1
Ficino's Commentary on the Eighth Book of the Republic

"Ultima Cumaei venit iam carminis aetas"

In the notable nineteenth expostulation in his Devotions , John Donne refers to God as a metaphorical God; and the Renaissance in general was enthusiastically attuned to the assumption that the world was itself a figure, a cipher. Necessarily the mathematical structures in the world were part of the divine figuration, and a sense of this figuration provided the foundation for both the methods and the goals of such learned disciplines as arithmology and numerology, astrology, iatromathematics, and musical therapy, the mathematical or at least computational arts that the age regarded as legitimate branches of learning and of proven utility. For the influential book of the Apocrypha known as the Wisdom of Solomon had proclaimed in a much-quoted text that God had made all things "in number, weight, and measure" (11.20[21]) as the architect of the world, as the heavenly geometer, as the musical master of a divine harmonics. And man in the divine image of God the Creator had been designed with a body of geometrical proportions, with a harmoniously balanced temperament, with a mathematical mind. The supreme ancient authority of this mathematical view of man as mathematician was Plato, spokesman for what was preeminently the Pythagorean tradition in which his own scientific studies had been nurtured.

Renaissance scholars were familiar with the report that the inscrip-

tion in the vestibule of the Academy had forbidden anyone unskilled in geometry to cross the threshold and seek initiation into the sacred mysteries.^[1] For geometry was a marvelous art that the Epinomis 990D had claimed was of divine not human origin, even though, as the Republic had argued at 6.511B ff. and 7.531D–534E, it was subordinate, like all its "sister" mathematical arts, to the "comprehensive" power of dialectic, "the coping stone" of the intellectual skills. Scholars were also aware that in the Timaeus , the dialogue on the Demiurge and his creation and the one most familiar to and most treasured by the medieval and the Renaissance West, Plato had advanced various Pythagorean notions—with what degree of seriousness it is now virtually impossible to say—on the harmonies governing the soul, and on the structure of the elements and the geometrical figures that constituted them.^[2] Although none of Plato's dialogues focus primarily on mathematics, several do contain significant loci mathematici . Apart from the Timaeus with its exceptionally important sections on means and proportions at 34B–36D and on the five regular polyhedra at 53C–56C, the Meno has two well-known passages on the duplication of the square at 82B–85B and on the measurement of areas at 86E–87B, the Theaetetus raises the issue of irrational or incommensurable roots at 147D–148B, and the Epinomis (which the Renaissance considered authentic) has an arresting section at 990C–991A on astronomy, geometry, progressions, the mean proportions, and the formation of numbers. Other dialogues contain mathematical references or observations: for instance, the Euthyphro at 12D, the Hippias Major at 303BC, the Philebus at 56D, the Charmides at 166A, the Statesman at 266AB, the Phaedrus at 274C, and the Laws 7 at 817E–820C.^[3]

[1] H. D. Saffrey, "Ageômetrêtos mêdeis eisitô: Une inscription légendaire," Revue des études grecques 81 (1968), 67–87, traces the evidence for the inscription (which is possibly apocryphal) back to a reference in an oration by the emperor Julian and to another in a scholion on Aelius Aristides, both from the mid fourth century A.D. ; see also D. H. Fowler, The Mathematics of Plato's Academy: A New Reconstruction (Oxford, 1987), pp. 197–202. The inscription was familiar to Ficino, however, from a number of later sources, including perhaps Philoponus's commentary on Aristotle's De Anima (Ioannis Philoponi in Aristotelis de Anima Libros Commentaria 1.3.406b25 ff., ed. Michael Hayduck, Commentaria in Aristotelem Graeca, vol. 15 [Berlin, 1897], p. 117).

[2] The Timaeus was known to the Latin West principally by way of Calcidius's Commentary. The authoritative modern edition is by J. H. Waszink, Timaeus a Calcidio Translatus Commentarioque Instructus (London and Leiden, 1962; 2d ed., 1975)—volume 4 in the series Corpus Platonicum Medii Aevi ; but see also that by Johannes Wrobel, Platonis Timaeus Interprete Chalcidio cum Eiusdem Commentario (Leipzig, 1876). For Plato's relationship to the Pythagoreans, see Cornelia J. de Vogel, Pythagoras and Early Pythagoreanism (Assen, 1966), pp. 192–207.

[3] For these and other references, see Paul-Henri Michel, De Pythagore à Euclide:

Contribution à l'histoire des mathématiques préeuclidiennes (Paris, 1950), pp. 75–76, 500–511.

More generally, the Parmenides is concerned throughout with the metaphysics of the one and the many, of unity and plurality; and the Republic 7.521C–531D outlines a mathematics curriculum in five parts beginning with arithmetic and ratio theory and thence proceeding to plane and solid geometry, and ending with astronomy and music. Finally, there are the complicated metaphysical issues of Plato's postulation, at least according to Aristotle in his Metaphysics 991b9, 1082b23–24, 1086a10–11, and De Anima 404b24, etc., of numbers as Forms, of the mathematicals as intelligible pluralities.

However, the most intractable or mystagogical of all Plato's mathematical speculations (depending on one's point of view) occurs in a passage towards the beginning of the eighth book of the Republic at 546A ff. Here Socrates refers to a mysterious geometric or "fatal" number in order to explain why it is that even perfectly constituted republics—those that do not contain within themselves the seeds of their own decay and ruin—decline nevertheless after the passage of many years into the first of four degenerate forms ending in a tyranny: into a contentious timarchy governed by the passionate pursuit of honor and "a fierce secret longing" for money instead of justice and the good. They are subject, it would seem, to some cyclical cosmic pattern, to an inexorable fate that overwhelms them despite their innate, their Platonic excellence. In the course of this baffling passage on the geometric number Socrates also argues for the necessity of state-planned eugenics. Citizens approaching parenthood must be adjusted to each other, like proportionate numbers, in order that they may breed good, tempered offspring and thus ensure the continuance of balance in the state. And the balance can indeed be maintained for a time: with Platonic planning and Platonic virtue men can work with Fate to ensure the continuance of their state's life or prosperity, as long, that is, as the fatal cycle of years has not yet been fulfilled. After that, no legislation by the magistrates, however wise and however rigorously enforced, can prevail against the inevitable, the periodic change. The eugenic theme is so prominent indeed that Plutarch, Nicomachus of Gerasa, Iamblichus, and Boethius, among others, did not hesitate to identify the fatal geometric number with the notion of a "nuptial" number,^[4] presumably because of the sovereign role it plays

[4] Plutarch, De Iside 56 and De Animae Procreatione 10 (Moralia 373F, 1017C); Nicomachus of Gerasa, Introductio 2.24.11; Iamblichus, In Nicomachi Arith. Introd. (ed. Pistelli, 82.20 ff.); and Boethius, De Institutione Arithmetica 2.46 (full citations of

these editions of Nicomachus, Iamblichus, and Boethius are provided below in nn. 74, 76, and 78 respectively).

J. Dupuis, ed. and trans., Théon de Smyrne, philosophe platonicien: Exposition des connaissances mathématiques utiles pour la lecture de Platon (Paris, 1892), p. 388, regards their identification of the nuptial number with the geometric as "impropre," while Auguste Diès, "Le nombre de Platon: Essai d'exégèse et d'histoire," Mémoires présentés par divers savants à l'Académie des inscriptions et belles-lettres (Paris), 14.1 (1940), 9, says that the commentators were correct, "la tradition n'a pas eu tort de lui donner ce nom."

in determining, for better or for worse, the fertility of a republic and thus the success of its marriages, begettings, and births.

Of particular importance for Platonic commentators is the fact that Aristotle commented upon this passage adversely in his Politics 5 at 1316a1–b26 in an arresting discussion and dismissal of Socrates' views on the causes of change affecting a perfect commonwealth, such as the hypothetical first state. Aristotle objects that Socrates "treats of revolutions, but not well, for he mentions no cause of change which peculiarly affects the first or perfect state. He only says that the cause is that nothing is abiding, but all things change in a certain cycle; and that the origin of the change consists in those numbers 'of which 4 and 3, married with 5, furnish two harmonies' (he means when the number of this figure becomes solid)."^[5] Aristotle is prepared to admit that at times nature may produce bad men who will not submit to education, "in which latter particular he [Socrates] may very likely be not far wrong, for there may well be some men who cannot be educated and made virtuous." Aristotle, who is insisting on the distinction between the "cause" of change and its actual "onset," then raises various objections, among them the following five: Why is "such a cause of change peculiar to his [Socrates'] ideal state, and not rather common to all states, or indeed, to everything which comes into being at all?" Is it merely attributable to the agency of time that "things which did not begin together change together?" Why postulate cyclical change and not merely change, since history furnishes us with many examples of one tyranny passing into another tyranny, not necessarily into another form of government entirely? Isn't it foolish to suppose that a state changes for the worse only because the ruling class begins to acquire

[5] "archên d' einai toutôn, hôn epitritos puthmên pempadi susugeis duo harmonias parechetai, legôn hotan ho tou diagrammatos arithmos toutou genêtai stereos " (1316a4–9). This text has engendered almost as much controversy as Plato's own. Saint Thomas Aquinas himself (or his continuator) complained of its being brief to the point of obscurity (In Arist. Pol. lib. 5, lect. 13). See W. L. Newman, The Politics of Aristotle, 4 vols. (Oxford, 1887–1902), 4:481–482; and Diès, Ess ai , pp. 12 and 60. Cf. nn. 10, 15 below. I am quoting from Benjamin Jowett's translation.

too much money? The causes of change are numerous, and yet Socrates mentions only one—the gradual impoverishment of the citizens—as if the citizens had been originally all equally well off. And why speak of revolutions in oligarchies and democracies, as though they each existed in only one form when in fact they exist in many forms?

In short, Aristotle marshals a sequence of powerful objections that charges Socrates with confusing the notion of a temporal cycle with that of temporal change and dismisses his conception of a historical cause as too naive or too simplistic. To anyone who believed in Plato's supremacy over Aristotle, or who was bent upon reconciling the two thinkers, these objections presented a formidable challenge, particularly given Aristotle's belligerent tone, his taking issue with an indisputably major dialogue, and his contentious impatience with the way Socrates had elected to present an important and influential Platonic theme, that of the ideal republic.

The mathematical enigmas in Plato's passage—along with Aristotle's objections—have occasioned speculative debate and intricate analysis since the fifteenth century when they were first rediscovered by the West. A number of "solutions" have been and are still being suggested, and translators have learned to approach Plato's veiled description of the geometric number with some wariness. In the past some have even declined to render it at all. One of the most distinguished of these was Victor Cousin (1792–1867), who footnoted his omission thus: "Ce qui me confond le plus dans cette phrase, d'une obscurité devenue proverbiale, c'est qu'elle n'ait pas plus tourmenté les philosophes grecs, venus après Platon, et qu'ils la citent, la critiquent, la commentent, en n'ayant pas l'air de n'y rien comprendre. . . . [J]e demeure très convaincu qu'une phrase écrite par Platon et commentée par Aristote, est fort intelligible en elle-même."^[6] Cousin assumed that an enhanced understanding of ancient mathematics and its terminology would assuredly lead to the untying of what he thought

[6] OEuvres de Platon 10:324, cited by Dupuis, Théon , p. 370. See also Dupuis's earlier study, Le nombre géométrique de Platon (Paris, 1881), pp. 11–12. On p. 8 of the latter Dupuis also adduces the refusal of Humblot, who edited an anonymous translation of the Republic in 1762, to render the mystery of the geometric number. When he came to the famous crux, Humblot substituted the clause, "dont il est inutile de vous expliquer le mystère parce qu'il est au-dessus de votre portée," and footnoted it thus: "Ici est le fameux nombre de Platon, que je n'ai point traduit, parce que je ne l'entends pas; je crois même qu'il est inutile de vouloir se rompre la tête à l'expliquer, personne n'ayant pu le faire avec succés jusqu'à présent."

of as "ce noeud embarrassé." The great Friedrich Schleiermacher before him had declared in 1828 that his inability to understand Plato's intentions here and his continually renewed and continually thwarted hopes of doing so had interrupted his work on translating the canon for twelve entire years.^[7] Eventually he had reluctantly decided that the value of the geometric number must be 216 (or its square), the product of 8 times 27, the first two "solids" at the two feet of the Platonic lambda as set forth in the Timaeus 35B ff., a text with a special role to play, as we shall see, in the launching of the modern, as well as the ancient, history of the number's interpretation. In our own day another great scholar, Francis M. Cornford, omitted the passage in his 1941 translation of the Republic .

The path of interpretation, moreover, is strewn with failures to calculate the value of this number convincingly for others, though most of these failures are themselves remarkable for their learning and ingenuity. The two preeminent twentieth-century interpreters are James Adam and Auguste Diès;^[8] and a shaky consensus arrived at by them and by other scholars has established 12,960,000 as the value that Plato may have had in mind.^[9] Even so, discussion continues.^[10]

[7] Dupuis, Théon , pp. 369, 394–395; Nombre , pp.10–11; and Diès, Essai , p. 87.

[8] James Adam, The Republic of Plato , 2 vols. (Cambridge, 1902), 2d ed. by D. A. Rees, 2 vols. (Cambridge, 1963), 2:201–209 (commentary), 264–312 (appendix); and Diès, Essai , pp. 5–9. For the preeminence of their solutions, see the note by Desmond Lee (H. D. P. Lee) to his newly revised translation of the Republic , 3d ed. (Harmondsworth: Penguin, 1987), pp. 360–361.

[10] For further references to the interpretative history of the geometric number

since the mid nineteenth century, see Adam, Republic , pp. 264–265, and especially Diès, Essai , chapter 4 (chapter 2 deals with the ancient interpretation tradition, and chapter 3 with the period from Faber to Carl Ernst Schneider, i.e., from 1506 to 1833). For more recent views, see Robert S. Brumbaugh, Plato's Mathematical Imagination (Bloomington, Ind., 1954), pp. 107–150; Rees's introduction to Adam, Republic l:xlviii–l; and W. K. C. Guthrie, A History of Greek Philosophy , vol. 4 (Cambridge, 1975), p. 529n.

The first modern contributor to the problem of Plato's geometric number, though he has not hitherto received appropriate recognition as we shall see, was Marsilio Ficino (1433–1499), the leading Florentine Neoplatonist of the Renaissance and the architect of Platonism's revival and European dissemination. His most formidable scholarly achievements were undoubtedly his Latin translations of the complete works of both Plato (Florence, 1484; 2d ed., Venice, 1491) and Plotinus (Florence, 1492); and he was recognized in his own age as the supreme interpreter and commentator on Plato. In 1576, nearly eighty years after Ficino's death, Jean Bodin for instance in his Les six livres de la République 4.2 refers to him as "(in mine opinion) the sharpest of all the Academikes."^[11] Not surprisingly then, the distinguished Florentine attracted the attention of J. Dupuis in a review of earlier attempts to decipher Plato's enigma that he included in an 1881 monograph, a monograph he subsequently revised and appended to his 1892 edition and French translation of Theon of Smyrna's Expositio .^[12]

Following in the footsteps of the great nineteenth-century editor of the Republic , Carl Ernst Christopher Schneider,^[13] Dupuis commences his doxology of post-ancient views with Ficino, "le plus ancien interpréte de Platon parmi les modernes." But both merely recall a passing remark in Ficino's argumentum for book 8 as it appeared in his 1484 and 1491 Plato editions—Dupuis uses the 1491—"Quid vero si in eiusmodi verbis plus difficultatis sit quam ponderis"; and this they take

[11] I quote from the lively 1606 translation of Jean Bodin by Richard Knowles, The Six Bookes of a Commonweale , facsimile ed. by Kenneth Douglas McRae (Cambridge, Mass., 1962), p. 458A (misprint for F).

[12] Nombre , pp. 4–16; Théon , pp. 388–399. The monograph itself went through several intermediate revisions—see Dupuis's own bibliographical note, Théon , p. 365n. The 1892 version begins not with Ficino but with Barozzi's 1566 treatise. For Theon's Expositio itself, see pp. 31–33 below.

[13] His edition of the dialogue is entitled, misleadingly, Platonis Opera Graece and consists of 3 vols. in 2 (Leipzig, 1830–1831). In two Commentationes written in 1821 but incorporated into the 1831 praefatio for book 8 of the Republic in the third volume of this edition (i.e., at 3:ii–lxxi), Schneider deals at length with Iacobus Faber Stapulensis (Jacques Lefévre d'Étaples), with Barozzi, and with others, though in an odd order.

to mean that for Ficino the passage contained "more of difficulty than of real substance." Schneider assumes that Ficino never followed through on his promise to write more fully on the matter in his Timaeus Commentary; and Dupuis concludes, "il n'indique aucun nombre."^[14] Interestingly, this joint dismissal merely echoes a comment made in 1581 by Jean Bodin: "Marsilius Ficinus . . . plainely confesseth himself not to know what Plato in that place ment, fearing lest it should so fall out with him as it did with Iamblichus, who seemeth to have been willing in three words not to have manifested a thing of it selfe most obscure, but rather to have made it darker."^[15] Bodin had already followed Ficino in his argumentum in mockingly observing that Aristotle "skippeth over this place as over a dich, neither doth here carpe his maister (as his maner is) when as for the obscuritie thereof he had not wherefore he might reprove him."^[16]

Ficino's argumentum , upon which these assumptions of Bodin, of Schneider, and of Dupuis are based, is not without interest. It is one of a number of prefatory argumenta or epitomes that Ficino prepared for each book of the Republic and the Laws and for the other dialogues. They were first published in his 1484 Plato edition and continued to appear in later editions of it and also in the three editions of

[14] Schneider, Platonis Opera Graece 3:iiii; Dupuis, Nombre , p. 5. Dupuis attributes a sentence to Ficino, incidentally, that is not in, and not like anything in, the argumentum !

[15] Again Knowles's translation, p. 458FG. Dupuis, Théon , p. 391, cites the French from Bodin's 1581 version in his Apologie de René Herpin , which was added to the 1583 French edition of Les six livres de la République (first published in 1576), f. 41 verso:

Marsille [sic ] Ficin, le plus grand Platonicien qui ait escrit, confesse qu'il ny entend rien, et non sans cause Ciceron disoit qu'il n'y avoit rien plus difficile que les nombres de Platon. Et Theon Smyrnean, des plus illustres Mathematiciens entre les Academiques, interpretant la Republique de Platon, n'a aucunement touché ce passage. Procle Academicien, ayant doctement interpreté les sept premiers livres de la Republique de Platon, est demeuré a l'huictiesme, où il est question de ces nombres. Et quoy que Jamblique se soit efforcé d'esclaircir ce passage, si est ce qu'il a encores plus obscurcy.

See also Schneider, Platonis Opera Graece 3:liii. In actuality, Bodin was merely following Ficino's review of the ancient authorities in his argumentum .

In his expanded Latin version of 1586 Bodin presents a lengthy exposition of Plato's passage in which, according to Diès, Essai , pp. 59 ff., he attacks the interpretation of Faber Stapulensis without naming him (see n. 34 below). Bodin himself proposed no solution on the grounds that Plato's presentation of it made no sense and that he must have been distracted at the time—"ut planum sit Platonem eo loco sui ipsius oblitum esse" (De Republica Libri Sex Latine ab Autore Redditi [Lyons and Paris, 1586], p. 417)! See Schneider, Platonis Opera Graece 3:xxxxviiii–lviii, and Diès, Essai , p. 56.

[16] Trans. Knowles, p. 457E ("Primus quidem Aristoteles locum hunc quasi vallum transiit, neque, ut solet, magistrum momordit, cum propter obscuritatem non haberet

que parte illum carperet," De Republica Libri Sex , p. 412). See Schneider, Platonis Opera Graece 3:liii; Dupuis, Nombre , p. 7; Diès, Essai , p. 13.

Ficino's own Opera Omnia (where the argumentum for the eighth book appears on p. 1413). Professor Paul Oskar Kristeller has argued convincingly that each argumentum was composed as Ficino completed his translation of the dialogue it was to preface, though the argumenta as a body were probably revised later and further crossreferences added.^[17] If he is correct, then the argumentum for the eighth book would date from the late 1460s, since the book itself is number 38 in the sequence of the dialogues as he translated them (counting each book of the Republic separately) and a draft of the sequence was completed during the rule of Piero, Cosimo de' Medici's son and successor, who did not die until 1469.^[18]

In the argumentum Ficino observes that it was not unjustly that Cicero had written that Plato's fatal number had become proverbial for obscurity—a reference to the Epistle to Atticus 7.13.5—and that Theon of Smyrna, otherwise the principal expounder of Platonic mathematics, had very astutely decided to omit all consideration of the number in his Expositio on the grounds that Plato's mystery was "inexplicable."

As the champion of Plato, Ficino has as his immediate goal, however, to refute Aristotle's objections to—what he characterizes as "calumnies" against—the views of Socrates concerning the cause of a perfectly constituted state's ultimate decline, the state that Ficino interprets Socrates as having already fully described in the first seven books of the Republic . Since this is one of the most prominent instances of disagreement between Plato and Aristotle, it naturally forced itself upon Ficino's attention.^[19]

[17] Supplementum Ficinianum , 2 vols. (Florence, 1937; reprint 1973), 1:cxvi–cxvii, cxlvii if.; see also his "Marsilio Ficino as a Beginning Student of Plato," Scriptorium 20 (1966), 41–54 at 46 ff., in answer to the arguments of Raymond Marcel in Marsile Ficin (1433–1499) (Paris, 1958), pp. 457–458, that Ficino only wrote the argumenta in 1475–1476 after he had finished translating all the dialogues. Marcel was reviving a hypothesis first put forward by Arnaldo della Torre in Storia dell' Accademia Platonica di Firenze (Florence, 1902), pp. 606–607. See also James Hankins, Plato in the Italian Renaissance , 2 vols. (Leiden, 1990), 1:318–321, 2:483–485.

[18] Kristeller, Supplementum 1:cil.

[19] For Ficino the most accessible and authoritative Latin translation of Aristotle's Politics , and the only Quattrocento one, was by Leonardo Bruni, who composed it between 1435 and 1437 (it was published in 1469); see Hans Baron, ed., Leonardo Bruni Aretino: Humanistisch-philosophische Schriften (Leipzig and Berlin, 1928), pp. 143, 175–176; Eugenio Garin, "Le traduzioni umanistiche di Aristotele nel secolo XV," Atti

e memorie dell' Accademia fiorentina di scienze morali "La Columbaria" 16 (1951), 55–104 at 67–68; and Gordon Griffiths in The Humanism of Leonardo Bruni: Selected Texts , translations with introductions, by Gordon Griffiths, James Hankins, and David Thompson (Binghamton, N.Y., 1987), pp. 113–115, also pp. 38, 109, 154–170. A glance at the relevant passage, however, clearly demonstrates that Ficino did not turn to Bruni, at least in this instance:

Inquit enim causam esse mutationis, quia sic natura comparatum sit ut nihil permaneat, sed in ambitu quodam temporis mutationem recipiat. Esse vero principium horum inquit quorum sexquitertium fundum quinario coniugatum duas exhibet harmonias, inquiens, quando numerus diagrammatis huiusmodi efficiatur solidus, utputa natura producente pravos, & meliores disciplina. Hoc ergo forsan inquit non male. (Cited from the 1542 Venice edition of Bruni's translation, f. 139v.)

Similarly, Ficino did not use the standard medieval translation of the Politics by William of Moerbeke (c. 1215–1286), which dates from around 1260 and was made perhaps at the request of Aquinas. See M. Grabmann, Guglielmo di Moerbeke (Rome,

1946, pp. 111–113; and Bernard G. Dod's chapter "Aristoteles Latinus" in The Cambridge History of Later Medieval Philosophy , ed. Norman Kretzmann, Anthony Kenny, and Jan Pinborg, with Eleonore Stump (Cambridge, 1982), pp. 45–79 at 49–50, 62–64, and 78. This version is preserved in some 107 MSS and was edited by F. Susemihl in 1872 for the Teubner series as Aristotelis Politicorum Libri Octo, cum Vetusta Translatione Guilelmi de Moerbeka . The relevant passage reads:

ait enim caussam [sic ] esse id, quod est non manere aliquid, sed in aliqua periodo transmutari, principium autem esse horum, quorum epitritus fundus quinario coniugatus duas harmonias exhibet, dicens, quando numerus diagrammatis huius solidus fuerit, tamquam natura quandoque producente pravos et meliores disciplina, hoc ipsum dicens forte non male. (Pp. 590.9–591.3)

Given Ficino's fierce scrutiny of the passage, we must conclude, therefore, that he probably turned to the Greek text of Aristotle directly and made his own translation from one of the many manuscripts of the work available to him (the first complete edition of the Greek Aristotle was about to issue in five volumes from the Aldine press in 1495–1498, but obviously too late for him to use here)

We might note incidentally that he may have dipped into Donato Acciaiuoli's commentary on the passage, a commentary written in the early 1470s and based on the lectures of Argyropoulos. Acciaiuoli quotes Bruni's Latin version as cited above (I have consulted the Venice 1566 edition Donati Acciaioli in Aristotelis Libros Octo Politicorum Commentarii , fol. 194r-v) and then argues that Aristotle is postulating a cube number resulting "ex sexquitertio et quinario"—which he identifies as nine, the origin of the Platonic great year!—and also that "diagrammatis" means "descriptionis." Still, his analysis is very brief and nothing suggests that Ficino would have found anything particularly useful or illuminating in it. See Hankins, Plato in the Italian Renaissance 1:124; and Arthur Field, The Origins of the Platonic Academy in Florence (Princeton, 1988), pp. 204, 226–229.

He counters the Stagirite's arguments by postulating two kinds of causes of change. The first is specific in that it occasions "the permutations alike of souls and of states from one form to another," the changes particular to an imperfect soul or state. But a perfect soul or state, such as that postulated here by Socrates, cannot be supposed to contain this kind of cause on the Platonic grounds that that which is perfect cannot degenerate. The second kind is a "common" or universal cause of change and it is to be identified, if not with Fate itself, then certainly with the "fatal order" that governs the temporal realm. For change in this realm is brought about by the shifting configurations, the "fatal order" of the celestial spheres and the planetary conjunctions and oppositions. Against the great cycles of Fate and its instrument, the stars, no sublunar form, perfect or imperfect, is immune. While men and states may possess the internal fortitude and virtue to endure for the full duration of their destined, their fatal time on earth, they must succumb eventually to change, not necessarily because of any innate defect—though most sublunar entities have such defects—but because of the universal condition of mutability. Interestingly, Ficino, the son of a physician and himself trained initially as a physician, suggests that we might think of the contrast as that between an endemic and an epidemic disease. Thus Ficino distinguishes between the minor "revolutions" that concern Aristotle and the great cycles of time that concern Plato.

The greatest astronomical cycle is the Platonic "great year," which is defined in the Timaeus 39D as the time it takes for the seven planetary spheres and the sphere of the fixed stars to return to the positions they had occupied at the beginning of the cycle—a "Pythagorean"

conception that can be traced back at least to Oenopides of Chios (fl. c. 450–425 B.C. ).^[20] The Platonists (and Stoics) entertained the corollary speculation that mankind too is governed by its own great year, which they identified as the time when history comes full circle and begins to repeat itself. The obvious question arises whether the two great years—that of the celestial spheres and that of mankind—are coterminous. Plutarch, for instance, had argued that they were in his essay De Fato 3 (Moralia 569A-C). When the heavens are restored to the state they were in at the beginning of the great year, then everything on earth including man will return to its first condition and history begin again; fate is thus both finite and infinite.^[21] But others had

[20] Cf. Timaeus 39D: "the perfect number of time fulfills the perfect year when all the eight revolutions, having their relative degrees of swiftness, are accomplished together and attain their completion at the same time." See Dupuis, Théon , p. 366, citing Theon's Expositio 3.40 (ed. Hiller, p. 198.14 ff.), for which see n. 67 below.

[21] Cf. Calcidius, In Timaeum 148 (ed. Wrobel, p. 206.12–24); see also Dupuis, Théon , pp. 366–367. Apokatastasis , in the sense of the return of the cosmos to its former state—apokatastaseis appears as a bad variant, we might note, for apostaseis in the Republic 8.546B6—is a term of some importance in the Corpus Hermeticum ,

in various Stoic and Neoplatonic treatises, and, given the reference in Acts 3:21, in Christian eschatological commentary, above all in Origen.

contended that the one great year was a multiple of the other. Proclus, for instance, had held that the great year of mankind was a multiple of the cosmic great year, whereas others had argued precisely the opposite.^[22] Moreover, the value of the cosmic great year was variously reckoned. Macrobius, for instance, had calculated it as 15,000 ordinary years,^[23] while the Neoplatonic and Ptolemaic traditions to which Ficino is here subscribing had determined upon 36,000 years.^[24]

In the Republic Plato does not actually say, however, that the period of the cosmic great year is measured by the perfect number or numbers, but declares rather at 546B3–4 that the perfect number presides over the period of "divine begettings." And though Theon of Smyrna for one had assumed that the cosmic great year was governed by a perfect number—in this case six, the first of such numbers—and was therefore indeed a "divine begetting,"^[25] nonetheless we must dis-

[22] Dupuis, Théon , p. 371, cites a passage from Proclus's In Timaeum book 4 (p. 271 in the Basel ed. of 1534). This is to be found in the standard ed. by Ernst Diehl, Procli Diadochi in Platonis Timaeum Commentaria , 3 vols. (Leipzig, 1903–1906), at 3:93.18–94.4, and in the trans. by André-Jean Festugière, Proclus: Commentaire sur le Timée , 5 vols. (Paris, 1966–1969), at 4:122—Proclus alludes, interestingly, to the Platonic Number at 93.23–25. However, Ficino almost certainly did not know Proclus's fourth book, since his exemplar, the Riccardiana's gr. 24, ends in the middle of the third book at the word sômasi (ed. Diehl, 2:169.4; cf. l:xi ff.).

[23] In Somnium Scipionis 2.11.8–13. Cf. Cicero, De Natura Deorum 2.20.51–53—but Cicero does not mention the fixed stars.

[24] See, for instance, Ficino's argumenta for Republic 10 and Laws 6 (Opera Omnia , 2d ed. [Basel, 1576], pp. 1431, 1505): "When he says the twelve parts of the city follow the circuit of the universe, perhaps too he means that the whole circuit of the eighth sphere is perfected in 36,000 years, which number indeed is perfectly completed by having three [periods] of twelve thousand years to the extent that in the first twelve thousand years you may understand from the opinion of the ancients the youthful habit of the whole world, in the second twelve the mature (virilem ) habit, in the third twelve the senile habit. But concerning these matters more opportunely in the Timaeus " (1505). In summa 20 of the Timaeus Commentary (Opera , p. 1468.2) he declares, however, that many think of this great year—"which is completed from the flood when all the circuits of the stars are complete around the world's center and their own center and all the stars together are brought back equally to the same part of the celestial longitude, latitude and height as they originally occupied"—as fifteen solar years, "while others measure it otherwise." This is presumably a reference to Macrobius. See Chapter 4 below.

For the associations of 36 as the sum of the first four even and the first four odd numbers, as the first number which is at once square and triangular, and as the number which is both the product of two squares (4×9) and the sum of three cubes (1+8+27), see Plutarch, De Animae Procreatione 13, 30 (Moralia 1018CD, 1027F).

[25] See Dupuis, Théon , pp. 376–377 and n. with references. For Ficino and the perfect numbers see Chapter 2 below.

tinguish in our own minds, at least initially, between the notions of the cosmic great year, of the perfect number(s), and of the fatal number(s), remembering that the Platonic number, which presides over "mortal" begettings, is a fatal number.

Ficino's position is this. The period of the great year necessarily contains lesser periods, and these are the periods of human engendering which are under the sway of the fatal geometric number. However, this number is itself subordinate to the perfect number that governs the divine cosmic creature which is the world (the "divine begetting"). The perfect number, not the fatal number, therefore is the ultimate determinant of celestial time, the world's time that is intermediary between terrestrial time and timeless eternity. But such a number eludes human intelligence, says Ficino, and is known to the gods, to God alone, for whom a thousand years, in the words of Psalm 90:4, are but as yesterday when it is past. If the Psalmist is to be believed, however, there emerges the possibility at least of an analogical relationship between God's measures and man's, and thus of our predicating on the basis of our circumscribed notion of a period (and thence of periodicity) the existence of divinely ordered periods that God has ordained should govern the world until the dawning of the great Sabaoth of His eternity.

However speculatively appealing, the task of actually measuring periodic time and its constitutive units, and therefore of establishing the basis for prediction itself, is utterly beyond man's reasoning powers. In the first place, the reason has no way of determining our position in a period (which may be part of a greater and even more mysterious period or cycle, and so on), and hence of determining when it began and when it will end. Thus it cannot know the number that governs our present period as its originating and therefore as its final cause. Yet such a cause, such a universal cause, and not particular and local causes, is precisely what Plato is concerned with. Accordingly, Plato does not resort, Ficino argues in this argumentum , to "the civil faculty" of the reason, like his calumniator, Aristotle, in order to measure the ultimate life of a state. Rather he has recourse to the faculty that transcends man's reason, to the suprarational, intuitive understanding (the mens ) that, insofar as it is concerned with the apprehension of time, is identical with "Apollo's prophetic art," or what Ficino refers to also as the "oracular" power bestowed on us by the Muses.^[26]

[26] "Quoniam vero eiusmodi causae assignatio praesentis civilisque facultatis terminos procul excedit, ideo Socrates vaticinio Musarum utitur, et profecto ita utitur ut et

nobis ad haec interpretanda opus sit Apollinis vaticinio"; noted by Schneider, Platonis Opera Graece 3:iv.

Ficino's account of prophecy has never been fully analyzed, nor for that matter has his conception of Apollo or the Muses; and it is part of his general theory, derived principally from the Phaedrus 244A–245C, of the four divine furies. We learn from an important section in his Platonic Theology 13.2 (completed, at least in draft, by 1474 but not published until 1482) that he viewed prophecy as culminating in the soul's ascension from the body and "comprehension of all place and time." At that moment the intuitive intellect is flooded with the splendor of the Ideas, the radiant Beauty that is the emanating light of Truth.^[27] But the prophetic "art" involves more than the initiatory rapture and then the intellectual skill and insight to interpret it correctly. In the argumentum , Ficino claims, perhaps extravagantly or facetiously, that the mysteries of the passage on the fatal geometric number and the mystery of that number itself not only defy interpretation by the process of normal discursive reasoning (the ratio ) and require intuitive or even mantic powers, but demand ultimately the descent of a god, of a divine and overwhelming force. Perhaps we should bear in mind a claim that Ficino had made elsewhere, namely that mathematics is the particular domain of the daemons and that skill with numbers is in essence a daemonic skill and the gift of the daemons,^[28] something that most of us have suspected since childhood.

Even so, the argumentum strikes a note of doubt. In the light of Theon of Smyrna's refusal to address the great mystery, despite his expertise in Platonic mathematics, Ficino wonders, as we have seen, whether there is "more of difficulty than of real weight" in Plato's reference to the fatal geometric number, especially given the reference at 545DE to the stupefying effect of the Muses' "tragically inflated" mode on a simple youthful soul. At this point he declines, furthermore, to address the technical difficulties of the passage or indeed to confront the mystery itself of the fatal number; and he suggests in-

[27] Ed. and trans. Raymond Marcel, as Marsile Ficin: Théologie platonicienne de l'immortalité des âmes , 3 vols. (Paris, 1964–1970), 2:205–214. On inspiration in Ficino, see my Platonism , chapter 1, and Sebastiano Gentile, "In margine all 'Epistola 'De divino furore' di Marsilio Ficino," Rinascimento , 2d ser., 23 (1983), 33–77.

[28] In the Platonic Theology 14.3 (ed. Marcel, 2:256) Ficino argues that "we live the life of the daemons when we engage in mathematical speculation," and contemplate cogitabilia , that is, mathematical matters. Cf. his epitome for Proclus's Republic Commentary 12, "Cogitabilia vero, id est mathematica, res in se quaedam speciesque sunt" (Opera , p. 942.1). In the Platonic Theology 6.2 (ed. Marcel, 1:228), he declares that numbers are spiritual entities or forms (spirituales ).

stead that the reader should turn to his Timaeus Commentary—his earliest commentary, we recall—for whatever is "more useful or opportune" in Plato's baffling discussion, though we should note that in that commentary Ficino does not take up the issue of the fatal number, despite his odd references to the pertinent passage in the Republic .^[29] The remaining sentences of the argumentum merely cull some "moral precepts" from the rest of book 8.

Clearly, at this stage in his Platonic career Ficino did not have the confidence to expatiate on an issue he had not yet resolved; indeed he was probably ambivalent, on the one hand suspecting that Plato was playing or joking with his reader, on the other believing that a divine inspiration was required for an interpreter to pierce through the cloudy veils with which Plato had encompassed the number to conceal it from the vulgar gaze.^[30] In either event, it was clear that Plato had hedged the passage around with apotropaic devices, with Pythagorean prohibitions, with learned silence. And not only to the young and uninitiated, and to those with the mere rudiments of geometry had he denied its resolution: Ficino himself felt compelled to wait upon some future inspiration, some descent of Apollo or his daemon.^[31] Having

[29] For the possible references, see the Index Auctorum et Nominum, below.

[30] Ficino seems, for instance, to have thought of Plato's concentration in books 8, 9, and 10 of the Republic on numbers ending in 8, 9, and zero respectively as an example both of veiling and of playing. He was fully apprised, nevertheless, of Plato's own reservations in the Protagoras 342 ff. and the Cratylus passim about the symbolic use of number, and of Plato's mercurial way with images and figures. See his references in the De Numero Fatali 15 to Plato's deployment of numerical metaphors in the Phaedrus . More generally, see his remarks in the Phaedrus Commentary, summa 25 (ed. M. J. B. Allen, Marsilio Ficino and the Phaedran Charioteer [Berkeley, Los Angeles, London, 1981], pp. 168–171), written just a few months before, and his contention in the dedicatory proem to his Plato edition that "Platonic jokes and games are much more serious than the serious discourse of the Stoics" ( = Opera , p. 1129). On Ficino and the varying semantic levels in Plato, see Hankins, Plato in the Italian Renaissance , 1:337–339, 344.

We might note that Ficino also thought of Plotinus as playing, and playing specifically "in the Egyptian manner," with regard to the theme of the possibility of the human soul's transmigration into the soul or the body of a beast: Opera , p. 1788.5 (the commentary on Enneads 6.7.6).

[31] The apollonian daemon is referred to in Ficino's Phaedrus Commentary, summae 10 and 30 (ed. Allen, pp. 139, 183). Given the reference in the Apology 23B to Apollo being Socrates' own god, it is associated with Socrates's warning voice and thus with his personal daemon. See my The Platonism of Marsilio Ficino (Berkeley, Los Angeles, London, 1984), pp. 21–22, 31, 33, 66–67.

In his Vita Platonis Ficino repeats the story that Plato was a son of Apollo. This Vita , which is deeply indebted to Diogenes Laertius's life of Plato in his Lives of the Philosophers , appeared as a preface for Ficino's 1484 Plato edition (succeeding the dedicatory proem to Lorenzo) and again as a letter addressed to Francesco Bandini in the

fourth book of Ficino's Epistulae (Opera , pp. 763–770; trans. in Letters [see n. 41 below], 3:32–48—the reference occurs at p. 770.3).

said this, we should note that Ficino did accept the scholarly responsibility of attempting a translation of the passage; and in doing so he relied upon his exemplar, the Laurenziana's 85.9.^[32]

However, the story of Ficino's involvement did not end here, as Bodin, Schneider, Dupuis, and others have too precipitately supposed.

Apparently, these scholars were familiar only with Ficino's argumentum , which offers no solution to the problem of the geometric number. They were obviously completely unaware, as all more recent

[32] This codex contains the complete works of Plato in a good text in the A family and with abundant marginalia by Ficino. See Sebastiano Gentile in Marsilio Ficino e il ritorno di Platone: Mostra di manoscritti, stampe e documenti (17 maggio—6 giugno 1984) , ed. Sebastiano Gentile, Sandra Niccoli, and Paolo Viti (Florence, 1984), pp. 28–31 (no. 22) and plate VII—hereafter Mostra —with further references to earlier studies by Marcel, Sicherl, Kristeller et al.; idem, "Note sui manoscritti greci di Platone utilizzati da Marsilio Ficino," in Scritti in onore di Eugenio Garin (Pisa, 1987), pp. 51–84 at 58. See also Aubrey Diller, "Notes on the History of Some Manuscripts of Plato," in his Studies in Greek Manuscript Tradition (Amsterdam, 1983), pp. 251–258 at 257; Paul Oskar Kristeller, Marsilio Ficino and His Work after Five Hundred Years , Quaderni di Rinascimento, no. 7 (Florence, 1987), pp. 72–74, 138; Gerard Boter, The Textual Tradition of Plato's Republic (Leiden, 1989), pp. xvii (the stemma where MS. 85.9 is assigned the siglum c ), 36–37 (its description), 137–139 (its relationship to MS. 59.1, which is assigned the siglum a ), and 270–278 (Ficino's use of it as his primary but not exclusive source); and James Hankins, "Cosimo de' Medici and the 'Platonic Academy,'" Journal of the Warburg and Courtauld Institutes 53 (1990), 144–162 at 157–158.

Diller and Gentile have both conjectured that Cosimo acquired the codex in 1438 during the Ferrara-Florence Council of Union not from the emperor but from one of the most distinguished members of his suite, the philosopher Gemistus Pletho. Its scribe has been tentatively identified by N. G. Wilson as Christophorus de Persona, who ended his career as papal librarian to Innocent VIII but who had probably studied under Pletho in his youth (see Hankins, "Cosimo," pp. 157–159, with further references).

Cosimo gave Ficino the codex to use in 1462. It contained, besides the complete canon of Plato (the Republic being on fols. 216–267), a number of other texts of great interest to him: the Definitiones and other Platonic spuria, the Golden Verses of Pythagoras, Alcinous's Epitome , Theon of Smyrna's Expositio , the "Life of Plato" by Diogenes Laertius, Albinus's Introductio , the De Anima Mundi by Timaeus Locrus, the De Animae Procreatione by Plutarch, and the Economics and Symposium by Xenophon; it also contained an oration by Aristides and another by Libanius. With regard to the Plato, scholarly opinion now holds that it was probably copied directly from the Laurenziana's 59.1 earlier in the fifteenth century—probably in Crete. For MS. 59.1 itself did not appear in Lorenzo's collection until it was brought to Florence by Janus Lascaris, who had bought it for him in Candia on 3 April 1492. It is extremely unlikely, therefore, as Diller and Gentile have recently argued (and Kristeller now agrees), that Ficino ever consulted MS. 59.1 for his Plato translations.

We should note that Boter is obviously unfamiliar with this newly skeptical view of MS. 59.1's importance for Ficino and therefore unwilling to dismiss it as one of Ficino's exemplars for the Republic translation. He also maintains that Ficino made occasional use of several other manuscripts, including one of Bessarion's (now the Marciana's gr. 187 [coll. 742]), and the Laurenziana's 85.7 (a member of the F family). Again, however, recent scholarship on Ficino has discounted MS. 85.7 as a likely source (see Kristeller, Ficino and His Work , pp. 74, 138); and the question of Ficino's access to the Marciana's MS has not been seriously raised before, but is of little bearing since that MS was itself copied from the Laurenziana's 85.9 (see L. A. Post, The Vatican Plato and Its Relations [Middletown, Conn., 1934], pp. 40 ff., and Boter's own p. 59). Interestingly, however, Gentile, "Note," pp. 69 ff., has proved that Ficino did collate MS. 85.9 with two MSS containing single dialogues or excerpts, the Laurenziana's Conventi soppr. 180 and the Vatican's Borgianus gr. 22; but neither includes the Republic .

Note that Dupuis's claim, Nombre , p. 9, that Ficino adopted the variants treis apokatastaseis for treis apostaseis, parechêtai for parechetai , and pempadôn for pempado is incorrect; he was almost certainly thinking of Barozzi's readings (see n. 35 below).

Finally, with regard to the question of Ficino's indebtedness to the three humanist Latin renderings that preceded his own, see Appendix 2 below.

scholars too have been unaware, of a major essay on the theme and its implications that Marsilio wrote some thirty years later in the early 1490s and published in 1496. There he takes up several of the many problems in some detail, and having insisted on the role of the diagonal numbers (diametri or diametrales ) as we shall see, he advances a solution consonant with Aristotle's gloss, namely, 12 to the third power. This solution apparently became generally accepted during the first half at least of the sixteenth century. It was adopted, for instance, by Raphael (Maffei) Volaterranus in the 35th book of his Commentaria urbana published in Rome in 1506 (though he proferred another solution in his 36th book!),^[33] and adopted too, more significantly, by Iacobus Faber Stapulensis (Jacques Lefèvre d'Étaples), again in 1506, in annotations to the last chapter of his commentary on book 5 of Aristotle's Politics, a commentary that was reprinted a number of times and exerted considerable influence in its day.^[34] It was also adopted, but with more detailed argumentation and annotation and

[33] Schneider, Platonis Opera Graece 3:lxix, Dupuis, Nombre , p. 6, Diès, Essai , pp. 73–74 (with errors).

[34] For the various editions, see the bibliography by Eugene F. Rice, Jr. in his edition, The Prefatory Epistles of Jacques Lefèvre d'Étaples and Related Texts (New York and London, 1972), pp. 553–554 (with the relevant preface on pp. 150–152). Faber's annotations amount to four pages; for an analysis, see Schneider, Platonis Opera Graece 3:lviiii–lxviii; and Diès, Essai , pp. 59–71. Diès claimed, given the absence for so long of Proclus's interpretation, that it was Faber who had first brought to bear the theory of the diagonal and the lateral numbers, an essential source for the modern solution of the Platonic enigma (pp. 66, 69). But credit for this must go to Ficino, or at least to Ficino's promotion of Theon's Expositio ; see his De Numero Fatali 5, and Chapter 2 below.

Interestingly, in the course of his argument Faber quotes verbatim from Ficino's translation of Plato's passage, but only as it appears at the head of the 1496 essay and not in the 1484 or 1491 Platonis Opera Omnia editions; for Faber's version includes the 1496 emendations, including the major emendation "quinitatis . . . comparabiles." This is irrefutable evidence in itself that he knew of Ficino's essay (though Diès, Essai , p. 61, incorrectly supposed that the translation was Faber's own). In 1496 Faber published an edition of the Arithmetica of Jordanus Nemorarius (d. 1237), along with an epitome of the arithmetical works of Boethius—a significant volume for Renaissance mathematics—and in 1503 another epitome of the arithmetical works of Boethius and of others, along with an introduction. In neither, however, does he deal with Plato's number.

again with an insistence on the bearing of the diagonal numbers, by the distinguished Venetian mathematician Francesco Barozzi in his Commentarius in Locum Platonis Obscurissimum published in Bologna in 1566.^[35]

The history, rich and curious in itself, of interpretative attempts before the twentieth century should therefore be rewritten to accord Ficino, and not Faber, the accolade of being the architect of the first modern interpretation of Plato's enigma and the first scholar since antiquity to confront a number of the major cruces and to address the issues and possibilities in the light of research into Platonic mathematics.^[36] We might note, incidentally, that Girolamo Cardano (1501–

[35] Dupuis, Théon , p. 389: "La dissertation de Barozzi est une des plus soignées. La version latine littérale du lieu est une des meilleures"; also Nombre , pp. 6 (though with incorrect dates for the 1566 edition and for Barozzi's death in 1604) and 20–47 passim. Barozzi's translation is to be found on fol. 12r, lines 13–28, and his solution on fol. 17v, line 32. For an analysis, see Schneider, Platonis Opera Graece 3:iiii–xxviiii, and Diès, Essai , pp. 69–85, who supposes that Barozzi's immediate debt was to Faber (for whom see pp. 66–67). We might note that Barozzi's Latin rendering is almost Ficino's revised version of 1496 (perhaps because Barozzi had his eye on Faber—see n. 34 above). His principal variants are "(in)effabilia," "quinarius," and "praelongiori" for Ficino's "(in)comparabilia," "quinitas," and "oblongiori(e)" respectively. Since Jean Bodin knew only Ficino's 1484 argumentum , we must suppose that he too was unaware of Faber's debt to Ficino's later essay.

[36] The opening of the third chapter of Diès's monograph in particular should be revised, since it is, and is likely to remain, the authoritative history of the attempts to interpret Plato's Number up to 1940. More research still remains to be done, however, on a variety of sixteenth- and seventeenth-century editions of, and commentaries on, the Republic . The following possibilities have been suggested to me by Professor James Hankins: Joachim Camerarius the Elder, Epistularum Familiarium Libri VI (Frankfurt, 1583), pp. 255 ff., 478–481, 525; Ulysse Aldrovandi, Bologna Univ., Aldrovandi MS. 124, vol. 56, fols. 270–277; Sebastian Fox Morzillus, Commentatio in X Platonis Libros de Republica (Basel, 1556); Muretus, Commentarium in Rempublicam , Vat. lat. 11591—published, misleadingly, in his Commentarii in Aristotelis X Libros Ethicorum ad Nicomachum (Ingolstadt, 1602), 712–740; Joh. Sleidanus, Summa Doctrinae Platonis de Republica et de Legibus , in Cl. Seyssel, De Republica Galliae et Regum Officiis (Strasbourg, 1548); and Ioannes Sozomenus, Divi Platonis de Republica Libri X (Venice, 1626).

1576), in his Opus Novum de Proportionibus (Basel, 1570), was to propose as a solution another number occurring in Ficino's analysis, 8128, the fourth in the series of perfect numbers; and that the disciple and friend of Descartes, Marin Mersenne (1588–1648), in book 2 of his Traité de l'harmonie universelle (Paris, 1627), was to propose a "lesser fatal number" that Ficino had actually entertained—since Plato himself had introduced it in the ninth book of the Republic —namely 729;^[37] and so forth. Clearly, early modern scholarship had not yet forgotten Ficino's role in the explication of Plato's refractory passage.

Ficino's essay takes the form of a commentary on book 8 of the Republic , which he first published along with others in 1496 (no earlier manuscript is extant). It therefore postdates the Plato editions of 1484 and 1491 and represents a renewed attempt by Ficino late in his life to come to grips with the value of Plato's geometric number. From the onset of his professional academic career he had committed himself to the task of extensive commentary on the Platonic dialogues. Even before he had learned Greek in the 1450s, he had written at length on the Timaeus , though he was to do so again on several other occasions—the Timaeus Commentary we now possess being the product of maturer explication.^[38] By 1469 he had already completed a fullscale commentary on the Symposium and written a substantial portion of another one on the Philebus (though this he never completed despite returning to it on at least two more occasions).^[39] In the following years, as he prepared his Plato translation for the press, he finished composing his epitomes and introductions for all the dialogues.^[40]

[37] Cardanus, De Proportionibus , p. 234, prop. CCV "super verbis Platinis de fine Reipub." (in the 1663 Lyons ten-volume edition of Cardano's collected works this occurs at 4:582); and Mersenne (pseudonym "le sieur de Sermes"), Traité , book 2, theorem XIII, part 1, pp. 424–436 at 430 (there is an acknowledgment, incidentally, to Ficino, Cardano, and Faber on p. 426). See Dupuis, Nombre , pp. 6–8; idem, Théon , pp. 392–393; Diès, Essai , pp. 85–86; and, for Cardanus, Schneider, Platonis Opera Graece 3:xxxviii–xxxx. Dupuis and Diès are incorrect in citing Cardanus's propositio as CCXV.

[38] For the history of the Timaeus Commentary, see Kristeller, Supplementum 1:cxxi, 78–79, and my "Marsilio Ficino's Interpretation of Plato's Timaeus and Its Myth of the Demiurge," in Supplementum Festivum: Studies in Honor of Paul Oskar Kristeller , ed. James Hankins, Frederick Purnell Jr., and John Monfasani (Binghamton, N.Y., 1987), pp. 399–439, esp. 402–403.

[39] See the introduction to my edition, Marsilio Ficino: The Philebus Commentary (Berkeley, Los Angeles, London, 1975; reprint 1979, with corrections), pp. 3–15, 48–56.

[40] On the 1484 edition and its preparation, see Kristeller, Supplementum 1:cxlvii–clvii; idem, "The First Printed Edition of Plato's Works and the Date of Its Publication (1484)," in Science and History: Studies in Honor of Edward Rosen , ed. Erna

Hilfstein, Pawel Czartoryski, and Frank D. Grande (Wroclaw, 1978), pp. 25–35; Gentile in Mostra , pp. 117–119 (no. 91); and Hankins, Plato in the Italian Renaissance 1:300–311, 2:465–482 (app. 18). For its later fortunes, see also John Monfasani. "For the History of Marsilio Ficino's Translation of Plato: The Revision Mistakenly Attributed to Ambrogio Flandino, Simon Grynaeus' Revision of 1532, and the Anonymous Revision of 1556/1557," Rinascimento , 2d ser., 27 (1987), 293–299. For the astrological significance of 1484 itself, see Chapter 3, n. 1 below.

Eventually in 1496 he assembled five long commentaries together with chapter breakdowns and summaries in one volume—those on the Timaeus, Philebus, Parmenides, Sophist, and Phaedrus (though two of them only can be said to be complete). To these he added a commentary focusing on the "fatal" number in the eighth book of the Republic , and dedicated the resulting collection to Niccolò Valori.^[41] It is

[41] Kristeller Supplementum 1:cxvii–cxxiii, clv; Marcel, Marsile Ficin , pp. 521–533; Gentile in Mostra , pp. 155–156 (no. 120). For Ficino's admiration in general for, and indebtedness to, the Valori family (Filippo, Niccolò's older brother who died in 1494, had been the financier of the 1484 Plato edition), see the Commentaria 's proem to Niccolò, f. lv (= Opera , p. 1136.2).

We know from a letter to Uranius of 18 Jan. 1493 that Ficino composed his De Sole in the course of the summer of 1492 and followed it with the De Lumine . Marcel claims that it was at this juncture that Ficino wrote the De Numero Fatali —"après la République , qui lui avait inspiré le De Sole et sans doute, dans le mème temps l'Expositio circa numerum nuptialem "—before taking up his Timaeus Commentary again on 7 Nov. 1492 (pp. 530–531). However, Ficino's references in the De Numero Fatali to the Timaeus Commentary suggest otherwise. Indeed, on the twin grounds that the De Numero Fatali is not referred to elsewhere in Ficino's writings (whereas the Timaeus Commentary is) and that it is printed in the Commentaria volume after the colophon, Kristeller has argued that it was written as late as 1496 or "a little before" (Supplementum 1:cxx–cxxi, cxxiii).

Corsi's Vita Marsilii Ficini , which has been edited by Marcel in his Marsile Ficin , pp. 680–689, cannot be relied upon, as Kristeller has demonstrated in his Studies in Renaissance Thought and Letters (Rome 1956; reprint 1985), pp. 191–211. Corsi's chapter 14 is particularly confusing: it asserts that in the last seven years of his life, "having finished [editis cannot mean 'published' here] what he had written on the fatal number of Plato in the eighth book of the Republic , and then the De Sole and the De Lumine , Ficino began new commentaries on the whole of Plato. . . . In this year [1492], after he had completed his extremely learned commentaries on the Parmenides and the Timaeus and also most of his commentary on Dionysius' Mystical Theology , he then wrote and finished his commentary on the Divine Names . But in that last period, besides commentaries on the Parmenides and Timaeus , he also wrote (edidit ) his commentaries on the Theaetetus, Philebus, Phaedrus, and Sophist " (I have modified the English translation given by the Members of the School of Economic Science, London, in an appendix to their third volume of The Letters of Marsilio Ficino (London, 1981), pp. 142–143—hereafter Letters ). Four volumes of this admirable series have appeared so far, rendering Ficino's books 1, 3, 4, and 5 respectively.

The unique codex of the De Numero Fatali —the Bayerische Staatsbibliothek's Clm 956b—was apparently made from the 1496 edition, since it is dated Nuremberg 1501 by the transcriber Hartmann Schedel (1440–1514), "doctor of arts and of medicine" and the author of the Liber Chronicarum of Nuremberg; see Kristeller, Supplementum 1:xxxv. I have included its readings in my apparatus in Part Two below.

(footnote continued on the next page)

Ficino's only full-length treatise devoted to the Republic , despite the work's prominence for him and in the Neoplatonic tradition; and it is remarkable that its subject should be the "fatal" number and not the allegory of the Cave, the myth of Er, the figure of the Divided Line, or the Idea of the Good—the "set pieces" of other more famous books. Nonetheless, the essay is an anomalous inclusion in the 1496 volume insofar as it is not a commentary upon an entire dialogue but rather a largely self-contained discussion of the issues raised by just a few lines in that dialogue. Perhaps Ficino felt he had covered the general territory of the Republic sufficiently in the course of his quite lengthy epitomes (no epitome exists for book 8, though the argumentum functions as such).^[42]

The 1496 volume was apparently in lieu of a deluxe revised edition of the 1484 Plato volume, which Ficino had envisaged before Lorenzo's death on 8 April 1492 and the expulsion of the Medici in the November of 1494, and which he had hoped would include even more extensive commentaries on many, if not on all, of the dialogues as well as revised translations and chapter breakdowns and summaries. Those for the five dialogues, incidentally, include further revisions for Ficino's Plato translations; and the volume concludes with a corrigenda list that occasionally corrects these revisions! In the event, the Commentaria in Platonem was to be the terminus of his specifically Platonic labors, since the last three or so years of his life were devoted to lecturing on and analyzing Saint Paul's Epistles and notably the Epistle to the Romans.^[43]

Even as late as 1496, however, Ficino was still uncharacteristically circumspect about Plato's intentions, as one can see from his prefacing expositio . He writes,

The prodigious enigmas of this chapter above [i.e., 546A–D] have terrified me and indeed other Platonists too for a long time from trying to explicate them. Nevertheless, the things in it that I am relatively sure about—having thought about the passage for many years—I will deal with first. At the end I shall take the plunge and deal with what is merely probable. The totally inexplicable I will omit altogether. For Plato wanted [only] certain things to be

[42] Opera , pp. 1396–1438. The dispositio that follows the proem to Niccolò (Opera , p. 1136.3) makes no mention of the De Numero Fatali in its list of the five companion commentaries and the order in which Ficino has chosen to present them. See Gentile in Mostra , p. 156.

[43] Opera , pp. 425–472; Kristeller, Supplementum 1:lxxxi–lxxxii. He only got as far as the first few chapters. For a study, see Walter Dress, Die Mystik des Marsilio Ficino (Berlin and Leipzig, 1929), pp. 151 ff.

explained. Words that men cannot understand, however, he justly attributed to the Muses—to the Muses at play—for what is hidden is something playful.

This is a revealing set of provisos and caveats. First it suggests that Ficino had carefully pondered the challenges of the "prodigious" chapter and deliberately postponed commenting upon it as long as possible, or at least until he had garnered a number of insights into its enigmas. In this regard we should note the emendations to his translation of the chapter for the 1484 Plato edition—particularly of the phrasing at note 16 of the apparatus criticus to Text 2 on p. 163 below—bearing in mind that his exemplar remained the Laurenziana's Greek manuscript 85.9.^[44]

Second, besides the "fatal" number, Ficino is predictably concerned with the number known in the Pythagorean manner, as we have seen, as the "nuptial" number because of its importance in Plato's advocacy of eugenics; and, in dealing with both numbers, he consciously prepares us to move from the certain, to the probable, to the inexplicable. Elsewhere, notably in the Vita Platonis which prefaces the 1484 Plato edition,^[45] and in the Platonic Theology 17.4,^[46] he had spoken of

[44] See n. 32 above. The Oxford text by John Burnet, which I shall refer to throughout, lists authoritative variants but does not take into account, since it does not really need to, the subsidiary manuscript tradition utilized by Ficino. It does not list in other words the 85.9 variants. See Appendix 1 below.

[45] Opera , p. 766.2 (the close of the section Libri Platonis ); trans. in Letters 3:38, "What Plato discusses in his Letters , in the books of the Laws , and in the Epinomis , using himself as a speaker, he means us to take as certainties; but what he says in the rest of the dialogues, when he uses Socrates or Timaeus or Parmenides or Zeno as speakers, he wishes us to understand as only resembling the truth."

[46] Ed. Marcel, 3:168–169, 174. Here Ficino argues that three pieces of evidence argue against Plato's having accepted certain Pythagorean tenets: first, he depicts the same people in the process of debating issues they had earlier pronounced on; second, he portrays an indecisive (ambiguus ) Socrates who is reporting on what he has heard (though he himself knows only that he knows nothing); and third, Plato never confirmed when he was old what he had earlier written on Pythagorean beliefs. Indeed, Ficino concludes, "in the Laws , written at an advanced age and the sole work where Plato speaks in his own person, he affirms nothing like these beliefs. Moreover, in the letter to the king Dionysius [i.e., the Second Letter at 314C], written when he was very old, he says that he had never written anything himself about matters divine and that he would never do so. It was as if he were not revealing his own mind to us but describing another's. In the letter to the Syracusans [i.e., the Seventh Letter at 341C], written afterwards when he was even older, he repeated the same opinions and added that no one existed then or would ever exist in the future who would know what Plato thought about such matters; and this was appropriate since he never wrote about them" (pp. 168–169). Ficino concludes the seventeenth book by saying that "Plato affirmed only those views about matters divine which he approved of in the Letters and the Laws ";

and that this did not include Pythagorean theories concerning the "infinite circuits of souls." What he described as the inventions of the ancient theologians in his other dialogues he must have considered "probabilities rather than certainties." Hence Ficino is moved to interpret Plato's words figuratively (longe aliter quam verba designent ). See Hankins, Plato in the Italian Renaissance , pp. 339–340.

Plato as habitually presenting us with the merely probable and as declining to promulgate certainties or dogmas. Only in the Laws (which for him included the Epinomis as an epilogue) and the Letters , the last works of Plato's career, does he see him prepared to commit himself publicly, and even then with regard to just three deeply held convictions: that Providence exists; that the soul is immortal; and that there is a scheme of reward and punishment in the afterlife for the good deeds we have effected or the sins we have committed in this life, in other words that a divine justice presides over all things.^[47]

Finally, Ficino makes an ambiguous reference to the Muses, something, significantly, that he elects to do again at the very end of his commentary: "But we have debated enough in the company of Plato and the Muses as they play with a serious and inextricable matter." While we might point to similar statements in the Parmenides Commentary for instance,^[48] in no other commentary do we find Ficino quite so candidly admitting that he has failed to unravel completely, or to his full satisfaction, the complexities of a Pythagorean-Platonic mystery, failed to penetrate to the core of the sapiential fruit. In none, moreover, do we find him more attuned to the seriocomic tone, to the presence of a mystagogic irony and obliquity in Plato's style and presentation. By way of explanation, he warns us in the prefatory expositio that we must remember that Plato had decided from the beginning to remain silent on certain issues: "certain things Plato himself chose not to unfold" ("quaedam noluit explicari"). The old Pytha-

[47] In his epitome for Laws 1 (Opera , p. 1488.2), in the course of arguing that Plato had "tempered" and reconciled Pythagorean contemplation with Socratic action and had made the one more humanly accessible and the other more adapted to converting us to matters divine and eternal, Ficino maintains that the Republic is more Pythagorean and more Socratic than the Laws , which is more truly Platonic. Plato wrote the latter "so that men who cannot climb the arduous mountain may at least not reject the path to the gentler foothills." For Plato's threefold debt to Heraclitus, to the Pythagoreans (including Parmenides), and to Socrates, see Diogenes Laertius, Lives of the Philosophers 3.8, and Ficino's Vita Platonis (Opera , pp. 764.1, 769.3; trans. in Letters 3:34,45); also my article, "Marsilio Ficino on Plato's Pythagorean Eye," MLN 97 (1982), 171–182.

[48] See my "The Second Ficino-Pico Controversy: Parmenidean Poetry, Eristic and the One," in Marsilio Ficino e il ritorno di Platone: Studi e documenti , ed. Gian Carlo Garfagnini, 2 vols. (Florence, 1986), 2:417–455, with further references.

gorean commitment to silence is assumed to be Plato's too, for all his volubility and eloquence.^[49]

Marsilio, however, was committed by his expository program to unfolding as much as he possibly could about Plato's most obscure passage in the Republic , and when he sat down he produced something that was for him—a constitutionally digressive and endlessly parenthetical and repetitive thinker—a passably compact, organized, and self-contained treatise. By the time he had reached his conclusion, moreover, he was convinced that he had resolved some at least of Plato's enigmas. Above all he had established a value for the fatal geometric number.

In the course of his inquiry, as we shall see, he also raised a number of questions of abiding interest to scholars both of the Platonic tradition and of Renaissance conceptions of man, of history, and of time, questions that as historians we are drawn to set against the backdrop of Florentine religion and politics at the close of the fifteenth century. For Plato's ideal city brought low by the fatal number prefigures a Florence inflamed by the Savonarolan reform movement with its apocalyptic predictions that an aeon was coming to an end. Ficino was certainly personally affected by the convulsive millenarianism of the 1490s, and brooding on the numbers of time and its dreadful passing was a preoccupation he undoubtedly shared with many of his friends and compatriots, quite apart from the professional astrologers and the self-appointed prophets, in those turbulent, unhappy years preceding the calamità .^[50]

More particularly, as a Platonist, he had by then been immersed in the canon for some thirty years and become thoroughly familiar with its allusions to a cyclical time in such works as the Statesman , the Timaeus , and the third book of the Laws . He had become convinced too that Plato had been a reformer and prophet, who had called for change in the polities of Athens and Syracuse, and had predicted, from the Neoplatonic viewpoint at least, the return of the age of gold.^[51] However, his acquaintance with Christian, and specifically with Au-

[49] The distinction between Plato's deliberate silence and Plato's playful veiling of mysteries is understandably somewhat unstable.

[50] In Lo zodiaco della vita (Bari, 1976), pp. 18–19—a marvelous study poorly translated by Carolyn Jackson et al. as Astrology in the Renaissance: The Zodiac of Life (London, 1983)—Eugenio Garin observes that the Quattrocento Italian Platonists were to insist on the fatal decline of republics even as they debated the possibility of renewal and return. But were republicans or Platonists more sensitive to this issue than others?

[51] See his Vita Platonis (Opera , pp. 769.3–770; trans. in Letters 3:45–46).

gustinian, historiography and with Joachimite prophecy had also exposed him to the contrary notion of a linear time with its successions: the reigns of nature, law, and grace; the four monarchies of Daniel 2:31–45 and 7:17–27; the six historical epochs as defined, for instance, in Augustine's City of God 22.30; the seven kingdoms of Revelation 17.10—the Jesse tree of durations, however numbered, in the history of man and his generations. How then to reconcile the two, since, given his Platonic (and we might add his humanist) assumptions, he was unwilling to accept Augustine's outright rejection in the City of God 12.14 of a cyclical dimension to time? I shall suggest in Chapter 4 that, a syncretist by temperament, he seems to have been drawn rather to the notion of a third temporal order as it were mediating between us and eternity: a spiraling providential time that governs alike the cyclical realm of the stars and the transitory linear history of the sublunar realm that gazes on and depends upon those stars.

Fundamental in this regard is the haunting presence in his mind not only of Hesiod's myth of the golden age and the possibility of its return^[52] —predictably so, given Plato's own allusion to Hesiod at the close of his description of the fatal number at 546E ff.—but also, and more importantly, of the myth of the Demiurge in the Timaeus and of the mathematical and musical formulas presented there for the composition of the World-Soul.^[53] For this creation myth, which problematizes for us the dualism of other prominent dialogues such as the Phaedo , presented Ficino with a Plato who was a visionary historian, an Attic Moses in Numenius's memorable phrase, whose intuitive, whose prophetic intelligence had been granted an insight both into the actual numbers of time, and thus into their concomitant geometrical figures and ratios, and into the numerical Ideas according to which the Demiurge and his sons had first fashioned a spatiotemporal reality in the image of the true and the good.

[52] Christian commentators often glossed this myth by adverting to the description of the statue with head of gold, breast and arms of silver, belly and thighs of brass, legs of iron, and feet of clay in Daniel 2:31–33.

[53] Since the Old Academy this had been a matter of debate, and the solutions formulated can be found in the following texts: Timaeus Locrus, De Natura Mundi et Animae 96A–C (ed. Marg, pp. 124–130); Plutarch, De Animae Procreatione in Timaeo 1027E ff. (ed. Cherniss, pp. 266–320); Calcidius, Timaeus (ed. Waszink, 81.19–103.12); Proclus, In Timaeum 2.167 .24–193.6 (ed. Diehl, trans. Festugière-Mugler 3.212–239). These are Bertier's references, p. 102, n. 5 (see n. 74 below).

In order to arrive at an understanding of Ficino's determination of the fatal number, we must eventually tread some unfamiliar mathematical ground. For an introduction to his approach to Platonic mathematics and to its close links with harmonics and therefore with music and astronomy, we cannot do better, however, than to turn to a concluding section of his epitome for the Epinomis . Ficino thought of this apocryphal dialogue—the author is probably Philip of Opus or another member of the early Academy^[54] —as Plato's authentic appendix to the Laws (as its name suggests), and therefore as being endowed with the singular and august authority he attributed to Plato's last work.^[55] It has a particular pertinence here in that earlier at 978B ff. the Athenian Stranger had been held to assert that the origin of our sense of numbers derives from our gazing up at the night sky and especially at the changing countenance of the Moon.^[56] The Epinomis epitome was probably written in the early 1470s and provides us with a general framework for an understanding of Ficino's more advanced treatment of individual topics in the commentaries on the Timaeus and eventually in the De Numero Fatali .

He is epitomizing the section (990C–991B) on the progression from arithmetic to geometry and then to stereometry.^[57] To begin with, he writes, numbers are "in themselves incorporeal" (990C), because they "are nothing other than the number 1 repeated" and 1 is indivisible and therefore without body. Following a Pythagorean formula (found, for instance, in Aristotle's De Caelo 1.1.268a7 ff. and De Anima 1.2.404b21 ff. and repeated throughout antiquity and the Middle Ages), Ficino proceeds to plot number geometrically as first a point, then a line, then a plane (superficies ), and finally a volume (profundum ). Hence there are three kinds of divisible numbers after the one as the indivisible point: linear, planar, and solid. Thus the doubling of 1 makes the linear 2, which in turn becomes the square 4 and eventually the cube 8 (991A).

[54] Diogenes Laertius, Lives of the Philosophers 3.37, writes, "It is said Philippus was the author of the Epinomis ." See Leonardo Tarán, Academica: Plato, Philip of Opus, and the Pseudo-Platonic Epinomis , Memoirs of the American Philosophical Society, vol. 107 (Philadelphia, 1976), pp. 133–139.

[55] See, for example, his Vita Platonis (Opera , p. 766.2; trans. in Letters 3:38), his epitome for the Epinomis itself (Opera , p. 1525.2—the heading), his epitome for the Laws I (Opera , p. 1488.2), and his Platonic Theology 4.1 and 17.4 (ed. Marcel, 1:165, 3:168–169).

[56] Cf. the Timaeus' s contention at 47A ff. that the sun, moon, planets, and stars were created in order to instill in us a sense of time and number (repeated by Diogenes Laertius in his Lives of the Philosophers 3.74).

[57] Opera , pp. 1529–1530. In the 1491 Venice edition of the Platonis Opera Omnia , the epitome (argumentum ) is found on fols. 323v–324v (sigs. R3v–R4v).

The perfect proportion or ratio^[58] is the double, and this "contains all the [other] proportions within itself." In effect, Ficino is concerned only with the three primary ratios that govern both music and the cosmos: those of the double (for us the ratio of 2:1), of the sesquialteral (i.e., of one and a half to one—the ratio of 3:2), and of the sesquitertial (i.e., of one and a third to one—the ratio of 4:3). These ratios he sees Plato deriving from the first four numbers, the Pythagoreans' tetraktys, which when added together make up ten. The four numbers, in short, encode two fundamental kinds of relationship: that of being arithmetically equal to and that of being geometrically proportional to. This is self-evident of course, but fraught with Pythagorean and Platonic implications, not least in the spheres of ethics and of politics.^[59]

With these primary ratios Ficino moves to the equivalent musical intervals of the diapason, the diapente, and the diatesseron, the "consonances" or harmonic ratios of the octave (2:1), the perfect fifth (3:2), and the perfect fourth (4:3) respectively. And this musical extension leads in turn: first, to the Pythagorean theory of the music of the spheres and the Sirens' song which Plato identified with it in the Republic at 616B–617E, where each Siren sings one of the eight notes of the octave; and, second, to the theory of harmonious proportions governing the cosmos and thus the distances between the Earth, the various planetary spheres, and the firmament of the fixed stars. Hence Ficino sees Plato postulating that "the interval" (with a play upon both the spatial and the musical meanings) from the Earth to the Sun compared to the interval from the Sun to the firmament of the fixed stars is in the proportion of 3:2 to 4:3, the first ratio creating the harmony of the diapente, the second that of the diatesseron. The diatesseron is also the harmony created by the interval between the Earth and the Moon.^[60]

[58] The distinction between the two terms is a matter of definition. See Chapter 2 below.

[59] Plutarch, Quaestiones Convivales 8.2 (Moralia 719A–B), notes that Lycurgus had expelled the arithmetical proportion from Lacedaemon as too democratic, and introduced the geometrical as more appropriate to "moderate oligarchy and lawful monarchy"! Antidemocrats could thus argue that their foes were arguing for a system that in fact equalized inequalities. See Plato, Republic 558C, Laws 6.757B, and Gorgias 508A ("geometrical equality is of great importance among gods and men alike"); and Aristotle, Politics 3.5.8, 3.9, 5.1.7, and Nicomachean Ethics 2.6.7.

[60] Ficino turns to the theme of musical and celestial "consonances" in his De Numero Fatali 12. The astronomy and mathematics of the Epinomis should be compared with that in the Laws 7.818–820. For the ratios governing both music and the cosmos, see Aristotle's criticisms in his De Caelo 290b12–291a28 and Macrobius's ac-

account in his In Somnium Scipionis 2.1.1–25 and 2.4.1–10—passages long familiar to Ficino, who thought of a harmony in the ancient sense as a proper sequence of sounds. See William R. Bowen, "Ficino's Analysis of Musical Harmonia ," in Ficino and Renaissance Neoplatonism , ed. Konrad Eisenbichler and Olga Zorzi Pugliese, University of Toronto Italian Studies, 1 (Ottawa, 1986), pp. 17–27; and in general S. K. Heninger, Jr., Touches of Sweet Harmony: Pythagorean Cosmology and Renaissance Poetics (San Marino, Calif., 1974), pp. 91–104, 115–132, with further references.

These summary remarks are sufficient for us to see the nature for Ficino of the inextricable links between number theory, geometry, harmonics, and Chaldaean-Ptolemaic cosmology. He had inherited these directly of course from Plato and then from the Neoplatonists, but also from the medieval tradition and more particularly from his youthful study of Calcidius's commentary on the Timaeus.^[61] The web of debts and influences may be a complicated one, but it is all of a piece.

The Epinomis epitome also emphasizes, as do many other passages in Ficino's commentaries, the Platonic significance of the number 12, 12 being the number of the world spheres—the eight celestial and the four elementary—in the Chaldaean system which Plato inherited.^[62] Under the World-Soul, Ficino writes, there are twelve souls for the twelve spheres, and within each sphere there are twelve orders of rational souls. In the eight celestial spheres we find the eight orders of souls of the constellations and stars; on earth, the one order of men (and we might add of the lowest daemons); and in the aether (fire), air, and water, the three orders of the higher daemons. From the onset, that is, there is a dramatic contrast between the fingers-and-toes world of 10 and the duodecimal world of the rational souls, divine, daemonic, and human, encompassing as it does the primary ratios and musical harmonies.

Before entering further into an account of the duodecimal mysteries Ficino saw at the heart of the Republic' s reference to a geometric number, I think it useful to conclude this opening chapter with a review of the ancient texts Ficino probably turned to for guidance, though none of them is a source as such, since none of them provided

[61] See my "Ficino's Interpretation of Plato's Timaeus ," pp. 404–408.

[62] The Epinomis does not discuss twelve as such, but it does treat of the cosmic nest of spheres: of the eight celestial spheres at 986A ff. and of the elementary spheres at 981C ff. Aether is defined in this dialogue as intermediate between fire and air—e.g., at 984B ff.—and Ficino does not count it as a separate substance. Aristotle's view was of course quite different.

him with "the answer."^[63] Ironically, the one Neoplatonic treatise he would surely have been most excited and convinced by, Proclus's thirteenth treatise in his Republic Commentary, was completely unknown to him and to his contemporaries, as we shall see; and the work he looked to most consistently for help with Platonic mathematics, a treatise by Theon of Smyrna, a Middle Platonist, has nothing whatsoever to say about Plato's great mathematical crux.^[64] In fact, Ficino's best guides remained the other texts of Plato himself, as our analysis of the Epinomis epitome has already in part indicated, though Auguste Diès has suggested, perversely, that Plato may have wanted to throw his readers off the scent by endowing technical terms here with different meanings than he had allotted them elsewhere.^[65]

As always with a medieval and Renaissance scholar, the question of "sources" is complicated; in Ficino's case particularly so, given his eclectic methods and wide scholarship, his continual reworking of ideas and motifs throughout his life, his recourse at times to secondary guides—compendia, epitomes, and digests—and on occasions his failure (or perhaps his refusal even) to identify his authorities, let alone his specific sources. One should add, however, that his scholarly standards, if we compare them with those of the majority of his contemporaries, were exceptionally rigorous.

We know by virtue of his explicit reference that he knew Theon of Smyrna's three-book (originally apparently five-book) treatise, Expositio Rerum Mathematicarum ad Legendum Platonero Utilium , an elementary work in Greek on arithmetic and the types of numbers, and on the theory of musical harmony and astronomy. It is valuable for its citations from a number of pre-Euclidean mathematicians, and notably for its long passages quoted verbatim from Adrastus of Aphrodisias and Thrasyllus. Indeed, John Dillon asserts that it is "essentially a compilation from these two immediate sources."^[66] Dating from the

[63] For a comprehensive index testimoniorum for 546A–D, see Boter, Textual Tradition , pp. 345–346. Perhaps half of these would have been unknown, however, to Ficino and his contemporaries.

[64] Diès, Essai , p. 11, remarks, "Il est presque étonnant de voir combien sont rares et maigres, en dehors de Proclus, les allusions ou commentaires des anciens relativement au Nombre de Platon." But see n. 63 above.

[65] Essai , pp. 132 ff., with a list of such terms.

[66] The Middle Platonists, 80B.C. to A.D. 220 (London and Ithaca, N.Y., 1977), pp. 397–399 at 397; Theon's "borrowings" have been known for some time. See also Michel, De Pythagore , pp. 119–120, and Diés, Essai , p. 28. This is not the Theon of the late fourth or early fifth century A.D. who was Hypatia's father (Vita Isid . fr. 104; see R. T. Wallis, Neoplatonism [London, 1972], p. 139).

first half of the second century A.D. , it is usually referred to by its Latin title simply as the Expositio .^[67] We can deduce, furthermore, from a notice in a letter Ficino wrote to Angelo Poliziano on 6 September 1474 or shortly thereafter,^[68] that sometime before that Ficino had translated the first part of the Expositio into Latin, though he never published the translation and had probably never intended to publish it. It has only survived, albeit anonymously, in the Vatican library's MS Vat. lat. 4530, fols. 119–151, and in Hamburg's Staats- und Universitätsbibliothek's MS cod. philol. 305, fols. 139–191v (a manuscript that was copied from the Vatican MS by Lucas Holstenius in the seventeenth century).^[69] Though anonymous, the Expositio follows in both manuscripts upon a Latin version, which has been convincingly attributed to Ficino, of Iamblichus's De Secta Pythagorica Libri Quattuor , a collection of four treatises consisting of the De Vita Pythagorica , the Protrepticus , the De Communi Mathematica Scientia , and the In Nicomachi Arithmeticam Introductionem .^[70] In Sebastiano Gen-

[67] It was edited by Eduardus Hiller for Teubner in 1878 and by Dupuis, with a French translation, in 1892 (see n. 4 above). The text has come down to us in bad condition; see J. Gilbart Smyly, "Notes on Theon of Smyrna," Hermathena 33 (1907), 261–279, who offers fourteen emendations to Hiller's edition. There is an English translation from Dupuis by Robert and Deborah Lawlor entitled Mathematics Useful for Understanding Plato (San Diego, 1979).

[68] Ed. Sebastiano Gentile, Marsilio Ficino: Lettere I: Epistolarum Familiarium Liber I (Florence, 1990), pp. cclix, 44–45 (no. 20); trans. in Letters 1:59–60 (no. 21). See also Kristeller, Supplementum 1:1.

[69] Kristeller, Supplementum 1:xxx–xxxi, xl–xli, cxlvi–cxlvii, 1, 3; idem, Ficino and His Work , pp. 90 (sub Hamburg, cod. phil. 305), 108 (sub Vatican, Vat. lat. 4530), and 136 (no. 37); Sebastiano Gentile, "Sulle prime traduzioni dal greco di Marsilio Ficino," Rinascimento , 2d ser., 30 (1990), 57–104 at 74n. The Vatican MS is written in the hand of Elia del Medigo with a few notes by Pier Leoni da Spoleto, Lorenzo's learned, bibliophile physician; it was once possessed and annotated by Giovanni Pico della Mirandola. The Hamburg MS has been missing, according to Kristeller, since World War II.

[70] Ficino probably worked from one or both of the Laurenziana's Greek MSS. 86.3 and 86.29 (which was copied from 86.3). See Kristeller, Supplementum 1:xxx, xl–xlii, cxlv–cxlvi; idem, Ficino and His Work , pp. 74–75 (sub Laur. 86.3 and 86.29), and 136 (no. 35); Martin Sicherl, "Platonismus und Textüberlieferung," in Griechische Kodikologie und Textüberlieferung , ed. Dieter Harlfinger (Darmstadt, 1980), p. 555; and Gentile in Mostra , pp. 32–34 (no. 24—the Laurenziana's 86.29). The contents-list of MS. 86.3 gives us the titles of five more treatises in what Iamblichus must have intended to be a kind of Pythagorean encyclopedia; see D. J. O'Meara, Pythagoras Revived: Mathematics and Philosophy in Late Antiquity (Oxford, 1989), part 1, esp. pp. 91–101.

Apart from MS Vat. lat. 4530 (and its Hamburg copy), Ficino's Latin version of the De Secta Pythagorica also survives in the Vatican's MS Vat. lat. 5953, which was copied by Luca Fabiani and owned by Pier Leoni; though lacking the Theon, it includes

Ficino's translation of Hermias and other texts and of his Philebus Commentary. See Kristeller, Supplementum 1:xli–xlii; idem, Ficino and His Work , pp. 108 (sub Vat. lat. 5953), 136 (no. 35); Gentile in Mostra , p. 33; idem, "Sulle prime traduzioni," pp. 73–74 and 80n (with a listing of the more glaring omissions in the De Vita Pythagorica rendering—first noted by H. Pistelli in 1893—and of some of the far fewer omissions in the renderings of the other treatises).

There is no modern edition, incidentally, either of Ficino's Theon translation or of his De Secta Pythagorica (which I consulted only in MS. 4530).

tile's words there is no doubting the Theon translation's "paternità ficiniana."^[71] Moreover, if Gentile is correct in arguing that the "translations" of the treatises constituting the De Secta Pythagorica show the telltale signs of being among Ficino's earliest attempts (being too literal and at the same time inexact) and that they were therefore probably written prior to 1464,^[72] then it would suggest a similarly early dating for the Theon translation, even though our first notice of it is in the Poliziano letter. I have placed "translations" in quotation marks, however, because my own cursory examination of Ficino's rendering of the In Nicomachi Arithmeticam encountered paraphrasing, summarizing, and some omissions (though not on the scale of that found in the De Vita Pythagorica ). Thus, we should probably think of the Iamblichus collection not just as an early but as a personal, working translation only; and this may also be true, as Gentile has suggested, of Ficino's work on Theon. The question awaits further investigation. Presumably, Ficino's copy text for the Expositio was the Laurenziana's 85.9, folios 12v–26r, part of the huge codex he had received from Cosimo de' Medici in 1462 containing the Plato text he was to use principally for his great translation.⁷³

Another parallel resource for Ficino might have been the better organized but less sophisticated treatise, again in Greek, by the Neopythagorean Nicomachus of Gerasa (who probably flourished also in the first half of the second century A.D. ), the two-book Arithmetica Introductio . Nevertheless, this too has nothing specific to say about Plato's number except for a passing allusion at 2.24.11 to the effect that some of the things Nicomachus has just discussed are best illuminated by Plato in the passage in the Republic (i.e., at 546A ff.).^[74] The

[71] Mostra , p. 30; also "Sulle prime traduzion," p. 74.

[72] Mostra , pp. 33–34; also "Sulle prime traduzioni," pp. 75–76.

[73] Kristeller, Ficino and His Work , p. 136 (no. 37); and Gentile in Mostra , pp. 29–30.

[74] Ed. Ricardus Hoche (Leipzig, 1866), p. 131.11–14. There is an English translation by Martin Luther D'Ooge, with a long and useful introduction under the head-

ing "Studies on Greek Mathematics" by Frank Egleston Robbins and Louis Charles Karpinski, entitled Nicomachus of Gerasa: Introduction to Arithmetic , University of Michigan Studies, Humanities Series, vol. 16 (New York, 1926; 2d ed., 1972). It contains some suggested emendations to the Hoche edition on pp. 158–166 and on pp. 37–43 a tabular comparison of Nicomachus's work with Theon's. Helpful too is the French translation and commentary by Janine Bertier, Nicomaque de Gérase: Introduction arithmétique (Paris, 1978); and George Johnson, The Arithmetical Philosophy of Nicomachus of Gerasa (Lancaster, Pa., 1916).

Introductio was translated into Latin by Apuleius, according to a notice by Cassiodorus, though the translation has not survived.^[75] The work was apparently unknown to the younger but still contemporary Theon, but was commented upon expansively by Iamblichus in one of his "Pythagorean" treatises, the In Nicomachi Arithmeticam,^[76] and therefore translated with the others by Ficino, as we have seen. It was also commented upon by Philoponus, by Sotericos, and by Asclepius of Tralles; and it was translated, paraphrased, expanded here and condensed there by Boethius in his De Institutione Arithmetica , and reproduced in part and more distantly by Martianus Capella, Isidore of Seville, and Cassiodorus.^[77] In fact, Ficino's vague allusion to Boethius at one point may be to the De Institutione Arithmetica in general or specifically to 2.46 (which is rendering Nicomachus's Introductio 2.24.11 and therefore refers to Plato's "nuptial" passage in the Republic 8); however, it could equally well be to Boethius's De Institutione Musica or to various passages in his many commentaries on Aristotle.^[78]

Finally, there is the possibility that he might have known the anonymous Theologumena Arithmeticae , which includes notice of Nicomachus's views.^[79] This is often attributed to Iamblichus but may indeed be by Nicomachus; for Nicomachus certainly wrote a treatise of

[75] In De Artibus ac Disciplinis Liberalium Litterarum , chapter 4, De Arithmetica . See Migne's Patrologia Latina 70.1208B.

[76] Ed. Hermenegildus Pistelli (Leipzig, 1894), rev. Udalricus Klein (Stuttgart, 1975). For a summary, see Robbins in D'Ooge, Nicomachus , pp. 126–131.

[77] See Robbins and Karpinski in D'Ooge, Nicomachus , pp. 132–142; also Bertier, Nicomaque , p. 9.

[78] The De Institutione Arithmetica and the De Institutione Musica were edited together by Gottfried Friedlein (Leipzig, 1867). The reference at 2.46 (p. 151.22–25) in the former reads: "Hoc autem facilius cognoscitur ex lectione Platonis in libris de republica eo loco, qui nuptialis dicitur, quem ex persona musarum philosophus introducit." In general see Michel, De Pythagore , pp. 134–136.

[79] Ed. Victorius de Falco (Leipzig, 1922). There is an English translation by Robin Waterfield entitled The Theology of Arithmetic: On the Mystical, Mathematical and Cosmological Symbolism of the First Ten Numbers (Grand Rapids, Mich., 1988).

that name.^[80] Interestingly, a manuscript containing the Theologumena appears in the Laurenziana as Plut. 71.30. It has notations by Poliziano (though these are not on the Theologumena , which appears on fols. 92–145) and was copied apparently from a manuscript of Bessarion's now in the Marciana as Marc. gr. 234 (667).^[81] The two manuscripts and others assuredly testify to the awareness at least of the text in Platonic circles.

We should also recall a tradition surely known to Ficino from Marinus's Vita Procli 28 to the effect that Proclus claimed to be the reincarnation of Nicomachus's soul, having been born 216 years after Nicomachus's death. Two hundred and sixteen years is the Pythagorean number assigned to the interval between lives, since it is the cube of 6 and also the sum of the cubes of the three numbers of the perfect Pythagorean triangle, i.e., of 3, 4, and 5.^[82] This would effectively invest Nicomachus with Proclus's authority, or at least validate his status as a Platonist-Pythagorean. Nevertheless, Ficino never mentions him anywhere in his Opera even though he must have known of him.

In the argumentum for the Republic book 8, having dismissed Theon, Ficino dismisses Iamblichus also, declaring that although Iamblichus had tried to unravel Plato's knot, he had only succeeded in making it the tighter. This is an explicit reference either to Iamblichus's In Nicomachi Arithmeticam 82.20–24, 83.13–18, or, more probably, to his De Vita Pythagorica 27.130–131, though in neither passage does Iamblichus determine Plato's number.^[83]

[80] Compare, for instance, De Falco's attribution to Iamblichus with Bertier's to Nicomachus (p. 9). Friedrich Ast, who edited the Theologumena in 1817 along with Nicomachus's Introductio , rejected Nicomachus's authorship without accepting Iamblichus's (p. 157); while Robbins in D'Ooge, Nicomachus , pp. 82–87, argued it was basically Nicomachean even if Iamblichus was its compiler.

De Falco's apparatus adduces numerous parallels between the Theologumena and the treatises and commentaries by Nicomachus, Theon, Iamblichus, and several others, including Proclus's In Timaeum .

[81] De Falco, p. v.

[82] John M. Dillon, "A Date for the Death of Nicomachus of Gerasa," Classical Review 19 (1969), 274–275. Note that the Theologumena claims in the section on the hexad that certain Pythagoreans had declared that Pythagoras himself was reincarnated every 216 years (trans. Waterfield, pp. 83–84). See Dupuis, Nombre , p. 54. In the Republic 8 at 546C6, "hekaton de kubôn triados ," Plato himself may be alluding to 216 (as the cube of 6 as 1+2+3); cf. Schneider, Platonis Opera Graece 3:xxxi. It is also of course the product of multiplying the first two cubes, 8 and 27, the "feet" of the Timaeus 's lambda.

[83] MS Vat. lat. 4530 renders neither of these passages into Latin. Indeed, Ficino is barely summarizing the larger contexts in which they appear.

In the De Vita Pythagorica 27.130–131 Iamblichus asserts that Pythagoras "constructed, as it were, three lines, representing forms of government, and connected them at the ends to make a right-angled triangle: one side has the nature of the epitritos, the hypotenuse measures five, and the third is in the middle of the other two. If we calculate the angles at which the lines meet, and the squares on each side, we have an excellent model of a constitution. Plato appropriated this idea, when he expressly mentioned, in the Republic, the first two numbers in the ratio of four to three which join with the fifth to make the two harmonies" (trans. Gillian Clark, Iamblichus: On the Pythagorean Life [Liverpool, 1989], p. 58). See Clark's note ad loc. on the confusions in the passage, particularly the identification of the side of 3 with "the nature of the epitritos": these may be precisely the confusions that Ficino has in mind. See also Schneider, Platonis Opera Graece 3:xxxiiii–xxxvii, and Diès, Essai , pp. 38–39.

We might note that there is insufficient evidence to determine Iamblichus's views on the Republic , which is mentioned only in passing in the two recent studies on Iamblichus as a commentator on Plato, namely, John M. Dillon's Iamblichi Chalcidensis in Platonis Dialogos Platonis Commentariorum Fragmenta (Leiden, 1973), and Bent Dalsgaard Larsen's Jamblique de Chalcis: Exégète et philosophe (plus supplement, Testimonia et Fragmenta Exegetica ) (Aarhus, 1972).

Iamblichus's De Communi Mathematica Scientia Liber , ed. Nicolaus Festa (Leipzig, 1891; rev. Klein, Stuttgart, 1975) also alludes in various places to Nicomachus, but not to Plato's passage.

Ficino probably scanned two other ancient authorities—both of them eminent Platonici in his genealogical tree of the Platonic wisdom—for their views on Plato's celebrated crux, though he only mentions one of them once and in passing in his De Numero Fatali .

In his notable essay, De Iside et Osiride 56 (Moralia 373F ff.), Plutarch (A.D. c. 46–c. 120) speaks of the right-angled scalene triangle so dear to the Pythagoreans, and observes parenthetically that "Plato seems to avail himself of this triangle in the Republic in order to form the nuptial figure (to gamêlion diagramma syntattôn ). In it the vertical side is worth 3, the base 4, and the hypotenuse, whose square equals the sum of the squares of the other two sides, is worth 5." It is "the most beautiful of triangles" to Plutarch (presumably because all three sides are rational whole numbers).^[84] This would have certainly confirmed Ficino's assumption, which he derived from Aristotle's gloss, that 12 was the secret key to the Platonic riddle. It also suggests, as Depuis notes, that Plutarch was unfamiliar with any comprehensive interpretation of the passage.^[85] Schneider, Dupuis, and others have

[84] Adduced in this context by Schneider, Platonis Opera Graece 3:xxxii; Dupuis, Nombre , p. 19; Adam, Republic , p. 267; and Diès, Essai , p. 25. Plutarch goes on to identify the 3, 4, and 5 respectively with the Egyptian triad of Osiris, Isis, and Horus.

[85] Théon , p. 372. Both Schneider, Platonis Opera Graece 3:xxxiii, and Diès, Essai , pp. 25–26, also cite Plutarch's De Animae Procreatione 10 (Moralia 1017C): "In the Republic , Socrates, when he begins to speak about the number that some refer to as the nuptial number, says: 'A divine object of generation has a period that is comprised by a

perfect number' [546B3–4], what he calls a divine object of generation being nothing other than the universe." But Plutarch then proceeds to define the tetraktys and at 1018C to treat of six as a perfect number which is called "marriage" because of "the commixture of the even and the odd."

adduced too a similar passage from the treatise On Music 3.23 by Aristides Quintilianus (probably third or fourth century A.D. ); but it is less likely though not impossible that Ficino had read it. It argues that "the sides of the triangle being 3, 4, and 5, if we take the sum of them, we obtain the number 12; . . . the sides at the right angle are in the relationship of epitritus [4:3], and it is the root of epitritus added to 5 that Plato is referring to [in the Republic ]."^[86] The observations here not only speak to the importance of the Pythagoreans' "beautiful" triangle but underscore the importance of the sum of its sides being 12, and the fact that the "root" of epitritus means 3 plus 4. We might note that other Plutarchan essays familiar to Ficino address a variety of related mathematical topics: these include the De Musica 22 on the harmonic means; the De E apud Delphos on the properties of the number 5; and, as we have seen, the De Animae Procreatione in Timaeo Platonis , especially chapters 11–20 and 29–30, on Plato's philosophy of numbers and the harmonic means and intervals.^[87]

The second authority, and the most problematic, was certainly Proclus (A.D. 412–485), the Platonist Ficino knew most thoroughly after Plato and Plotinus and to whom he was deeply indebted throughout his career. Indeed, Ficino must have at one time turned to the Successor as his best hope. For he first encountered the opening half of Proclus's huge commentary on the Republic in 1492, after Janus Lascaris had purchased a manuscript of the first twelve treatises in Greece, probably in Crete, and sent it in excellent condition to Florence to Lorenzo's library, where it eventually became the Laurenziana's 80.9.^[88] Ficino must have borrowed it almost immediately,

[86] Schneider, Platonis Opera Graece 3:xxxiii, and Dupuis, Théon , p. 372, citing from the edition by Meibom in his Antiquae Musicae Auctores Septem (Amsterdam, 1652), 2:1–164 at 152 (i.e., in the more recent ed. by R. P. Winnington-Ingram, De Musica [Leipzig, 1963], p. 124.25–26). See also Adam, Republic , p. 267, and Diès, Essai , p. 39.

[87] Ed. and trans. Cherniss, pp. 262–321. Michel, De Pythagore , pp. 151–153. See also Schneider, Platonis Opera Graece 3:xxxii.

[88] This is in fact the first half of a single codex of more than four hundred leaves dating from the ninth or tenth century. It was later divided, and by the fifteenth century the first half belonged to a certain Armonios or Harmonios of Athens. See the edition by Wilhelm Kroll, Procli Diadochi in Platonis Rem Publicam Commentarii , 2 vols. (Leipzig, 1899–1901), 1:i–vii; also the French translation by A.-J. Festugière, Proclus: Commentaire sur la République , 3 vols. (Paris, 1970), 2:7–8. Kroll's first volume con-

tains the first twelve treatises, his second the last five. The only allusion in the first twelve treatises to the crucial passage in the Republic 8:546A1–D3 occurs in the seventh treatise at 1:219.10 ff., where Proclus asserts that a state is undermined when its magistrates neglect not so much gymnastics as mousikê , "as Socrates declares in the Sacred Discourse of the Muses."

for we have a note attesting to his loan dated 7 July 1492.^[89] By as early as 3 August 1492 he had gathered some "flowers" from its "delightful meadows" which he epitomized in a letter to his close friend Martinus Uranius (alias Prenninger) and later published in 1495 in the eleventh book of his Letters .^[90] However, from Proclus's massive treatise Ficino received in fact no illumination. For the meadows he had wandered in treat only of the first seven books of the Republic , and Proclus does not deal with the Discourse of the Muses in book 8 until his thirteenth treatise, the Melissa . But this Ficino and his contemporaries could not have known, since the second half of Proclus's commentary—now the Vatican's MS Vat. gr. 2197—did not arrive in the West until years later (how many exactly I cannot discover)^[91] and was for all intents and purposes hidden from the scholarly world until the appearance in 1886 of Richard Schoell's edition.^[92] Thus, notwithstanding his erudition, Schneider was completely unaware of its existence in 1830, and even more tellingly Dupuis was unaware of it as late as 1881. As Diès observes, "les recherches sur le nombre géométrique de Platon durent se poursuivre, même après la Renaissance,

[89] Kristeller, Ficino and His Work , pp. 73 (sub Laur. 80.9), 125–126; idem, "Proclus as a Reader of Plato and Plotinus, and His Influence in the Middle Ages and the Renaissance," in Proclus: Lecteur et interprète des anciens (Paris, 1987), pp. 191–211 at 203; Gentile in Mostra , pp. 151–152 and plate XXXIV (no. 117—the Laurenziana's 80.9); Viti in Mostra , p. 189 (no. 160).

[90] Opera , pp. 937.2–943.1. See Gentile in Mostra , p. 152; Marcel, Marsile Ficin , pp. 524 ff. For another partial translation, that by Nicolaus Scutellius of Trent in 1526, see Paul Oskar Kristeller, Iter Italicum , 6 vols. (London and Leiden, 1963–1991), 1:409.

[91] It came to the Salviati in Florence (certainly long after Ficino's death) and eventually passed into the hands of the Colonnas, and thence in 1821 to the Vatican. See Diès, Essai , pp. 53–54 (but for "via Cologne" on p. 53 read "via the family of the Colonna"); and John Whittaker, "Varia Procliana," Greek, Roman, and Byzantine Studies 14.4 (1973), 427–428.

[92] Procli Commentariorum in Rempublicam Platonis Partes Ineditae (Berlin, 1886). Schoell edited the only version available to him at the time, a mid-seventeenth-century copy (now the Vatican's MS Barberinianus gr. 65) made by Lucas Holstenius (and in part perhaps by Leo Allatius) of a portion from the Salviati's MS. Two years later J. B. Pitra edited the Vaticanus gr. 2197 directly but badly. It was not, therefore, until Kroll's second volume of 1901 that the last half of Proclus's work was authoritatively edited and reconstructed.

comme si Proclus n'eût pas existé."^[93] Furthermore, even had Ficino been able to gain access by some stroke of fortune to this second half, his interpretative skills would have been challenged to the utmost, for its leaves, and notably those containing the Melissa , had probably already sustained some at least of their present damage.^[94]

While he did not know the pertinent treatise of Proclus's commentary on the Republic , however, he was certainly well acquainted with Proclus's Timaeus Commentary and its detailed analysis of the loci mathematici in that dialogue. Also, it is just possible he had skimmed through Proclus's commentary on the first book of Euclid where there are some obvious references to Plato's passage. The prologue, for instance, declares first that "matters pertaining to powers (dunameis ) . . . whether they be roots or squares . . . Socrates in the Republic puts into the mouth of the loftily-speaking Muses, bringing together in determinate limits the elements common to all mathematical ratios and setting them up in specific numbers by which the periods of fruitful birth and its opposite, unfruitfulness, can be discerned"; and then again that the Republic 's "geometrical number" is "the factor that de-

[93] Diès, Essai , p. 54 (cf. p. 4). Hence Bodin's comment in his 1581 Apologie de René Harpin , f. 41 (and repeated in his 1586 De Republica Libri Sex , p. 412), that Proclus had learnedly interpreted the first seven books of the Republic but demurred before the difficulty of the eighth book where it was a question of numbers—cited in Dupuis, Théon , p. 391, and Diès, Essai , p. 54: "Proclus septem quidem Platonis de Republica libros satis accurate interpretatus octavum attingere noluit, rei difficultate, ut opinor, revocatus."

[94] In 1640 Holstenius was already lamenting its condition, which he blamed on the Salviati, perhaps unjustly. In general all the upper borders are in very bad condition, and accordingly Kroll's text is full of lacunas or conjectural reconstructions. With regard specifically to the thirteenth treatise, missing are the introduction, and the first eight and a half paragraphs of forty-five, though Kroll was able ingeniously to reconstruct the initial two leaves of the introduction from sixteenth-century copies (see his 2:vi, 473, and apparatus for pp. 1–4). Missing, in other words, are the very paragraphs which must have glossed the twenty-three lines (in Burnet's edition) of the Republic 's 546A1–D1 (though it is just possible they were still there in the late fifteenth century). Proclus's detailed commentary currently begins at 2.4.24, glossing 546D2's "ouk euphueis oud' eutucheis paides esontai ."

For the bibliographical complexities, in addition to Kroll's remarks, see Festugière, Proclus: Commentaire sur la République 2:7–8, 105n, 108n; and, more speculatively, Carlo Gallavotti, "Intorno ai Commenti di Proclo alla Repubblica," Bollettino del Comitato per la preparazione dell'edizione nazionale dei classici greci e latini 19 (1971), 41–49. Gallavotti, pp. 45–47, believes that Proclus's commentary consists of six heterogeneous treatises written at different times and with different aims and later collected together.

For accounts of aspects of the Melissa 's exceedingly complex exegesis, which is variously and ambitiously arithmetical, geometrical, musical, astronomical-astrological, cos-

mological, psychological, and dialectical, and for which Proclus was probably indebted in part at least to earlier exegetes, see the appendix by Friedrich Hultsch (himself an important figure in the history of the modern interpretation of Plato's Number) in Kroll's second volume, pp. 384–415; also Diès, Essai , pp. 28–51, and especially 40–51. I have not seen A. G. Laird's Plato's Geometrical Number and the Comment of Proclus (Madison, Wis., 1918). We should note that Proclus has enriched our understanding of the ancient interpretative tradition for this passage by referring to a number of authorities necessarily unknown to Ficino, as Diès's second chapter of his Essai , "La tradition antique et Proclus," demonstrates.

termines whether births will be better or worse."^[95] However, in the analysis of proposition 47 near the very end of his commentary, having noted that the hypotenuse and side of an isosceles right triangle cannot both be expressed in rational numbers, Proclus turns to the Pythagoreans' "beautiful" scalene, where indeed the "square on the side subtending the right angle is equal to the squares on the sides containing it," and boldly declares, perhaps echoing Plutarch, "Such is the triangle in the Republic , where sides of three and four contain the right angle and a side of five subtends it."^[96] By contrast, as we shall see, Ficino will take up the isosceles triangle, not the exemplary scalene, as the key to Plato's mystery. Characteristically, moreover, he will fail to mention Proclus at all in his De Numero Fatali ,^[97] except to say once, at the end of chapter 7, that Plotinus and Proclus had proven "most subtly that numbers exist in the prime being itself as the first distinguishers there both of beings and of ideas."

Indeed, given Ficino's profound, acknowledged, and lasting indebtedness to Plotinus, and given that he had just finished translating and analyzing the Enneads in their entirety—his Plotini Enneades being published in 1492—we might have expected certain Plotinian treatises to be in the forefront of his mind; and notably perhaps 6.6 [34 in the chronological order] entitled "On Numbers," one of the great meditations of Plotinus's maturity. But Plotinus's concerns here are exclusively ontological, and he gives no indication of being influenced by, or interested in, the arithmological tradition as developed

[95] Ed. Gottfried Friedlein (Leipzig, 1873), pp. 8.12 ff., 23.21 ff.; trans. Glenn R. Morrow as Proclus: A Commentary on the First Book of Euclid's Elements (Princeton, 1970), pp. 7, 20.

[96] Ed. Friedlein, pp. 427.25 ff.; trans. Morrow, pp. 339–340. Cf. Schneider, Platonis Opera Graece 3:xxxii–xxxiii (his sole reference to Proclus since he did not know of the Melissa !); Dupuis, Nombre , p. 20; Adam, Republic , p. 267; and Diès, Essai , p. 77.

[97] See my Platonism , pp. 249–255.

by the Pythagoreans.^[98] For him, as apparently for the later Plato, ordinary quantitative numbers are merely images of the ideal numbers, which, he argues, on the basis of his metaphysical conviction that the One is above Being, are in Intellect but higher than other Ideas. These ideal numbers are thus at the very apex of the intelligible world and serve as the principles of being, as the highest level of Ideas, as the measures of all reality. Indeed, according to Porphyry's Life 14.7–10, Plotinus seems to have dismissed the preoccupations of ordinary mathematicians as irrelevant to the philosopher, though he was well acquainted with Plato's various mathematical concerns and alludes to the account in the Timaeus 39BC and 47A ff. of the origins of man's idea of number in his exposure to the alternation of night and day. Indeed, despite the De Numero Fatali and various disquisitions of his own on the musical proportions, Ficino probably willingly embraced this Plotinian dismissal, sanctioned as it was by such passages in the Republic as 7.529CD where Socrates insists that genuinely philosophical astronomy is concerned with "true" number and figure and not with the visible motions of the heavenly bodies. Be that as it may, the larger underlying issues of the passage in the Republic 8, namely the nature and function of the celestial circuits and their role in the providential plan, and the question of man's freedom of choice in the midst of a sensible reality governed by destiny, are very much Plotinian issues and figure prominently in 3.2–3 [47–48], the late treatise on providence, in 2.3 [52], the even later treatise on astrology, and in 3.1 [3], the early treatise on destiny. Nonetheless, despite his fundamental Plotinianism, one does not sense here the presence, or at least the pressure, of Plotinian texts, except perhaps in his concluding chapter on astrology.^[99]

[98] See A. H. Armstrong's comments in the headnote to his translation of the treatise in the seventh and last volume of his Loeb Plotinus (Cambridge, Mass., and London, 1966–1988), pp. 6–8: "he does not seem to have been very much interested in or affected by the Pythagorean or Pythagoreanizing numerologists." Armstrong adduces Janine Bertier's comments to the same effect in her edition of the treatise, along with translation and commentary (Paris, 1980), introd., pp. 9–10. He does, however, draw our attention to Plotinus's discussion of Aristotle's account of ideal numbers in the Metaphysics A, M, and N. We should note that Plotinus had a full knowledge of the subject and recommends its practice; see his Enneads 1.3 [20].5.5–10.

[99] The mention of Plotinus and Proclus at the conclusion of chapter 7 already cited above is Ficino's only allusion in the De Numero Fatali to the problem of the Platonic mathematicals and their status vis-à-vis the Ideas.

We should recall that for the Greeks in general the unit is outside the number series, Aristotle defining number as "a plurality of units" or alternatively as a "limited plural-

ity" (Metaphysics 1039a12, 1053a30 versus 1020a13; see Guthrie, History of Greek Philosophy 5:439n). The Phaedo 101B9–C9 declares that there are Forms of numbers in which individual numbers participate (see Guthrie, History 4:523)

Given the absence of any unequivocal mention of an intermediate class of mathematicals in the dialogues themselves (see Guthrie, History 4:343–345, citing Cherniss), Ficino's primary source for the Platonic status of mathematicals was, ironically, Aristotle, who declares in his Metaphysics 987b14–18 and ff.: "Besides the sensibles and the Forms Plato posits mathematical objects in between, differing from the sensibles in being eternal and unmoved, and from the Forms in that there are many alike, whereas the Form itself is in every case unique." Again at 997b1 he writes of "the Forms and the intermediates, with which they say the mathematical sciences are concerned." For these and other such passages in Aristotle, see Adam, Republic 2:160, and Guthrie, History 4:343, 523.

While mathematicals are nowhere named in the Republic , later interpreters have argued they are part of the upper half of the Divided Line at 509D. While the matter of what Plato himself believed has been the subject of endless controversy among modern scholars, Ficino was in no doubt of their existence, given the Neoplatonic tradition and above all the testimony of Proclus.

The Republic 7.521C–531D deals with the mathematical education of the guardians (i.e., Ficino's magistratus ); see F. M. Cornford, "Mathematics and Dialectic in Republic VI and VII," in Studies in Plato's Metaphysics , ed. R. E. Allen (London, 1965), pp. 61–95.

In short, having found no guidance earlier in the Platonic tradition, and having wandered earnestly in the "delightful meadows" of the first twelve treatises of Proclus's Republic Commentary that had come to his attention as late as 1492 and still found nothing, Ficino must have gradually concluded that he would have to attempt an independent explication of the geometric number. For the mathematical treatises of Theon, of Nicomachus, and of Iamblichus, the extant philosophical treatises of his two most revered Platonic authorities, Plotinus and Proclus, the essays even of Plutarch—all had maintained a judicious Pythagorean silence. The sources of Ficino's wider knowledge of astronomy, judicial astrology, and harmonics are of course another matter, but would include Ptolemy, Calcidius, Macrobius, Martianus Capella, Proclus again, Boethius, and a number of medieval figures, along with medieval epitomes and handbooks.

Thus the starting point for him clearly remained: first, the contentious passage in the fifth book of Aristotle's Politics ; and second, what Plato had to say about the cosmological significance of numbers and their proportions in the Timaeus^[100] and Epinomis . These texts—

[100] Indeed, as Diès, Essai , p. 20, notes, the Timaeus remained a point of departure for exegesis until the twentieth century: "Toutefois, les exégètes du Nombre platonicien trouveront leur bien dans ces calculs sur la composition de l'âme du monde, et le double quaternaire, pair et impair, 1, 2, 4, 8, 1, 3, 9, 27, continuera, au moins à partir du XVI siècle jusqu'à nos jours (Dupuis, Hultsch, etc.), d'être invoqué pour expliquer les 'accroissements à trois intervalles et quatre termes.'"

along of course with the Platonic lemmata of 546A-D^[101] —account for the musical and astronomical-astrological cast of the argument throughout Ficino's De Numero Fatali , and for its concern with why a perfectly constituted state must necessarily decline along with all other things after what is a finite term, however vast, however indeterminable it may seem in the darkened glass of our understandings. At stake, as the last chapter testifies, is the status of astrological disposition and influence in the providential order, and thus the problematic relationship between man's divinely ordained freedom and the motion of the stars—the relationship, that is, between transitory human time and what the Timaeus 40C calls the intricate "choric dances" of celestial time.

[101] There were, incidentally, no scholia particularly useful for Ficino here except possibly for the longish gloss on 546B3's "theiôi men genêtôi, " and the diagrams of three "perfect" scalene triangles (with sides of 3–4–5, 9–12–15, and 27–36–45 respectively) keyed to 546C4's "promêkei de, " representing a Proclan explanation of the phrase "at the third augmentation." See William Chase Greene, ed. Scholia Platonica, American Philological Association Monograph 8 (Haverford, Pa., 1938; reprint Chico, Calif., 1981), pp. 256–257; and Boter, Textual Tradition, pp. 345–346.