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M. Semikolennykh (Saint-Petersburg)
JEAN DE MEUNG’S „MARVELLOUS TRIANGLE”
AND TRANSFORMATIONS OF GEOMETRICAL FIGURES IN NICHOLAS OF CUSA’S „DE DOCTA IGNORANTIA”
At the beginning of his famous study of divine truths through geometrical figures, Nicholas of Cusa gives a kind of „historiography of the question” in which he demonstrates traditionality of the chosen method and thus justifies its legitimacy: „The most devout Anselm compared the maximum Truth to infinite rectitude. Following him, let me have recourse to a figure of rectitude that I picture as a straight line. Other leading theorists compared the triangle consisting of three equal right angles with the Super-blessed Trinity. Since such a triangle inevitably has infinite sides, as will be proved, let us call it an infinite triangle. Yet others who attempted to befigure the infinite unity spoke of God as an infinite circle. But those who considered the most actual existence of God affirmed that He is an infinite sphere”1. Having argued that the infinite line is simultaneously a straight line, an infinite triangle, a circle and a sphere, Nicholas suggested that none of his predecessors had been mistaken and that all their theories might have been true.
In fact, assuming that one side of a triangle is infinite, the other two would then inevitably be endless and since there is only one infinity, the infinite triangle can only consist of one infinite side and „it will be required that the one infinite line be triple and that the three lines be infinitely simple”2. Let us imagine an infinite straight line ab that is a side of an infinite triangle abc; the arc bc formed by the movement of the line ab will then also be infinite, „maximally straight and minimally
1 Nicholas of Cusa. De docta ignorantia (I, 12). English translation by Jasper Hopkins (Minneapolis, Minnesota: The Arthur J. Banning Press, 1981).
2 Ibid., I, 14. The same reasoning can be applied to the angles of the infinite triangle.
curved”, i. e. the maximal arc of a circle or the maximal circle itself.3 The arc of the maximal circle is also the arc of the maximal sphere and the circumference of the maximal circle, the circumference of the maximal sphere. On the other hand, the maximal sphere is formed by rotation of the maximal circle.
How can a triangle, circle or sphere then be symbols of the Divinity? In an infinite triangle, all the three sides are the same infinite line and each of the three angles is turned out an infinite line; this infinite line is therefore triple. But since there could be no two infinities, this trinity is oneness4. Furthermore, all the three angles of the infinite triangle are maximal and infinite and each of them is the maximum, and all of them together are the only one maximum. „Moreover, the truth of a triangle requires that no one angle be the other and likewise, in the illustration the truth of the oneness of the most simple essence requires that these three angles not be three distinct things but be one thing”.5 Thus, relations between the sides and angles in the infinite triangle mirror the relationship between the three indivisible but separate Personae of the Deity.
The infinite circle with its infinite circumference that is the infinite line and also the diameter (that is also the infinite line) coincides with the infinite triangle;6 this circle is „a perfect figure of oneness and simplicity”. This perfect unity is not only spatial but also temporal; in the infinite circle, „the past is none other than the future and the future is none other than the present; rather, they are the most one duration, or eternity, without beginning and end”.7 The maximal circle is actually the absolute maximum coinciding with the absolute minimum.
3 Ibid. I, 13.
4 Ibid. I, 19.
5 Ibid.
6 Two infinities cannot exist simultaneously; therefore two infinite straight lines — the diameter of the infinite circle and its circumference — must coincide. The infinite diameter has the infinite centre, which is also coinciding with the infinite line (the diameter and the circumference of the infinite circle). Thus, in the infinite circle, the centre coincides with the circumference and the diameter, the absolute minimum — with an absolute maximum, and the beginning with the end.
7 Ibid. I, 21.
The same unity (the unity of the centre, the diameter and the circumference of the maximal circle) is a distinctive feature of the infinite sphere that is „the ultimate perfection of figures”.8 If the infinite triangle is a symbol of the triplicity of God and the infinite circle is the symbol of His unity, then the infinite sphere is a symbol of fullness of His effectual existence. The infinite sphere is an action of the infinite line, infinite triangle and infinite circle that is the realization and actualization of the absolute maximum. In God, potentiality and actuality are one; every motion is coming from Him, existing in Him and through Him, and in Him it is over: „For He is maximal rest, in which all motion is rest”.9
During his Mediterranean Sea trip in 1437, Nicholas of Cusa experienced a mystical revelation that made him look for new and rather unusual ways of speaking to God. His meditations are undoubtedly extremely original and he certainly started a new period in the history of philosophy but naturally, he had not only followers but also predecessors. However, his sophisticated geometrical models and especially his „geo-metry of infinity” are generally considered his own inventions that laid the foundations of a new mathematical logic typical for the Modernity.10 That is why the following lines written almost two centuries before De docta ignorantia hold a surprise for the reader: „Only the Virgin who had conceived in the womb understood this mystery better than Plato. She knew for certain that the divine Father breathed life into her womb. This marvelous triangle cannot have any limits, its centre is everywhere, it is impossible to comprehend it; it has three angles and the one angle in three, this triangular circle or circular triangle in the Virgin’s womb. Plato did not know it, and could not grasp a unity in the simple Trinity. He had not seen the incarnated Deity”.11
8 Ibid. I, 23.
9 Ibid.
10 Cassirer E. The Individual and the Cosmos in Renaissance Philosophy / Translated by A.G. Gadzhikurbanov // Cassirer E. Selected works: the individual and the cosmos. Moscow, 2000. P. 42.
11 Guillaume de Lorris et Jean de Meun, Roman de la Rose (19127-19143) // E. Langlois (ed.), Société des anciens Textes français. 1914. V. 1. P. 255-256:
Car plus que Platon, c’est certain,
En dut-elle soudain apprendre
These words on the mystery of the Holy Trinity were written around 1260 by Jean de Meung, one of the authors of the famous „Romance of the Rose”. This image is so reminiscent of Nicholas of Cusa’s complex mathematical models that the question arises immediately, is there any common tradition or any source common for both thinkers? To answer this question, let us turn out attention to the „Romance of the Rose” and to Jean de Meung himself.
The allegorical „Romance of the Rose” in which courtly ideals of knighthood coexist with didactics and encyclopaedic knowledge typical of urban literature, is a story about a handsome youth dreaming of the beautiful Rose symbolizing love, harmony and divine grace. The French trouvère Guillaume de Lorris, who wrote the first part of the novel in the 1230s, left his desperate young hero at the moment when he lost the grace of his beloved through machinations of Shyness, Fearfulness, Chastity, Jealousy and Slander. Several decades later, Jean Klopinel of Meung completed the poem by adding another 17,000 verses.
Unlike his predecessor who presumably belonged to the court circles, Jean de Meung, also known as the „Voltaire of the Middle Ages”, was
Lorsque vit son ventre se tendre.
Alors elle comprit, sentant A grand confort battre son flanc,
Qu’il était la sphère infinie,
Source de l’éternelle vie,
Qui son centre lance en tous lieux Sans que son tour frappe nos yeux,
Car c’est le merveilleux triangle Don’t l’unité fait le triple angle,
Lesquels trios collectivement N’en font qu’un seul tant seulement.
C’est le cercle triangulaire Et le triangle circulaire Qui dans la Vierge se logea.
Platon ne sut voir jusque-là,
Ni la déité souveraine Incarnée en la peau humaine,
Il ne vit la triple unité En cette simple trinité.
Russian translation from the Old French by. N. V. Zababurova based on word-for-word translation by D. N. Valjano (Rostov-on-Don, 2001).
a well educated cleric whose erudition was truly amazing. There is hardly any need to mention all the names of ancient and medieval authors that he quoted in his works. However, two important points should be made. First, he was undoubtedly familiar with the teaching of Aristotle but preferred Plato who was treated in medieval universities with some caution since, to quote Albertus Magnus, he had „a bad method of presentation” and ”his writings were full of metaphors”. Second, Jean de Meung was well acquainted with the writings of Alain de Lille.12
It is obvious that Jean de Meung belonged to the Neoplatonic tradition in which mathematics played a crucial role. Let us not forget that it was within this tradition that the cycle of seven liberal arts had been formed. Thus, Aurelius Augustine whose authority to theologians of the medieval West was indisputable, wrote in his treatise De ordine of three kinds of things within which rationality exhibits itself: things consisting of intentional actions (ethics), speaking (grammar, dialectic and rhetoric) and pleasure (mathematical sciences of quadrivium).13 Augustine considered the Logos, i.e. mind as well as relation of numbers and measures to be a foundation for all sciences of trivium.14 This idea apparently goes back to Plato who in the dialogue „Filebus” refers to a tradition according to which „all things that are said to have ever existed sprang from one and many and have the finite and the infinite inherent in them”.15 This postulate enables us to carry out any investigation by measuring unity and plurality. Mathematical sciences are a „well-organized way” of guiding one’s mind towards „a blissful contemplation of divine things themselves”.16
12 More information about the acquaintance of Jean de Meung with the works by Alain de Lille see: Lukacs E. A., La métaphore de la sphère dans les œuvres d’Alain de Lille, Jean de Meun et Vincent de Beauvais, diss. (Budapest, 2008).
13 Aurelius Augustinus. De ordine, II, 12, 35.
14 Ado I. Free arts and philosophy in ancient thought. Translated from French by E.F. Shichalina (Moscow, 2002). P. 136.
15 Plato. Philebus, 16c—19c. It is interesting that in Plato the numerical relations between sounds and letters are set by the Egyptian god Teut (Tot) which is identified with Hermes.
16 Aurelius Augustinus. De ordine, II, 14, 39.
It is worth noting that the idea of mathematical harmony of the Universe is expressed in the Bible. The Book of Wisdom reads, „You are all arranged by measure, number and weight” (Wisdom 11:21). When describing creation of the world in the Book of Proverbs (Proverbs 8:27-29), God appears to be a geometer. There are many images of creation in which God is depicted holding compasses and scales in His hands.17
Therefore, what roles do the infinite circle and the infinite triangle play in this ordered space? What do these symbols mean?
The image of an infinite sphere, or that of a circle, appears rather ancient; it is rather tempting to search for their roots in the works of antique philosophers. On nature, a poem by Parmenides, mentions that One is like a sphere, i.e. the most perfect of geometrical shapes. Also worth noting are the idea of sphericity of God in Xenophanes, Meliss’s thesis on the infinity of existence, a description of the Universe by the Pythagoreans and Plato, divine sphere embodying everything else in Cicero’s De re pu-blica18, the goodness in Plotinus and finally, the „amazing circle of divine simplicity” in Boethius whose Lady Philosophy refers to the dictum of the aforementioned sage from Elea.19 Similarly, the doctrine of a triangle as the basis for the Universe can be traced back to the Pythagoreans and Plato20, while Plato’s pupil Xenocrates illustrated his demonology by comparing equilateral, scalene and isosceles triangles; the equilateral one is named „divine” due to its perfection21. In the Middle Ages, Thierry of Chartres made an equilateral triangle a symbol of God. While trying to explain the mystery of the Trinity to their illiterate parishioners, preachers often used different images of geometric shapes consisting of three elements, e. g. shamrock, equilateral triangles, triangles inscribed in a circle (or vice versa), triquetrae, triskelions, etc.22
17 E. g., Zaitsev E. A. The Meaning of Early Medieval Geometry: From Euclid and Surveyors’ Manuals to Christian Philosophy. Isis, vol. 90, no. 3 (Sep. 1999). P. 535 ff.
18 Cicero. De ordine (VI, 17).
19 Boethius. Consolatio Philosophia (III, 12).
20 Plato. Timeus 53c-55c.
21 Dillon D. Middle Platonians / translated by E. V. Afonasev. St. Petersburg, 2002. P. 126.
22 Rowlatt U. Popular Representations of the Trinity in England, 990-1300. Folklore. Vol. 112. N 2 (Oct., 2001). P. 204.
Much has been written on the symbol of the infinite circle; suffice it to mention the book by Jean Poulet called Les méthamorphoses du cercle23 that was published in 1961 and followed his paper Le symbole du cercle infini dans la littérature et la philosophie.24 Like many other researchers,25 Poulet believed that the definition of God as an infinite sphere with its centre everywhere and circumference nowhere was introduced in the „Book of the twenty-four philosophers”, a small anonymous treatise written presumably in the 12th century and attributed to Hermes Trismegistus.26 In this short piece whose origins are still unknown for certain, twenty-four philosophers gave twenty-four definitions of God including „God is an infinite sphere with its centre everywhere and circumference nowhere” (II)27 and „God is a sphere that has as many circumferences as it has centres” (XVIII).28
23 Poulet G. Les méthamorphoses du cercle. Paris, 1961.
24 Poulet G. Le symbole du cercle infini dans la litterature et la philosophie. Revue de Métaphysique et de Morale. № 3 (1959). P. 257-275.
25 E. g.: Mahnke D. Unendlich Sphaere und Allmittelpunct. Halle, 1937; Nicolson M. H. The Breaking of the Circle. Northwestern University Press, 1950.
26 Liber XXIV Philosophorum // Baeumker C. «Das pseudo-hermetische Buch der vierundzwanzig Meister (Liber XXIV philosophorum). Ein Beitrag zur Geschichte des Neupythagoreismus und Neuplatonismus im Mittelalter», Studien und Charakteristiken zur Geschichte der Philosophie, inbesondere des Mittelalters. Gesammelte Aufsätze und Vorträge (Beiträge zur Geschichte der Philosophie des Mittelalters, 25 / 1-2. Münster, 1927). S. 194-214. On the Book of twenty-four philosopers see: Denifle H. Meister Eckeharts lateinische Scgriften und die Grundanschauung seiner Lehre. Archiv für Literatur- und Kirchengeschichte des Mittelalters. 1886. N 2. S. 427-429; Hermetica philosophica. Appendix I. Liber XXIV philosophorum, in Catalogus Translationum et Commentariorum: Medieval and Renaissance Latin Translations and Commentaries, I. Washington, 1960. P. 151-154; Hudry F. (ed.), Liber viginti quattuor philosophorum. Turnhout, 1997; Lucentini P.
Il Libro dei ventiquattro filosofi. Milano, 1999; Lucentini P. Liber viginti quattuor philosophorum nei poemi madievali: il Roman de la rose, il Granum sinapis, la Divina commedia. Poetry and Philosophy in the Middle Ages. A Festschrift for Peter Dronke, ed. by J. Marenbon . Leiden; Boston, 2001. P. 131-153 and especially; Flasch K. Was ist Gott? Das Buch der 24 Philosophen. München, 2011.
27 Deus est sphaera infinita cuius centrum est ubique, circumferentia nusquam.
28 Deus est sphaera cuius tot sunt circumferentiae quot puncta.
For the first time, the infinite sphere had become a symbol of Divinity.29 It is impossible to distinguish between Personae of the Trinity in the infinity of this sphere where only Divinity is cognizable through nothingness. This infinity is not to be interpreted as that of a circle coinciding with itself but as a kind of philosophical destruction of the concept of circle; the infinite sphere has no centre or else its centre is in any random point that immediately creates a new sphere. God is not a centre of an infinite sphere but a whole infinite set of points; He is simultaneously present in each of these points in all His integrity and inseparableness. This uncertainty is accentuated by the sixth maxim of the Book, „Comparison to God’s substance is an accident, and accident is nothingness”.30 God in Himself loses substantial distinctiveness, as if dissolving into infinity of His own divinity. However, the fact that the infinite and indefinable Deity is everywhere means that everywhere, i.e. the created world, is also infinite. „The infinite unity with its infinite number of centres creating infinite number of circumferences replaces a strict hierarchy”.31
It was from „The Book of the twenty-four philosophers” that Alain de Lille borrowed the eighth principle of his Theologicae regulae, that of the comparison between God and the infinite sphere. Even more interestingly, in his sermon on the intelligible sphere32 he juxtaposes intelligible sphere with a triangle, „But while all the other spheres, i.e. sensible, imaginary and rational ones, cannot be reduced to a square or a triangle, the intelligible one shows the properties of an equilateral triangle”.33 The difference between four spheres lies in their geometrical properties; unlike the intelligible sphere whose centre is everywhere and the circumference nowhere, the three other spheres have the moving circumference everywhere and immovable centre nowhere. Alain calls the centre of the intelligible sphere
29 Kurt Flush on whose comments we rely while considering the „The book of twenty-four philosophers” emphasizes the novelty of the image especially if we take into account a trend of the time to avoid infinity in philosophical and theological meditations and keen interest in finite being, its kinds and species, substances and accidents (see.: Flasch K. Ibid. S. 31).
30 Deus est cuius comparatione substantia est accidens, et accidens nihil.
31 Flasch K. Ibid. S. 33.
32 Alain de Lille, Règles de théologie suivi de Sermon sur la sphère intelligible, introduction, traduction et notes par F. Hudry (Paris, 1995).
33 Lukács E. A. Ibid. P. 136.
„opus mundanum, id est universitas rerum” (the Universe, that is the totality of [all] things); therefore this sphere embraces the whole world. The centre of the intelligible sphere is everywhere because the number of created things is unlimited; its circumference is nowhere because the infinity of the Deity can be neither excluded from the totality of all things nor included in it. God is therefore in the world and not a being in it. The centre of the sphere moves since all created things are variable. The circumference is fixed because the divine infinity is not liable to changes.34 The Trinity is symbolized by an equilateral triangle; the centre of the triangle in its uniqueness is the foundation of the Trinity while three peaks are spatial expansions of the point, the One and the centre. The triangle is not just „inscribed” into the divine sphere, but „is imprinted” (impressit) into the world, which makes presence of God in the world „even more evident and unique”.35
Consequently, by the 12th century the image of a triangle coinciding with a circle or sphere became more than a mere illustration for the Trinity doctrine; it had acquired some philosophical and theological grounds.
It is interesting that this symbol was often used by medieval poets. A good example is a Latin hymn written in the early 13th century and attributed to Philip the Chancellor (Philippus Cancellarius, ca. 1160-1236). Speaking of the mystery of the Incarnation, the author weaves a sophisticated network of contradictions and incompatible definitions.36 Another
34 Ibid. P. 135.
35 Ibid. P. 136.
36 Analecta Hymnica Medii Aevi, Hrsg. Guido Maria Dreves. Vol. 20. Leipzig, 1886. P. 88:
1. Centrum capit circulus 2. Concordem discordiam Quod est majus circulo, Rerum parit novitas,
In centro triangulus Vestem texit variam
Omni rectus angulo, Fecunda virginitas,
Sedfit minor angulus Matrem vocat filiam
Unus de triangulo, Facta caro Deitas,
Dum se mundi figulus Osculatur sociam
Inclusit in vasculo. Vanitatem veritas.
(1. A circle has a centre that is bigger than the circle; in the centre, there is a triangle that is right in respect to all its angles but one corner of the triangle is becoming smaller until the Creator of the world does not enclose Himself in the womb.
2. The novelty generates concordant discordance of the things, fertile virginity weaves multicoloured cloth, the Deity who becomes flesh calls His mother daughter, the truth is kissing the lie, its ally.)
example, although a less obvious one, can be found in the Divina Com-media.37 It is worth mentioning that Dante was familiar with the work of Alain de Lille.
The „marvelous triangle” in „The Romance of the Rose” certainly represents the Trinity. These quotations are from the speech of Nature complaining about human wrongdoings38. Citing Plato, Nature talks about the incommensurability of her power and that of God who alone can give mind to a person and „sees triple temporality in a moment of eternity”.39 Incomprehensible to Plato, but known to the Blessed Virgin, God confers immortality to man, a gift he cannot receive from the Nature. A kind genius echoes Nature’s speech describing the wonderful Garden of Eden and the Spring of Life hidden it in; waters from this spring give immortality. „In the spring is ... amazing carbuncle that is more beautiful than all the precious stones in the world. It is round but also triangular and it is placed so high that it shines over the whole garden. Neither rain nor wind can extinguish its ray. The source of the stone’s power is in equality of all its facets. Each of them is beautiful but no-one can separate them; if somebody tries to unite them, they will remain distinct”.40
Thus, the „triangular circle” does not appear by chance in Jean de Meung’s Romance of the Rose; it is used more than once, is enriched and made more complex. It is difficult to say with confidence whether Nicholas of Cusa was familiar with the work of „Voltaire of the Middle Ages”. However, when speaking of God, he used the same symbols so it is feasible that both authors shared an old tradition. Unlike Nicholas of Cusa, Jean de Meung did not provide a rigorous geometrical basis for the „magic” of his image, the one that removed all doubts on the originality of Krebs’s thought. However, it might just be possible to trace a certain continuity of ideas in the traditions of medieval Neoplatonism.
37 Dante Alighieri. Divina Commedia (Paradiso, XXX, 100-123; XXXIII, 115-138).
38 This speech corresponds to the famous Deplanctu Naturae by Alain de Lille.
39 Guillaume de Lorris et Jean de Meun, Roman de la Rose (19072-19076): „Elle n’est rien près la puissance / De Dieu, qui voit en sa presence / La triple temporalité / Dans un moment d’éternité”.
40 Guillaume de Lorris et Jean de Meun, Roman de la Rose (21241-21266). Vol. IV. Paris, 1879.
