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Squaring the Circle: Paradiso 33 and the Poetics of Geometry

Published online by Cambridge University Press:  29 July 2016

Ronald B. Herzman  and
Gary W. Towsley
Ronald B. Herzman
Affiliation:
SUNY Geneseo
Gary W. Towsley
Affiliation:
SUNY Geneseo
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Extract

The last canto of Dante's Paradiso brings the Commedia to an appropriately climactic end: to a point of closure matched by few or—as most Dantists would probably be willing to put it—any other works of art. One explanation for this is that what the ending reveals all but forces on the reader a retrospective look at the vast terrain that has come before. The richness of the end emerges from and folds back into the richness of the entire work. Thus the vision at the end constitutes an intense paradox. It is the climax of the journey but also its ground, its final cause but its formal cause as well. Everything that has come before can only be fully understood in terms of that final vision. Indeed, it has now become something of a commonplace in Dante studies to say that to finish reading the Commedia is finally to be able to begin to read it.

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Articles
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Traditio , Volume 49 , 1994 , pp. 95 - 125
Copyright
Copyright © 1994 by Fordham University 

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References

1 We would like to acknowledge the considerable help we have received from two undergraduate research assistants, Lisa Lucenti and Laura Sythes; from our colleague, Graham Drake; and from the Dartmouth Dante Project, a data base which made much of the research for this work possible.Google Scholar

Among attempts to deal with the very end of the poem, the most important is probably that of John Freccero, “The Final Image: Paradiso 33.144,” in Dante: The Poetics of Conversion, ed. Rachel Jacoff (Cambridge, Mass., 1986), 245–57. Freccero's essay elegantly confronts the difficulty of dealing with the ending of the poem. Moreover, his understanding and appropriation of the Platonic tradition for the final image has a great many ramifications for the image immediately preceding, that is, for the image of squaring the circle which is the subject of this essay. James Chiampi's recent essay, “Dante's Paradiso from Number to Mysterium,” Dante Studies 110 (1992): 255–78, deals with the way that geometry is used in the Paradiso, pointing toward the end of the poem, though he deals with the image of squaring the circle only indirectly.Google Scholar

Quotations of the Commedia are from The Divine Commedia, ed. and trans. Singleton, Charles S., Bollingen Series 80 (Princeton, 1970–1975), hereafter cited as Singleton. The following abbreviations are used throughout: Par. = Paradiso, comm. = commentary.Google Scholar

2 Dante, , The Banquet. trans. Christopher Ryan, Stanford French and Italian Studies 61 (Saratoga, Calif., 1989), 70 (emphasis added): “Si che tra punto e lo cerchio si come tra principio e fine si muove la Geometria, e questi due a la sua certezza repugnano; chè lo punto per la sua invisibilitade è immensurabile, e lo cerchio per lo suo arco è impossibile a quadrare perfettamente, e pero è impossibile a misurare a punto.” Another reference to the problem of squaring the circle in Dante occurs in Monarchia 3.3.2 (trans. Schneider, Herbert W. [Indianapolis, 1957]): “There are many things about which we are ignorant but which are not subjects of dispute: the geometers do not know how to square the circle, but they do not dispute the question….” (“Multa etenim ignoramus de quibus non litigamus; nam geometra circuli quadratum ignorat, non tamen de ipsa litigat….”) Dante is of course saying that we know what it means to square the circle, even though we do not know how to do it.Google Scholar

3 Dante, , The Divine Comedy, vol. 3: Paradise, trans. Mark Musa (New York, 1984), 394.Google Scholar

4 Singleton, 584.Google Scholar

5 For a contrasting perspective, see the commentary of Grabher (1934–36), who describes the situation and the language in more emotional terms. For example, he writes that “Il geometra qui lo vediamo non nella fredda meditazione di un problema, non nella sua pura ricerca di scienziato, ma con lo smarrimento dell'uomo che, perdendosi nella contemplazione di quell'uniforme cerchio, ha dinanzi qualcosa che lo trascende. È problema intellettuale fatto sentimento e da esso trasfigurato” (La Divina Commedia col commento di Carlo Grabher [Bari, 1964–65], comm. on Par. 33.133–38).Google Scholar

6 Though it should also be pointed out that Singleton's notes on the subsequent lines are more congenial to the lines of inquiry of this study. See the notes on lines 136–38 (585).Google Scholar

7 “Simultaneously with the gradual evolution of the Elements, the Greeks were occupying themselves with problems in higher geometry; three problems in particular, the squaring of the circle, the doubling of the cube, and the trisection of any given angle, were rallying points for mathematicians during three centuries at least, and the whole course of Greek geometry was profoundly influenced by the character of the specialized investigations which had their origin in the attempts to solve these problems.” (Thomas Heath, A History of Greek Mathematics [New York, 1981], 1: 218).Google Scholar

8 The Thirteen Books of Euclid's Elements, trans. Thomas Heath (New York, 1956), 3: 371.Google Scholar

9 Archimedes, , On the Measurement of the Circle. Proposition 1, trans. Heath, Thomas L. in Great Books of the Western World (Chicago, 1952), 6: 447.Google Scholar

10 A more detailed presentation of this tradition may be found in the epilogue of Wilbur Knorr, The Ancient Tradition of Geometric Problems (Boston, 1986), 367–70.Google Scholar

11 A Commentary on the First Book of Euclid's Elements, trans. Morrow, Glenn R. (Princeton, 1970), 40.Google Scholar

12 Dante himself refers to the Timaeus in Convivio 3.5.6 and to the speaker of the Timaeus in Par. 4.49. It is the only Platonic dialogue mentioned by name by Dante.Google Scholar

13 Plato, , Timaeus 36b–39c in Plato's Timaeus, trans. Cornford, F. M. (Indianapolis, 1959), 27–32.Google Scholar

14 Plutarch, , Moralia (Cambridge, Mass., 1961) 9: 119–21: “It was for this reason that Plato himself reproached Eudoxus and Archytas and Menaechmus for setting out to remove the problem of doubling the cube into the realm of instruments and mechanical devices, as if they were trying to find two mean proportionals not by the use of reason but in whatever way would work.”Google Scholar

15 Most important is the discussion of education as a preparation for dialectic in Republic 7, especially 527b–528e. See The Republic, trans. Grube, G. M. A. (Indianapolis, 1974), 178–80.Google Scholar

16 Phaedrus 274d–275b in Plato: The Collected Dialogues, ed. Edith Hamilton and Huntington Cairns, trans Hackworth, R., Bollingen Series 71 (Princeton, 1961), 520. Bruno Nardi (Dante e la cultura medievale [Rome, 1985; orig. 1942], 11–12) has called attention to the influence of the Phaedrus on Dante, suggesting that the rhetorical concerns of the thirteenth century seem to be an unfolding of the concerns of the Phaedrus, and then more specifically noting a parallel between the central thesis of the Phaedrus and the words of Dante to Bonagiunta in Purgatorio 24.52–54: “… l'ultima tesi del Fedro, esser la vera eloquenza espressione dei sentimenti dell'animo, trova perfetta risonanza nei versi del poeta: ‘I'mi son un, che quando / amor mi spira, noto, ed a quel modo / ch'e ditta dentro vo significando’ ” (12).Google Scholar

17 Phaedrus 274d (Collected Dialogues, 520).Google Scholar

18 Plutarch, Moralia, 121.Google Scholar

19 Aristotle, , On the Heavens 1.2.269a19–23, in The Complete Works of Aristotle, ed. Jonathan Barnes, Bollingen Series 71.2 (Princeton, 1984), 1: 449: “For the complete is naturally prior to the incomplete, and the circle is a complete thing. This cannot be said of any straight line: not of an infinite line, for then it would have a limit and an end; nor of any finite line, for in every case there is something beyond it, since any finite line can be extended.” (All references to Aristotle are taken from Barnes.)Google Scholar

20 Euclid's Elements (n. 8 above), 1:349.Google Scholar

21 The propositions of the Elements come in two forms: “theorems,” which express a relation between geometrical objects and quantities, and “problems,” which are given in infinitive form and describe the construction of a geometrical object satisfying certain conditions. A survey of the Elements shows that only books 5, 7–9, and 12 lack propositions that justify constructions. These books are concerned with proportion, number theory, or irrational numbers, topics which require only constructions that Euclid has presented in earlier books.Google Scholar

22 More precisely, through extant works of Euclid and Archimedes, the fact that there exists a square equal in area to a given circle was known. The fact that special curves such as the quadratix and the spiral could effect squarings of the circle came to western Europe chiefly through the writings of Neoplatonists such as Proclus and Plutarch. If Dante knew of such squarings, however, then he also knew that they were viewed as being somewhat suspect in their use of these “mechanical curves.”Google Scholar

23 “Über die Zahl π,” Mathematische Annalen (1882): 221–25. William Dunham has pointed out that Lindemann succeeded in his proof by “translating the issue from the realm of geometry to the realm of number” (Journey Through Genius: The Great Theorems of Mathematics [New York, 1990], 23). Dunham also presents a synopsis of Lindemann's proof (226). It is interesting to note that the commentary of Lombardi (1791), after a careful mathematical statement of the nature of the problem, states that it is perhaps impossible to find a solution: “e forse impossibile a trovarsi.” (La Divina Commedia, novamente corretta, spiegata, e difesa da Fra Baldassare Lombardi, minore conventuale [Rome, 1791–92], comm. on Par. 33.134–135.) Commentators in the twentieth century assume that it is impossible to solve, as for example, Sapegno, who calls it “insolubile problema” (La Divina Commedia a cura di Natalino Sapegno [Milan/Naples, 1957], comm. on Par. 33.134–35).Google Scholar

24 Par. 10.109–114: “The fifth light, which is the most beautiful among us, breathes with such love that all the world there below thirsts to know tidings of it. Within it is the lofty mind to which was given wisdom so deep that, if the truth be true, there never rose a second of such full vision.”)Google Scholar

25 Par. 13.86–87: “… that human nature never was, nor shall be, what it was in those two persons.”Google Scholar

26

… non per sapere il numero in che enno

li motor di qua sù, o se necesse

con contingente mai necesse fenno;

non si est dare primum motum esse,

o se del mezzo cerchio far si puote

trïangol sì ch'un retto non avesse.

Par. 13.97–102: “… not to know the number of the mover spirits here above, nor if necesse with a contingent ever made necesse; nor si est dare primum motum esse; nor if in a semicircle a triangle can be so constructed that it shall have no right angle.”

27 Par. 13.121–26: “Far worse than in vain does he leave the shore (since he returns not as he puts forth) who fishes for the truth and has not the art. And of this Parmenides, Melissus, Bryson, are open proofs to the world….”Google Scholar

28 Il codice cassinese della Divina Commedia … per cura dei monaci benedettini della badia di Monte Cassino (Tipografia di Monte Cassino, 1865), comm. on Par. 13.125. The full text is as follows: “Parmenide et Melisso et Brisso, qui tres phylosophi non distinguentes reprobati fuerunt per Aristotilem in primo physicorum et in primo post silogizantes falsa forma syllogistica ponendo falsum simpliciter quod per interiectionem solvendum erat vel secundum aliquid et sic solvendum erat per distinctionem.”Google Scholar

29 The Cassinese codex (ibid.) is, as it turns out, misleading if not somewhat inaccurate, suggesting a reading of the text of Aristotle that is simply not there. The lines “tres philosophi non distinguentes reprobati fuerunt per Aristotilem in primo physicorum et in primo post silogizantes falsa …” suggest that the three philosophers are grouped together by Aristotle (“tres philosophi non distinguentes”). Though in some sense this is true-that all three are in fact reproved by him in his work taken as a whole—the quotation makes it seem that Aristotle himself has grouped the three together, and that we therefore might expect to find all three of them similarly and simultaneously criticized in the first book of the Physics. But, in point of fact, we do find there a critique of Parmenides and of Melissus, but not of Bryson. (Aristotle, Physics, 1.2.185a–185b18;1.317–18).Google Scholar

30 The text of the Monarchia (n. 2 above) talks about two ways of being a bad philosopher, both of which were exemplified in Parmenides and Melissus: “Et quia error potest esse in materia et in forma argumenti, dupliciter peccare contingit: aut silicet assumendo falsum aut non sillogizando; que duo philosophus obiciebat contra Parmenidem et Melissum dicens: ‘Quia falsa recipiunt et non sillogizantes sunt’ ” (3.3.4).Google Scholar

31 L'Ottimo Commento della Divina Commedia. Testo inedito d'un contemporaneo di Dante (Pisa, 1827–29), comm. on Par. 13.124–125: “Fue Melisso filosofo in quello medesimo tempo che Parmenide; de’ quali Parmenide e Melisso dice il Filosofo, nel primo libro della Fisica, ch'elli affermavano, che tutte le cose ritornavano in una cosa, sì come da una procedeano.”Google Scholar

32 Benvenuto de Rambaldis de Imola Commentium super Dantis Aldigherij Comoediam, nunc primum integre in lucem editum sumptibus Guilielmi Waren Vernon, curante Jacopo Philippo Lacaita (Florence, 1887), comm. on Par. 13.124–126: “Et hic nota quod Parmenides et Melissus fuerunt duo philosophi, qui conantes investigare principia rerum naturalium nimis enormiter erraverunt; quorum rationes philosophus improbat primo physicorum saepissime, et alibi saepe.” Among more modern commentators, see Ernesto Trucchi, Espozione della Divina Commedia (Milan, 1936), comm. on Par. 13.124.129: “Parmenide e Melisso furono due eleatici.”Google Scholar

33 Lombardi (n. 23 above), comm. on Par. 13.125: “Melisso, filosofo di Samo, erasi tra gli altri errori, messo a sostenere, che ralmente moto veruno non si desse, ma che solamente sembrasse.” See also the commentary of Portirelli, who repeats Lombardi verbatim. (La Divina Commedia di Dante Alighieri illustrata di note da Luigi Portirelli [Milano, 1804–05], comm. on Par. 13.125. [The notes for Paradiso in this commentary are in fact the work of Ferrario, G.])Google Scholar

34 L'Ottimo says that Bryson was “filosofo al tempo di Ciro re predetto …” (comm. on Par. 13.124–125.) The commentary of Bosco/Reggio states: “filosofo greco di Eraclea, figlio di Erodoto, lo storico greco, e discepolo di Socrate e di Euclide” (La Divina Commedia a cura di Umberto Bosco e Giovanni Reggio [Florence, 1979], comm. on Par. 13.125). This commentary also says that Parmenides, Melissus, and Bryson were names found together in the Physics of Albertus Magnus. This is true only in that they are all mentioned by Albertus, but as is true with the texts of Aristotle, they are not mentioned together. Albertus talks of Bryson in book 1, and then subsequently spends several lengthy chapters on Parmenides and Melissus considered together. (See nn. 38, 39 on Albertus, below.) Bryson the sophist, son of Herodotus, is mentioned twice by Aristotle in his History of Animals (6.5.563a7, 9.11.615a10), but in neither case, nor in a reference to Bryson in the Rhetoric (3.2.1405b8–10), is he associated with squaring the circle, which may raise the possibility that there are two Brysons.Google Scholar

35 Arist. An. Post. 1.9.75b40–41, Barnes 1:123.Google Scholar

36 The texts are as follows: “But Bryson's method of squaring the circle, even if the circle is thereby squared, is sophistical because it does not conform to the subject at hand …”; and, “Thus, e.g., though the squaring of the circle by means of the lunules is not contentious, Bryson's solution is contentious….” (Arist. Soph. El. 11.171b16–17, 172a2–4: Barnes, 1:291). Bryson, called “Bryson the sophist,” is mentioned three other times in the Aristotelian corpus, but never in the context of squaring the circle, which explains why these other references are not noted by the commentary tradition, but also perhaps supports the suggestion that there are two Brysons (see n. 34 above).Google Scholar

37 “Questi che m’è a destra più vicino, / frate e maestro fummi, ed esso Alberto / è di Cologna, e io Thomas d'Aquino” (Par. 10.97–99).Google Scholar

38 The purpose of Albertus's Physics is clearly stated in its opening sentences: “Intentio nostra in scientia naturali est satisfacere pro nostra possibilitate fratribus ordinis nostri nos rogantibus ex pluribus iam precedentibus annis, ut talem librum de physicis eis componeremus, in quo et scientiam naturalem perfectam haberent et ex quo libros Aristotelis competenter intelligere possent.” Alberti Magni Physica, Pars 1, Libri 1–4, ed. Paulus Hassfeld (Münster, 1987), 1.1.1 (“Our purpose in natural science is to satisfy as far as we can those brethren of our order who for many years now have begged us to complete for them a book on physics in which they might have a complete exposition of natural science and from which they might be able to understand correctly the works of Aristotle.” [Trans. Ashley, R. M. in “St. Albert and the Nature of Natural Science,” Albertus Magnus and the Natural Sciences, ed. Weisheipl, James A. O.P. (Toronto, 1980), 78.]) The relationship between Dante and Albertus Magnus has been studied extensively by such Dantists as Paget Toynbee and Bruno Nardi. For Toynbee, see Dante Studies and Researches (London, 1902), 42–47, 88–89. For Nardi, see his Dante e la cultura medievale (n. 16 above), passim. See also Mahoney, E. P., “Albert and the Studio Patavino,” in Weisheipl, Albertus Magnus, 541, 562.Google Scholar

39 Physica 1.2.1, 17: “Fuerunt autem duo male quadrantes circulum, quorum unus fuit Brisso, qui quadravit ex principiis geometriae incidendo circumferentiam in quattuor portiones aequales et inveniens quadratum aequale illis portionibus et putabat sequi, quod illud quadratum aequale esset circulo. Descisiones autem circuli portionum vocavit lunulas, eo quod sunt similes lunae semiplenae. Non sequitur autem quod Brisso dixit, quia portiones non consumunt totum circulum. licet consumant totam circumferentiam, et ideo non sequitur, si quadratum est aequale portionibus, quod sit aequale circulo.”Google Scholar

40 Tozer, H. F., An English Commentary on Dante's Divina Commedia (Oxford, 1901), comm. on Par. 13.125.Google Scholar

41 Physica 1.2.1, 17: “Est autem artificium Brissonis, quod ipse convertit cordas portionum in arcus et postea accepit lineas aequales arcubus et ex illis composuit quadratum et putavit esse probatum, quod ex quo latera illius quadrati sunt aequalia circumferentiae, quod totum quadratum esset aequale circuli toti. Et hoc nihil sequitur propter diversam proportionem cordae ad arcum et quadrati ad circulum…. Et fuit Brisso subtilis, sicut patuit ex sua probatione, quoniam valde est latentis deceptionis.”Google Scholar

42 L'Ottimo (n. 31 above), comm. on Par. 13.124–125.Google Scholar

43 Benvenuto (n. 32 above), comm. on Par. 13.124–126.Google Scholar

44 Lombardi (n. 23 above) writes: “Brisso, filosofo antichissimo, di cui fa menzione Aristotile nel 1 Libro Posteriorum Analyticorum, al capo 9 dove si rapporta e si biasima la sua maniera di provare la quadratura” (comm. on Par. 13.125). The same phrase is used by Portirelli (n. 33 above).Google Scholar

45 Par. 13.101–102: “o se del mezzo cerchio far si puote / trïangol sì ch'un retto non avesse.” This is of course not to say that there is anything wrong with solving mathematical problems as such. Rather, it is wrong for Solomon and for Bryson in different ways. For Solomon it is a question of the kind of knowledge appropriate to a ruler; for Bryson it is a question of solving mathematical problems by illegitimate means. Thus Solomon, in not spending his time solving such problems, provides the antitype to Bryson. For a discussion of the wisdom of Solomon as it relates to the various kinds of knowledge in the circle of the sun, see Etienne Gilson, Dante the Philosopher, trans. David Moore (London, 1948), 253–57. An interesting “gloss” on Solomon's wisdom as it is understood in the Commedia can be found in a late fifteenth-century fresco cycle in the Palazzo Publico of Lucignano, near Arezzo. Among its full-length depictions of heroes from antiquity, the figure of Aristotle—holding an open book that states that Prudence is the road of right conduct in life—stands directly above the figure of Solomon as an exemplum of the virtue of Prudence.Google Scholar

46 For a comprehensive study of these cantos, see Teodolinda Barolini, “Dante's Heaven of the Sun as a Meditation on Narrative,” Lettere Italiane 40 (1988): 3–36.Google Scholar

47 The idea of commensurability is of course an important mathematical concept. In Definition 1 of Book 10 of the Elements, Euclid defines commensurable magnitudes: “Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.” Commensurability is a relation between like magnitudes, that is, between two lengths or two areas or two volumes. Two line segments A and B, are commensurable if there is a third line segment C that measures each. C measures A if A equals a whole number multiple of C in length. One imagines marking off a whole number of copies of C along A and precisely exhausting A with these copies of C. For example, let A be a segment of length 3⅓ inches and B a segment of 5¼ inches. A segment C of length inch will measure both A and B. A is 40 copies of C while B is 63 copies of C. Thus A and B are commensurable. Two segments are incommensurable if no such common measure exists. The diagonal and the side of a given square are incommensurable magnitudes. The discovery of this fact is usually attributed to the Pythagoreans of the fifth century B.C. The proof of incommensurability almost always involves a reductio ad absurdum, assuming the existence of a common measure and deriving a contradiction. In modern terms, two segments are commensurable (incommensurable) if the ratio of the lengths is a rational (irrational) number. From this one can see how commensurability provides a vocabulary for talking about the mystery of the Incarnation: what common measure can there be between the human and the Divine?Google Scholar

48 This point is argued by Piero Boitani, “The Sibyl's Leaves: Reading Paradiso 33,” in The Tragic and the Sublime in Medieval Literature (Cambridge, 1989), 223–49 at 238. The essay is on the whole an illuminating reading of the final canto, but the evidence that Boitani cites to make this particular point is at best inconclusive. He cites Peter Dronke's analysis of Alan of Lisle's “Rhythmus de Incarnatione,” a poem about the Incarnation in which Alan does indeed talk about the failure of the geometer, and about the relationship between circle and square. From this, Dronke concludes, not unreasonably, that a direct link between Alan and Dante “seems extremely probable.” But to say that Alan is an extremely probable source for Dante is not to say that the problem of squaring the circle is in fact used by Alan himself as an analogue to the mystery of the Incarnation. This is nowhere explicit in the lines that Dronke quotes, nor is it a claim that Dronke himself makes. (Peter Dronke, Fabula: Explorations into the Uses of Myth in Medieval Platonism [Leiden/Cologne, 1974], 152–53.) The lines of Alan are as follows:Google Scholar

Suae Artes in censura

Geometra fallitur,

Dum immensus sub mensura

terrenorum sistitur;

In directum curvatura

Circuli convertitur,

Sphaeram claudit quadratura

Et sub ipsa clauditur.

(ed. d'Alvernay, Mélanges H. de Lubac [Paris, 1964], 2.126–28.)

We would argue that these mathematical ideas are surely present in a tradition that Dante drew from. Indeed Dronke's point in his analysis of Alan is to show the wealth of geometric conceits that he uses (151), and this strengthens our implicit argument about the continuity of the Platonic mathematical tradition. But the available evidence suggests that the direct connection between squaring the circle and the Incarnation is Dante's.Google Scholar

49 La Divina Commedia di Dante Alighieri commentata da Manfredi Porena (Bologna, 1946–48), comm. on Par. 33.133–138: “… esatto rapporto tra il raggio del cerchio e il lato del quadrato equivalente. Questo è il principio di cui il geometra indige.”Google Scholar

50 “A single moment makes for me greater oblivion than five and twenty centuries have wrought upon the enterprise that made Neptune wonder at the shadow of the Argo.”Google Scholar

51 De Consolatione Philosophiae 2.8.28–30: Boethius, Tractates, De Consolatione Philosophiae, trans. Stewart, H. F., et al., (Cambridge, Mass., 1973), 226. Fallani, whose observations about the relationship between microcosm and macrocosm at the end of the poem are apt and insightful, makes the connection between the Consolation and the end of the Paradiso (La Divina Commedia a cura di Giovanni Fallani [Messina/Florence, 1965], comm. on Par. 33.145). The connection is also noted by Peter Dronke, “L'Amor Che Move Il Sole e L'Altre Stelle,” in The Medieval Poet and His World (Rome, 1984), 439–75 at 439. In this essay Dronke traces the “origins and development of the theme of cosmic love” (441) to its Boethian and Aristotelian strains.Google Scholar

52 Torraca (1905) brings together relevant material from Thomas on the meaning of desire and will and their relationship (La Divina Commedia di Dante Alighieri nuovamente commentata da Francesco Torraca, 4th ed. [Milan, 1920], comm. on Par. 33.145).Google Scholar

53 La Divina Commedia commentata da Isidoro del Lungo (Firenze, 1926), comm. on Par. 33.145. See also La Divina Commedia commentata da Attilio Momigliano (Florence, 1946–51), comm. on Par. 33.145.Google Scholar

54 Par. 10.7–21: “Lift then your sight with me, reader, to the lofty wheels, straight to that part where the one motion strikes the other; and amorously there begin to gaze upon that master's art who within himself so loves it that his eye never turns from it. See how from there the oblique circle which bears the planets branches off, to satisfy the world which calls on them: and were their pathways not aslant, much virtue in the heavens would be vain, and well-nigh every potency dead here below; and if it parted further or less far from the straight course, much of the order of the world, both above and below, would be defective.”Google Scholar

55 It is a complementarity at oblique angles, with circles crossing each other.Google Scholar

56 Par. 14.28–33: “That One and Two and Three which ever lives, and ever reigns in Three and Two and One, uncircumscribed, and circumscribing all things, was thrice sung by each of those spirits with such a melody as would be adequate reward for every merit.”Google Scholar

57 Chiampi, “From Number to Mysterium” (n. 1 above), 255, 270. Though, as we have already observed, this study deals only glancingly with the image of squaring the circle in itself (265, 271), it does deal specifically with ways in which the geometry of the poem looks forward to the image of squaring the circle, as these images “make it clear that in some way it [squaring the circle] will be less an isolated image than the culmination of an encounter with perfect human being” (271).Google Scholar

58 We are told as much at the beginning of the circle of the sun, when the poet accounts for all that moves in mind or space as an outpouring of the love which is the relationship, the complementarity, between the Father and the Son (Par. 10.1–6). Thus, in another kind of complementarity, the end of the circle of the sun circles back to the beginning. For a discussion of the harmony of the Trinity as a subject in the circle of the sun, see Kenelm Foster, “The Celebration of Order: Paradiso 10,” in The Two Dantes and Other Essays (Cambridge, Mass., 1977), 120–36.Google Scholar

59 Freccero, “Final Image” (n. 1 above), 245–57.Google Scholar

60 Timaeus 37c–38c.Google Scholar

61 Timaeus 47a (Cornford [n. 13 above], 44): “Sight, then, in my judgment is the cause of the highest benefits to us in that no word of our present discourse about the universe could ever have been spoken had we never seen stars, sun, and sky. But as it is, the sight of day and night, of months and the revolving years, of equinox and solstice, has caused the invention of number and bestowed on us the notion of time and the study of the nature of the world; whence we have derived all philosophy, than which no greater boon has ever come or ever shall come to mortal man as a gift from heaven.”Google Scholar

62 The Almagest. trans. Catesby, R. Taliaferro, in Great Books of the Western World (Chicago, 1952), 16: 12. Alfraganus, the ninth-century Arabic commentator on Ptolemy, whose work was translated into Latin in 1142 (Alfragani Elementa Astronomica, or, alternatively, Liber de Aggregatione Scientiae Stellarum), is mentioned twice in the Convivio directly: 2.12.11 and 2.4.16. In addition to these two direct references, Dante is indebted to Alfraganus and draws on him as an authority for a great many of the astronomical ideas of the Convivio. According to Paget Toynbee, “Dante was evidently familiar with the Elementa Astronomica and studied it closely …” (A Dictionary of Proper Names and Notable Matters in the Works of Dante [Oxford, 1898], 25–26). See also Paget Toynbee, “Dante's Obligations to Alfraganus,” Romania 24 (1895): 413–32.Google Scholar

63 Sacrobosco [John of Holywood], The Sphere of Sacrobosco and Its Commentators, ed. and trans. Lynn Thorndike (Chicago, 1949), 119, 123, 125. On Dante's knowledge of Sacrobosco, Orr, M. A. has written: “It is rather curious that there is no trace in Dante's writing of acquaintance with the great Roger Bacon, or with Sacrobosco, especially as Brunetto Latini knew the former personally, and Cecco d'Ascoli wrote a commentary on the latter, but Dante may have known them…. Those we have described elsewhere [Alfraganus and Aristotle] would, however, be enough to supply him with all the astronomical data, except one, which we find in his writings” (Dante and the Early Astronomers [London, 1913], 161).Google Scholar

64 Schnapp, Jeffrey, The Transfiguration of History at the Center of Dante's Paradiso (Princeton, 1986), 77. Here, Schnapp is quoting Hugo Rahner, “The Christian Mysteries and the Pagan Mysteries,” in The Mysteries: Papers from the Eranos Yearbooks, ed. Joseph Campbell (Princeton, 1971), 372. Among other observations relevant to this study, Schnapp shows that in the circle of Mars Dante wishes “to emphasize the cross's all-comprehending embrace of the universe, its centrality and its stabilizing role, the figurative act of binding it performs on the great scroll of creation” (78).Google Scholar

65 Timaeus, 47c (Cornford, 45): “The god invented and gave us vision in order that we might observe the circuits of intelligence and profit from them for the revolutions of our own thought, which are akin to them, though ours be troubled and they are unperturbed; and that, by learning to know them and acquiring the power to compute them rightly according to nature, we might reproduce the perfectly unerring revolutions of the god and reduce to settled order the wandering motions in ourselves.”Google Scholar

66 Freccero, “The Firm Foot on a Journey Without a Guide,” in Dante: The Poetics of Conversion, ed. Rachel Jacoff (Cambridge, Mass., 1986), 30.Google Scholar

67 Chiampi, “From Number to Mysterium” (n. 1 above), 258.Google Scholar

68 Freccero, “Firm Foot,” 30 ff.Google Scholar

69 The commentary of Bosco/Reggio (n. 34 above) draws attention to this citation, in emphasizing the Incarnational aspects of the end of the poem (comm. on Par. 33.145). Seeing the poem as beginning and ending with the figure of Paul is of course no gratuitous interpolation on our part. Paul is explicitly present in the beginning of the Commedia as a model for the pilgrim's journey, the pilgrim who ironically asserts, “I am no Aeneas, I am no Paul” (“Io non Enea, io non Paulo sono.” Inf. 2.32), even as he is beginning the process of learning exactly how much he is like Paul. And the final consummation, the vision at the end which is the subject of our analysis surely has as its most important exemplar the Paul of 2 Corinthians, who speaks of “a one caught up to the third heaven…. That was caught into Paradise and heard secret words, which it is not granted to man to utter” (2 Cor. 2–4). An informative, thorough, and lucid discussion of Paul's role throughout the Commedia is that of Rachel Jacoff and Stephany, William A., Lectura Dantis Americana: Inferno 2 (Philadelphia, 1989), 57–83.Google Scholar

70 Confessor, Maximus, Chapters on Knowledge. 2.21 and 2.25, in Selected Writings, trans. Berthold, George C. (New York, 1985), 152–53. Maximus is an especially appropriate example to use in that he was the key figure in insuring the orthodoxy of Pseudo-Dionysius, by interpreting the Dionysian corpus in more clearly Christocentric terms. As Jaroslav Pelikan has written in the introduction to this volume, he was responsible for a “restoration of a balance between Neo-Platonism and Christian Orthodoxy” (7). That balance is of course what we would likewise claim for the ending of the Paradiso. Google Scholar

71 La Divina Commedia di Dante Alighieri, riveduta nel testo e commentata da Scartazzini G. A., 2nd ed. (Leipzig, 1900), comm. on Par. 33.145. See also the commentary of Momigliano (n. 53 above), which describes the end in terms of a return to the initial motif of “La gloria di colui” of Par. 1 (comm. on Par. 33.145).Google Scholar

72 Par. 1.1–3: “The glory of the All-Mover penetrates through the universe and reglows in one part more and in another less.”Google Scholar

73 Fratris Johannis de Serravalle Ord. Min. Episcopi et Principus Firmani Translatio et Comentum totius libri Dantis Aldigherii, cum textu italico Fratris Bartholomaei a Colle eiusdem Ordinis, nunc primum editum (Prato, 1891), comm. on Par. 33.133–145. “Nam qui facit circulum cum circino, si incipit ab uno puncto et sic circulariter movendo et trahendo circinum, sive sextum, redit ad illum eumdem punctum, a quo incepit, circulus est perfectus, et non ante.”Google Scholar

74 Ibid.: “Nota etiam quod quando anima humana fit, a die primo suo esse, quod est punctus, incipit circulum, qui tam diu gyret, quousque homo vivit; et si sic recedens a Deo per creationem, adversatur ipsi Deo per peccatum, errat nimis. Dicente Beato Augustino super Genesim ad litteram: Creata intellectualis substantia, adversa Deo, scilicet per peccatum, fluctuat informiter; formatur autem per conversionem ad allud ineffabile, bonum, lumen, quod est Verbum Dei in excelsis. Quod considerans auctor, se a Deo processisse per creationem, [et quod] peccavit tamen, ideo ivit ad Infernum, videns ibi tot mala, pervenit ad Purgatorium, ubi peccatores purgantur a peccatis, et fuit per purgationem factus dispositus venire ad Deum”Google Scholar

75 Ibid., comm. on Par. 33.133–138: “At length he came through the spheres of heaven to this place, that is, to divinity and to the highest good, and thus he completed his circle.”Google Scholar

76 This point is convincingly made in various ways by Chiampi, “From Number to Mysterium” (n. 1 above), passim.Google Scholar

77 Par. 30.89–90: “così mi parve / di sua lunghezza divenuta tonda.” See Jacoff and Stephany, Lectura Dantis Americana: Inferno 2, 54 n. 64: “The Jordan, which, as we have seen, was associated both with the rivers of Paradise and with Oceanus, implies the double linear/circular form of the rivers of Purgatory, as well as the poem's other important fiumana, the river of light in Paradiso 30….” For the river of light, see also Albert Rossi, “ ‘A l'ultimo suo’: Paradiso 30 and its Virgilian Context,” Studies in Medieval and Renaissance History, n.s. 4 (1981): 29–88. Jacoff and Stephany also have some very suggestive observations about the relationship between what they term external and affective motion. Their discussion of the relationship of one to the other in their analysis of Inferno 2 is charged with implications for the entire poem.Google Scholar

78 In addition to seeing the ending of the poem as an explicit and self-conscious fulfillment of what was adumbrated in Par. 1.1–3 with respect to cosmology, the ending of the poem also brings to a fitting conclusion the inexpressibility motif—the inability of the poet to capture in words an experience that is beyond words—which is also an explicit and crucial concern at the beginning of the Paradiso. Among other commentators, Momigliano (n. 53 above) describes the return of the poem to these two initial motifs.Google Scholar

79 As the discussion of the Phaedrus, above, reminds us, the verbal analogue to squaring the circle by the correct use of mathematical principles is the correct use of memory. A proof that Dante has “constructed” his poem according to correct principles is that he has correctly constructed it as the book of memory.Google Scholar