UC BERKELEY MASTER NEGATIVE STORAGE NUMBER ”0-3.1 (National version of master negative storage number: CU SNOOO30.1) MICROFILMED 2000 UNIVERSITY OF CALIFORNIA AT BERKELEY LIBRARY PHOTOGRAPHIC SERVICE REPRODUCTION AVAILABLE THROUGH INTERLIBRARY LOAN OFFICE MAIN LIBRARY UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720-6000 COPYRIGHT The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted materials including foreign works under certain conditions. In addition, the United States extends protection to foreign works by means of various international conventions, bilateral agreements, and proclamations. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or reproduction is not to be "used for any purpose other than private study, scholarship, or research." If a user makes a request for, or later uses, a photocopy or reproduction for purposes in excess of ”fair use," that user may be liable for copyright infringement. University of California at Berkeley reserves the right to refuse to accept a copying order if, in its judgment, fulfillment of the order would involve violation of copyright law. Adam, James The nuptial number of Plato: its solution and significance London 1891 BIBLIOGRAPHIC RECORD TARGET University of California at Berkeley Library Master negative storage number: 00-30.1 (national version of the master negative storage number: CU SNOOO30.1) FORMAT: BK LEVEL: r ISBN: GLADIS#: 50397819C LANG: eng CNTRY: enk LCCN: MOD: 000116/MO ME: Adam, James, 1860-1907. TI: The nuptial number of Plato: its solution and significance lM: London, C. J. Clay, Cambridge University Press Warehouse, 1891 co: 79 p. illus. 24 cm CALL: B398.N8.A4 Microfilmed by University of California Library Photographic Service, Berkeley, CA FILMED AND PROCESSED BY LIBRARY PHOTOGRAPHIC SERVICE, UNIVERSITY OF CALIFORNIA, BERKELEY, 94720 DATE: 5/00 REDUCTION RATIO: 8 a PM-1 31/2"x4" PHOTOGRAPHIC MICROCOPY TARGET NBS 1010a ANSI/ ISO #2 EQUIVALENT W3 E 2-5 ” LO .5. mug "mg L7 "I"; "HIE .. mug: E mg mug H E u E wuu I” — ”m |.8 "NILE“ "I"; .A /\// \ €57 .3 . re» a, 0/9 00 <3 ’5“ 0 THE NUPTIAL NUMBER OF PLATO: ITS SOLUTION AND SIGNIFICANCE. #61 5‘1 0 3101111011: O. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE AVE MARIA LANE. CAMBRIDGE: DEIGHTON, BELL AND CO. _ LEIPZIG: F. A. BROCKHAUS. NEW YORK: MACMILLAN AND,CO. wwwygml ,5» ' #mbww-m / as“ , {Tw’x ‘vr [3] THE NUPTIAL NUMBER OF PLATO: ITS SOLUTION AND SIGNIFICANCE. BY JAMES ADAM, MA, FELLOW AND TUTOR 0F EMMANUEL COLLEGE, CAMBRIDGE. “ Every divisor is a. gift of God.”——PLAT0. C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE. 1891. [A ll Rights reserved] (bl/\OTTAATQZI CDIAOTTAATQN. Qtamhringez PRINTED BY 0. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS. 47570 Who laid the corner stone thereof, when the morning stars sang together, and all the sons of God shouted for joy? Or who shut up the sea with doors, when it brake forth as if it had issued out of the womb ; when I made the cloud the garment thereof, and thick darkness a swaddling band for it, and prescribed for it my decree, and set bars and doors, and said “Hitherto shalt thou come, but no further: and here shall thy proud waves be stayed? ” Job xxxviii 6—11. The World’s great age begins anew, The golden years return, The Earth doth like a snake renew Her winter weeds outworn; Heaven smiles, and faiths and empires gleam Like wrecks of a dissolving dream. SHELLEY: Hellas. PREFACE. THE present essay claims to be a complete solution of the Number of Plato. 'If the results at which I have arrived are correct, the Number is not merely in itself a simple and elegant mathe- matical problem, but forms an organic and essential part of the argument of the Republic, and furnishes us with the right point of View from which to study the cosmology of the Timaeus. It is also full of interest for students of theology, as well as of ancient astronomy, embryology, and music. The Muses in the Nuptial Number connect the creation of the human child with that of the divine, and measure the lifetime of the World; and the splendid harmonies of sound and style are worthy of the Muses and their theme. I desire to thank Mr Neil of Pembroke College for reading through my proof-sheets and making various criticisms and suggestions. EMMANUEL COLLEGE, October 30, 1891. Msmwwwwmm sum». ., a . , “an w THE NUPTIAL NUMBER. "‘P'I'opos'ltum meum est cum Platonic in octowo de Rep. locum ewpouere, qu'i cum omnium, quae sub kumanum cognitionem cadere possum, obscurissimus sit: hactenus a nemine (quod pace omm'um dicam) non mode non 'recte declaratus sed uequldem perceptus est: win—w trétum dude proverbium dpud Autiquos ortum fuz't, quod uempe numerls Platonicis nihll obscurlus.” INTRODUCTION. THE words above quoted were written by Barocius in the preface to his attempted solution of the Nuptial Number in the year 1566. I choose them to introduce my own solution, because they are almost as true in 1891 as when Barocius wrote them. “Almost,” I say, not “altogether,” for the nature of the problem has been grasped (perceptus) by Hultsch, and to some extent by Tannery and Dupuisl, though it has by no means been solved, as I think they will themselves allow. Human nature is prone to regard the Unsolved as the In- soluble: and so it comes to pass that while there are some who think the secret is lost for ever, others remind us that “the Muses are speaking in jest,” and that “when Plato is in the mind for a mathematical joke, he is not to be taken too strictly au pied de la lettre.” Perhaps: but to Plato mathematics were 1 I wish I could say the same of 10: 1rv0pnfiv: and I owe him no small Dr Grow, who has written on this sub- gratitude for the “base four-three.” ject in the Journal of Philology (x11 Mr Monro’s article in the same Journal pp. 91—102): but he has altogether (vm pp. 275—289) is cautious, and failed to appreciate the conditions of puts the points well, but makes no the problem. The one phrase which advance on previous discussions of he has very nearly explained is é1rl'rpL- the subject. WW j: 1.." n; 10 THE NUPTIAL NUMBER. no joke, but the corner-stone of his Academy: and duovaos‘ is the man who thinks the Muses joked with difficulty. If by obscurity is meant want of clearness, whether in- tentional or otherwise, then the reputation of the Nuptial Number for obscurity is undeserved: it is extremely difficult in its language, but not obscure. Cicero’s unfortunate numero Platonis obscurius, by passing into a proverb, has had the most injurious effect upon the criticism of the Number. Writers have for the most part vied with one another in maintaining the credit of the proverb: it is hardly overstating the case to say that it is more difficult to master the theories of others, than to discover the truth for oneself. But Cicero is no more an oracle than Plato is oracular: and we have learnt many things about Plato that Cicero never dreamed of in his philosophy. Some of the later Greek writers understood the Number, in part, at all events, as I shall shew ; and Aristotle treats it quite as an ordinary piece of mathematics‘. Had Aristotle thought the Number obscure, does any one think he would have omitted to say so? It is not his way to let “dear Plato ” off so easily. The fact that Aristotle found no difficulty should have warned all critics to look for the difliculties in the Greek, and not in the mathematics, but instead of this they have played games 1 “Nicht bloss dem Aristoteles sind scurity “greater than that of the Sibyl’s diese Worte noch vollkommen klar, leaves”: while Ficinus declared that sondern auch Nikomachos, Plutarchos, Iamblichos, Proklos, und Aristeides Quintilianus sind noch nicht im min- desten zweifelhaft iiber ihre Bedeut- ung” Susemihl, Aristoteles’ Politik II p. 370. Compare Schneider III p. ii. I have little doubt that the passage was perfectly intelligible to Aristotle and Aristides at least, and possibly to Iamblichus; but I do not think Nicomachus or Plutarch understood it fully, and Proclus, I fear, did not : see note on p. 11. Theo Smyrnaeus, in his Expositio rerum mathematicarum ad legendum Platonem utilium, dis- creetly embraces the Number from afar. Melancthon groaned over an ob- nothing short of “Apollinis vatici- nium” could make the Number out; and Schleiermacher (as Dupuis in- forms us) interrupted his translation of Plato for no less than twelve years in the vain hope of finding the right solution. The “ smart and facetious Thracian maid” who scoffed at Thales when he fell into the well obs Ta ue‘v éu oripavq'i 1rp00u,uoi‘ro eiéév (11., rd 6’ 5p.- 1rpoa'06v aii'rofi xal 7rapa 1r65as havfio’woc azirc'w (Theaet. 174 A) might well have said the same of many a writer on the Nuptial Number, ets ¢péar¢i re Kal radar droplet]! e'mrhrroura (ibid. 174 0). INTRODUCTION. with numbers, prostrating themselves before their Sphinx with indiscriminate offerings of multiplication, addition, squaring, cubing, extracting the square and cube roots, till she has turned away her head in very shame, being in truth no Sphinx, but a self-respecting Muse. Where doubt is possible, I will elucidate the meaning of each word from Plato’s use of the word else- where, except in the case of two words, for which I will cite parallels in Greek writers on mathematics. And when the arithmetic is stripped of its Greek dress, it will be seen that there is nothing in it to alarm “a gentleman and a free- man,” who “may be expected to know as much arithmetic as an Egyptian child.” « When reference is made to the following writers, it is, unless otherwise stated, to those of their works whose titles are here printed after their names. Schneider (Platonis Opera Graece, 111 pp. ii—lxxxii): Monro (Journal of Philology VIII pp. 275—289): Gow (Journal of Philology XII p. 91 H.) : Hultsch (Zeitschrift fur Mathematih u-ncl Physik XXVII, Historisch-literarische Abtheilung, pp. 41—60). I have read besides: Donaldson in the Proceedings of the Philological Society Vol. 1p. 81 ff.: Martin in the Revue Arche- ologique XIII p. 257 ff: Dupuis (Le Nombre Ge’ome’trique ole Platon, Interpretation N ouvelle, Paris 1881 and Seconde Inter- pretation, Paris 1882): Tannery in the Revue Philosophique I p. 170 ff., XIII p. 210 if, XV pp. 567 ff.: Gow in the Academy no. 522: Demme in the Zeitschriftfur Math. uncl Phys. XXXII, Historisch—literarische Abth. pp. 81—99 and 121—132: Hultsch ole numero Platonis a Proclo enarrato disputatio in Schoell’s Procli commentariorum in remp. Platonis partes ineditae1 (1886) 1 Not the least interesting point in connexion with the criticism of the Number is the publication for the first time in 1886 of Proclus' commentary upon the passage. It is disappointing to find that Proclus does not explain the meaning of the words, but wanders ofi‘ into astrological and other vagaries: “ freilich findet sich ” says Hultsch “ bei Proklos keine direkte Erlauterung der dunkeln und vieldeutigen Worte Pla- / tons; allein immerhin ist es ein Gre- winn zu betrachten, dass die‘meisten der von Platon gebrauchten schwieri- gen Ausdriicke vgn dem Neuplatoniker wiederholt un {in den Bereich seiner! eigenen Spekulatlon aufgenommen wor- den sind.”} It is clear that Proclus either would not or could not explain the passage: when he commits him- self to a definite view, he is for the most part demonstrably wrong] , adage. :4.sz manage «« lizauirsaaramaswsmeax . ,A . , 12 THE NUPTIAL NUMBER. pp. 140—148: J owett in The Republic of Plato translated, 1888, p. cxxx ff.‘ I have refrained (except where absolutely necessary) from discussing the views of these writers, because it does not seem necessary to refute them in order to prove my own View, most of their solutions being confessedly more or less tentative. It will be my duty to discuss them when they claim to be right. To the scholarly insight of Schneider as well as to his careful collection of passages from the later Greek authors, I owe most. Monro’s article first introduced me to the subject. From Gow I obtained the key to the meaning of one difficult phrase. There are valuable suggestions in several of the articles which I have read, and especially in that by Hultsch, my relation to whom will be more fully explained as we proceed. ‘ THE TEXT, AND PLAN OF THE DISCUSSION. FOR the sake of convenience I quote the passage (Rep. VIII 5435 0—547 A) in full. I also divide the words with which we are more immediately concerned into five sections, A, B, C, D, E, to facilitate reference. I I 9 (Pepe 'rowvv, 171/ 8’ e’vyai, wetpailtega ke’vyew, Tlva Tpéwov I 9 a A Ttpoxparla ryeuow’ all (if dpta'roxpa'rlas. 7) 7-636 péu dwkovv, (I A I I , , A A )I \ , I 0'” 7ra0'a. wom'reta ueTaBahhe‘t sf au'rov 70v exov'ros 'ras apxae, I A bray e’u ail-re") 7013790 onto-(,9 e’tyvyévn'rat- dam/001111709 36', mil! I 9 9 A A warm oMryov :r, d315va'rov Icw'nHm/at ; "Eon rydp 0570). Han ’ I 9 A cup 317, swap, (3 Thad/cow, 6 77'6th wimp Icwnflfiae'rat, [cal wfi I - a'Taa'taO'ova'w 05 e’7rllcovp0t [cal 05 dpxou‘res 7rp69 dkkfikovs T6 \ \ Q I ‘\ I ’I (I I ’ 5 Kat 7rpos‘ eav'rove ; 17 Bovkeo, (00'7er Omypos, evxw/Leea rate I a A A A Mova'ats' ewrew fipw, 371109 31) 7rpc07'ou anions gawea'e, Ical A $ \ A A A Gamay“ av'ras‘ 'rpa'yuccos‘, (59 wpds‘ watb‘as‘ 75/1419 waté’ozia'ae [cat I A e’pea'xnkova'as‘, 059 31) 0-770sz; Nevada-a9 dxknkoko'yovpévae Xé- A Q‘ l A ryew ; Hum; 036 77(09' xakewdu [téu Kaunanvat wéhw 05700 A ’ Eva-Taaau- ant, e’7rel ryevopévcp Tram-l (pdopd early, 0133’ 1; I I \ (I A I , \ I Totav'm Eva-Tame 7'01! anal/Ta [LGVEt xpovou, akka kvdnae'rat' M5019 3c 7386. 013 ,aévov gbv'rols‘ e’ryryelots‘, dud [cal e’v e’m'yelom 1 Some account of the enormous, in Schneider and in Dupuis (Interpre- literature of the Number will be found tation Nouvelle). INTRODUCTION. 13 {95009 (papal Kai 0Z¢0pla \[rvxfis 76 [cal awpd'rwv rylryuov'ral, bray 'n'ept'rpovrai exam-0L9 m’ucMov wept¢opd9 Ewan-Twat, Bpavalote Iaév Bpaxv'n'épove, évav'rlms 3e e’uawlas' oyévovs 8e dpe'répov stiryovlas 're Icai dcfioplas‘, Kal'rrep b'v'res aodml, oi)? fiveaévas 71-6).er e’qratb‘edaaaae, 0:333:11 ”,5th Kayla-p.53 ,ue-r’ ale-9750mm Tedfov'ral, ciMwi wdpela'w afirofiq [cal ryevvfiaovat 'n'ai‘b‘a’s were 013 Béov. I é'a'q't 3% gelcp ,aév 76111117793 weptoSos‘, '31; a’ptdudq wetha/L- A Ba’wet Téhetos, I dvdpwvret’tp 33: 6’11 9') wpai'rrp afifrfa'ets vadlteval B 76 Kai. Suvaa'revé/Levat, 'rpeZs‘ dwoa'rda'ets‘, 'ré'r'rapas‘ 5?: Spam; hafiofiaat ripotoziu'rcov 76 [cal (ivopotodv'rwv [cal ad‘g’éwwv Kai ¢9w6wwv, wall/Ta wpoafiryopa [cal [Sn—rd 727069 c’z'Mmka dwétpnvav‘I (51/ évrlrpw'os‘ rue/uh 7T6/L71'é86 a'vé’v'yeis 8130 dppom’as wapéxe'ral. C 'rpls‘ afifndelg‘, I 71)!) ,aév lam! teams, élca'rov Too-avTa/cts‘, | T7)” 83: D, E Zoe/Mix” psi) 73'}, 7rpomfim) Sé, Exa'rdu pév dptdpciiv rim-d Btapé'rpwu fmm’iv wean-d309, Bee/tévcou eves é/caa'rwv, dppn’rcou 3e 3u02vb,baue2u mss. élca'n‘w 8e xfifiwv Tptd309.| flaw-ac 3% 015709, dptdpde «yew/1.67m- K69, Towdrov nuptos‘, duewéku 76 [cal xetpéuwv ryevéaewv, ('29 liq-av a’ryuofiaau'res dplv of (balance avvomlé’waw wimpas yup,- dnfoos‘ wapd. Katpév, 0131c eddwels 0135’ eJ'rverS‘ walSee é'aom'at. (5v Ica'raa'rfiaov'rat pév '1'on aplo'rovs‘ 05 wpdrepoa, 3/1409 3?: 311769 dudftoo, 659 70k 1131/ 'n'wrépwv ad SUI/(541,618 e’hfléwee, 15,1431} 'n'péi'rov (ilpfom'at apekelv (békalces 311769, 7rap’ é’xa'r'rou 700 Séou'ros firmed/1.61101, Ta Movamfis‘ 36157613011 33: n2 'yvavam-ucfis' $0611 dpovaé'repm ryevriaov'rat 15/1131) 05 uéot. Ex 33: 7015va dip- Xov'res‘ 013 wdvv (fivkalcucoi Ica'raa'rfiaov'ral 71'de 7'6 Soxtpdgew Tel tHa'té'o‘ov ‘TE Kai Ta crap; 15/1/51! yew), xpvaofiv Te Kai. a'pryvpofiu Ical. xaNcofiv [cal ULSnpofiu, o'aofi 8e ptryév'ros‘ atdnpoi} dpryvpc‘?) [cal an/cofi xpvag?) dvopotéms‘ éryvyevfia'e'rat [cal depahla oil/appea— 7'09, (‘2 ryevépeva, 013 Zip éryryévn'rat, dei Tl/c'rel fiche/4.011 xai é’xfipau. Tad'ms 'TOt ryevei'is xpfi (palm elvat ardaw 371-011 (31/ vyllymrrat a’ei. The plan of my discussion is as follows: Part 1. The Solution of the Nuptial Number. Chapter I. Preliminary remarks on the Pythagorean tri- angle‘. Chapter II. The words from dudpwwelgo 3%. to élca'rc‘w 8?: xéBaw TptdSas‘. 1 It will be noticed that I postpone oetou 700077-61: to Part 2. The reason the discussion of the reploaos of the will appear later. 14 THE NUPTIAL NUMBER. The numbers will be discovered by explaining :—(1) sections C, D, E, from a’w ém’rpwos‘ to TpLdSOS‘, the sections being taken in the reverse order, viz.: E, D, C: (2) section B, from a’vflpw- area'cp 8% t0 a’méqbnvav. Part 2. The Significance of the N uptial Number. Chapter I. The translation. Chapter II. The significance of the Numbers both of the “Divine and Human Creatures.” PART 1. THE SOLUTION OF THE NUPTIAL NUMBER. CHAPTER I. THE PYTHAGOREAN TRIANGLE. IT is no part of my purpose to give an account of the Pythagorean trianglel—the right—angled triangle, whose sides are 3 and 4, whose hypotenuse is consequently2 5, and whose r""'---'------------'-‘-“f;l area is 4 3 3 = 6 (see Fig. i)3. But it is necessary to mention the evidence which , there is for believing that Plato made 3 use of this triangle in his N uptial Number. I will afterwards shew how Aristotle himself implies beyond dispute that the Pythagorean triangle was the basis of the number in the section which I have called B. Besides Aristotle we have the testimony of at least three authors (cited by Schneider‘), viz. Plutarch (ole Is. at Os. p. 373 F), where he says 7:81! Tpuyaivwv «rt xtha‘Tov—g') [cal de'rwu 6’11 733 wokorelt‘z 30x62 wpoa/cexpfioflat, 7'6 yapfiktou Selig/pay.- 1w, a'vv'rd'r'rwu. é'xel. Bé e’xei‘uo 76 Tpt’rywuov 'rpm'iv 'rfiu 7rp69 dpeds, Icai, TeTpoaw Tfiv Bda'w Ical 7réu're will fiwoveo’vovaav 1 See Grow History of Greek Mathe- 3 ABC, the right-angled triangle in matics, p. 155. which A023, and AB: 4, is half the 2 By Euclid I 47—3. proposition 4x3 1 BDG, t —= . which is said to have been discovered rectang e A i' e. 1 is 2 6 by Pythagoras, and which was cer- 4 p. xxxii. Compare also Iambli- tainly familiar to Plato: see Tim. 54 B. chus, Vit. Pythag. § 131. 16 THE NUPTIAL NUMBER. i’a'ou 'ra'iq weptexozio-atc Suvape’unv: Proclus (in Euclid. p. 428, Teubner edition) in these words: 7-?) 6’1) WOXL'TGlfq, Tpirywvov, or? 71):; dpefiv wepte'xovaw 3 'TE TpL'a. (so. dpcflpds‘) [cat 6 'réa'a'apa: and Aristides Quintilianus, who (De Musica, ed. Meibom, p. 152) remarks: a5 8% will 6p0fiv weptéxovaao 3777\0001. 761/ e’7rl'rpo'rov. Tori-ray 81) Kai, Ilka'xrcov gbna'iv éwirpo'rov wvfipéva wep- mib‘i avé‘vyév'ra. We are therefore prepared to meet with the Pythagorean triangle—in whole and in part—its area and its sides—throughout the N uptial Number. CHAPTER II. THE WORDS FROM duflpw'lret'tp Bé TO élca'rdv 8% 1615,80»! TpLaiSOS‘. ACCORDING to my plan, I will discover the numbers by ex- plaining (1) E, D, C, and (2) B. We begin with E, viz. the words will 3% Zaopnf/cn ,uév Ti}, wpopfim; Bé, é/cwrov ,uév (ZpLH/ua'iu dwo Sta/.LéprV [Sm-(511 wean-(2309, Sequel/coy évds‘ élcdorwv, cip- p17er 36 Such, e/ca-rov 36 Ichwu Tptao‘os‘. I will work backwards, beginning with the clause é/ca'rou Be Ichwv TpLaSOS‘. These words mean “of, on the other hand, one hundred cubes of three ” = 100 x 33 = 2790. Next comes the clause: é/caxrdv ,uév a’ptflud‘w a’vro Siapé'rpwv [5717611 wepwddos‘, Seopévwv 6'de élcaiarwv, (ippfi'rwv 5% 3120211. “Hoc loco,” says Schneider‘, “maxima gratia debetur Ba— rocio, qui primus verborum obscurissimorum sensum perspexit et aperuit ...... Quamquam ne ipse quidem portum cepit, quum in aliis partibus ofl‘endisset, hunc quidem scopulum superavit felicissime.” There is no doubt whatever about the meaning, and the interpretation which I shall give is accepted by every scholarly critic with the single and most unfortunate exception of Hultschz. 1 p. xxv. Schneider (pp. iii—xviii) for rejecting the usual interpretation gives a very full account of Barocius’ of these words. If he had only in- tract, which is now exceedingly rare. terpreted this part rightly, I think it There is a copy of it in the British not unlikely that he would have solved Museum. The Italian form of Baro- the number: as it is, his article is a cius’ name is Barozzi. striking confirmation of the old proverb 2 p. 48. Hultsch assigns no reason dpx7‘7 1% 76 17mm; 1rav16s. See p. 39. A. 2 THE NUPTIAL NUMBER. The “rational diameter of five’ number to the real diameter of a square 1 is the nearest rational FIG. ii. whose side is five‘, i.e. to J50 by Euclid I 47 (see Fig. ii). Now the nearest rational number to JR) is 7 =~/4_9. Therefore [aural drape-rpm. wean-d8“ means simply “sevens.” dpiflpol (in-6 means “numbers representing the area of squares constructed upon” (0277-6 B = “from,” so. as a base), or in geometrical language “squares of.” This use of (in-6 is regular among Greek mathematical writersg, and is also found in Plato, e.g. Meno 85 B 0271'?) 7139 Sta/iérpov c711, 059 0'0 (fifis‘, (5 7m? Méuwvoc, I 3 «A \ I I vyuyuow av ‘TO dtwkaa'iou XCOPLOV. The words é/ca'rou ,aéu (iptdacfiv 0271'?) Slaaé'rpwu [50770311 relu- 7rd809 therefore mean3 “of, on the one hand, one hundred squares of sevens,” i.e. 100 x 72 = 4900. It remains to explain Seam-Evan; eyes é/cdarcov, and dppfiq-wu 8e Buoiv. The first means simply “ each a’padao’s- or square wanting 1,” i.e. minus 1. The logical expression would be Seoae’uov éudq 1 The evidence for this is Theo Smyrnaeus, p. 43 fl. Cf. Gow, Gk. Math. p. 96. This is (I think) the only passage in Plato where “rational diameters” are mentioned. In Theaetetus 147 D ff. he merely distinguishes rational from irrational roots or surds: while a care- ful study of Polit. 266 A—B shews that the passage is in no way parallel E D to ours except in the use of Btduerpos. But there is nothing harsh or difl‘i- cult, from the Greek point of View, in the expression “rational diameter.” The geometrical construction is very simple. Let AB=J50, i.e. the irra- tional diameter of 5, and consequently ABDE=the square of (oi-Ire) A3250; we have only to insert in ABDE the largest square of a rational number which it will contain, say AFH G, and AF will be the rational diameter of 5, i.e. that part of the diameter of 5 which is rational. 2 See Euclid, passim. _3 It may be asked, Why did not Plato simply say “of 4900,” and above “of 2700”? Why express himself in this apparently cumbrous fashion? In order, of course, to express the numbers in terms of sides of the Pythagorean triangle, 5 and 3. The number 100 is also significant, as we shall see. Compare p. 71. THE SOLUTION OF THE NUPTIAL NUMBER. 19 éICdO'TOU, but this would be cumbrous and obscure, and Plato is neither. .He therefore writes as he would speak, Seoaévwv ($1269 endemu, letting é/cdo-rov assimilate itself to olptflha'iv. We can now interpret the whole expression é/cardv aév ciptgya'iv Lin-6 Sta/iépru fnrrcSV henna/509, Seoaévwv «51/69 é/ca’c'rwu. It is equivalent to (72 x 100) — (1 x 100) = 4900 —- 100 = 4800. dppnraw 8% 8vo'iu is superfluous—the number could be made out if—e’o 76 min e’mrzfawq—the two words had been left un— spoken. They merely give another way of arriving at 4800. The translation is: “or, if you take irrational diameters of 5, wanting 2 each.” The construction is dppé'rwv 8é <31apé'rpwv 360péuwv> Bvoiu <é/cda'rcov>. The meaning, ex« pressed in figures, is simply this: “or, if you prefer it, of (7%)? x 100 — (2 x 100) = 5000 — 200 = 4800.” Those who know ancient Greek as a living language will read the sentence out aloud, laying the proper emphasis on é/ca'réu, [3777612, éVéC, dppfirwu, Bvoi‘u, and pronouncing the words dppn'raw and Bquu slowly and distinctly, and the meaning will at once appear. 86' gives an alternative, as in name.» 8é=vel potius. Plato in— serted the words a’ppfi'rwv 8% 81/0212 for the benefit of those who might be puzzled by the somewhat technical, and, in his day, perhaps novel expression, “rational diameter”: so little did he wish to be obscure. So far we have reached1 two numbers, viz. 4800 and 2700. It remains to see what use is to be made of them. That they stand to one another in some kind of contrasting relation is clear from the [.téu and 86/ in é/ca'rdv ,u.e'v and é/ca'rdv 36’. The words 71):; 3e Zo-omfxn ,aév 73'), wpoan/cn 86’ make clear the nature of the relation. They denote, as Hultsch2 has seen, a rect- 1 Proclus understood that these unusual order throws emphasis on 75, two numbers were 4800 and 2700, as appears from Schoell’s Procli partes ineditae p. 25, 21 ff. But he wanders far astray very soon, and does not try to explain how the two numbers can be got out of Plato’s words. 2 p. 46. Hultsch changes 1;? into my (sic), wrongly, I think, for £77 is not sufficiently precise. The somewhat and serves to bring idoflfikfl and 1rpo- Mm; into sharp contrast The sense would be complete if Plato had written only Thu 66‘ npoafilcn, but lac/min] ,ue‘v Ti} connects the rectangle closely with the square, which it is worth while to do, because their areas, as we shall see, are the same. 2—2 20 THE NUPTIAL NUMBER. angle—the figure which is “ in one direction equal in its lengths, while in another it is not.” “This rectangle” (says Plato, literally translated) is, “on the Fm. iii. one side, of 4800, on the other, of 2700,” i.e. its adjacent sides are 4800 A and 2700. The area is therefore 0 4800 x 2700 = 12,960,000 ff (see Fig. iii). ‘” One question remains, viz., why is D c, the rectangle called a nip/Layla? Fully to answer this now would lead me into the part of my discussion dealing with the Significance of the Number. But the mathe- matical meaning may well be explained at the present stage. The rectangle is a dppom’a because the two sides, which enclose and make it, are derived by Plato from a certain ratio by means of an 5067979 ryewperpmn—the kind of 5067779 which Icat ($1) 06029 Kai. e’v dvapai'n'om [Lérya Eduaratl. Thus: the two sides are 4800 and 2700, and Plato derives them from the ratio 48 :27, by means of the Zaé'rns‘ ryewperpucfi 4800 : 2700 = 48 : 27, in other words, by multiplying both terms of the ratio 48 : 27 by 100. . Armed with the number which we have just wrested from E, we next proceed to encounter D viz. the words 71):! ,uéu 1/0071! fad/cw, Exam-611 'z'oa'av'rd/cu)‘.~ But it is necessary, as a preliminary step, to interpret the word wapéxe'raa. Let us for brevity call the subject of wapéxerat the number as, then “ac provides us with two cip/tom’at” may mean “the number or may be made to assume either of two shapes, each of which is a dpnom'a.” Of course wape’xe'ral. may also mean as furnishes us with two tippom’aa, of which it is itself the sumzz which of these two meanings it bears here, (“375 Beifea. We will assume till we are undeceived that wape’xe-rat bears 4800 THE SOLUTION OF THE NUPTIAL NUMBER. 21 the former meaning‘. One of the two shapes which as assumes we have seen to be a rectangle of 2700 X 4800, and thus we have discovered the value of w to be (2700 x 4800) = 12,960,000. We have therefore to see whether this number will assume the shape of a second rip/tomb, described in the words 7-1511 ,uéu i’a'nv Za'd/cts‘, Ema/1'61! TOO'aUTd/CLS‘. i’an Zed/cw can only mean a square, as (I think) every writer except Grow2 admits. The expression is a regular one in Plato: compare Theaet. 147 E 761/ pév Smut/1.611011 i’aov iodine 'ylf'yvea'fiat 1'93 're'rpwyaivgo 7'6 axfina (insomnia-auras 're- Tpdrywvéu TE [cal Zaéwkevpou wpoaelvropeu: i.e. the number which is able to be made equal an equal number of times is a square, e.g. 36, which can be made 6 six times. The words éxardv TOO'aU'TulICLS‘ mean literally “ so many times 100”: this harmony is therefore a number which can be made “equal an equal number of times, viz. so many times” (i.e. a certain number of times") “ 100.” But how many times 100 ? Call the number of times .90, and the harmony may be expressed as 10006 x 100w, i.e. (100 x at)”, exactly as 36 = 6 x 6. Will then 12,960,000 assume the form of “(100 x (6)2”? It resolves itself at once into (100 x 36)”, i.e. 36002 (Fig. iv). As in the case of the rectangle 2700 x 4800, so here, I must postpone my final answer to the question, “Why 3600 is 36002 a épnom’a?” For the present it is enough to note that when Plato says that the sides of the square are D 0 each “one hundred multiplied by :0,” he is deriving them from the ratio wzw by means of the yew/Lemmy) 5067719 100a: : 100m=w : x, in other words, by multi- plying each term of the ratio a: : w by 100. The parallel with the rectangle is thus complete: in the one the 3067079 ryew- FIG. iv. 3600 1 Gary. 508 A, where see Thompson’s note. a:b::c:danda:b2:b:care both examples of 'yewuev-pm'r‘; 1061-2”: the former Aristotle calls duaho‘yla diminuéun, the latter drake—yin a'vuexns. See Eth. Nic. v 6. 1131a 31 fl. and cf. Nicom. Inst. Arithm. II p. 141 ch. 24 (Ast). 2 I think the use of the middle is somewhat against this view: cf. Phaed. 110 C. 1 So also Hultsch, p. 46. Of. infra, p. 40. 2 p. 94. Gow actually thinks that the words can denote a. cube. 3 With this use of Toaav'rdms cf. Phaedrus 271 D dvoi'yxn eldévm tpvxn b'cra. ei’dn é‘xet. é‘o‘rw 0171/ 76011. Kai Tcia'a. Kai 1020. Kai 102a. and Laws IV 721 D {n/uofia'fiw ,uéu Ka'r’ émau-rbv 7'6ng Kai 7609). The antecedent is in reality the unknown quantity. \ 22 THE NUPTIAL NUMBER. ,ue'rpucfi involved is 4800 : 2700 :: 448': 27; in the other, since w = 36, it is 3600 : 3600 :: 36 :36. E and D have yielded us their secret and we rise to C— _ the words (51/ 6’7ri7'pb7'09 7rv0,un)z/ wep'n'a'Sl, a'vé’v'yeis‘ 3150 rip/Louie»? wapéxe'rai '7'th 111357719609. These words are somewhat more diffi— cult than anything in D or E, but they can be interpreted with absolute certainty. Let x (as before) represent the whole Subject of wapéxe'raiz Plato says that a? wapéxe'mi 5150 dppovias‘. N ow we have found that a: is 12,960,000: and the whole subject of wapéxe'rai is (51/ é'n'i'rpi'roq wvfimiu wepwddi avé’vyeis '7'th aflfnflslqi these words therefore must mean 12,960,000. Let us take them singly. c511=“0f which” has for its antecedent the word 1113500619 in B. In order not to anticipate my inter- pretation of the words in B, I will ask the reader to believe now, what I shall prove afterwards (p. 33 ff), that the process denoted by av’ffiaeoc gives us~the elements Which make up the number contained in section B, and that we may therefore, for practical purposes, regard that number as the antecedent of 151/. It is therefore necessary to discover that number before we can interpret 151/. For the reason which I have stated, and also for another which will presently appear, I will not now wring the secret out of Plato’s words, but from Aristotle. In the Politics (V 12. 1316“) Aristotle criticises Plato’s account of the transition from the best to the second-best .woktreia in these words—6’11 3% 733 71'0M'reic‘t he’rye'rat aéu 7T€pt 7031/ ,ue'rafioka'iu 1571-0 701') Zwlcpd'rovs‘, 013 #45va kérye'rai xaMSs‘ 7139 TS ydp dpi’o'rm 7TO>\.LT€L/a.9 Kat WpalTflS‘ 01’2'0'179 013 kéryel. 71):) perafiokfiu 551109. (1)1702 rydp ai’nou Gil/at 7'0 ,uxr) ,uéuew ,unaév d7t7t’ é‘lll 'TlJ/L wepiddrp ,ue'rafldhkew, dpxfiu 3’ Given. 7015er c511 é7ri'rp1'ros 7116,1101) wean-(1'81 avé’vryets‘ 3150 cip- pouias‘ wapéxeraz, )térywu 0711.11 15 +013 Starypa'ppa'roe Lipid/1.09 70111-00 veil/77711.1. arepeo’s‘, (59 1'69 (#1506169 7076 (in/0150179 ¢a15>tovs~ Kat eri'r'rovs‘ 7139 watBeias‘. 7-0070 péu 0131/ mini Kéywu 70109 01} Icalca'ie' e’vo‘éxe'rai rydp eivai Twas 039 watSevgfiuaL Kai, 761/150-0011. awovdar’ovc iii/Spas d5151/a7'0u. 05A)», as???) n’ 521/ {3109 61'?) [1.67018on) 71'7‘9 157r’ e’lceivov keyope’vm dpi— 07-179 “IroMTeL'as‘ ,uc'ikkou i} 7181/ d’kkwu wad-c811 Kat 7-0311 ryuyvo- ,uéuaw 7112127101}; This is not the place to discuss the relevancy THE SOLUTION OF THE NUPTIAL NUMBER. 23 of Aristotle’s criticism as a Whole: I will only remark, for the sake of those who think Plato is only fooling, that Aristotle not only takes him seriously, but admits that the number of Plato actually does give a correct reason for the change of constitutions in general. Plato, he says, ought to have given the specific reason for the change from the best to the second- best constitution in particular. This will help us to discover, when we come to Part 2, what facts or supposed facts in life Plato meant his number to represent. Meantime the words that concern us now are those which I have printed in spaced type, viz. from (1)770}. rydp to rye'unral. arepeés‘. They have been grievously misunderstood by Monro and Gow, who think they refer, not to the section which I call B, but to the subsequent sections. Gow, indeed, seems to have felt prickings of heart when he wrote the following: “The word 031/ seems to me hardly capable of translation in Aristotle and is retained only to identify the quotation. In Plato of , - course it is very important‘.” The correct translation is this: “for he says the fact that nothing abides but everything changes Within a certain period, is the cause” (sc. of change from Aristocracy to the Laconic state), “while the: beginning (so. of change) “is of“ (i.e. belongs to, comes from) “those things” (i.e. the adfn’aets‘), “the 4 : 3 base of which, yoked with 5, furnishes two harmonies—meaning, when the number of this diagram is made solid.” That is, the beginning of change is the number which is virtually the antecedent of 13v, and that number is the number which you obtain by making the “number of this diagram solid.” Now what is “this” diagram? The antecedent of 7015700 is e’m’rpwoe mama: and e’7r15'rprros‘ 71-1/0- ;uiv is the ratio 4: : 3, considered as a base of something—in this case of the number contained in B. Virtually, therefore, Aristotle 1 p. 93, note 1. 2 I first explained the words dpxfiu 6’ elmu 7'01?er (.511 as =dpx’hv 5’ 611/011 Toivrwu (so. change, or the influences which make for change) c311 _(i.e. those things, viz. the aflg'aets of which 810.). On looking at Schneider, ' who alone of the writers whom I have read, interprets Aristotle aright, I found that he definitely rejected the construction which I thought right. The meaning is the same in either case; but though I now agree with Schneider, I think it worth while to state my earlier View, in case it should better satisfy the ear of some readers. TIIE NUPTIAL NUMBER. calls e’vrl'rpuroq (wvfimiv) a diagram. Put e’vrl'rprros‘ in the form of a diagram. Let A0 = 4: and AB = 3 be at right angles FIG. v. m -————--__ - - to one another: join B0 and your diagram is complete. Then 302:32+42=9+16=25; 30:5. 7'6 701'} Hvdwyépov‘ efipn/ca. It is pleasing to welcome his triangle again. The area or number of this triangle, as we have seen, is 6; make it solid, as Aristotle bids us, i.e. cube it, and 63:216. I have thus shewn from Aristotle, without making any use of Plato, that the number contained in B is 63 = 216. At last we are in a position to interpret (312. The word means “of 63.” N CW 63: 33+43+53, and we shall presently see that Plato himself expresses the number of B as the sum of the cubes of three numbers. What then is the ém’rpuros‘ 7rv0,un§v of 33+43+ 53? I quote from Gow‘, who has come very near to the true explanation of this phrase. “The most common mathematical use of wvdpfiv is in the sense of ‘type,’ ‘simplest form,’ ‘lowest terms,’ or something analogous to these. Thus, according to Theon (De Musica, c. 29, p. 80, of Hiller’s ed), the ratio 3 : 4 is the wvdmfv of the ratio 6 : 8, 12 : 16, etc. Similarly, Apollonius of Perga, for the pur— pose of his new system of multiplication which Pappus (Bk II) describes, used to call 7 the wvdlmfiv of 70, 700, 7000, etc., and 5 of 50, 500, 5000, etc. (cf. wvdpeve'iv, wudyevmcfis).” The words of Theo are (l. c., p. 81) fipLonv ,uéu Niven) 7rpd3709 Icai 7rv0,u.1)v 5 7151/ 7' 77'de “rd ,8', éWtTplva 5e, 6 7031/ 3' 7rp59 7', Mai, ém're-rdpv-wv 6 7ch e’ 7rpc‘29 8'. We may reasonably inferlthat the wvdmfiu of 33 + 4:3 + 53 is 3 + 4 + 5, which = 12. It is called e’7rl'rpo'ros because two of its terms are 4 and 3: it is really e’vn'rpt'ros [cal e’vra'ré'rap'ros‘, but it is not necessary to add “real 1 p. 96. THE SOLUTION OF THE NUPTIAL NUMBER. 25 ém'ré'rap'roq,” because, if you have the sides 4 and 3 of a right— angled triangle given, the hypotenuse can be found by ex- tracting the square root of the sum of their squares: Plato still has the Pythagorean triangle in his mind. It is worthy of notice by those who think Plato meant to be obscure that he should have further defined wvdmfv by e’7rl'rpt'ros, when five/mil! by itself would have expressed his meaning fully. Therefore (51/ e’7rl'rpuros‘ wvdufiu=12 wemra'uSL avé’vryels‘ is “yoked with five,” i.e. multiplied by five. In avaryels‘ there is a hint of what I may call the matrimonial metaphor, not in- appropriate in a Nuptial Number. The union of 12 with 5 produces. 60. A few minutes more, and the problem will be solved. 60 'rpte afifndele means 60 raised to the fourth power. This is perhaps a shock——the first, I hope—but I can prove it, to the hilt‘. First, let us clear the ground. By every writer whom I have read, the words are taken as meaning cubed, or else the multiplication of some three factors: Weber, however, as I learn from Monro”, saw this much—that the words denote three separate processes of multiplication. I will take Monro and Gow as types. The former observes3: “The phrase 'TplS‘ align— der’q may be translated ‘raised to the third dimension,’ since it may imply either ‘solid’ numbers (products of three factors) in general, or the cube, which is the solid number par excellence. For the former use, see Rep. 528B; for the latter, Rep. 587 D. Aristotle (1.0.) paraphrases 'rpis afifndels‘ by the words 257m; 6 7'01") Stavpdmuwroq dpzdpds‘ 7015702) ryémrrat a'Tepeés‘.” In Gow’s article4 we read: “are/360's” (so. in the passage quoted from Aristotle above) “ seems to be equivalent to and explanatory of . "rpis‘ azifndels‘ (of. Plato, Rep. VII 528 B).” That is, they would regard 60 'rpis adfqydelq as equivalent to 603, not 60“. (I say “would regard,” because Gow does take 'rpis afi‘g‘ndelq as mean- ing merely the multiplication of three numbers, which in this 1 It is not, of course, necessary to 2 p. 284. ’ . - ~»-~,_«;~»-.N_ supposethatPlato conceived ofa fourth 3 p. 280. . . ‘ . ~ 1;:‘31, dimension: for 604 may be represented 4 p. 93. by a solid figure which is 3600 x 60 x 60. i . =5 'j‘ 31' E .\ I“. r) . j)" *9, e-» 5v ;, ’ ¢ (' f” . . a...) a... - x ~--:-~- * c"- Ac ‘ ‘ ' ,' ' ,. flaw” 1. """ “Ms-K. \ v . . g ‘3 THE NUPTIAL NUMBER. case, he thinks, are different from one another (viz. 15, 20, 25), but Monro admits‘, and Grow would not deny, that the three factors may be identical. The fact is aflfnflelq means simply “multiplied”: if the multiplier is not stated, it means “multi- plied by itself” ,otherwise the multiplier must be expressed.) In support of their explanation they quote, in the first place, the words of Aristotle (may o 701) 8LarypalLL/La'ros‘ apL0/L09 70v— 70v rye’mrraL o-Tepeos), but I have already shewn that these words were never intended to explain 'TptS‘ avfnaeiq—Tpie 11125176659 Aristotle does not so much as quote: they were in- tended to explain the number in the section which I have called B. In the second place, they refer to Plato, Republic VII 528 B and 587 D. I cite these two passages in full. The first is as follows: 1167a 6’71'L'7r680u, 17v 3 €705,131} wepLgbopeL ov 1’7'317 arepeov )taBov'res', 7rpw av'ro [cad avTo kaBeZV' oprS‘ 51-: e'xeL effic ,LLe'rLi Bears/pay av'Env 'TpL'T’ml kapBavew. e'a'TL 86’ 710v 'rov'ro wepL 71)1/ 7-1311 K153111112 1113'an Icai 71) [811190119 [Leréxou (528 B). The second runs thus: Ira—rd 31% 819mm“) 1m). 'rpL’Tnu (11’5an 8131011 31) a’wé- 0701061! 001711 dc/Jea'rn/cws ’YL'YVGTab (587 D). The first passage occurs whele Plato says that the study of solids by themselves should precede the study of solids év nep1¢opii, i. e. astronomy: after the second “increase,” says Plato, we ought to take the third. Let us see what this means. A point (=unity) has no “increase”: a line (say 3) has one: a rectangle (say 3 x 41) has two (Sevrépa av’fjn): a solid figure (say 3 x 41 x 5) has three. A solid figure is therefore rightly said to be or have TpL’i-n av’fn, because your reckoning begins from the point, which has no dimension. Heaven forbid that‘I should deny this! The second passage deals with a case, not of solids in general, but of cubes. The number in question is 9— which is (says Plato) the measure of the distance separating the tyrant from true pleasure: “and how far removed the tyrant is according to the square and the third increase, is manifest.” But the third increase of what ? Certainly not of 9 (thoughvbelong- ing‘ to 9), for 9 is itself already one increase, viz. of unity—but the third increase of unity in that special case Where 9 is its 1 p. 280. THE SOLUTION 01” THE NUPTIAL NUMBER. 27 first increase. The phrase TpL’Tn 11173317 (sc. of unity) has already become a stereotyped phrase for third dimension—third, i. e. reckoning from the unit or point. Second: we may arrive at the right meaning by a study of the word ab’Enan. Aristotle1 distinguishes «ye’ueoLq from 111751)- O’LS‘ in these words : ¢auep6y 31)—37L 0131c é’a-rw 1) 11172317019 para/30).?) élc vadpeL ,uevyéfious‘, éurekexeic‘z 8;: [11751511 é’xov'ros pé76009...e'n 3% if 'ye Tomlin) peTaBoM) 0131c afiffiaews‘ i’3L09 (into) ‘ A . ryeve'acwq' 1) qdp 1117317059 éa’rL 7'01) éuv'n'dpxov'roq ,ue'yé- 00m; e’er’SoaLc, 1) 3e (Mia-L9 neiwan. Compare with this another passage of the same treatise2 : e’u new rydp 793 ryL’ryuwflai 'rL tin-Rah: 1’) (peeipeaeaL 013x 157r0/LéveL, (31/ 8e 71‘?) d7t7t0L01'300aL r’) afiEdveaGaL 1’) (bah/GL1! drape/um «rb 111371171) adgavénevovi) dkkowflpeuov' th’ 3116a, ,uev To 71-62009, 35’an 3e 71‘) ,uéryeflos‘ 76 11.1371) 013 néve L. In other words, the process which calls a thing into being out of nothing is (in-M} ryéuewq, not aiz'fno-Lg : in 1113317an the original size is not lost, but increased. N OW apply this to the number 60. Increased once, what does it become? Certainly not 60, which it already 15: else what of the words 'TOU GVUWGPXOV'TOC 6771800719, and UWO/LGVEL TO av'ro 'TO av- Eauonevov,...ev0a 56 ‘TO [LE/76009 'To (turn ou ,LLéueL? Does the process of 111751717“, when applied to a number, scare it away first, and afterwards tempt it back again? Credat J udaeus Apella. 60 once “increased” (i. e. accordingto the Greek idioms, multiplied by itself), is 602; twice “increased” it is 603; thrice “increased” it is 60“. Apply the reverse process to 603, and the absurdity of the usual interpretation will at once appear. If 608 is 60 three times “increased,” then 602 is 60 twice “in- creased,” 60 is 60 once increased, and nothing—or rather, unity, which is the point‘,———is 60. Shylock will “make money for others and not for himself” if he even takes “ doit of usance” by this scale: “Give me my principal and let me go; I pray you give me leave to go from hence; I am not well.” R 1 De Gen. ct Corr. A 5. 320b 25 if. as a number and had no symbol for it: 2 ib. 3213 22 ff. 3 supra, p. 26. Cantor, Vorlesungen zur Gesch. der 4 The Greeks did not regard ‘nought’ Mathem. p. 144. ' 28 THE NUPTIAL NUMBER. When Aristotle says, in the passage above cited, 37m! 6 701') Sawypa'mua'roq dptgpdc 70157-01) rye'vn'ral. arepeée he means that 6, which has already one dimension, by getting two more becomes 216. The number 216 we may then call either an a’pofiués‘ 'rpie ndfnpéuoé, i.e. a number which is thrice increased (sc. from unity, by means of 6), or we may call it 6 3‘5 dptfipds 51‘s nfi‘g‘nue’uos‘, because 85 Sic afifnfiet’s produces it. Third : I think I have fully proved my point, but as this is the keystone of the whole fabric of the Nuptial Number I make no apology for presenting the drift of my argument in a . tabulated form : unity 60 = unity once multiplied by 60 602: unity twice multiplied by 60 603 = unity thrice multiplied by 60. 60 602 = 60 once multiplied by 60 603 = 60 twice multiplied by 60 60“ = 60 thrice multiplied by 60 II. The words (31/ 6’7TL/Tpl/TOS‘ wvdufiu we,u7rai8t avé’v'yeis '7'th adfndet's‘ accordingly mean 60‘, i.e. 60 x 60 x 60 x 60 = 12,960,000, which can be, as we have seen, resolved into two harmonies, viz. 36002 and 44800 X 2700. (2) The Number of B. The first rampart has now been carried, and we may pause to collect our forces before we assault the second. We have shewn that Aristotle explained the number of B as the cube of the area of the Pythagorean triangle, viz. 63 = 216. We have shewn that e’7rl-rpo'ro9 7ru6’my'u is 3+4+ 5, which is the sum of the sides of the Pythagorean triangle. It remains to shew that the processes described by the words of B give us the number 216, expressed in terms of the Pythagorean triangle. THE SOLUTION OF THE NUPTIAL NUMBER. 29 So far as I can discover, except as regards the words 81mi- peuai ‘TE [cat Suvair'revépevaa, which have been partially explained before, my explanation of this section of the Number is al- together new. But I should not be surprised if it were some day found among the writings of the Neoplatonists or N eopythagoreans. Every word can be interpreted out of Plato’s own mouth, with the sole exception of Suvaarevépevat, which is fully explained by Proclus: and I am far from thinking that the Neoplatonists, who spoke the language of Plato, and taught his doctrine, blended indeed with much that is new and strange, were all of them baffled by what is after all no more than an extremely difficult piece of Greek. The words are: ’ I \ , ? ’ " I , avapw'n'eup 3e 61/ (p wpwrtp avfnaetq Suvdpevat’ 7'6 [cal Suvaarevéuevat, “rpe'is‘ a’m'oa'raiaebs, TéTTapas‘ 5% 3povc kaBofi- C out. ouctoriu’rmu 'TE Kai (ire/tatoziv'rwv Kai 0.135611er Kai, ¢0wdu- I I \ l \ \ 9’ ’ I 'raw, mun-a. wpoonryopa. [cat pm'a 7rp09 aMwflta avrecbnuau. I will take the words in the order in which they occur. The construction of dvgpam'et’qo 33: 6’11 (.3 7rpai’rgn is a’u0pw7reitp 3% 6’11 (‘5 wpai-rcp, which is itself an abbreviated expression for dvdpwvrer’cp 8% 5’11 93 wpaircp. The words mean this : “while the number of an dvfipaimswu ryeumrro’v is the first number in which.” I cannot, Without anticipating the results of Part 2, fully explain the meaning of the word 7rpai'rq): it will suffice at present to say that it means “the first number,” i.e. “the first number after unity,” which fulfils the conditions herein described. I will now explain the words afifn'aerq SUI/(iuevar’ 'TE Kai Suvaareuéuevat. It will be allowed, after the discussion on p. 25, that afi’g’fia-ets‘ means “multiplications.” But of what factors? I will shew that the factors are explained by the words Suvdpeuat' TE Ical, Suvaorevépevao. First let me take the word Suvduevat. It is clear from more than one passage in Plato that the mathematical? sense of Snuaadat “be equal when squared to” i.e. “be the square root of” was not in his day fully developed. In." Theaetetus 147 E———'Tdv dthudv 7r(iv'ra 35901. SteXdBO/Lev' 76v pév Sumt— 30 THE NUPTIAL NUMBER. per/0v Z'O'ov Zed/cos vyi'yveagat 7'93 Te'rpa'yaivcp 7'6 a'xfiaa a’vreucéaau'res‘ Te'rpa'rycovéu 1'6 Kai iaéwkevpou wpoqeiwopev— it is arsquare number which is said to be 8vvdaevos~ (viz. i'aos‘ (mime rylryvea'flai), while in 148 B—bo‘at aév rypaaaai 761/ irféwkevpov [cat €7rt'7r630v dptdludv 're'rpavycovté’ovm, poi/cos aipia-(i/Lefia, {fa-at 5% 7-61) é'repO/ué/cn, Suva/1.61.91, (59 [Mi/eel, psi/pi} Evaaérpovs email/01.9, 7-029 3’ €7rt7re'50t9 ('2 deav'rat—it is the roots which are Suvaaeua (sc. to produce squares), as in Euclid ' X Def. 11 real ai’ Suva/1.61mi. adrd (i'horyoi. But on com- paring these two passages from Plato, we note that, while Suva/Levon is not used absolutely in the sense of a square, but requires to be explained by the words 3009 Zodiac ryt’ryueoflai, 5151/aIITaL, where it is used absolutely, means “ are the roots of ” We infer that vadaeuat in our passage refers to roots and not to squares. Our inference will be confirmed as we proceed. Next I will explain the word Suvaarevéaevaa It is fortunate that Proclus2 should have expressly alluded to this part of Plato’s Number. At the outset of his commentary on the first book of Euclid he endeavours to shew that the ’dpxat' of the Universe of things—76w 6'11er awdvrwv—are also the u’pxai’ of Mathematics. One of his examples, that from Svudaeis, is as follOws”: Kai 230a Ica'rei Ta? Sundae-rte duagbat’ue'rat 711301.11 daoiws '7T'P00'flllce‘t 7029 paflfiaam, 'ra'iv aév Suvaaéuwv, 7:51) 3?: Suvaareijoae’uwv. ('1‘. 31) Mai de’v wokt'rei'a mepdrns Tats ,aov’aats‘ ziglrnko— )tovyovaévais‘ dvéflnxev, "rd Icowa war/Twp 7651/ ,uaamta'rméiv 7&6wa e’u we’paa'w aipwae’uots weptkafia‘w [cat wpoarna'daevos‘ (31/ 7029 eipmte'vow a’ptfiao'is‘, dd), (31/ 81) Kai «rd 'aé'rpa 7'69 76 eziryovtag [cat "rife évavn’as wpds Tadrnv a’youias Ica'radaat've-rat. The first sentence means that “powers” play a part in every department of Mathematics as well as in Nature and in Life—— “some” (i.e. roots) “having power, while others” (i.e. squares) THE SOLUTION OF THE NUPTIAL NUMBER. 31 “are. subject to power.” For example 3 is Swami/09, because -it has power (viz. over 9—to make 9): 9 is Suvao-Tevéaevos‘, because it is subject to power (viz. of 3—to be made by 3). It will not be denied that Suvaareu’oaat is intended by Proclus as the passive of Suva/tail. Now Mum-at, said of a root, means Savarat Terpdvywvov woteZv. The passive of this, said of a square number, is Sui/arm. Terpd'ycouos‘ ryt'ryvea‘dat (315va'rai i'aos‘ iodine ryirvadat in Theaet. 147 E). In the case of the active, it was found possible to drop Terpdrywvov watch: but if, in the passive, "re'rpdvywuos vyt’ryveo-fiat is discarded, at least the passivity must not. For this reason Séuarai becomes Suva- or—reifie'rai—if indeed it was not so before, by reason of the passive infinitive? It has thus been shewn that Suvaarevéaevat in our passage refers to squares. But before I interpret the expression as a whole, it is necessary to discuss a passage of Alexander Aphro- disiensis, which has not unnaturally been quoted in connexion with section B, since it seems to be the only other passage in which Suvaa'reéerat occurs in mathematical surroundings. The words are3 dummy Bé (baaw 1572-6 7081/ Hvfiaryopet’cou Kéyeaflat will wequBa, 70070 3% 571 71311 6p€oywviwv Tptryaiku 7131/ (570511er fm‘ras‘ 7&9 wkevpds wpcfirév e’an 'ra'iv weptexova'a'iv 6p0fiu ryawe’av wkevpcfiu i) aéu Tpta'iu 1i 5% TeTpowv, ii 3% ti'n'o'rei- vova‘a wév're. e’7rei, Toiuvv i) dworet’uovaa i’aou 315va'rat da¢oré— pate Etna, 8“}. 70070 if aév Suvaaém] xake't‘rat, ai 5e Suvao'revé- pal/at, Kai. (,E/O'TL mime. 7771/ TS weuTaSa auua’az/ é’ke'you (69 It“) vucwae'vnv ax?» dnrrnrou Icai Icpa'rovaav. The general drift of the passage is that the Pythagoreans called the number 5 “Never-say-die,” because it is the hypo- tenuse of the first right-angled triangle with rational sides— the one 3, the other 4:. As the hypotenuse is equally powerful‘ with both the other sides, it is called Suvaaéwy, the others 1 The word fivvdaets is here con- fined to irrational roots, but this is a. limitation introduced by Theaetetus. Theaetetus in fact proposes to confine the word awaits” to surds, and to use ,ufixos for the rational roots. The usual meaning of swam in Plato’s mathe- matics is “squaring” or “the square,” e.g. Rep. IX 587 D Kara 6e (Suva/Mu Kai rplrnu (11757711 dfikou 61‘) d1r60'rao'w b’a’nu d¢earnmbs 'yi'yverai. See on this sub- ject Gow’s History of Greek Mathe- matics, p. 78 note 1. 2 In Euclidem I (Teubner edition), p. 8. 3 In Eucl. l. c. 1 The use of the passive is like that in Tiuoxparefafiai, duaoxpareia'oai, regno'r and the like. The words (Strata-0a; and Bwaa'reizea’flat had also a special astro- logical sense in Proclus’ time : see Schoell’s Procli partes ineditae, p. 28. 2 Cf. Lucretius I 1045 dum veniant aliae ac suppleri summa queatur. 3 In Arist. Met. A 8. 990“ 23. 4 i.e. is equal when squared to the sum of the squares of the other two sides. ”we. 9% .4. may...» .s»m....m.m.r...s.wwgw . . A. an , fi=mwwg~ WW4.-. I «was... ~u.t.....~...¢;, ‘- ' WW- mamgfioaafifif 3‘) THE NUPTIAL NUMBER. H 8vvaa'rev6 6?th. It is “ never-sa -die” because it remains un- /-‘ , conquered and prevails. . Suvapémy here means “powerful,” “prevailing”: vaaa'revé- Mel/ac “ subject to power,” “ prevailed against.” Our sympathies being with the hypotenuse, because the odds are against him, we call him conqueror even although the battle is a drawn one. The only bearing of the passage on our text is this: it uses Suvaa'revopéun as a passive of Suvapévn. But whereas, in Proclus, Buzzao-Tevéaeva means “what can be produced by roots ” (i.e. squares), and Sundaeva “what can produce squares” (i.e. roots), here vaapévn means “equal, or rather greater in power” (viz. the hypotenuse), and Suvaa—revéaeuao “prevailed against” (viz. the sides). It is evident that the words are used by Alexander less in a technical, than in a metaphorical sense, and with no reference to their occurrence in the Nuptial Number—to which indeed he makes no reference at all. The interpretation of Suvaarevéaevat in Plato is given us, not by ‘Alexander, but by Proclus‘. We are now in a position to interpret fully the words afifn'aeoc vadaeual Te [cat Bvuaa'revé/Levar. They mean “root- and-square multiplications,” i.e. multiplications of roots by squares,—in other words, cubings. Take for example w x 202, w x as", y X 92. Each of these, taken by itself, is an az‘J’Eno-rg Suva/Lem; ’TE [cat Suvaarevopéun: collectively, they are afifn'aetg Suva/Laval 'TG Kai. Suvaa-Tevo/revar. The awkwardness of the English expression “ root-and-square multiplications ” is escaped 1 I have treated the words of Alex- Kai Km" (11’sz TETd'YMél/Ol’ arorxei‘ov, 6 ander seriously, because there is no aififip, km; Tal’le Kai cbaaérws é‘xwv a, priori reason why the Pythagoreans 6careke’i, vellcous Kai ,ue'rafo‘okfis e’u rois should not have called 5 a’wmia, or fnr’ az’rrbu firapxéurwu a’m‘; aeMuns ,uéxpc indeed almost anything else in heaven Jy'fig, dM’ (511 rd 1rpu’arw—ra 5La¢épovra or earth. But those who know what Kai max Emma. 700 dpLGaofi 6150 61677, confusiOn has arisen from the simi- o’ipnovml 1repm-6v, az’n-bs cbcrowei é¢c>xtwae larity of vowel in Ileuc- and um- may Kai a'vufianO'e KT)\. Megillus is quoted have their suspicions aroused when to the same effect a few lines lower they read the following from the down, and Ast in his note adds further Theologumena Arithmetica (p. 26 ed. references. Zeller4 I p. 369 note 4 Ast): Kai dueuciav 7rpoa'7776pevov Thu regards duuda as more original than 7rep.1roi6a, oz’; milieu, e’7ret57‘1 7'6 7rémrrov o'wemia. THE SOLUTION OF THE NUPTIAL NUMBER. 33 by the Greek idiom, because Suva/1.9m and vaaa'revéaevat are participial adjectives. It may be asked, “Why did not Plato express the idea of cubings in a single word?” The reason will appear in Part 2, where the full force of this most exact and elegant expression will be seen. I come now to the words 'rpeic a’wroo-Tdarew, Terrapae 88: Spam; kaBofm-ao. These words seem to have been misunderstood by every writer on the Nuptial Number. Let us first determine the meaning of KaBoDa-ac. If Béos kadeveL #5 (Laws III 699 C), I am afraid, if émdvm’a. (C'm'to 52 B), I desire—if 01.175770“, what happens to me? afifdvoaar. It is in vain that “ cubings” lay hold on me, if I am not cubed 1. Next, What are Tpeic dwaardaets‘, Té'r'rapas‘ Bé bpovq? Plato himself shall tell us. In the Republic (IX 587 D, quoted above on p. 26) we find: Ica—rd 83-: deaaw [cat 'rpt’mv ai’z'fnu Sfihou 81) d'n'éo-Taaw 5507712 (224)60777/“59 rylfryve'rar. In the T imaeus (43 D) 'TdS‘ 1'00 3L7T7taa't’ov [cat prrkaow'ov TpeZS‘ é/ca-re’pac d'n'oo-"rdaets‘. The meaning of dwéaraaos‘ in the first place is “distance from ” (so. true pleasure): 'rpwrkam’ov d’pa, says Plato, 'rpmkcia'wv a’ptapgfi dknfiofis 753012139 a’gbé- urn/cs Tripavuos‘. In the second the word means “the distances of 2 from 1, 45 from 2, 8 from 4” (7&9 7013 Sowkam'ov (2271-0- O’TdO‘ElS‘), and “the distances of 3 from 1, 9 from 3, 27 from 9 ” (7&9 7013 'rpwfitam’ov dwoardaetg), as is manifest from Timaeus 35 B. rpeic d7TOO'TCtO'GLS‘ means therefore “three distances from.” But distances from what? We shall see in Part 2 that d770- ardaerc means distances from the moment of conception, which is regarded as the unit or pointg. The? (i’n'OO'TdUGLS‘l are limited 1 The word kafiofiaai, which is per- I am not sure that Nicomachus was haps the simplest in the whole passage, not on the way to discover the mean- cost me more labour thanany other ing of this word, but he did not under- single word. Nicomachus(Inst.Arithm. stand the passage as a whole. p. 143 ed. Ast, quoted by Schneider on 2 dvréaracns is used in the same arith- p. xxxi) induced me to believe at first metical sense by Nicomachus, ed. Ast that it was an obscure mathematical p. 5, where he says (speaking of the term for “ multiplying,” but I persisted, unit) éwréaazs yap (iv ant/$17623 dfl'OO'TdO'GO'W refusing to believe that Plato is obscure, 7’7‘ 67réaas (Stu afigfiay 062v wpéaw “mice and the meaning flashed on me at last. KT)“ A. 3 34 THE NUPTIAL NUMBER. by four 3pm, each a’vréa—raaaq being only a aefiépwv‘; so that 'ré'r'rapac 5povs‘ means “four limits ”—four 25pm. T031) Btaa'rnpd- 'raw (Phil. 17 D). Here the limits are points: a5 awry/rat 7131; [167606311 3pm”. 36’ in TéTTapa9 3% Zipove implies a latent ,uév after 7706593. ‘ Now let AB, BC, 0D be three d'n-oa'rdaecq (Fig. vi), and A, B, C', D the 4 3pm which confine them—what then are AB, BC, 0D? We recognise our old friend in repose. Take him up tenderly, lift him with care, and 10:! (Fig. vii). FIG. vii. Three, four, and five are the only three numbers the sum of whose cubes will produce the number 216, which we have seen to be the whole number of B. There remain the words onoaoéurwu 're Icai duoporouvrwv Kai aliféin'wv Kai (fifibvéu'rwv. These words are not here used with any geometrical or arithmetical meaning, and it will be shewn in Part 2 that they denote months and nothing more. They tell us in what the d'n'oa'Tda'eLs‘ consist: it is “distances of months,” “distances in months” with which we have to deal, for the three dwoardaem, as we shall see, are thethird, fourth, and fifth months after a child has been conceived. The indi- vidual words will be explained on p. 54: in the meantime I will only remark that the intransitive use of aix’ (0 seems not to owccur elsewhere in Plato, though it is frequent in Aristotle, as a' glance at the Index Aristotelicus will shew. In connexion 1 @011 66‘ of) wdurwu, Lbs é‘ouce, ‘rc’bu 2 Cf. Arist. Met. N 5, 1092b 9. 6'11er b’pos 6pm 1rpoa’m'ymfis, (ii/\N of: €011 3 Cf. Theaet. 181 D 660 67‘) M70) uefiéptov, Tofiro éu ,uéa’cp 6pm! 7rp6‘rep0u 701510) 61577 Kw'éa'ews, dkkoiwaw, 1'7‘7» éKaTéptp 7,0007%th 'yi'yvorr’ 62v dyqfioiu 5% (popdv. ,uwaézfi Laws IX 878 B. THE SOLUTION OF THE NUPTIAL NUMBER. 35 with (Mimi, the intransitive use of ai’Ew is not unnatural, and harmonises with the lofty poetic diction of the Muses. I do not propose to comment on 'n'ciw'ra wpoawiqopd '11-: [cal ,5an till Part 2, further than to say that these words refer to some kind of a'ppow’a. As according to the Pythagoreans every cube is a dpnouia‘, a dppom’a may well be supposed to result from the sum of two or more cubes. Just so we are told by Plutarch2 that the Pythagoreans called the number 35 a rip/wom’a because 35 is equal to the sum of two cubes, viz. 23 and 33. The meaning of the section which I have called B is there- fore as follows: The period of gestation for a human creature is measured by the first number which contains w x w", w x .172, and y X 3/2; w, w, and y denoting three distances in months from the time of a child’s conception, and these multiplications rendermg all things conversable and rational towards one another. Nov: we have seen that 216 is the Whole number of B; and w x w , w x w”, y x y”, are contained in 216, if we put w = 3, a; =4, y ; 5, since 216 = (3 x 32) + (4 x 4“) + (5 x 5‘). The foregoing discussion may be summed up in these equations, in which w, a; and y are 3, 4 and 5, the Sides and hypotenuse of the Pythagorean'triangle: (1) w3+w3+y3=z=216. (2) [(w + 50+ 3/) x 5]4 = 3600‘2 = 4800 x 2700. It will be observed that in the left-hand member of both equations no number occurs which is not the number of a Slde in the Pythagorean triangle, and further,lthat 1n thesecond equation the numbers of the sides of this trlangle v1rtually occur twice over, viz. first in w + a; + y, i.e. 3 + 4 + 5, and second, when we multiply'the sum of these terms, whlch are 3 1n number, by 5, and raise the product to the power denoted by .4. Thus are the symbols on the Sibyl’s leaves de01phered: 1n Part 2 their meaning will be read. 1 See Cantor Gesch. der Mat-them. 2 7repl 77‘]: év Twang gbvxo'yovias‘ XII p. ’140. 1017 F. PART 2. THE SIGNIFICANCE OF THE NUPTIAL NUMBER. THE plan of this part of my discussion will be as follows : Chapter I. Translation of the passage. Chapter II. The Significance of the Nuptial Number. Chapter II is divided into the following sections: Section 1. Introductory. 2. The point of view discovered. 3. The meaning of the words from xahevrbu ,u.éu to ryevufia'oval, waZSdS‘ rrro're ou’ Séou. The wept/0809 of the 662011 ryevmrro'v. The wepc’odos‘ of the dvdpaivrewu ryeumrréu. The Diapason. The Number 36000. Conclusion. CHAPTER I.‘ TRANSLATION. 5455 D. HOW then, Glaucon, said I, shall our city be moved, . and in what way will the auxiliaries and the rulers become seditious towards one‘ another and towards themselves? or shall we pray like Homer to the Muses to tell us “ how first sedition entered,” and shall we, like a tragedianl, picture2 them speak— ing in a lofty style, as if they were speaking seriously, whereas they are playing with us, like one child with another, and making banter? How? In some such style as this: Difficult it is, indeed, for a city so compounded to be moved; but since for everything that is born there is destruction, neither will such a compound as ours last for all time, but be dissolved. The dissolution is as follows: Not on y to plants within the ‘ ground, but also among animals upon the ground, cometh fertility and infertility of soul and bodies, as often as their revolutions make the circumferences of the respective ciffies complete their course3, faring a short way in those whose life is short, and the‘ reverse in the reverse. Now as touching your kind‘, clever though the leaders of the city be whom you saw educated, none the more will they by calculation together with perception attain to due fecundity and barrenness, but it will escape them, and they will one day beget children when they ought not. Now for a divine creature there is a period which . 1 Tpoi'ymc’bs. Homer was Tam KaMw 'circumferences of their circles”—sc. to drdvrwv 1-013er 1-631! Tpa'ymav «pom the point where they began to revolve 6L6dom>x6s 're Kai inqudw according to —“for short-lived &c.” Plato (Rep. X 595 o). ’ 4 i. e. human kind: it is the Muses 2 Read Haney. who are speaking. 3 Lit. “ revolvings join for each the 38 THE NUPTIAL NUMBER. is comprehended ‘by a number that ism 3,131 for a humanfihe number is the first in whiz-h multiplications of root by square, having laid hold on three distances, with four limits, of that which maketh like and unlike and waxeth and waneth, have rendered all things conversable and rational with one another: whereof the base, containing the ratio of four to three, yoked with five, furnishes two harmonies when thrice increased: the one equal an equal number of times, so many times ahundred, the other of equal lengths one way, the other way unequal ;— on the one side, of one hundred squares rising from rational diameters of five diminished by one each, or if from irrational diameters by two; on the other, of one hundred cubes of three. The sum of these, a number measuring the earth, is lord of better and worse births, which not knowing, when your guar- dians marry brides to bridegrooms out of season, children of ill nature and ill fortune will be born: whereof the best their predecessors will indeed make rulers; nevertheless, being un- worthy, when they have succeeded to their fathers’ offices of power, us1 they will first begin to heed not though they are our guardians, having set too little store by music first and second by gymnastic”, and so our‘°’ children will grow up with- out us. And the rulers who succeed them will have little of the guardian in them to prove Hesiod’s races and your own,— races of gold and silver and copper and iron. And, iron mixed with} silver, and copper with gold, unlikeness will arise within, ariI-vtnequality wherein there is no harmony, which evermore breed war and enmity, whereinsoever they arise. This surely is the pedigree of Sedition, wherever she arises, evermore. I 1 i.e. the Muses. excuse for Madvig’s distressing change 2 The reading of A is right. Full to oeérepd 7'6 'vaaarmfis. well do the Muses know that he who 3 A is right again; and there is no despises ,uovmm} will hate 'yU/waanlcfi more tender touch in Plato’s writings. ere long, for 'yvuvaanm'] as well as Whoso sees it not, let him go and l ;wva'ucfi educates the soul and not the “sacrifice to the Graces.” body (Rep. III 410 c if). 548 c is no I » was my surpris CHAPTER II. THE SIGNIFICANCE OF THE NUPTIAL NUMBER. Section 1. Introductory. I HAVE endeavoured throughout the foregoing discussion to confine myself strictly to interpreting the Greek, because we cannot hope to understand what Plato meant untll we first know what he said. Wherever the full discuss1on of a word or phrase would have led us to inquire into .the. Significance of the Nuptial Number, I have postponed it, till the present chapter, as in the case of the words wpai‘rg), avEnoefqbvu‘apevat 'T€ Kai, Suvaarevopeuab, wail/Ta ‘wpoofiryopd ire [cab- pn’ra 7rpos‘ d'hhflta. dwédmvav, and (Emmi/lag. But as 11: is certain that Plato meant something when he wrote the number, and something moreover which contributes to the harmony of the Repubhc, we must attempt, to shew that the results which we. have reached do but swell the music of the whole. If we fail, the previous discussion is in no way affected, for it can only be. shaken by shaking its foundation—my interpretatlon of the Greek. Thanks to the abundance of collateral ev1dence which we possess, it is fortunately easier to discover Platos meanmg than it was to elucidate his language. . . ' Before I enter on this section of the discuss1on, it is neces- sary to say something of the interpretation proposed by Hultsch. I had reached the present stage in my 1nvest1gat1ons when I first read his article. Grow1 had led me to beheve that Hultsch’s solution of (the Second Number was 216,000: what e and delight to find that it was really 12,960,000, 1 p. 99. 40 THE NUPTIAL NUMBER. and so far absolutely identical with my own I I read his article with the avidity of Socrates 5when he devoured the 'n'epl ([3190er of Anaxagoras 1, and wondrous was the hope from which I was cast adrift, ewe-137) wpotaiv [cal duavyuyvaia'lcwv dpd) (iii/Spa. 71,3 ,uév 1/93 0:333:11 xpaipeuou 0656' 7'1.an alriaqz éwatrtaiueuou 659 rd BLaKOO'MGZII rd wpdrypa'ra, a’épag 83: [cal aldépas‘ Ical UBara. airtai/ievou Kai, dkka wokkd [cal dro'n'a, after he had said, With no uncertain voice, («is dpa 1/009 6,071.11 5 Sta/coaua'iu 're Kai wdvrwv ai’rtoq. The first number, 216, he obtains, with Dupuis, not from the Pythagorean triangle, but from 23 x 33, the smallest number in which each member of the two scales 1, 2, 4, 8 and 1, 3, 9, 27, is contained as a factor. ' This he hardly attempts to prove himself, referring us to Dupuis for the different steps: “Alle diese Beziehungen werden von Herrn Dupuis, zum Theil nach dem Vorgange friiherer, so zweifellos dargestellt, dass ich dar'uber nur zu referiren hatte 3.” Hultsch’s compliment so worked upon Dupuis that the latter gave up what he had so “zweifellos dargestellt” before the year was out 4.. Men have been known to sing a palinode when there was no need, but Dupuis did wisely to recant: he would have done more wisely had he done no mores. In dealing with the Second N umber Hultsch arrives at 12,960,000 by multiply- ing 36‘2 by 1002 (so he interprets Tfiu ,uéu lam; lad/us, é/cwrc‘w roaavrdms‘), and he equates this with the rectangle (77):; 53: loaf/him; ,uéu 71-116, wpoun'm} 5(5) by making the sides of the latter 100 X 7 JE and 100 X 33 JE. The é'n'lrpt'ros‘ wvapfiu rep.- del a'vé‘v'yei; "rpis‘ (1135776659 he explains as (3 + 44 + 5) X 3. “ So sind wir ganzi sicher bis zur Zahl 36 vorgeschritten.” Then in some mysterious way this number 36 furnishes two harmonies, the one 362 x 1002 and the other the rectangle which I have just described. I know not whether Hultsch believes this explanation, but for my own part the impression which his article has left upon my mind is‘exactly what it left upon 1 Phaedo 97 c. 5 Besides his Seconde Interpréta— 2 i.e. a study of/the Greek. tion, Dupuis has also written on the 3 p. 43. same subject a Troisieme Mémoire, 4 Cf. Seconde Interpretation 10. 9 ff. which is now out of print. with Nouvelle Interpretation p. 24 fl. 6 He changes rfi into my. , y ‘ «t ,- " THE SIGNIFICANCE OF TIIE NUPTIAL NUJIIBEIL’. 41 Dupuis’: “L’e’minent professeur dit en effet: l’interprétation, qui est dans l’obscurite’ depuis deux mille ans, n’est pas encore trouve’e: mais quelques nouveaux points de vue se sont offerts qui paraissent propres a donner une solution prochaine 1.” I venture to think that the illustrious metrologist made up his mind what the number ought to be from his familiarity with the Babylonian sexagesimal system, and afterwards tried to find it in the Greek. The present solution of the Nuptial Number proceeds 0n the opposite lines : but it owes no little to the man who has shewn us, more clearly I think than any of his pre- decessors, where to look for confirmations of our result 9'. So much was due to the di quibus imperium est pelagi, quorum aequora curro. Section. 2. The point of view discovered. We shall first obtain our point of View from Plato himself, and, from it, afterwards interpret the expressions in detail. Our city, says Plato, will be moved, when a’rda’ts‘ appears in the two higher classes. The Muses shall tell us how ardats entered first. The cause of our city’s being moved, they say, is that everything created is liable to destruction. The process of destruction (Mime) is when the leaders of the city vyévovs duerépov ezi'yovlas‘ TE Icai a’qbopias—oziSév uthov redfovrai, dMMi wdpeww (1137009 [Cal ryevmia'ovat waZSds‘ were or} Béov. Whenever,‘in ignorance of ‘ better and worse births,’ oi (inflame? a'vuouclé’wat lief/Abate vvuqSL’ow 71'an Icatpév, 0131c €d¢v629 0133’ edrvxei‘s‘ 7ra23€9 é'oovrat. In the next generation the dissolution has already gone so far that iron breeds with silver, and copper with gold. Accordingly the Mine of the ideal state is the begettiug of children when children ought not to be begotten, or briefly, the begettiug of children out of season 3. _ 1 Seconde Interpretation p. 4. The day beget children when they ought words are practically translated from not to’ and may refer, not simply to Hultsch himself; see p. 56. unions at the wrong season, but also 2 p. 51 ff. to the unions of wrong couples, or the 3 The words 'yewn’aovo-L raiéds wore like. That Plato here means unions or? 6éou mean literally ‘they Will one ‘out of season’ is made probable by 42 THE NUPTIAL NUMBER. This is confirmed by Aristotle‘, who after stating that the number which we have seen to be 216 is, according to Plato, the dpxn’ of change, adds the explanatory remark (59 7739. (pv'aeais‘ 7ro're (36120150779 (fiaékovs‘ Ical Icpelfwovs‘ 7139 watSelas. It receives additional support on a reference to those passages in Book v where it is ordained that bridegrooms and brides are to be brought together at certain fixed times. Thus in 458 D when the male and female archons fall in love, they are not allowed drrdlc'rws‘ nlyuvoflat, but marriages are to be celebrated iepol 659 Suva/1.1.11 25w ,udMa-ra. In 459 E it is said: 013x001) 82) éop'ralf Tom-:9 vonofle'rn'réao, (311 at; Evvdfopeu 'ra'e 7e vii/1.43m; [cal 7-009 vvmfia’ovs‘, [call Qua—{at Ical iipwm. wowrréoz. 7-029 ripe'répow noun-ate 7rpé7rovr€€ 7-029 ryuyvonévow «ydpooc. And at 461 A it is reckoned a sin against God and man to produce a child for the state mix 1571'?) 01101611 0135' 1577" efixdiv (jails (1‘9 ed), élcda'romvoi‘s Wei/woe silfovrao zeal; [épeoat Kai. iepe'is Ical Eli/ureter; 7; 776M? KT)». Section 3. The meaning of the words from xakevrbu péu to ryeumiaovow 7Ta23d9 wave or} Séov. We have thus obtained the point of view from which the whole passage is to be interpreted. While the cause of change from the best to the second-best commonwealth. lies in the perishability of everything which is created, the process which leads to change is the begetting of children out of season. Let us now unravel Plato’s meaning by this clue. Plato deals first with the process leading to change (Mime). To plants and animals, he says, cometh FIG. viii. production or non—production (a crop or A no crop) of soul and bodies, whenever \ revolutions join for each the circumfer- ences of their circles, these circumferences faring a short way for the short—lived, but the reverse for the reverse. That is to say, plants and animals have fixed periods of gestation, which may be represented by shewn to be certain. 1 See above p. 22. 7rapd Katpév below as well as by the words of Aristotle (to be presently referred to): in the sequel it will be_ m» . .4», .. ., «a- .3»..va ” - 3 \ l, ’ Ll'fiflfl’fi‘mxaauxmnuxw,rc... . taxman“... . THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 43 circles whose circumferences revolve (Fig. viii). Every time that the fixed point _A is reached, there is (pope). \lrvxfiq 7'6 [cal mommy, if the seed was sown on the last occasion when the same point of the circle was at A, and if it has come, without accident, to maturity; if however the seed was not then sown, or, though sown, has not .come to maturity, there is d¢op5a \[rvxfis‘ 'TG [cat a'wpa'rwv. The singular ‘i’vX’i is used because soul, viewed as the principle of life, is one in all plants, in all animals, and in both 1. Why are the circumferences long in the case of long-lived animals, and short in the case of short- lived? Simply because animals that live long have long periods of gestation, and conversely. Aristotle takes note of the same general rule in his Problematazz 81.02 71’ 7d név 'va-ré/ca n31} {930012 tic-Ti, 7031/ 82: wokvxpévws 75 Iczimns‘; 1‘} 371 "rd fur/epo- ,BLai'repa BpaSurc-tpov wécfiv/ce Telecofiafiao ; Earn, 5% Bpan'rd/ca 7d ,ualcpoBta. ‘ ‘ V The meaning of the words from Mung 8% 1386 to e’vawrr’aq may therefore be summed'up in the sentence: In all plants and animals the period of gestation is fixed by nature. Now as man is the animal with whom in the ideal state we are con- cerned, we are prepared by this exordium for the mention of the period of gestation in the human race. It will come in due time. \ Plato proceeds to narrow the case down to man: “Now as touching your kind (i.e. mankind), clever though the leaders of the city be whom you saw educated, none the more will they by calculation together with perception obtain” (literally, hit the obtaining of) “good offspring and no offspring, but this” (i.e. etiryovc'a 'TG [cal dgbopia) “ will escape them, and they will one day beget children when they ought not.” Several points in this require to be explained. First? in‘place of repeating gbopd [cal ddmplfa Plato writes eziryouiae 76 [cal quopL’as‘, because it is not enough for the prosperity of the ideal state, merely to produce children—the children must be good in quality. The word dcfiopi’aq is full of meaning; it is the duty of the rulers to render, if possible, illicit unions unproductive, mil/7a Stamekeu- ‘ GdflGVOl, wpoev/Lei‘a'aao, ,mikw'ra péu ,me’ 659 (1)639 elects/pew 1 Of. Tim. 77 B. 2 x 9 p. am 25. 44 THE NUPTIAL NUMBER. 161577;“; [1.7736' ry’ é’v, e’du ryémrrat, e’du Sé 'Tt Bade-nun, oil'rw Heel/at, (59 0131c 013/0779 Tpogbfic 793 Towu'rcp (V 461 0). Second: how does ‘calculation together with perception’ help the archons in their efforts to obtain good offspring and avoid bad? It is by the process which these words describe that the archons try to learn at what times the unions should take place. ' In the pages which follow, the truth of this proposition will be made manifest: but a few words are necessary now by way of caution and of explanation. It may be thought that the words horytaacfi ,ueT’ alafln'aewe refer to the faculty or process by which the rulers select the couples whose offspring will be good, not to that by which they determine the time of mating. This is not so, for, as far as choosing the right parents - is concerned, the rulers have already abundant means at their disposal. Ex hypothesi, being as yet Te'keoa (inflame, they I thoroughly know the quality of every member male and female of the two higher classes, and Plato has provided them with the ‘medicinal lie ’——the xkfipol “TU/69 Kopmlro‘l (V 459 0—460 A)— for bringing together the best men and women. But if it be allowed that the season of our begetting may have seemed to Plato to affect Our destiny, we should expect him to provide his archons with some means of determining the right season, so far as the laws of nature will permit; and no such instrument has hitherto been put into their hands. We have indeed been told that state-marriages are not to take place at random, but on certain holy festivals, with every circumstance of pomp and splendour, but we have not been'told by what method the dates of these festivals should be fixed. The method is Rory/Lanes ae'r’ aladfiaewe, and in the end it fails, from no fault of our rulers, but, as we shall see, Std 76 I“) pe’vew ,unb‘éu (i706 c'lvrav'ra é’v 7'th weptéSgo aerafidxxew, because “the great world spins for ever down the ringing grooves of change.” So much by way of caution. By way of explanation I would say this. Plato may well have wondered (see for example Protagoms 319 A——320 0) why the children of good parents are often bad, and conversely, and it is not unreasonable to suppose that he attributed this in part to their having been begotten at wrong seasons. In the face of the unfathomable mysteries of human THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 45 fate, who shall say that he was wrong? Not Hesiodl, whom he continually follows, nor, I imagine, Hippocrates or Aristotle, or the Asclepiads of to-day. It is one thing to hold that we may safely disregard as an unknown quantity, if indeed it is unknown, the influence which the season of their concep- tion exercises on human creatures, and another to say that it'haslnone: the man who is wise because he knows his ignorance will stop short of the dogmatic ‘no.’ For us, however, the interesting point to notice is that Plato thought we are in some measure affected by the season of our begetting. As surely as Nature has appointed for all animals, including man, a fixed period of gestation, so surely has she ordained times and seasons for their procreation, and they who disobey her, steering their lives by 75801»; rather than by aicpékmov, shall surely pay the penalty, if not in their own persons, in the persons of those by whom they fain would make themselves immortal 2, and finally in “red ruin and the breaking up of laws.” Such was the doctrine of Plato when he wrote the Republic, and such it remained to the end 3. Section 4. The weploBos‘ 0f the 662011 «yew/77761;. We come now to. the words é’an 3e delay ,aév 7mm; weptob‘oq 7'51! dpLQ/toq wepLXa/Lfiduel Téketos‘. Plato has stated that all é'gfia have a fixed period of gestation: he now proceeds to specify the periods of (1) the 065012 «yam/777611 and (2) the dvdpcl'rrelov, beginning with the 665012, on the principle 6% Ala; a’pxailuea'da. Four expressions require to be discussed before we can arrive at Plato’s meaning—via: #6950309, wepLKa/tfiduet, dptflpoq Tékewc, and 962012 «yer/12177611. The word weplooog means nothing more than ‘way round.’ - One complete revolution of any circle is a weploSos‘: two or more of the same circle, or one (or more) of one circle and one (or more) of another or others, are mptoam. This will not be denied by any one who will take the trouble to study side by 1 Works and Days 780—794. 3 See Laws v1 773 A ff., 783 D ff. 2 Symp. 208 A if. 46 THE NUPTIAL NUMBER. side the examples quoted in Ast’s Lexicon of the use of wepfodos‘ in Plato) In the present passage the ‘way round’ is that des- ., cribed above in the words grauvreprrpovral eerie-"row led/dual) weplcpopele Euva'nr'rwa'l: the wept’oBoe of a 96201! ‘ryeumrrév is there— I fore fulfilled bray wept-rpo'm) Belg) 7611117)ng KdKKOU wepupopdv fwdwry. Now it has been shewn that the words from 370w wepnpo'n'at’ to EUVdW'TwO’L refer to periods of gestation: we are therefore forced to the conclusion that the wepl’ob‘oq of a 65201) eyemméu is nothing more or less than the period of gesta— .: tion which ends in the birth of a Divine Creature‘. ' This is at i first sight startlingly anthropomorphic, and will appear to some readers about as unlike Plato as anything well could be, but let them reserve their judgment for a little. Were it ever so an- thropomorphic, as I shall shew that it is not, I would rather believe that Plato’s View of the Divine was as gross as that of his countrymen whom Xenophanes satirised so well”, than hold him guilty in one of the most finished passages which he ever ; wrote, of using a word which he has just explained, in a sense 'i other than his explanation gave to it, and then in the very I next sentence, which hangs so closely with the preceding that he actually does not think it necessary to repeat the word, giving to it again the meaning which he has already said that it A ~ n. Jera-Aw wmvw’NcVJ n.» -.——. w v sat-w. «V... m... bears. The word wepma/lfldvel means ‘comprehends.’ If a num- ber is represented by a rectangle, its sides, or factors, are said to ‘comprehend’ it, as in Theaet. 148 A, 7612 "roll/v1; ,ue'rafd Tod-rov, (51/ [cal 7d 'rpl'a Ical "rd 7rém'e [cal was 39 ddea/roe 2'0'09 lad/cw ryevéadat...pel§wu 3% Kat 63ther del wkevpei azi'rdu weplkap- Bail/eun-rrpopfi/m dplQ/adv e’xahéo-apev. In the present case, we are dealing, not with a number, but with a weploBos which is comprehended by a number, and. that number weplkanfidvel 1 For 1rep£o§os in the sense of ‘period 73‘ way/ac xelpeam Kai é‘p'ya ‘relxéi‘v d1r€p of gestation,’ cf. Aristides Quint. De du6pes, Musica, p. 143, 7112‘: “raw éwranfivwv i’1r1roc ,uév 0’ ’L’mrowc, [366: M 7'6 fiovalu 1r€ plééocs. Aristidesthoroughlyunder- 6/.wlas stood this part of the number. Kai KE 0651! laéas é‘ypacpou Kai a’a'wa'r’ “ Mullach’s Frag. Phil. Grace. Vol. érolovv I p. 102 70La90’, 0761i 1er mini 5é,uas elxov 6/4020». 6008 ei’rm xe’ipés 7’ e’l‘xou [366$ 73:3 Mower, l THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 47 wepr'oBou which gives the time that the revolution takes to accomplish. The period of gestation of a Divine Creature is therefore expressed by a ‘ final number.’ I will now discuss the words 'réMlos dpldpés‘. It is well known that a fperfect’ number meant to Euclid1 and Greek arith- meticians generally anumber \Which is equal to the sum of its diVisors, e.g. 6 = 1 + 2 + 3; 28 = 1 + 2 + 4: + 7 + 14. But there ISIDOt a trace of such a meaning in Plato, nor in the fragments of Philolaus’, and even among the later Pythagoreans numbers are often called ‘perfect,’ although they are not equivalent to the sum of their factors’. The 're’keloq dptb’pds par excellence was 10 according to the Pythagoreans: Oewpeiv 862 7d eprya Ical Tclu e’a'ow’av n3 alpLH/m'i [cm-Tell! Mud/aw, fine éa‘Tlv e’v TL; Sexdc‘l' perydka rycip Ical 7rav7€7\n)9 Mal wavroepydq Icai 660/60 [60:2 ozipam’a) ,BL’a) Ical dudpww‘t’ua) dpxd Ical (iryeaaiv Ical Icoomf'relpa, ai Stir/ante ai 7&9 56Md8094. But 10 was called by them wavrekfis‘ 0r Téhemq Simply because, as the basis of their system of calculation, which was a decimal one, it may be regarded as the ‘consummating’ or ‘all-ending’ number, the numbers above ton being considered merely repetitions of the first ten5. Plato was perfectly at liberty to call any other number 'rékelos‘ which ‘ends’ or ‘brings ' a consummation? Let us see whether he does so. In the szaeuf', p. 39D, we find the words: é’a'TI, 3’ 55an 0:338:11 7777-01) [ca—rayonc'm 3vz/a7'o'u, (69 3 eye Tékeloq dpld/Lbs‘ xpo'z/ov To‘v Tehelou emav'rdu wknpoi’ 7676, 57m) aim-avail! 71311 o’lc'rai 7repc63w1/ 7a ’7?de dhknka funnepavdévra Trix?) axfi Icecpakfiy 7'63 'roz') Tav'roi) Ical 6,140le dvaMETpnde'VTa xii/eke). It is rightly held that we have here a reference to a Great Year—the period , 1 E’uclides Elem. VII def. 23 réxews Demme (referred to on p. 11) pp 84“ aptfiuos‘ e’o’nv 6 TOYS éavToU [iépeaw i’o’os (.311 85. The rest of Demme’s artihle is 2 Some of Philolaus’ fragments, radically unsound throughout whether genuine or not, are at all 4 Philol.Frag. 13 in Mullach 11p 4 events tolerably early. 5 See Zeller Phil. der Griech 4 I p. 3- e.g. 3 and 9: see the Theolog. 367 note 4, and Aristotle quoted .there: Amthm. pp. 13, 58 (ed. Ast). The 6 Cantorinhis VorlesungenzurGesch number 3 is on p. 13 said to be 'réxecos der Math. p. 142 agrees with me in L’dtal'repou 7611 dew, implying that denying that ‘perfect number’ here other numbers may also be réxewi, means ‘a. number equal to the sum of though in a less specific sense. Cf. its divisors.’ 48 THE NUPTIAL NUMBER. within which all the eight spheres1 revolving around the earth, simultaneously reach the point from which they started at the commencement of our cycle, and at the end of which (as will be afterwards seen2) Plato, like many other philosophers, said that the Universe was not indeed annihilated, but in part at least dissolved. From the fact that the same words, Te’ketos‘ dptflao’s‘, are used both of the number in the Timaeus and of the number of the 662011 ryevmrrév, we may provisionally infer that these two numbers, the one expressing the duration of the World, and the other the period of gestation for a Divine Creature, are identical. This will be fully established later". There remain the words Help «yawn-rd. The ancients‘ ex- plained these words, with perfect justice, as referring to the Universe: odpavos i) [odd/1.09 7’} [cat th0 3 Ti 7ro're dvoaaé’éaevos‘ pdkw’r’ d1) 8éx0£TO, T0176, nail! aivopdodw (Tim. 28 B). In the Timaens there is abundant evidence that Plato regarded the World as a Divine Creature; 761/86 761/ [coal/.011, he says, £9301) é'axlrvxou 5111201111 76 7'3; dkné’eiq. did 71):) 701'} 0601') «yew-50001.1. 1 viz. the circle of the Fixed stars, creature,’ and so includes under it Saturn, Jupiter, Mars, Mercury, Venus, the creations of the created gods of the Sun, Moon: see Rep. x 507A fl. There Ttmaeus—e.g. man, birds and beasts is a good definition of the Great Year —-everything in fact which has a cycle: in Macrob. Somn. Scip. II 11, 10. but Plato does not here call these 2 p. 68. 3 p. 69. divine. Proclus is nearer the truth 4 See Plutarch 1repl 'rfis éu Twain: ybvxo- when he observes (In Tim. 254 D): 'youlas X 1017 c, and the references in é‘n Tolvvu e’Kelvov (so. 7'00 ct: dxnfidrs THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 49 wpéuocav‘: with which compare the words of Proclus (in Tim. 89 D)f draw 38: é’axfivxov 61.1376 Kai é’vuovv i599, 0651/ (1137-6 Kaleb-6L9, grep 6 th'rwu €11 Hoht'reia péu 06201! 761111777611, e’m‘afiga. 8% 069111 eddaiaoua wpoaelvre'iu 76v ICO’O’MOIJ rift/coats. The Universe is 0620:», because it is a God; ryewméu, because it is created?- The 06201! vyevmrréu is therefore the World, and we have thus reached this (at first sight) somewhat curious result. The period during which the World is in the womb is of the same duration as the period during which the World lives. In other words, the Universe takes as long to make as to unmake: the time of its making is expressed by an dptflaég which fulfils (whereas its growth, the time of its unmaking by the self-same number fulfilling its destruction. It needs no mythical inter- pretation to enhance the splendour of this idea, the full sig- nificance of which will be explained in Section 6. Section 5. The nap/50809 of the dvdpaiwetov (yawn-1'61). . ' The general statement, with which we started, that every llving thing has a fixed period of gestation, has now at last been narrowed down to man. The period of gestation for a human creature, says Plato, is the “first number in which root-and~ square multiplications, having laid hold on three distances, etc., will make all things ” (i.e. all things in the development of the Schneider. The scholiast (apparently Proclus; see Schoell p. 21) whom he quotes gives to the words Below yawn-6V a somewhat wider sense than I think the Timaeus will allow us to assign to them: 0650:! 'yew'rrrou mi Toll dhov (157701 Kéa'yov, el Kai wponyovuévws 'roii'rov... dhhd mix! 76 deLKan'rov Kal weptqtepéaevou. But if such a view is right, then an dufipcérewv yet/11777611, since it also is a 1r€pt¢ep6aevom is a 06301: ‘yevum'éu, Where- as Plato here opposes the two 761/11an to each other as if they were distinct. The scholiast’s view rests really on a mistranslation: he takes delay 'yewm‘bv as ‘created by gods,’ not ‘divine xpévov) rd ,ué'rpa mil/Ta neptéxovros évo- etdc’fis, Ka0’ 61‘ mi Tais xpuxa'is ai weplodoi KaZ 702‘s a'dmaaw e’mrehofiuraz Kal To €11 ,aérpov 7739 (Days droxaraa’rdoews (é‘a‘n yap 06lov yawn-rail neplodos, fir dpLB/Aos‘ neptha/Lfiduet Téhetos, dis #9119776“! 6 év Hohtrela Zprdrns), ofi'ros‘ 6 xpéuos 6L0- pLa'T‘LKOIS e’an ‘rc’bu aé-rpwv raw éu Tut‘s lpvxai‘s 1)‘ awaanKai‘s (topa'is Kai gbpovpnTLde. It is true, and this is the whole point of the Nuptial Num- her, that the replodos of the 66201! 'yewn- 16V seemed to Plato to control the 76V- \ 1:an which are within it, but this is a difierent thing from saying that the 0620:! 76111177761! is way To wept¢ep6aeuow _ child within the womb) “conversable and rational towards one ; another.” We have seen that the three distances are 3, 4, and 5: and 216 is the number in which 3 X 32, 4 x 42, and 5 x 52 are found. Plato declares that 216 is the shortest number of days that can elapse between conception and the birth of a fully 1 303: cf. 30 D, 32 D, 34 A, 34 B (et- 5al,u.oua 0661/ 0.137611 e’yewfiaaro), 37 C. 2 2813 'yé‘yoveu' opa'ros quip air-1'63 1'6 é‘a'n Kai (Tana éxwu, wdura 6e Ta ToraO-ra ala’fin'rd, 7'6. 5’ ala‘finrd, 56$?) repair/r731. ,uer’ ala‘fifia’ews, ywuéaeva Kai yeuunra warm. Whether these words are to be taken in their literal meaning or A. not, it will be allowed that if Plato can call the world yeuum-éu in the Timaeus, he may do so with equal justice in the Republic. 3 Tekecoi is used with the same mean- ing in the Theol. Arithm. p. 58 (Ast): Kahei‘TaL dé aw?) (sc. éweds) Tehw¢dpos, rehewi‘ 66‘ rd. éwedmyra. 4s 50 THE NUPTIAL NUMBER. developed child and this is the full meaning of wpairw. It is fortunate that Aristides Quintilianus, in the passage to which we have already referred2 as full of allusions to the Nuptial Number, helps us here again. (Speaking of the Pythagorean triangle, he says. all 62 [ML 1131' wkevav endow-nu Icarra Badoq .avfnaaL/Lev' (30.6119 «yap 17 a'ai/La'ros ¢UO‘LS‘) "Ironic-away av 701/ BLa/coa-La Belcaéf, Za-apLanv o'v-ra. a'vveryfyvs 7-9) 'ch1/ éwrapfiku. war/Aw 82: 7719 'rp629 e271" dkkfikovs' [card Bdaos 'n'omoavres‘, Ica. 713 wpoeLpnpévw 7rpoa'0év'7'69 (i..e [3x4x5]- +216: 276), 701/ 7:31) evvea/Lnuwu a'vu'rtdepev 5LalcocrLa (5,8- 80/Lnlcov'raéf. As 210 and 270 days were generally legarded as the periods respectively of a seven and a nine months’ child, he adds: e’v ap¢orépom Be 0 ESE WGPLTTGUGL, rya/LL/cos Tvryxavwv 8L aurwws. It is curious that Plato should have selected the period of the seven months’ child as the period ordained by nature, but on this. subject the opinions of the ancients differed very widely, as we may see from Censorinus‘, who observes “quoto post conceptionem mense infantes edi soleant, frequenter agitatum inter veteres nondum convenit.” In Book V (461 D) he recognises both the longer and the shorter periods, and we ‘ shall see that room is made for both of them in a later part of the Nuptial Number. The fact that the sum of the sides of the Pythagorean triangle when they have been cubed 1s nearly »- equal to the period 111 question, may well have determined Plato to choose the number 216. It should also be noted that 216 (= 23 x 33) 1s the ploduct of the cubes of the first even (female) and odd (male) number (cf. '.Macrob Somn. 801310. I 615—16), and cubing makes body: ,Badog ryap 7} a'w/La'ros gbv’aLc, as Aristides says. 216 i 63 is also the area (6) of the Pythagorean TIIE_SIGNIFIC'ANCE OF THE NUPTIAL NUMBER. 51 triangle cubed: it is likewise 6 X 36, and 36 is not only the sum of the Pythagorean Te'rpalc'rés (1,2, 3, 4, 5, 6, 7, 8), but, as being the number of the Samar/oi or (Lipoa'lcmrm, was considered to be peculiarly important in generation]. But it should be remembered that Plato’s object is not so much to determine when children should be born as when they should be begotten. He was firmly convinced that Nature has appointed for man, as she has for many of her other children, a fixed season for beget- ‘ting good and healthy offspring. In deciding what this season is, he had as little to guide him then as any man who helds similar views has now, or perhaps even less. What he did do was probably something of this kind. Taking the shortest period of gestation as his unit of measurement, viz. 216 days, be divided a woman’s life into periods of 216 days, from the day when first she was able to conceive a child. It is this process ‘ of calculation, or something like this, which is meant by Roma'- #6? ,ue'r’ aL’adn’o-ewq. (170077an is observation by the archons of the time when each individual girl reaches the age of puberty, ' and )toryLo-péq is the calculation by which they divide her life into periods. There is no reason for holding that the words mean “calculation together with perception of the heavenly bodies,” for it would be a waste of labour to deduce from the stars’what a perfect organisation within our ideal-city could of itself supply, even if Plato admitted practical astrology into his city, of which there is no evidence? During the time when Plato~allowed a woman to bear childrena, she would, so far as the claims of maternity or other circumstances would permit her, unite with a bridegroom on the first day of each of these cycles, and possibly'on other days within the cycle in which, the 1 p. 151 (ed. Meib.). 2 p. 16. 3 Cf. Theol. Arithm. p.40 (Ast): e’1rel _ 6é 6 a’md 7'00 67' K1530; a'urr’ 'yiveTaL, 6 é7rl érTa/Lfiuwv 'yéumos xpévos, avuapcflpov- ,uéuwv Tais find 701! «$5 fiuepc’bv, 6’11 01?: d¢pofiTaL Kai 51124519066 aréppcwos hay.- fla’uet 7'6 o‘7rép,u.a. The first Six days were not credited with any great effi- cacy, as appears from an unpleasant story in the beginning of the Hippo- cratean treatise 1rep2 ¢170Los “Law. The ai‘ria mentioned by Aristides is that 6 is the product of the first even (ie. female) and the first od‘d (i.e. male) number: 2px 3:6. This may or may not have occurred to Plato: it probably did occur, but I think he set no great value on it. 4 De die Natali VII 2, where a num- ber of different views are mentioned: cf. Gellius Noct. Att. III 16. 1 Compare note on p. 53: and Cantor Gesch. der Math. p. 86. See also‘ Procli partes ined. (Schoell) p. 32. 2 The only use of astrology would be (like the Roman spectio) to fortify the archons 'when they said “Don’t,” but they are tolerably secure already. The Nuptial Number being Kzfiptos dya- véuwy Te Kai xetpo’uwv 'yeuéo'ewu, afforded ample scope for the astrological enthu- siasm of the Neoplatonists: cf. Schoell’s Procli partes ined. p. 27, 34 ff. 662‘_ Toiuvu 701): 16311 702,1va Kuplous 16V Kmpdv av’TcSV Bnpdu Karol ,uéu 'rd drhaufi 510?. 7'6 7'01! dipoa/cé'lrwv Kai 1551! 701510” 1rap- ava-rehhéwwu darépwv TE Kai fieKaVL'BV. Proclus proceeds to illustrate this at length. See also Thompson on the Phaedrus 252 E. 3 This was (in the Republic) between the ages of 20 and 40: see v 460 E. 4—2 52 THE NUPTIAL NUMBER. number 6 predominated (e.g. on every thirtieth or thirty-sixth day). This point will be briefly touched upon again. I will now explain what is meant by afiEfio-ets‘ Suvdpevai 7'6 Kat Suvaorevéueuat, rpe'i's‘ (iwoa'rdo-ets‘, Té'r'rapas‘ 3% 55pov9 7m- Bofidat duowém‘wv Te [cat duopowfivrwv Kai. aiifév'rcpv Kai. gbBwém-wv. It is manifest that these words denote in some way the gradual growth of the child within the mother’s body. The Tpeiq dwoardo—ece, which we saw to be 3, 4, and 5, are the third, fourth, and fifth months from conception, the four 3pm being the beginnings of these three months together with the end of the fifth. We know from Censorinus1 that a peculiar importance was assigned by the Chaldaeans to these three months in the development of the embryo, and as we shall see that Plato is following the Chaldaean reckonings in the next sentence, we may assume that he has them in his mind here also. “Itaque eum” (i.e. the sun), says Censorinus, speaking of the Chaldaeorum ratio, “qui stellas ipsas quibus movemur per- movet, animam nobis dare qua regamur potentissimumque in nos esse moderarique, quando post conceptionem veniamus in lucem; sed hoc per tres facere conspectus.” He then explains the word conspectus as follows. The circle of the Zodiac, which is traversed by the sun in the course of a year, is divided into 110619113219 rum: 1 De die Natah' VIII. THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 53' twelve equal parts (Fig. ix). In each part the sun tarries about a month. Seat signum quodlz'bet cum cetem's singulis habet mutuum conspectwm, non tamen uniformem cum omnibus: nam vali- diores alii, infirmiores alii habentur. At the time of conception the sun must be in some one sign, and in some part of it: the name of this part is the locus conceptionis : they are in all 360, 30 in each sign: and they are called by the Greeks poipac, eo videlicet quod deas fatales nuncupant Moeras et eae particulae nobis velut fata sunt‘. NOW let us suppose that a child is begotten when the sun is at the point A in our figure and let us follow his development. When the sun passes into the next sign, viz. OD, locum illum conceptionis aut imbecillo videt con— spectu aut etiam nec conspicit: nam plures proximantia sibimet zodia invicem se videre omnino negaverunt. This means that in the second month the child makes no considerable advance. But when the sun enters on the third sign and the child conse- quently on the third month, tune primum illum locum unde profectus est videre dicitur, sed valde obliquo et invalido lumine ; qui conspectus vocatur Kant éfdvycovov, quia sextam partem circuli subtendit (i.e., by joining every alternate sign, a hexagon " can be inscribed within the circle). In the third month there- fore the child makes at all events some progress. When the sun reaches E, and the child thus enters on the fourth month of its development, the sun videt [ca-rd Terpdrywuov, for a similar reason. When the fifth sign and month are entered, [ca/rd "rpo’rywuou aspicit: nam tertiam signiferi partem visus ille me- titur: quae dude oisiones, adds Censorinus, tetragom' et trigom' perquam efiicaces incrementum partus multum adminiculant. The conspectus from G, when the sixth month begins, 0mm} caret efiicientia: eius enim linea nullius polygoni efficit latus. But 1 Cf. Aristides Quint. p. 152 foll. where all this is worked out in detail, and Stobaeus I § 470 ff. It appears from these writers that each sign was further divided into 3 equal segments, the whole circumference being thus split up into 36 portions of equal length, which were called Bexauoi or ctpoaxérm or perhaps dipovo'uoc (for 6/10- uo’aoc in Aristides is probably corrupt), and made use of in casting horoscopes. Sext. Emp. Adv. Astrolog. p. 728 ff. (ed. Bekker 1842) gives abundant in- formation on these points: cf. August. de Civ. Dei v cc. 1—7. See also Sir G. C. Lewis’s Ancient Astronomy p. 306 ff. and Proclus in Schoell p. 32 referred to above. 54 TIIE NUPTIAL NUMBER. from the seventh sign, quod est contrarium, plenissimus poten- tissimusque conspectus quosdam iam maturos infantes educit, qui septemmestres appellantur, quia septimo mense nascuntur. Seventh month children are accordingly said 11113011111702 31d- ,uev'pou. The Pythagoreans, as we learn from the same authority‘, thought the child was born precisely at H, i.‘e. after 210 days: Plato held that it remained in the womb for 6 days longer, and as 216= 6 x 36, it is reasonable to suppose that the number of the Swan/01' (36) affected his calculations? We can now see the reason Wh y Plato called cubings “multi— plications of root by square.” It would seem that he, or those whom he is here following, held that the potency which the child receives at the moment when it enters on the third, fourth, and fifth months respectively results in each case by the end of the month in an increase which is equal to the original potency made solid. The sun, as the giver of all 1ncrease3 ,may well give increase to the embryo, and greater will be the in— crease, the more fully as the months roll on he flames upon the poZpa of the child’s conception. It is the business of Nature to receive his potencies which may be measured by 3, 4, and 5 in. the three months that make the child, and (Br/i009 ryc‘1p 75 0115,1101- 709 4115019), making them holid by adding to them two dimensions, ' create the body of a living child“. I have stillto explain why the words 6,110101512711112 76 mai ' 3 I I I ' avopowvwwv Kai 11135011er 116012 91561110117101) are sald of months. The rolling months‘ make like and unlike,’ because days and nights, nay, spring and summer and winter5 are their offspring, 1 Ch. x1. seems however to be no trace of the THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 55 ever like themselves and unlike one another. They wax and they wane with the phases of the moon that bore them‘, even as their own children, night and day : 5Z9 6 wa'rép, 77112569 8150 [cat Bélca' 711312 5?: é/cda'rcp ' Icovpa1 651711011711 311111319311 62309 é’xovaaL' a'1‘ [116V Rev/c111 e’aa—w 18611511? 8 aU’Té‘ ,uéka1va1, dddva'ral, 56' 7' 60110111 dwo¢01vvdoua1u awaaa12. I have now to deal with the words mil/Ta 7rpoo-n'ryopo1 [cat porra qrpoq deflta dwe’dmvau. It is curious that these words find 1 an echo in a fragment attributed to Philolaus3 on the virtue of the number 10. The resemblance may be interpreted as an indication either of the spurious or of the genuine character of the fragments of Philolaus,'but I am inclined to think that it is slightly in favour of the genuineness of this particular fragment, for it is not unlikely that in a passage so full as this is of Pythagorean influence there should be some verbal indications of the source whence Plato drew something of his inspiration. However this may be, we are told by Censorinus4 that the Pythagoreans thought the development of the embryo pro— ceeded according to the proportions of the harmony Or Octave: eos vero numeros, qui in uno quoque partu aliquid adferunt mutationis, dum aut semen in sanguinem aut sanguis in carnem aut caro in hominis figuram convertitur, inter se conlatos rationem habere earn quam voces habent quae in musica' 015,11- (tan/o1 vocantur. How they worked the idea will appear from these words5: quorum prior ac minor (so. partus,‘i.e. the seven months’ child, to which they assigned a life of 210 days within the womb) senario maxime continetur numero. Nam quod ex 2 See note on p. 53. 3 Rep. VI 509 B Tau fihcou Tois 6pm- ,uéuots 015 willow, alum, 77‘71/ 7-017 6p6‘10'0a1 5611111111! 7rapéxew gin’ydeLs‘, dRML Kai 77‘111 yéueow Kai 0.13.5171! Kai 'rpoqifiu. 4 Proclus (see Schoell p. 28) appears to give to the words aiié'éaets duvd/reual 7'6 Kai duuachevéneuaL a further astro- logical meaning, interpreting (Swat/1611011 of the stars that prevail, and 6vua0'7'eud- Mei/111 of the stars which are prevailed against 61/ 7012s anopiuats Lb'pdts. There astrological meaning of 611111100111 and dot/0107615660011 as early as Plato. 5 Aristotle even went so far as to say that the moon—infirm the maker of months—produces a winter and summer in the months as the sun does in the year: see De Gen. An. Iv 2. 767a 5 6 #éu'ydp iiNos e’v 6M.) 71,3 e’mavrgfi' rate? xemc’bua Kai 0épos, 7'7 6% “MW e’v 7-13 ,quL. Compare Schoell’s Procli partes ined. p. 29, 1, and Sir G. C. Lewis’s Ancient Astronomy p. 310. Proclus explains 6110100qu 16 ml. due/10101711711 as astrological terms equivalent to «rt/1911101101 and dafimpwua: here again he seems to be ante—dating the superstitions of his own age. 1 Hesiod ( Works and Days 771) calls the first half of the month 6165611611“, and (ibid. 796) the second ¢0£uwv. 2 Cleobulus ap. Stob. I 240. I shall have occasion to refer to this epigram again. 3 Mullach II p. 4 111711 66‘ 01370: (BC. dpLO/Mis) 7ro1'wa $11de dppéfwu 0110013061 1ro’wra 'yuwaTd Kai 1rol'ro'ryopa oth- Xfikou‘ Nani yvu’mouos «Maw 0’11rep'yd- fen“. It was Tannery who first drew my attention to this remarkable parallel. 4 Ch. IX ad fin. Cf. also Arist. Quint. p. 142, and Plut. wept Tfis év TL/Latlp Ipvxo'yovias XII 1018 B. 5 Ch. XI. THE NUPTIAL NUMBER. semine conceptum est, sex, ut ait (sc. Pythagoras), primis diebus umor est lacteus, deinde proximis octo sanguineus: qui octo cum ad primos sex accesserunt, faciunt primam symphoniam 84d Tea-odpwu. (That is, the fourth, which is 8 : 6 or 4 : 3) Tertio gradu novem dies accedunt iam carnem facientes: hi cum sex illis primis collati sescuplam faciunt rationem et sectindam symphoniam 8w} 7rév're. (That is, the fifth, which is 9: 6 or 3: 2.) Tum deinceps sequentibus duodecim diebus fit corpus iam formatum: horum quoque ad eosdem sex collatio tertiam 3rd waaa‘w reddit symphoniam duplici rationi subiectam. (That is, the octave, which is 12: .6 or 2 :vl.) Now 6 + 8 + 9 + 12 = 35, and as we saw on page 35 that 35 is a a'paom'a, 210 which is 6 x 35 contains 6 a'paoufat or octaves. It is impossible not to admire the ingenuity of this, whether it rests on any foundation of fact or not, but the question of interest for us is, How did Plato picture to himself the ‘ harmony ’ in the development of the child? So far as I have observed, he has not expressly told us in any of his writings, but it is worthy of remark that the making of Soul in the Timaeus likewise proceeds according to the proportions of the octave‘, and that the Universe is con- stituted, as Plato thought, in the same way 2. I have little doubt that if he speculated on the subject at all, he followed closely the views of the Pythagoreans, framing the Microcosm, as his manner is, on the lines of the Macrocosm, for although he made3 the period of gestation 216 (i.e. 6 x 36) rather than, with the Pythagoreans 210 (i.e. 6 X 35), yet, 216 is only 210 + 6, and 6 is itself a significant number‘, being the union of the first male THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 57 Let us now sum up our explanation of the wept/0809 of the dvflpaivrewu 'ye‘vmrrév. It is the shortest time within which a child can be perfectly fashioned within the womb, viz. 216 days, and it is during the third, fourth and fifth months after con- ception that the harmonies of a living organism are fulfilled. Section 6. The Diapason: or the words from (51) évrr’rporoq wvdafiv to é/ca'rou 32: Iczifiaw TptdSOS‘ explained. We now approach the diapason-—the cycle of the Universe. I will shew that the words which we are about to discuss give us both the WEpfOSOS‘ of the 0620p «yew/17761; and the Great Year—— the duration of which is one and the same. I will farther esta- blish yet more surely what we have already seen to be probable, that the period of the world’s making was conceived by Plato as of the same duration as that of its unmaking; in other words, the Greatest e’mav'rée, which, as we shall see is 72000 years, falls into two periods, each of 36000 years, in one of which the] world is made, as in the other it is unmade. With Plato, as with Heraclitus, the Universe is eternal, not because it always is as we see it now, but because for evermore it is born and dies, each successive body housing its immortality in time : KéU/LOV 761/35 761/ av’Tdv dwdurwu oiI'Te TL? @6031! 013,76 dvdpai'rrwz/ 6,706,770'6, (DOC 771/ aiei [cal é’a'n Kai é'a'rat 770p deb/{(0015 dWTéaevou ,aé'rpa Icai. dwaafiewdaeuou ,aé'rpa‘. interesting remarks are made on the (p. 151) that the sum of the sides 3 correspondence of the numbers 210 and and 4:7 (the repiodos, in months, of and female numbers5. 1 Tim. 35 B. In the Timaeus 44 D ff. the creation of man’s body by the created gods is described. It is reason- able to suppose that the embryo was thought by Plato to develop on the same lines as those on which the gods first made the human body, and a minute study of the Timaeus from this point of view might possibly yield some in- teresting results. In Tim. 91 D Plato sums up the development of the embryo in the words aéxpc 1r6p d‘u—dys els a'povpaw 1'1‘11! ,afi‘rpau dépm'a {11rd a’mxpérn- Tos Kai. ddrdnhaa'ra {93a Karaareipau‘res Kai rdhw 6LaKpluaVTes‘ ,ae‘ydha e’urbs éxfipéxpwu'rac Kai [1.er 70610 sis ¢€2s dya'yo'wes ('de dWOTGKéa'wa'L 'yéveaw. 2 Rep. x 617. 3 See above, p. 54. 4 p. 50 above. We shall see that Plato recognises the period of 210 days» and also the longer period of 270 in the sequel. 5 In Aristides pp. 142 foll. many - w~mmw~~>w~r 1..ew-.«w»wv,.m—Wmum~m . ,v—..av..»m..u.- 1 216 and their parts with the pheno- mena of gestation. Thus in 35 days (the sum of 6, 8, 9 and 12), the child, he says, is formed: and 35 x6:210, which is the repiodos Twat! é'IrTapLfiI/wv huepna't'a: the sum of 1, 2, 3, 4 (in which are involved the i’aos, BurMaiwv, fiatéhws, and érlrptros ratios) =10, added to 35, yields 45, m0’ 811 «paw. ,uopgboil‘a'fiat 'rai. e’weo’mnua: and 45x6 =270, which is the number of a nine months’ child. It is also important for our purpose when in speaking of the Pythagorean triangle he notices the emawos), of 4 and 5:9, of 3, 4, and 5:12 (the number of signs in the zodiac), and that (as we have already noted) the sum of the cubes of the sides :216, and 216 + (3 x 4 x 5):276 which is about the period of the nine months’ child. The Pythagoreans were certainly single-minded and whole- hearted in their devotion to the octave and their favourite triangle. See also Plutarch 1repi 177$ e’u TL/Adlq’ Ipvxo'youias x11. 1 Heracl. Frag. 20 (ed. Bywater). I --_" i x \ f: _, " a. L ,7 3, . , .5, 7?, ”g .. 31‘ 2‘.- .r: 3g 2; . it") THE NUI’TIAL NUJIIBER. \f It will be convenient to begin by justifying more fully than has hitherto been done the statement that The period during which the World is in the womb is equal in duration to the period during which the World endures‘. _ We are here concerned primarily with the myth of the Politicus (268 13—274 E), the Timaeus 29 E to the end (especially 41 A—44 D), and to a less extent with the speech of Aristophanes in the Symposium (189 0—193 E). The myth in the Politicus is so important for the right understanding of the wepioBoq of the world’s creation, that some account of it is absolutely necessary here. I will analyse it as succinctly as I can, quoting the words of the original where they are important for my immediate purpose, which is to prove that, in the Politicus, the period during which God accompanies (av/uroSnryeZ Icai ovyxuxkei) the Universe upon its way is in fact the period during which the Universe is being ' fashioned in the womb of the Creating Soul“. At one time, says Plato 3, God himself accompanies and helps to wheel the revolving world, at another, when the times are fulfilled, he lets it go, and the Universe begins to roll back again spontaneously, {(93012 31/ [cal qbpbvncrw eiMyng e’rc 'roi} avuapuéaawroq Ica'r’ dea’q. The reason for the reversal of the world’s motion is that the Universe, since it partakes in body, is not exempt from change, but being more nearly so than is aught else corporeal, 72):! dua/cv'lcxno-w eixnxev, 1 I have no wish to mingle in the wordy war which has raged from the time of Aristotle on the question whether Plato really held the world to be 'yevmrrbs Karol xpo’vou, and not merely Ka'r’ e’1riuoww (see Zeller3 II p. 666 it). It does not affect my argu- ment or results in the slightest degree, because those who interpret Plato in this matter allegorically may simi- larly interpret me: they have some- times to be so interpreted themselves. Our first business is to interpret aright what Plato said: when we have done this (as we have not done yet), it will be time to inquire whether he meant what he said to be literally understood or not. See also page 68, note 2. 2 In the Orphic verses this identical metaphor is used: see Proclus twin. 94Bz1rdu'ra quip e'v Znuds ne'ydhou raids debaa'rt Kei'rat Kai vad s 6’ éw‘. 'yaa'répt 0191! [5a necbéxet, and again 95 E roo’vexa any 143 1rav-rl Aids 7rd)\w évro‘ s é'réxfln Aidépos eiipeins 7’76’ au’powoi} d'ykaou iz’ipos KTh. This whole section in Proclus is most instructive, as well as sections 99-100, whether Plato followed the Orphic doctrines, or they him. ' 3 Polit. 2690 E. I ma;- ismuammfiwmmw ..- THE SIGNIFICANCE OF THE NUI’TIAL NUMBER. 59 37-1, out/cpo'rd'rny 'rfis‘ adrofi Icwn'a'ecos 'zrapdkhafw. Thus Plato declares that the Universe 707% néu zi'rr’ £15)»an o-vpxrroSn- 76200at’ 3650.9 aiTias‘, 75 2:61) ndhw énmraiuevov Ital haufiduovra ddavaoiau én‘tolcevaa'rrfiv 71'an 7013 37}- utovp'yofi, 7079: 3’ bray til/603?, 51.’ éauroi} av’rdu iévat 1. At present the universe is rolling back (dualczi/ckna'ts‘): in the reign of Cronus it rolled forward 2. The end of the backward is the beginning of the forward movement, and when the forward ends, the backward begins, and the limits of both are marked by destruction among animals and men. When the backward movement ends, and the forward begins, a few men are left surviving, and these suffer change in sympathy with the whole. The old grow middle-aged and young again till at last they dwindle to a point and disappear: fresh generations are born, not from one another, but from the earth: for those that lie dead within the earth come to life again and in their turn are born old, grow young, and vanish, hoovs‘ ,un) 0669 admin eis‘ 81%th ,uo'ipav end/Macy. In those days, when God ruled the rolling world, and divine shepherds kept their flocks, no creature preyed on any other, nor was there any war nor strife. God was himself the shepherd" of the earth—born: they had no 7roM-reZat nor Ic'nfaew ryuvamév 'Icai. nut/3am, being born by resurrection from the ground. The earth, their mother, fed them with abundant fruits, and they toiled not, neither did they spin. Whether they were happier than we depends entirely on whether they used their manifold advantages as means to help them to attain unto Wisdom: tradition says they did not. When the forward movement ended, and ueTaBokfiu 3561. ryi'yveadat Icat 31) mai 76 ryn'ivov 17577 717312 dun’kwro 'ye'vos, naia'as endow-779 7739 \[rvxfis‘ 7&9 ryeuéa'ets dwaSedwxvias, baa 77V éxdary npoaraxdév, Toa'afficr'a eis‘ 7731/ ane’pua’ra necrotic-rig, then the pilot of the Universe oiov myBaM’wv oZ’a/toq d¢éueuoq 659 7731/ (1.157013 newton-nu d’fl'éd‘T’l). Thereupon began the backward movement. At first there was o-ewuo’e within the world, attended by destruction among all kinds of living things. The few who survive, ceasing to become young, grow 1 270 A. 2 271 D, 269 A. 6O . THE NUPTIAL NUMBER. old, while those just born with hoary hair die and return to the earth from whence they have just come. Fresh genera- tions are no longer born from the earth, but even as the world is now left to itself, so also are all its parts, and each race breeds offspring from its kind 1. After the shock of turning, wpoekdév'ros [Icon/of) xpovov, dopquu ‘TE [cal Tapaxfis 7’7'Sni7rav6- ,uevos‘ [cal 705;; decoy/.0311, yakfiuns‘ Erbkafiépevos‘ El]? 76 Toy eiwfio’ra Bpo'pou "roll éavroz’) Kara/coopozlpeuoe fist, éwopékemu A t A \ f n \ A [cal Icpdros é’xaw (1.13769 Twp e’u av’Tcp 76 [Cat eav'rov, 7171/ "rov_ A I 2 Smutovp'yov Kai wa'rpds‘ (Zwoavnaouev'wv 5L5axfiu €59 Suva/.Lw . Gradually the Universe became less and less accurate in its A \ A I movements: 7-013va Sé (£2379) 7'6 awaa’roeaaes‘ 71/9 avry/cpaaews‘ . / «‘ ai’wov, 75 7'69 wdkat 7ro7'é (1)150er fuquogbou, bu} 7ro7t7tns. A I ’ 3. 1311 ,ueréxou draffas 77-pin €59 76v vvv [coo-p.01! acfiuceo-Qat . . . , . . a! it Is 1; é/proa'deu (2"ch to Which 1s due 3001. xakewd [cal aduca. e’u odpauo") ryl'yue'rao. As time rolls on, the Chaos of the past. makes itself felt more and more till the world is at last in .danger 'of perishing for ever with all that it contains. There- upon God, careful lest his Universe should vanish 659 761/ 769 a’voaoré'rnros‘ d’we-Lpou o'u'ra 7677011, takes the helm again, and reversing the motion of the world, d0dvaxroz/ aflrov Ical, dryq'fiwu dwepvydé’eTaL 4. It would be an interesting inquiry to investigate the sources from which Plato drew the materials for this myth. That it embodies many echoes of the early cosmogonies, there can be no doubt. The essence of the whole story is contained in two lines - of Hesiod, who, in speaking of the end of the fifth or iron age in which we live, observes 3 h 3 I Z609 3 dkéaeo [cal TOUTO 'yévoq pepéwaw avdpwwwu, 5' 3 9\ I I I 5 an av ryewoaevor woktoxporagbor. Tekeflwa'w . Plato would have interpreted this to mean: the present age will end when the age of rynryeuei‘q begins. Now as many 1 273Efi. Polit. 273 E, where 707m}. «Infill-rot of 2 273 A—B. course means “born with grey hair.” 3 273 B. See also Heraclitus Frag. 78 (ed. By- 4 273 E. water) : the germ of the myth is to be 5 Works and Days 178—179: cf. found there also. ‘ , ; -,- - a‘5WW‘U‘MLJW'a'éu/w-udc‘m. m m... a, _ THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 61 of Plato’s predecessors and notably Empedoclesl held that the world was in the process of being formed when she put forth the earth-born creatures, it is probable that the ‘children of earth ’ in the Politicus are likewise intended as a feature of the making of the world. . That it is the creation of the world which Plato means by the ‘forward movement ’ in this myth can be proved I think, not only by a study of the language taken by itself, but also by comparing the whole myth with the Timaeus, and illustrating it by the Symposium. Let us first look at the passage by itself. The language of 270 A2 seems to me conclusive. While the world is being guided by God, it is To @3311 7rd7tw 6’7”- -K'Tai,uevoz/ Kat Kapfldvov'ra ddavaob’av e’wwxevaa'rn'u. The present participle can only mean that the world is acquiring life again, that is, is being recreated, refashioned in the womb of the Divine, while God accompanies it. Once let go 3, the world iS‘described as £98011 51/ [cal ¢p6myaw eiknxde 6% 701') avuappoa'av'ros‘ az’yro Ica'r’ dpxa’s‘ (269 D): and to What can cvuapaoa'av'ros‘ refer if not to the dp/Low'a of creation ? Nor is it possible to read the description of the life of the 7777612629 without being driven to think of the golden age of Hesiod 4, when our world was being made and the children of the Earth like Gods é'é’mov dxnb‘éa eupou é’XOVTee midday c’i'rep 're 71'6va Ical ol‘é’vos' ov’b‘é 'TL 360.61; ryfipas‘ e’m'y‘v, aid 3% 7768039 Ical erpas‘ littoral.5 Tépvrov'r’ e’v Hakim” [ca/«311 é’lc'roa'geu (i'n'dv'rwu gufia'lcov 8’ (59 5717/9) Sedpnpéuot' éa‘dkd 3E waiv'rrf 7020'“! (197V Kap'rrov 3’ é’¢6pe é’et’Swpos c’l'povpa ad'roludrn 7ro7t7t6v 76 Karl c’fcbeovov. 1 Mullach I p. 9 vv. 306—309, 313 5 See Symp. 189 E. I strongly sus- —316. . 2 See above, p. 59. 3 I do not think it is by accident that Plato selects the word dueflfi in 270 A, for timévat means sometimes “to put forth offspring”- (e.g. Rep. 111 414 E), but of course no stress can be laid on this. Works and Days 109 ff. pect that Hesiod also conceived of the golden race as round, and never grow- ing old, but young (01756‘ TL Beahov 'yfipas‘ évrfiu), and dying like a sleeping child (fiufiakou 5’ Lbs ihrvgu Bedamxévot). A thorough study of the influence of Hesiod and Hesiodic notions on Plato would amply repay the trouble. 62 THE NUPTIAL NUMBER. Further: the words of 273B foll. harmonise well with my theory. The world when it is let go fares ill, becauseof the slumbering elements of chaos, in which it partook before it came into its present chapoc—Viz. when it was in the making and ' God accompanied its movement. In like manner, the creation of the body of the world in the Timaeus is nothing but the reducing of chaos into order: see for example 30 A “may 5001/ 13v dpa—roy wapakdBdw (so. 6 (9669) mix fiavxiau d'ryoudkh‘z [cu/015- uevou whnmtekék‘ Kat (i'rd/c'rcos‘, €139 TdEw c1137?) fitya'yev 6% 7739 oi'raEL'as‘, fiyna'dpevos‘ é/ceZuo 7015701.) 7TdJJ'TwS‘ duewou. The pre- sence of the Deity coerced the chaoswhile he made the world, i but when the universe has issued from the womb, the trouble reappears. . We have now to compare the myth of the Politicus with the making of the world in the Timaeus, and the speech of Aristo- phanes in the Symposium. No one can read the Politicus and Symposium [side by side without being struck by the likeness between these two splendid efforts of the cosmological imagination. The spherical creatures of the Comic Muse are none other than the «yon/611639 of Philo- sophy, who e’ryéwwv [cat é’Tucrov mix 6:39 dkkn’hove dm’ 659 761/ (307er of TéTTL'yES‘I. The slicing of the sphere asunder and the consequent creation of creatures like ourselves is the comedian’s description of the moment of dua/cémym, while the yearning of the two halves to reunite and make é‘u é/c 3110212 symbolises the longing of the iron age for the ancient realm of gold. More might be made of this, but the Timaeus offers us an even more fruitful field of discovery. In the earlier part of that dialogue we have abundant references to the convulsions which attend the changes in the world’s movements". It is true that the Timaeus is less precise on this point, as for example when the change is called simply mpdmagm, and not, as in the Politi- cus, further defined as duamfixknms‘, but this is because Plato’s purpose in the earlier part of the Timaeus is to describe, not the ‘ moment of the world’s rebound, but the course of its forward movement—in other words, its generation. The myth of the 1 Symp. 191 B. Cf. Polit. 271A. 3 220. Cf. Polit. 269E. 2 Egg. Tim. 21 D, 22c—D, 23E. THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 63 Politicus is inkfact the circle of which the Timaeus is one half While the one Sketches in majestic outline the whole pro ress of -the Universe from the moment of, its inception to its defline the other describes in detail how the Macrocosm and Microcosrri were constructed. In the half-circle which is common to both we shall find that they agree. It Willbe remembered that in the Timaeus, after God had ' created the world, and beforerwmortal bodies were fashioned he “abode in his own nature”: Kale aév 31) (Straw-a. ‘Tafi'Ta 8m- “Tafas g/LGVEII 6’11 7g?) éav'roz'} xa’rd 'rpé7roz/ 7,},6661. What is this but the retiring to his watch-tower? 7-671; 82) 700 war/769 6 hey Ichepwy'Tm‘, oIov undakt'wv oi’ar/coq dqbé/Leuos‘, 629 7-1)!) aii'roi} 77'6- ptwmfiv cinch-7712. The moment for the creation of man has come and the created gods fulfil their part3. This it was no part of Plato’s purpose to describe in the Politicus, because the Timaeus fills the gap. But the younger gods do not create the soul but only the human body: whence then do they get the soul 7 ,God had created it already, and sownit in the earth, and theiother 1nstruments of time: 3076th rods win 659 761/, 7-01); 8’ 659 ask-6117711, "rods 3’ 659 'rd'Mta, o'pryava, xpéuov“. Did the seed lie dormant after it was sownfi? Here the Politicus fills the gap6 ' ism-61,81) «ydp wail/Tam Tov'Taw xpéuoc e’TeKeaifln Icai neraflokfiu rebel. 75712606011, nut 81} [cal 75 (MEI/oz) T7187) way a’wfkw'ro ryévos‘, wa’a'ae f/é/cdatrns 7779 Ilrvxfie 7&9 ryevéaets‘ (2708630)- ICULftQ, no? 61/ e'lcda'ry wpooraxde‘v Tooafi'ra 659 7131/ 07:16pMa'ra wee-0210779, then God left the helm. God had told'8 the souls that they must beisown into their allotted instru- ments of time before they were born as men, and He soweth not 1n valn. Full many a time the seed bore fruit in the children of the earth, and sun, and moon, as their hoary locks grew black and manhood faded into infancy and fell asleep”. 1 42 E. 3 Polit. 272 D~E. 2 Polit. 272 E. y 7 'i.e. sowings. 3 42E foll. 8 Tim. 41 E. 4 ‘ i 5 film. 42 D. . , 9 '0yfia'Kou 5’ dis {hm/(p 5e§,un,uéuoz as . ato wavklfez, or Paul wkarwutfi‘a Hesxod says. See also Symp. 190A '51) 1n 1 Cor. 15. 35 it: compare Eusebius as 6rd; rafira Tpia rd 'yéun Kai TOLaU‘ra Pr. Ev. XI 28. 17 fl"., and August. Civ. 8'11 76 ,uéu dppeu ‘31! 100 'r'fiu’ov 77‘71/ dva‘yu’ Dezgvrll 9 fl’. é‘x'yovov, Ta 6e 07%!) T779 7739, To 66 d,u.¢or 64 THE NUPTIAL NUMBER. Fully to discuss all the passages in the Timaeus which ex- plain or are explained by the myth of the Politicus would lead us too far, but it is necessary to mention a few more points. The well-known crux in 36 0—D resolves itself at once, if we take the Politicus as I take it. In creating the world God makes the circle of the same to revolve from left to right, that is from west to east, whereas now it revolves from east to west. The simple explanation is that the world was created before the a’va/ciichnms‘, which reversed the revolutions of the same like every other revolution human and divine. If I am right in this, it is a signal instance of the necessity for expounding one! dia- logue of Plato by another‘. Tépwv [lETéXOV 11'): “Mm 81!. ml 7) admit/7] du¢orépwu Meréxeu 7r6pt¢epfi 86 id?) 7):! Kai a676, Kai 1'7 1rop6£a abn’bv 6Ld. To 21029 ‘yovefia‘w éfloi‘a ell/cu. There is per- haps the merest hint of this rept- ¢6p1‘1s raped: in Tim. 44 D—E i’v’ ofiu p.67 kvhwdoé/Levov 6’1rl 'yfis iii/In 7'6 Kai Baffin wawodard éxobans dropo? KTX. 1 In the De Caeloyu 2, Aristotle argues that the Universe being"’é‘;/.ipu- xos‘imust not only have a right and left] of its own, as the Pythagoreansl believed, but also an up and down, a front and a back (d’vw Kai prco'u'w, 6p.- 1rpoo’06v Kai dma'fiev): we may con- ceive, he says, of the right and left of a sphere, by regarding the sphere as pu round something which has a right and left of its own (as for ex- ample a human being). Further, Aristotle holds that the right is the side from which Vmovement begins: 665L611 yap éKota'rov hé'youev, 6661/ i7 dpxi) 77’): Karen 761m» tar/fiestas: see also de Incess. Anim. ch. 4), and also that the right of the Universe is the east. In order then to explain why the Universe re- volves from its own right round again to its own right, he is compelled to assert that the South pole is 16 dug), and the North pole Tb mine, that in fact (as we should express it) the Universe stands on its head. If Plato had held that the Universe has a right and left of its own, then this passage of the Timaeus might possibly refer to the present movement of the Universe (conceived as standing on its feet) ' from its own left round again to its -- own left, but it is certain that he did not (see Tim. 620 ff.), but meant by / I I o “ motion to the right ” Simply motion to the right of the spectator. And if in the face of his own definition (Laws v1 760D) of right as “towards the east,” Plato could use the words é1rl . 565w; here in the sense of “towards the west,” he is nothing but a framer of riddles, and poor at that. Neither Martin (Vol. II p. 42), nor Bockh (Uber das kosmische System des Platon, , p. 29) offer any explanation worth considering: but Bockh at least re- cognises the difficulty. The words mean “toward the east,” and refer to the motion of the Universe before God left the helm: compare the story about the change in the direction of the sun in the time of Atreus, at the beginning of the myth in the Politicu3‘(269 A): To 7r€pl 'rfis ue‘rafiohfis 615176ch 1'6 Kai o’warohfis 1'7Mov Kai 76w a’iwav da'rpwu, Lbs 6,001. 6061/ p.611 dram-00‘s; ufiu, 6i: 106701! 7616 16y 'rb‘lrou 6615670, (ivé- THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 65 'Again: in the Politlcus we hear of the convulsions which accompany the commencement of the backward movement of the heavens, when our present cycle came into existence. Now the circle of the same exists not only in the world-soul but also 1n the soul which God had already made for human creatures‘: reverse the motion in the one, and it is reversed in the other. The moment of the reversal in the soul which is to belong to man, is precisely the moment when that soul 1s enclosed in a human body : and the convulsions which happen at thls moment are in the Timaeus described in language which recalls the words of the Politlcus. c SThe shock of, the rebounding Universe is thus describedZ: o, 6 neragrpedmaevoe [Cal Evpfidhkwv, dpxfis‘ 7'6 adv-$611769 filial/Tlau op/uyv (Splwlyae‘L/S‘, 061,071,611 wokuv 6,1! éow'ra'} 7mm?!) aMtnu a5 qSflopciv é’giaw 7761.1170le dweaprya'aa'ro. Ofithe souls we read": elei'n'o'nmdv 6v86062cmo wolf/u oz’i'r’ éxpdrovv ci’J'T’ 6Icpal'rofiv'ro, Bic} 8’ écbépovro [Cal 6'¢>6pov: sensations 61/ 'ch napoun whet/077711 Ical [LG’YliO'TnlJ wapexdpeval. Iclmya'w, nerd mi} piOIJ’TOS‘ 611361696639 dxe'roi} Icwofiaal Icai a¢63pa adovaat "role T779 \Irvxfis‘ 7r6pL680v9, 76v ,uéu Tali-ref} wav'rd'n'aa'w 6’7r6’- ,‘na'au éi/av'rlw av’rfi fiéova'al, Ical éwéa'xov c’lpxova-av Kai Love-av, 727v 8’ at? darépov 3Léaelaav—(the revolution of the same must needs be checked before it will turn back). While the coflmmotion lasts, Taflrdv 7'90 Icai 9d'rep6u 70v Tdvavrla 7661/ a’My- Haw npoaa'yopefiovaal \IJ‘GUSGZS‘ Ical dvén'roo ryevyéyaaw (sc al wepl¢opal)‘, and the revolutions think themselves victorious Tehke 6’ 6K 7'06 éuaurlov, 1676 66 67) ,uaprvp'éa'as film: 6 0663 ’A'rpei‘ ,ueréflaheu (1.1de é1rl 1'6 ufiu a-xfi/m. In the Electra 726 fi. (quoted by Campbell) Euripides alludes to the same storyrin a way which shews that it was interpreted by some as the mythical expression of an actual permanent change in the movement of the heavens: see also Orestes 1101 fl. 6661’ em: 76 T6 wrepwrou dMou pceréfiaheu oi'paa—érrawépov 7'6 (Spit/lama. Hehaddos 65s 66611 d'AhaV ZEI‘JS , Me‘rafidhka, and compare Hdt. II 142, and Sir G. C. Lewis’s Ancient Astro- A. nomy, pp. 69, 133. The airy way 1n which the myth of the Politicus is so frequently said to be “not serious ” (see Zeller3 II 1, pp. 668 and 685), before even an attempt has been made to interpret it, is unworthy of modern scholarship: the ancients were more wise, and some of them (to judge from Proclus in Tim. 88 D if.) appear to have taken the right view. 1 Tim. 41 D if: cf. 340 if. 2 Polit. 273 A. 3 Tim. 43 A ff. 4 44 A. 66 THE NUPTIAL NUMBER. when they are beaten (xparozfiaevat repaTeZv Boxofio't). This is why a soul a’ivovq ryt’ryve'ral. Ta 7rpc37'ov, when it is imprisoned in a mortal body. Now observe what happens when the waves abate. The Universe, says Plato‘, wpoekdéuroe {Kai/of) xpévov, dept/3a)!) 're Ital Tapaxfiq 1’7’817 wavépevoq Ira A A I ’ i 7am O'ew'pwv, yaknvns‘ 67n- . haBépeI/oq ei’c 7'e 7'61) eiwt967'a. 8p6uov 76v éavrofi lea/ra- xoa/tozfipevoq fist, i.e. the Universe begins to go back the way it came. Of the soul in man it is said: bray 3e 76 7739 (1135779 [cat Tpo¢fi9 é’ka7'7'ov e’7rt’y bed/ta, wdkw Be at weptob‘ot RauBavé- ,uevat yakfivns 7' Mallow erbovroe 7'06 xpévov, 7):) e'av'ra'iv 686v ’t'wa'l. [cat xadwra’ivrat \ 7'67'6 7’7’31} 77de 7'6 Ica'ra (inflow Zéwwu axfma e'mia'rwu 'réiv Iczikav at 7rept¢opat Icwrevdvvé— ,uevat, 7'6 7'6 GdTepov Mat 76 7'azi7'dv wpoadryopeiiova'at 1cm" dpdc’w, é'ucfipova, 7'61) excl/Ta adrde tyu'yvéuevov d7ro7'e7toi30'w2: when the ‘ periods’ becoming calm begin to go back the way they came, then the revolutions of the respective circles being forced into the form (not the reality) of that which goes its natural way, straightly name the Other and the Same and put their possessor on the road to reasons. 1 Polit. 273 A. ‘3 Tim. 44 B. 3 The parallel with the Polittcus seems to me to make it probable that 7rd.)\w means ‘backwards.’ Ast is per- fectly right in saying that 7ro’0uv in Plato is “retro (cum vv. eundi et ducendi est redire, reducere)” ; and for an “again,” in which there is nothing of “backward,” Plato gene- rally uses afiBLs. But some may think that mix“! goes with haufiauéueuat, as I think Archer-Hind must have taken it, though he does not translate the word: “and the revolutions calming down go their way and become settled as time goes on.” I do not think this is right: but esto. If it is while they are getting calm a second time that the periods go their way, when did they get calm a first time? There is no possible answer which does not bring in the theory of the Politicus; for the emphatic position of rdkw seems to exclude the possibility of connecting it only with 'yanns in the sense of “becoming calm again ” so. just as they were calm before the shock. But even if rdkw did mean “again” and not “backwards,” my argument would not be affected: for the words of the T'imaeus clearly denote the same motion as the words in the Poli- ' ticus, that is, the backward movement. The emphasis on axfiua, the care- fully selected expression 16 T6 fiat-repay Kai 7'6 Tat/Tow wpoaayopeuovdat Ka'r’ 6p06v, which by no means implies that the soul’s circles themselves revolve accord- ing to nature, only confirm my View. I know that Archer-Hind’s translation “the orbits are reduced to theform that belongs to the several circles in their natural motion” differs toto caelo THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 67 The subsequent history of the Universe is parallel to that of man, but there is a difference. The Universe goes from good to bad, and from bad to worse : so therefore does the mace of man. This is the ai’wov of the degeneration of the ideal state—(final «ydp ai'nou eivat 71‘) m) ,uévew [weep (370», 31/ 70116 ,7rep708zp aeraBdAkew‘. Be our archons never so perfect the ageing world will make their state decay. But as for, the 1nd1v1dual, it rests with him, or rather with his educators what hls \fate shall be: (’21) ,uév 0131/ 81) Ital fuvewohapfidvn'rat' 7't9 ope? Tpod») watseuaewc, dhoxknpoc dywis‘ 7'6 7ra1J7'e7tc39, Tfiv ,ueryw'ITnv d7ro¢vryaiv véo'ov, ryt’ryveTat, xarayekn’aas‘ 3e, xwkfiv :ov Btov Sta'n'opeveeis‘ é’wfiu, dTeXfis‘ Kai (ii/677709 eZQ'IAt3ov 7rd>tw epxeratz’. Vain though the aspiration be, while he is yet alive let lum aspire 7&9 7repi 7'73); ryéveaw 6’11 733 Ice¢a7tfi Stegbfiap/téuag, nuwv‘ weptéSovs e’fopdofivra 5M2 7'6 Ica'rapavédvew 7729 7'06 won/7'09 a'p/tovt’ac 7'e Kai weptduopcis 7'93 xaTavoov/téuw 76 Katmai/001312 e’fopotaiaat Ica'rd 773v dpxat'av ¢1§atv3" let hlm seek to bring the movements of the Same and Other to their original direction. So when the “strong deliv’ress” comes and he knows Death, “the sacred knowledge of Death” whom he has wooed so long, he “returns into his dwelling- place within his kindred star“,” and in due time “unless God has set his lines in other places“,” learns what he had learnt before, that “ to grow old in Heaven is to grow young.” Thus 1t is that the myth in the Polittcus carries us to the very roots of Plato’s philosophy. It is essential to understand X, .1f we would understand the full meaning of the Number. torlssltlzzile hlmself connected the two, as I shall now proceed In the passage to which I have already frequently referred6 occur these words": Ical Std '76 7'06 xpéz/ov, 31.’ 31/ level. mil/7a: from mine, but apart from the fact that this is inconsistent with the Poli- ticus, it can only be got out of the Greek, I think, by doing violence to the language. 1 Arist. Pol. v 12. 1316a 4. 2 Tim. 44 B. 3 Tim. 90 D. 4 Tim. 42 B. 5 Polit. 271 c. 6 Pol. v 12. 1316“ IE. 7 I take Bekker’s text, changing only (with Susemihl) 76 into ye: but I think that 1'00 xpéuov should be 761! xpouou or 5—2 68 ‘ THE NUPTIAL NUMBER. perafidkkew, [cat 70‘. m) duct dpfdpeua vireo-fiat dpa peraBdXXet, 0501/ 613 73'} wporépqt fiae’pq e’ryéuero 7139 Tpo'n'o'y's‘, iipa. d’pa peTaBdKXet. “And by means of the time, by reason of which he says that all things change, those things also which did not begin to be born at the same time are changed at the same time, for example if a thing has been born the day before the turning, it consequently changes at the same time, (so. as something born at a different time from it).” The words are a criticism of the ai’wou ,ueTaBohfie assigned by Plato in the Nuptial Number, not of the Mime or process of dissolution—— Plato’s account of which Aristotle in fact commendsl. Now Plato’s ai’rtou perafiokfis may be regarded as the number which is the measure of our present cycle: this therefore is what is meant by 6 prvos‘, 3t, 31/ Xéryet wail/Ta, ,uerafldkkew. Aristotle’s objection then amounts to this. In that case a thing born one day before the end of the cycle changes at the same .time as a thing born, say, on the day when the cycle began: but if you hold that the wept’ob‘og of the whole is the cause of change, it should be fulfilled for each individual thing before it can cause that thing to be changed: so that if you call" the wept/0509 say 36000 years, a thing born in the year 1 will be changed in the year 36000, while another born in 2 will be changed in 36001 and so on. In brief, Aristotle argues that change cannot be explained by the theory that the world will be changed at the end of a certain cycle: for he himself believed. in the non—generation of the Universe, and continually attacked Plato for saying that it was generated? The "rpm-17’ else (more probably) BL’ by he changed marduat Kai drahzielu or’zééu dhkotb’repov to 5L’ of) (with Spengel). This however rarely early 7? To KaTaa’Kevdfi‘ew ai’rrbu does not aflect my argument. 1 101370 ,ue‘u ofiu ab‘rb hé‘ywu i’cws mi KaKc'Zzs (e’vééxerou yap dual Twas oils mu- 5ev0fival Kai 'yeuécb’cu c7rou6a£ovs (Six/6pm déi’wa'rou). 2 De Caelo I 10. The words of Aris- totle in this passage, whether they were intended to refer to Plato or not, ex- press in a single sentence Plato’s statements on the generation of the world (280a llff.) 7a 6’ évaMdE a'u- didtov ,u.e‘v ded MeTaBdAAOV-ra, 7'7‘71/ ,u.op- ¢7§u, (5'07er ei’rts e’K ratébs d’vau 'ywo'ueuov [(412 6’5 duo/36$ 7ra2‘6a 676‘ ,ue‘v ¢66lpeadal 67% 5’ eTum ofot-ro. Com- pare Politicus 270 E, 273 E. Plato’s view is put briefly and emphatically by himself in the Timaeus 31 B sis 555 uovo‘yevhs obpaubs 'ye'youu’as éa’rl ‘re Kat é‘T’ é‘a'TaL i.e. this universe, one and only-begotten, has been born and shall be born hereafter. Compare page 58. THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 69 in Aristotle is the ”Woml of Politicus 270 D 370w 7; 7739 vfiu leadeornxvt’ae e’vav'rt'a ryt'ymrrat "rpo'mj. We have now shewn that the period during which God accompanies the Universe in the Politicus is the period during which the Universe is being fashioned, and thereby justified our identification of this period with the wepr’oo‘og of the 062011 ryeu- mrrév. Let us now determine the duration of that W's/250809. That the forWard movement of the Universe was conceived by Plato to last for the same period of time as the backward move- ment, is certain; for the march of the Universe being regarded as progression and retrogression along one and the same circle, the forward'revolution has to traverse the same space as the reverse, and nothing is said of any difference in the speed of the. two revolutions‘. In other words, the creation of the Universe - is completed in (so to speak) one revolution of a great circle : in the other it travels on to dissolution. We have thus confirmed the result which we reached on p. 48, where we inferred that the making of the World lasts as long as its unmaking, by inter- preting the 'réketoq éptdao'q of the Timaeus as the same number with the Téketoe _dpt6/b69 of the Republic. Each is in a certain sense a Great Year: the sum of the two we may call the Greatest Year—the cycle which sees the Universe begun, com- , pleted, and ended. Each is one ‘journey round’ or wepo’oSoe, and it is therefore with perfect justice that Plato uses this word to denote the time which it takes to fashion the Divine Creature. " The 7rept'0309 of the (962011 ryevmrrév is thus identical with the Great Year. Let us now see how Plato defines its duration in days. Plato builds up the number of the Macro- cosm from the number which expresses the gestation of the Microcosm, Man. Taking the sum of the 3 numbers, 3, 4s, 5, the sides of the Pythagorean triangle, he multiplies them by the hypotenuse 5, and raises the product to the power denoted by 4:, which is the number of the remaining side of the triangle. The wepiodoq of the 06201! vellum-612 is therefore produced by 1 In 271A we find wepupopoi applied to the forward movement: and in 273 E reploaos is said of the backward. The life of the Universe is thus pic- tured as the revolution of a single circle. The plural 7r€pt050t in 2690 and in 270A does not refer to the cycles of the Great Year, but merely to the revolutions of the world on its own axis. 70 THE NUPTIAL NUMBER. manipulating the numbers of this magical figure, just like the period of man’s gestation, viz. 33 +43 + 53. [(3 + 4: + 5) x 5]4 yields 12,960,000, which is the revolution “measured by the circle of the same’” i.e. expressed in days. The life of a child in the womb was also defined in days. .The number 12,900,000 will assume two harmonies, none of Wthh is 36002, the other 4800 x 2700. What are these two dpaoviat? What but the 'a'ppom’a of the World’s life, and the ap/Lom’a of her creation? The latter is sung by the Sirens2 as their spheres roll onward, its volume swelling till the da when man is born: y “From harmony, from heavenly harmony This universal frame began: From harmony to harmony Through all the compass of the notes it ran The diapason closing full in man.” , Theformer is the harmony sung while the spheres roll back, waxmg feebler and more feeble till God’s organ3 answers to His touch again“. THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 71 the Pythagorean perfect number 10. We are now able to express the period in years: it is 1 96§§—0—°=36000 years. Further, 36002 is 3602 x 100. Now we know from the Republic1 that Plato reckoned the duration of human life as 100 years, i.e. 100 x 360:36000 days. It follows that the 06201) «yer/11077611 lives one year for every day in the life of the dvflpaivreoou vyewnrévz. In the second harmony the number 100 predominates again. It is “of 100 squares of the ’rational diameter of 5, minus one each, and of 100 cubes of 3.” Now 4800 x 2700 = (480 x 10) x (270 X 10) = (480 x 270) x 102. 270 is the Pythagorean period of gestationifor a nine months’ child, and 4180, which = 210 + 270, is the sum of the periods of gestation for children born after seven and after nine months’. The period of gestation for the Universe may therefore be denoted by a rectangle whose sides are respectively the longer period and the sum of the two periods in the race of man, after it has been multiplied by the square of the Pythagorean perfect number 10. As the Universe is a ‘magnus homo,’ and man a ‘brevis mundus‘,’ it ' The two harmonies will repay a closer study. Let us begin w1th the first. We know from the Laws5 that Plato counted 360 ‘days’ in the year: the Great Year which is 36002 or (360 x 10)“’=3602 x 102 days, is therefore the square of the number of days in the ordinary year multiplied by the square of 1 Tim. 39 D, cf. 39 B. 2 Compare Rep. x 6173 é1ri 6é Td‘w Klfikhwv m’rrofi dwwfieu 颒 éKda‘Tov [36,817- Kéuou Zetpfiua avareptdzepO/rérnu, qbwm‘yu ,wfaw iefaau, é’ua 'rbuou' éK raac’bv 5é 6nd) 000551! plow dp/Loulau £vp¢wueiw 3 “Dorylaus scripsit esse mundum organum Dei.” Cens. XIII. 4 Cf. Polit. 273 B. This is the full and only adequate meaning of dpaom’a in this passage. It has been shewn that the Nuptial Number as a whole is to be interpreted from the myth in the Politicus, and it is only by the forward and backward revolutions that we can explain why the harmony is said to be twofold. The mathe- matical meaning has been explained on pp. 20, 21. 5 v1 758 B. The number of Senators in the Laws is 360: these are to be divided into 12 sections of 30 each, and each section is to administer the state for one month. The number 60 with its multiples and divisors is the dominant number throughout the Laws. 360 ‘days’ is of course only an ideal division of the year: see page 73. Plato recognises (with Philolaus) 364% days in Rep. In 587 E, where he makes out that the king is 729 times happier . than the tyrant, meaning that he is happier every day and every night of his life (cf. 588 A). 1 x 615 B. Sir James Crichton- Browne in an address on old age (see the Times of Oct. 2, 1891), recently said that “ he thought it a good work- ing hypothesis that the natural life of man was 100, and that in so far as it fell short of that, it was ‘curtailed of fair proportion.’ He would especially exhort medical students to start with a resolution that they would not be content with a duration of life shorter than that either for themselves or for their patients.” 2 That it was the custom to follow up discussions de die natali by treat- ing of the epochs and duration of human life, may be seen from Cens. XIV and Macrob. Somn. Scip. I 6. 65—75. 3 That Plato has previously reckoned the gestation of a seven months’ child at 216 (and therefore, by implication, that of a nine months’ child at 276) need not, I think, create a difficulty. He is dealing here with round num- bers, solely with a view to arrive at the period of the World’s gestation: , and Aristides Quintilianus, in the passage where he is alluding to the Platonic number, recognises that 210 days is the period for a seven months’ child, explaining the number 216 (the sum of the sides of the triangle after they are cubed), as 210+the number which denotes marriage, viz. 6.} Cf. also Theolog. Arithm., p. 40 (Ast), Cens. XI and Macrob. Somn. Scip. I 6. 15—16. Tannery (Rev. Phil. I p. 179 note) agrees with me in think- ing that in 2700 there is a reference , to the nine months’ gestation. 4 Macrob. Somn. Scip. II 12. 11. The ' same expression is used by Philo: see Zeller3 III 2, p. 397. 72 THE NUPTIAL NUMBER. is reasonable that the times which regulate the birth of children should also determine the creation of the Universe. It will be seen that I take the first harmony to represent the backward revolution of the Universe, i.e. our present cycle, and the second as expressing the revolution forward, or, in other words, the weptoBos‘ of the 062012 ryeumrréu—the making of the world. Section 7. The number 36000. We have thus seen that the Universe endures for 36000 years, which is likewise the period of its incubation. Before we proceed to explain a’ptfiuo‘s‘ ryewpe'rpucés, 7060157011 mipooq, duewdeu 7'6 [cat Xe‘tpéyaw ryevéa'ewv, let us briefly explain the system upon which Plato’s reckoning is based. The number 36000 rests upon the Babylonian sexagesimal system‘, which made 60 the unit, and multiplied it by the factors of itself. This mode of reckoning, which to the present day divides our hour into 60 minutes, and our minute into 60 seconds, was Widely spread in very early times, and there are traces of it as far west as Italy. It survived in the Latin use of sescenti for an indefinitely large number, and in the period of 6000 years, which was the duration of a dynasty of Etruscan gods. Among the Greeks we find traces of the sexagesimal system as a measure of time as early as Hesiod2 and Cleobulus3, and Herodotus expressly tells us that the Greeks borrowed from the Babylonians the division of the day into 12 parts“. It is therefore hardly necessary to suppose that Plato borrowed his reckoning directly from the Babylonians, even although, if Berosus may be trusted, 36000 years was THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 73 actually the duration of a Babylonian cyclel. What it is of importance to note is, that thesexagesimal system was very commonly used in calculating long periods of time, from the notion that the year could be divided into 360 equal parts corresponding to the 360 degrees of the circle yearly traversed by the sunz. Thus among the Indians 360 years was ‘a year of the gods,’ 3600 a ‘cycle of Brihaspati,’ 216000 a ‘cycle of Prajapati, 4,320,000 an ‘age of the gods,” and the ‘kalpa’ 1000 ‘ages of the gods’ or one ‘day of Brahma,’ while twice this number, or 8,640,000,000 years was ‘a day and a night of Brahma3.’ Let us now see how the number 36000 is connected with other Greek cycles, and with other indications in Plato of the cycle in which he believed. It does not appear that Anaximander, Anaximenes, Diogenes of Apollonia, or Anaxagoras defined the period during which the world endures, although theyheld the Universe to be ¢€ap7694. According to Stobaeus5 the Great Year of Heraclitus was 18000 years, that is, one half of Plato’s. Schuster’s conjecture“, that the time from one e’mnipwo-Lq to another was reckoned by Heraclitus at 36000 years, 18000 being the 6869 metro), and 18000 the 6869 Biz/co, is thoroughly in harmony with the whole tone of Heraclitus’ philosophy, and brings Heraclitus very near to Plato7. 1 Cf. Hultsch, p. 51. Full infor- mation on this system will be found in Brandis, Das Miirnz- Mass- and Ge- wichtswesen in Vorderasz'en, pp. 7——21, and in Cantor, Gesch. der Math, pp. 67—94. 2 Works and Days 562, 764 et al. I do not of course deny that there must even in Hesiod’s time have been some way of making this division cor- respond with the solar year. See Ideler, Handbuch der Chronologie I p. 257 it, for more evidence on the subject. 3 If the epigram quoted on p. 55 is genume. 4 Hdt. II 109. It would appear that for astronomical purposes the Baby- lonians divided the day into 60 parts: see Cantor, p. 82. 1 I take this from Brandis, Das Miinz- etc., p. 11. Compare Sir G. C. Lewis’s Ancient Astronomy, p. 4001i. The Greek and Egyptian cycle of 36525 years (ibid. pp. 282, 389) is reached by a similar calculation, viz. by multi- plying the number of days in the year (taken as 3651) by 100. Lewis’s ex- cellent and learned work is a mine of information (see p. 256 if.) on the part played by the numbers 60 and 360 in the astronomical reckonings of the ancients. 2 Martin, Rev. Archéol. XIII p. 287 ff. 3 Martin, p. 286. Martin interprets the verses of Hesiod beginning éwéa TOL feta 'yeueds AaKépvg‘a Kopu'wn oil/opal! medal-raw (Plut. de def. 07'. 415 c) by taking 400 years as the life of the Kopdwn, and thus assigns 43,200 (:3600x12) years to the phoenix, whose appearance was generally sup- posed to herald some kind of new era, and 432,000 years to the nymphs. 432,000 years was according to the Chaldaeans the period from the crea- tion to the deluge. 4 Stob. I 417: cf. Zeller4 I pp. 213, 230, 247. . 5 I 264. Cens. XVIII 11 assigns 10800 (:30 x 360) years to Heraclitus’ cycle. 6 Zeller4 I 640 note 2. 7 The 6669 Kim leads to the forma- tion of the world, and the 6669 am 74 THE NUPTIAL NUMBER. The nearest approach to the doctrine of a Great Year in Empedocles is the theory that the wicked Barium/69 are con- demned “to wander away from the blessed for thrice ten thousand seasons‘.” Zeller rightly observes that this in no way determines the duration of the world, since the Samar/69 must have lived-before the beginning of their wanderings and will live after they are done. A comparison with the Great Year of Philolaus will not yield any satisfactory result, because, as we know from Gen-- sorinus”, he counted 364:},— days in the year. We can only say that had he counted 360 days in the year, then according to the method of reckoning which he employs, his great year would have been 59 x 360 = 21240 years, which is {36% of Plato’s cycle —and Philolaus (as well as Oenopides) recognised a smaller cycle of 59 years”. Aristotle is hostile to the idea of a Great Year, and the only passage which could possibly be otherwise construed is in the first book of the Meteorologica 14 p. 35221 28 ff. 4; but the most that can be made out of his words is an assertion of the periodical recurrence of partial floods. Of the later physicists, it is enough to mention the Stoics, whose great year was 365 X 18000 years, i.e. @315 times the TIIE SIGNIFICANCE OF THE NUPTIAL NUMBER. 75 It will be seen that Plato’s Great Year was arrived at in the same way as that of many of the other Greek philosophers. Nor is there in the whole compass of his works a single note which is not in unison with it. True it is that in the Phaedrus1 the cycle of transmigrations (including periods for punishment and reward) for a human soul is fixed at 10000 yearsz, but as the soul selects a new body only every 1000 years 3, let her clothe herself in body as often as she will, she can do so but 36 times before the cycle closes. And even supposing the philosophic soul, which is winged after three incarnations, or 3000 years‘, chose still to die the death that others may live, for her too there is rest, after 36 lives have been spent in the service of mankind. But these are they who (ti/cu awud'rwv £1301, 7'6 wapd'n-au 6139 761/ é’neo'ra xpauov, Kai. €59 ohmic-etc é’n 7015er xakM’ovs‘ dgbucuofiv'rat, its 0576 559131.01! 8 A, A 9/ c I 9 \ a n ' 5 ’7] (Udall OU'TG O XPOVOS‘ mar/09 61! 7'9) wapov'n . Section 8. Conclusion. The sum of the two harmonies, counted in days, is (36002) + (4800 X 2700). This is a ryewue'rpucae dicta/1.69 6, for great year of Plato, and the astronomer Ptolemy, Whose great cycle, like Plato’s, was 36000 years5. to its dissolution. It is the same way, now up, now down. Just so (in the Politicus) the repiodos is the same, now forward, and now backward. The parallel is so exact that one might feel inclined to hold that Plato him- self regarded the whole period of the Universe as 36000, falling into two halves: but this will not account for the 6150 dpuoviai, 36002 and 4800 x 2700, and lands us in many other difficulties. 1 Mullach I p. 1: cf. Zeller“ I p. 706. 2 Ch. XIX. See also Tannery Rev. Phil. XIII p. 213 fi. 3 Cens. xvm 8: cf. Stob. I 264. 4 This explains the remark of Gen- sorinus in Oh. XVIII 11, as Usener has pointed out (Rhein. Mus. XXVIII p. 392 fi.). 5 See Hultsch, p. 57: “Vielleicht kniipfte der grosse Astronom an die oben ermittelte geometrische Zahl Platons an, deren Betrag ihm wohl bekannt sein konnte.” In Barocius’ ’ Cosmographia I p. 6 (Venetiis, 1598) I find these words said of the move- ment of the ninth heavens: qui pro- fecto motus complet unam perfectam revolutionem spatio 36000 annorum iuxta Ptolemaei opinionem; iuxta au- tem Albategnii, spatio 23760 annorum; iuxta vero Alphonsi, et quorundam aliorum sententiam, 49000 annorum; quad utique (i.e. whatever its duration is) tempom's spatium vacant magnum Platonicum annumt This is an in- teresting confirmation of Hultsch’s conjecture, as well as of the present discussion: but a still more convincing proof is furnished by the Sphaera of Johannes de Sacro-Bosco (ed. Burgers- dicius, 1639) p. 12: orbis nonus cente- nis quibusque annis juxta Ptolemaeum unum gradum proprio motu confi- cit, totamque periodum peragit unnis 36000 (quad spatium magnus annus appellari sclet, aut annus Platonicus), subiectasque sphaeras una secum cir- cumducit. . The work from which this is quoted was a regular text-book of Astronomy till the Copernican theory 'prevailed over the Ptolemaic: and 36000 years could hardly have come to be called the annus Platonicus in a text-book of Ptolemaic Astronomy un- less Ptolemy or some of his prede- cessors or commentators had under- stood the Nuptial Number, for there is no other passage in Plato which gives the duration of the Great Year. 1 248 E ff. I reserve for another oc- casion a full discussion of the con- nexion between the cycles of the Phae- drus and the Great Year of Plato. 2 Apparently 11000 according to Rep. x 615 A—B. 3 Phaedr. 249 B. 4 Phaedr. 249 A. 5 Phaed. 114 c. 0' See note 1 on p. 20. It is worth while to remember that the vague- rpcm‘y usaérnslwas the symbol of et- vantagamong the Pythagoreans and Platonists: see Proclus in Tim. 238A. 76 THE NUPTIAL NUMBER. 2700: 3600 :: 3600 :4800i.e. 9 : 12 :: 12 : 16 But w h not yet sounded the full depth of meaning in the wofd ave Memo/€09. It is the climax of the whole—a climax notfyiih- 3:53}? of the Muses. The whole number, 25,920,000 days or b years, is the measure of the Earth—from the day of her thit-tgrflglflll the day she issued from the womb, and from the For 3600 er birth till her soul returns to God who gave it. 0 years she waxes : through 36000 years she wanes 1 , We have now to explain the words 70l0197'00 mlpws d i - 110ku T6 [cal xelpéuwu ryeve’aewv. , [well He fiestsellisglhtere 1n the Republic”, Plato ascends but to descend. cyde of th Urom the 7repL0809 of the human rye/11177611 t0 the d e niverse: from the cycle of the Universe he now escends to that of man. This whole number measurin the prime life of the World from the moment when ,it begins t?) the of 1:21;:hwg::rlt emlis, is, 1n the fullest sense, the final ninnber, sayss: 62,ch y\ ot rer is a part. Of this number Proclus 7‘51, 6’” W609 (yap lea: Tan/Rev 7-029 fiwkavéal Icwfiaewu 5350):) Ical éfigwéq 665:: (177'th 70:9 612 ovpaz/cji Icwov/léuow (Edam/0'39 7’) , 9 vex/saw 17 [1.er 060159, ad 7031/ 6’11 7029 1571-0 THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 77 dekfivm/ uaxpovropw'répwv 1') Bpawiopwrépwv wepléb‘wv ¢0pc8u ‘TG [cal drpopla'iv e’a'ri wepnt’n'n'rucés' 5L0 Ical 701') dquwvret’ov ryévovS‘ 7779 7T6pl680U xdplés‘ éarrw. Seeing therefore that the cycle of the Universe ‘comprehends the cycles of all her children, when she has run her course, so likewise have they. It is in this sense that the number 36000, which measures a single weptoSoq of the World, is the cause of change : and the remark of Aristotle‘, that the Xpévoc, 85 81/ kéryel (sc. Tlxdrwu) nah/Ta perafidmtew, is no specific cause of the degeneration of the ideal state in particular, is most just. Only for an Ideal City there can be no cause of change within itself : nothing but the growing age of Heaven can shake it. Thus the number 36000 expresses the dvdryicn that compels Our city to decline, but why is the dpldubs ryewue-rpucés, 72000 years, mipws duewévwv 'TG Kat xelpévwv ryevéaecov? The reason is this. If children are begotten at times which correspond-— si parva li‘cet componere magnis—to the times when the Uni- verse was begotten, ryevéaelq will be duelvoveq; if at other times, xelpovee. Now at what times is the Universe begotten? At the beginning of every 72000 years. How was this number reached ? 1 yewuerpmfi in Gory. 508A has the same twofold meaning: «pawl 5’ oi aogbol, c3 KathheLs, Kai or’zpuybu [ml 75):! ml 0600: Kai dufipu’movs 7'7‘7u Kowwvlau o’uvéxew KT>\. 0’1‘26é [1.0L doxei‘s of; 7rpoa'éxew 'rbv V001! Tourots‘, Kai Tafira 7092509 c311, dhhu héhnfié 0's 6’7; 1‘7 £06717: 7] yewueTpL/ci) Kai 6’1! 0602‘: Kai 6’11 du- Gpu’nrocs ué‘ya ézfiuaraa, i.e. has great power in heaven as well as on earth. Cf.Proclus in Bekker’s Scholia,p. 83: 10 ,ue‘u 06V “trapddeL'y/La 10V dual/Ta aic’bud €0"7"L, ,ué'rpou 5’ 0 was ain’w' ,uéTpou dpa Kat 6A1rfis pruos. oiMx’ 0 ,uéu 7% 7-06 11071700 fwfis ,uéTpou, 0 Bé 7'77: 70066 7'00 Kbauov {0177s, 0 xpc’wos: and in Tim. 270E 0 prvos ,ue'rpe? 17‘71/ 5km Klu'rja'w Ifal 7'0 're'hos afn’fis‘ éma'rpécpa 1rp0s 77‘11/ apxfiv' 6L0 Kai 0 dptb‘ubs érououdg‘e'rar Kid Téxews and ibid. 270]? ue-rpet 6’ ow‘ 0 films pruos 6 é'yxéamos T‘l‘)!’ ,ulow farm! 100 wan/1'63 and 271 A—-B. The dptdués is in fact 793 61"” yewuerpt- Kés: compare Rep. v1 51113 Tbs furo- fiéa'as woroéuevos 00K dpxds, ded. 1'93 b’un bwofiéaecs (“ad-sumptions”), olov émfio’w’ets 7-6 ml (Spud; 2 Book VI 511 3—0. .3 In Tim. 271 B. Speaking of the dlfierence between the Tékaos dpL0/.L6$ of the Timaeus, and the a’pifiubs 'yew— ,uerpmés of the Republic, Proclus re- marks: 1r>\1‘)u rpoaxeladw Tois elpnuévots, 87': 1017701! 7'01! Téhetou dptflubu e’Keluov 7'02 61’ 110)“ng 0770661709, 65‘ Tip! 71111170: TOU Bela!) 'yewn'roii repLhaufldueL replodou, olnréou 6La¢épew, ,uepuccb-repou 6111a Kal ,ubuwv 1651! 0nd) repcédwu, dromraara- nxéu. Then follow the words quoted in the text above. Proclus expresses him- self a little awkwardly, in that at first Sight he seems to be comparing, not the dpcfiubs yewuerpmés, but the dpLB/Lbs réxeaos of the Republic with the rékeios By building it up from the expresses in days the wept’oSo throughout his entire philosophy, wvdmiu of the number 216, which 9 of a human creature. Here, as the disciple of Socrates inter- prets the Universe from man, but only to turn round and say: “ Learn by studying the harmonies and revolutions of the All, 7'93 xaravoovuévrp 1'0 [canal/00011 éfouméiaal [cm-d 'rfiv dpxalav épcfluo’s of the Timaeus. But a closer scrutiny of his words shews that he did not make this mistake. For the (iPLO/L03 réketos of the Republic does not “ embrace the period 1rav70s 1-00 Heiov 76111177100,” but only 06lov 'yeuum’oi}, and Proclus goes on to say that the number of the Timaeus is smaller than the number of the Republic, meaning thereby the concluding number, 701'} dudpw'lrelou yévovs 7979 weptédou Kbptos =10L0ifi-rov Kbpws, duewévwv T6 Kai xetpévwv 'yevéaewv in Plato), i.e. the dpieubs yewuerpmbs, which alone is larger—being in fact, as we have seen, exactly double either of the two others. If Proclus had understood Belay 'er- vn-réu aright—“ Divine Creature” and not “Creature created by the Divine,” he would have expressed himself more accurately: but even as it is, we can see that it is the dplfiuds yewuerpmds. which be interpreted as embracing and fulfilling every other weplobos in heaven and earth. 1 Pol. l.c. 78 THE NUPTIAL NUMBER. (pic-w 1.” Logically speaking, this is what men call reasoning in a circle : but practically it is not. F or when man discovers, or thinks he discovers, that the conditions which regulate his own nature are the laws that rule the whole, he realises, far more surely than before, that the conditions of his own nature are likewise laws, not to be violated without insult to the harmonies of heaven”. The categoric imperative “thou shalt” is derived by Plato from the doctrine of man’s unity with Nature. To take for example the case which concerns us now, when Plato declares that the Laws of Generation for the Universe are but the Laws of Generation for man writ large, we recognize that we are part and parcel with the Whole, to which in this as in every other thing it is our duty toconform. As surely as this goodly Universe is begotten once every 72000 years, and after a period of incubation represented by multiplying the longer period in man into the sum of both the human periods and multiplying the product by the square of ten 3, issues forth a created God, the only-begotten son of the Creator4, so surely are there times and seasons for begetting man: for what is man but the Universe epitomized5? And just as the moment of the World’s conception is discovered from its period of incubation, 36000 years, so the right season for begetting children is to be determined from the period of gestation among mankind, 216 days, by means of ‘calculation together with- perception,’ in the way already described 6. But though our archons are, as far as may be, perfect guardians whose calculation errs not, yet shall our state decay. Though it was founded upon a rock, and for many ages “the 1 Tim. 90 D. - 2 Heraclitus Frag. 91 Tpé¢OPTaL ‘ydp 1ro’w-res oi dvfipcbwem uéuor inro‘ émis 'roii fleiow KparéeL yap ‘roaofirou (micron édéket Kai éEapKéeL ram Kai 1r6pL‘yL'1/e1'at. 3 {270 x (210+ 270)} x 102 : seeabove, p. 71. 4 Tim. 31 B. 5 This is the fundamental idea of Plato’s Anthropology, and indeed of all the Greek anthropological inquiries. It is well expressed by Aristotle in his Physics (VIII 2. 252b 24 E): El 6’ 6’» {9590 701770 duvartiy 'yevéa'dat, 'rL' Kwhziet T?) (11376 avnfifivat Kai Kai-d. 7'6 ray; cl yap éu ,umpgf; Kéa'pgu 'yiuemt, Kai éu us‘ydkgu. See Zeller3 II 2 p. 488 and III 2 p. 397 and Stein’s essay on Mikro- und Makro- kosmos der Stoa in his Psychologie der Stoa I pp. 205—214. 6 On page 51. THE SIGNIFICANCE OF THE NUPTIAL NUMBER. 79 rains descended, and the floods came, and the winds blew,” and beat upon our city, and it fell not, what of the rock meanwhile? Slowly but surely it was crumbling into sand. Our ideal city degenerated into timocracy, timocracy into oligarchy, oligarchy into democracy, democracy into the “last and worst disease of cities tyranny 1,” because the race of man degenerates as the World grows old and weary of child—bearing: Tristis item vetulae vitis sator atque vietae temporis incusat momen caelumque fatigat: nec tenet omnia paulatim tabescere et ire ad capulum spatio aetatis defessa vetusto 2. The irresistible strength of dydrym; forces the World onward to her dissolution in “much fire 3,” and as she moved farther from her Maker and forgot Him 4, our Perfect City passed away—~but not for ever. When the Divine Child dies and is born again, there shall descend from the Heaven where she is stored 5, the perfect Athens 6, ever new and ever old, eternal as the Universe and man, because for ever she is born to die and dies but to be born—‘Ekdeac é’pewpa, xkewat ’Adfiyat, 5m- ;Lémov 7r’roMet9p01/7. “A brighter Hellas rears its mountains From waves serener far; A new Peneus rolls its fountains Against the morning-star. Where fairer Tempes bloom, there sleep Young Cyclads on a sunnier deep. Another Athens shall arise, And to remoter time Bequeath, like sunset to the skies, The splendour of its prime; And leave, if nought so bright may live, All Earth can take or Heaven can give.” 1 Rep. VIII 544 c. 5 Rep. IX 592 A. 2 Lucretius II 1171—1174. 6 Tim. 23 c. Plato’s Ideal City is but 3 Tim. 22 D. a patriot’s dream of Athens. 4 Polit. 273 c. 7 Find. Frag. Dithymmb. 4. @amhrihge : PRINTED BY 0. J. CLAY, M.A. AND SONS , AT THE UNIVERSITY PRESS. @amhtihge : PRINTED BY 0. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS. END OF TITLE