Posidonius and the sources of pythagorean arithmology

Fr. E. Hobbens, Posidonius and the sources of Pythagorenn arıtlanolagy, p. 309-322. | L'exameu de Ada. math. vu, 91 ss. ne prouve pas que Posidonius soit la source principale de Sextus Empiricus : la comparaison d'Anatolius, Theon, Philon et Lydus fail remonter è une source unique qui a été aussi celle de Posidonius. _: | 149 + pe ROBB Classical Philology Vorums XV October 7920 NumBER 4 POSIDONIUS AND THE SOURCES OF PYTHAGOREAN ARITHMOLOGY! Br FranK EGLESTON RoBBINS It has become the practice in recent years, especially in Germany, to regard Posidonius’ Commentary on the Timaeus as the source of scientific pronouncements of various kinds in the ancient authors. Although this commentary was undoubtedly a most important and comprehensive work, it nevertheless seems somewhat strange, to mention at once the matter with which these pages are concerned, that the Greeks invariably went thither for information about Pythagorean number-theories at a time when the literary world was flooded with books, pseudonymous or otherwise, written by the Pythagoreans themselves. Posidonius became recognized as the chief ancient authority upon the Pythagorean lore of numbers chiefly through the arguments of A. Schmekel,? who used as the basis of his theory Sextus Empiricus Adv. math. vii. 91 ff. Since that time his conclusion has 1 * Arithmology" has been defined as "ce genre de remarques sur la formation, la valeur et l' importance des dix premiers nombres, où se mêlent la saine recherche scientifique et les fantaisies de la religion et de philosophie" by A. Delatte, ‘Études sur la littérature pythagoricienne,” Bibl. de l'École des Hautes Études, fasc. 217, Paris, 1915, p. 139. 2 Die Philosophie der mitileren Stoa in ihrem geschichtlichen Zusammenhange dargestellt, Berlin, 1892, pp. 403 ff. [CLASSICAL PHILOLOOY XV, October, 1920]

Classical Philology VorumE XV October I 920 NUMBER 4 POSIDONIUS AND THE SOURCES OF PYTHAGOREAN ARITHMOLOGY! By FRANK EGLESTON ROBBINS It has become the practice in recent years, especially in Germany, to regard Posidonius’ Commentary on the Timaeus as the source of scientific pronouncements of various kinds in the ancient authors. Although this commentary was undoubtedly a most important and comprehensive work, it nevertheless seems somewhat strange, to mention at once the matter with which these pages are concerned, that the Greeks invariably went thither for information about Pythagorean number-theories at a time when the literary world was flooded with books, pseudonymous or otherwise, written by the Pythagoreans themselves. Posidonius became recognized as the chief ancient authority upon the Pythagorean lore of numbers chiefly through the arguments of A. Schmekel,? who used as the basis of his theory Sextus Empiricus Adv. math. vii. 91 ff. Since that time his conclusion has 1“ Arithmology”’ has been defined as “ce genre de remarques sur la formation, la valeur et l’ importance des dix premiers nombres, où se mêlent la saine recherche scientifique et les fantaisies de la religion et de philosophie” by A. Delatte, “ Études sur la littérature pythagoricienne,” Bibl. de l'École des Hautes Etudes, fasc. 217, Paris, 1915, p. 139. 2 Die Philosophie der mittleren Stoa in ihrem geschichtlichen Zu gestellt, Berlin, 1892, pp. 403 ff. [CLASSICAL PHILOLOGY XV, October, 1920] hange

generally been accepted as final.! More may be drawn from the study of Sextus, however, than Schmekel took account of. In the first place, Schmekel used the passage designated almost to the exclusion of another in the same work, iv. 2 ff., which, since it agrees practically word for word with vii. 91 ff. throughout much of its length, is obviously ultimately derived from the same source. A proper diagnosis of the situation surely demands, first of all, that the two passages be examined side by side and their meaning analyzed;? hence one may infer whence each came and for what purpose it was written. If we do this, and take iv. 2 ff. first, we find that its outline is as follows: At the very beginning the general theme of the passage is stated, which is simply that the Pythagoreans ascribed great power to numbers in the government of the universe; this caused them to use the expression ‘‘all things are like to number,” and in their well-known oath by Pythagoras to speak of the tetraktys as the “source that holds the roots of everlasting nature.” The following sections develop the implications of this epithet and are therefore thoroughly consistent with the central theme of the passage; the numbers of the tetraktys, 1, 2, 3, 4, in their capacities as point, line, surface, and solid, respectively, represent all bodily natures, and, furthermore, soul; for since 1, 2, 3, and 4 contain the fundamental concords they control the harmony whereby the world is ordered and the soul constituted. This passage, then, is thoroughly consistent within itself, its main thesis is kept in view throughout and supported by argument, and its logical structure is admirable. The other, vii. 91 ff., treats an entirely different theme, the criterion. It may be summarized thus: Anaxagoras declared that the criterion was XöYyos, using the term in a general way; but the Pythagoreans restrict it to the mathematical Adyos, because, as Philolaus said, if it takes cognizance of the universe it must be akin 1Cf. G. Borghorst, De Anatolii fontibus, Berlin, 1905; G. Altmann, De Posidonio Timaei Platonis commentatore, Berlin, 1906; Gronau, Poseidonios und die jüdischchristliche Genesisexegese, Berlin, 1914, p. 197, n. 1. 2 Zeller, Philosophie der Griechen, III, 1, p. 514, n. 5, concludes from such a comparison that the Pythagorean element in Adv. math. vii, 92 ff. is not from Posidonius. Against this view Schmekel argues, op. cit., p. 407, n. 1. A proper comparative analysis of the two passages has not, however, been made.

to the universe, on the principle of the perception of like by like; Empedocles, who is quoted,! clearly states this principle. And as the light is perceived by the kindred sight, so Posidonius says in his Commentary on the Timaeus, and sound by the hearing, which is akin to air, so the universe ought to be apprehended by a Aóyos of the same nature. But number is the source of the universe, wherefore the Aóyos that is the criterion of all, not without its share in this, would be called number. It is to declare this that the Pythagoreans often say “all things are like to number” and use their accustomed oath. Its author was Pythagoras; the tetraktys is 10, or 1+2+3-+4, which is the “source of everlasting nature,” first, because it contains the basic concords; second, because it contains point, line, surface, and solid; furthermore, because, all things being numerable, number is fundamental. This is the outline of the passage; I have more carefully reproduced the first portion in order to make clear the logical connection; but the last part, from the mention of the Pythagorean sayings, is that which is closest to iv. 2 ff., and no argument is needed to show that ultimately the two must have been derived from the same source. Now it must be equally clear, it seems to the writer, that there are signs of disjointedness and of compilation in the second, but not in the first, passage. For in the former there is one theme only, and that sustained consistently throughout; in the latter we begin with the criterion and then abandon it for a detailed discussion of certain Pythagorean theories about number, the transition being of the most formal character. The last portion of vii. 91 ff. leaves the impression of being an argument, already existent and ready to hand in the work of some earlier author, awkwardly forced into a new context by the writer of the part about the criterion, and betraying its alien origin by the noticeable lapse in logical connection. This conclusion commends itself the more strongly because we know iv. 2 ff. also; having the latter, and seeing in it none of the marks of patching together, but on the contrary an admirable unity of construction, we must conclude that in iv. 2 ff. we have Sextus’ quotation of certain material in its original form and used as it was 1 Fragment 109, Diels; cited also by Aristotle De anima A 2 40468; Metaphysica B 4 1000 b 5; and by Chalcidius in Plat. Tim. c. 50.

originally intended to be used, whereas in vii. 92 ff. he has employed a passage of another author who had himself cited this same material. There is still further proof that the Pythagorean part of vii. 92 ff. existed in another form (i.e, that of iv. 2 ff.) before it was taken into its present context; not only that, but we may infer the original purpose of its composition. This is afforded by the discovery of a probably identical origin for Adv. math. iv. 2 ff. and two passages of Theon of Smyrna and Anatolius respectively. Sextus Empiricus Ad». math. iv. 2ff.: Theon, p. 99, 11. 8-23 :! Anatolius, p. 29, 1-10:? kaféhou uèy olv ol ard THY uabnudrwy Hvdayopıkol peyadyny Arovenovor Öbvayır rots dpgots ws Tis Toy Sdwr ploews Kar’ abrobs Ötotkovutons. 80ev Kal ael 6 6€ kai éx TÔv TETpakruwv tobrwy ovoTàs Köonos éorat <TeXeios > hppoopeévos kara yewperpiay Kal àpuoviav ù Öekáòos Kal trav eros wore érepévouv TO ‘‘ àpu0po dE re mévr’ éréoukev,”? dpvdovres où pévoy Tòv dpvOpdv &AX& Kal Tov ÚrroBelkavra aûroîs Toürov IIvdayspav ds Beòv Sta Thv dv Apbpnricyg Büvapty, Aéyovres “Où pa rèv âperépa ux mapaddvra rerpaKTúv, IInyùv deváov pioews Prlépar’ Exovvay’? rerpaxrds òë rpoonyopebero wap’ abrois 6 &k av wpdtev 8 áplpôv kai Apıduov, Övväner Tepie- Angas râcar apuod Pbow way Te ueyedos Kal wav cÔua âmA\oîv Te Kai aüvderov, TE eds Te, éwerd)} Ta mávra bey Tobrov uépn, aùròs ôè aùríjs apıdusv pois uvupia uèv mepıexeraı kal émideikpuor KaddAn Tots Ökvöcpkws TÔ v@ kabopâv TA Touadra ôvrauévois. Soa uêv obv éay oldv re 7 Atkonev Ep’ Exdorov TÔv âpuÔv, rocodrov Öt mpoA&yonev viv, obdevós. 816 mpdte TÔ elpnpéva 6pkw ot ITu0ayoptkol éAdyovro?.... Kal GprOpd dE re wave’ éméouke. Kai Toûro elvat TÔ copwrabre Tov’ wavTa pev yap Tov GpOpov els SexdSa Hyayou, éreSh trip Sexdda oùdels éoriv aprOpds, év ri advice mad Huôv mávra apipov els Béka dvfjyov, trip ôéka 6€ oùBels Erı AprOpds, Ev racy of Ilwayopetou TÒv avéfoe médww pay ém- 1 Expositio rerum mathematicarum ad legendum Platonem utilium, ed. Hiller, Leipzig, 1878. The work is generally acknowledged to be a compilation. Schmekel, op. cit., p. 410, believes this passage to come from Thrasyllus; so also Switalski, Des Chalcidius Kommentar zu Platos Timaeus, Beiträge z. Gesch. d. Phil. d. Mittelalters, III, 6, Münster, 1902, pp. 85-86; Borghorst, op. cit., pp. 17 ff., assigns it to Moderatus, and Altmann, op. cit., pp. 19 ff., to Adrastus. 2Ed. J. L. Heiberg, Annales internationales d’ histoire, Congrès de Paris, 1900, 6e section, histoire des sciences, Paris, 1901. The title is wept dexddos Kai r&v Evrös adrñs ápuôv. Many excerpts from it appear in the anonymous Theologumena Artthmeticae, ed. Ast, 1817. 3 Theon has already quoted the oath given by Sextus in p. 94, ll. 6-7, a passage which contains probably more of the material found in Adv. math., IV, 2 ff. other citations of this oath cf. Hiller, ad loc. Delatte, op. cit., p. 250, note. evident textual difficulties at this point see Hiller, ad loc. On the

Theon: Anatolius: àäpiôués : bmoorpebövrwv &ml povaorpebövrov a’ yap Kai ß’ kal y’ kal 5” VU ylverar® ds gore Teheéraros Apıduös, &melmep &m’ aùròv dHláoavres madi dvakúoper &ml Thv povada kal &€ bmrapxfjs movoúpeda da Kal Sud5a Kal Toùs éEfjs hv St Sexdda ml rerpéôa ouvloracdar év yap kal B’ kal y’ Kal 8 éor 1’, pera 1d ouxmAnpoüoôar mâcav SexaSa° dAAà kal bre ke rerpéôos orvvlorarar tj Sexads els ra pddiora Thy Terpaxkrdv érluwr. ras Oewpetabar. Sextus: avykelpevos 1’ ’ ápuphoes. ryy#r @ore Toùs ÖvvarwTáTous apıduobs évrds THs TeTpddos dm povdda KTA.. . . . eipnkacıv abrov bua. 7d Kar’ abroùs dv abrö Tov Aöyov TÔv dmrávrwv Ketodart ovoráoews, olov ebOéws Tod Te owparos Kai Ths Wvxis . . . . That all three of these passages occur in different contexts is the reason why there is divergency among them. No two contain exactly the same material, yet by pairs they show close agreement, and the same thought is implicit in all three. The close similarity of the latter parts of Anatolius and Theon leaves no doubt that both go back to the same ultimate source,! and their differences are to be ascribed to the position of the passage in Theon, as a link between the discussion of the tetraktyes and that of the first decade, and to Anatolius’ tendency to condense, noticeable throughout his whole treatise. He has here left out all mention of the Pythagorean oath and the other saying,? but his own last words, “they honored the tetraktys especially,’ its inclusion in both Sextus and Theon, and the fact that his statements about the Decad have point only as an explanation of the oath by the tetraktys, make it hardly doubtful that this is his own arbitrary omission. As for Sextus, his passage contains all the thought of the other two—the two Pythagorean sayings: the notion that 10 is made up of 1, 2, 3, and 4, the tetraktys; 1If there were any doubt, the verbal correspondence of Theon and Anatolius in the other passages dealing with numbers would dispel it. 2 This, however, is due only to the desire to condense, for the saying is found in a collection of excerpts from Anatolius, in this form: örı Tr &puôunruxr ob pdvos eriua Ivbayépas, dda Kal of robrov yvwpmuor Emi\éyovres ’ApiOuGd de Te wart’ Eméouer (see F. Hultsch, Heronis Alexandrini Geometricorum et stereometricorum reliquiae, Berlin, 1864, p. 279). 3 This phrase, and the concluding phrase of Theon, should be compared with the end of the passage of Sextus, which is quoted below, p. 318.

that 10 includes all the numbers (he says that it is “most perfect’’), because we find no new ones in counting above it. The order of the last two is reversed in Anatolius and Theon as compared with this, and in what follows the part quoted. Sextus elaborates further the powers of the numbers of the tetraktys. In general, we may conclude that we have here two versions of the same thing, a longer one represented by Sextus, and a condensed one in Theon and Anatolius, the latter having still further condensed on his own account.! Possibly more material that originally belonged in this context and is still seen there in Sextus is to be found in Theon’s foregoing discussion of the tetraktyes.2 To make the relationship even more probable, we shall see later that in another section Anatolius agrees word for word with Sextus in the latter’s continuation of the citation above. As to the original nature of the passage, the clue, I believe, is given by Theon and Anatolius. In each it appears as the introduction to a discussion of the numbers of the first decade; in Theon this is not so apparent, since it also happens to conclude the section about the tetraktyes, but Anatolius’ employment of it should be taken as the proper one. This conclusion will recommend itself if one simply reads over the Theonian context and notes the plain evidence of its disjointedness, and furthermore that in citing these words Theon is led to make an awkward repetition of statements already introduced relative to the first tetraktys.* We have, then, in this passage the introduction to a Pythagorean treatise on the first ten numbers, and surely nothing could be more appropriate for this purpose, from the Pythagorean standpoint, than a forceful exposition of the general powers of number in the cosmos, as epitomized in 10 and the tetraktys. 1 There are other versions of the material presented in the three passages quoted. The original arithmological document, of which I assume these to cite the preface, apparently repeated most of the statements later, in the chapters devoted to 4 and 10, and from these probably are derived, as far as any connection exists, the reports of Philo De mundi opificio cc. 15-16, Lydus De mensibus iii. 4, iv, 64, Wünsch, Chalcidius in Tim. c. 35, Hierocles in carm. aur. c. 20, Martianus Capella, Favonius, etc. 2 See p. 312, n. 3, above. 3 Cf. Theon, p. 99, 11. 21-23, with p. 93, ll. 19 ff., Hiller.

In order to clarify the situation still further, the words of Sextus immediately after what has been quoted may be compared with the following: Sextus Adv. math. iv. 4 (p. 722, 15 Bekker.): Philo De mundi opificio 16: Anatolius, p. 32, 3 Heiberg: wey oby povds dpxh Tus bmoreiraı Tis TÔv a&d\d\wv apıdusv amepyacrixh ovoracews, ÔË Suds unkous éort Amepyacrıry. Kabamep yap Emi TÔv yewperpik@v &px®r Ürreelfauerv! TpÈTOr ris &orıv n orvyuh, elta ner’ abri ypauuÿ uûjkos àm\arès TvyXávovoa, Tov aùròv Tpórov Kal &mi Tod mapóvros uèr povds Tòv THs oTuyuñs Emexa Aöyov, N 5é duds rdv Ths ypauuñs Kal rod pnKous* rode yap mod épépero 6 vods, kal mpooriWeuevns TH TpaTn yap arn Tr Tod orepeod hour take, Trav mpò abrñs dpiOuev rots dow uáTots dvakeéevwvy. KaTa bev yap TO &v rarrera 7d Aevyouevov Er Yewuerpla elvat omuetor, Kara òë Ta dbo ypauuú, Sidre pboew ëk onuelov ovvlorarat. % ypauun Òé gore uijkos Amar mr\árous Öë Tmpooyerouévou mpÔros take thy orepeod blow: Kata uMkos dtacrace Tihs kara mA\áros Òvaordoews émipávera voetrat, AAA Kav Emdewpnon Tis TH Tpidde TerapTny Hovâda, TOUTÉOTUV Téraprov onuelov, yiverat Tupauis, orepeòv cpa Kai oxijua’ Kal yap ufjkos exer kal mAáros Kal Bálos: Sore & TÔ réooapa apiOug rov Tod oœuaros srepikxeodar Aoyov. yiveraı émipdvea, ù Térak- Tat karà Tpidda’ émipavera 6& mpôs Tr orepeod Pho évès Öetrart rod Bábovs, 6 mpooredev TH Tpıadı Yiveraı TETPAS. . 2... 6 öl un ovpueis TO Aeyöuevov Ex Tivos mratâs elveraı avy ovvnbovs. ot Kapvarifovres eisdacı Tpia Ev émrimréôg onuetoy yap, elra ypauuú, elta éripévea, elra orepeöv, à &orı c&ua. a A Todro TÜV kKapvaruórruv a 2 mad, motodoa oxua mvpautöos. mporıdevres Kapua émipéper Ev, oxfua mupanoeöts àToyevvûvres. TO uèr olv &v éruméô@ Tpiywvov loraraı nexpı Tpıädos, TO Öë Emıredev Terpáöa pev Ev Apıduois, & dé axnuacı mupaulda yerva, orepeöv Hin cÔua. The passage of Anatolius comes, not from the introduction, but from the chapter on the tetrad, and there is a close parallel to it in Johannes Laurentius Lydus De mensibus iv. 64 (Wünsch).? There can be no doubt that Philo, Anatolius, and Lydus ultimately go 1 This is probably a reference to Adv. math. iii. 19 ff. 2 mp@ros oby rerpéywvos &piOpds odros Kai rerpaxrbs, &\AG ur Kal mpôros eke Thy orepedy iow omuetor yap, elra ypauu, era &mıpäavea, tra orepedv, B tore oûua. Theon, p. 101, 11 Hiller has so abbreviated his notice of the number 4 that little can be made of the parallel passage: 4 5& rerpds orepeod &orıv ei, mpôrós re âpiuòs [kai] Terpayuves tori &v aprious, Kr).

back to an identical source; the former, however, apparently used it in an unabridged form, while Anatolius and Lydus both drew from an epitomized version. Without delaying to speculate upon these matters, in support of which further arguments could be stated, let us note that, although Sextus treats of similar topics, the form of his statement is very different from that of the other three. This is probably because he drew from the introduction, the others from the body, of the common source. Comparing the immediately succeeding words of Sextus, again, with others of the arithmological writers, a somewhat different situation appears: Sextus Adv. math. iv. 6ff. (Bekker, p. 723): Anatolius, p. 32, 15 Heiberg: Philo, De mundi opifieio 15: Chalcidius,Comm. in Tim. ce. 35: kal phy Kal rov rijs wuxis ws yap Tv Sdov kKÓopov Kara dppoviay Néyouat ôtouxetobar, oÚrw Kal TO CÔov Yuxovodaı. boxed 6& N TEA os Gppovia dv Tpiot ovudwviaıs Aaupßavew Tv Òmóoraouw, TH Te bua TEerräpwv kal TH dca mevre Kal TR bua macôv. % uèv oby bia Teooûpov Ev Emırpitp Keitat Noy, ù dé da mévre ev molle, 7 6& bia mac®r ey Ôtwraciom. emirpiTos? 6& Dé€yerar pipes 6 &£ Sdov où uövov dt Tov Tod Tepréxe den Terpäs symphoniae quodœuaros émréxe X6yov tv àpiôuoîs reTpás,! dAAÀ Kal rov THs Wvxíjs: ws yap Tov Sdov kKÓopOv paaoi kara dppoviar Övouketodar, odtws Kal TO (Gov Wvxodoat. dokel Öë TEXetos dppovla év Tpioi ovudwvlaıs Òpeordvar, TH dia 5’, ris & émirpirm Ketrar Aoyw, TH did €’ & HuwoNiw, tH da maov év dirdaclov. Kal robs Aöyous TÔv Karà MOVOLKNV oupque ratio ex eorundwrıöv, Tis Te dua Terrápwv Kai dua Tévre kal và Tacav kal mpooerı bis dia racûv, & av obornpa TÒ Tekerórarov amoyeväaraı = Tis pev yap dua terrapov 6 Abdyos &mirpiTos, THs 6 dud mévre AuıöAuos, durAdovos öde Tÿs 61a mracôv, Terpamd\dotos Ôè THs bis à macv. oùs äravras Terpas Exer TepiAaßodea‘ Tov uev yoûv émirpiTov tv TÔ Técoapa dem numerorum qui decimanum numerum conplent quasi quodam fonte demanat: siquidem ex his epitriti et sescuplares et duplices et triplices et quadruplices numeri sonique nascuntur. epitriti quidem ut quattuor adversum tres. habent enim totum numerum trientem et eius tertiam partem, id est 1 Note that this parallels the final words of Sextus in the last passage quoted above. It is a very significant parallel. 2 The following section, containing matter explaining certain arithmetical terms, since it is not paralleled by Anatolius, might be regarded as an addition by Sextus; on the other hand it could as easily be an instance of an omission by Anatolius for the sake of brevity. Chalcidius presents similar details, without, however, the distinctive marks of close relationship present in Sextus and Anatolius.

Sextus: Tod dbo Kal 6 dbo rijs povdbos éort durdaolov, tv & &elro } bua racy ovubwvia, 6 6& rpla rod öbo Hubwos (Kai yap abròv rdv dbo Chaleidius: unum. sescuplares uero, ut tres aduersum duo; habent enim mpös & } Teooapa tres mpos dbo, röv Öë TeTpamdkáowov & TÔ Téooapa mpös év. merum dualem et eius partem dimidiam, id est unum. duplices uero ut sunt duo aduersus unum. triplices totum nuporro ut sunt tres tov dvo" dis yap Tv aùròv TepLeoxnKev. ad yap Tobrwv otrws E&xövrwv, Kal kara thy Apxiider brddeow recodpwv Övrov appar, rod Te évds kal dbo Kai tpla Kal réooapa, ev ols éXeyouey Kai Thy ris Wvxijs iôear mepıexeodaı = Kara Tov évapuóviov Aóyor, 6 uèv Téooapa mpès Tpla, rov dé ùutbhvov ev TÔ Tpla mpös dbo, röv dé ÖLmAdoıov tv TÔ dbo Philo: Anatolius: Twos &puôuoD auveomykos Kal ëk rod tpirov pépous éxelvou, ws Exe. 6 ÖkTù mpös Tov € Kal yap abrov rov EE meprëoxnxe kal rd Tplrov abrod, rovreorı Tih ôvéôa. modos Öë kadetrat, Stray mepıexn aprOpds Apıdpov Kal TO Auov éxeivou, ds Exe 6 evvea mpds tov EE ouvéoTyke yap ëk Tod @& kad éx Tod uioeos abrod, rovréore Tov TptÔv. durraciwy Öë mpooayopeberat à övoiv äp@uots Toos, ds 6 Teooapa mpòs övrwv dt dpuÔv Terrápwv TÔv pur Tov a’ B' y’ ò', & tobros Kal ù Tis puxijs löta mwepréxeTat Kara Ty évappéviov Adyor, à wey 5’ rod B’ Kai 6 ß’ tov a’ Sudaotos, &v @ keiraı } bua mao&v ovudwvia, à dé xy! Tod B’ jyrdruos, mepiexwv abrov Kal TÔ uov, Thy bua TÉVTE ovudwviav broBadre, à dé 5’ aduersus unum. et quadruplices ut sunt quattuor adversus unum. epitritus autem in calculando idem est qui. diatessaron dicitur in canendo. sescuplaris uero idem est qui diapente dicitur in canendo. duplex uero qui diapason dicitur in canendo,! quadruplex qui didiapason dicitur in canendo. 1 At this point Wrobel would add “triplex qui diapason et diapente dicitur in canendo,”’ following Fabricius and thinking that Chalcidius omitted the words, which appear neither in the MSS nor the first edition, through error or forgetfulness. however, that the passage agrees with Theon, p. 58, 13 ff., as it stands. Note,

Sextus: Anatolius: mepléoxnKe Kai TO mov Tobrov, 60ev kal Ty dua Tévre ouupwrlar dréBan- Tov y’ Emirpıros, & ® 7 5a 5’ cupdwvia. el dé & 7G 5’ apiOue TO way Ketrar ëk Yuxäis Kal oœuaros, adnbés äpa kat örı ai ovudwviaı mâcat kar’ abröv TeAodrrat! dev), 6 5& réooapa To rpla &mirpıros, Umexeıto Öë Kal & tolrm M 6ua Teocâpov ovupwrla. @ore elkórws ov Téooapa äpuôudy mapa tots Ilvôayopixots eipñolar myviv devaov dicews pitwpar’ Exovoar. Philo: Chalcidius: In Theon’s arithmology, p. 101, 12 Hiller, he merely says Kai ai ovudwria Òë maou Kar’ abrov avumAnpodvrau, ws ébelxOn. This reference leads us back to p. 93, 17: &reön mávres of Tay cuugwriav ebpeßmoav dovyou, Kaba SéSecxrar, ev rH THs Sexdbos rerpakrlı, Kal Tepi ToUTWY mpórepov ÀEKTÉOY. THY wey Yap Terpakrüv ouvéornoev 7 Òekás. & uèv yap kat B kat Y Kal à ı" a BY 6. & dé robrous Tots Gpıduots écri ÿ Te à Tecodpwr ouupuwria Ev Emirpirw OY Kal 7 5a mére Ev nuodiw kal % dia macwv Ev durdaciw Kal rn Sis ba macv Ev rerpankaciw' EE @v auumAnpodraı TO àmeraBoNor Öbypapua. And in turn the reference here is to p. 58, 13: sráoas 5€ Tas ouubwvias Tepiéxe ÿ Terpaxrbs. ouvéornoe uev yap abriv a’ Kai ß kal y kalò. év dé rovrous Tots àpuôpots éori ÿ TE dia Teoodpwv ovubwvia Kal n ba Tévre kal ù bia rar, Kal 6 érirpiros NOYos Kal HuıöAıos Kal dirdAdovos Kai Tpur\áotos Kal TeTpaTAdoLos. Among these passages, there certainly exists an agreement between Sextus and Anatolius which is indicative of a common source; the others do not share this verbal similarity, and, besides, 3 A 2 A ’ A 3 4 x LA , ve A they include among the harmonic ratios the diapason, making four, while Sextus and Anatolius report only three. Chalcidius and Theon, here as elsewhere,? show marks of close relationship; they 1 The concluding sentence of Anatolius is obviously modeled after what is seen in Sextus, but modified to suit its new context. 2 This alliance is manifested by several significant minor identities of phrase; for instance, of the triad, Theon says mpwrn Atyeraı mévra elvaı and Chalcidius

both mention the triple ratio, and each regards the number four as the tetraktys. The conditions may be explained by concluding that Sextus and Anatolius for this passage depend on the introduction of the arithmological treatise, Philo, Theon, and Chalcidius upon its chapter on 4. The latter two, however, are probably derived from some common ancestor between the ultimate common source and themselves; this will explain their slight disagreement with Philo. The likeness between Sextus and Anatolius, however, is a most important confirmation of the conclusion reached above with regard to the first set of passages cited. Thus the passage of Sextus is seen to be connected most intimately with a lost book, the existence of which, however, is as certain as that of the Itiad—namely, the common source of Anatolius, Theon, Philo, and Lydus, which determines the form of the first two throughout the whole length of their arithmologies,! and from which Philo and Lydus excerpted lengthy fragments identifiable by their close likeness to Theon and Anatolius.? Besides these, Chalcidius? was certainly affected by it, probably also the Theologumena Arithmeticae,* Varro,® Hermippus of Berytus,® and indirectly Macrobius, tria . . . . dicta sunt omnia; but Anatolius and Lydus (ii. 8) both have ért pros Td mävra onuaive, Otherwise the passages are much alike. Again, in the chapter on 7, Chalcidius inserts a block of material much resembling a similar block inserted by Theon, which disagrees with the corresponding parts of Anatolius, Philo, and Lydus. 1 It is to be identified with the “grand traité d'époque alexandrine ” mentioned by Delatte, op. cit., pp. 140, 207. ? See Philo De mundi opificio cc. 3, 15-16, 20, 30-42; Lydus De mensibus ii. 7, 9, 11, 12; iii. 4; iv. 64 (all in part). The connection of other passages of both authors with this source is debatable. 3 In his commentary on the Timaeus ce. 35-37. 4 An anonymous treatise first edited by Ast in 1817, thought by most to have been compiled by Iamblichus; its chief sources are Anatolius and the Theologumena Arithmeticae of Nicomachus of Gerasa (this latter is otherwise known only through Photius, Bibliotheca, cod. 187). Probably Nicomachus’ Theologumena also was influenced by the anonymous arithmologist. 5 In the first book of the Hebdomades or Imagines, ap. Aulus Gellius Noct. Att. iii. 10, and in Tubero or De origine humana, ap. Censorinus De die natali 9, 1 ff. $ Clement of Alexandria, Strom. VI. xvi (see especially sec. 145, 2) quotes his mepi éBdoud60s.

Martianus Capella, Favonius, and Isidore of Seville.! By comparisons among these and other writers it would still be possible to reproduce a large part of this source in its original form. It consisted of an introduction and ten chapters dealing with the ten numbers? The study of its transmission, which would be too lengthy to attempt here, would show that it circulated in several versions and epitomes,? and that its influence was spread still farther by those who derived their arithmology from it. But although it was the most generally quoted of all ancient arithmologies, its very name and that of its author are lost and probably will never be known. It was probably written by at least 100 B.c., for Varro seems to have been influenced by it; Philo,* in the early first century A.D., certainly was. With regard to the question of the share of Posidonius in promulgating Pythagorean lore, the conclusion must be that even if the whole of Sextus Adv. math. vii. 91-109 is his, the Pythagorean part of it must nevertheless be a citation on his part from the introduction of an already existent work, which was known to Sextus also from another source, that used in iv. 2 ff. This treatise, and not Posidonius, was the ultimate source of information used by the long list of writers just named. In conclusion, let us consider briefly the question whether or not the citation of vii. 91 ff. belongs to Posidonius, reviewing the arguments that connect him with the arithmological traditions. First, it has been claimed? that the mention of Posidonius’ name at 1 The sources of these writers are not so easily determined. Macrobius in his commentary on the Somnium Scipionis (i. 6) parallels extensive passages of the Theologumena Arithmeticae. 2 There is a possibility that it formed a part of a large Pythagorean work dealing with the mathematical sciences generally. 3 Anatolius, Theon, and others seem to have used epitomes; Philo, discussing the number 7 in De mundi opificio 30-42, and Sextus, quoting the Introduction, apparently had access to the unabridged work. As has been seen above, Anatolius also quoted a section of this Introduction. ¢ Philo was the author of a book, now lost, called rept dpuÔu@v, mentioned by title in Vit. Mos. iii. 11, Quaestiones et solutiones in Genesim iv. 110, 151, and less definitely in De mundi opificio cc. 16 and 43, Quaest. et sol. in Gen. ii. 14 and iii. 49. Apparently he collected here all that is said of the numbers in his extant works, and the treatise was doubtless affected, as they are, by the anonymous arithmologist. 5 By Schmekel, loc cit.

the beginning marks him as the author. This is a double-edged argument and by no means conclusive; for ancient writers quite as often concealed the name of their primary source as they revealed it, and frequently they veiled their action by mentioning freely the authorities named in their real source! Furthermore, we have now seen that in any case Posidonius himself must have been quoting; and this will also apply to the purely subjective argument that attempts to decide, from the nature of the passage, that the whole belongs to him? More satisfactory in some respects is an argument based on the commentary of Chalcidius, e. 50, where the following occurs: uult igitur animam sensibilis mundi tamquam permissa usurpandi licentia nasci, cognitricem tamen rerum omnium, quae sunt tam intellegibiles quam sensibiles. est porro Pythagoricum dogma similia nonnisi a, similibus suis comprehendi. quod etiam Empedocles sequens ait in suis uersibus: terram terreno conprendimus, aethera flammis, humorem liquido, nostro spirabile flatu, pacem tranquillo, litem quoque litigioso. haec quippe constituebat elementa et initia uniuersitatis, ex quibus animae quoque censebat constare substantiam. The thought of this passage is parallel to Sextus Adv. math. vii. 91 ff., and the citation of the same verses of Empedocles, and the possible reference to Philolaus, are remarkable. The reappearance of all these matters in conjunction could be offered as evidence that the passage in Sextus is a unity, and from the pen of Posidonius; it has, in fact, been argued that Chalcidius (probably through Adrastus) and Sextus both depend upon him.? Even if this is so, however, it does not prove that the Posidonian quotation in Sextus extends over the arithmological sections as well, of which in any case he could not have been the author. 1Cf. Eyssenhardt’s introduction to his edition of Martianus Capella, p. xxxii, and Hiller in Rheinisches Museum, XXVI (1871), pp. 582 ff., cited by Switalski, op. cit., p. 62. ? For the controversy over the exact extent of the Posidonian quotation, cf. Zeller and Schmekel, cited above, p. 310, n. 2, and R. M. Jones, The Platonism of Plutarch, Chicago dissertation, Menasha, 1916, p. 77, n. 21. The latter agrees with Zeller that the Pythagorean material does not belong to Posidonius. It may be noted that at the conclusion of the debated passage Sextus (vii. 110) appends the words radra pév of IIvdayopıroi. 3 Borghorst, op. cit., p. 60.

Finally, there is some interest in the citation of Posidonius by Theon in his arithmology, p. 103, 16 ff.: éréuevos de TH bloe Kal 6 Tdrov é£ érrà dpiBu@v ouvéornoe thy Wvxiv & TS Tiuaiw. muépa yap kal vil, ós nor 6 Wooerôwvios, dpriov Kal repitrod pbow Exovan. wv Òë kaf’ éBôouédas révoapas ovur\npodrat, TH uèr mpwrn EBdoudde ÖLxorógov THs TeAHVNS òpwuérns, kA. At this point, it may be noted, begins a block of material! which, unlike most of the chapter on the number 7, seems to be taken from a source different from that of Anatolius—that is, probably not from the anonymous arithmology of which we have spoken.? Opinions as to the extent of the Posidonian citation may, of course, differ; there seems to be no reason, however, to think that it did not at least include the sentence describing the influence of the number 7 on the moon and the lunar month. Nevertheless, all that can be held proven is that Posidonius was interested in Pythagorean arithmology, and that he may have influenced some lines of its tradition, not that he was the author of any part of the general source of Theon, Anatolius, Philo, and the rest. At most, he may have been responsible for introducing alien elements into the descendants of this arithmology. UNIVERSITY OF MICHIGAN 1 Especially p. 104, ll. 1-5, but in all the rest of the section about the hebdomad there are striking differences of detail between Theon and Anatolius. 2 This special source, however, was probably itself influenced by the Anonymous.