. Cf. Nicholas of Cusa, De docta ignorantia, II, 6, 123, ed. P. Wilpert – H. G. Senger, p. 44; and Thierry of Chartres, Super Arithmeticam, II, 41 (1173-1185), ed. I. Caiazzo, pp. 197-198. On the hexad and heptad, see Nicholas of Cusa, De coniecturis, II, 7, 106-111, eds. J. Koch – K. Bormann, pp. 102-108. 162. On geometrical points, see Nicholas of Cusa, De coniecturis, I, 8, 30, eds. J. Koch – K. Bormann, pp. 36-37; and ibid., II, 16, 168-169, pp. 170-171. On divisibility, see ibid., I, 10, 44-45, pp. 47-49. On proportions, see ibid., I, 2, 81-83, pp. 78-81. 163. See Nicholas of Cusa, Idiota de mente, X, 126-128, ed. R. Steiger, pp. 178- 181. Cf. C. Rusconi, “Grandeur et multiplicité: Les catégories de l’alterité dans le De coniecturis (1440-1445) and le De mente (1450),” in: H. Pasqua (ed.), Identité et différence dans l’œuvre de Nicolas de Cues (1401-1464), Louvain 2011, pp. 35-49. 164. See Nicholas of Cusa, Idiota de mente, VI, 88-96, pp. 132-145. 165. See ibid., III-IV, 72-75, pp. 108-115; cf. ibid., XV, 157-158, pp. 213-216.

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Cusanus states in De mente that (1) mind measures through the point and line (magnitude) as much as through number (multitude).166 (2) Because points function as geometrical limits, they must be indivisible.167 (3) As Boethius states, a point added to a point adds nothing, but remains a single point, just as unities added to unity result in unity.168 (4) Lines are not only the unfolding (explicatio) of points but also their unrolling (evolutio) into space.169 (5) The point is the terminus, perfectio, and totalitas of the line.170 (6) The point enfolds the line just as unity enfolds number.171 All of these ideas from De mente have definite parallels in Thierry’s Arithmetica. As impressive as this evidence is, there is even more to consider in later works. Nicholas’s De theologicis complementis (1453) is a sophisticated theological meditation on points, lines, angles, and curves, as if the cardinal were composing his own roundabout gloss on De arithmetica. In the compendious Dialogus de ludo globi (1463), Cusanus collects some of his past Pythagorean ideas but also introduces new discussions of rithmomachia and rotunditas – two concepts which also appear in Thierry’s Arithmetica.172 Caiazzo’s rediscovery of Thierry’s Arithmetica commentary will have profound effects on our understanding of Christian Neopythag- oreanism. The work was foundational for Thierry’s development, setting the agenda for his De trinitate commentaries and perhaps for

166. See Nicholas of Cusa, Idiota de mente, IX, 116, ed. R. Steiger, pp. 171-172; cf. Thierry of Chartres, Super Arithmeticam, II, 4 (230-232), ed. I. Caiazzo, p. 163. 167. See Nicholas of Cusa, Idiota de mente, IX, 117, ed. R. Steiger, pp. 172-173; cf. Thierry of Chartres, Super Arithmeticam, II, 4 (241-256), ed. I. Caiazzo, p. 164. 168. See Nicholas of Cusa, Idiota de mente, IX, 118, ed. R. Steiger, pp. 173-174; cf. Thierry of Chartres, Super Arithmeticam, II, 4 (280-296), ed. I. Caiazzo, pp. 165-167. 169. See Nicholas of Cusa, Idiota de mente, IX, 118-119, ed. R. Steiger, pp. 173-174; cf. Thierry of Chartres, Super Arithmeticam, II, 4 (239-241), ed. I. Caiazzo, p. 164. 170. See Nicholas of Cusa, Idiota de mente, IX, 120, ed. R. Steiger, pp. 174-175; cf. Thierry of Chartres, Super Arithmeticam, II, 4 (236-241, 260-275), ed. I. Caiazzo, pp. 164-165. 171. See Nicholas of Cusa, Idiota de mente, IX, 121, ed. R. Steiger, p. 175; cf. Thierry of Chartres, Super Arithmeticam, II, 4 (230-232), ed. I. Caiazzo, p. 163. 172. See Nicholas of Cusa, De theologicis complementis, in: Nicolai de Cusa Opera omnia iussu et auctoritate Academiae Litterarum Heidelbergensis, Vol. X/2a, ed. H. D. Rie- mann – K. Bormann, Hamburg 1994; and id., Dialogus de ludo globi, in: Nicolai de Cusa Opera omnia iussu et auctoritate Academiae Litterarum Heidelbergensis, Vol. IX, ed. H. G. Senger, Hamburg 1998. On rithmomachia and rotunditas, see Thierry of Chartres, Super Arithmeticam, Accessus (53-56), ed. I. Caiazzo, p. 93; II, 24 (550-571), pp. 176-177; and II, 29 (761-769), p. 184.

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Tractatus as well. It was also foundational for Nicholas, leaving a deep imprint on De docta ignorantia and perhaps becoming even more influential by the 1450s and 60s. In light of its stature, we should only expect that the Arithmetica commentary would shed light on the most puzzling and controversial link between Thierry and Nicholas, namely the Fundamentum naturae treatise discovered by Hoenen. Like the Arithmetica, this treatise purports to be another missing source of De docta ignorantia that might have better claim to some heretofore ‘Cusan’ concepts than Cusanus himself. Given our current understanding of how Nicholas handled his various Chartrian sources over time, we can now ask how the Arithmetica commentary inter- acted with Fundamentum naturae within the economy of Nicholas’s collected Chartrian texts. If Caiazzo’s discovery were found to support our emerging picture of Cusanus as collator and editor of these sources, this would add new weight to Hoenen’s proposal. In a recent monograph I have traced Cusanus’s evolving interaction with his dossier of Chartrian sources, including Fundamentum naturae, over the 1440s, 50s, and 60s. I defend four theses about Thierry’s complex influence on Cusan thought, broken into four stages from 1440 to 1464.173 (1) In the first stage, the composition of De docta ignorantia (1440), Nicholas constructed a synthesis of two groupings of Chartrian sources: Tractatus, Commentum, De septem septenis, and Commentarius Victori- nus, on the one hand; and Fundamentum naturae, on the other.174 These two bodies of texts often overlap in their vocabularies, but occasionally disagree on doctrines. Cusanus may have detected the tensions, but it is not clear he understood the force of the Fundamentum treatise’s critique of Chartrian theology. Roughly speaking, Nicholas deployed the first group in the theology of Book I, and the second in the cosmology of Book II. In the Christology of Book III, he tried to reconcile them through his mystical theology of Incarnation. (2) Next, given the abiding tensions between his sources, Cusanus attempted a second, alternative synthesis, now in terms of Neopythagorean

173. See D. Albertson, Mathematical Theologies, chapters 7-10. 174. The discovery of Thierry’s Arithmetica commentary might eventually occasion a reappraisal of the evidence for Tractatus and Commentum as sources for De docta ignorantia, but only after the sequence among these three Chartrian works is better established.

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number theory rather than Christology. Here the theme of multitude and magnitude as the twin foundations of the quadrivium comes to the fore. This second stage is visible in sermons after 1440 and especially in De coniecturis (1440-1445), the lengthy philosophical sequel foreshad- owed in De docta ignorantia. For the rest of the decade, Nicholas pur- sued both projects at once. He developed his Christology further in short theological works like De filiatione dei (1445) and De dato patris luminum (1446). But he also initiated a series of attempts to prove the quadrature of the circle in purely mathematical works, beginning with De transmutationibus geometricis (1445). (3) In the critical third stage, Cusanus finally overcame the tensions between Fundamentum naturae and other Chartrian sources. This occurs in the major treatise Idiota de mente (1450) and in subsequent works enabled by its new perspective, particularly De theologicis complementis (1453). Somehow Nicholas found his way around the impasse by means of newfound access to Thierry’s second and third commentaries on De trinitate (Lectiones and Glosa). There are strong indications that Nicholas had not read Lectiones or Glosa by the time he finished De docta ignorantia. But in De mente Nicholas quotes Glosa side by side with De septem septenis; he overturns the critique of Thierry’s four modes of being voiced by Fundamentum naturae; he dramatically expands his use of reciprocal folding; and he highlights several themes from Boethian arithmetic, especially the four arts of the quadrivium, number in the divine Mind, and the categories of multitude and magnitude. The new epistemological outlook of De mente, along with expanded access to Proclian sources in the 1450s, informed new lines of questioning in De beryllo (1458) and De principio (1459), even as Nicholas ended his program of geometrical proofs in the same years. (4) By the turn of the 1460s, Cusanus entered a distinct fourth stage as he began to apply the insights of his third stage to the subject matter of the first stage, especially its Christology. This points to Dialogus de ludo globi (1463) as an important but underappreciated benchmark of Cusan development among the cardinal’s late works, alongside other classics like De possest (1460) or De venatione sapientiae (1463). This four-stage model of Cusan development represents my best attempt to reconstruct the cardinal’s dynamic interpretation of his

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Chartrian sources, given our current knowledge of Thierry’s works and readership. It does not answer every question we have about Nicholas’s access to particular texts (where great mysteries still abound), but it stands as the most comprehensive account of the textual evidence, and provides a basic heuristic model for evaluating new evidence, such as Caiazzo’s Arithmetica discovery. Nevertheless, there are two unanswered questions posed by this model, two riddles which up until now I have been unable to explain, given the textual evidence in our possession. First, as we have seen, Cusanus discusses folding in Book II of De docta ignorantia. This occurs, however, not only in chapter II, 3, but also in chapters II, 9 and II, 10. Per Hoenen’s theory, these are passages in which Nicholas is adding his own exposition after having copied paragraphs from Fundamentum naturae verbatim.175 In those paragraphs, the author of Fundamentum makes a lopsided appeal to folding. The Aristotelian author is happy to countenance the complicatio of all forms in God, but castigates explicatio as a foolish error of the Platonists.176 Cusanus does not object when he cites his source’s words. But when he elaborates on Fundamentum naturae in his expositions, his own treatment of complicatio and explicatio is far more balanced. Rather than praise complicatio and blame explicatio, Nicho- las pairs them together reciprocally, as Thierry did, and indeed as Nicholas already had done in chapter II, 3.177 This suggests that Cusanus had access to some account of Chartrian folding which would have allowed him to identify and redress the skewed version presented in Fundamentum naturae. Yet it is highly unlikely that Cusanus possessed (or if possessed, read) Thierry’s Lectiones and Glosa commentaries in 1440. This seems improbable because the author of Fundamentum naturae contradicts, precisely and vociferously, Thierry’s doctrine of the four modes of

175. See Nicholas of Cusa, De docta ignorantia, II, 9, 143-147, ed. P. Wilpert – H. G. Senger, pp. 66-72; and ibid., II, 10, 151, pp. 78-80. Cf. M. Hoenen, “‘Ista prius inaudita’,” pp. 461-471. 176. See Fundamentum naturae, fol. 8r, ed. M. hoenen, p. 468 = Nicholas of Cusa, De docta ignorantia, II, 9, 150, ed. P. Wilpert – H. G. Senger, pp. 76-78; and Fundamentum naturae, fol. 9r, ed. M. Hoenen, p. 472 = Nicholas of Cusa, De docta ignorantia, II, 10, 155, ed. P. Wilpert – H. G. Senger, p. 84. 177. See e.g. Nicholas of Cusa, De docta ignorantia, II, 9, 143-45, ed. P. Wilpert – H. G. Senger, pp. 66-70; cf. D. Albertson, Mathematical Theologies, pp. 189-190.

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being, which only appears in Lectiones and Glosa. Despite his palpable enthusiasm for other Chartrian concepts throughout De docta ignorantia, Nicholas utters no complaint when the Fundamentum treatise condemns Thierry’s signature modal theory. By contrast, by the time of De mente in 1450, Cusanus has evidently discovered the four modes of being in Lectiones and Glosa. For then in a dramatic volte- face, he works directly from these late commentaries, focusing on the details of Thierry’s modal theory, and in particular on the second of the four modes, the very mode which the author of Fundamentum naturae had singled out for special ridicule. If this is what it looks like when Nicholas is aware of Lectiones and Glosa, clearly he did not possess them during the composition of De docta ignorantia. On the other hand, if he was not acquainted with Lectiones and Glosa in 1440, how did Cusanus manage to formulate a sound response to the imbalanced interpretation of complicatio and explicatio in the Fun- damentum treatise, articulating a doctrine that more closely mirrors Thierry’s own? After all, reciprocal folding does not appear in Tractatus, Commentum, Commentarius Victorinus, or De septem septenis. Are we to think that Cusanus first extracted the basic notion of folding from Fundamentum naturae, but then, still working within this source, was able to counteract its author’s bias against Thierry, all of his own powers? Or did Cusanus have another, unknown source at his disposal? This is the first riddle. Its lack of solution and the inconsistencies it augurs pose one of the last significant challenges to Hoenen’s theory of Fundamentum naturae as source for De docta ignorantia. The second riddle concerns the nature of Cusanus’s breakthrough in the third stage of development, namely, the composition of De mente. According to my account, by 1450, Nicholas essentially overcame the rejection of Thierry’s four modes of being by Fundamentum naturae that he had unwittingly repeated in De docta ignorantia. Thierry’s four modes effectively clarify how the disciplines of theology and mathematics can work together. Thus Nicholas’s new acceptance of them could be seen to entail, at a theoretical level, a greater open- ness to mathematics in theology. That said, the degree of the cardinal’s excitement about mathematics in De mente – and more precisely, about Boethian arithmetic – far surpasses the mere recognition of a new theoretical possibility. Never before, not even in the Neopy- thagorean De coniecturis, had Cusanus celebrated the quadrivium of

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Boethius with such energy and in such comprehensive detail. Before 1450, Nicholas had only contemplated the theological meaning of multitude and magnitude in two sermons; in De coniecturis he hardly mentions the quadrivium at all. But suddenly in De mente, Cusanus gives the theological meaning of the quadrivium his undivided attention, and now he declares that the reciprocal folding of unity and number is the very essence of both the human mind and its divine exemplar. It is easy to forget that these two coincident events in De mente – Cusanus reconciling his Chartrian sources, and Cusanus celebrating the Boethian quadrivium – have little to do with each other prima facie. What does Boethian arithmetic have to do with the elevation of Glosa above Fundamentum naturae? None of these quadrivial themes in De mente appear in either of those Chartrian sources. Moreover, why would the theological potential of the quadrivium be more highly esteemed in De mente than in De coniecturis or indeed De docta ignorantia, where Nicholas discusses the four arts at length? For these reasons there seems to be another, unidentified factor at work that drives Cusanus to link a speculative theology of the quadrivium, on the one hand, to his recuperation of Thierry’s Glosa commentary, on the other. The two are not obviously related. These are the two riddles that hang over Nicholas of Cusa’s use of Chartrian texts from 1440 to 1450. Where does Nicholas learn of reciprocal folding in 1440, when all he has is Fundamentum naturae? What accounts for his sudden emphasis on the quadrivium in 1450, when he rejects Fundamentum naturae? Remarkably, Thierry’s lost Arithmetica commentary solves both of these problems in one stroke.178 First, as Nicholas sought to explain the lopsided folding doctrine of Fundamentum naturae in Book II of De docta ignorantia, he must have drawn on his familiarity with complicatio and explicatio derived from Thierry’s Arithmetica commentary, just as he had in chapter II, 3. Thierry’s Arithmetica is the secret source that helped Cusanus grap- ple with Fundamentum naturae in 1440. Yet by the same token the Arithmetica commentary could not possibly have unveiled the anonymous author’s attack on the four modes for Cusanus to see, since

178. Cf. D. Albertson, Mathematical Theologies, pp. 366-367, n. 110.

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Thierry had not yet formulated that doctrine at the time of its writing. That is, Caiazzo’s discovery explains both what Cusanus understood well about Fundamentum naturae and what he understood poorly. Second, when Cusanus returned to the four modes in 1450, he appears to have had two revelations. For the first time he grasped that Glosa was intimately linked with the other Chartrian sources in his collection, and that it countermanded and preempted the criticisms of Fundamentum naturae. Therefore he must have also grasped that Glosa was linked specifically to his valuable Arithmetica commentary. Perhaps he noticed that they share the exemplar doctrine in common.179 This would have validated Glosa and quickly verified its doctrine of four modes over the objections of the Fundamentum treatise. But given the substance of De mente, it apparently also drove Cusanus back into Thierry’s Arithmetica once again, so that to accept Glosa was not only to diminish Fundamentum naturae, but also to elevate the Arithmetica commentary, making it the most important single work connecting his various Chartrian sources to each other. This explains the mathematical orientation of De mente. The two major lacunae in my four-stage model are thus transformed from hermeneutical problems into grounds for affirming that Cusanus used Thierry’s Arithmetica commentary extensively. That work provided the cardinal with a consistent touchstone from De docta ignorantia and De mente through De theologicis complementis and Dialogus de ludo globi. In this way, Caiazzo’s discovery of the Stuttgart Arithmetica and Hoenen’s discovery of the Eichstätt Fundamentum naturae are not only compatible, but mutually reinforcing. This is a remarkable fact that merits further attention in Cusanus scholarship. As Klibansky wisely counseled in his edition of De docta ignorantia, a deflationary reading of Nicholas of Cusa – one that exposes his dependence on sources, distributes his token concepts to others, and restricts the scope of his innovations – can nonetheless potentially reveal the cardinal’s significance at a more profound level than one that only praises his singularity. For if Cusanus is even more depend- ent on Thierry of Chartres’s writings than previously thought, this

179. Cf. Thierry of Chartres, Glosa, I, 38, ed. N. M. Häring, p. 267; and id., Super Arithmeticam, I, 1 (343-350), ed. I. Caiazzo, p. 114.

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only redoubles his status as an indispensable conduit of Christian Neopythagoreanism to the Renaissance, much as Boethius himself renounced any claim to novelty when he translated Nicomachus of Gerasa for the Middle Ages in De arithmetica.

David Albertson University of Southern California Dornsife College of Letters, Arts and Sciences 825 Bloom Walk, ACB 130 Los Angeles, CA 90089-1481 USA [email protected]

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