2
Figured Numbers and the Fatal Number
"Magnus ab integro saeclorum nascitur ordo"
In order to understand Ficino's unraveling of Plato's mathematical mystery in his commentary on the Republic 8, we must first familiarize ourselves briefly with aspects of the basic terminology of traditional Pythagorean arithmogeometry, arithmology, and the lore of figured numbers, as Ficino himself had become acquainted with them earlier in his career by way of Theon of Smyrna's Expositio . We must bear in mind that his mathematical explanations and excursions here are oriented towards one particular goal: the interpretation of perhaps the most riddling passage in the Plato canon. Certainly, he never intended his commentary to serve as a counterpart to, or even as a compendium of, the various ancient introductions to mathematics, notably those by Theon himself and by Nicomachus and his commentators. Portions of his own earlier Timaeus Commentary had to a degree already served that purpose, especially with regard to promoting a Platonic understanding of musical proportions and harmonics and of the crucial role they had played in the Creator-Demiurge's structuring of the material world and of the World-Soul and other souls.[1] It is the Timaeus indeed, not Aristotle's Politics , that provides us with our starting point.
From his earliest years as a scholar, the Timaeus up to 53C was fa-
[1] See especially chapters 28–34 (Opera , pp. 1451.2–1460.2).
miliar to Ficino in the Latin translation embedded in Calcidius's great commentary;[2] and he had learned to interpret it initially through the Middle Platonic, or possibly Neoplatonic, spectacles of that commentary.[3] Subsequently he mastered the Greek original and then turned to study the other great Timaeus commentary extant from antiquity, and for him the more authoritative of the two because unquestionably and profoundly Neoplatonic, the massive and difficult work by Proclus, though once again he only had access to a manuscript containing the first half.[4] As a consequence, his own Timaeus Commentary seems to have passed through a number of drafts as he became more and more adroit or confident in handling the dialogue's profusion of ideas and images: it is one of his very first Platonic labors and also one of his last, and it incorporates several chronological layers of interpretation.[5]
When Socrates observes in book 8 of the Republic that the geometric number is a "human" and imperfect number, and that it has four terms and three intervals related to each other in certain proportions, to elucidate his meaning seems to require, at least for a Neoplatonist
[2] Milan's Biblioteca Ambrosiana has a MS (S. 14 sup.) containing this commentary on ff. 4–98v with abundant marginalia and bearing the arms of Ficino—two stars on either side of an upright sword—and a note on f. 172r that he had copied out the whole MS during February and March of 1454 (Florentine style) when he was only twenty. See Raymond Klibansky, The Continuity of the Platonic Tradition during the Middle Ages (Munich, 1981), p. 30; Kristeller, Supplementum 1:liv; idem, Iter 1:342; idem, Ficino and His Work , pp. 93–94; and Gentile in Mostra , pp. 7–8 (no. 6).
[3] In the introduction to his edition, Timaeus a Calcidio Translatus (see Chapter 1, n. 2 above), Calcidius's distinguished modern editor J. H. Waszink has argued for the influence on Calcidius (whom he assigns to the first half of the fourth century) of the Timaeus Commentary by Plotinus's leading disciple and biographer, Porphyry (c. 232–305 A.D. ). Stephen Gersh, Middle Platonism and Neoplatonism: The Latin Tradition , 2 vols. (Notre Dame, Ind., 1986), 2:421–492, also argues for the influence of Porphyry and of the Neopythagorean Numenius and assigns Calcidius's activity to the late fourth and early fifth centuries (p. 424n). Dillon, Middle Platonists , pp. 401–408, on the other hand, disputes the presence of anything other than Middle Platonic sources, while admitting that linguistic considerations would suggest the fifth rather than the fourth century for its composition (p. 402). Ficino was obviously not in the position of being able to assess Calcidius in these terms. While he speaks of him as a Platonist and lists his commentary among the Platonic books to be found among the Latins in a letter to Martinus Uranius (alias Prenninger) of June 1489 (Opera , p. 899), he refers to him only twice in his own Timaeus Commentary—in chapters 19 and 42 (Opera , pp. 1446.1, 1463.2)—and only rarely elsewhere. Certainly he never accorded him the stature he accorded Plotinus, Proclus, and the Areopagite.
[4] MS Ricc. gr. 24—this ends at sômasi in the middle of the third book (ed. Diehl, 2:169.4)—see Chapter 1, n. 22 above. For Ficino's debts to the Proclus commentary, see Gentile in Mostra , pp. 109–110 (no. 87), and my "Ficino's Interpretation of Plato's Timaeus ," pp. 422–426, 431–434.
[5] Allen, "Ficino's Interpretation of Plato's Timaeus ," pp. 402–403 and n., with further references.
committed to a synoptic view of the canon, recourse to the wellknown argumentation of the Timaeus at 35B ff. and 43D. Here Timaeus deals with the generation of the first two cubes of 8 and 27 by way of the two quaternary sequences 1–2–4–8 and 1–3–9–27, which commentators since antiquity have visualized as a lambda, the eleventh letter in the Greek alphabet. In the process he invites us to examine the proportional relationship between 8 and 27 in terms of two means, 12 and 18,[6] and thereby establishes a set of fundamental proportions or what we now think of as ratios (though Euclid and Nicomachus, for instance, had insisted that "proportion" should be reserved only for a relationship between at least three terms embracing two ratios).[7] The Pythagoreans and the Platonists found it significant
[6] At 31B–32B Timaeus had noted that, while square numbers such as 4 and 9 require only one geometric mean (in this case 6), cube numbers require two means; cf. Ficino's Timaeus Commentary 19 (Opera , p. 1446.1), and Euclid, Elements 8, props. 11 and 12. With these means, the lambda, which signifies 30, is transformed, like the tetraktys, into a triangle, into a delta signifying the all-important 4! Even so, 6 remains its key in that the products of the three descending steps as it were of the lambda—that is, of 2x3, of 4x6x9, and of 8x12x18x27—are all powers of 6, 216 being its cube, and 46,656 being the square of that cube (or 6 x6 or 6 x6 x6 or 6 ).
[7] Euclid, Elements 5, defs. 3–5, 8 (and in general defs. 1–18), and 7, def. 20; Nicomachus, Introductio 2.21.2–3, 2.24; cf. Aristotle, Nicomachean Ethics 5.3.1131a31 ff. See Michel, De Pythagore , pp. 366–369, and Fowler, Mathematics of Plato's Academy , pp. 16–21. In his Expositio Theon takes logos to mean "the relationship of proportion or ratio" or the "ratio of proportion" (2.18,19), though he also speaks of the proportion as defining "the relationship of ratios with each other" (2.21); Adrastus, he says, had claimed that the geometric mean alone is a "true proportion" (2.50) (ed. Hiller, pp. 72.24–74.7, 74.12–14, 106.14–20).
Ficino's access to and knowledge of Euclid has yet to be investigated. The Elements were rendered into Latin from Arabic translations by Adelard of Bath (fl. 1116–1142), apparently in three distinct versions, by Hermann of Carinthia (fl. c. 1140–1150), and by Gerard of Cremona (c. 1114–1187); and directly from the Greek by an anonymous twelfth-century Sicilian scholar enjoying the patronage perhaps of the Admiral Eugenius, Emir of Palermo (1130–1203). However, the standard medieval if thoroughly "scholasticized" redaction was done by Campanus of Novara in 1255–1259 (and eventually published in Venice in 1482, and again in 1486 and 1491). A "free reworking" of earlier translations from the Arabic, including Adelard's, it included books 14 and 15 and sought to elucidate the axiomatic structure of the Elements by emphasizing arithmetical rather than geometrical proof. It was revised and reissued by Luca Pacioli (Paciuolo) in 1509 in response to the new humanist version by Bartolomeo Zamberti published in 1505; and the two competing versions were then issued in 1516 by Faber Stapulensis in a composite volume. The editio princeps of the Greek text by Simon Grynaeus did not appear until 1533. Earlier, the distinguished mathematician and astronomer Regiomontanus (1436–1476) had set about revising the Campanus version in the 1470s, though this revision has been lost since 1625; however, a copy of Adelard's Latin Euclid that belonged to him does survive, dating from 1459 and containing annotations up to the seventh book. For further references, see Heath's introduction to his translation of the Elements (1:92–101); Paul Lawrence Rose, The Italian Renais-
sance of Mathematics (Geneva, 1975), pp. 50–52, 77–79, 81, 93, 106, and 144; and H. L. L. Busard, The First Latin Translation of Euclid's "Elements" Commonly Ascribed to Adelard of Bath (Books I-VIII and Books X.36-XV.2), Pontifical Institute of Mediaeval Studies: Studies and Texts, 64 (Toronto, 1983), pp. 2–15 (Busard, incidentally, has now edited all the pre-Campanus Latin versions).
We might note that in his entire works Ficino never alludes to the Elements , and that the references to "Euclides" in his Opera , pp. 764 and 1008, are to Euclid of Megara, the philosopher who lived c. 400 B.C. However, from the Middle Ages to the end of the sixteenth century, translators, editors, and commentators frequently confused the two Euclids (though Lascaris, for one, had correctly distinguished them); see Heath, Elements 1:3–4. Ficino too may therefore have thought them one and the same.
that the proportions between 27 and 18, 18 and 12, and 12 and 8 are all in the same ratio of 3:2.[8] But between the two cubes 27 and 8 exist the two squares 16 and 9, with 12 mediating between them by way again of the same ratio, this time of 4:3.
It was precisely these two ratios of 3:2 and 4:3 that Ficino was to bring to bear on his elucidation of the crux in the Republic , more particularly since they appear to be underscored by the important testimony of Aristotle's Politics at 5.12.8. For here Aristotle argues that Plato had established "the origin of change" in a hitherto perfect state in a number with a "root" in the ratio of 4:3, and that this root "when joined to the five gives two harmonies." By "two harmonies," Aristotle concludes, Plato had meant "when the number of this diagram—or [in Acciaiuoli's and Ficino's rendering] the description of this figure—becomes solid."[9] The "fatal geometric" number will therefore be a "solid," and specifically a cube, and the clue to its discovery will lie in the understanding of number sequences, of square and cube numbers, and of the nature of certain primary proportions. We therefore need to be acquainted with the basic categories, as Ficino understood them, of what the Pythagorean tradition had presented as figured or figural numbers.
I. We should begin with Ficino's working assumptions about, and definitions of, the kinds or classes of numbers. These will be largely familiar to those scholars already acquainted with the ancient Pythagorean mathematical tradition as fully described by Paul-Henri Michel, for instance, in the monumental study already cited.[10] However, this
[8] Cf. Ficino's Timaeus Commentary 19 (Opera , p. 1446.1).
[9] "'quorum sexquitertia radix coniuncta quinario duas exhibet harmonias,' dicens videlicet quando numeri huius descriptio fiat solida"—as rendered by Ficino towards the end of his De Numero Fatali 12 (see Chapter 1, n. 19 above). For "diagram" (diagramma ), see n. 12 to Text 2, p. 166 below.
[10] De Pythagore , esp. part 2, chapters 1 and 2.
Pythagorean dimension of Ficino's work and intellectual background has remained up till now entirely unexplored, perhaps even unsuspected, by scholars of Renaissance Platonism; and the terrain as a whole is rather forbidding. In the following analysis the references in parentheses are to the chapter and line numberings of my edition of the text (Text 3 in Part Two below).
A. Odd numbers Ficino thinks of as male, as indivisible, and as incorporeal, since they derive "from their own root or seed" (6.63–64), an assumption that necessarily follows given the Neoplatonic status of the 1 as their "mean and center" (8.38–39). They have "greater kinship with oneness"; and they "abound" with it, beginning with, ending in, and converting to it (8.36–39). The even numbers by contrast are female, divisible, and corporeal, the 2 being the first "fall" from the 1 and thus the first instance of division and diversity. Ficino refers to the 2 as being like indeterminate matter, citing Archytas as supposing the 1 is the Idea of the odd numbers while the 2 is the Idea of the even (8.19–22).[11]
Odd numbers possess the one as "the bond" or "hinge" of themselves, and exist about the 1 as their center; while even numbers once divided are "torn apart" and none of their parts survive, the odd numbers once divided continue to exist with the 1 in their parts as the "indivisible link" (8.39–43). Hence they seem to be "unfolded" rather than "divided." Or, to use a traditional emanative metaphor, while the even numbers flow in the initial procession out from the 1, the odd numbers are at the second stage—they turn back towards the 1, the 1 which is "like the world's maker" in that it creates "order" for them and is their "measure" and "principle" (8.3–19, 46–47, 60–64).
The first number as such is the 3, the 2 being not so much a number as the "first fall from the one," "the first "multitude" (6.46–47; 8.79–80). This situation Ficino declares "is like the mystery of the Christian Trinity" (8.22–23). The "fate" of the first number 3 is thus
[11] Cf. Euclid, Elements 7, defs. 6–11; Aristotle, Metaphysics 986a22 ff., with the Pythagorean table of ten opposites—limited and unlimited, odd and even, one and many, right and left, male and female, rest and motion, straight and crooked, light and darkness, good and bad, square and oblong; idem, Physics 203a10: "The Pythagoreans identify the unlimited with the even"; and Iamblichus, De Vita Pythagorica 28.156: the right is the origin of the odd numbers and is divine, the left is the symbol of the divisible even numbers.
See Michel, De Pythagore , p. 332: "l'arithmologie des Pythagoriciens accorde à l'impair une sorte d'avantage sur le pair"; also Dillon, Middle Platonists , pp. 3–5.
paradigmatic of the "fates" of all numbers, 3 of course being the number of the Fates themselves (15.10–11).[12] Three is as it were at the third perfective stage in the emanative cycle, the return to the 1 (6.77–79) where it "abounds" in it as in "its head and bond" (9.5–6). Because of this abundance or "copiousness," the 3 is called masculine.
If the male odd numbers abound, the female even numbers by contrast suffer from "dearth," "partition," and "fall" (6.80–81). Ficino acknowledges that such a view runs counter to the "human and moral praise" we usually extend to the even numbers because they can be equally and therefore justly distributed (if we are thinking, that is, of enacting justice among equals). But, he argues, "the more sacred and divine praise" is directed towards the odd numbers such as 3, 7, and 9; for they "comprehend" the even and are "hinged" upon the 1 as their "mean," "center," and "god," the 1 which is the source of equal distribution and "the principle of the world's order" (6.82–87, 95–97). Clearly, Christian trinitarian assumptions are reinforced by such definitions.
Despite his acceptance of 3 as the first number proper, there are times when Ficino thinks of 2 and indeed of 1 also as numbers; for all numbers look to the 1 as their source according to the ancient tag that they are 1 multiplied.[13] Strictly speaking, 1 is both odd and even, but the Pythagoreans thought of it more as odd on the grounds that, while evens are divided and destroyed and thus torn apart from the 1, the odds are "unfolded" from it and retain it as their center (8.39–43). The 1 as odd is thus the ultimate principle of identity and likeness and as such resembles God.
"Simple" or uncompounded numbers are those which Ficino thinks of as "consisting of" and "being measured by" the 1 alone, such as 3, 5, 7, 11, 13, 17, and so on. They are the prime numbers, and Ficino, following the Euclidean tradition, describes them as the "prime unequals."[14] "Compound" numbers therefore are those which are products of factors other than 1, as 6 is the product of 3x2.[15] Compounds that are odd and therefore compounded by factors that
[12] As in Plato's Republic 10.617C ff. (with Ficino's epitome, Opera , p. 1434).
[13] Theon, Expositio 1.7 (ed. Hiller, p. 24.23): "as for the 1, it is not a number, but the principle of number." See Michel, De Pythagore , p. 332. Ficino usually thinks of 2 not 4, however, as the first even number (but cf. 3.40–42).
[14] Two is not considered as belonging to this category; cf. Theon, Expositio 1.6 (ed. Hiller, pp. 23.6–24.8). See Michel, De Pythagore , p. 330.
[15] Theon, Expositio 1.7 (ed. Hiller, p. 24.16–23). See Michel, De Pythagore , p. 331.
are both odd—15 for instance as the product of 5x3—are said to be "oddly odd"; whereas compounds that are even are said to be either "oddly even" if just one of the factors is even—10 for instance as the product of 5x2—or "evenly even" if both of the factors are even—8 for instance as the product of 4x2.[16] Various individual numbers clearly fall into more than one class; 12, for instance, we can think of as the product either of 6x2 or of 4x3, and therefore as either "evenly even" or "oddly even." These definitions are crucial, given Ficino's wrestling with the lemma at 546C3 "isên isakis " ("aequalem aequa-liter").
B. There are three important related categories of numbers that the Pythagorean tradition characterized as either "perfect," or "abundant," or "deficient."[17]
First is the category of truly perfect numbers. Though 10 is thought by the Pythagoreans to be a perfect number and 1 is perfect in power,[18] a truly perfect number is exceedingly rare, since it is identical with the sum of its own factors, its aliquot partes (4.18–19).[19] Six is the first of such numbers, being the sum of 3+2+1; 28 is the second, being the sum of 14+7+4+2+1; 496 is the third and 8128 the fourth (17.29–31). There are still higher perfect numbers, but Ficino never mentions them.[20] He perceives a mystical significance in the fact that the last digits of these first four perfect numbers alternate be-
[16] See Michel, De Pythagore , pp. 333–334—citing Euclid, Elements 7, defs. 8–11 [10]: "evenly even" (8), "evenly odd" (9), "oddly even" (10), "oddly odd" (11)—and p. 336—citing Theon, Expositio 1.8–10 (i.e., ed. Hiller, pp. 25.5–26.13). In using "oddly odd" as a term for compound odd numbers Ficino seems to be following Euclid rather than Theon, who reserves the term for the prime odd numbers—see Expositio 1.6 (ed. Hiller, p. 23.14, 16, 21). However, he seems to be ignoring Euclid's, or an interpolator's, finespun distinction between "evenly odd" and "oddly even," a distinction also noted by Theon in 1.9 and 1.10. Heiberg and Heath both in fact reject def. 10 as the work of an interpolator; see Heath's note on def. 9 in his translation of the Elements (2:282–284).
[17] See Michel, De Pythagore , pp. 342–346.
[18] Theon, Expositio 2.39–40 (ed. Hiller, pp. 99.17–20, 99.24–100.8); Nicomachus, Introductio 1.16.8–10 (on one). For the Pythagoreans' praise of 10's perfection, see Aristotle, Metaphysics 1.5.986a9–11.
[19] Cf. Euclid, Elements 7, def. 22; 9, prop. 36; Theon, Expositio 1.32 (ed. Hiller, pp. 45.9–46.3); and Nicomachus, Introductio 1.16; also Plutarch, De Animae Procreatione 13 (Moralia 1018C). See Michel, De Pythagore , pp. 342–346.
[20] Iamblichus, In Nicomachi Arith. Introd. 33.20–23 (ed. Pistelli), alludes to the possibility of the fifth perfect number, that is, 33,550,336. See Heath's table of the first ten perfect numbers, Greek Mathematics 1:74–75.
tween 6 and 8 in what he calls "a marvelous vicissitude";[21] and he also finds it significant that only one such number occurs below 10, one again between 10 and a 100, one between a 100 and a 1000, and one between a 1000 and 10,000 (17.31–35, 45–47).[22] Six is doubly perfect for Ficino because it has the perfect ratio of 2:1 within itself in that 6 equals 4+2, and 4:2 is the double ratio of 2:1 (4.23–26). The perfect numbers "contain the circuit itself of divine generation"—"as rare as is the perfection, so rare is the divine progeny that proceeds" (17.42–43, 47–48).
Most numbers, however, are "deficient," that is, their factors add up to a sum less than themselves: for instance, 8, the first of such numbers, has as its factors 4, 2 and 1, but they only add up to 7 (4.28–30).
Finally, an "abundant" or "increasing" number is one where the sum of its factors exceeds itself. Twelve is the first of such numbers, the sum of its factors of 6, 4, 3, 2, and 1 being 16 (4.32–35). Twelve is also abundant because it is the product of the "twinning" of 6, the perfect number (3.57–58).23
C. Still another category interests Ficino, that of the "circular" numbers, numbers whose powers happen to end in the same digit. Besides being the first of the perfect numbers, 6 is also an example of a circular number in that both its square of 36 and its cube of 216 also end in 6. Another example of such a number is 5 with its square of 25 and its cube of 125. However, 4 has its circularity "intercepted in the plane" of 16 even though its cube of 64 ends in 4; it is thus an example of a "lesser" circular number (17.8–15).[24]
[21] This alternation does not pertain for higher perfect numbers, though Ficino might have supposed it did following Nicomachus, Introductio 1.16.3,7 and Boethius, De Institutione Arithmetica 1.20.
[22] Nicomachus, Introductio 1.16.3, mentions this distribution whereas Theon only mentions 6 and 28. See Robbins in D'Ooge, Nicomachus , p. 52, who notes it was incorrect of Nicomachus to imply both that a perfect number can be found in each order of the powers of 10 and that all such numbers alternately end in 6 and 8.
[23] Theon, Expositio 1.32 (ed. Hiller, p. 46.4–14), and Nicomachus, Introductio 1.14,15, not Euclid, provided Ficino with the definition of both abundant and deficient numbers.
There is a fourth category, incidentally, that Ficino never alludes to here, namely of "friendly numbers"—where one of two numbers is equal to the sum of the aliquot parts of the other. Iamblichus, In Nicomachi Arith. Introd. 35.3–5 (ed. Pistelli), attributed their invention to Pythagoras, and the only such numbers known to antiquity seem to have been 220 and 284; magical powers were attributed to both. See Michel, De Pythagore , pp. 343, 346–348.
[24] For circular numbers, see Theon, Expositio 1.24 (ed. Hiller, pp. 38.16–39.9);
also the Theologumena 36.17 (ed. de Falco), and Nicomachus, Introductio 2.17.7. Cf. Ficino's Timaeus Commentary 17 (Opera , pp. 1444.4–1445): "etsi superiores numeros ratione ortus sui alios quadratos alios oblongos nominant, tamen ratione casus atque finis nuncupant circulares." Thus 5 and 6 are the "prime roots" of the circulars and thus the "circular body of the world" is divided into five or six elements; cf. ibid. 20 (Opera , p. 1446.2), which states that 5 and 6 "are in accord with the circular shape of the world" ("circulari mundi figurae congruere").
D. A "spousal" or "nuptial" number is the product of two adjacent numbers: for instance, 6 is the product of 2x3 and 12 of 3x4. Indeed, 6 is the first "spousal" number because it is the product of the first odd (and therefore male) number multiplying the first even (and therefore female) number—and was so denominated by the Pythagoreans. Ficino specifically says that 12 is the second "spousal" as the product of 3x4; and presumably, 20 would be the third spousal as 4x5, 30 the fourth as 5x6, 42 the fifth as 6x7, and so on. Multiples of factors differing by more than 1 cannot be called spousal (4.41–45).
Interestingly, the heading of Ficino's expositio in all the texts speaks of "the nuptial number" whereas the heading of his commentary proper speaks of "the fatal number." Antiquity had often identified the two numbers on the grounds that Plato's fatal number must be especially regarded by the magistrates when they set about orchestrating public mass marriages.[25] But Ficino obviously intended us to keep the idea of a nuptial number quite distinct from that of a fatal number, since his two proposed fatal numbers are not products of adjacent male and female numbers, but rather numbers with cube roots, though admittedly one of these roots is indeed a spousal number. In short, from Ficino's perspective Plato was concerned in the Republic with at least three kinds of mystical numbers: with the fatal numbers that signal the onset of a perfect constitution's decline; with the nuptial numbers that signal the best opportunities for marriage and begetting in a state that wants to resist a decline before its fatal time; and with the truly perfect numbers that betoken and preside over the divine births Plato writes of at 546B3.
II. Let us now turn to various conceptions of numbers as products. Of these there are three kinds: linear, plane, and solid and the terms can be used in the Pythagorean tradition of sums as well as of products.[26]
[25] See Chapter 1, n. 4 above. Ficino's most likely sources were Iamblichus, In Nicomachi Arith. Introd. 82.17–24 (ed. Pistelli), and Plutarch, De Iside 56 (Moralia 373F).
[26] Cf. Timaeus 31C ff. and Ficino's Timaeus Commentary 19 (Opera , pp.
(footnote continued on the next page)
A. All numbers when seen as the products of 1 or that have no factor other than 1 are called "linear." But when numbers are the products of two numbers other than 1, then such composites are called "plane," and this is so whether they are the products of the same factor multiplying itself (in which case the plane will be a square) or of one factor multiplying another. Thus 4 as 2x2, 6 as 3x2, 8 as 4x2, 9 as 3x3, 10 as 5x2, and 12 as both 3x4 and 6x2 are all planes. Obviously, planes multiplying either linear numbers or other planes will always produce further planes as the case of 12 above illustrates: and the multiplication of two planes that are squares, for instance, 4 and 9, will always produce another square, in this case 36 (11.8–10).[27] Equally obviously, as both 12 and 36 demonstrate, a number can be a plane in different ways: 12 is either 6x2 or 3x4; 36 is either 6x6 or 9x4 or 12x3 or 18x2.
"Solid" numbers are the products of three factors greater than 1, whether of the same factor multiplying itself twice (in which case the solid is a cube), or of a factor multiplying itself and another factor, or of three different factors multiplying each other.[28] Again Ficino and the Pythagorean tradition were particularly interested in "solid" products that were cubes. Thus 8, 27, 64, and 125 are the cubic solids of 2, 3, 4, and 5 respectively. Obviously, cubes multiplying cubes will always produce further cubes: for instance, the two prime cubes 8 and 27 multiplied together produce 216, the cube of 6 (11.14–15). Solid numbers are known Platonically as "of the three" (5.44–45).[29]
Clearly, some numbers that are products can be viewed as linear, or as plane, or as solid: 8, for instance, can be seen as 8x1 or as 2x4 or as 2x2x2. But their solidity will be their most characteristic or important mode.[30] Moreover, we should constantly bear in mind that both plane and solid numbers seen as products can also be viewed as sums, and this is particularly important if they are sums in one of the three primary series I shall outline later.
B. Among plane and solid numbers there are three kinds of products; and these Ficino designates, following Plato's Theaetetus 147E–148B and Theon's Expositio , as "equilateral" (Theon's "equally equal"),
(footnote continued from the previous page)
[1445] 3–1446): "Triplices esse numeros apud Pythagoricos alibi declaravimus, lineares, planos, solidos." See Michel, De Pythagore , pp. 298–299.
[27] Euclid, Elements 7, defs. 16 (plane), 18 (square); and 9, prop. 1.
[28] Euclid, Elements 7, defs. 17 (solid), 19 (cube); and 9, props. 3–7.
[29] Because of Aristotle's gloss on 546C6? See n. 9 above.
[30] See Michel, De Pythagore , p. 298.
"unequilateral" or "oblong" (Theon's "unequally unequal"), and "diagonal."31
i. An "equilateral" is the product of any number multiplying itself—either once to produce its square or twice to produce its cube. Uniquely as always, the 1 too is an equilateral insofar as it is the square and the cube of itself (6.30–32; 8.51–55). All other equilaterals resemble it in their "equality and straightness" since it is their "seed" (8.48–53). Accordingly, the first series or succession of equilaterals as products is the regular succession of square numbers: [1], 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on.[32] In a way these equilaterals are like God, muses Ficino, in that God "acting with Himself [multiplying Himself as it were] procreates others" (8.70). But, he continues in an extension of the analogy, cube numbers are like God when He uses the prime being as His means.[33] Thus if we think of God as 2, then His first created being will be 2x2, and all subsequent beings He creates directly will be 2x2x2. But if we think of God using the prime being as His means to create them, then we must replace 2x2x2 with 2x4, 4 (as 2x2) being the prime being whom God multiplies. Similarly 27 can be viewed either as 3x3x3 or as 3x9 (8.64–72). It is difficult to gauge the force of these distinctions for Ficino.
An equilateral number of peculiar importance is the "universal" number 100 (as 102 ) and the multiples that immediately "teem" from it and from 10 its root: 1000 (as 103 ), 10,000 (as 1002 ) and 1,000,000 (as 1003 ) (3.75–77).
[31] Expositio 1.11–12 ("un/equilateral"), 31 ("diagonal") (ed. Hiller, pp. 26.14,18–19; 42.10–44.8). See below.
[32] See Michel, De Pythagore , pp. 304–305.
[33] Ficino is thinking here Neoplatonically. He is equating God with the first hypostasis, the One, and the prime being with Mind, the second hypostasis. From Mind proceeds Soul, the third hypostasis, and thence Quality and the realm of Body. In Christian terms the realm of Mind can be equated, in part at least, with the angels generically conceived as Angel or as seraphim, the highest of the angels. Hence Ficino is suggesting that we can think of God either as the immediate Creator of everything in the universe or as the Creator by way of the angels, who themselves created, or assisted Him in the creation of, the lower realms. However, insofar as the Neoplatonic hypostasis Mind also suggests God in His immanence (God in His transcendence being identified with the first hypostasis, the One), then we must think of the prime being as signifying the Son. It is not clear what Ficino intends.
For the problems besetting the attempt to accommodate Ficinian metaphysics with trinitarian theology, see my "The Absent Angel in Ficino's Philosophy," Journal of the History of Ideas 36 (1975), 219–240, and "Marsilio Ficino on Plato, the Neoplatonists and the Christian Doctrine of the Trinity," Renaissance Quarterly 37 (1984), 555–584, with further references.
ii. An "unequilateral" plane, on the other hand, is the product of two different numbers, and an unequilateral solid of three. Unequilateral planes are in turn subdivided into "long" (heteromêkês ) and "oblong" (promêkês ), though, strictly speaking, the "long" are a special class within the general class of "oblong."[34]
An unequilateral is "long" when it is the product of two numbers differing only by one[35] —differing by 1 being a privileged difference given the unique status of the 1. For instance, 6 is the product of 3x2, 12 of 3x4; or, in the case of solids, 12 is the product of 2x2x3, 18 of 2x3x3. The "long" series is thus [2], 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, and so on. These unequilaterals also constitute from another perspective the sequence of spousal numbers, that is, of 1x2, 2x3, 3x4, 4x5, and so on.
An unequilateral is "oblong," however, when it is the product of numbers differing by more than one, as 15 is the product of 3x5 or 24 of 12x2 or 8x3 or 6x4 (6.18–20). Here the possibilities are endless and of little interest to the Pythagorean tradition. Of course, with the possible exception of 6, which is uniquely long (except as 6x1), all unequilateral long products can also appear as oblong: for example, 12 is both 3x4 (long) and 6x2 (oblong), 20 is 4x5 (long) but also 10x2 (oblong).[36]
An equilateral multiplying another equilateral produces an equilateral: for instance, 4x9=36 [22 x32 =62 ], 16x25=400 [42 x52 =202 ]. But a long unequilateral must multiply an adjacent long unequilateral to produce another long unequilateral: for instance, 6x12=72 [(2x3)x (3x4)=8x9]. An oblong unequilateral multiplying another oblong, or an equilateral multiplying an unequilateral of either kind, both produce an oblong: for instance, 8x15=120 [(2x4)x(3x5)=10x12], 4x6= 24 [(2x2)x(2x3)=4x6], 4x15=60 [(2x2)x(3x5)=6x10]. But a long multiplying an oblong can sometimes produce a long, for instance, 6x15=90 [(2x3)x(3x5)=9x10], or even an equilateral (if the long is itself one factor in the oblong and the other factor is an equilateral), for
[34] Theon, Expositio 1.17 (ed. Hiller, p. 30.12–18), had insisted on this more restrictive meaning for "long." Again cf. Plato, Theaetetus 147E ff., and Nicomachus, Introductio 2.17.1, 2.18.2. See Michel, De Pythagore , pp. 311–321 (pp. 318–319 deal with the various senses of "long"—heteromêkês ).
[35] Theon, Expositio 1.13 (ed. Hiller, p. 26.21–22)—a definition of heteromêkês .
[36] Ibid. 1.17 (ed. Hiller, pp. 30.8–31.8)—a definition of promêkês . Cf. Ficino, Timaeus Commentary 19 (Opera , pp. 1445.4–1446), "Dicimus et oblongos qui ex ductu numeri admodum minoris in numerum longe maiorem conficiuntur, qualis est denarius."
instance, 12x108=1296 [(3x4)x(3x4x32 )=362 ]. The multiplication of a cube by an unequilateral never produces another cube, but a cube multiplying a cube always produces another cube, for instance, 8x6=48 and 8x15=120, but 8x27=216 [63 ].37
iii. Finally, there is a category of diagonal (or "diametral") products which Ficino doubtless derived principally from Theon's Expositio 1.31 and which involve the problem of certain irrational numbers and their status.[38] The products are described in chapter 5 as those which are alternatively greater or less by one than double the squares of the sides in a particular sequence of geometric squares. As described by Theon, we obtain the sequence of the squares by adding the value of the diagonal to the side while adding the value of twice the side to the diagonal. For instance, if we start with a side of 2 and a diagonal of 3, then the next such square will have a side of 5 [3+2] and a diagonal of 7 [3+(2x2)], the next a side of 12 [5+7] and a diagonal of 17 [7+ (2x5)], the next a side of 29 [12+17] and a diagonal of 41 [17+ (2x12)], and so forth. The diagonals will be 3, 7, 17, 41, 99, and so on; and each when squared—9, 49, 289, 1681, 9801—will equal double the squares of the sides provided we give or take 1 in alternation. That is, they will equal 8 (as 2x22 ) plus 1, 50 (as 2x52 ) minus 1, 288 (as 2x122 ) plus 1, 1682 (as 2x292 ) minus 1, 9800 (as 2x702 ) plus
[37] Cf. Nicomachus, Introductio 2.24.10,11 (with a specific reference to the Republic 's passage on the number "they call the marriage number").
[38] For the irrational or incommensurable numbers, Ficino would have turned most obviously, apart from Theon, to Plato's Theaetetus 147D–148B, Hippias Major 301D–303C, Laws 7.817E–820B, Parmenides 140B–D, Epinomis 976A–977B, 990C–991A, and, of course, Republic 8.546BD; to Aristotle's Prior Analytics 1.41a23–30, 46b26–37, 50a37–38, 65b16–21, and other texts; and also to Euclid's Elements 2, propositions 9 and 10, and Elements 10, passim. Because, we recall, he had access only to a manuscript containing the first twelve treatises, Ficino cannot have been influenced here by the analyses in the thirteenth treatise of Proclus's Republic Commentary (ed. Kroll, 2:24.16–29.4—see the note by Hultsch on pp. 393–400; trans. Festugière, 2:130–135).
In general see Michel, De Pythagore , part 2, chapter 2 passim, esp. pp. 427–430 (on Theon), 433–441 (on Euclid's two propositions in his second book and on Proclus), 441–455 (on Euclid's tenth book), and 500–511 (on three of the Platonic texts). See also Charles Mugler, Platon et la recherche mathématique de son époque (Strasbourg, 1948), pp. 191–249 (especially pp. 226–236 on the problem of the rational and irrational diagonal); and Fowler, Mathematics of Plato's Academy , pp. 166–192 (on Euclid's tenth book), 192–194 (an appendix on the use of the terms alogos and (ar )rhêtos in Plato, Aristotle, and the pre-Socratics), and 294–308 (on the discovery and role of the phenomenon of incommensurability, with a list on pp. 294–302 of the source material in Democritus, Plato, Aristotle, and so on).
1, and so forth. Or, put another way, the square constructed on the diagonal will always be now smaller by 1, now greater by 1, than double the square constructed on the side.[39]
Since this plus-or-minus-one alternation is perfectly regular, from the Pythagorean-Ficinian viewpoint, adaequatio or compensatio emerges in the long run; that is, the "power" of the diagonal as a genus, as distinct from the powers of individual diagonals, maintains a ratio to the "power" of the side of 2:1. Incidentally, the successive values of the accompanying sides—2, 5, 12, 29, 70, and so on—constitute the "lateral" numbers.
Diagonal numbers are defined Platonically as "of the 5" because in the very first instance of the series the side is 2 and the diagonal 3, and the sum of 2+3 is 5 (hence the primacy of the harmony diapente) (5.43–44). For, with the alternating plus-or-minus-1 rule, a side of 2 produces a diagonal of 3 in that (2x22 )+1=32 , a side of 5 produces a diagonal of 7 in that (2x52 )-1=72 , a side of 12 produces a diagonal of 17 in that (2x122 )+1=172 , and so on. With all these diagonal and lateral powers the 1 is called the "equalizer" (5.37–38). We might note that if the sides are even then 1 has to be added to them, but if odd then subtracted from them.
The diagonal numbers were already known to Plato, for in the celebrated passage in the Republic which is our primary concern here he gives 7 as the "rational diagonal" (diametros rhêtos ) of a square with the side of 5. The issue turns on the Pythagorean-Platonic distinction between a rational and an irrational root. While 9 and 49 for instance have rational roots of 3 and 7 respectively, 8 and 50 by contrast have irrational roots of 2.8284271 . . . and 7.0710678 . . . respectively. Nevertheless, 8 and 50 can be said to have rational roots, in the Pythagorean sense, of 3 and 7 in that 9 and 49 are their proximate powers, the nearest squares (equilaterals) to them. Thus 8 and 50 can be said to have both irrational and rational roots, the latter being primary. Accordingly, Ficino followed Theon and what he took to be the Pythagorean-Platonic tradition in postulating both rational and irrational roots for the product of twice the square of the side and then
[39] Cf. Theon, Expositio 1.31 (ed. Hiller, p. 44.14–17): "The square constructed on the diagonal will be now smaller now greater by one than double the square constructed on the side in such a way that these diagonals and sides will always be rational [that is, whole numbers]." See Michel, De Pythagore , p. 428. Put algebraically the problem is to find a series of positive, integral solutions for the equation y[2] = 2x[2] &x1771 where y = the diagonal (diameter) and x = the side.
assigning primacy to the rational root. In this way he could arrive at a rational value for the diagonal and hence resolve to his own satisfaction part at least of the infamous crux at 546C4–5, "with individual comparable diagonals requiring one, but those which are not comparable requiring two," comparabilis being his rendering of Plato's rhêtos (rational),[40] in that an expressible ratio derives from a comparability between the power of a diagonal and that of the side.
C. Some products, finally, are "similar," others "dissimilar."[41] The similar are those that the Greeks had traditionally defined as the products of two proportional factors. While equilaterals are always similar (whether as squares or cubes), unequilaterals are similar only when their "sides" or factors are proportional; for example, 6 is similar to 24 in that, as 3x2 and 6x4 respectively, both contain the ratio of 3:2; again 18 and 8 are similar in that, as 6x3 and 4x2 respectively, both contain the ratio of 2:1.[42] All other unequilateral products are dissimilar; for example, 18 and 24 are dissimilar in that, as 6x3 and 6x4 (or 12x2) respectively, they do not share the same ratio (4.46–55).
III. Let us now turn to the figural or geometrical importance that Ficino associates, like the ancients before him, with certain fundamental number series we generate not by multiplication but by addition; that is, to numbers viewed as sums and not as products.[43] This tradition is now largely unfamiliar to us but once held an esteemed place for the Pythagoreans, who were accustomed to conceptualizing sums as extensions in space. The chief authorities for Ficino would have been, as we have seen, the treatise of Theon, and perhaps that of Nicomachus and of his commentators, all of them Neopythagorean works that were probably preserving or amplifying a tradition concerning figured sums and summing stemming from earlier, perhaps even from primitive, Pythagoreanism. Clearly, Ficino was aware from
[40] See n. 17 to Text 2, p. 167 below.
[41] Ficino renders Plato's phrase at 546B6–7, "of those that make like and make unlike"—homoiountôn te kai anomoiountôn , as similantium et dissimilantium , but he interprets it to mean simply "like" and "unlike." See n. 7 to Text 2, p. 165 below.
[42] Cf. Euclid, Elements 7, def. 21, and Theon, Expositio 1.22 (ed. Hiller, pp. 36.12–37.6). See also Michel, De Pythagore , pp. 321–322 on Euclid, Elements 9, prop. 1—the remarkable proposition that the product of two such similar plane numbers is a square number: 6x24=144 (i.e., 12 ); also pp. 341 and 507.
[43] See Michel, De Pythagore , p. 297: "dans l'ancienne arithmo-géométrie—et chez les auteurs qui en conservent la tradition—la préférence est accordée aux nombressommes."
the onset of his career that an understanding of such figured sums was a key to the secrets of Platonic mathematics, and this must have been the principal reason behind his decision to work through Theon's Expositio and Iamblichus's Pythagorean treatises. We should note that he was only concerned with figured sums; for such unfigured sums or random additions as 7+2+11 had as little interest for him, and for the Pythagorean-Platonic tradition he was rediscovering, as random products.[44]
Leaving aside the special case of linear numbers seen simply as the sums of ones (or as the products of nx1),[45] let us concentrate on three kinds of "plane" sums. Ficino refers to these likewise as equilateral or unequilateral, or as triangular, and he concentrates on the three paradigmatic series, those resulting from: a) the summing of the regular sequence of odd numbers; b) the summing of the regular sequence of even numbers; and c) the summing of the regular sequence of both odd and even numbers. From his reading of Theon, he was certainly aware of other derivative arithmetic series, those resulting, for instance, from summing by 3's, 4's, 5's, 6's, and so forth.46
A. The "equilateral" series of numbers viewed as sums is the result of adding or "composing" the odd numbers in their regular sequence, starting with 1: 1, 1+3=4, 4+5=9, 9+7=16, 16+9=25, 25+11=36, and so on (6.30–40).[47] Ficino finds it important that the successive sums alternate between odd and even numbers and that the series begins with the addition of 1 and 3. He thinks of 1 as "the leader of the odd and of the equilateral numbers," because it is also, mysteriously and uniquely, the square and the cube of itself (6.40–41, 47–49; 9.31–33). Ficino treats of this addition series first precisely because it generates a category of especial interest to the Pythagorean tradition, and to himself in his analysis of the Plato passage, the category of sums that are also, from another perspective, square products or, to put it another way, that have rational roots. For the successive equilateral sums—1,
[44] Ibid., p. 299.
[45] And so defined by Theon, Expositio 1.6 (ed. Hiller, p. 23.11–14): "arithmoi grammikoi ."
[46] Theon, Expositio 1.26–27 (ed. Hiller, pp. 39.14–41.2).
[47] Cf. Theon, Expositio 1.15 (ed. Hiller, p. 28.3–15), and Nicomachus, Introductio 2.19.1–2. Following Theon, Ficino thinks of the sums in the series as being the result of the addition of the next odd number to the sum and not to the succession of the preceding odd numbers, e.g., 16=9+7 and not (5+3+1)+7, etc. See Michel, De Pythagore , pp. 304–311.
4, 9, 16, 25, and so on—coincide with the square products of the regular sequence of odd and even numbers—12 , 22 , 32 , 42 , 52 —numbers whose root is known, as chapter 10 explains.
B. The "unequilateral" series of numbers as sums begins with 2 as "the leader of the even numbers" and is the result of adding the even numbers to each other in regular sequence thus: 2, 2+4=6, 6+6=12, 12+8=20, 20+10=30, 30+12=42, and so on (6.40–44, 53–58).[48] The sums of this series—2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, and so on—are also, from another perspective, the unequilateral "long" products (i.e., products of immediately adjacent numbers—1x2, 2x3, 3x4, 4x5, 5x6, 6x7, 7x8, and so on). They are also, from a third perspective, the spousal numbers.
As products these unequilaterals are always even (and therefore feminine), because they result from an even number multiplying an odd (6.59–61). This is only true of course of the "long" and not the "oblong" unequilateral numbers, that is, of the products of two numbers differing only by 1 (1 having, I repeat, a unique and mystical significance that also pertains to numbers that differ by 1). With "oblong" numbers—for instance, with 35 as the product of 7x5—the situation is obviously different, since a third of the products will be odd (those which are the result of odd numbers multiplying odd) while two-thirds will be even (those which are the result of evens multiplying either evens or odds) (6.70–73). Ficino must nevertheless be concerned with the oblong numbers insofar as Plato's geometric number, being inclusive of all kinds of numbers, also contains this kind. In particular, he will have to confront 7x100 as we shall see.
Following Theon's Expositio 1.16,[49] Ficino then relates the equilateral to the unequilateral series by arguing that the equilaterals "contain" or "bind fast" the unequilaterals just as the odd numbers contain the even (9.15–25). His argument hinges once again on the notion of ratio. Between, say, 9 and 4, both of which are equilaterals, the mean is 6, an unequilateral; and the ratio of 9:6 and of 6:4 in each case is 3:2. Again between 16 and 9, both equilaterals, the mean is 12, an
[48] Cf. Theon, Expositio 1.13 (ed. Hiller, p. 27.7–13), and again Nicomachus, Introductio 2.19.1–2. Once more Ficino thinks of the sums in this series as being the result of the addition of the next even number to the sum and not to the succession of the preceding even numbers, e.g., 20=12+8 and not (6+4+2)+8. See Michel, De Pythagore , pp. 316–317.
[49] Ed. Hiller, pp. 28.16–30.7.
unequilateral; and the ratio of 16:12 and of 12:9 in each case is 4:3. In both these instances the equilaterals are said to contain or "bind fast" the unequilaterals, since the ratios between the mediate unequilateral and its two bracketing equilaterals are identical. But if we reverse the situation and regard the equilateral as bracketed by two unequilaterals, for instance 9 as bracketed by 12 and 6, then the unequilaterals do not contain the equilateral, because the accompanying ratios are not identical—12 to 9 being 4:3, but 9 to 6 being 3:2. Thus Ficino argues, the equilaterals "encompass and bind fast" the unequilateral (9.34–35). Again, note that, for the Pythagoreans, the 1 is an equilateral, although it is simple and all the other equilaterals are compound, while 2 is an unequilateral.
C. The third primary series, logically perhaps the first but treated third both by Theon and by Ficino and whose discovery the tradition attributed directly to Pythagoras, Ficino refers to as the trigon or triangle sequence.[50] He describes this in chapter 7 as the series of sums that results from the adding together of the odd and even numbers in regular succession, starting with the 1 since it bears "the trigonic power in itself" and is "as it were a trigon in power": 1, 1+2=3, 3+3=6, 6+4=10, 10+5=15, and so on. If we add adjacent trigons together, we arrive at sums that are also the successive square products of the whole numbers: 1+3=4, 3+6=9, 6+10=16, 10+15=25, and so on, the products one obtains, of course, from summing in the equilateral series.
IV. Throughout Ficino works with a set of terms concerned with ratio (logos ), or what he consistently thinks of as proportion (analogia ), though proportion, strictly speaking, involves at least two ratios and three terms, as we have seen. In the forefront of his mind are musical proportions and the resulting intervals, a subject he had already dealt with in some detail in his commentary on the Timaeus and elsewhere,[51] and the principles of which he had derived from Plato himself, from the commentaries on the Timaeus by Calcidius and by Proclus, and in all probability from the musical treatises of Augustine
[50] Cf. Theon, Expositio 1.19,23 (ed. Hiller, pp. 31.13–33.17, 37.7–38.15), and Nicomachus, Introductio 2.8.1–3. See Michel, De Pythagore , pp. 299 ff.
[51] For instance in his Philebus Commentary 1.28 (ed. Allen, pp. 264–269). See n. 1 above.
and Boethius, and this is to leave aside the possible mediating roles of various medieval sources. We should recall, furthermore, that Plato himself uses the harmony of the musical scale as a symbol of the harmony of the state in the Republic 4.443D and that both Theon and Boethius situate their studies of proportions in their accounts of music.[52]
Ficino's starting point for an examination of the cosmic and musical ratios is the celebrated passage, as we have seen, in the Timaeus on the lambda at 35B–36B and again at 43D. Between the prime cubes 27 and 8, he observes in chapter 3, Plato had postulated three intervals with 18 and 12 as the geometric means, the three "proportions" between 27 and 18, 18 and 12, and 12 and 8 being all in the same ratio of 3:2, that is, of one and a half to one.[53]
However, between these two prime cubes also exist the two "equilateral planes," 16 (42 ) and 9 ([3] ) from the equilateral sequence. And between these two equilaterals appears the unequilateral 12. Since the equilaterals contain the unequilateral, the proportions between 16 and 12 and between 12 and 9 are both in the same ratio of 4:3, that is, of one and a third to one.
Thus Plato's lambda of numbers implicitly links the two prime cubes by way of the two geometric means 18 and 12 and by way of the constant ratio of 3:2. But 16 and 9—the equilateral planes—are each linked to 12 in the ratio of 4:3 and thence in Ficino's terms "bind" it in.[54] The ratios pertaining to 18, however, Ficino ignores because of the primacy and importance of 12. For 12 is the sum of the "prime foundations" of the two ratios governing Plato's lambda, namely 4:3 and 3:2, in that 7 is the "foundation" or "root" of the ratio of 4:3, and 5 of the ratio of 3:2, the two roots together adding up to 12 (3.22–26). These two ratios are especially esteemed because they and the ratio 2:1 "accord with the perfection and steadfastness of things" (9.36–38). Moreover, Ficino argues, the sesquialteral is in accord with the ratio of 2:1 in that 2:1, 3:2, 2:2, 2:3, 1:2 form a se-
[52] Theon, Expositio 2, and Boethius, Institutio Musica 2. See Michel, De Pythagore , pp. 358n, 362–364.
[53] Sesquialter means half as much again, sesquitertius a third as much again, and so forth. But Ficino clearly thinks of the situation musically, that is, in terms of sesquialteral and sesquitertial ratios, and therefore in the form 3:2 and 4:3 (to parallel the double ratio of 2:1) and not in the form 1 1/2 to 1 or 1 1/3 to 1. Hence I have rendered the terms as ratios throughout. For the complex situation, see Michel, De Pythagore , pp. 348–362 (on the various relationships of inequality) and 365–411 (on proportions and means).
[54] Cf. Ficino's Timaeus Commentary 19 (Opera , p. 1446).
quence of what he refers to as "overcoming" and "overcome" ratios; and similarly with the sesquitertial ratio and 3:1 (4.3–16).[55]
The two ratios of 3:2 and 4:3 are also especially esteemed by Platonists because the one produces the musical consonance of diapente , the interval of the perfect fifth; the other that of diatesseron , the interval of the perfect fourth. From them is produced the universal harmony known as the diapason , the interval of the octave, "the most celebrated of harmonies" (3.26–31), which Ficino thinks of traditionally as a "double proportion" in that (4x3):(3x2) gives us the ratio 2:1. Once again 12 is the important number, for it contains 5 and 7, the "roots" of the diapante (as 3+2) and of the diatesseron (as 4+3), and it is the sum of their being "compounded" that is added together. From Ficino's Pythagorean viewpoint 12 also contains the two ratios internally in that, having dissolved the one root of 5 into 3 and 2, we can double the 3 and then double it again to produce 12. Similarly, having dissolved the other root of 7 into 4 and 3, we can triple the 4 to again produce 12. Thus 4, 3, and 2, the "parts" of 7 and 5, when all "mixed together," produce 12. Accordingly, both by addition and by multiplication 12 contains the 7 and 5. Additionally, it is the result of multiplying the first two prime numbers 3 and 4 together—1 and 2 we recall are not numbers, 1 being the source of all numbers and 2 being "a confused multitude" (3.40–42). Ficino sees the presence of a great "mystery" here in that 7 is the number of the planets, and 5 the number of the zones of the world—the four zones of the four elements and the zone of heaven (3.59–60).[56] Five is also the "prime origin" of the perfect shape of the circle in that, whether squared or cubed, it ends with the number 5 and therefore is the first circular number (3.61–64).
V. Before turning to Ficino's attempted solution of Plato's account of the geometric number, let me end this review of his arithmological assumptions and his presentation of material from Theon's Expositio , by addressing briefly the traditional core of arithmology, the symbolic associations of the first ten numbers, the decad. Ficino does not present us with a schematic account here (or as far as I know elsewhere), but
[55] This provides us with a clue to Ficino's interpretation of Plato's phrase in the Republic 8.546B5, "dunamenai te kai dunasteuomenai ."
[56] And not, we might note, of the two frozen and the two temperate zones and the one torrid zone, the five climatic zones of the earth. Cf. Ficino's Critias epitome (Opera , p. 1486).
the decad figures prominently at times in his thinking, and he certainly assumed that Plato took its Pythagorean dimensions seriously. The nature of his debts to particular ancient or medieval numerologists I leave to others to explore,[57] but he was obviously familiar with the sections on the decad in Theon's Expositio 2.40–49.
ONE, chapter 8 argues, is the principle of all numbers and dimensions and therefore most resembles the principle of the universe itself, the One,[58] since it too remains entirely eminent and simple even as it procreates offspring. All the even numbers proceed from the 1 and the odd numbers turn back towards it. All dimensions issue from it as from a point. It is the substance of all numbers in that any number is 1 repeated. Hence 1 is the "measure" of all numbers whether odd or even, simple (that is, measured by the 1 alone) or compound (that is, also measured, i.e., divided, by a number other than 1 as 4 by 2). The 1 is like the maker of the world who imposes form on the 2 as on indeterminate matter.[59] Indeed Archytas, Ficino recalls, had supposed that the 1 was the Idea of the odd numbers and that the 2 was therefore the Idea of the even numbers. Yet the 1 "is both none of the numbers because of its most simple eminence, and all the numbers because it has the effective power of all numbers." Hence it has no "parts" and is neither even nor odd; for added to an even it makes it an odd and thus appears odd, and added to an odd it makes it an even and thus appears even. Ficino refers to Aristotle's lost work The Pythagorean here in affirming that the Pythagoreans preferred the 1 to be odd because the odd, male number (unlike the even, female number) changes the number to which it is added, making the previously odd into an even, the previously even into an odd; as such the 1 is consid-
[57] However, I doubt that he knew the most detailed of the "Platonic" arithmologies, the Pseudo-Iamblichean Theologumena . It contains, however, precisely the kind of Pythagorean number lore he was interested in, and I have cross-referenced it for that reason.
[58] Cf. Theon, Expositio 2.40 (ed. Hiller, pp. 99.24–100.8). The Theologumena 14.7 and Proclus in his Republic Commentary 13 (ed. Kroll, 2:21) both declare that the monad is sacred to Zeus (but for Proclus, see n. 38 above).
[59] Cf. Ficino's Philebus Commentary 2.1 (ed. Allen, pp. 386–387): "God is the measure of all things" who as the infinite limit imprints the limit in them—Ficino is contrasting (and reconciling) the Parmenides ' description of God as the infinite with the Philebus 's as the measure and limit (p. 389). The One "encloses all, forms all, sustains all, circumscribes all" (p. 387). By contrast, the infinite Plato is speaking of in the Philebus (as in the Timaeus ) is matter itself, which is formed "by a certain beneficent glance of the divine countenance" (p. 389). This matter is the "necessity" that is formed by intelligence, the pure potentiality that submits to God's act (p. 391). In general see the Theologumena 1.3–7.13 (on the monad) and 7.14–14.12 (on the dyad).
ered the prime equilateral. The 1 is indivisible, for when it appears to be divided it is in fact being miraculously doubled. Thus as the principle of "identity, equality, and likeness" it again resembles God. It has a "marvelous likeness" to Him also in that however much you multiply or divide it by itself, you neither increase nor diminish it; for, without altering itself, it is the square, the cube, and all other greater powers within itself.[60]
TWO the Pythagoreans wished to be indeterminate, says Ficino (6.46–47). As the dyad it is the "first multitude" (8.77–80) but not exactly the first number, though it is the begetter of the even, female numbers and as such is the prime unequilateral (as 2x1). Even so, it is the "principle" of no one figure, as the 1 is of the circle, or the 3 of the triangle and hence of all rectilinear figures. Archytas, as we have seen, had supposed 2 to be the Idea of the even numbers; but in Neoplatonic ontology, the dyad is identified with the "infinite" or "indefinite" (to apeiron ).[61] The two's negative connotations carry over for Ficino, following Plato in the Laws 4.717AB, to all the even numbers and make them subordinate to the odd.
Given the unique status of the 1 and the 2 as in a way nonnumbers, the THREE, Ficino declares, again citing Archytas, is the first number proper; as such it is made from the 1, and from the confused multitude, the otherness, the degeneration from the 1 that is the 2 (8.77–80). As the first number, it is necessarily the principle of all rectilinear figures (6.49–51); for the 3 is the first trigon, and the triangle is the first of the rectilinear figures (and there are of course three types of triangle—the equilateral, the isosceles and the scalene—and three types of angle—the obtuse, the acute, and the right angle). God "rejoices" in the 3 since it is "hinged" upon the 1 as 1+1+1. Therefore, it is the first of the three preeminent male numbers, the other two being 7 (3+1+3) and 9 (3+1+1+1+3) (6.88–92). It is of course the number of the Christian Trinity and thus of the "footsteps" (vestigia ) of that Trinity in all creation.[62]
[60] The analogies between the monad and the One as the ultimate metaphysical principle were as integral to the Christian philosophical tradition as to Pythagoreanism; see Theon, Expositio 1.3–4 (ed. Hiller, pp. 18.3–21.19) and the Theologumena 3.1 ff. Ficino had expatiated endlessly on the theme from his earliest involvement with Platonism; see, for instance, the earlier versions of the Philebus Commentary 2.1 (ed. Allen, pp. 394–395) dating from the 1460s.
[61] Again see Ficino's Philebus Commentary 2.1–4 and unattached chapter 3 (ed. Allen, pp. 384–425, 430–433). Among his sources were Proclus's Platonic Theology 3.7–9, In Timaeum 1.176 (ed. Diehl), and In Parmenidem 1118.9–1124.37 (ed. Cousin, 2d ed., Paris, 1864). Cf. the Theologumena 7.3–13, 8.5 ff., 12.9 ff.
[62] Augustine's De Trinitate 9–15 was assuredly Ficino's principal source here.
Among notable threes for Ficino and his contemporaries were the biblical three patriarchs, three magi, and three Marys, and the pagan three Graces, three Fates, the various triform or triple-headed deities such as Hermes, Cerberus, and Hecate, and the many threefold invocations and libations in classical literature. For the latter cf. the final summa of Ficino's Philebus Commentary (ed. Allen, p. 518), which declares that Socrates at 66D is offering up sacrificially "the third libation to the savior Zeus (Jovi conservatori )"; for "the ancient priests customarily poured the libation bowl thrice, declaring that we need god [Jupiter] the savior not only in the beginning of our affairs and of our life, but in the middle too and end." Ficino was also drawn to the Chaldaean trinity of Ohrmazd, Mithra, and Ahriman (Oromazes, Mithras, Areimanius) which he encountered in Plutarch's De Iside 46 (Moralia 369E). In general, see Theon, Expositio 2.42 (ed. Hiller, pp. 100.13–101.10); the Theologumena 14.13–19.20; Ficino's own De Amore 2.1; and Pico, Heptaplus 6, proem; also Edgar Wind, Pagan Mysteries in the Renaissance , rev. ed. (New York, 1968), pp. 241–255.
FOUR is the first square number and the first number to have two means. It is the number of man's ages, of the humors and seasons, and of the elements; and it refers, says Ficino, to the "revolution" or "commutation" of the four elemental spheres which is "in a way intercepted" in the plane (17.14–15, 18–20). Hence 4 is not a wholly circular number like 5 or 6. In Pythagorean lore the "sacred quaternary" is the four numbers 1, 2, 3, and 4—the tetraktys—since together they add up to 10.[63]
FIVE, the proportional arithmetic mean of the decad according to Theon (2.44), is the first of the fully circular numbers and refers to the "period," that is, to the circuit in general, of the planets (17.17–18). For the planetary region is the fifth, the celestial region above the four regions of the elemental spheres, the region made from Aristotle's fifth element, the aether. In the Timaeus 55C Plato had identified the fifth regular solid, the dodecahedron, the figure significantly with twelve pentagonal faces, with the world.[64] Aristotle says
[63] For disquisitions on the 4, see Ficino's Timaeus Commentary 20–24 and 26 (Opera , pp. 1446–1450); Theon, Expositio 2.38 (ed. Hiller, pp. 93.25–99.16)—on the eleven quaternaries; and the Theologumena 20.1–30.15; also Dupuis, Théon , pp. 385–386. Ficino makes no mention here of many other associations: with the tetragrammaton, the Evangelists, the corners of the world, the winds, the kinds of animals and of plants, the rivers of paradise and those of the underworld, the cardinal virtues, the horses of the Apocalypse, etc.; and with such fourfold triplicities as those of the angelic orders, the tribes of Israel, the zodiacal signs, and the apostles. See Heninger, Touches of Sweet Harmony , pp. 82–83, 151–156.
[64] Cf. the Theologumena 31.4–7, 32.17–20, 34.11 ff. and Ficino's Timaeus Commentary 44 (misnumbered 41) (Opera , p. 1464v). Ficino, erroneously, sees the twelve pentagonal faces as each divided into five equilateral triangles, which are in turn each divided into six half-equilateral triangles, making 360 such triangles in all. This equals the number of degrees in the zodiac, while 360x10 equals the number of the great year. His source here was almost certainly Plutarch's Platonic Questions 5.1 (Moralia 1003CD).
that the Pythagoreans called 5 (as 3+2) the marriage or wedding number,[65] though they also associated it (and 4 and 8!) with justice.[66]
SIX is called by the Pythagoreans the "spousal number" because "in its conception a male number joins with a female" (4.41–42).[67] As 3x2 it is thus the first product of the first even number and the first odd (given the 1's unique status). However, as the first of the perfect numbers, the first to equal the sum of its factors, and as the higher of the first two purely circular numbers (as 6-36-216), it is also a symbol of divine generation and "contains the circuit itself of divine generation" (17.3–17).[68] This refers, I take it, to the six days of Creation, to the six intervals between the seven planetary spheres,[69] and to the "circuit of the firmament," that is, to the revolution of the sphere of the fixed stars above the fifth region of the planets which is in turn above the four elementary regions. Working with the Platonic order of the planets (or rather with its Porphyrian variation),[70] Ficino uses 6 to plot certain critical astronomical durations or distances in the sense of spans. Thus in six steps from the firmament we reach Venus, from Saturn we reach the Sun, and from the Moon we reach Jupiter. Since
[65] Cf. Aristotle apud Alexander of Aphrodisias, In Metaphysicam 39.8 (fr. 203)—see G. S. Kirk, J. E. Raven, and M. Schofield, The Presocratic Philosophers , 2d. ed. (Cambridge, 1983), p. 336 (no. 436); also the Theologumena 30.19, 41.12–14. But see SIX (as 3x2) below. Five and six are linked insofar as one is 3+2 and the other 3x2, and insofar as they are the first circular numbers.
[66] Cf. Proclus, Republic Commentary 13 (ed. Kroll, 2:22); also the Theologumena 35.6 ff., 40.5–9 (and, in general, 30.16–41.20) and Iamblichus, In Nicomachi Arith. Introd. 16.11–20. Again, other associations—with the five senses, the five precipitations, the five books of Moses, the five wounds of Christ, and so on—are not adduced. For disquisitions on the quinarium , the five elements and the five universal genera (from the Sophist ), see Ficino's Timaeus Commentary 24 and 28 (Opera , pp. 1449.1, 1451.2–1452).
[67] Cf. Theon, Expositio 2.45 (ed. Hiller, p. 102.4–6); and the Theologumena 43.5–9, "it is also called 'marriage' in the strict sense that it arises not by addition, as the pentad does [i.e., 5=3+2], but by multiplication" (trans. Waterfield, p. 75). In his Timaeus Commentary 12 (Opera , p. 1443.2) Ficino observes that Pythagoras had agreed with Moses "probans senarium numerum genesi nuptiisque accommodari, unde et Gamon appellat propterea quod partes suae iuxta positae ipsum gignant similemque reddant genitum genitori." Six is also the mean between the prime square numbers 4 and 9.
[68] On the perfection of the hexad, see the Theologumena 42.19–43.3 (and, in general, 42.18–54.9), and Augustine, City of God 11.30, and De Genesi ad Litteram passim (the most authoritative for Ficino of many hexaemeral commentaries).
[69] Cf. the Theologumena 48.10–14, 50.5–6.
[70] The Porphyrian order goes: Moon, Sun, Venus, Mercury, Mars, Jupiter, Saturn, whereas the strictly Platonic order reverses the positions of Venus and Mercury. See my Platonism , pp. 118–119, n. 17, for further references.
these are the three "vivific" planets, the 6 is therefore associated with propitious periods in our lives, particularly the sixth month of gestation, and, even more importantly still the sixth, the jovian, year and its multiples.[71] The 6 is the key number to be taken into account when determining the opportune time for marrying and begetting and for embarking on any kind of project, material or intellectual. Even so, the 6 pertains more to the class of the "divine" than to that of the human; for, like the divine, the 6 is neither wanting (deficient) nor overflowing (abundant) and as such it is tempered equally and consists in, in the literal Latin sense of stands firm in, its parts and powers (17.57–62). Finally, it is the key to the ratios governing the Platonic lambda, being the first of the geometric means.[72]
SEVEN is the number of the planets, which constitute the fifth, the celestial, world zone; and the 7 added to the 5 composes 12. There are seven terms in the lambda (associated in the Timaeus 34B–37A with the planets and thus with the harmonies of the soul), and each of the two progressions in the lambda has four terms plus three intervals. There are also seven planetary modes.[73] Along with the 3 and the 9,
[71] And we should recall the importance of the cube of 6 in determining the period between reincarnations. See Chapter 1, n. 82 above.
[72] See n. 6 above. An Orphic verse familiar to Ficino from Plato's Philebus 66C reads, "With the sixth generation ceases the glory of my song" (ed. O. Kern, Fragmenta Orphicorum [Berlin, 1922], frg. 14; cf. Plutarch, De E apud Delphos 15 [Moralia 391D], and Proclus, In Rempublicam 2.100.23). He cites this verse in his Philebus Commentary 1.27, 2.2, and summa 65 (ed. Allen, pp. 257, 405, 518), in the first instance quoting the Greek and arguing that "Nature's progress stops at this, the sixth link of the golden chain introduced by Homer." In summa 65 he entertains the possibility that Plato's reference to the Orphic "sixth generation" may be to the Good Itself; for, he concludes, "Orpheus orders the hymns to end at the sixth generation, at the ineffable [Good] as it were. We see that the number 6 is celebrated by Orpheus as the end, as it was by Moses. For in the first ten numbers, this is the perfect number, being compounded from its parts disposed in order, that is from 1, 2, and 3." These comments point not only to Ficino's subtle sense of Platonic play but also, given the summa's heading which reads "Jupiter is the savior of the sixth grade or level," to his early association of Jove with 6; and this is despite the fact that in his ontology he identifies Jove with Soul (or with Soul in Mind), not with the Absolute Good that is the One. See my Platonism , pp. 125–128, 238–240, 244–245, 250–251, 252; and Chapter 4 below.
We should also recall that Ficino saw Plato as the sixth, and therefore as the most perfect, in the line of the six ancient theologians, the prisci theologi : Zoroaster, Hermes Trismegistus, Orpheus, Aglaophemus, Pythagoras, Plato. See, for example, his Philebus Commentary 1.17, 26 (ed. Allen, pp. 181, 247), and his Platonic Theology 6.1 (ed. Marcel, 1:224).
[73] See Ficino's De Vita 3.21. Wind, Pagan Mysteries , p. 268, observes that among the seven planetary modes "only the Dorian, associated with the sun and thus with
Apollo, is pure. Placed in the centre of the planets it secures the symmetry of the other six by dividing them into two triads."
the 7 is one of the three conspicuous male numbers because it exists on either side of its 1 as 3+1+3 (6.91–92). But Ficino also accepts the idea that it is sacred to Athena, being the first virgin number to succeed the first spousal number; and as such it is endowed with a marvelous property, as Theon says, since alone in the decad it has no multiple and no divisor.[74] There are of course seven days to the week, and a lunar quarter spans seven days. In ancient and medieval numerology there are seven orifices in the head and seven viscera; and Ficino refers elsewhere to the topos of man's seven ages[75] and to the seven metals associated with the seven planets.[76] It was also an important number, as he knew, in embryology, in human development theory with its climacterics, and in fever theory.[77] In the Bible it is particularly prominent in the Book of Revelation,[78] and it is traditionally the number of the deadly sins, of the virtues, and of the blessed sacraments.
For the Pythagoreans, EIGHT is a number associated with egalitarian justice, like 4 but for different reasons (6.93–94). But its primary significance for Ficino is its status as the number of the celestial
[74] Expositio 2.46 (ed. Hiller, 103.1–5): "thaumastên echei dunamin ." Cf. the Theologumena 54.11, 71.3–10; Macrobius, In Somnium Scipionis 1.6.11; Plutarch, De Iside et Osiride 10 (Moralia 354F); Proclus, In Timaeum 168C (ed. Diehl, 2:95.5); also Vergil, Aeneid 1.94: "O terque quaterque beati."
Plutarch, on the other hand, in his De E apud Delphos 17 (Moralia 391F), says that 7 is consecrated to Apollo, though Apollo is usually identified with the One (as for instance in the De Iside 10 cited above). We might note that there were traditionally Seven Sages and that the Theologumena 70.24, 71.10–12 says that 7 is the number of kairos , of the appropriate time (cf. Aristotle fr. 203 apud Alexander In Meta . 38.16 as cited by Kirk-Raven-Schofield, p. 331).
[75] Opera , pp. 1919–1920 in his translation of Proclus's In Alcibiadem 90–91 (ed. Westerink). Cf. Theon, Expositio 2.46 (ed. Hiller, pp. 103.19–104.19); and the Theologumena 55.13–56.7.
[76] See Ficino's argumentum for the Critias (Opera , p. 1486): gold for the Sun, silver for the Moon, lead for Saturn, electrum for Jupiter, iron and bronze for Mars, orichalcum (copper or brass) for Venus, and stagnum (an alloy of silver and lead) for Mercury.
[77] See Ficino's Platonic Theology 17.2 (ed. Marcel, 3:155), De Vita 2.20.1–17 (ed. and trans. Carol V. Kaske and John R. Clark, Marsilio Ficino: Three Books on Life [Binghamton, N.y., 1989]), cf. Theon, Expositio 2.46 (ed. Hiller, p. 104.9–12) and the Theologumena 55.5–7, 61.5–63.5, 64.17–67.2, 67.14–71.3 (with further references).
[78] As the number of the churches of Asia (1:4), of the spirits before God's throne (1:4), of the golden candlesticks (1:12), of the stars in the right hand of the Son of Man (1:16), of the angels of the churches who are symbolized by the candlesticks (1:20), of the number of the seals, of the angels with the seven trumpets and the seven plagues (5:1–8.2, 15:1), of the horns and eyes of the Lamb (5:6), of the heads of the beast
upon which sits the whore of Babylon (17.3), and of the world kingdoms (we are now in the sixth age) (17:9–10). Jude's Epistle declares that Enoch was seventh in the line from Adam (1:14).
We should recall that Pico's Heptaplus (1489) is a sevenfold narration of the six days of Creation and is divided into seven books which are subdivided into seven chapters. On God's resting on the seventh day, see Ficino's Timaeus Commentary 17 (Opera , p. 1445.1) and Augustine's City of God 11.31.
spheres and the octave, and as the first of the "solid" numbers, being the cube of 2.[79]
NINE is the number sacred to the Muses,[80] and there are nine heavenly choirs in Christian angelology.[81] In the Proclian tradition it is the number of the "like" and the "same," since it is the square of the first odd number,[82] and there are of course nine months to gestation.
Besides being the number of the commandments, TEN is the universal number as the sum of the sacred quaternary of 1, 2, 3, and 4 (and thus a trigon). It is the origin of the other universal numbers, that is, of 100, 1000, 10,000 (the myriad), 1,000,000, and so on.[83] Theon says that 10 is imperfect, even though it is sometimes thought
[79] Cf. Ficino's Republic 4 epitome (Opera , p. 1403) which refers to the Pythagorean 8, the Orphic 8, and the Egyptian 8. In general see Theon, Expositio 2.47 (ed. Hiller, pp. 104.20–106.2); and the Theologumena 72.1–75.4.
[80] See Ficino's argumentum for the Ion (Opera , p. 1283) and Platonic Theology 4.1 (ed. Marcel, 1:164–165); also my Platonism , pp. 28–30.
[81] For medieval Christianity it was the authority of the Pseudo-Dionysius's On the Celestial Hierarchy that had fixed the ranks and orders of the angels at three hierarchies of three choirs each; cf. Ficino, De Raptu Pauli 6 (ed. Marcel, in Ficin: Théologie 3:352), and Pico, Heptaplus 2, proem.
[82] Cf. Proclus, In Rempublicam 2.80.23–26 (ed. Kroll); and the Theologumena 78.14. Plato was said to be ninth in succession from Pythagoras. In his Phaedrus Commentary, summa 24 (ed. Allen, pp. 165–169), Ficino addresses the problem of the nine lives introduced in the Phaedrus 248C–E; see my Platonism , pp. 174–179.
[83] Cf. Ficino's Phaedrus Commentary, summa 25 (ed. Allen, pp. 168–171). On the universal nature of the decad, see Ps.-Aristotle, Problemata 15.3.910b23 ff.; and the Theologumena 79.16–81.3, 82.10–85.23 (this includes an important fragment by Speusippus [fr. 4, Lang] on the Pythagorean numbers). For the myriad, cf. Pico, Heptaplus 3.6, paraphrasing Daniel 7:10: "Ten thousand stood before him [the Ancient of days], and a thousand thousands ministered unto him."
The Republic 10.615AB declares optimistically that 10x10 is man's natural life-span; and Pythagoras is said to have lived for almost a hundred years (see Iamblichus, De Vita Pythagorica 36.265). In a brevis annotatio inserted between his translations of books 5 and 6, Pier Candido Decembrio had argued that Plato's ten books represent the ten decades of this natural life (see Hankins, Plato in the Italian Renaissance 1:133, 135 and n.). However, in his epitome for book 1 (Opera , p. 1396), Ficino supposes, presumably on Pythagorean grounds (see n. 84 below), that Plato had elected to write the Republic in ten books because ten is "the most whole" (integerrimus ) of all the numbers.
of as a perfect number by Pythagoreans,[84] who referred to it as "the unity of the second rank," the 1 being the unity of the first rank, the 100 the unity of the third rank, and the 1000 the unity of the fourth.85
VI. While traditional arithmology treats of the decad alone, there is one other number of particular importance to Ficino here, given his obsession with the Timaeus 's lambda as a model, and given the role he assigns to the harmonic ratios in preparing us for an understanding of the fatal geometric number. For set over and against the ordinary world of 10 and its multiples is the duodecimal world governed by 12, the number presiding over Plato's last book, the Laws , where, Ficino maintained, Plato had spoken for the first time, apart from his Letters , in his own person.[86]
Though the second of the spousal numbers as 3x4,[87] 12 is the first and "prince" of the "abundant" numbers (4.38–40), meaning, as we have seen, that the sum of its factors—that is, of 6+4+3+2+1—exceeds itself, in this instance by as much as a third again. As such, it designates fertility. Ficino claims that 12 is the number "secretly" venerated in the eighth book of the Republic , while it is openly venerated in several other dialogues (3.42–44). In the twelve books of the Laws (at 5.745B–E,746D, 6.771A–C, and 8.848CD), 12 is the number into which the Athenian Stranger divides the state's capital city, and then divides and subdivides its surrounding agricultural districts and its 5040 citizens; and each twelfth portion of territory and people is dedicated to one of twelve gods—the ideal state has in fact a duodecimal structure. In the Phaedo 110B ff. Plato affirms that the globe resembles a ball made from twelve pieces. In the Timaeus at 55C4–6 he speaks of the fifth "combination" or figure which later Platonists iden-
[84] Theon, Expositio 1.32 (ed. Hiller, p. 46.12–14) and Nicomachus, in his Introductio 2.22.1, say that the Pythagoreans thought the 10 perfect (as the sum of 1+2+3+4 and not as the sum of its factors). But Theon goes on to declare later at 2.49 (ed. Hiller, 106.7–10) that it "contains in itself the nature of both even and odd, of that which is in motion and that which is at rest, of good and of evil."
[85] Cf. Iamblichus, In Nicomachi Arith. Introd. 88.22–26 (ed. Pistelli).
[86] See his epitome for Laws 1 (Opera , p. 1488).
[87] In an interesting passage in his Commentary on the First Book of Euclid's Elements 1.36, Proclus observes that the dodecad is the product of the triad and the tetrad and that it "ends in the single monad, the sovereign principle of Zeus, for Philolaus says that the angle of the dodecagon belongs to Zeus, because in unity Zeus contains the entire number of the dodecad." Ficino had access to this passage, though he does not cite it.
tified with the mysterious dodecahedron that the Demiurge had used for "the delineation of the universe" with the twelve zodiacal signs, as we have seen; and at 58C–61C he speaks of the twelve world spheres and the twelve parts of the elemental spheres (each of the four spheres being divided into a higher, middle, and lower zone). In the Phaedrus 246E–247E Plato describes the twelve orders of the gods and their ascent as charioteers to the outer convex rim of heaven, there to gaze upwards at the supercelestial place. And in the Critias 109B ff. and 113BC he refers to the ancient prediluvian division of the earth by the gods into twelve allotments (at least in the Neoplatonic interpretation). In short, Ficino associates 12 with several major texts, including as we saw earlier the Epinomis , and assumes it to be a number fraught with especial significance for Plato quite apart from its being the number of the chief Olympian deities, of the signs of the zodiac, of the months, and so forth.
Ficino writes that 12 is the governor "of the universal world form, of the human form, and of the form of the state," because it is the number that presides over the increase and mutations of all things, being the double of 6, the first of the perfect numbers (3.53–58). Yet 12 is also the sum of 7 and 5–7 being the number of the planets and the "root" of the sesquitertial ratio as we have seen, and 5 being the zones of the world and the root of the sesquialteral ratio. Hence 12 is the most "accordant with" the world's orb and is the traditional number of the world spheres in Chaldaean-Ptolemaic cosmology. Twelve is particularly associated with the planets presiding over life: the Sun, Jupiter, Venus, and the Moon—the founts of vitality—whose orbital paths significantly are measured in twelves, the Sun and Venus completing their orbits (around the Earth) in twelve months, Jupiter in twelve years, and the Moon waxing (and then waning) in twelve days in her course through her twenty-eight mansions, and thereby establishing twelve as the months in the year (3.64–68). No wonder, writes Ficino, that such a number was observed by the "Prophets and in sacred writings."[88]
We are now in a position to follow Ficino's unraveling of the mystery of the fatal geometric number.
[88] Apart from there being twelve tribes of Israel and twelve apostles, in the Book of Revelation twelve is the number of the gates of the New Jerusalem inscribed with the names of the twelve tribes, each having 12,000 people (and thus 144,000 [12 x10 ] is the number sealed to be saved) (7:4–8). Revelation also speaks of the woman with the crown of twelve stars (12:1) and of the twelve kinds of fruit on the tree of life (22:2).
VII. The Republic book 8 speaks of a human or imperfect number that has four terms and therefore three intervals or distances, the terms being related to one another in certain ratios or proportions. In addressing the challenge of identifying this number, Ficino plays with the circumspect notion, given his knowledge of book 9, that there may perhaps be several such "fatal" numbers—not to be confused necessarily as we have seen with "spousal" numbers.[89] As numbers that have an "immense power to produce both good and not good progeny" (13.35–37), the fatal numbers must all be contrasted with the perfect (or perfecting) number that Plato had begun the discussion with and that presides over divine and therefore wholly good progeny, a number that was itself either the first in, or at least one of, the select class of such numbers.
Plato is most drawn, Ficino argues, to two particular fatal numbers, since they "best agree with the universe" and "embrace the consonances," that is, the universal Pythagorean harmonies that govern the motions of the nest of its spheres (3.108–109). Following Aristotle, Ficino finds it especially important that Plato had arrived at a cubed number, since cubing constitutes raising to the "highest" power, raising to even higher powers being merely an imitation or complication of cubing—trinitarian assumptions are obviously to the fore here as are the associations of three with the classical Fates and specifically with Plato's presentation of them.
A. In chapter 3 Ficino first considers the candidature of 729, a number ending in 9 which is celebrated from Ficino's viewpoint significantly and not coincidentally in book 9 of the Republic at 587E ff., the only other passage in the entire work which casts a light, however dim, on the problem of the computation of the fatal number alluded to in book 8.[90] This "great and fatal" number 729 is seen as the product of cubing 9—the number symbolizing the nine celestial spheres, those, that is, of the planets, of the fixed stars, and of the primum mo-
[89] Compare the heading for the Expositio with that for the Commentarius as a whole (though neither heading may be Ficino's).
[90] In his epitome for book 9 (Opera , pp. 1426–1427), Ficino merely observes, "Inter haec casu quodam nescio quid interserit mathematicum, cuius declarationem ex commentariis in Timaeum accipies opportunius" (p. 1427). The fact that the number begins with 7, the number of the planets, and ends in 29, the number of days in which it takes the Moon to catch up with the Sun, would not have been lost on him. It is, incidentally, the seventh term in the series 1-3-9-27-81-243-729; and in the De animae procreatione 31 (Moralia 1028B), Plutarch declares it is the number of the Sun.
bile (3.77–80; 14.6–11). It is described in the ninth book as a measure of the interval separating the king from the tyrant and therefore as "an overwhelming expression of the distance that separates the just from the unjust in regard to pleasure and pain." "A true calculation," it is "a number which nearly concerns human life," being one less than the total number of days and nights in a year. Moreover, it is a circular number in that 9 is both its beginning (its root) and its end (its last digit), though, like 4, it is intercepted in its plane of 81.[91] Finally, while it has a cube root of 9, it also has a square root of 27; momentously it is endowed, in other words, with two roots that are themselves powers of 3.
B. Nonetheless, for Ficino the "principally fatal number" is not 9 raised to the third power or its equivalent 27 raised to the second power, but rather 12 raised to the third power, namely 1728 (which is also the product of 8x12x18, the first three numbers at the base of the Platonic lambda).[92] Since the last digit of 1728 is 8, 1728 is appropriately the subject of the eighth book (728 being one less than 729, the subject of the ninth book!). Ficino also finds it mathematically witty that Plato had begun his presentation of this fatal number in the eighth book by adducing 6, the first of the perfect numbers and the lambda's key, and had then ended the number with 28, the second of the perfect numbers (15.42–45). He is postulating, in other words, the presence not only of the manifest parts of a number, namely its factors, but of what he calls the "hidden" or secret parts also, those that constitute, as we shall see, its beginning, middle, and end.
The cube of 12 is both "fatal," that is, concerned with the marking out and the governing of time, and "universal" or compendious in that it embraces odds and evens, equilaterals and unequilaterals (both longs and oblongs), planes and solids, laterals and diagonals, and the better and worse consonances (13.31–35). Its compendiousness renders it "a discordant concord," appropriately so since it presides over
[91] Plato's reported death at the age of 81 is therefore numerologically significant, not least because 81 is the square of the months of gestation in the womb; see Seneca, De Senectute 5, and Dante, Convivio 4.24. In his Vita Platonis (Opera , p. 770.2; trans. in Letters 3:46), Ficino observes that the Magi saw 81 as "the most excellent number," being 9x9. Plotinus, incidentally, died at the age of 66. Pythagoras, Plato, and Plotinus thus offered three options for a philosophical life-span and Ficino selected 66 for himself!
[92] One is led to speculate whether the sesquitertial ratio between the two roots of 1728 and 729, namely 12:9, was of any significance to him (note wording of 3.77–82).
the discordant concord of the realms of quality, generation, and decay (15.33–36). Moreover, because its root of 12 is the first of the increasing or abundant numbers, 1728 is abundantly abundant—is the increasing number's increase to the third degree, to the absolute degree of increase. And, because its root contains the two harmonies of the perfect fourth 4:3 and the perfect fifth 3:2 in the sense that added together they make 12, the number of the diapason, 1728 too contains them. Indeed, it "extends" them still further, Ficino writes, and therefore "best agrees with the universe" (3.93–97). After 1728 years (and Ficino ignores the possibility that the number could apply to any set of temporal units—days, months, seasons, centuries), the circle of Fate reaches its turning point, and we enter upon a period of decline. Accompanying such a turning point are the various signs and wonders that augur the eventual end of a duration double that span of 1728 years, a duration of 3456 years, though neither Plato nor Ficino mentions such a duration.[93] Both durations are obviously considerably less than the cycle of the great year as defined by Plato in the Timaeus 39D as the time taken by the Sun, the Moon, and the rest of the planets to return to the same relative positions, a cycle governed by "the perfect number of time" (D3–4) and to which, as we have seen, Ficino had already assigned the traditional value of 36,000 years.[94]
Ficino asks why the fatal number is referred to as "proportional" and as "geometric" (13.35–36). It is proportional because it contains the musical proportions contained in 12, but it is geometric because it is the cube of the sum of the sides of the Pythagoreans' beautiful rectangular scalene with its sides of 3, 4, and 5 (and 5 can be defined, as we have seen, as the "root" of the ratio 3:2 and 7 as the root of the ratio 4:3). Since it contains the musical proportions, and since as a cube it is the beautiful triangle triangled as it were, it has an immense power to "abound" with temporal progeny good and bad, to abound with opportunities and occasions. Since numbers in the world ages and human ages should be carefully observed, writes Ficino at the end
[93] Of course, it might not be that symmetrical, and no one actually declares that 1728 is halfway through a cycle. Theoretically it might instead take double time, or triple or quadruple or whatever, to decline from the "abundant" point, or even some fractional span like time and a half, time and an eighth—the possibilities are limitless but unlikely.
A remote possibility is 46,656, the product we recall from n. 6 above of the numbers at the base of the Platonic lambda, since 8x12x18=1728=12 and 1728x27=46656 (i.e., 12 x3 , or 216 or 6 x6 , or 6 x6 x6 , or 6 ).
[94] See Chapter 1, n. 24 above.
of chapter 13, and because a praiseworthy number signals the opportunity for fecundity, an unpraiseworthy one the occasion for evil and sterility, this number, which is praiseworthy and unpraiseworthy equally, must be observed before all others. It is the most sublime and the most terrible instrument of Fate, is indeed the threefold number of the three Fates. For Ficino finds it significant: first that Plato had elected three components for the fatal number, second that he had concealed three different kinds of a hundred in it, and third that he had made it the third power of another number. In sum, Ficino thinks of Plato's fatal number as itself fatally—that is, triply—threefold, a fatal companion if you will of the 9 and its threefold powers.
C. Let us now turn to what Ficino calls the three "hidden" parts of 1728, the second of which involves him in some ingenious extrapolation.
i. The first such part is 1000. Ficino had argued earlier that 12 embraces 10 in the ratio of 6:5 in that 12=(5x2)+2; and thus "best agrees with the universe," 10 being the first of the universal numbers as the product of the first four numbers, the Pythagorean tetraktys (3.82–89). Hence 12 raised to the third power must likewise embrace the universe betokened by 10 raised also to the third power.[95] The 1000 is therefore the universal number raised to the fatal third or solid power.[96] In chapter 14 Ficino speculates that "perhaps" it signifies the firmament "hidden in a way in (among?) the stars," the stars themselves being the fabled myriad of the 10,000, i.e., 104 (14.13–14; cf. 3.97). Ficino knew the myriad had been celebrated particularly in the Phaedrus at 248E ff.;[97] but here he adduces it not only as the number of the "numberless crowd" of the stars—"numberless" because difficult to number, not because the stars are infinite in number[98] —but also as "the more ample" number, the limit, if you will, of the realm of planes just as the million is the limit of the realm of solids (14.18–22). The myriad can be unfolded, chapter 15 will subtly ar-
[95] Would Ficino have seen any numerological significance in Moses giving ten commandments to the twelve tribes of Israel?
[96] It is also the tenfold penalty described in the myth of Er that a man must pay for crimes during his life of a hundred years (cf. n. 83 above).
[97] See Ficino's Phaedrus Commentary, summa 25 (ed. Allen, pp. 170–171). See my Platonism , pp. 177–178.
[98] "Myriad" indeed can mean a huge crowd—the emotional and not just the numerical ten thousand.
gue, in three "secret" ways: as 1002 ; as the number that contains 100 diagonal powers; and as the principal factor of a million. Moreover, it has an "unequal dignity" as 10x1000 (10x103 ) but an equal dignity as 100x100 (102 x102 ). As the latter it is "the century of centuries," and through it "not only republics but all ages may be measured" (15.52–53). It is thus the number signifying the absolute temporal measure of the spans of all "compounded" things viewed in their species and kinds, while its constituent roots of 100 and 10 serve by implication as measures of the lesser spans, the centuries and decades, apportioned to individual entities (15.53–55).
ii. Ficino next turns to the second "hidden part" of 1728, namely to the 700. This was much more difficult to extract from Plato's conundrum than the 1000, and he was forced to delve more deeply into the series of cruces at 546C which declares that two harmonies result from the coupling of a base of four thirds to a root of 5 at the third augmentation, the one being "the product of equal factors and of a hundred multiplied the same number of times," the other being "of equal length but very oblong." This latter is enigmatically described by Plato as the 100 "of numbers from comparable diagonals of the 5, with individual diagonals requiring one, but those which are not comparable requiring two."[99] Since the text immediately goes on to mention that "the 100 of the cubes is of the three," Ficino interprets it to mean that, as will be the case with the myriad, Plato is presenting us with the 100 thrice, the 100 being the second in the "order" of universal numbers stemming from the 10, but the first equilateral, the first power, in that order (14.16–18). We recall that it is the 100 which is celebrated, again appropriately, in book 10 of the Republic at 615AB in the climactic account of the myth of Er, a hundred years being reckoned there as we have seen as the ideal length of a man's life. Clearly the 100 as a century—the basic unit of an age—is the governing paradigm, for the 100 is described as the "brood" or "fruit" of the 10 (15.13–14), and hence as the brood of the temporal decade, the leading of the decad—and thus of the tetraktys—to itself.
Plato's opening phrase at 546C3 describes the first harmony as
"equally equal," as 100x100. This Ficino calls "the first denomination," and he takes it to be referring to the 100, itself an equilateral, as the root and therefore as the "producer" of the ampler equilateral of the 10,000, which as 102 x102 is "equally equal," the square of a square as we have seen.
Plato's second harmony is "of equal length" with the first and must therefore be for Ficino 100 long. But its "width" is measured by a number that is the diagonal of a square with sides of 5—Ficino's interpretation of the phrase "of the pempad." The irrational diagonal of such a square is the square root of twice the square of its side, and therefore
Finally he refers to the "third denomination" of the 100 when it signifies the cube root of a million (14.35–38; 15.30–31), a million being the value he sees Plato having intended by the phrase at 546C6, "But the hundred of the cubes is of the three," "of the three" meaning raised to the third power (5.44–45). For, he observes, Plato had deliberately extended "the fatal numbers to the solid as to the highest [point or power], so that hence he might show, having reached the highest already, that little by little all are brought back to the opposite" (3.111–113). This million refers to all the heavenly bodies, seen and unseen in the firmament,[101] and beneath it is situated presumably the realm of the innumerable planes.
In short, Ficino interprets what he sees as Plato's triple reference here to 100 as follows: the 1002 refers to the 10,000 visible beings in the firmament (itself symbolized by the 1000); the 100x7 refers to the seven planets, the most obviously visible of heavenly beings beneath
[100] See n. 17 to Text 2, p. 167 below. There are many difficulties.
[101] For the theory of countless "unseen" beings in the heavens, which has good Ptolemaic authority, see Chapter 4 below.
the firmament; and the 1003 refers to the 1,000,000 of the totality of the heavenly beings unseen as well as seen.
iii. The final hidden part of 1728 is 28. If 1000 and 10,000 betoken the firmament and the visible stars, and 700 beto-kens the seven planets, then 28 must betoken the Moon specifically (3.98, 103; 14.41). For, apart from being the number of days in the lunar month and of her mansions, 28 is 7x4, that is, the number of the planets times the number of the elements. It is thus a singularly appropriate product to symbolize the planet that mediates between the planetary and the elemental spheres. As the seventh and nearest planet—and Ficino says that 6 betokens the six higher planets (3.105)—the Moon has no harmony or proportion with the firmament except by way of the six higher planets, which have "a similar proportion to the stars as the Moon to them" (14.41–45). What Ficino surely has in mind here is not some numerical proportion[102] but rather the fact that 28 is the second perfect number after 6, and as such "brings the second perfection to things subject to fate," that is, to sublunar generation (3.106–107).[103] Once again this raises the possibility that the second perfection must depend in some way on the first, just as 12, the cube root of the fatal geometric number and the second spousal number, also depends on 6, the first spousal that had constituted Plato's "exordium." At all events, as 4x7, an oblong number that is also the sum of its parts and therefore a perfect number, 28 is particularly appropriate for the Moon and her mansions and for the power she exercises over all beneath her sway.
D. To conclude, the fatal number had as its hidden parts the three numbers associated with the firmament, with the planets and with the Moon, even as its most prominent "unhidden" part, its cube root, its trinitarian root if you will, constituted the number of abundance, and thus the number associated with the months, with the zodiacal signs, with the Olympian deities, with the books of Plato's Laws and the books of Vergil's prophetic Aeneid , with the tribes of Israel, with Christ's Apostles, with the gates of the Apocalypse's New Jerusalem,
[103] In what does this "second perfection" consist?
and so forth.[104] It was thus a cornucopia enveloped in the same kind of mystery that had long enveloped the Pythagoreans' tetraktys. Indeed, from a musical viewpoint the mystery was the very same in that the fatal number and its cube root also contained the three universal harmonies,[105] as Ficino himself had already pointed out in his epitome for the Epinomis .[106] Finally, whatever the remaining cruces in Plato's crux laden description of the number, 1728 was in wonderful accord both with Aristotle's gloss and with the cosmological numerology of Plato's Timaeus .
Nonetheless, the origins of 1728 and its cube root lie, Ficino was convinced, in the perfect number 6, and clearly not in the quotidian 5. For Plato had intended us to look beyond the realm of Fate that 12 signifies to the higher realm of Providence; and to set another, a hexadic, time, God's time, over and against both the decades, the centuries, and the millennia that we measure by 10 and its multiples, and the dodecadic time of the Sun, the Moon, and the stars, and the calendars we base upon them. He had intended us, that is, to set a perfect, golden time over and against both the iron time, the clock time, of nature's and of man's present imperfection and mutability, and the silver time of the celestial spheres. However, the poets' superficially simple and nostalgic notion of a "golden" time and its generation or regeneration has complex, far-reaching mythological and philosophical implications for the Neoplatonic tradition that Ficino inherited and revived, and to some of these we must now turn.
[104] As the reference in the De Numero Fatali 12 indicates, 144 as the square of 12 was also in Ficino's mind as signifying a time when "a great permutation occurs among men." We recall that 144,000 is the number of the saved in Revelation 7:4–8.
[105] Bowen, "Ficino's Analysis of Musical Harmonia ," pp. 21–23, points out that the two scales described by Ficino, the Pythagorean and the syntonic diatonic, both derive from the tetraktys. See also Heninger, Touches of Sweet Harmony , pp. 93–97.
[106] See Chapter 1, p. 29 above.