Text 3:
De Numero Fatali

The Commentary on Plato's Passage from the Eighth Book of the Republic Concerning the Republic's Mutation through the Fatal Number

The Exposition of Marsilio Ficino Concerning the Nuptial Number in Book 8 of the Republic

For a long time the prodigious enigmas in the preceding chapter [i.e., 546A–D] have terrified us and other Platonists from devoting ourselves to their explication. The enigmas I will deal with first, however, are those that have struck me, having thought about it for a long time, as very certainly interpretable. Eventually, I will append those I can very probably explain and ignore those that I cannot. For Plato himself did not wish certain enigmas to be unfolded. Indeed, discourse inexplicable to men deservedly he attributed to the Muses, but to the Muses at play, for there is something in a fable which is hidden from us.

Chapter 1. On Circles, Conversions, Revolutions; and by What Opportunity the Lower May Be Led by the Higher.

At the onset he names the substances themselves of the world spheres (but principally of the celestial spheres) "ciclos," that is, circles or rings.^[1] Then he calls the absolutely circular motions, which the celestial spheres and any of the fixed stars as it were complete around their own centers, "peritropai," that is, conversions.^[2] Moreover, the circuits, which all the planets enact in addition from east to west and back to the east or in alternation, and likewise from south to north and the reverse, again forwards and backwards, upwards and downwards—these he calls "periphorai," that is, revolutions or ambits.^[3] Such planetary revolutions or ambits are ruled by the spherical conversions and accord with things earthly. The planetary revolutions also accord with and fit the spherical conversions to things earthly.^[4] Thus life and fertility and [their] opposites among things earthly are measured by way of things heavenly, but according to the law that declares that particular species of plants or animals are subjected to and guided by particular measures. For one revolution of Saturn (or one

Commentarius in Locum Platonis Ex Octavo Libro de Re Publica de Mutatione Rei Publicae per Numerum Fatalem^[*]

[149r][1414] Expositio Marsilii Ficini^[1] Circa Numerum Nuptialem in VIII de Re Publica.

Aenigmata in capite praecedenti^[2] prodigiosa iamdiu ab explicationis studio nos et Platonicos alios absterruerunt. Sed quae diu cogitanti mihi certiora succurrunt imprimis adducam; denique probabilia subdam, [5] inexplicabilia praetermittam. Nam et ipse Plato quaedam^[3] noluit^[4] explicari, sermonem vero hominibus inexplicabilem merito Musis attribuit, sed ludentibus, quia later^[5] aliquid fabulosum.

De Circulis, Conversionibus, Revolutionibus, et Qua Opportunitate a Superioribus Inferiora Ducantur. Cap. 1.

Principio ciclos, id est, circos vel circulos, nominat substantias ipsas mundanarum sphaerarum, praecipue vero caelestium. Appellat deinde [5] peritropas, id est, conversiones, motus simpliciter circulares quos sphaerae caelestes et stellae quaelibet quasi fixae peragunt circa propria centra. Vocat praeterea periphoras,^[1] id est, revolutiones vel ambitus, ipsos circuitus quos planetae omnes insuper agunt ab ortu ad occasum rursusque ad ortum vel vicissim; item a meridie ad septentrionem atque [10] contra, rursus ante et retro, sursum atque deorsum, sive vicissim. Revolutiones ambitusve eiusmodi a conversionibus illis reguntur quidem, ac rebus terrenis accommodantur. Ipsae quoque revolutiones terrenis conversiones^[2] accommodant atque coaptant. Vita igitur et fertilitas et opposita in rebus terrenis per caelestia mensurantur, sed ea lege ut [15] aliae species plantarum^[3] vel animalium aliis subiiciantur mensuris atque ducantur. Aliis enim pro ipsa naturae suae vel specialis vel singularis proprietate convenit una Saturni revolutio vel insuper dimidia vel^[4]

* Titulum in Z (p. 1413.2); "Marsilii Ficini Florentini Platonici Expositio De Numero Fatali" in M (fol. 150r); in Y deest .

[1] Marsilii Ficini om. Z

[2] praescedenti Y sequente M

[3] quidem M

[4] voluit Z

[5] satis M

[1] peripteforas Z

[2] conversionibus M

[3] planetarum Z

[4] vel om. Z

and a half, or two, three, or four revolutions) accords with some things, according to the property of their own nature (whether that of the species or that of the individual). However, one revolution of Jupiter accords with others or many revolutions accord (and similarly of Mars, the Sun, Venus, Mercury, and the Moon). For still others a fixed number of days is proper, or one day, or [merely] hours. There are those too that may be measured by one conversion of some star around the center or by many conversions, though they are unknown to us.^[5] The time is also unknown to us which is absolutely meet for some effect, namely the time in which the spherical conversions necessary for this effect unite with the planetary revolutions necessary for the same effect, and finally combine with, and are adapted to, the preparation itself of things earthly. For then fate coincides with nature and executes the effect that is favorable or adverse to fertility or sterility, or that pertains to other matters.^[6] Not only the dispositions of bodies in general but also in a way the varieties of feelings, of mental dispositions (ingenia ),^[7] and of habits and humors are led by this particular fatal and natural law. But certain divine (rather than human) dispositions prosper in certain ages, having been produced, that is, by the spherical conversions and the planetary revolutions that are determined by and known to God alone. These the perfect number measures—the number, I repeat, of centuries, or years, or months, or days, and known likewise to God alone. But more concerning this number elsewhere.^[8] However, the condition of human dispositions is thought to be subject to different conversions and revolutions, and these are computed by numbers that are also different.

Chapter 2. How There are Various Durations of Things.

Plato multiplies such a perfect number as it were to the numberless^[1] in order to be able, with the whole of such a number, to measure the whole life of the world, or its reformation from deluge to deluge, or the great year. But with the parts of such a number he measures also the lesser durations pertaining to private or public form.^[2] Take as an analogy the number seven which also measures many things: seven years the changes in life, seven days the greater changes in diseases, seven hours the lesser changes, now into the good, now into the bad.^[3] However, when number reaches six,^[4] which is perfect, it designates the perfect condition (habitus ). When it reaches eight, which is defi-

duae vel tres vel quatuor; aliis autem una Iovis aut plures similiterque Martis aut Solis, Venerisque atque Mercurii aut Lunae; aliis certi^[5] dies^[6] [20] vel dies una vel horae. Sunt et quae mensurentur conversione alicuius stellae circa centrum una vel pluribus sed nobis incognitis. Incognitum quoque nobis est tempus ad effectum aliquem penitus opportunum, quo scilicet conversiones ad hunc necessariae^[7] concurrunt cum revolutionibus necessariis ad eundem, ac denique cum ipsa terrenorum praeparatione [25] conveniunt atque coaptantur. Tunc enim fatum congruens cum natura effectum peragit vel prosperum vel adversum ad fertilitatem vel sterilitatem vel ad alia pertinentem. Non^[8] solum vero dispositiones corporum omnino sed quodammodo etiam affectuum ingeniorumque et morum varietates fatali hac et naturali quadam lege [30] ducuntur. Proveniunt vero quibusdam seculis divina quaedam ingenia potiusquam humana, producta videlicet a conversionibus revolutionibusque soli Deo certis atque destinatis. Quas sane numerus metitur perfectus, numerus inquam vel seculorum vel annorum vel mensium atque dierum similiter soli Deo [149v] notus. Sed de hoc numero alias [35] aliquid. Humanorum vero ingeniorum conditio conversionibus revolutionibusque aliis per alios^[9] quoque numeros computatis subiecta putatur.

Quomodo Rerum Durationes Variae. Cap. II.

Multiplicat vero Plato numerum eiusmodi quasi ad innumerabile ut toto eiusmodi numero metiri possit totam mundi vitam^[1] vel reformationem eius a diluviis ad diluvia vel annum magnum. Partibus vero numeri durationes^[2] quoque minores ad privatam formam vel publicam [5] pertinentes, sicut^[3] etiam septenarius multa metitur: per annos quidem mutationes in [1415] vita, per dies autem mutationes in morbis maiores, per horas vero minores, tum in^[4] bonum, tum in malum. Iam vero quando numerus pervenit ad sex qui perfectus est designat perfectum habitum; quando ad octo qui deficit partibus forte deficien- [10]

[5] certe Z

[6] dicitur Z

[7] necessarie YM necessario Z

[8] Ac Z

[9] alias Z

[1] machinam M

[2] duratioris Z

[3] sicuti M

[4] videlicet M

cient in parts, it designates perhaps the deficient condition—unless the thing can be balanced by the solidity of the number or for another reason.^[5] When it reaches twelve, which is abundant [in parts], it designates fertility.^[6] When it arrives at unequilateral numbers, it designates inequality;^[7] when at equilateral numbers, equality;^[8] when at solid numbers, constancy and plenitude.^[9] But more will be said about these matters in the ninth book.^[10] Likewise, with diagonal powers (diametrales ), when it arrives at the proportion of being less than double, it signifies sterility. However, with the same powers, when it arrives at the proportion of being greater than double, it signifies fertility. But this will be discussed a little later.^[11]

Chapter 3. On the Prime Solid Numbers and on the Number Twelve. How Twelve Contains Consonances within Itself and When Thrice Multiplied Unfolds Them to the Full.

Let us return to the numeral order first posited by Plato.^[1] Plato affirms that he is speaking of the numeral order in which, for the first time, there are four terms and three intervals. It is clear from the Timaeus^[2] that this order is between the prime solids, that is, between 8 and 27, whose proportional means are two, namely 12 and 18. Thus far the terms are four, and the intervals among them are necessarily three: the first being from 8 to 12, the second from 12 to 18, the third from 18 to 27. But the proportion is everywhere alike among these terms. For the proportion of 27 to 18 is in the ratio of three to two. For it contains the whole [i.e., 18] and a half besides [i.e., 9]. The proportion is similar from 18 to 12, and from 12 to 8. But between the prime solids, that is, 8 and 27, are the two equilateral planes, that is, 9 and 16, which envelop an unequilateral plane between themselves, namely 12. For just as from 16 to 12 the proportion is in the ratio of four to three—for 16 contains the whole [i.e., 12] and a third part besides [i.e., 4]—so from 12 to 9 the proportion is discovered to be in the ratio of four to three.^[3] Therefore, since in the numeral order taken up initially the [prime] solids are connected by way of the two means [i.e., 12 and 18]^[4] —both with the proportions to the solids in the ratio of three to two—but since the planes [i.e., 9 and 16] are joined by only the one mean [i.e., 12] with the proportions in the ratio of four to three, it is appropriate that Plato, having seized the occasion here, should bring to our attention the prime foundations of such propor-

tem, nisi res soliditate numeri vel ratione alia compensetur; quando ad^[5] duodecim qui abundat, fertilitatem; quando ad numeros inaequilateros, inaequalitatem; quando ad aequilateros, aequalitatem; quando ad solidos, firmitatem atque plenitudinem. Sed de his in nono^[6] dicetur. Proinde quando in diametralibus pervenit ad proportionem dupla [15] minorem, sterilitatem; quando vero in eisdem ad proportionem dupla maiorem, fertilitatem. Sed de his paulo post agetur.

De Primis Solidis Numeris et de Duodenario, Quomodo et Intra se Continet Consonantias et Ter Multiplicatus Explicat eas in Amplum. Cap. III.

Redeamus ad numeralem ordinem primo positum a Platone. Affirmat Plato se loqui de ordine numerali in quo primo sint termini quatuor et [5] intervalla tria. Manifestum vero est ex Timaeo hunc ordinem esse inter solida prima, scilicet inter 8 atque 27 quorum sunt media proportionalia duo, scilicet 12 et 18. Hactenus sunt termini quatuor inter quos necessario intervalla sunt tria: primum ab 8 ad 12, secundum a 12 ad 18, tertium a 18¹ ad 27.^[2] Inter hos vero terminos similis est utrinque [10] proportio. Nam ab ipso^[3] 27 ad 18 sexquialtera proportio est. Continet enim totum insuperque dimidium. Similis ab hoc ad 12⁴ , similis a 12 ad 8 proportio. Inter prima vero solida, scilicet 8 atque 27, sunt plana aequilatera duo, scilicet 9 atque 16. Haec planum quoddam inaequilaterum, scilicet 12, inter se convinciunt. Nam sicut ab ipso 16 ad 12 [15] sexquitertia proportio^[5] est—continet enim totum tertiamque insuper eius partem—sic ab ipso 12 ad 9 sexquitertia proportio reperitur. Cum igitur in ordine numerali imprimis adsumpto solida quidem per media duo plana^[6] sexquialteris proportionibus colligentur, plana vero uno dumtaxat medio proportionibusque sexquitertiis vinciantur, merito [20] Plato hinc^[7] nactus occasionem prima fundamenta proportionum eius-

[5] ad om. Z

[6] Novo Y bono Z

[1] tertium a 18 om. Z

[2] 17 Z

[3] ipsis M

[4] 22 Z

[5] reportio Z

[6] plana] delendum

[7] hunc M

tions, namely the 7 and the 5. For the first instance of proportion bearing the ratio of three to two is between 3 and 2; whence the number five is called the prime root of such a proportion. The first instance too of that proportion bearing the ratio of four to three is between 4 and 3. Therefore the number seven is called the root or foundation of that bearing the ratio of four to three. But Plato especially esteems these two proportions, because the proportion bearing the ratio of three to two generates the consonance diapente ,^[5] and that bearing the ratio of four to three produces the consonance diatessaron .^[6] He esteems these most because from them is produced the universal consonance that consists in that double proportion which they call the harmony diapason , the most celebrated of harmonies.^[7] Hence therefore Plato cultivates the number twelve preeminently as the first of the means among the [prime] solids. For it is constituted from the two roots of these proportions and consonances, namely from the numbers five and seven by way of composition^[8] (as we said); and likewise by a way of mutual commixture, namely among the parts.^[9] Resolve the number five into 3 and 2. Twice 3 is 6, and likewise twice 3 twice is 12 or twice 6 is 12. Resolve 7 into 4 and 3. Thrice 4 is 12. Thus not only do 7 and 5 added make 12, but when the parts in both are mixed together, that is, are multiplied, they also make 12. Twelve is also made from the first numbers multiplied together, that is, thrice 4 is 12. For if two is not a determined number but a confused multitude, then the first numbers are 3 and 4,^[10] the elements of the number 12, and should be celebrated on this account. But Plato venerates the number 12 not only secretly here but also openly in the Laws , the Phaedo , the Timaeus , the Phaedrus , and the Critias . In the Phaedo with twelve [as] the number of the forms he describes the globe.^[11] In the Critias , in referring to the twelve regions, he is describing the ancient reigns before the flood.^[12] In the Laws he uses the same number to arrange the city and fields.^[13] In the Phaedrus he adduces the twelve orders of the gods.^[14] In the Timaeus he forms the world with twelve faces both because of the twelve spheres of the world and the twelve signs and divinities in the zodiac;^[15] and likewise because of the twelve parts [or zones] of the [four] elemental spheres, since each is divided into three, namely into a superior, an inferior, and a middle zone. But this is sufficient. We have already talked about it in the commentaries on the Timaeus , in the arguments for the Laws , and in the Theology .^[16] Wherefore Plato judges this number twelve to be the governor of the universal world

modi adducit in medium, septem videlicet atque quinque. Prima enim sexquialtera inter tria nascitur atque duo. Unde quinarius prima dicitur proportionis^[8] eiusmodi radix. Prima quoque sexquitertia inter quatuor provenit atque tria. Quocirca septenarius radix vel fundum^[9] [25] dicitur sexquitertiae. Plato vero proportiones eiusmodi magni facit,^[10] quoniam et sexquialtera consonantiam generat diapente^[11] et sexquitertia consonantiam procreat diatesseron. Quas ideo plurimi facit, quoniam ex his conflatur universalis consonantia illa in^[12] dupla proportione consistens quam diapason harmoniam vocant summopere [30] celebratam. Hinc igitur Plato colit magnopere duodenarium ceu primum inter solida medium, quoniam ex duabus radicibus illis proportionum consonantiarumque eiusmodi constituitur, quinario videlicet atque septenario per modum compositionis (ut diximus), item quodam mutuae commixtionis modo videlicet inter partes. Resolve quinarium [35] in 3 scilicet atque 2. Bis 3 = sex. Item bis 3 bis = 12, vel bis sex = 12. Resolve 7 in 4 atque 3. Ter 4 = 12. Non solum ergo 7 et 5 composita faciunt 12, sed in utroque partes invicem mixtae, scilicet multiplicatae, 12 quoque conficiunt. Fit etiam 12 ex primis numeris in se invicem multiplicatis, scilicet ter quatuor = 12. Si enim duo non sit [40] determinatus numerus sed multitudo confusa, primi numeri sunt 3 atque 4 elementa duodenarii ob hoc etiam celebrandi. Non solum vero clam hic, sed etiam palam in Legibus, Phaedone, Timaeo, Phaedro, Critia duodenarium veneratur. In Phaedone quidem duodenario formarum numero describit orbem. In Critia vero plagis duodecim antiqua [45] ante diluvium regna describit. In Legibus eodem numero civitatem agrosque disponit. In Phaedro duodecim adducit ordines divinorum. In Timaeo duodecim faciebus format mundum, etiam propter sphaeras mundi 12, signaque^[13] et numina in zodiaco 12, item partes elementorum duodecim siquidem quodlibet in tria dividitur, in plagam videlicet [50] superiorem, inferioremque et me[150r]diam. Sed de his quidem satis. In commentariis in Timaeum et argumentis Legum et Theologia iam a nobis est dictum. Quapropter Plato numerum hunc universalis formae mundanae vel humanae atque civilis^[14] gubernatorem esse iudicat,

[8] proportionis om. M

[9] fundamentum Z

[10] magni facit] magnificat M

[11] diaxente Z

[12] in om. Z

[13] signatque Z

[14] civilibus Z

form, of the human form, and of the form of the state.^[17] He judges it to accord most with the propagation or mutation of things, since, as we shall show later, it is the first of the increasing and abundant numbers.^[18] Twelve is made from the number six twinned, from six the perfect number as we call it. In other words, twelve is more than perfect. Nor does it want mystery in that in its composition Plato elects 7 and 5. For 7 corresponds with the 7 planets, and the number 5 with the regions of the world, that is, with the 4 elementary regions and with heaven. Likewise 5 is the prime origin of the perfect circular number. For if you lead it through the plane to itself [i.e., square it] it makes 25, and if you lead it back through the solid to itself [i.e., cube it] it makes 125. And each is a circular number in that each commences from the number 5 and ends in the number 5.^[19] Hence the number 12 accords most with the world orb. But compared with the rest [of the planets] it accords [most] with the Sun, Venus, Jupiter, and the Moon, the fountains of life.^[20] The Sun and Venus each complete their orbits in 12 months, Jupiter in 12 years.^[21] Daily the Moon passes through 12 degrees "in middle motion," and she has her [28] mansions of 12 degrees;^[22] and she enacts 12 months with the Sun. Not without weighty cause has this number been observed by the Prophets and in sacred writings.^[23] Now I leave aside the fact that 12 twinned completes the day and much similar.

[ii] However, since Plato had chiefly posited four terms in that prime numeral order—8, 12, 18, 27—and since he wished to arrive thence at the fatal and great number, he could not choose a number less than the 12. For the ten is not contained in a lesser number—the ten that is in a way the universal number and the origin of the universal numbers insofar as from it teem 100, 1,000, 10,000, 1,000,000.^[24] Under the preeminently fatal number it must needs be too that the somewhat lesser fatal number, 729, should be comprehended—729 which is celebrated in the ninth book of the Republic^[25] and produced from 9 thrice increased. But 9 is contained under 12—not only under a greater number as it were, but also in a certain proportion, namely in the ratio of three to four. But the number twelve embraces ten not only in amplitude but also in proportion. For 12 has to 10 the proportion in the ratio of six to five. When [in terms of this ratio] the twelve seems to divide the ten into 5 parts, and to add to the ten the two, that is, the fifth portion of the ten, then it completely remakes the ten. For if you lead [i.e., multiply] the two to the five, you will immediately make the ten. Therefore 12, when it embraces and remakes

plurimumque rerum propagationi vel mutationi congruere, quoniam, [55] ut in sequentibus ostendemus, primus est crescentium abundantiumque numerorum. Fit ex geminato senario numero ut dicemus perfecto, videlicet ipse plusquam perfectus; neque mysterio caret quod in eius compositione 7 elegit atque 5; nam septem planetis 7 competit, quinarius quinque mundi plagis, scilicet quatuor elementis et caelo. [60] Item quinque origo prima est perfecti numeri circularis. Sive enim per planum in se ducas, efficit 25; sive per solidum in se reducas, facit 125; uterque vero circularis existit, incipiens videlicet a quinario desinens in quinarium. Hinc duodenarius orbi maxime congruit, prae ceteris vero cum Sole, Venere, Iove,^[15] Luna, vitae fontibus. Sol Venusque duodecim [65] percurrit mensibus,^[16] Iupiter annis duodecim. Luna quotidie gradus peragit duodecim motu medio, suasque duodecim graduum mansiones habet, ipsaque cum Sole menses agit duodecim. Nec^[17] sine gravi causa hic numerus est a prophetis sacrisque eloquiis observatus. Mitto nunc quod duodecim geminatus implet diem multaque [70] similia.

[ii] Cum vero Plato in primo illo ordine numerali quatuor praecipue terminos posuisset, 8, 12, 18, 27, velletque illinc ad fatalem magnumque numerum pervenire,^[18] non poterat minorem eligere quam 12. Nam in minori non continetur decem, qui quodammodo [75] universus est numerus, numerorumque universalium est origo quatenus ex eo pullulant centum, mille, decem milia, mille milia. Oportebat quoque sub hoc imprimis fatali numero compraehendi fatalem illum aliquanto minorem, scilicet 729 in^[19] nono de Re Publica celebratum a nove[1416]nario ter aucto procreatum. Continetur autem [80] 9 sub 12 non solum quasi^[20] sub maiore verum etiam proportione quadam, scilicet sexquitertia. Numerus vero 12 ipsum decem non solum amplitudine sed etiam proportione complectitur. Nam proportionem sexquiquintam habet ad decem; atque dum dividere videtur ipsum in partes quinque, binariumque quintam denarii portionem^[21] [85] denario superaddere, tunc maxime^[22] reficit ipsum decem. Si enim binarium duxeris in quinarium, decem profecto conficies. Itaque 12, dum numerum universum proportione complectitur atque reficit, ad universum maxime pertinere videtur.

[15] sunt Z

[16] mentibus Z

[17] Neque M Haec Z

[18] provenire Z

[19] in om . M

[20] 9 M

[21] proportionem Z

[22] maximum M

the universal number [10] in this proportion [of 6:5], seems to pertain completely to the universe.^[26]

[iii] Moreover twelve, just as it contains those two harmonies, the elements of the diapason,^[27] within itself, so when it is increased twice— namely 12x12—it bears these same harmonies within itself and fully unfolds them under the plane and equilateral number, namely 144.^[28] Again, when it is increased thrice—namely 12x12x12—with itself it also extends these same two harmonies even further under the solid and equilateral number that is created by such a multiplication, namely 1728. And this number indeed most accords with the universe. For 1000 accords with the firmament, but 700 with the 7 planets. To these is added 28 to represent the lunar circuit; for this circle expedites and perfects fate. Indeed the return of the Moon to the same point of the zodiac is designated by the number 28. But the return of the Moon to the Sun is expressed precisely by 29;^[29] and this is declared in the Republic book 9.^[30] The number 28 accords with the Moon for another reason too, namely because she has 28 famous mansions.^[31] Six is the prime perfect number; but the second perfect number is 28 because it is made from its own parts as 6 is. For 6 accords with the 6 higher planets, but 28 accords with the Moon. After the first perfection that comes [to us] from the six higher planets, she brings to things subject to fate the second perfection.^[32]

[iv] Plato chiefly accepts the numbers, however, that can accord with the universe and embrace [its] consonances in order to show, by way of certain numbers and measures, that the good fortunes of lower things depend on the universe and especially when they are in accord with these numbers and measures. But he extends the fatal numbers to the solid as to the highest point,^[33] so that hence he might show, when this highest point has already been attained, that little by little all are brought back to the opposite [the lowest point]. For the condition of mobile nature does not suffer it to remain for a long time in the same or in a similar disposition (habitus ).

Chapter 4. On Increasing and Decreasing Numbers, and Those That are Like and Unlike.

In the first numeral order, which proceeds from the solid 8 to the solid 27, and similarly in the numbers produced from it, the number of overcoming augmentations is equal to those overcome, as Plato says.^[1] For everywhere the half corresponds to the double, the third to

[iii] Praeterea duodenarius, sicut intra se duas illas continet harmonias [90] ipsius diapason elementa, ita quando bis augetur, scilicet duodecies duodecim, secum profert easdem explicatque in amplum sub numero plano aequilateroque, scilicet 144.^[23] Rursus quando ter augetur, scilicet duodecies duodecim duodecies, easdem harmonias secum latius quoque diffundit sub numero solido atque aequilatero^[24] qui eiusmodi [95] multiplicatione creatur, scilicet 1728, qui sane numerus maxime convenit universo. Nam mille quidem congruit firmamento, septies vero centum planetis 7, additum vero est 28 ad lunarem circuitum exprimendum. Hic enim circuitus fatum^[25] expedit atque perficit. Reditus quidem Lunae ad idem zodiaci punctum 28 numero designatur, reditus [100] autem eiusdem ad Solem 29 prorsus exprimitur; quod in nono de Re Publica declaratur. Convenit numerus 28 Lunae alia etiam ratione, quoniam Luna 28 mansiones habet insignis.^[26] Senarius quidem primus numerus^[27] est perfectus; secundus vero perfectus est 28 quia suis partibus constat ut ille. Ille igitur convenit cum superioribus sex planetis; [105] hic vero cum Luna, quae post primam illinc perfectionem ipsa secundam fatalibus adhibet.

[iv] Accipit vero Plato numeros potissimum qui cum universo conveniant consonantiasque complectantur, ut ostendat inferiorum eventus per certos numeros atque mensuras ab universo pendere praecipue [110] quando consonant ista cum illis. Producit autem numeros fatales ad solidum velut ad summum ut hinc ostendat, ubi ad summum iam perventum est, paulatim in oppositum omnia relabi, quippe cum in eodem vel simili habitu diutius permanere mobilis naturae conditio minime patiatur. [115]

De Numeris Crescentibus et Decrescentibus, Similibus^[1] Atque Dissimilibus. Cap. IIII.

In primo autem illo ordine numerali ab 8 solido usque ad 27 solidum procedente^[2] similiterque in numeris inde productis, quot sunt augmentationes superantes totidem superatae, ut Plato inquit; ubique [5] enim duplae respondet subdupla,^[3] triplae quoque subtripla. Item quot

[23] 144 scripsi 164 Y 169 Z om . M

[24] scilicet 144. . . . aequilatero om . M

[25] factum Z

[26] insignes Z

[27] primus numerus tr . M

[1] similibusque M

[2] procedentes YZ

[3] subtripla YZ

the triple.^[2] Likewise among the manifold proportions, the overcoming ones as it were are immediately matched by those that are being divided—those that have been in a way overcome yet remain entirely congruent. Thus the sesquialteral proportion [of 3:2] accords with the double proportion [of 2:1]. For just as the greater here doubles the lesser—for instance, 4 doubles 2—so the sesquialteral proportion, when it divides, distributes the lesser number as it were into two and gives us the ratio of 6 to 4.^[3] For over and beyond the fact that the sesquialteral proportion contains the whole once, it seems to divide in a way and to add the half to the whole. Similarly, the sesquitertial proportion [of 4:3] seems to accord with the triple proportion [of 3:1], and the sesquiquartal proportion [of 5:4] with the quadruple [of 4:1]. And successively multiples endlessly augment as it were the number [to be divided], but those that do the dividing diminish as it were the result.^[4]

[ii] Furthermore, the increasing and decreasing numbers are named here by Plato. For a certain number is said to be perfect because it is constituted exactly from its parts, namely from its several parts placed in their successive order; for instance the six is constituted from one, from two, and from three.^[5] These indeed are truly parts of the six; for any one of these parts taken up several times makes the six, and likewise arranged (as I said) successively—1, 2, 3—the parts constitute 6 exactly. Hence the number 6 customarily is called perfect. It is also perfect for another reason: it is made exactly from a double proportion which it contains perfectly within itself, namely the proportion of the four to the two; but four and two together equal six. You may find this in other numbers only with great difficulty. But the perfection of the six as a half is referred to the 12 as the whole.

[iii] However, a number is customarily called deficient because its several parts thus simply arranged do not make up the whole. Take 8. Its parts indeed are 4, 2, and 1.^[6] But these arranged make 7. The like goes for 9 with regard to its parts.

[iv] A number is judged abundant, however, because its parts when so arranged swell to something bigger than itself. Take 12. Its parts are: the half—6, the third—4, the fourth—3, the sixth—2, the twelfth—1. But these parts added together eventually make 16.^[7] But they increase happily. For they rightly proceed from the unequilateral [12] to the equilateral [16] with the proportion preserved, for the proportion of 16 to 12 is in the ratio of 4:3, as is that of 12 to 9, nine being also an equilateral.^[8] Therefore twelve accords most with the

sunt proportiones ipsae multiplices quasi superantes totidem subinde sunt partientes quodammodo superatae sed penitus congruentes. Nam duplae respondet sexquialtera. Sicut enim dupla minorem numerum geminat, ut quatuor geminat duo,^[4] ita sexquialtera dividendo minorem [10] quasi partitur in duo, ut sex ad quatuor. Praeter enim id quod totum semel continet, videtur quodammodo distribuere dimidiumque addere super totum. Similiter sexqui[150v]tertia quidem triplae, sexquiquarta vero quadruplae respondere videtur atque deinceps sine fine multiplices quidem augent^[5] quasi numerum, partientes vero minuunt [15] quasi continuum.

[ii] Praeterea nominantur hic a Platone numeri crescentes^[6] atque decrescentes. Aliquis enim numerus dicitur perfectus quoniam ex suis partibus, scilicet aliquotis deinceps ordine positis, constat ad unguem, ut senarius ex uno, duobus, tribus. Hae sane revera sunt senarii partes; [20] quaelibet enim earum aliquotiens sumpta senarium complet. Itemque dispositae (ut dixi) deinceps 1, 2, 3, ad unguem 6 efficiunt.^[7] Hinc senarius numerus perfectus appellari solet. Est etiam alia ratione perfectus, quia constat ad unguem proportione dupla quam intra se proxime continet, haec autem est quaternarii ad binarium, sed^[8] quatuor [25] simulque duo = sex. Id in aliis numeris vix invenias. Perfectio vero senarii velut dimidii refertur ad 12 tanquam totum.

[iii] Aliquis vero numerus nominari deficiens consuevit, quia partes aliquotae simpliciter ita dispositae non implent totum, ut 8. Nempe partes eius sunt 4, 2,^[9] 1; hae^[10] vero digestae septem faciunt. Similiterque [30] 9 se habet ad partes.

[iv] Aliquis vero numerus iudicatur abundans, quia partes eius ita compositae in maiorem excrescunt, ut 12. Partes huius sunt dimidia quidem 6, tertia vero 4, sed quarta 3, sexta^[11] 2, duodecima 1. Partes autem hae congestae 16 postremo conficiunt. Crescunt vero feliciter. [35] Nam ab inaequilatero ad aequilaterum recte procedunt proportione servata, quoniam 16 ad duodecim sexquitertiam proportionem habet sicut 12 ad 9 etiam aequilaterum^[12] habuit sexquitertiam. Itaque maxime convenit universo et fertilitatem incrementumque significat, praesertim quia primus est et^[13] princeps abundantium numerorum. [40]

[4] ita duo add . Z

[5] auget M

[6] decrescentes M

[7] efficunt Y

[8] scilicet Z

[9] 8 Z

[10] Haec Z

[11] septa Z

[12] aequilateram Z

[13] est et] esset M

universe and signifies fertility and increase, especially because it is the first and the prince of the abundant numbers. Furthermore, the Pythagoreans called 6 the spousal number,^[9] because in its conception a male joins with a female, that is, an odd [number] with an even—2x3. But 6 is the first of the spousal numbers and 12 is the second (in the twelve's conception 3 mingles itself with 4–3x4=12). But where even and odd are distanced by intermediary numbers, they do not seem to unite as spouses.^[10]

[v] Furthermore, Plato introduces here certain similar and dissimilar numbers.^[11] Said to be similar among themselves, and preeminently so, are equilaterals with regard to equilaterals, cubes with regard to cubes. But those unequilaterals are [also] similar whose sides are proportional. Take 6 and 24.^[12] The width of 6 is 2, the length 3. Twice 3 is 6. But the width of the number 24 is 4, the length 6; for 4 times 6 is 24. But the same ratio exists between 6 and 3 as between 4 and 2. Therefore, the same ratio exists between the width of 24 and the width of 6 as between the length of 24 and the length of 6. For this reason they are called similar. But those that do not accord with such proportions are adjudged dissimilar.

Chapter 5. On Numbers Associated with Sides and with Diagonals.

Unity itself, as it is the principle of numbers and of figures,^[1] so it is the principle of the side and of the diagonal, and has the power for each. Take therefore this unity A here, but that unity B there. Indeed A, while it stays alone, makes no line at all and therefore makes neither the side nor the diagonal. If A proceeds to its twin, then it will make the line, which can become the side of the future square. Again, if it has proceeded so far as to make the diagonal—and because the diagonal is necessarily greater than the side—then it has proceeded at least to the three. Wherefore, just as you have brought the unity A forth to the two, so you will have brought the unity B forth to the three, so that A signifies the side of the future square, but B the diagonal.^[2] The square that is generated from the binary A led to itself is undoubtedly four. But the square that comes from the ternary B similarly led to itself becomes nine. Therefore the square made from the diagonal [compared] with the square generated from the side is greater by one than double.^[3]

Praeterea Pythagorici 6 sponsalem numerum vocaverunt, quoniam in eius conceptu mas cum femina coit, scilicet impar cum pari—bis 3. Sex est autem sponsalium primus, secundus vero^[14] 12 (in cuius conceptu 3 cum 4 se commiscet—ter 4 = 12). Ubi vero par et impar per media distant, congredi non videntur. [45]

[v] Introducit hic insuper Plato numeros quosdam similes atque dissimiles. Similes quidem inter se dicuntur aequilateri plurimum aequilateris, cubi cubis. Inaequilateri vero invicem illi sunt consimiles quorum latera proportionalia sunt,^[15] ut 6 atque 24. Latitudo senarii est 2, longitudo 3; nempe bis 3 = 6. Numeri vero 24 latitudo 4, longitudo [50] 6; quater enim 6 = 24. Quemadmodum vero se habet 4¹⁶ ad 2, ita 6 ad 3. Itaque sicut se habet latitudo numeri 24 ad senarii latitudinem, ita longitudo illius ad senarii longitudinem. Qua quidem ratione similes appellantur. Qui vero proportionibus eiusmodi non conveniunt dissimiles iudicantur. [55]

[1417] De Numeris Lateralibus Atque Diametralibus. Cap. V.

Unitas ipsa, sicut numerorum figurarumque^[1] principium est, ita lateris et diametri, atque ad utrumque potentiam habet. Expone igitur hic quidem hanc unitatem A, ibi vero unitatem illam B.^[2] A quidem, dum [5] sola manet, nullam efficit lineam, igitur neque latus neque diametrum. Si ad geminum A processerit, lineam iam efficiet, quae possit latus fieri futuri quadrati. Si rursus adeo processura sit ut faciat^[3] diametrum—quoniam diameter necessario est latere maior—saltem processura est in tria. Quapropter, sicut unitatem A ad binarium produxisti, sic unitatem [10] B^[4] producturus es ad ternarium, ut A quidem significet quadrati futuri latus, B^[5] vero diametrum. Quadratum quidem quod ex A binario procreatur in se ducto est proculdubio quaternarius, quadratum vero quod ex B ternario similiter in se ducto fit novenarius. Itaque quadratum hoc ex diametro factum ad quadratum illud ex latere procreatum [15] unitate maius est quam duplum.

[14] mas cum . . . secundus vero om . Z

[15] sunt om . M

[16] 4 om . Z

[1] figuramque Z

[2] B] 6 Z

[3] faciet M

[4] B] 6 Z

[5] B] 6 Z

[ii] If you wish to bring forth greater squares again from the sides, and similarly from the diagonals, add to the side of two that diagonal of three. Now you will have five for the side. Also add to that diagonal of three twice that side of two. You will now have seven for the diagonal. Therefore make the square from the side of 5 led to itself. The square will be 25. Do likewise with the diagonal 7 and the square will be 49. This diagonal square will be less by the one than double that lateral square [of 25]. For this is the ratio of 49 to 25.^[4]

[iii] Again, in order for you to make bigger squares, add to the side that was 5 the diagonal that was 7. You will have 12 for the side. In turn add to the diagonal that was 7 twice that side of 5. The diagonal will be 17. From that side of 12 led to itself you will obtain the square 144. From the diagonal of 17 led to itself, however, you will have the square 289, which is greater by the one than double the square [of 144] made from the side.^[5]

[iv] However, in increasing the squares, why must we add the earlier diagonal by itself to the earlier side, and yet add both the earlier sides to the diagonal? Because twice the power of the side can equal only once the power of the diagonal.

[v] But compensation must in general be made.^[6] If you proceed in increasing the squares to 100 and beyond, at length adequation will be accomplished, now in the outcome being less by one, now in turn being more. All told, therefore, the result will be the double proportion [of 2:1].^[7] Accordingly, Plato says that such numbers need the 1 as the equalizer, the incommensurables singly but the commensurables together.^[8] But more of commensuration in what follows.^[9] But perhaps Plato is talking about two incomparable [relationships], because in the first constitution of the squares—where the diagonal was 3 to the side of 2—he had proportion,^[10] but in the second constitution where it was 7 to 5, he did not. Similarly in the third constitution, where it was 17 to 12, he was lacking proportion. But he calls the diagonal numbers "of the five,"^[11] because in the first instance the side was 2 and the diagonal 3. He names solids "of the three,"^[12] because triple replication makes solid numbers, and triple dimension makes the solid body. But preeminently he calls those solids "of the three" which he produces from the nine (which is resolved into the three).^[13]

[ii] Si cupis quadrata rursus maiora producere ex lateribus similiter atque diametris, adde lateri quidem illi binario diametrum illum ternarium. Habebis quinarium iam pro latere. Adde etiam illi diametro scilicet ternario bis latus illud binarium, habebis iam septenarium pro [20] diametro. Fac ergo quadratum ex quinario latere in se ducto; erit quadratum viginti quinque. Fac similiter ex 7 diametro; erit quadratum 49. Quadratum hoc diametrale erit ad illud laterale^[6] unitate minus quam duplum. Ita enim 49 ad 25 se habet.

[iii] Iterum ut ampliora quadrata conficias, adde lateri quod erat 5⁷ [25] diametrum quod fuit 7, habebis pro latere 12; vicissimque diametro quod erat 7 bis latus illud scilicet 5, erit diameter 17. Ex illo quidem latere scilicet 12 in se ducto reportabis^[8] quadratum 144. Ex diametro autem hoc 17 in se ducto habebis quadratum 289, quod est unitate maius quam duplum ad quadratum ex latere factum. [30]

[iv] Sed curnam oportet in augendis quadratis^[9] priori quidem lateri addere diametrum prius^[10] unum, diametro vero latera priora duo? Quia videlicet quantum latus bis valet tantum diameter potest semel.

[v] Omnino^[11] vero compensatio facienda. Si in augendis quadratis ad centum^[12] et ultra processeris, tandem adaequatio fiet, tum unitate [35] deficiente, tum excedente vicissim, ut summatim resultet pro[151r]portio dupla. Ideo Plato inquit eiusmodi numeros indigere uno scilicet aequatore, et incommensurabiles quidem singulatim, commensurabiles vero summatim. Sed de commensuratione in sequentibus. Inquit vero duos incomparabiles forte, quoniam in prima quidem horum [40] constitutione 3 ad duo proportionem habuit, in secunda vero 7 ad 5 non habuit, similiter in tertia 17¹³ ad 12 proportione carebat. Appellat autem diametrales numeros quinitatis,^[14] quia in primo latus quidem fuit 2, diameter vero 3. Solidos nominat trinitatis, quia trina replicatio numeros facit solidos et trina dimensio corpus solidum. Sed [45] praecipue illos solidos vocat^[15] trinitatis quos producit ex novenario qui resolvitur in ternarium.

[6] latere M

[7] 5] v. quinque Z

[8] reportatis Z

[9] quadritis Y

[10] prium Y primum Z

[11] Omnis Z

[12] centrum M

[13] 97 YZ

[14] quinitates YZ (vide Rempublicam 546C4–5; nota lectiones "pempados," "pempadôn" )

[15] vocant Z

Chapter 6. Plane and Solid Numbers, Also Equilateral and Unequilateral, Even and Odd, Feminine and Masculine Numbers.

Plato calls plane numbers those numbers which are generated by prime [i.e., simple] multiplication, as 2x2=4 or 3x3=9, and so on similarly. He calls solids, however, those which are born not only from prime multiplication but from triple replication, as 2x2x2=8, 3x3x3=27, and so on similarly.^[1] In both categories are equilaterals and unequilaterals. Equilaterals indeed are created from any number multiplied by itself; of this kind are those we have just spoken about. But unequilaterals arise from the multiplication of one number by another, as in the planes 2x3=6, 3x4=12, and so on similarly, and as in the solids 2x3x2=12, or 2x3x3=18.^[2] Therefore unequilaterals are called either "those which are longer by one part" or "oblongs."^[3] Those which are longer by one part for the sake of brevity I shall more often refer to as "longs." They are generated from the leading of any one number to the next, as 2x3=6, 3x4=12; and in them the greater number to which the lesser is led is greater than the lesser only by one. But "oblongs" are generated from the leading of a number to a more distant number, as 2x4=8, 3x5=15; for here the greater number exceeds the lesser by a distance greater than one.^[4]

[ii] Thus far these numbers—plane or solid, equilateral or unequilateral, long or oblong—are made by multiplication either of some number by itself or of some number by another—in both cases by reason of commixture and of generation.^[5] Furthermore, they can also be made by way of composition [i.e., addition]. To constitute them, a number is added either to the one or to a number successively. Equilaterals are constituted when odd numbers are added to odd, starting with the one; unequilaterals, when even are set to even, starting with the two. But let us begin with equilaterals.^[6]

[iii] The odd numbers in sequence are 1, 3, 5, 7, 9, 11. One, as the first equilateral, is a square; for once one is one. If you add 3 to this as to an odd number, you will make the squared equilateral 4. This will be two-footed equally in breadth and in length. The next odd number is 5. If you add this like a workman's square to the preceding square [of 4], you will get 9. This square is similarly an equilateral, whose sides will each be three-footed. The next odd number is 7. Now if you move it to 9, you will make 16, four-footed equally in length and breadth, for 4x4=16, and so on similarly.^[7] In these, plainly the odd

Numeri Plani et Solidi, Item Aequilateri et Inaequilateri, Pares, Impares, Feminae, Masculi. Cap. VI.

Numeros appellat planos qui prima multiplicatione numeri procreantur, ut bis 2 = 4, vel ter 3 = 9, similiterque deinceps; solidos autem qui [5] non solum multiplicatione sed etiam terna replicatione nascuntur, ceu bis 2 bis = 8, ter tria ter = 27, deincepsque similiter. Utrobique vero vel aequilateri vel inaequilateri sunt. Aequilateri quidem ex numero quolibet per se in se ipsum multiplicato creantur, quales sunt quos modo narravimus. Inaequilateri vero ex multiplicatione numeri alterius [10] per alterum oriuntur, velut in^[1] planis quidem bis 3 = 6, ter 4 = 12, similiterque deinceps; in solidis autem bis 3 bis = 12, vel bis 3 ter = 18. Proinde inaequilateri vel altera parte longiores vel oblongi dicuntur—altera quidem parte longiores quos brevitatis causa saepius appellabo longos. Illi sunt qui ex^[2] ductu numeri alicuius^[3] in proximum [15] procreantur, ut bis 3 = 6, ter 4 = 12, in quibus maior numerus in quem minor ducitur hoc ipso minore unitate dumtaxat est maior. Oblongi vero ex ductu numeri in remotiorem numerum generantur, ut bis 4 = 8, ter 5 = 15; hic enim maior numerus minorem longiore spatio quam unitate superat. [20]

[ii] Hactenus hi numeri—plani vel solidi, aequilateri vel inaequilateri, longi vel oblongi—multiplicatione fiunt vel numeri alicuius per se ipsum vel numeri alterius per alterum, utrobique quadam commixtionis generationisque ratione. Confici praeterea possunt quodam compositionis modo, quando videlicet ad eorum constitutionem unitati [25] deinceps vel numero numerus additur: aequilateri quidem quando impares imparibus unitate duce numeri adhibentur, inaequilateri vero quando pares paribus duce duitate subduntur. Sed ab aequilateris ordiamur.

[iii] Sunt autem consequentes^[4] impares: 1, 3, 5, 7, 9, 11. Unum [30] quidem quasi primum aequilaterum quadratum est; semel enim unum existit unum. Huic tanquam impari si addideris 3, quadratum facies aequilaterum, scilicet quaternarium, quod et latitudine et longitudine pariter erit bipes. Consequens impar 5. Hunc si praecedenti quadrato addideris ceu normam, reportabis 9, quadratum similiter aequilaterum [35] cuius latus quodlibet erit tripes.^[5] Consequens impar 7. Nunc^[6] si ad-

[1] in om . Z

[2] sex Z

[3] numeri alicuius tr . M

[4] consequenter Z

[5] tripes scripsi triples YM triplex Z

[6] Hunc M

number is always put to the preceding number, which is either an odd number, or at least constituted from two odd numbers, starting always with the one.^[8] For just as the one is the leader of the odd and equilateral numbers, so the two is the leader of the even numbers, of those numbers composing as it were the unequilateral figure.^[9]

[iv] For the two is as it were the first unequilateral, since it is the first to descend from the one, the most equal of all. Therefore twice 1 is 2. This two is 1 in breadth but 2 in length. But truly the Pythagoreans wanted duality to be something indeterminate, to be the principle of no one figure.^[10] For they suppose that 1 is the principle of the circular figure, because it is converted to itself—for 1x1 or even 1x1x1 only exists as 1. But they suppose that the trinity is the principle of the rectilinear figures. For the three is the first trigon; and the triangle is the first of the rectilinear figures, triangles indeed composing squares and all the rest.^[11]

[v] But of this elsewhere. Let us return to our suppose. Therefore the various even numbers are expounded in order: 2, 4, 6, 8, 10, 12, and the rest similarly. Compound [i.e., add] 2 with 4 and you will make 6. Likewise compound this 6 with 6 and you will obtain 12. Add 8 to 12 and you will make 20. Therefore in sequence the long numbers will be 6, 12, 20; and with those that follow, the same reasoning will pertain.^[12] The equilateral numbers were successively even and odd—4, 9, 16, 25. But the unequilateral, that is, the long, numbers are everywhere even numbers—6, 12, 20, and the rest similarly, because in creating them the even number multiplies the next odd number or the reverse.^[13]

[vi] That these numbers have been constituted either even (when they are also called females) or odd (when they are adjudged males) derives, however, from their own particular root as from their seed. For the fact that four is accounted equal and feminine follows from the fact that 2 is similarly even and feminine; and 2 is the seed of 4, for doubling in itself, namely 2x2, it generates 4. Similarly 9 is both odd and masculine on account of the 3 that is its seed and root. Furthermore, if an even is in the root, when it multiplies an odd or the reverse, it makes an even, as 2x3=6, 3x4=12; likewise 3x6=18, which is oblong. Therefore, all unequilateral long numbers for this reason are both feminine and even, being constituted from feminine evens; but the oblongs are both even and odd (but exceedingly unequal).^[14]

[vii] The odd numbers naturally excel the even, however; for the

moveris novenario, conficies 16 longitudine pariter et latitudine quadrupes.^[7] Quater enim 4 = 16, similiterque deinceps. In his plane semper impar subditur praecedenti vel impari numero vel saltem ex duobus imparibus constituto, semper unitate duce. Haec enim ita dux est [40] imparium aequilaterorumque numerorum, sicut duitas parium^[8] quidem numerorum figuram vero velut inaequilateram componentium.

[iv] Est enim duitas quasi primum inaequilaterum, siquidem primus est discessus ab unitate omnium aequalissima. Itaque [1418] bis unum existit duo. Haec sane duitas latitudine quidem unum est, longitudine [45] vero duo. Re autem vera Pythagorici duitatem indeterminatum aliquid esse volunt, nullius figurae principium. Nam circularis quidem figurae principium esse putant unum, quia convertitur in se ipsum—semel enim unum vel etiam semel unum semel dumtaxat existit unum—trinitatem vero rectilinearum principium figurarum. Est enim primus trigonus [50] ipse ternarius et triangulus rectilinearum figurarum prima, trianguli vero quadrata reliquaque componunt.

[v] Sed haec alias. Redeamus ad institutum. Exponantur ergo pares quilibet deinceps numeri: 2, 4, 6, 8, 10, 12, ceterique similiter. Compone cum 2 4, efficies inde 6. Item cum hoc senario compone [55] senarium, inde 12 reportabis. Item 12 et 8 compone, facies inde 20. Erunt igitur consequenter longi numeri 6, 12, 20, eadem quoque ratio in sequentibus. Aequilateri quidem numeri consequenter pares erant et impares: quatuor, 9, 16, 25. Inaequilateri vero, scilicet longi, sunt ubique pares^[9] numeri: 6, 12, 20, ceterique similiter, quia in eis [60] creandis par^[10] proximum imparem vel e converso multiplicat.

[vi] Quod autem hi numeri constituti vel pares sint, qui dicuntur et feminae, vel impares, qui et masculi iudicantur, id ex radice quadam sua velut semine provenit. Nam quaternarius in ratione pari feminaque sequitur binarium parem similiter atque [151v] femininum, qui et [65] semen est quaternarii, nam geminans^[11] in se ipso—scilicet bis 2—generat quaternarium; similiter novenarius et impar et masculus propter ternarium semen eius atque radicem. Quinetiam si in radice sit par, multiplicans imparem vel e^[12] converso facit parem, ut bis 3 = 6, ter 4 = 12,^[13] item ter 6 = 18 qui est oblongus. Hac itaque ratione omnes in- [70]

[7] quadrupes scripsi quadruples Y quadrupres M quadruplos Z

[8] partium Z

[9] erant et impares . . . ubique pares] rep. M

[10] par om. Z

[11] germinans M

[12] e om. Y

[13] 16 YM

even seem to be like corporeal and divisible things, but the odd like incorporeal and indivisible things. Again, the first even, namely 2, is the first division and diversity, and the first fall from the 1. But the first odd, that is 3, is as it were the return to the one and to [its] principle;^[15] it abounds in the one more than the even [2] does, and on account of this obvious copiousness it is called masculine. But the even [2], on account of [its] dearth, partition, and fall, appears as it were to be feminine. The human and moral praise is given to the even numbers insofar as there is a just distribution in their partition on both sides. But the more sacred and divine praise is extended to the odd numbers, since in the even number justice has been broken up as it were and has no hinge on which it might depend. But in the odd number there is always the one: it is the mean between the number's even parts on either side. It is as it were the center and the god by whom equal distribution is governed and to which it is referred as to its end.^[16]

[viii] Among all odd numbers 3, 7, and 9 seem to be eminent. For in the three the one exists equally on either side around the three's mean, that is, the one, just as the simple and divine beings exist around the divine being or God. Therefore God rejoices in the 3. In the 7 the 3 (which is consecrated to the divine) exists on either side of the one, which is divine. Finally, in the 9 the one, as the divine so to speak, inserts itself as a mean into justice, that is, into the 8. For 8 is named justice by the Pythagoreans because of its perfectly equal distribution.^[17] In short, the odd number, because of [its] mean, possesses the bond of itself within; because of [its] center, is circular; and because of the relationship of [its] extremes to [its] mean, is the principle [or cause] of the universal order.^[18]

Chapter 7. The Trigon Numbers, Which are Composed from Even and Odd Numbers Successively. And How the Square May Be Made from Trigons.

They call the numbers trigons which are composed from both odd and even numbers arranged in succession.^[1] Thus, if you add the even two, like a workman's square,^[2] to the one, as to an odd number possessing the trigonic power in itself, you will make the trigon, that is, the triangle, namely the three. If then you add the three—the next number to follow—straightway you will obtain the trigon six. Again when the four has been added, the ten will be generated, itself a

aequilateri scilicet^[14] longi et feminae sunt et pares, ex paribus videlicet feminis constituti, oblongi vero sunt pares^[15] et impares sed nimium inaequales.

[vii] Simpliciter autem impares numeri praestant paribus. Pares enim rebus corporeis atque dividuis, impares autem incorporeis individuisque [75] similes esse videntur. Item par primus, scilicet duitas, est divisio diversitasque prima casusque primus ab uno. Primus autem impar, id est ternarius, est quasi reditus ad unum atque principium; atque ultra parem abundat uno, ob quam plane copiam masculus appellatur. Par autem ob inopiam, partitionem, casum quasi femininus [80] apparet. Parium quidem numerorum humana laus est atque moralis quatenus in eorum partitione iusta^[16] utrinque fit distributio. Imparium vero sacratior laus est atque divina, siquidem iustitia in ipso pari quasi dissoluta est, nec ullum habet cardinem quo nitatur. Sed in ipso impari semper ipsum unum: inter partes^[17] numeri utrinque pares est [85] medium, quasi centrum atque numen quo aequa distributio^[18] regitur^[19] et ad quod refertur quasi^[20] finem.

[viii] Inter omnes vero impares 3, 7, 9 eminere videntur. Nam in 3 circa medium eius, id est unum, utrinque extat pariter unum, quasi simplicia et divina circa divinum sive Deum. Ideo Deus ternario [90] gaudet. In septenario circa numen unum utrinque ternarius existit numini consecratus. Denique in novenario ipsum unum quasi numen iustitiae, id est octonario, se medium inserit; nam octo propter aequalem ad ultimum distributionem a Pythagoricis iustitia nominatur. Summatim vero impar et propter medium sui ipsius vinculum in se [95] possidet, et propter centrum circularis existit, et propter comparationem extremorum ad medium universi ordinis est principium.

Numeri Trigoni Qui Ex Paribus Deinceps Et Imparibus Componuntur, et Quomodo ex Trigonis Fiat Quadratum. Cap. VII.

Numeros vero trigonos nuncupant qui ex imparibus simul atque paribus consequenter dispositis componuntur. Itaque, si unitati velut [5] impari virtutemque in se trigonicam possidenti subdas duitatem parem velut normam, efficies trigonum, id est, triangulum ipsum, scilicet

[14] scilicet om. Z

[15] sunt pares] et pares sunt M

[16] iuxta M

[17] pares YZ

[18] distributo Y

[19] requiritur M

[20] quas M

trigon. These then are the trigons in the order of succession—3, [10] 6, 10, and so on similarly.^[3] But as in figures two triangles make one square, so in numbers also two [adjacent] trigons make a square number. Thus the one (a trigon in power as it were) along with the three (itself a trigon) make the square 4. Similarly 3 and 6 (trigons both) generate the square 9. In the same way the trigons 6 and 10 together make the square 16.^[4] For if you explore diligently, everywhere you will discover that the powers and properties of numbers are preserved in planes and in figures. So why be amazed that the same powers and properties extend by gradations through planes to solids, and thus that all bodies come into being and are moved by their numbers? Wherefore Plato here and everywhere attributes all things to numbers.^[5] And Plotinus and Proclus prove most subtly that numbers exist in the prime being itself as the first distinguishers there both of beings and of ideas.^[6] Consequently it is not to be wondered at that lower things too are distinguished through numbers, and that, just as the species of things all wield their particular powers, their prerogatives and privileges as it were, so do the species of numbers do the same.

Chapter 8. The One, the Odd and Even Numbers, and the Equilateral and Unequilateral.

The one, the principle of numbers and dimensions, seems most like the principle of the universe, because, while it procreates all its offspring, it stays meanwhile most eminent and most simple. From the one, however, dimensions proceed from a position as it were of the point and of points; and numbers flow on as if with their own particular motion, although the even numbers flow more in procession, the odd mostly in conversion.^[1] Nevertheless, the one, which depends on the One, is the substance of numbers insofar as each number perhaps is nothing other than the one repeated so many times.^[2] Furthermore, the one is the measure itself of numbers. For 1x2 is the two; 1x3 similarly is the three; and so forth with the rest of the numbers similarly. Moreover, just as incorporeals and bodies alike are made from the one principle of things, so the odd and even numbers are made from the one. Likewise, just as simple things and composites are made from the one principle, so simple and compound numbers are made from the one. The simple numbers are those which simply consist of and are measured by the one—as 3, 5, 7, and the like; but compound numbers are those which are measured additionally by a number smaller

ternarium. Si deinde trinitatem subieceris consequentem, mox senarium trigonum reportabis. Rursus addito quaternario denarius et ipse trigonus generabitur. Hi sunt igitur deinceps trigoni consequentes: 3, 6, 10, similiterque deinceps. Quemadmodum vero in figuris trianguli duo quadratum unum^[1] efficiunt, sic et in numeris duo trigoni numerum quadratum faciunt. Itaque unitas quasi quidam virtute trigonus simulque ternarius et ipse trigonus quadratum conficiunt quaternarium. Similiter 3 et 6 ambo trigoni quadratum procreant [15] novenarium. Eodem pacto 6 et 10 trigoni quadratum 16 simul faciunt. Enim vero si diligenter exploraveris, comperies ubique numerorum^[2] vires^[3] proprietatesque in planis figurisque conservari. Quid ergo mirum easdem per plana gradatim in solida pervenire, atque ita corpora suis quaeque fieri numeris atque moveri? Quapropter Plato hic [20] et ubique numeris omnia tribuit. Et Plotinus Proclusque subtilissime probant numeros in ipso ente primo tanquam primos distinctores entium illic^[4] idearumque existere, ut non mirum sit per numeros inferiora quoque distingui, atque sicut et rerum sic et numerorum species omnes suis quibusdam viribus quasi praerogativis privilegiisque [25] pollere.

Unitas, Impares Paresque Numeri, Aequilateri et Inaequilateri. Cap. VIII.

Ipsum unum numerorum dimensionumque principium videtur principio universi simillimum, quoniam, dum sua omnia procreat, eminentissimum interea permanet atque simplicissimum. Ab hoc autem et dimensiones [5] quasi quadam puncti punctorumque positione procedunt, et numeri quasi suo quodam motu profluunt, tametsi pares quidem potius processione quadam, impares autem conversione potissimum. Interea unitas ab uno dependens est et substantia numerorum, quatenus unusquisque numerus forte nihil aliud est quam unitas totiens [10] repetita; est insuper et [1419] numerorum unitas ipsa mensura. Semel enim duo est ipsa duitas; semel tria similiter^[1] est ipsa trinitas; ceterique similiter deinceps numeri. Praeterea, sicut ab uno rerum principio incorporea fiunt atque corpora, sic ab unitate impares atque pares. Item sicut^[2] ab illo simplicia compositaque, ita et ab hac numeri simplices et [15]

[1] unum om. M

[2] numerum Z

[3] iures Y

[4] illhinc M

[1] similiter om. M

[2] sicut] si ut Y

than themselves—as 4 by the 2, 6 by the 2 and the 3. The one is like the maker of the world; but the two is like indeterminate matter, as Archytas says.^[3] Archytas wishes the one to be the idea of odd numbers, the two of even; and the two to be not so much a number as the first fall from the one. The first number he wishes to be the three.^[4] This is like the mystery of the Christian Trinity. Moreover, the one is not one of the numbers because of [its] most simple eminence; and it is all the numbers because it has the effective power of all numbers.^[5] For this reason, therefore, it has no parts and it is neither an even nor an odd number. Insofar as it adds itself to a number already born an even, and renders it odd, it seems odd itself. Again, insofar as it accommodates itself to a number born an odd and makes it even, it appears even again.^[6] This Aristotle says in the Pythagorean ,^[7] although the Pythagoreans [themselves] were more willing to call the one an odd.^[8] For it is proper for an even number not to change the number it is added to: if it is added to an even, it preserves it as an even; if to an odd, it preserves it as an odd. When the one meets an even number, on the other hand, it makes it an odd; and when it meets an odd, it makes it an even. In the same way, the odd number—as the male and effective number—changes the number it meets: out of an even number it makes an odd, out of an odd it makes an even. However, the even number—as the female—does not change; rather it is itself changed and itself suffers. Therefore the odd numbers seem to have greater kinship with the one, and this is because they are indivisible in a way, and yet they abound: they always have the one in themselves as [their] mean and center, and from the beginning they end in and are converted to the one.^[9] Finally, after you have divided an even number, it seems entirely torn apart, nor does anything of it survive among its parts. But when you study to divide an odd number, the one exists among that number's divided parts as its indivisible link, so that the odd number seems to be unfolded rather than divided.^[10] But the one is entirely indivisible. For what is divided is cut into lesser parts. But the one cannot be cut into anything less than one. On the contrary, when it appears to be divided, it is doubled rather. But the one is the principle of identity, equality, and likeness, and in these with some justice it is able to resemble God.^[11]

[ii] Wherefore squares, which are always equilaterals,^[12] are more like the one than unequilaterals because of [their] equality and straightness, the [attributes] most closely associated with the one. For in squares both the lines and all the angles have equality, mutual likeness,

compositi: simplices qui unitate simpliciter constant atque mensurantur, ut 3, 5,^[3] 7 atque similes; compositi vero qui insuper quodam minori numero mensurantur, ut 4 binario, 6 binario atque ternario. Unitas quidem similis est opifici mundi, duitas vero materiae indeterminatae, ut inquit Archytas, qui unitatem impa[152r]rium ideam esse [20] vult, duitatem vero parium;^[4] et hanc non tam numerum quam primum ab uno casum, numerum vero primum esse ternarium. Mysterium Christianae trinitati simile. Iam vero unitas et propter simplicissimam eminentiam nullus^[5] est numerorum, et propter virtutem omnium efficacem est^[6] omnes numeri. Qua igitur ratione nullas habet partes, [25] nec par est nec impar; qua vero se adhibet numero pari iam nato imparemque reddit, videtur impar; qua rursus impari genito se accommodans facit parem, par rursus apparet. Id quidem Aristoteles inquit in Pythagorico , quamquam Pythagorici unum libentius impar appellaverunt, quia paris proprium sit non mutare numerum cui additur. [30] Nempe si addatur pari, parem^[7] servat; si impari, imparem. Unum vero contra obvium quidem pari, facit^[8] imparem; obvium autem impari, reddit parem. Similiter et numerus impar tanquam mas et efficax numerum accessu mutat: ex pari quidem facit imparem, ex impari vero parem. Numerus vero par ceu femina non mutat,^[9] sed permutatur et [35] patitur. Numeri ergo impares maiorem cum unitate cognationem^[10] habere videntur, quia et quodammodo sunt individui nihilominusque abundant, et unum ipsum semper habent in se medium atque centrum, et ab initio in unum desinunt atque convertuntur. Denique postquam numerum parem diviseris, videtur omnino divulsus^[11] nec [40] inter eius partes eius aliquid extat; cum vero imparem distribuere studes, inter partes eius digestas existit unum, eius insolubile vinculum, ut explicatus potius videatur quam divisus. Unum vero est prorsus indivisibile. Quod enim dividitur, in minora secatur; unum vero in aliquid uno minus secari non potest, immo vero^[12] cum videtur dividi [45] potius geminatur. Est autem unum identitatis et aequalitatis similitudinisque principium, in quibus Deo simile videri non iniuria potest.

ii] Quapropter quadrata semper aequilatera similiora sunt uni quam inaequilatera propter aequalitatem et rectitudinem uni quam

[3] 6 Z

[4] partium Z

[5] nullius MZ

[6] et Z

[7] parem] scilicet parem M

[8] faciet M

[9] mutatur Z

[10] cognitionem Z

[11] dividuus M

[12] immo vero] uno vero in aliquid uno minus secari non potest, uno vero per homoioteleuton M

and straightness.^[13] But the excellence in equilaterals is other;^[14] for it is by the gift of the one that they overcome the unequilaterals. For the seed of the equilateral is the one;^[15] and, while the seed remains in its unity or doubles,^[16] from it sprouts the square. Thus the two duplicated by way of itself makes 4. The three multiplied by way of itself creates 9, and the 9 is that much more excellent than the 4 in that the 3 that is its seed is more outstanding than the 2. But the seed of the unequilateral has been divided into two and does not remain but is transferred from one [number] to another: thus 2 multiplied by 3 or the reverse makes the unequilateral 6.^[17]

[iii] Finally, the one itself for the same reason too has a marvelous likeness to God, the absolutely most simple, because, however much you try to multiply and say 1x1 or again 1x1x1, you never divide or diminish or increase the one itself. In numbers too there is a likeness to God Himself. For any one number working with itself generates a number, for example, 2x2=4, 3x3=9; and after it has given birth to the number, it generates another by way of this generated number; for example, 2x4=8, 3x9=27. Furthermore, the number that is the author, by using itself alone—and not using as its instrument [this generated number]—can produce the same number that it produced when using the instrument; for instance, 2x2x2=8, 3x3x3=27.^[18] From this it appears that God acting in Himself procreates other things.^[19] And in fact, if He uses the prime creature as the means to produce other effects, He can nonetheless procreate the same effects without this means, acting likewise in Himself. There are many other likenesses, but these may presently suffice for us, if I refer, that is, to the Pythagorean saying: As all things after God consist of a property of God Himself along with a degeneration from Him, and consist moreover of the same and difference and of unity and multiplicity, so too are numbers with regard to the one.^[20] Wherefore as the first number—not, I repeat, as the first multitude but as the first number—three is made from the one and from the two (i.e., from the two as a degeneration of the one, as otherness, as confused multitude). Similarly the rest of the numbers seem to follow this fate of the first number.^[21]

proximam. In his enim et lineae et anguli omnes aequalitatem et similitudinem [50]invicem habent atque rectitudinem. Est et alia aequilateris excellentia, ipsius videlicet unitatis munere quo inaequilatera superant. Nempe semen aequilateri unum est, ac, dum in sua permanet vel geminat unitate, pullulat inde quadratum. Ita duitas per se duplicata facit quatuor; trinitas per se multiplicata creat 9. Ipseque novenarius tanto [55] est excellentior quaternario quanto ternarius eius semen praestantius est binario. Inaequilateri vero semen divisum est in duo neque permanet, sed alterum migrat in alterum. Ita 2 in 3 multiplicatum vel converso senarium inaequilaterum^[13] efficit.

[iii] Denique ipsum unum mirabilem hac quoque ratione similitudinem [60] habet ad Deum simpliciter simplicissimum, quia quantum-cumque multiplicare contenderis, dicens semel unum item semel unum semel,^[14] nunquam vel dividis vel minuis vel auges ipsum unum. Est etiam in numeris similitudo quaedam ad ipsum Deum, quilibet enim numerus agens secum ipso generat numerum, ut bis 2 = 4, ter [65] tria = novem; et postquam genuit per genitum numerum^[15] generat alium, ut bis 4 = 8, ter 9 = 27. Potest quinetiam ille numerus auctor sine hoc instrumento eundem per se numerum producere, quem hoc instrumento produxerat, ut bis duo bis = octo, ter 3 ter = 27. Ex his apparet Deum secum ipso agentem alia procreare; necnon si creatura [70] prima utatur ut media ad effectus^[16] alios producendos, posse nihilominus eosdem sine hoc medio procreare agendo similiter secum ipso. Sunt et aliae multae similitudines, sed hae^[17] nobis in praesenti sufficiant, si retulero videlicet Pythagoricum illud: Quemadmodum post Deum omnia ex quadam ipsius Dei proprietate una cum quadam illinc [75] degeneratione constant atque ex eodem simul et altero et unitate atque multitudine, ita numeri se habent ad unum. Quapropter ternarius tanquam primus numerus—non inquam multitudo prima sed numerus primus—ex unitate fit atque duitate quadam unitatis degeneratione atque alteritate confusaque multitudine.^[18] Similiter ceteri numeri [80] hanc numeri primi sortem sequi videntur.

[13] inaequilaterum scripsi quadratum YMZ

[14] semel om. Z

[15] genitum numerum] genitus numerus M

[16] affectus Z

[17] haec Z

[18] in multitudine Z

Chapter 9. Odd Numbers Comprehend the Even. Likewise the Equilateral Contain the Unequilateral.

The odd numbers are not comprehended by the even, but rather the odd comprehend the even. For instance, the three contains the two in itself in that the one, which is the mean in the three and so to speak its head and bond, contains the two around itself. Plainly in the three there are three terms or grades, and two intervals are included in the three. Similarly, the four is in the five; for twin twos are on either side of the one, the five's mean, and between the five terms are four intervals. Similarly, the six is contained in the seven. And any even number preceding an odd number in the [numerical] order is comprehended by that next odd number as in [its] whole or end. Indeed, no order ever appears at all except by way of the odd terms: in them the one is the mean, the hinge so to speak, and the terms are even and the intervals are even on either side of it.

[ii] Just as the odd numbers contain the even, so the equilateral numbers, which are all compounded from the odd numbers, comprehend the unequilateral, which are all procreated from the even.^[1] The first equilateral compounded is 4, the second 9. The proportional mean between these is the unequilateral 6. For the proportion from 9 to 6 is in the ratio of 3:2. The like proportion also pertains from 6 to 4. The third equilateral is 16, for it is the result of 4 led to itself, just as 9 is the result of 3 [led to itself], and 4 of 2. Between 16 and 9 the proportional mean is 12, which is unequilateral; for it is the result of 3 led to 4. But just as the proportion between 16 and 12 is in the ratio of 4:3, so between 12 and 9 it is also in the ratio of 4:3. Therefore in these the unequilaterals seem to be enclosed by the equilaterals.^[2] But this is not the case with the contrary situation. Certainly 6 and 12 are unequilaterals. The mean between them is the equilateral 9. Yet this does not have the like proportion to the two extremes; for 12 to 9 has the proportion in the ratio of 4:3, but 9 to 6 that in the ratio of 3:2. Therefore 9 is not bound fast by these [its two unequilateral extremes]. In subsequent numbers the like reason also prevails.^[3]

[iii] I said a little earlier that the equilaterals are compounded. Moreover, among the Pythagoreans the one is equilateral, although simple; for 1x1=1. Between 1 and the equilateral 4 is the unequilateral 2. For just as from 4 to 2 the proportion is in the ratio of 2:1, so is it from 2 to 1. Therefore the equilaterals [1 and 4] encompass and bind fast the unequilateral [2].

Impares Numeri Compraehendunt Pares. Item Aequilateri Inaequilateros Continent. Cap. VIIII.

Impares numeri non compraehenduntur a paribus sed compraehendunt, ut ternarius binarium in se continet, siquidem in ternario unitas [5] quidem media quasi caput et vinculum binarium circa se continet. Tres plane in ternario termini sunt vel gradus; intervalla duo contenta ternario. Similiter in quinario quaternarius; nam et circa medium eius unum geminus est hinc et inde binarius, et inter quinque terminos intervalla sunt quatuor. Similiter in septenario senarius continetur. Et [10] par quilibet ordine praecedens imparem in^[1] proximo impari tanquam toto vel fine compraehenditur.^[2] Iam vero nullus usquam apparet ordo, nisi per terminos impares, in [1420] quibus unus sit medius quasi cardo et utrinque pares termini et intervalla sint paria.

[ii] Quemadmodum vero impares numeri pares continent, sic aequilateri, [15] qui omnes ex imparibus componuntur, [152v] compraehendunt^[3] inaequilateros, qui omnes procreantur ex paribus. Primus quidem aequilaterus compositus est 4, secundus vero 9. Proportionale inter istos medium est senarius inaequilaterus; nam ab ipso 9 ad 6 sexquialtera^[4] proportio est. Similis quoque proportio a 6 existit ad 4. [20] Tertius aequilaterus est 16; fit enim ex 4 in se ducto, sicut 9 ex tribus et 4 ex duobus. Inter 16 atque 9 proportionale medium est 12 qui inaequilaterus est; fit enim ex tribus ductis in 4. Sicut vero proportio inter 16 atque 12 sexquitertia est, ita inter 12 atque 9 est sexquitertia. In his igitur apparet inaequilateros ab aequilateris^[5] contineri, neque [25] vero fit vicissim. Nempe 6 et 12 inaequilateri sunt. Inter hos aequilaterus medius est 9. Neque tamen est hinc^[6] ad extrema proportio similis; nam 12⁷ ad 9 proportionem sexquitertiam habet, sed 9 ad 6 sexquialteram. Ipse igitur 9 non devincitur ab illis. In sequentibus quoque ratio similis. [30]

[iii] Dixi paulo superius compositos aequilateros. Praeterea unum apud Pythagoricos est aequilaterum, licet^[8] simplex, semel enim unum = unum. Inter hoc et 4 aequilaterum inaequilaterus est binarius. Sicut autem a 4 ad 2 proportio dupla est, sic a duobus ad unum. Sic igitur aequilateri inaequilaterum continent atque devinciunt. [35]

[1] in] et in M

[2] compraehendit Z

[3] comprehendent M

[4] sexquilatera Z

[5] ab aequilateris om. Z

[6] hic Z

[7] 12] ad 12 M

[8] scilicet M

[iv] But how the double proportion in the ratio of 2:1 along with the proportions in the ratios of 3:2 and 4:3 are all in accord with the perfection and steadfastness of things, this we have described in [our] introductions for the Laws and in the Epinomis .^[4]

Chapter 10. How the Diagonal is or is not Commensurable to the Side.

In squares Plato says that the diagonal is and again is not commensurable to the side. It is commensurable in power, for the power of the diagonal is adjudged double the power of the side. For were you to derive an equilateral square from the diagonal, it would consist of double the square already derived from the side. But the diagonal does not seem to be commensurable to the side in act or in [having] a determinable root. For if the square from the two-foot side is 4, the square produced from the diagonal will be 8. It seems (as I said) that the diagonal is proportional to the side in power. For a different reason, however, the diagonal is adjudged not proportional; this is because the root of the 4 is known, namely 2, but the root of the 8—the 8 as a plane and equilateral number—is undeterminable.^[1] For no one number led to itself once makes 8. Similarly, if the square made from a three-foot side is 9 (namely 3x3=9), then the square of its diagonal will be 18. For the power of the diagonal is double the power of the side. In this condition [i.e., of power] they seem commensurable, as I was just saying. Yet they are not commensurable in act, in root, in line.^[2] For the root of the 9 is known, that is, the 3 led to itself. But the root of 18 is unknown. For no one number led to itself makes 18. Likewise, if this [the square of the side] is 16, then that [the square of the diagonal] will be 32. The former's seed is certain, namely 4. But the latter's is unknown. The root and seed of a number, however, is properly called that [lesser] number which—having been multiplied by itself and striking root and sprouting as it were^[3] —generates the greater number.

Chapter 11. On the Mutual Multiplication of Even Numbers and in Turn of Odd, of Equilateral, of Unequilateral, and of Solid Numbers.

If an even number multiplies an even, either itself or another, an even always arises—2x2=4, 2x4=8. Again if an odd multiplies an odd, either

[iv] Quomodo vero proportio dupla, sexquialtera, sexquitertia perfectioni et perseverantiae rerum conveniant, diximus in argumentis Legum et Epinomide .

Quomodo Diameter Sit Lateri Commensurabilis Vel Non Commensurabilis. Cap. X.

Plato in quadratis ait diametrum esse commensurabilem lateri rursusque non esse. Esse quidem virtute; nam virtus diametri ad virtutem^[1] lateris dupla censetur. Si enim aequilaterum ex diametro quadratum [5] duxeris, duplum erit ad quadratum iam ex latere constitutum. Non tamen actu vel certa radice commensurabilis lateri diameter esse videtur. Si enim quadratum ex latere bipede factum sit 4, quadratum ex diametro productum erit 8. Qua quidem virtute videtur (ut dixi) diameter lateri proportionalis, sed altera quoque ratione proportionalis [10] non iudicatur quatenus radix quidem quaternarii nota est, scilicet duo, radix autem octonarii ut numeri plani et aequilateri est incerta. Nullus enim numerus semel in se ductus facit 8. Similiter si quadratum ex latere tripede constitutum sit 9, scilicet ter tria = 9, mox quadratum ex huius diametro ductum erit 18. Potentia enim diametri ad lateris potentiam [15] dupla est, qua quidem conditione commensurabilia haec videntur, ut modo dicebam. Non tamen commensurabilia sunt actu, radice, linea. Radix enim novenarii nota est, scilicet ternarius in se ductus. Radix autem ipsius 18 est ignota; nullus enim numerus in se ductus 18 constituit. Item, si illud sit^[2] 16, hoc erit 32. Semen illius [20] certum est, scilicet quatuor; huius autem est ignotum. Radix autem semenque numeri ille proprie numerus appellatur qui per se ipsum multiplicatus, quasi coalescens atque germinans, numerum generat ampliorem.

De Mutua Multiplicatione Parium Invicem et Imparium, Aequilaterorum, Inaequilaterorum, Solidorum. Cap. XI.

Si par numerus parem multiplicet, aut se ipsum aut alium, par semper exoritur: bis duo = quatuor, bis quatuor = octo. Rursus, si impar im- [5]

[1] virtem M

[2] sit om. M

itself or another, everywhere it generates an odd—3x3=9, 3x5=15. But if an even multiplies an odd or an odd an even, everywhere it produces an even—2x3=6, 3x4=12. For this reason surely when an equilateral multiplies an equilateral, either itself or another, an equilateral is born—4x4=16, likewise 4x9=36. And when an unequilateral multiplies an unequilateral, an unequilateral always arises—2x6=12, 6x10=60.^[1] But when an equilateral multiplies an unequilateral or the reverse, an unequilateral always arises—4x6=24, likewise 6x9=54.^[2] Moreover, if a solid number [i.e., a cube] multiplies a solid, either itself or another, it too will create a solid—8x8=64, 8x27=216.^[3] But if an unequilateral multiplies a solid or the reverse, a solid will never be procreated—2x8=16, likewise 8x6=48.

Chapter 12. On the Proportions in the Powers of the Soul; and on Spirits, Celestial Influences, and the Causes of Immense Mutations.

Plato often says that some powers of the soul should be diminished, others increased; and he signifies that all in turn should be composed in musical proportion. Such are the rational, the irascible, and the concupiscible powers.^[1] But the reason is twofold—speculative or practical. The former is called the intellect, the latter properly the reason. Therefore, from the onset men should be so educated through discipline that, if we opt for the golden race,^[2] the proportion of the understanding to the reason (as 4 to 3) should be in the ratio of 4:3, that of the reason to the irascible power (as 3 to 2) in the ratio of 3:2, and that of the irascible power to the concupiscible (as 2 to 1) in the ratio of 2:1. However, if we opt for the silver race, men should be so educated that the proportion of the reason to the understanding should indeed be in the ratio of 4:3 but reversed, with the reason being 4 but the understanding 3. The musical consonances are contained in these proportions—the diatesseron, diapente, and diapason.

[ii] Similarly through nutrition and the entire diet, the spirit, which comes from blood, should be so composed that in it the air should exceed the fire by the ratio of 4:3, the fire the water by that of 3:2, and the water the earth by that of 2:1.^[3]

[iii] Furthermore, if we consider the principal members [i.e., organs], the heart is hot and dry, the liver hot and wet, the brain cold and wet. Heat and wetness are the elements of life. These therefore should overcome the cold and the dry in good measure, but overcome

parem multiplicet vel se vel alium, imparem ubique generat: ter^[1] 3 = 9, ter 5 = 15.^[2] Sin autem par imparem aut impar parem, gignit utrobique parem: bis 3 = 6, ter quatuor = duodecim.^[3] Hac utique ratione ubi aequilaterus multiplicat aequilaterum, vel se vel alium, nascitur aequilaterus: quater quatuor = sexdecim, item^[4] quater 9 = 36. Ubi autem [10] inaequilaterus inaequilaterum, semper inaequilaterus^[5] oritur: bis 6 = 12, sexies decem = 60. Sed quando aequilaterus inaequilaterum vel vicissim, semper inaequilaterus oritur: ut 4 sex = 24, item sexies 9 = 54. Praeterea, si solidus solidum multiplicaverit, sive se sive alium, solidum quoque creabit: octies octo = 64, octies 27 = 216.^[6] Si vero inaequilaterus [15] solidum vel converso, nunquam solidus procreabitur: bis^[7] 8 = 16, item octies sex = 48.

De Proportionibus in Viribus Animae et Spiritibus Influxibusque Caelestibus, et de Causis Ingentium Mutationum. Cap. XII.

Plato saepe iubet alias quidem animae vires extenuandas alias augendas, omnesque significat invicem proportione musica componendas. [5] Eiusmodi vires sunt rationalis, irascibilis, concupiscibilis. Sed ratio duplex, speculativa vel practica. Intellectus illa, haec proprie ratio nomi[153r]nantur.^[1] Sic igitur ab initio per disciplinam instituendi sunt homines: si aureum genus optamus, ut intelligentiae ad rationem^[2] quasi quatuor ad tria sit proportio sexquitertia, rationis autem ad irascibilem [10] velut 3 ad 2 sit sexquialtera, irascibilis ad concupiscibilem ut duo ad unum [sit] dupla; [1421] si autem genus optamus argenteum,^[3] [ut] rationis proportio ad intelligentiam sexquitertia quidem sit sed converso ut ratio quidem sit ut 4, intelligentia vero sit ut tria. In his proportionibus consonantiae musicae continentur, diatesseron, diapente, [15] diapason.

[ii] Similiter^[4] per nutritionem omnemque dietam spiritus qui fit^[5] ex sanguine componendus ut in eo aer^[6] ignem sexquitertia superet, ignis aquam sexquialtera, aqua terram dupla.

[iii] Praeterea, si praecipua membra consideremus, cor calidum est [20] et siccum, iecur calidum humidumque, cerebrum frigidum atque humidum. Calor et humor vitae sunt elementa. Haec igitur frigidum sic-

[1] ter om. YZ

[2] 25 Z

[3] 16 YM

[4] ter Z

[5] inaequilaterum semper inaequilaterus om. per homoioteleuton M

[6] 2016 YM

[7] Dies Z

[1] nominatur M

[2] orationem Z

[3] argumentum Z

[4] Similiterque Z

[5] sit [?] Z

[6] aerem Z

coldness more than dryness.^[4] For coldness is opposed to heat, dryness to wetness. But heat is more preeminent for life than wetness—it is the craftsman of, or a form so to speak for, wetness.^[5] Therefore, when all has been computed, heat in us should perhaps exceed coldness in total power by the proportion of 2:1, wetness exceed dryness by that of 3:2, and heat exceed wetness by that of 4:3. For unless heat were to exceed wetness to a degree, it would never act itself vitally on wetness, nor daily cook the things consumed by us, nor withstand external conditions. Furthermore, in [any] salubrious place, or climate, or season—in all the parts measured together—perhaps heat should exceed coldness in the ratio of 2:1 and wetness in the ratio of 4:3; and wetness should exceed dryness in the ratio of 3:2. Add to these arguments what we have said about such proportions in the introduction to the Epinomis with regard to the intervals of the spheres, and the generation of things and the humors of our body.^[6] One can find the same tempering too in the planets. But of this in the third book of the De Vita .^[7]

iv] If such proportions are dissolved either in us or in the air, either death soon ensues or sudden suffocation threatens. If, because of multiple conjunctions [such proportions] are at variance in the heavens, marvelous fires and floods ensue. On account of these things [and] for the same reason, the favor of the heavens must be captured as far as we are able, so that the influence of Jupiter on us with regard to Venus may be as 4 to 3, the influence of Venus with regard to the Sun as 3 to 2, and the influence of the Sun with regard to the Moon as 2 to 1. For the Sun and the Moon simply bestow life; [whereas] Jupiter and Venus bestow prosperity and increase of life and a profusion of good things.^[8] In elections the Moon must be observed therefore as the 1.^[9] Then it must be directed to Jupiter in 4 degrees if that is possible, to Venus in 3, but to the Sun in 2.^[10]

[v] As long as all proportions and harmonies of this kind prevail among mankind, then a good habit^[11] endures in bodies, spirits, souls, and states. But when they fail, that habit also becomes exhausted, and at length the republic changes for the worse. Discipline^[12] can do much, but the fatal order^[13] seems to determine that when the number 12—the number in which the said proportions and harmonies are first unfolded and which has been destined for the universe—has been changed into its plane [i.e., its square] of 144, then among men a great mutation occurs, which is for the better if our discipline endures; but that when 12 arrives at its solid [i.e., its cube] of 1728, as at its

cumque non parum superare debent, magis vero frigus quam siccum. Frigus enim calori opponitur, humori siccum. Calor sane ad vitam praestantior est humore tanquam artifex vel forma quaedam (ut ita [25] dixerim) ad humorem. Itaque omnibus computatis calor in nobis virtute summatim superare forte debet frigus proportione dupla, humor siccitatem sexquialtera, calor humorem^[7] sexquitertia. Nisi enim calor aliquanto excedat humorem, neque aget ipse vitaliter circa humidum, neque quotidie nobis adsumpta concoquet, nec resistet^[8] externis. Praeterea [30] in loco, aere, anno salubri, omnibus summatim partibus computatis, calor forte debet dupla frigus excedere, humorem sexquitertia, humor siccum sexquialtera. His adde quaeque^[9] de huiusmodi proportionibus in argumento Epinomidis circa sphaerarum intervalla rerumque generationem et humores corporis nostri tractavimus. Eandem [35] quoque temperiem in planetis invenire licet. Sed de his in libro De Vita tertio.

[iv] Si proportiones^[10] eiusmodi in nobis aut aere^[11] dissolvantur, vel^[12] brevi resolutio^[13] sequitur, vel suffocatio imminet repentina. Si propter multiplices coniunctiones in caelo dissideant,^[14] incendia mira [40] illuvionesque sequuntur. Quas^[15] ob res eadem ratione favor caelestium pro viribus est captandus ut influxus Iovis ad Venerem ceu 4 ad 3 sit in nobis, Veneris ad Solem velut 3 ad 2, Solis ad Lunam sicut duo ad unum. Sol enim et Luna simpliciter vitam praestant; Iupiter Venusque prosperitatem et incrementum vitae bonorumque affluentiam largiuntur. [45] Observanda igitur in electionibus Luna est ut unum; dirigenda deinde ad Iovem gradibus si fieri potest 4, ad Venerem tribus, ad Solem vero duobus.

[v] Quamdiu proportiones harmoniaeque omnes huiusmodi in genere hominum plurimum perseverant, permanet in corporibus, spiritibus, [50] animis, civitatibus bonus habitus. His autem^[16] deficientibus, ille quoque fatiscit, tandemque in deterius res publica permutatur. Prodest quidem disciplina multum, sed fatalis ordo destinare videtur ut, quando numerus 12, in quo primo^[17] proportiones^[18] harmoniaeque huiusmodi explicantur et qui destinatus est universo, in suum planum [55] fuerit permutatus 144, magna quaedam in hominibus permutatio fiat, et haec quidem disciplina perseverante sit^[19] in melius; quando vero pervenerit ad solidum velut summum finemque suum 1728, res pub-

[7] humorum Z

[8] resistit Z

[9] quaeque scripsi quae quod YM quae quoque Z

[10] proportionem YZ

[11] in aere Z

[12] vel om. Z

[13] res solutio Z

[14] dissideat Z

[15] Quod Z

[16] autem om. M

[17] prima Z

[18] proportio YZ

[19] fit YM

highest end, then the republic, the state itself—if the discipline has endured thus far—also attains to its highest end; and that thereafter gradually it declines by the fatal law to a worse condition, even as the discipline by the same fate also degenerates little by little. However, before these limits have been reached, if the discipline fails through our negligence or infelicity, then the public form totters that much earlier, brought low not only by a particular fate but also by our imprudence.

[vi] In the fifth book of the Politics Aristotle briefly described the cause of such a mutation without entirely denying it. He writes:

The cause of mutations, Plato says, is because nothing endures, but all are changed in a certain cycle. He says that the principle of mutations is among those things "whose root in the ratio of 4:3 when joined to the 5 furnishes two harmonies." He is saying in effect "when the description of this number becomes solid," since nature produces at times worse or better men than discipline produces. In fact, perhaps this has not been badly said.^[14]

These are Aristotle's words. Among those things . . ., that is, among either the classes of numbers or the compounds of things.^[15]Whose etc. . . ., that is, among the numbers in which those proportions, which are contained in the twelve, supply two harmonies (the kind we have said), the elements of the diapason.^[16]He is saying in effect "when . . ., that is, the beginnings of the mutations occur when the 12 by its multiplication attains first the equilateral which is its plane [i.e., 144] and then reaches all the way to [its] solid [i.e., to 1728]. These matters and the rest have been explained in earlier chapters.

Chapter 13. On Good or Bad Offspring through the Observance of Numbers and of Figures.

The Pythagorean and Platonic view is that from two good parents is born an entirely good offspring, from two bad an utterly bad; from a bad and good together an offspring that is not wholly bad indeed, but never good.^[1] Likewise the view is that the odd numbers are in the order of the good and should be called males and bridegrooms and fathers (especially because of the strength which they possess in their middle knot, namely the one); but that the even numbers, when compared with the odd, are in the class of the bad and should be called females and brides and mothers—if, that is, they are joined to the odd numbers. For within each class too numbers can be called in a way grooms or brides, since a more outstanding even number can be

lica et ipsa civitas illuc usque disciplina durante summum suum finemque consequatur, deinde sensim in peius fatali lege labatur, disciplina [60] quoque^[20] interim eodem fato paulatim degenerante. At vero, si ante hos terminos per negligentiam nostram infelicitatemve disciplina defuerit, longe etiam prius forma publica non solum fato quodam verum etiam imprudentia^[21] nostra labascit.

[vi] Eiusmodi mutationis causam Aristoteles in quinto Politicorum [65] ita breviter enarravit nec omnino negavit:

Plato mutationum^[22] causam^[23] esse ait quod nihil maneat, sed omnia in quodam circuitu permutentur; principium vero mutationum esse penes illa "quorum sexquitertia radix coniuncta quinario duas exhibet harmonias," dicens videlicet quando numeri huius descriptio fiat solida, quippe cum natura quandoque [70] deteriores vel meliores disciplina producat. Hoc ipsum quidem forte non male dictum.

Haec Aristoteles. PENES ILLA,^[24] scilicet vel genera numerorum vel composita rerum. QUORUM et cet.,^[25] id est in quibus proportiones illae, quae duodenario continentur, duas (quales diximus) harmonias [75] constituunt, ipsius diapason elementa. DICENS^[26] VIDELICET QUANDO, id est, exordia mutationum fiunt quando duodenarius multiplicatione^[27] sua primo quidem ad aequilaterum suum planum, deinde ad solidum usque pervenerit. Haec et reliqua in superioribus sunt exposita. [80]

De Stirpe Bona Vel Mala per Observantiam Numerorum Atque^[1] Figurarum. Cap. XIII.^[2]

[153v] Pythagorica et Platonica sententia est ex duobus bonis nasci prolem omnino bonam, ex duobus malis prorsus malam, ex malo simul et bono non omnino quidem malam nunquam vero bonam. [5] Item numeros^[3] impares esse in ordine boni vocandosque masculos et sponsos atque patres, praesertim propter robur quod in nodo sui medio, scilicet uno, possident; pares autem in genere mali, si cum imparibus comparentur, nuncupandosque^[4] feminas et sponsas atque matres, videlicet si cum imparibus conferantur. Nam etiam in utroque [10] genere sponsi quidam vel sponsae quodammodo nominari possunt,

[20] quoque om. M

[21] prudentia Z

[22] mutationem Y

[23] causa Z

[24] ea Z

[25] Quorum tres Z

[26] dicemus YM

[27] multipluticione M

[1] atque om. Z

[2] XII Y

[3] numerus Z

[4] nuncupandasque Z

called the groom for an inferior even number, and an inferior odd number can be called the bride for a superior one.

[ii] Therefore, since equilaterals are made from the odd numbers (with the one leading), but unequilaterals are born from the even numbers (with the two leading), the equilaterals are certainly deemed the children of the good, the unequilaterals the children of the bad. But trigons, since they arise from the even and odd numbers compounded together, are thought to be not the worst offspring and yet not good offspring. Similarly, from equilaterals multiplied either by themselves or by each other, as from couples already good, good offspring are born. Bad offspring, however, are born from unequilaterals. From solids [i.e., cubes] doubtless come good offspring. From the mixture of unequilaterals with solids good offspring are never born.

[iii] Pythagoras and Plato seem to use these metaphors especially in the propagation of men; and this Iamblichus and Boethius indicate.^[2] Plato chooses in generation to have the most choice parents on both sides, those who possess, like the one, an utterly unitary power, and are, in the manner of the odd numbers, indissoluble, strong, well ordered, fertile.^[3] Plato also wants them to have, in the manner of equilaterals, an equable and virtuous complexion so that a most honorable progeny might thence arise. For from parents who find themselves in the contrary condition Plato supposes there arises a base offspring; and from mixed parents there springs a stock that is not honorable.

[iv] Therefore, since Plato had here adduced the great number—the number wherein exist the odd and the even numbers, the equilaterals and the unequilaterals, the oblongs, planes, solids, and lateral and diagonal numbers, and the better and worse consonances too (if the [better] diapente is compared to the [worse] diatesseron) and likewise the best harmonies (if the diapason is united with them)^[4] —then it is proper for him to have also added that this universal geometric, that is, proportional number has an immense power in itself to produce both good and not good progeny. Over and beyond such metaphors, however, we should observe the likeness of numbers in the ages of the world and in human ages. For, as I have signified elsewhere,^[5] in these ages when we arrive at the praiseworthy number or at its opposite,^[6] then comes the opportunity for good, or the occasion for evil.^[7] From the former springs fecundity and good propagation; from the latter sterility or bad issue.

siquidem par praestantior ad deteriorem parem^[5] sponsus dici potest, atque deterior^[6] impar ad potiorem sponsa.

[ii] Cum igitur ex imparibus unitate duce aequilateri fiant, ex paribus autem duce binario nascantur^[7] inaequilateri, nimirum illi quidem [15] filii boni, hi vero mali censentur. Trigoni^[8] autem, quoniam ex paribus simul imparibusque compositis oriuntur, proles quidem non pessimae neque tamen bonae putantur. [1422] Similiter ex aequilateris vel per se vel invicem multiplicatis tanquam sponsis^[9] iam bonis proles bonae nascuntur, inaequilateris autem malae. Ex solidis proculdubio [20] bonae; ex mixtura inaequilaterorum cum solidis nunquam bonae.

[iii] His utique translationibus Pythagoras et Plato in propagatione hominum uti videntur; quod Iamblichus Boethiusque significant. Optat Plato electissimos utrinque in generatione parentes, qui instar unitatis [25] vim habeant prorsus unitam, et more imparium indissolubiles sint, robusti, ordinati, fecundi; aequilaterorum quoque conditione aequalem et rectam complexionem habeant ut generosissima inde progenies oriatur. Nam ex aliis qui opposita conditione se habeant pravam oriri stirpem putat;^[10] ex mixtis autem pullulare non probam.^[11] [30]

[iv] Cum igitur magnum hic Plato numerum adduxisset—in quo impares sint et pares, aequilateri, inaequilateri, oblongi, plani, solidi, laterales, diametrales, consonantiae quoque potiores atque deteriores (si diapente ad diatesseron comparetur), item harmoniae potissimae (si cum his diapason conferatur)—merito subiunxit universum hunc numerum [35] geometricum, id est proportionalem, magnam in se vim habere ad prolem bonam atque non bonam. Ultra vero translationes eiusmodi observare oportet numerorum similitudinem in temporibus mundi aetatibusque humanis. Nempe, ut alibi significavi, quando in his ad laudatum pervenitur numerum vel ad oppositum, aut opportunitas [40] est ad bonum, aut occasio fit ad malum; et illinc quidem fecunditas et propagatio bona, hinc autem sterilitas vel successio mala.

[5] partem Z

[6] deteror Z

[7] nascuntur Z

[8] Trignoni Y

[9] spontis Z

[10] putant Z

[11] probant Z

Chapter 14. How the Numbers Here Assigned by Plato are in Accord with the Firmament, the Planets, and the Elements.

In the ninth book of the Republic Plato reveres the 3 as divine; likewise the square made from the 3, namely 9; and again the solid conceived from it, namely 27. Finally he reveres that great and fatal number, namely 729.^[1] This is because it has the prime root 3, the second root 9, and finally the third root 27. For it is made on the one hand from 9 increased by itself thrice, and on the other from 27 increased by itself twice, and both these numbers are resolved into the three. Furthermore, 729 is solid and circular^[2] and in accord, as we say, with the celestials.^[3] But in this eighth [book] Plato is about to signify a greater destiny. He takes up the greater number 1728, which is procreated from the 12 thrice increased. Perhaps he wishes the 1000 hidden away in this number to signify the firmament hiding in a way in the stars. Then from that great number which is the multiple of twelve thrice increased, namely from that number 1728, in the first place he chooses, and chooses openly, the 100 celebrated in the tenth book of the Republic ,^[4] because that equilateral 100 is procreated from ten, from the universal number as it were led to itself. Similarly, he leads the 100 to itself, multiplying the 100 a hundred times. The result is that squared equilateral number of 10,000 celebrated in the Phaedrus .^[5] For from the ample equilateral [of 100] the still more ample equilateral is thus produced. Either square [i.e., 100 and 10,000?] corresponds to the stars—the strictly fixed stars—which are in the firmament, so that not unjustly it [the equilateral?] was chosen at the onset.^[6]

[ii] From that great number accepted previously there remains, therefore, 728, which is unequilateral and therefore not (only) long but oblong (besides).^[7] For 700 is incontrovertibly oblong and indeed totally so, since its width is 7 and its length 100. If you add 28 to this oblong, it will still be an oblong. But having chosen this oblong, Plato straightway selected a twin 100 from it, the one being diagonal, the other solid. For anyone is permitted to suppose 100 diagonal and equilateral numbers in order, also 100 other numbers in order, solid ones.^[8] In the meantime however he increases this number [of 100] to the numberless crowd,^[9] having the reason which we declared from the beginning. Certainly he increases the diagonal numbers to the numberless crowd,^[10] and the solids similarly (if from increasing solids you make solids in succession).^[11] But if in the succession of numbers

Quomodo Numeri Hic a Platone Assignati Conveniant Firmamento et Planetis Atque Elementis. Cap. XIIII.^[1]

Plato in nono de Re Publica ternarium colit quasi divinum; item quadratum ab eo factum, [5] scilicet 9; rursus solidum ab ipso conceptum, scilicet 27; denique magnum illum numerum et fatalem, scilicet septingenta 29,^[2] quia primam radicem habet tres,^[3] secundam vero novem, tertiam denique 27. Fit enim partim quidem ex 9 per se ter aucto, partim etiam ex 27 bis per se aucto,^[4] qui in ternarium resolvuntur; et solidus est atque circularis caelestibusque conveniens ut dicemus. [10] Sed in hoc octavo ampliora fata^[5] significaturus, numerum accipit ampliorem 1728 ex duodenario ter aucto procreatum, in quo quidem ipsum millenarium latenter inclusum forte vult firmamentum ipsum in stellis quodammodo latens significare. Mox vero ex magno illo numero multiplicato per duodenarium ter auctum, scilicet ex numero [15] illo 1728,^[6] palam seligit imprimis centenarium unum in decimo de Re Publica celebratum, quoniam ex denario, quasi universo numero in se ipsum ducto, aequilaterus procreatur.^[7] Ipsumque centenarium ducit similiter^[8] in se ipsum, multiplicans videlicet centum centies, unde conficitur quadratus numerus aequilaterus decem millia celebratus [20] in Phaedro . Sic enim ex amplo aequilatero aequilaterus amplior procreatur. Quadratus uterque stellis proprie fixis quae sunt in firmamento respondet ut non immerito^[9] selectus^[10] principio fuerit.

[ii] Restat igitur ex magno numero prius accepto 728 qui inaequilaterus est. Nec solum propterea longus, sed insuper est oblongus; [25] nam septies centum extra controversiam est oblongus et quidem maxime, quippe cum latitudo quidem eius sit septem, longitudo vero centum. Si^[11] huic oblongo addideris 28, nihilominus oblongus erit. Sed cum elegisset hunc oblongum, mox ex illo centum excerpsit geminum: unum quidem diametrale,^[12] alterum vero solidum. Cuilibet [30] enim licet excogitare numeros ordine centum diametrales^[13] et aequilateros, centum quoque alios ordine solidos. Sed interim in turbam innumerabilem^[14] numerus hic excrescit^[15] ratione quam ab initio diximus habita: diametrales quidem ad innumerabilem^[16] proculdubio, solidi similiter, si ex solidis crescentibus solidos deinceps efficias. Sin autem [35]

[1] XIII Y

[2] septingenta 29] septingenta novem Y

[3] triam YZ 3 M

[4] partim etiam . . . se aucto] rep . M

[5] facta Z

[6] 1782 Z

[7] procreatum Z

[8] simpliciter Z

[9] merito Z

[10] sed electus Z

[11] Si vero M

[12] diametralem Z

[13] diametrale Z

[14] innumerabilium Z

[15] excresit Y

[16] numerabilem M

you make from any one number its solid, then after you have arrived at the solid made from the 100, you will obtain 1,000,000 (since the plane from the 100 was 10,000).^[12] But let us return now to earlier matters.

iii] That number 700 agrees with the 7 planets subsequent to the firmament.^[13] But 28 principally agrees with the Moon, which follows the planets. For the firmament scarcely has proportion with the Moon, or in turn.^[14] Therefore, we observe the aspect of the Moon not so much with regard to the sublime stars as to the planets.^[15] But even more diligently we should observe the aspect of the planets to the highest stars; for the planets have a similar proportion to the stars as the Moon to them, and as the humor of the elements to the Moon, and [their] heat to the Sun.^[16] But principally Saturn has been allotted the gifts of the sublime stars. The equilateral and the even numbers,^[17] signified by way of 100x100, are in accord principally with the firmament because of its even, simple, and absolutely circular motion. The unequilateral and the odd numbers, however,^[18] are in accord with the planets and the elements because of [their] odd and multiple motion. Among the planets, however, the oblongs^[19] accord with Mars, Mercury, and the Moon, the authors of the most and the greatest motions; and among the elements they accord with fire, mainly for the same reason, and with water.^[20] Oddness I have placed both in the planets, if we compare the planets to the firmament, and in the elements, if we compare the elements to the planets. Otherwise it is the even numbers which accord with the great spheres of the planets,^[21] and with the Sun and the aether because of the evenness of [their] motion, and also with Jove, Venus, and the middle air because of the tempering of [their] qualities.^[22]

[iv] Furthermore, certain plane and solid numbers are indicated here in the planets.^[23] The planets are called solid which have the fullness of their class and are not "referred" in this to another, as is the case with the Sun, Saturn, and Jupiter.^[24] Therefore the Sun has the highest fertility of life absolutely. Saturn, however, has the same fertility but in a life that is incorporeal, separate, and divine. Jupiter too has the same fertility but in life and action that is corporeal and human. In the present context Plato is speaking of both kinds of fertility, that is, of bodies and of souls. In the Cratylus he calls Jove the fountain of human life, but Saturn the pure and full understanding.^[25] In the Laws too he declares Saturn the true master of those who have understanding.^[26]

in successione^[17] numerorum ex quolibet solidum suum facias, postquam perveneris ad solidum factum ex ipso [154r] centum, mille millia reportabis, siquidem planum ex ipso centum fuerat decem millia. Sed ad priora iam revertamur.

[iii] Numerus ille 700 planetis septem convenit sequentibus firmamentum, [40] sed 28 praecipue congruit Lunae sequenti^[18] planetas. Firmamentum enim vix proportionem habet^[19] cum Luna vel vicissim. Ideo non tam observatur aspectus Lunae ad stellas sublimes quam ad planetas. Aspectus autem planetarum ad stellas altissimas diligentius observandus. Sic enim illi ad illas proportionem habent sicut ad illos Luna, [45] atque elementalium humor ad Lunam et calor ad Solem. Praecipue vero sublimium stellarum munera sortitus est Saturnus. Numeri quidem aequilateri et aequales per centies centum significati praecipue conveniunt firmamento propter motum eius aequalem, simplicem, simpliciter circularem. Inaequilateri vero et inaequales^[20] planetis et elementis [50] propter inaequalem multiplicemque motum. Oblongi autem inter planetas Marti, Mercurio,^[21] Lunae, mutationum^[22] auctoribus plurimarum atque maximarum; inter elementa igni ob eandem causam potissimum atque aquae. Inaequalitatem in planetis posui—si comparentur ad firmamentum—et in elementis—si ad planetas—alioquin aequales [55] numeri sphaeris magnis competunt planetarum et Soli atque aetheri propter motus aequalitatem, Iovi quoque et Veneri^[23] aerique medio ob temperantiam qualitatum.

[iv] Designantur hic insuper in pla[1423]netis plani quidam^[24] numeri atque solidi. Solidi^[25] quidem planetae dicuntur qui plenitudinem [60] sui generis habent atque in hoc ad aliud minime referuntur, ut Sol et Saturnus et Iupiter. Sol igitur summam vitae fertilitatem simpliciter habet. Saturnus autem^[26] tenet eandem sed in vita quadam incorporea, separata, divina; eandem quoque Iupiter, sed in vita et actione corporea potius et humana. De utraque fertilitate Plato loquitur in praesentia, [65] corporum scilicet atque animorum. Iam vero in Cratylo Iovem appellat humanae vitae fontem, Saturnum vero intelligentiam puram atque plenam, quem in Legibus etiam verum dominum iudicat eorum qui mentem habent.

[17] successiones M

[18] sequentibus Z

[19] proportionem habet tr . M

[20] inaequalis M

[21] Mercurii Y

[22] motionum Z

[23] Veneris Z

[24] quadam Z

[25] solidae Z

[26] enim Z

[v] Among the planets there are four, however, which are planes so to speak insofar as they are "referred" to the solid planets. Mars indeed and the Moon minister to the solidity of the Sun—the Moon to its light, Mars to its heat. Mercury moves with the ingenious gift^[27] of Saturn or accompanies it or executes it. Venus agrees with Jove in office.^[28] Also among the plane planets, some are called lateral, some diagonal. The diagonal does not have a perfect proportion with the side, rather it doubles the power of the side. Thus the Moon undoubtedly relates to Venus. For Venus begins and stimulates birth, but Lucina [i.e., the Moon] bears the power.^[29] Mars similarly relates to Mercury, for he rouses and inflames Mercury's motion. But it is not novel for Platonists to indulge in such metaphors; to the contrary, the Timaeus and Phaedrus inform us it is necessary.^[30] But Plato warns us to observe such influences as these in making judgments and choices; and we have taught the same in the third book of the De Vita .^[31]

Chapter 15. The Observance of Certain Particular Numbers in the Great Number.

It is worth considering why from that great number, 1728, Plato thrice chooses 100. First, he chooses the 100 as the producer of the equilateral [10,000], that is, insofar as it is led to itself. Second, after he has accepted the unequilateral and oblong number, namely 728,^[1] he chooses the diagonal 100 (in the first instance as equal to itself, in the second as a plane^[2] ). Third, he chooses the solid 100, by name the cube.^[3] Why does he also signify the 1,000 and the 10,000? And why did he wish for three terms in describing the fatal number: first the 1000, second the 700, third the 28? Certainly, he meant the three^[4] to signify the Fates, appointing the beginnings and ends and middles of things.^[5]

[ii] He rejoices perhaps in the 3 as in the first [number], certainly as in the most sacred of all [numbers]. Moreover, he rejoices in the 100 as in the brood of the universal number, that is, of the 10; for 10x10 makes 100. He also introduces this number, the 100, in the tenth book of the Republic as if it were life's particular end and the term of judgment.^[6] Moreover, in the Phaedrus especially he delighted in the 1000 as in the body of the 10, for 10x10x10 makes its own solid the 1000.^[7] Again, he rejoices in the 10,000 openly in the Phaedrus and secretly here, because it results from 10 and 1000 (in both books meanwhile he reports the unequal dignity);^[8] likewise

[v] Quatuor vero sunt inter planetas quasi plani quatenus referuntur [70] ad solidos. Mars quidem atque Luna Solis soliditati ministrant: Luna^[27] lumini, Mars calori. Mercurius ingeniosum Saturni munus movet vel sequitur vel exequitur. Venus cum Iove in officio convenit. Sunt etiam inter planetas^[28] planos aliqui laterales diametralesve nominati. Diameter proportionem cum latere consummatam quidem non [75] habet, sed lateris duplicat potestatem. Sic utique Luna quidem se habet ad Venerem.^[29] Nam Venus partum incohat atque stimulat. Lucina vero fert opem. Mars similiter ad Mercurium, nempe motum eius acuit et accendit. Eiusmodi vero translationibus indulgere Platonicis non est novum, immo et necessarium esse Timaeus, Phaedrus que docent.^[30] [80] Eiusmodi autem influxus in iudiciis et electionibus observandos Plato monet, et nos in tertio^[31]De Vita docuimus.

Observantia Certorum Numerorum in Numero Magno. Cap. XV.^[1]

Consideratione^[2] dignum est cur Plato ex illo^[3] numero magno 1728 ter eligat centum: primo quidem centenarium aequilateri productorem, quatenus videlicet in se ducitur; secundo, post numerum inaequilaterum [5] et oblongum, scilicet 728, acceperit, centum diametrale (et primo pariter et secundo planum); tertio vero solidum nomine cubum. Cur etiam significet mille atque decem millia et quare in fatali numero describendo terminos tres voluerit: primum quidem mille, secundum vero 700, tertium 28. Profecto tres^[4] voluit parcas^[5] significare, [10] principia rerum ac fines et media^[6] destinantes.

[ii] Gaudet quidem ternario forte tanquam primo, certe velut omnium sacratissimo. Gaudet insuper centenario quasi foetu universi numeri, id est denarii. Decies enim decem^[7] facit^[8] centum. Introducit etiam hunc numerum, scilicet centum, in decimo de^[9] Re Publica quasi [15] quendam vitae finem et iudicii terminum. Delectatur^[10] quinetiam millenario praesertim in Phaedro tanquam denarii corpore, decies enim decem decies mille solidum suum efficit. Delectatur rursus palam in Phaedro et hic clam decem millibus, quoniam et^[11] ex decem atque

[27] Lunae Z

[28] planetas om . M

[29] autem tenet eandem sed . . . eandem quoque Iupiter] e summo folii praecedentis sui (i.e. f. 180v [14.63–64 supra]) rep. et del . M

[30] dicent YZ

[31] in tertio] interio Y

[1] XIIII YM

[2] Conditione Z

[3] illorum Z

[4] tertius Z

[5] parens Z

[6] medici Z

[7] deces M

[8] facit om . YM

[9] est Z

[10] Delectatus Z

[11] et om . Z

10,000 is made from the 100 led to itself as to an equal. Here he rejoices secretly in the number 1,000,000 as in the body [or solid] of the 100. This number exists as the hundredth cube in the order of numbers,^[9] and can also be called "the cube of the trinity."^[10] For, after you have accepted the ten as the line, then the 100 is like the surface made from the ten, and finally the 1000 is like the body produced from the 100. From these three terms you can immediately extract the middle, namely the 100, and take that as the line which you may lead forth to the surface of 10,000, and finally to the solid of a thousand thousand.^[11] For thus Plato in the first denomination of the 100 had arrived as far as the surface, when he said "100x100."^[12] But in the third [denomination], when he said "the 100 of the cubes," he arrived at the solid made from that surface. In the second denomination, however, that is, of the diagonal numbers, he wandered through innumerable planes.^[13]

[iii] However, he mixes evens with odds, both because the discordant concord^[14] of qualities moves and generates all things, and also because he is investigating here not only the generation (genitura ) but also the death of things, and exploring fertility and sterility equally. He mixes planes with solids, both because solids are resolved into planes and planes are brought back into solids, and because, as I was saying above, one can discover all these,^[15] each with its particular property, in the celestials and in the elements, by whose^[16] powers and motions individual things compounded beneath the Moon are borne along, for better or for worse.

[iv] Not without mystery, and signifying the fatal increment of things, Plato led forth^[17] from the 12, which is the first of the increasing numbers. Choosing the entrance and the exit, he drew the perfect exordium out from the first of the perfect numbers, out from the 6 doubled.^[18] Then at the end he arrived at the second perfect number, namely at 28, the term of the fatal number.^[19]

[v] Finally, over and beyond that great number^[20] which is the multiple of the 12 led to and led back to itself—that is, over and beyond 1728, Plato secretly multiplies [i.e., unfolds] this innumerable number.^[21] He presents it first as a hundred times a hundred;^[22] then as a hundred lateral and diagonal numbers successively arranged and increasing^[23] — that is, as a hundred squared numbers derived from squared;^[24] and then again as a hundred cubes—that is, solids—that come from cubes that are ever increasing in amplitude.^[25] He does this so that not only republics but all ages may be measured by this most

mille resultat, sed inaequalem^[12] dignitatem interim utrinque^[13] reportans; [20] item fit ex centum in se ducto velut aequale. Gaudet hic clam ipso numero millies mille tanquam corpore centenarii, qui et centesimus cubus existit ex ordine numerorum, et trinitatis cubus dici potest. Postquam enim decem ut lineam accepisti, deinde centum ut superficiem ex eo factam, postremo mille ceu corpus inde productum.^[14] [25] Statim ex his tribus terminis medium, scilicet centum, excipere potes tanquam lineam quam^[15] producas in superficiem decem millia, postremo in solidum mille millia. Sic enim Plato in prima centenarii denominatione ^[16] ad superficiem usque pervenerat ubi dixit centies centum; in tertia vero ubi dixit centum cuborum pervenit ad solidum ex [30] illa superficie^[17] factum; in secunda vero nominatione,^[18] sc ilicet diametralium, per innumerabilia plana [l54v] vagatus.

[iii] Miscet autem aequalia inaequalibus, cum quia ipsa^[19] qualitat um concordia discors omnia movet et generat, tum etiam quia non solum genituram hic rerum investigat sed et interitum;^[20] fertilitatem [35] pariter sterilitatemque indagat. Miscet et plana solidis, quoniam solida resolvuntur in plana atque haec referuntur ad solida, et quoniam, ut supra dicebam, haec omnia sua quadam proprietate in caelestibus elementisque reperire licet, quorum viribus motibusque singula infra Lunam composita ad melius deteriusve feruntur. [40]

[iv] Nec^[21] sine mysterio fatale rerum significans incrementum ex numero 12 crescentium primo produxit, optansque principium exitumque perfectum a numero perfectorum^[22] primo deduxit exordium, scilicet ex geminato senario; pervenit insuper ad perfectum postremo secundum, 28, fatalis huius numeri terminum. [45]

[v] Denique ultra magnum illum numerum multiplicatum^[23] ex duodenario ducto in se ipsum atque reducto, scilicet 1728, Plato numerum hunc^[24] innumerabilem clam multiplicat (scilicet per [50] centum centies; item per numeros centum laterales diametralesque deinceps dispositos atque crescentes, quadratos videlicet ex quadratis; rursus per centum^[25] cubos, id est solidos, ex cubis crescentibus semper in amplum) ut amplissimo numero quasi seculo seculorum non res pub-

[12] inaequale YM

[13] utrumque M

[14] productam Z

[15] qua Y

[16] dominatione YM

[17] superfice Y

[18] denominatione M

[19] ipsum Z

[20] in tertium Z

[21] Hec M

[22] profectorum Z

[23] multiplicam Z

[24] hinc YZ

[25] centrum Z

ample number, by the century of centuries as it were; and so that a term may exist which things compounded cannot surpass,^[26] and single things in the meantime, all closed in their own measures,^[27] may be distinguished by way of the certain parts of such a great number.^[28]

Chapter 16. On the Habit, Age, and Time Span of the Body for Begetting; and on Their Accommodation.

However, for the sake of happy offspring Plato orders unions to be made from good parents on both sides. Accordingly, I draw attention to the fact that the dispositions (ingenia ) of each parent should indeed be good; but they should not be in the same condition of good, nor absolutely equal and alike, but rather good for each other, insofar as we adjudge this needful for good progeny, as Plato argues in the Statesman and in the Laws .^[1] All this is in order that fiercer dispositions may be united with gentler ones, and the more vehement may be tempered by the more relaxed, otherwise progeny may emerge which is either exceedingly ferocious or exceedingly cowardly. But both dispositions should be, to their utmost capacity, the most equal in their class,^[2] to their utmost capacity the choices. In the zodiac such signs as are male seem joined successively with feminine signs. Such is the union of the Moon with the Sun, and of Venus with Mars. Such seems to be the union under heaven of the higher wetness with heat, an aethereal heat, and of the lower wetness with cold.^[3] The result in the compounded body is the discordant concord such as we find among musicians, when the temper of low-pitched with high-pitched voices is everywhere observed; yet both kinds indeed, although they are uneven, must be accepted in song. Thus too from unlike proportions, namely from the diatesseron and the diapente, is produced the diapason, the most equal of all.^[4] Moreover, even habits are generated from odd numbers (as in the case of squared numbers); but odd habits are generated from even numbers.^[5]

[ii] The opportune time for public marriages requires evenness [i.e., calmness] in the air and solidity in each body's habit, desire (affectus ), and age, and in all else. Likewise it requires the power of the Sun, who is solid, and of Venus, who is even, and of Jove, who is vigorously both, and also of the Moon (her aspect according with them).^[6] But in his republic Plato requires that all these things mustbe observed by the magistrates when particular matters are publicly regu-

licae^[26] solae sed etiam omnia secula mensurenter, sitque terminus quem composita praeterire non possint, sed interea per certas tanti numeri partes singula suis quaeque mensuris clausa distinguantur. [55]

De Habitu Corporis, Aetate, Tempore ad Generandum Accommodatis. Cap. XVI.^[1]

Quod autem Plato iubet felicis geniturae gratia ex utrisque bonis coniugia facienda, sic accipiendum moneo ut bona quidem utriusque parentis sint ingenia, sed non in eadem conditione boni nec aequalia [5] prorsus atque simillima, sed ita invicem bona quatenus ad bonam stirpem necessarium iudicatur. Id autem est, quemadmodum in Politico Legibus que disputatur, [1424] ut acriora ingenia mitioribus copulentur, vehementiora remissioribus temperentur, ne alioquin ferocissima vel ignavissima progenies oriatur. Sed pro viribus aequalissima utraque [10] vero ingenia in suo genere esse debent, pro viribus electissima. Talia in zodiaco signa tanquam masculina femininis^[2] signis deinceps coniugata videntur. Tale Lunae cum Sole coniugium est, Venerisque cum Marte; tale sub caelo humoris quidem sublimis cum calore quodam aethereo, inferioris autem humoris cum frigore consortium esse videtur. Talis in [15] composito corpore resultat concordia discors, tale apud musicos gravium ubique vocum cum acutis temperies observatur, quae quidem, etsi sunt impares, utraeque^[3] tamen accipiendae sun canore. Sic etiam ex proportionibus dissimilibus, scilicet diatesseron atque diapente, diapason omnium aequalissima procreatur. Sic insuper, velut in quadratis [20] numeris, aequales^[4] quidem habitus ex imparibus; inaequales autem ex paribus generantur.

ii] Tempus autem publicis connubiis opportunum exigit aequalitatem in aere, et in ipso cuiusque habitu corporis et affectu^[5] aetateque et in cunctis solidetatem; item potestatem Solis (qui solidus est) et [25] Veneris (quae aequalis) et Iovis (qui valet utroque) Lunae quoque^[6] ([quae habet] competentem ad haec aspectum). Observanda vero haec magistratibus Plato mandat in re publica sua ubi publice singula

[26] Pca M

[1] XV. YM

[2] foeminis Z

[3] uterque Z

[4] aequalis Z

[5] affectus Z

[6] quo Z

lated and when many brides are joined with their spouses together in public rites.^[7]

[iii] In the sixth book of the Laws he tempers in union the more vehement passions with the more gentle and [requires that] both passions be moderate and constant in the hour of copulation, so that the child thence conceived—to use his words—may be generated even, stable, and solid—which are mathematical terms.^[8] Not only this, but by marriage he joins the more powerful men in the republic with the less powerful, and the rich with the poor, so that from these odds the whole state may emerge even; and—to use his example—that from being exceedingly potent wine it may emerge, after being mixed with water, a tempered drink.^[9]

[iv] However, he chooses that a man should enter upon marriage between the ages of 25 and 35. Here he is observing an equilateral, namely 25, which is created from the 5 led to itself (25 is also circular in that starting from the 5 it ends in the 5). Likewise, he is immediately looking forward to and approving 27, the solid procreated from the 3. Finally, when he introduces 35, he is explicitly recognizing both a long number and an oblong.^[10] But he is implicitly recognizing 36 also, to which, as to [its] higher term, 35 seems to arrive, 36 being an equilateral (produced from the 6 led to itself), and also a circular (for beginning from the 6 it ends in the 6).^[11] Such indeed must be the details you should observe in commentaries, even if they seem trivial, when once you have undertaken to be a mathematician.

[v] Consequently, in the fifth book of the Republic he measures the complete span in a man for giving himself over to the begetting of children as being from 30 years to 55, but in women as being from 20 to 40. He supposes that during this span especially men are lively and strong in mind and body alike for the office.^[12]

[vi] Why in the Laws does he begin the span in a man at the 25th year, but in the Republic at the 30th? Because in the Republic the most perfect is everywhere desired, and man's rational soul is more perfect and more peaceful in the 30th year. In the Republic he attributes 25 years to man as the span of generating, because this number is equilateral and circular. But the age for a woman begins from the unequilateral [20], and the interval of [her] conceiving similarly spans the unequilateral, namely 20 years. For the female is inferior herself and is deemed inferior in the office of generating. The better things, however, are rightly signified in Plato by the better figures and signs.

dispensantur, sponsaeque multae cum sponsis simul in sacris publicis copulantur. [30]

[iii] Sed in sexto Legum non modo vehementiores affectiones cum mitioribus coniugio temperat^[7] et utrasque in ipsa congressionis hora sobrias atque constantes, ut^[8] conceptus inde—ut eius verbis utar— aequalis stabilisque et solidus generetur—quae verba mathematica sunt; verum etiam potentiores in re publica cum minus potentibus et [35] divites cum egenis connubio copulat, ut tota civitas ex imparibus fiat aequalis, atque—ut utar eius exemplo—ex validiore quodam mero simul et aqua potus quidam temperatus evadat.

[iv] Eligit autem virum ad matrimonium ineundum ab annis 25 ad^[9] 35, observans videlicet aequilaterum, scilicet 25, ex quinario in se [40] ducto creatum, et circularem, a quinario videlicet in quinarium desinentem; item mox sperans et approbans 27 solidum ex ternario procreatum. Denique ubi inducit 35, longum palam intelligit et oblongum. Subintelligit autem 36 ad quem velut terminum superiorem^[10] 35 pervenire videtur tanquam ad aequilaterum ex senario in se ducto [45] productum atque etiam circularem, nam incipiens a senario desinit in senarium. Talia quidem in commentariis observanda sunt, etsi videntur levia, ubi semel mathematicus esse ceperis.

[v] Proinde in quinto de Re Publica tempus totum procreandis liberis indulgendum in viro quidem metitur ab annis 30 ad annos 55, [50] in mulieribus autem a viginti ad quadraginta, existimans in hoc praecipue tempore homines ad hoc officium tam animo quam corpore vegetos validosque existere.

[vi] Sed curnam in Legibus in viro incipit ab anno 25, [55] in Re Publica vero a 30? Quoniam in Re Publica perfectissimum ubique desideratur, in anno vero 30 perfectior est et [155r] pacatior animus. Tribuit autem in Re Publica viro spatium generandi annos 25, quoniam hic numerus aequilaterusest atque circulars. Mulieris autem aetas ab inaequilatero^[11] incipit, similiter et concipiendi spatium per inaequilaterum, scilicet 20, producitur. Femina enim et deterior ipsa est et in [60] officio generandi censetur inferior. Melioribus vero figuris et signis apud Platonem meliora rite significantur.

[7] temperant M

[8] ut om . Z

[9] ad annum M

[10] superiorem scripsi superior YMZ

[11] aequilatero Z

Chapter 17. On the Perfect Number, on Divine Generation, and on the Observation of Celestials.

Thus far [I have dealt] with the generating that is called human, but now something must be said about divine generation, whose circuit is contained by the perfect number (as Plato says). The perfect number, I repeat, is either known to God alone, as we said from the onset, or perhaps it is 6 and numbers like it (those which are composed from their parts). But 6 is the prime perfect number for the reasons we gave earlier. Moreover, men add to the praises of the 6 the following: that led to itself it makes the plane circle, namely 36; led back to itself it enacts the solid circle, namely 216. But these numbers are called circular because, beginning from the 6, they end in the 6. Furthermore, they also contain twin circles below themselves, one from the 5, another from the 4. For 5x5=25 and likewise 5x5x5=125; likewise 4x4x4=64. But the circle we should produce from the 4 has been intercepted in the plane; for 4x4 does not end in the same number [i.e., in 4].

[ii] Therefore the circle from the 6, because of its perfection, refers to the circuit of the firmament. But that from the 5 refers to the period of the planets; for this is a fifth region above the elements. But the circle from the 4 refers to the revolution or mutation of the four elements which is in a way interrupted.

[iii] You know, I think, the Platonic order of the planets: Saturn, Jupiter, Mars, Mercury, Venus, Sun, Moon.^[1] Therefore, when you arrive at the sixth, you will have arrived for the most part at what is good and life-giving. If you begin from the firmament, you will arrive at Venus; if from Saturn, at the Sun; and if from the Moon, at Jupiter. If you start at the onset itself of conception from Saturn, in the sixth month you will be led to the Sun. If you number the years from birth, beginning from the Moon, you will arrive in the sixth year at Jupiter; and so on similarly. It is not without mystery, therefore, that Moses proposed that the world was perfected on the sixth in the number of the days.

[iv] Remember, moreover, that below 10 the perfect number is 6, below 100 it is 28, and below 1000 it is 496; and below 10,000 there also exists one perfect number, 8128.^[2] Here a marvelous vicissitude must be observed: the perfect numbers, beginning from the 6 and then arriving below 100 at the 8, below 1000 revert to the 6, and below 10,000 return again to the 8, and so on similarly.^[3] But enough

De numero Perfecto et Generatione Divina et Observatione Caelestium. Cap. XVII.^[1]

Hactenus de genitura quae nominatur humana, nunc vero de divina genesi^[2] nonnihil est dicendum. Huius circuitum numerus (ut inquit Plato) perfectus continet, perfectus inquam vel soli Deo notus (ut ab [5] initio diximus) vel forte senarius atque similes qui partibus suis constant. Sed 6 primus est perfectus rationibus quas in superioribus assignavimus. Accedunt haec insuper ad senarii laudes, quod in se ductus circulum facit planum, scilicet triginta sex, in se reductus circulum agit solidum, scilicet 216. Dicuntur vero circulares, quoniam incipientes a [10] senario desinunt in senarium. Continent^[3] insuper infra se circulos quoque geminos, alterum quidem ex quinario, sed ex quaternario alterum. Nam quinquies quinque = 25,^[4] item quinquies quinque quinquies = 125, item quater quatuor quater = 64. Sed circulus qui producendus est ex 4 interceptus est in plano; nam quater 4 non desinit in eundem. [15]

[ii] Circulus igitur ex ipso 6 propter perfectionem refert^[5] firmamenti circuitum; qui autem ex 5, periodum planetarum—est enim haec super elementa quinta quaedam regio; sed qui ex 4, revolutionem vel commutationem 4 elementorum quodammodo interruptam. [20]

[iii] Scis ut arbitror Platonicum ordinem planetarum: Saturnus, Iupiter, Mars, Mercurius, Venus, Sol, Luna. Perveniens igitur in senarium, plurimum in bonum vivificumque perveneris: si a firmamento inceperis in Venerem, si a Saturno in Solem, si a Luna in Iovem. Si ab ipso conceptionis exordio exorsus fueris a Saturno, sexto mense perduceris [25] ad Solem. Si a nativitate annos numeres incipiens a Luna, anno sexto consequeris Iovem, similiterque^[6] deinceps, ut non absque mysterio Moses senario dierum numero mundum velit fuisse perfectum.

[iv] Memento praeterea perfectum numerum infra 10 quidem sex existere, infra centum vero 28, sed infra mille 496, at vero intra decem [30] millia unum quoque perfectum existere, 8128. Ubi vicissitudo mirabilis observanda per quam perfecti numeri a senario incipientes, et mox infra centum pervenientes ad 8, iterum intra mille ad sena[1425]rium revertantur,^[7] rursusque^[8] intra decem millia ad 8 re-

[1] XXVI Y X6 M

[2] generatione Z

[3] Continet YM

[4] quinquies quinque = 25, item om . M

[5] infert M

[6] Simulteque [?] Y

[7] revertuntur MZ

[8] -que om . Z

of this. What I am now going to say about the 6 suppose said of the rest of the subsequently perfect numbers.

[v] We arrive at the 6 either through its parts or through the whole. Its parts are 1, 2, 3. We approach it through the 1 when we say once 6 or 6x1, through the 2 when we say twice 6, and through the 3 when we say thrice 6; we approach it through the whole when we reckon 6x6 or 6x6x6.^[4] Therefore it seems meet that we look to almost all the numbers of this kind, exactly as we do to the 6 itself, when Plato says that the perfect number contains the circuit of divine generation. This is similarly true of 28 and 496, and likewise of 8128 and the rest of the numbers that are perfect for a similar reason. These are indeed most rare. For just as there is only one such number under 10, so there is in turn only one under 100, and one under 1000, and then just one under any 10,000. As rare as is the perfection, so rare is the divine progeny that comes forth.

[vi] Let us return to the 6. How should we observe either the 6 itself or such multiplications of it? Let me briefly reply [that we should observe it] in the years of the century, or in the centuries of years, and in the life span (aetate ) of man; and hence in the time that is opportune for marriage and conception, for the onset of education and instruction, and for trying to capture auspicious [moments] to embark on projects and the like.^[5] It is difficult enough to explore these matters with regard to the 6, but quite impossible with regard to the other perfect numbers more ample than the 6, especially those beyond 28, the number second in perfection.

[vii] Allegorically the 6 (and each perfect number) seems to pertain to the divine class. Nothing is wanting or overflowing to this divine class—as is the case similarly with the 6 and numbers like it arranged by way of their members [i.e., parts]. This divine class neither lacks nor exceeds anything, nor does anything flow away out of it or flow into it, nor does it need outside assistance; but it is equal and tempered, and it depends on, and stands firm in, its parts and powers.

[viii] Therefore, having lighted on the occasion, I am disposed to debate for a little while with the astrologers. The perfect number [6] seems to signify constancy, equality, temperance, and therefore a particular complexion for man—tempered, sufficient to itself, and constant (which is most rare indeed, like that number). Likewise it seems to signify Jupiter, who, among the celestials, possesses this complexion to the greatest degree;^[6] or again to signify the whole harmony of the celestials when he/it accords with us thus.^[7] Therefore we must choose

meant;^[9] similiterque deinceps. Sed de hoc satis. Quae vero nunc de [35] senario dicam, de ceteris subinde perfectis dictum existimato.^[10]

[v] Pervenitur autem ad 6 vel per partes suas vel per totum. Partes eius sunt, id est,^[11] 1, 2, 3. Per unum acceditur quando dicimus semel 6 vel sexies unum; per duo quando bis 6; per tria vero quando ter 6; per totum autem quando computamus sexies 6, vel sexies 6 sexies.^[12] [40] Omnes igitur eiusmodi numeri ferme perinde atque senarius ipse observandi videntur, ubi dicitur a Platone perfectus numerus ipsum divinae geniturae circuitum continet; similiter quoque circa^[13] 28, atque circa 496, item circa 8128,^[14] atque ceteros simili ratione perfectos qui profecto rarissimi sunt. Sicut enim unicus infra decem, sic unicus [45] deinceps infra centum, unus infra mille, deinde unicus intra quodlibet decem millia. Tam rara perfectio est, tam rara^[15] progenies divina prodit.

[vi] Sed ad senarium revertamur. Quonam pacto vel ipsum 6 vel huiusmodi multiplicationes eius observare debemus? Ut breviter respondeam [50] in annis seculi vel in seculis annorum, in aetate hominis atque hinc in opportunitate connubii conceptionisque, in educationis et eruditionis exordio, in captandis operum auspiciis atque similibus. Difficile quidem est haec circa senarium explorare, impossibile vero circa perfectos alios senario ampliores, praesertim ultra 28 perfectione [55] secundum.

[vii] Allegorice vero senarius et quisque perfectus ad divinum genus pertinere videtur cui sicut senario similique per sua membra digesto neque deest neque superest, nec deficit nec excedit quicquam, nec effluit nec influit aliquid, nec alienis indiget adminiculis,^[16] sed aequale [60] temperatumque^[17] est, et suis partibus viribusque nititur atque consistit.

[viii] Hinc igitur occasionem nactus^[18] parumper cum astrologis confabulari libet.^[19] Perfectus numerus constantiam, aequalitatem, temperantiam significare videtur, ideoque complexionem quandam hominis [65] temperatam sibique sufficientem atque firmam, quae quidem sicut ille numerus rarissima^[20] est; item Iovem inter caelestia maxime talem; rursus totam caelestium harmoniam quando nobiscum ita consonat.

[9] remeat M

[10] existimatio Z

[11] id est, om. M

[12] sexties YZ

[13] circa om. Z

[14] 8129 Z

[15] perfectio est tam rara om. Z

[16] adminiculus Y

[17] -que om. Z

[18] nactis YM

[19] licet Z

[20] carissima Z

this complexion because of its suitability for marriage and conceiving. But Jupiter in general is designated through the 6, both because of the reasons we have just talked about, and also because for us he is sixth among the celestials. Nor must we neglect those conjunctions of Jupiter with Saturn^[8] in which Jupiter, who is happily disposed [towards us], by a certain closeness or familiarity conciliates Saturn, who is otherwise discordant to us. But they say that the influence of that league flourishes for twenty years, until they are joined together for a second time elsewhere.^[9] Perhaps Saturn acts in the first year after the conjunction, Jupiter in the second, Saturn in the third, Jupiter again in the fourth, and so on in succession. That conjunction must therefore be chosen for the advantages it offers us. Also we must choose that year in which Jupiter is active and especially the sixth year, the twelfth, and the eighteenth (for these two accord with the sixth). Meanwhile we should choose the sextile aspect of Jupiter to Saturn, or the trine aspect, which is composed from the double sextile.^[10] For these particular [aspects] have or represent the benign nature (affectio )^[11] of the sixth and perfect number. Furthermore, the Moon, when she mixes her quality rightly with the quality of the Sun, performs the six-like temperance, equality, and constancy of Jupiter:^[12] first, if she is in the center of the Sun, which is briefest indeed;^[13] and second, if she is in the sextile or trine aspect to the Sun, for thus she makes [her] quality most jovian and like the 6. And because the 12 is the first of the increasing numbers, remember that for propagating offspring most happily the Moon should be chosen when she is increasing in light.^[14] Perhaps too in acting the Moon alternates daily with the Sun in the same manner as Jupiter does with Saturn over the years: thus the Moon possesses the second day after her union with the Sun—for the Sun possesses the first day after the union—and so on until they come into union again. Therefore it seems we should choose the Moon on each day following, when she is tempering the Sun for us. We should also observe Jupiter when he is ascending or otherwise potent; and observe Venus as a lesser Jupiter;^[15] and observe likewise the day or hour of Jupiter or of Venus.

[ix] We should inquire into all these things and reflect upon (comparanda ) them to our utmost ability, so that, having acquired temperance and stable prosperity in our spirits and bodies, we may then acquire the power suitable for contemplations from Saturn (the patron of understanding) by way of Mercury (in this office the servant of Saturn).^[16] In this way the Saturnian ages may return to us some day,

Haec igitur ad connubii conceptionisque opportunitatem sunt optanda. Omnino vero Iupiter per senarium designatur, tum^[21] propter [70] ea^[22] quae modo diximus, tum etiam quia^[23] nobis est inter caelestia sextus.^[24] Neque praetermittendae sunt Iovis cum Saturno coniunctiones illae in quibus feliciter affectus Iupiter Saturnum alioquin nobis dissonum quadam [155v] familiaritate conciliat. Tradunt vero foederis illius influxum annos viginti vigere donec iterum alibi coniungantur.^[25] [75] Forte et^[26] anno dehinc primo Saturnus agit, secundo Iupiter, tertio Saturnus, iterum quarto Iupiter, vicissimque deinceps. Eligenda^[27] igitur est ad opportunitates nostras illa coniunctio; necnon annus ille in quo Iupiter operatur, praesertim sextus annus et duodecimus decimusque octavus (nam hi duo cum sexto conveniunt). Optandus est [80] interea sextilis aspectus Iovis^[28] ad Saturnum aut trinus ex^[29] gemino sextili compositus. Haec enim singula senarii perfectique numeri affectionem habent vel repraesentant. Praeterea Luna qualitatem suam cum Solis qualitate recte commiscens senariam Iovis temperantiam et aequalitatem^[30] agit atque firmitatem:^[31] primo si in^[32] centro Solis sit^[33] —quod [85] quidem est brevissimum; secundo si in aspectu ad Solem sextili vel trino, sic enim maxime Ioviam conficit qualitatem senario similem. Et quia duodenarius primus est crescentium, memento Lunam lumine crescentem eligendam^[34] esse ad prolem felicius propagandam. Forte etiam Luna cum Sole eam in dies agendi vicissitudinem agit quam [90] Iupiter cum Saturno per annos. Itaque Luna a coitu Solis secundum obtinet diem, primus^[35] enim inde Sol tenet atque ita deinceps donec rursus congrediantur. Luna igitur sequenti quoque die Sole nobis temperans eligenda videtur. Observandus quoque Iupiter ascendens aliterve^[36] potens, aut Venus quasi minor Iupiter; item dies vel hora [95] Iovis aut Veneris.

[ix] Haec investiganda sunt omnia et pro viribus comparanda, ut, temperantiam firmamque prosperitatem adepti spiritibus et corporibus,^[37] inde vim contemplationibus^[38] aptam a Saturno intelligentiae fautore^[39] per Mercurium ad hoc Saturni ministrum adipiscamur, ut [100]

[21] cum M

[22] propterea M

[23] qui M

[24] sexus M

[25] coniugantur YZ

[26] etiam M

[27] Eligendo Z

[28] ad Iovem Z

[29] et Y

[30] qualitatem Z

[31] infirmitatem Z

[32] in om. MZ

[33] fit M

[34] religendam Z

[35] primas Z

[36] Alive Z

[37] et corporibus om. Y

[38] contemplationis Z

[39] favere Z

and our dispositions (ingenia )—as Plato fervently wishes here—may be transformed from iron into silver and gold.^[17]

[x] Finally, Plato seems as it were to have prophesied that in those ages and times which arrive at, or return to, the perfect number, certain divine men will arise; and to them the ends of those ages will be known.^[18] Perchance the following lines refer to this:

Now comes the last age of the Cumaean song.
The great order is born from the whole of the generations . . .
Now the new progeny is dispatched from heaven on high.^[19]

But these matters issue indeed from that dispenser who has arranged all things in number, weight, and measure.^[20] But we have debated enough in the company of Plato and the Muses as they play with a serious and inextricable matter.^[21] The end.

quandoque secula^[40] nobis Saturnia revertantur atque (ut Plato hic vehementer optat) ingenia ex ferreis in argentea et aurea transformentur.

[x] Plato denique quasi^[41] vaticinatus videtur^[42] in his seculis et temporibus, quae ad numerum perfectum veniunt vel referunt, divinos quosdam homines exoriri in quos fines seculorum pervenerunt.^[43] Huc [105] tendit forsitan illud:

Ultima Cumaei iam venit carminis aetas.
Magnus ab integro seclorum nascitur ordo
Iam nova progenies caelo dimittitur alto.

Sed haec illo quidem^[44] dispensatore proveniunt^[45] qui omnia numero [110] et pondere mensuraque disponit. Nos autem una cum Platone Musisque in re seria inextricabilique ludentibus satis confabulati sumus. Finis.^[46]

[40] specula Z

[41] quas M

[42] videatur Z

[43] provenerunt M

[44] quidem om. Y

[45] provoniunt Y

[46] Scripsi Ego Hartmannus Schedel artium & utriusque medicine doctor Anno domini MCCCCCI In Nuremberga. Laus Deo. add. colore rubro M