The Anonymus Arithmologicus and its philosophical background

The Anonymus Arithmologicus and its philosophical background Leonid Zhmud 1. Early studies in the Anonymus An anonymous Neopythagorean treatise devoted to the marvellous properties of the first ten numbers (further Anonymus) undoubtedly belongs to the most influential but understudied pseudo-Pythagorean works. Not only was it the originator of the genre of arithmological writings which were compiled till the end of Antiquity, it also has left numerous traces in philosophical, theological and encyclopaedic literature in Greek and Latin. Being once a subject of intensive research, since the early 1930s the remains of the Anonymus have not attracted special attention, therefore a concise overview of earlier studies seems quite fitting. As this is often the case, at the beginning this issue has been dealt with only tangentially. When Alfred Schmekel discussed in his influential book on Middle Stoicism a division of Neopythagoreanism that combined Platonism with Stoicism, he leaned towards seeing its origin in what he considered Posidonius’ (ca 135 – ca 50 BC) commentary on Plato’s Timaeus preserved in Sextus Empiricus. 1 To reconstruct the relevant part of the commentary he compared four texts on the wonderful properties of numbers, namely, Varro (in Censorinus and Aulus Gellius), Macrobius, Theon of Smyrna and Philo of Alexandria. Printed in parallel columns, they clearly showed traces of common origin, and because two of them, Macrobius and Theon, mentioned Plato’s Timaeus, Schmekel identified this source with Posidonius’ commentary on this dialogue to which Sextus Empiricus allegedly referred. 2 Schmekel has not yet used the word ‘arithmology’ that first appeared in its normative meaning in Arman Delatte’s 1915 book on the Pythagorean literature. 3 According to Delatte, arithmology has been created by the ancient Pythagoreans, whereas the direct source of the later arithmological lore was “un grand Recueil d’époque alexandrine, qui fut utilisé par une foule d’auteurs de la decadence”, among them Philo, Aetius, Sextus Empiricus, Theon, Hippolytus, Macrobius. Having dated this Recueil in the third–second centuries BC, 4 Delatte refrained from analysing it, being more interested in the preserved arithmological texts, some of which he first published, while others thoroughly studied. I would like to thank Constantinos Macris, Joel Kalvesmaki and Federico Petrucci for their helpful suggestions on an earlier version of this article and Tobin Auber for improving my English. 1 Schmekel A. Die Philosophie der mittleren Stoa. Berlin, 1892, 403ff. Schmekel relied on Sext. Emp. Adv. phys. 2.281f., Adv. math. 4.2–9 and especially Adv. math. 7.92f. 2 ǠǑǝʐǗ ɟ ƻǙǝǏǓǎʗǗǓǙǜ ǞʒǗ ƻǕʋǞǣǗǙǜ ƾʑǖNjǓǙǗ ȲǘǑǍǙʕǖǏǗǙǜ (Adv. math. 7.93 = F 85 E-K); cf. Theon. 103.16–104.1 = F 291 E-K. Schmekel. Op. cit., 424f. 3 Delatte A. Études sur la littérature pythagoricienne. Paris, 1915, 139. He mentioned Schmekel only once, in a footnote denying the identification of Posidonius with the source of Pythagorean arithmology. Ibid., 139, 140 and n. 1, 232f., 253 and n. 2.

The next and decisive step was taken by Frank Robbins, 5 who employed the same method of parallel columns as Schmekel did but in a more general and efficient way. To begin with, he analysed more systematically a wide range of sources on which Delatte drew, namely, Varro, Philo, Moderatus, Nicomachus, Theon, Sextus Empiricus, Anatolius, Ps.-Iamblichus’ Theology of Arithmetic, Chalcidius, Macrobius, Hierocles of Alexandria, Martianus Capella, Favonius Eulogius, and John Lydus. Further, he persuasively showed that Posidonius was not the original source of arithmology. The Stoic, so Robbins, only quoted this source, which is to be identified with Delatte’s Hellenistic arithmological collection and dated shortly before Posidonius, in the last part of the second century BC. 6 This Pythagorean arithmology consisted of an introduction and ten chapters dealing with the numbers of the decade; such structure is fully preserved in Anatolius’ short work On Decade and Ps.-Iamblichus’ Theology of Arithmetic and is presupposed in most other arithmological works. 7 The original Pythagorean treatise, though widely read and quoted in all later arithmologies, has been lost; neither its title, nor the name of its author are known. At the end of his second paper on the topic Robbins published a diagram showing dependence of the arithmological writings studied by him from the Urquelle S. He divided all such sources into two families, Philonian and Theonian, the privileged witnesses of the first being Philo and Lydus (the latter very close to but not directly dependent on Philo), because they preserved the richest textual material from the original treatise, as well as Anatolius and Martianus Capella. The second, Theonian family, included Varro, Theon, Nicomachus and Ps.-Iamblichus and was regarded by Robbins as dependent on Posidonius as an intermediary source. Simultaneously with Robbins Karl Reinhardt published his famous book on Posidonius, where he scorned Schmekel’s idea that the Stoic wrote a special commentary on Timaeus and suggested another solution: the Pythagorean part of Sextus Empiricus’ exposé of theories of criterion (Adv. math. 7.92–108) comes from Posidonius’ work On the Criterion and presents not so much Pythagorean as his own views. 8 The two theories of the origin of Sextus’ accounts of Pythagorean number philosophy were debated most of the previous century; 9 both still have 5 Robbins F. E. Posidonius and the Sources of Pythagorean Arithmology, CPhil 15 (1920) 309–322. Robbins F. E. The Tradition of Greek Arithmology, CPhil 16 (1921) 97–123. See also Robbins F. E. Arithmetic in Philo Judaeus, CPhil 26 (1931) 345–361. 6 Robbins. Tradition, 97f. 7 Robbins. Posidonius, 320. 8 Reinhardt K. Poseidonius. Berlin, 1921, 414ff., 416 n. 4, 419f. 9 See below, 55 n. 12. The last serious but unsuccessful attempts to revive Schmekel’s thesis were made by Burkert W. Lore and Science in Ancient Pythagoreanism. Cambridge (Mass.), 1972, 54ff. (German original 1962) and Mansfeld J. The Pseudo–+LSSRFUDWLF 7UDFW ȆǼȇǿ µǼǺǻȅȂǹǻȍȃ &K –11 and Greek Philosophy. Assen, 1971, ch. 6. Cf. below, n. 30, 33.

followers, though Schmekel’s theory is now more widely rejected as outdated than Reinhardt’s The last significant contribution to the study of the common ancestor of arithmology as a genre has been made by Karl Staehle in his doctoral dissertation that aimed to give the most precise picture of Philo’s lost treatise ƻǏǛʐ ȢǛǓǒǖ̆Ǘ. 11 Because Philo several times refers to ƻǏǛʐ ȢǛǓǒǖ̆Ǘ as his most detailed exposition of arithmological subjects, Staehle related to this treatise all arithmological passages to be found in Philo’s oeuvre, in the first hand in De opificio mundi, and published them. Inside of chapters devoted to the first ten numbers Staehle organized material by individual topics, such as that the Monad is the beginning of number (5a), equal by its nature to God and nous (4a-c, h), that it generates all the other numbers but is not generated in itself (5e), that the Dyad ‘flows’ from the Monad (8) and is the first even number (9), and so on. Each numbered topic was accompanied by an extensive collection of parallels from the authors considered by Delatte and Robbins, among them the Aristotelian commentators Alexander, Asclepius, and Syrianus. In his introduction Staehle pointed out to the Early Academy – and not the Pythagorean school, as it was usual before and after him – as to the real birth place of arithmology and to Plato, Speusippus and Xenocrates as its creators. 12 The arithmological compilation on which Philo relied belonged to the Platonic tradition and has been written at the turn of the second and first centuries BC, the time, when interest in Pythagoreanism aroused again and many Neopythagorean apocrypha appeared. 13 Depending much on Robbins, Staehle disagreed with him on several important issues. Thus, he believed that Robbins overestimated Posidonius’ role in transmission of the arithmological tradition, for, though the Stoic did use in On the Criterion the original arithmological source, his presence in Sextus Empiricus is limited only to Adv. math. 7.92–93. 14 Staehle also disputed Robbins’ division of the sources into two separate groups and his idea that the Anonymus circulated in different versions and abridgements. 15 Some Staehle’s particular conclusions, for example, Lydus’ reliance on Philo, were rightly challenged 16 yet on the whole his study in Philo’s arithmology and its source was accepted and 10 Though Sedley follows Reinhardt’s thesis, he does not even mention arithmology: Sedley D. Sextus Empiricus and the Atomist Criteria of Truth, Elenchos 13 (1992) 21-56, at 30ff. 11 Staehle K. Die Zahlenmystik bei Philon von Alexandria. Leipzig; Berlin, 1931. 12 Ibid., 4-7. 13 Ibid. 15-16. 14 Ibid., 13-15. Stoic influence on this source predates Posidonius. 15 Ibid. 17-18. Cf., however: Mansfeld. The Pseudo–Hippocratic Tract, 172ff. 16 Boyancé P. Études Philoniennes, REG 76 (1963) 64–110, at 91f.; Burkert. Op. cit., 249 n. 51; Huffman C. A. Philolaus of Croton. Pythagorean and Presocratic. Cambridge, 1993, 334-339; see already 44 B 20 DK. Runia D. Philo of Alexandria. On the Creation of the Cosmos according to Moses. Leiden, 2001, 298f., 303 returns to Staehle’s position concerning Lydus, without sufficient ground, in my view.

endorsed by the subsequent scholarship. 17 But since Philonian scholars focused on the usage of arithmology as Philo’s exegetical tool, the study of its principal source has not been further advanced. The Anonymus ceased to attract scholars’ attention and no special studies of this text appeared. ,Q ǾROJHU 7KHVOHII¶V FROOHFWLRQ RIWKH pseudo-Pythagorean writings ‘arithmology’ though presented in the subject index 18 refers to the arithmological passages or fragments from the other works ascribed to Pythagoras and the Pythagoreans; the Anonymus itself is not even mentioned. David Runia in his commentary on Philo’s De opificio mundi offered the most judicious and helpful analysis of arithmological parallels to Philo, starting from late Hellenism. He, however, questioned Staehle’s method of reconstruction of ƻǏǛʐ ȢǛǓǒǖ̆Ǘ as being “totally flawed” (though Staehle did not attempt to reconstruct this work) and hypothesized that Philo took over his material from several already existing arithmological collections. 19 In his recent book on number symbolism in Platonism and early Christianity Joel Kalvesmaki examines several important Neopythagorean texts of the first century BC yet passes the Anonymus over in silence, returning in fact to the view of Schmekel: “The tradition of handbooks of number symbolism began possibly with Posidonius”. 20 This is an obvious step backward, because by the end of the previous century at the latest it became clear that no evidence of Posidonius’ commentary on Timaeus exists and that vicinity of his citation with Neopythagorean number doctrines in Sextus Empiricus cannot prove the Stoic’s role as the transmitter let alone the originator of arithmology. 21 2. The remains of the Anonymus The revival of the interest in the Pythagorean pseudepigrapha provides an opportunity to go beyond Thesleff’s collection and highlight the crucial role of the Anonymus in formation of the arithmological genre. But beforehand we have to delineate the limits of our knowledge. The 17 Boyancé. Op. cit., 83f.; Collins A. Y. Numerical Symbolism in Jewish and Early Christian Apocalyptic Literature, ANRW II.21 (1984) 1256f.; Moering H. Arithmology as an Exegetical Tool in the Writings of Philo of Alexandria, J. P. Kenney (ed.) The School of Moses: Studies in Philo and Hellenistic Religion. Atlanta, 1995, 141-176; Kraft R. A. Exploring the Scripturesque. Jewish Texts and their Christian Contexts Leiden, 2009. Ch. 13; Wyss B. Philon und die Pentas. Arithmologie als exegetische Methode, Feldmeier R. et al. (eds.). Alexandria. Tübingen, 2013, 361–379. 18 Thesleff H. The Pythagorean Texts of the Hellenistic Period. Åbo, 1965, Index IX, 1: Mathematics in general, arithmology. 19 He criticized Robbins (and thus Staehle) for “excessive use of the method of the Einquellenhypothese”: Runia. Philo, 27–28, cf. 264. “Arithmological handbooks” (191). On this, see below, 5. 20 Kalvesmaki J. The Theology of Arithmetic. Number Symbolism in Platonism and Early Christianity. Cambridge (Mass.), 2013, 9 n. 4. This view is wrongly ascribed to Robbins. 21 Adv. Math. 7.93 = F 85 E-K. See e.g. Edelstein L., Kidd I. G., eds. Posidonius. Fragments. Vol. 1. Cambridge, 1972, 337ff. Long A. The eclectic Pythagoreanism of Alexander Polyhistor, in Schofield, ed. Aristotle, 139–159, at 145: “I. G. Kidd (in his commentary on Posidonius) has convincingly shown that there is no reason to extend Posidonius’ presence in Sextus’ text beyond that single statement”. Cf. above, n. 9 and below, n. 55.

popularity of the treatise allowed it to produce copious offspring, yet none of those who directly or indirectly made use of this nameless and untitled work felt obliged to explicitly refer to it or to quote literally. Consequently, whereas most of the Anonymous’ topics are reconstructable, there is very little that can be legitimately presented as a fragment of this work. The longest paraphrases – surely, with an additional material due to Philo’s verbosity – are preserved in his lost work On Numbers, many fragments of which are contained in his other writings. The seven occupies most space, the four is treated much shorter, the six still shorter, while eight and nine are just touched upon. In cases when there are textual parallels between Philo and Lydus we can come closer to the original text, but since Lydus’ catalogue-like treatment, on the one hand, is much denser than Philo’s, and on the other, covers many topics which Philo omits, the possibilities to reconstruct the text, and not just thoughts, are very limited. Varro’s fragments deal with several chosen numbers, mostly with seven. Anatolius presents a complete but very short arithmology, his longest chapter, on the number seven, takes only three pages against forty paragraphs in Philo’s De opificio mundi. Extracts from Nicomachus’ lost Theology of Arithmetic, preserved in Ps.-Iamblichus’ homonymous treatise, textually are farther from the Anonymus and contain many late layers. To a large extent this also applies to Theon’s text. Given all this, it is reasonable to focus on the content and principal concepts of the Anonymus, as well as on its intellectual provenance and immediate influence. The first question to ask is whether originally there was only one such treatise or several. Indeed, the existence of Neopythagorean apocrypha with similar or identical doctrines is well attested, yet the writings themselves were different. To postulate two anonymous arithmological ‘handbooks’ one would need to show differences between them, which has not been done. As any ancient author could and did use the Anonymus for his own use without acknowledging his debt to it, what would be the reasons to produce another Anonymus? Theophrastus’ doxographical compendium once went through the stage of an anonymous handbook, the so-called Vetusta placita, but all the subsequent versions had real (Aetius) or made-up (Plutarch) authors. All other pseudo-Pythagorean arithmological writings have their ‘authors’ too. 22 Against a purely theoretical possibility that the Anonymus was not alone speaks the uniqueness of this writing which, besides its doctrines, is visible in a peculiar sequence of its chapters and sections and exact textual parallels to be found in various texts dependent on it. To adduce just one example, let us compare two early borrowings from the Anonymus: Varro’s Hebdomades as quoted by Aulus Gellius (book 3.10) and Philo’s De opificio: Varro Philo 22 Archytas (21.1f.), Lysis (114.13f.), Megillos (115.15f.), Opsimos (140.27f.), Philolaus (see below, 28), Proros (154.19f.), Pythagoras (164.1f.), Telauges (189.10f.). The pseudo-Pythagorean texts are quoted, if not otherwise indicated, by page and line of Thesleff’s edition (see above, n. 18).

Seven stars of the Great Bear (10.2) Seven stars of Pleiades (10.2) Seven planets (10.2) Seven heavenly circles (10.3) Equinoxes occur in the seventh month (10.4) Moon’s cycle is 4 weeks × 7 days = 28, which is equal to the sum of its parts (10.6) The formation of flesh in woman’s womb occurs in 7 days (10.7) Moon’s cycle is 4 weeks × 7 days = 28, which is equal to the sum of its parts (101) Seven heavenly circles (112) Seven planets (113) Seven stars of the Great Bear (114) Seven stars of Pleiades (115) Equinoxes occur in the seventh month (116) The formation of flesh in woman’s womb occurs in 7 days (124) A block of five identical astronomical items (Aul. Gell. 3.10.2–4) has a slightly different arrangement in Philo (De opif. 112–116); as follows from his other work, where the sequence is planets – Great Bear – Pleiades – phases of the moon (De spec. leg. 2, 58), the moon also belonged to this block, but has been removed to another place (101). 23 In Gellius’ condensed exposition of Varro an embryological item immediately follows astronomical ones, in Philo it comes a bit later (124). In view of this considerations we can safely assume that the Einquellenhypothese as successfully explains the origin of the arithmological genre from the Anonymus as the origin of doxography from Theophrastus’ ǀǟǝǓǔ̆ǗǎǦǘNjǓ. 24 3. Arithmology as a genre Some theoretical explication is needed regarding the notion of the arithmological genre which is an abstraction based on a natural grouping of similar writings. 25 What are the acceptable limits of similarity which would allow us to establish boundaries of arithmology that are neither too blurred nor too rigid? We can define arithmology as a literary genre of popular philosophy, originated in the framework of Neopythagoreanism, that systematized various speculations on the generation, properties and extra-mathematical significance of the first ten numbers in their relations to each other. 26 This definition is exclusive, not inclusive. Not every speculation on any individual significant number up to ten, say, three, four or seven can be taken as arithmology but only such where a complete system of the first ten numbers is observable or detectable. In such a system every number from one to ten acquires its meaning as a 23 Cf. De leg. 1, 8: planets – Great Bear – phases of the Moon. See Zhmud L. Die doxographische Tradition, H. Flashar, D. Bremer, G. Rechenauer (eds.). Grundriss der Geschichte der Philosophie. Die Philosophie der Antike. Vol. 1. Basel, 2012, 150–174. 25 For more detail on the origin of arithmology as a genre, see Zhmud L. Greek Arithmology: Pythagoras or Plato? A.-B. Renger, A. Stavru (eds.). Pythagorean Knowledge from the Ancient to the Modern World. Wiesbaden, 2016, 311–336, at 312–315. 26 Cf. the original definition of arithmology: Delatte. Op. cit., 139.

member of the arithmetical progression: two is the first female number, three is the first male number, five (or six) is the number of marriage, seven is the maiden-number and so on. In any arithmological text, either short or long, we find close mutual relationship between numbers of the decade; the same pattern is presupposed in arithmological fragments and extracts, even if they deal with individual numbers. 27 What remains beyond the boundaries of arithmology is a vast and diffuse field of number symbolism that deals with individual significant numbers – three, seven or nine – which are conceived neither as generated nor as mutually related. 28 Number symbolism is a universal phenomenon going back to preliterate times, in Greek culture it is amply documented already in Homer and Hesiod, in religion, especially Apollo’s cult, and later in the early Greek philosophy and medicine. Not only the ancient Pythagoreans with their reputation as chief proponents of number symbolism but also such perfectly rational philosophers as Aristotle and Theophrastus revealed predilection for some traditionally significant numbers such as three and seven. 29 Therefore, we cannot relate to arithmology passages or works where such individual numbers are highlighted, praised or extolled, but are not arranged in a system of mutually related numbers of the decade, be it Solon’s elegy on the seven-year ages of man’s life (fr. 24 West), Hippo’s embryological calendar based on numbers seven and three (38 A 16), or the Hellenistic pseudo-Hippocratic treatise On the Sevens. 30 Building on these considerations it is possible to narrow the dating of the Anonymus as the first specimen of arithmology down to half a century. The first secure traces of this work are to be found in the Vetusta placita (Aët. 1.3.8) dated in the mid-first century BC (it comes probably from the school of Posidonius, the last philosopher mentioned in it) and in two writings of Varro (116–28 BC), Tubero, or De origine humana (published ca 40 BC) and Hebdomades, or De imaginibus (published 39 BC); 31 Varro also used the Vetusta placita. Thus, the middle of the 27 As e.g. Proros’ ƻǏǛʐ Ǟ˜ǜ ȳnjǎǙǖdžǎǙǜ (154.19f.), on which see Macris C. Proros de Cyrène, DPhA 7 (2018) 1698f. Mansfeld suggested a common source of Philo and Proros (The Pseudo–Hippocratic Tract, 169 n. 59) that we can identify with the Anonymus. 28 For relationship between number symbolism and arithmology, see Zhmud L. From number symbolism to arithmology, in: L. Schimmelpfennig, R. Kratz (eds.). Zahlen- und Buchstabensysteme im Dienste religiöser Bildung. Tübingen, 2019, 25–45. 29 Roscher W. F. Die Hebdomadenlehre der griechischen Philosophen und Ärzte, Abhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften 24.6 (1906) 97f.; Zhmud. From number symbolism, 26, 28, 31f. 30 Roscher W. F. Die hippokratische Schrift von der Siebenzahl in ihrer vierfachen Überlieferung. Padeborn, 1913. Mansfeld. The Pseudo–Hippocratic Tract, ch. 6 dated On the Sevens, ch. 1–11 after Posidonius (cf. below, n. 33) and referred it to the arithmological genre. The reverse chronological order seems more plausible (Zhmud. Arithmology, 314 and n. 13), so that the tract should be dated in the late second century BC at the latest. 31 Dahlmann H. M. Terentius Varro, RE Suppl. 6 (1935) 1178. See Aul. Gell. 1.20 (the cube of three equals to the moon’s circle, i.e. 27), 3.10 (on the hebdomad, see above, 5), 14.3–7 (quotations from Solon and Ps.-Hippocrates); Cens. DN 9.1 (“opinio Pythagorica” on gestation in seven or ten months); Serv. Ad Verg. Ecl. 8,75 (number three is perfect, it comprises the beginning, middle and end, cf. Staehle. Op.

first century BC may be regarded as terminus ante quem. On the other hand, Varro’s older contemporary Posidonius and the tract On the Sevens quoted by the Anonymus, do not reveal influence of arithmology in the defined above sense; their discussions of the number seven belong to the traditional framework of number symbolism. Therefore, time around 100–90 BC can be taken as a convenient terminus post quem for the Anonymus. To be sure, Posidonius was acquainted with new developments in pseudo-Pythagorean literature, for he was willing to infer about Pythagoras’ doctrine on the soul from the writings of his students and followers. 32 But our arithmological treatise, as it seems, has not reached him yet or left no traces in his oeuvre. 33 Nor there is any secure evidence of its earlier existence. 4. Platonic number philosophy and the perfect ten Theoretical foundation of arithmology was laid down in the Early Academy. As Aristotle noted, while criticizing the Platonists: “Mathematics has come to be identical with philosophy for modern thinkers, though they say that it should be studied for the sake of other things” (Met. 992a33, tr. W. D. Ross). This philosophy is presented only to a certain extent in Plato’s dialogues, especially in his Timaeus. 34 More often than not scholars reconstruct it from Aristotle’s reports and criticisms of the unwritten doctrines of Plato and from a few fragments of Speusippus and Xenocrates on this account. Plato was the first to separate incorporeal things such as numbers and geometric principles (points, lines, planes, solids) from the sensible world and to attach an ontological status to them. This transformed Ǟʊ ǖNjǒǑǖNjǞǓǔʋ into independent entities which, similar to physical objects, can be generated. According to the Academic doctrine, ontological priority resides with that which can exist without another. Solids are less substance than planes, planes than lines, lines than points, and points than units, 35 since “a unit is substance without position, while a point is substance with position”, which is to say that the latter contains an additional property. 36 Thus, units, i.e. numbers, are by nature first. 37 Respectively, the line is derived from the point (or FLWʋ 17, 18a-b). More on Varro’s arithmology, see Mansfeld. The Pseudo–Hippocratic Tract, ch. 6; Palmer R. The Archaic Community of the Romans. Cambridge, 1970, 19ff. 32 Posid. F 151, 165 E-K; Zhmud L. The papyrological tradition on Pythagoras and the Pythagoreans, in: C. Vassallo, ed. Presocratics and Papyrological Tradition: A Philosophical Reappraisal of the Sources. Berlin; Boston, 2019, 111–146, at 136ff. 33 Cf. Mansfeld. The Pseudo–Hippocratic Tract, ch. 6: Posidonius in his Comments on the Timaeus developed the Early Academic arithmology; On the Sevens depends on Posidonius; the Anonymus revised and completed Posidonius’ arithmology and used On the Sevens; Varro first (in Imagines) used Posidonius but later (in Atticus) adduced the Anonymus. 34 For evidence of Plato’s dialogues on his number philosophy, see Tarán L. Speusippus of Athens. Leiden, 1981, 13ff. 35 Arist. Met. 1002a4–12, 1017b6–21, 1018b37–1019a4; De bono, fr. 2 Ross. 36 Arist. APo 87a34f., cf. Met. 982a26–28. A point as a monad having position is an Academic formula. Alex. In Met. 55.20–27 = Arist. De bono, test. and fr. 2 Ross.

produced by a moving point, Arist. De an. 409a3–7), the plane from the line, and the solid from the plane, and this derivation sequence is closely connected to the first four numbers. Speusippus, for example, associated the point with one, the line with two, the plane with three, and the pyramid with four (see below, 10). Aristotle attributed to Plato the derivation of line, plane, and solid “after numbers” or even from numbers; different schemes of generation of magnitudes are attested also for Speusippus and Xenocrates. 38 Yet numbers, according to Plato, were not the ultimate level of reality, they themselves derive from the pair of the highest principles, the Monad and the Indefinite Dyad. 39 From these two principles ten ideal numbers, or Forms-Numbers, are derived, such as Twoness, Threeness, etc.; their generation serves as a model for the generation of all other numbers. When Aristotle refers to the theory of the ten archetypal numbers, he obviously has Plato in mind, 40 and at Phys. 206b27–33 he directly names Plato (ǖʍǡǛǓ ǍʊǛ ǎǏǔʋǎǙǜ ǚǙǓǏ˪ ǞʒǗ ȢǛǓǒǖʓǗ). This is why the decad was counted in the Academy as the perfect, or complete number. 41 The doctrine on the decad as ǞʍǕǏǓǙǜ ȢǛǓǒǖʓǜ was fully formulated in Speusippus’ lost treatise On Pythagorean Numbers. The first part of the book dealt with different kinds of the socalled figurate numbers (linear, plane, solid, etc.), continuous and discontinuous proportions, and the five regular solids. 42 The second part, better known to us thanks to a two-page quotation from it (fr. 28 Tarán), was devoted to the marvellous properties of the decad: Ten is a perfect number, and it is both right and according to Nature that we Greeks and all men arrive at this number in all kinds of ways when we count, though we make no effort to do so; for it has many special properties which a number thus perfect ought to have, while there are many characteristics which, while not special to it, are necessary to its perfection (tr. I. Bulmer-Thomas). The most conspicuous feature of Speusippus’ exposition is that he focuses not on the correspondences between numbers and things but on numbers and geometric magnitudes and the interconnections between them. Such an emphasis is perfectly understandable insofar as numbers constitute the first layer of beings for Speusippus, with geometric magnitudes coming after 38 Plato: Arist. De an. 404b19–24; Speusippus: F 28 ad fin., 51–52, 65 Tarán; Xenocrates: F 99–100, 117, 195 I. P. See also Theophr. Met. 6a23-b16 = Speus. fr. 59 Tarán = Xenocr. fr. 100 I. P. 39 The evidence, mostly from Aristotle, is conveniently collected in Gaiser K. Platons Ungeschriebene Lehre. Stuttgart, 1963: 474ff. (nn. 22–34). See also Burkert. Op. cit., 21f.; Dillon J. The Heirs of Plato. Oxford, 2003, 18f. 40 1073a17–22; 1084a12–b2: ǚǏǓǛ̆ǗǞNjǓ ǎ˔ ɻǜ ǞǙ˹ ǖʍǡǛǓ Ǟ˜ǜ ǎǏǔʋǎǙǜ ǞǏǕǏʑǙǟ ɢǗǞǙǜ ȢǛǓǒǖǙ˹ (a31); 1088b10–11. On Plato’s teaching on the decad, see e.g.: Dillon J. The Middle Platonists. Ithaca, NY, 1996, 19ff.; Erler M. Grundriss der Geschichte der Philosophie. Die Philosophie der Antike. Bd. 2/2. Basel, 2007, 427f. 41 To be sure, in Plato’s dialogues ǞʍǕǏǓǙǜ ȢǛǓǒǖʓǜ refers either to the so-called nuptial number or to the great year (Res. 546b–d; Tim. 39d3–4). In mathematics, a perfect number is equal to the sum of all its divisors, e.g. 6 = 1+2+3, but this meaning is not attested before Euclid (El. 7, def. 22; 9, 36). 42 Most of these things go back to Pythagorean mathematics. If the title of the work is Speusippian, which is not certain, it most probably referred to mathematical material used in this work (Tarán. Op. cit., 263).

them. 43 He does not seem to be primarily motivated by traditional number symbolism: the numbers three, seven, or nine do not interest him as such, and the other numbers insofar as they add up to ten. Thus, playing with the sides, angles, and edges of different plane and solid figures, he several times adds six to four in order to get ten. The decad is perfect not because it corresponds to a natural phenomenon, but because it comprises all the other numbers: Furthermore all the ratios are in 10… and so are the linear and plane and solid numbers. For 1 is a point, 2 is a line, 3 is a triangle and 4 is a pyramid; all these are elements and principles of the magnitudes like to them. In these numbers (1, 2, 3, 4) is seen the first of progressions… and they have 10 for their sum. The primary elements in plane and solid figures are point, line, triangle, pyramid, they contain the number ten and are limited by it (tr. I. Bulmer-Thomas). Speusippus’ second most important number is four; he tirelessly connects it with the ten, being very enthusiastic about the transformation of the tetrad into the decad: 1 + 2 + 3 + 4 = 10. In the late Hellenistic filiations of Platonic number philosophy the fact that the sum of the first four numbers makes up ten acquires special significance (see below, 16). We know from Aristotle that the Academics (Plato, Speusippus or Xenocrates?) matched various types of cognitive activity with the first four numbers: ǗǙ˹ǜ is one, ȲǚǓǝǞʏǖǑ is two, ǎʓǘNj is three (the number of the plane), and NjɒǝǒǑǝǓǜ is four (the number of the solid). 44 Xenocrates identified ǗǙ˹ǜ with Ǟʒ ȷǗ and with god, Speusippus also maintained ǗǙ˹ǜ to be god. 45 The doctrine that the dyad is the first female number and the triad the first male number seems to originate with Xenocrates, who assigned such predicates as ȦǛǛǏǗ–ǒ˜Ǖǟ and ǚǏǛǓǞǞʒǗ– (ȦǛǞǓǙǗ) to his first principles ƷǙǗʋǜ and Ưǟdžǜ. 46 All these ideas have been absorbed and creatively reworked by the author of the Anonymus, who put them in a framework of the treatise of ten chapters devoted to each number of the decad. But Speusippus’ deliberate focus on relations between the first ten numbers and geometric magnitudes was too radical and refined to be directly followed in a popular philosophical genre. The originator of arithmology took a considerable step backwards by incorporating number symbolism into the conceptual scheme created in the Early Academy. What we observe in 43 He rejected the theory of Forms and replaced the ideal numbers with mathematical ones (Arist. Met. 1083a23 = Speus. fr. 34 Tarán; 1075b37f. = fr. 30; 1080b11f. = fr. 33). 44 Arist. De an. 404b18–24 = On Philosophy, fr. 11 Ross. Cf. Hicks R. D. (ed.). Aristotle. De Anima. Cambridge, 1907, 222 (Plato); Isnardi Parente M. Per l'interpretazione di Aristotele, De An. 404B18 sgg., in: Palmer R.B., Hammerton-Kelly R., eds. Philomathes. Studies and Essays in the Humanities in Memory of Philip Merlan. The Hague, 1971, 146–170 (Speusippus); Tarán. Speusippus, 459f. (Xenocrates). The latter seems the most plausible candidate. 45 Xenocr. fr. 213 I–P, cf. Pl. Tim. 47e; Speus. fr. 58 Tarán. Dörrie–Baltes. Op. cit., 192ff. Dillon. Middle Platonists, 99ff. 46 Aët. 1.7.30 = fr. 213 I–P; Dillon. Middle Platonists, 102ff. – In the ‘Pythagorean’ table of opposites (Arist. Met. 986a24-26) that has a clear Academic origin (Zhmud. Pythagoras, 434f., 449f.) we also find ǚǏǛǓǞǞʒǗ–ȦǛǞǓǙǗ and ȦǛǛǏǗ–ǒ˜Ǖǟ.

arithmological texts is, as it were, Speusippus ‘lite’: they are not so heavily, metaphysically loaded and contain much entertaining material (see above, 5). In the Anonymus each chapter was organized according to the Platonic division of the world into ǗǙǑǞʋ and NjɎǝǒǑǞʋ. 47 In the realm of ǗǙǑǞʋ that always comes first, the arithmologist treated specific features of each number and all its possible connections with the others numbers of the decad (sometimes also outsides of the decad), with plane and solid figures and harmonic intervals. As for realm of NjɎǝǒǑǞʋ, Speusippus’ treatise had very little to offer: though he called the decad the divine model of the cosmos, 48 no other references to the extra-mathematical world are to be found in the only preserved fragment. Therefore, the arithmologist had to turn here to the much older and richer tradition of number symbolism with its favourite numbers three and seven. 5. The Anonymus and kindred Neopythagorean writings The most obvious shortcoming of the earlier study of the Anonymus is that it was considered isolated from the main body of the Neopythagorean apocrypha which began to spread in the first century BC, most probably from Alexandria.49 The bulk of this literature is constituted by Doricized treatises having titles and bearing names of various known, unknown and fictional Pythagoreans. There is a group of non-Doric texts that constitute an exemption from this pattern and is akin to the Anonymus in several important respects. 50 In terms of form, these anonymous and mostly titleless works are doxographical or, in a wider sense, historiographical. In content, they expound specific doctrines similar or identical to the Anonymus, which are either derived from their common source, as for example, on the Monad and Indefinite Dyad, or directly borrowed from the Anonymus, as for example, on the tetractys. The earliest of these writings, the Pythagorean Notes (D. L. 8.25–33), a titles and anonymous doxography transmitted by Alexander Polyhistor (ca 100 – ca 35 BC), seems to be prior to the Anonymus or at least not dependent on it. ƻǟǒNjǍǙǛǓǔʊ ɫǚǙǖǗʏǖNjǞNj is not a proper title; though referring several times to the authority of unnamed Pythagoras (ǎǙǍǖNjǞʑǐǏǓ, ǠǑǝʑ, but ǔNjǕǙ˹ǝǓ 8.7), these notes lack a proper author. 51 The Anonymus Photii is a Neopythagorean bio-doxography summarized by Photius from an anonymous njʑǙǜ ƻǟǒNjǍʓǛǙǟ, probably, of the 47 Staehle. Op. cit., 10f. See, e.g. ȲǗ ǖʌǗ ǙɰǗ ǞǙ˪ǜ ǗǙǑǞǙ˪ǜ Ǟʒ ȢǔʑǗǑǞǙǗ ǔNjʐ ȢǚNjǒʌǜ ȲǚǓǎǏʑǔǗǟǞNjǓ ȳnjǎǙǖʋǜ, ȲǗ ǎʌ ǞǙ˪ǜ NjɎǝǒǑǞǙ˪ǜ ǖǏǍʋǕǑǗ ǔNjʐ ǝǟǗǏǔǞǓǔǣǞʋǞǑǗ ǎʕǗNjǖǓǗ (Phil. De opif. mundi, 101). 48 ǚNjǛʋǎǏǓǍǖNj ǚNjǗǞǏǕʍǝǞNjǞǙǗ Ǟ̇ ǞǙ˹ ǚNjǗǞʒǜ ǚǙǓǑǞ˝ ǒǏ̇ ǚǛǙǏǔǔǏǓǖʍǗǑǗ (fr. 28 Tarán). 49 Zeller E. Die Philosophie der Griechen in ihrer geschichtlichen Entwicklung. 3 Vols. 6th edn. Leipzig 1919. Vol. III.2, 113f.; Zhmud L. What is Pythagorean in the pseudo-Pythagorean Literature?, Philologus 163 (2019) 85 with n. 71. 50 Cf. Burkert. Op. cit., 53ff., 57ff. 51 Long. Op. cit.; Laks A. The Pythagorean Hypomnemata reported by Alexander Polyhistor in Diogenes Laertius (8.25–33), in: G. Cornelli, R. McKirahan, C. Macris, eds. On Pythagoreanism. Berlin, 2013,

late first century BC – early first century AD. 52 Pythagoras’ biography takes here only two out of 23 paragraphs, whereas doxography presents the teaching of the Pythagorean school as similar to that of Plato and Aristotle, who figure as the ninth and tenth diadochos of Pythagoras (235.5–7). Besides direct borrowings from the Anonymus, this text reveals a marked interest in number symbolism. 53 Finally, this group includes four anonymous accounts of ‘Pythagorean’ number philosophy in Sextus Empiricus, belonging to two kindred but different versions. 54 They derive from the first-century BC sources, 55 based, in turn, on the Neopythagorean pseudepigrapha. Sextus regularly refers to the Pythagoreans 56 and only once to Pythagoras himself (10.261, cf. also 9.366). Our earliest witness for the Anonymous, Varro, also speaks of the Pythagoreans. 57 This common tendency to avoid Pythagoras as an author is understandable if we recall that by the first century BC it was widely believed that he did not leave anything in writing, 58 so that the Neopythagorean pseudepigrapha were attributed either to his students, or to further unspecified “Pythagoreans”, in which case they remained anonymous. Another important salient feature of the Anonymus and the group of the pseudo-Pythagorean texts akin to it is Stoicized Platonism, in which the originally Platonic and Early Academic theories and ideas are modified under influence of Stoicism. A crucial link between these two schools and emerging Neopythagoreanism was Antiochus of Ascalon (ca 135/130 – ca 68 BC), who taught for some time in Alexandria. 59 Antiochus definitively turned from the skeptical New Academy to the Old Academy that included, in his view, not only Speusippus, Xenocrates and other Platonists but also Aristotle and his students (Cic. De fin. 5.7), fusing this kind of 52 Phot. 438b–441b = 237.4–242.9 Thesleff. Bibliography: Macris C. Pythagore de Samos, DPhA 7 (2018) 752f. On dating, see Zhmud. Pythagoras, 72 n. 48. Theiler W. Philo von Alexandria und der Beginn des kaiserzeitlichen Platonismus, in: K. Flasch, ed. Parusia: Festschrift für J. Hirschberger. Frankfurt a.M., 1965, 199–218, at 207ff. has demonstrated many important parallels between the Anonymus Photii, Philo, and Sextus Empiricus’ sources on Pythagoreanism, yet his attribution of this work to Eudorus is not shared any more. 53 Borrowings: e.g. tetractys (237.23–238.1), see below, 18. Numbers: 3 ways to improve a man, 12 colours, 7 tastes, 5 senses, 12 zones in the heaven, 4 causes, 4 seasons, the sun is 100 or 30 times bigger that the earth, 4 elements, 8 cognitive faculties, 3 elements of learning, 3 meanings of the word ‘heaven’. 54 1) Adv. math. 4.2–10 treats more briefly and with some variations the same subject as Adv. Math. 7.94109; 2) PH 3.152–157 is a short variant of Adv. math. 10.249-284. See below, 15. 55 Isnardi Parente Ɇ. Sesto, Platone, l’Accademia antica e i Pitagorici, Elenchos 13 (1992) 119-168, at 146, 150-152, 157 and n. 49. Tarrant H. Agreement and the self-evident in Philo of Larissa, Dionysius 5 (1981) 66–97 derived the account in Adv. math. 7 from Antiochus of Ascalon via Aenesidemus, Sedley. Op. cit. from Posidonius via Aenesidemus, while Theiler. Philo, 208f. related Adv. math. 10.248–283 to Eudorus (cf. above, n. 52). 56 ƻǟǒNjǍǙǛǓǔǙʑ, ƻǟǒNjǍǙǛǓǔ̆Ǘ ǚNjljǎǏǜ, Ǚɏ ǚǏǛʐ ƻǟǒNjǍʓǛNjǗ, Ǚɏ Ȣǚʒ ǞǙ˹ ƻǟǒNjǍʓǛǙǟ, ȿ ǞǙǓNjʕǞǑ Ǟ̆Ǘ ƻǟǒNjǍǙǛǓǔ̆Ǘ ǝǞʋǝǓǜ. 57 See above, 7 n. 31. See also Staehle. Op. cit., 11f. 58 D. L. 8.6 (Sosicrates of Rhodes); Posid. fr. 151 E-K; Philod. De piet. B 24, p. 66 Gomperz (from Stoic doxography of the second century BC); see Zhmud. The papyrological tradition, 134f. 59 Görler W. Antiochos aus Askalon und seine Schule, in: Flashar H., ed. Grundriss der Geschichte der Philosophie. Die Philosophie der Antike. Vol. 4.2. Basel, 1994, 942f.

Platonism with contemporary Stoicism. 60 The next move, decisive for Neopythagoreanism, was to conceive ‘Pythagoras’ as one of Plato’s principal teachers (thus returning to the view of Aristotle and the Peripatetics), 61 for Socrates could not account for the dogmatic part of Platonism. Among the (admittedly, scanty) evidence for this is Antiochus’ opinion that Pythagoras originated Plato’s bipartite (originally tripartite) division of the soul into rational and irrational, the latter including the affections (ǚʋǒǑ). 62 Antiochus’ coeval Posidonius also shared this opinion, 63 but since he inferred Pythagoras’ view on ǚʋǒǑ in the soul from the writings of the Pythagoreans (F 151 E.-K.), which is to say, from the Neopythagorean apocrypha, he obviously reacted to them rather than influenced them. (The tripartition of the soul has been ascribed to Pythagoras in the Pythagorean Notes 64 and in many other pseudo-Pythagorean writings. 65) What is lacking in Posidonius is Antiochus’ new biographical perspective called to reinforce the doctrinal link between Plato and Pythagoreanism: Plato came to Italy and Sicily in order to meet the Pythagoreans and to appropriate their dogmata, of which Socrates had not even wanted to hear; he became acquainted with Archytas, Echecrates, and Timaeus of Locri, got access to Philolaus’ book, learned all the Pythagorean teaching, first of all their mathƝmata and the doctrine of the soul, and made it more argumentative; out of love for Socrates, however, he ascribed this Pythagorean sapientia to his teacher. 66 In such a framework it was easier to interpret – not necessarily by Antiochus himself – a Platonic pair of principles, the Monad and the Indefinite Dyad, as a Pythagorean teaching, the more so as the precedent for this could be found in Aristotle and Theophrastus. 67 Stoic elements are most visible in the Pythagorean Notes that combines a Platonic system of principles, the Monad and the Indefinite Dyad (D. L. 8.25), with a largely Stoic body of cosmological and physical doctrines (see above, n. 51). The doctrine of the Monad and the 60 See Sedley D., ed. The Philosophy of Antiochus. Cambridge, 2012. Zhmud L. Pythagoras and the Early Pythagoreans. Oxford, 2012, 436ff., 452ff. 62 Cic. Tusc. 4.10. For its Antiochean provenance, see Bonazzi M. Eudorus’ Psychology and Stoic Ethics, in: M. Bonazzi, C. Helmig, eds. Platonic Stoicism-Stoic Platonism. The Dialogue between Platonism and Stoicism in Antiquity. Leuven, 2007, 133–148, at 121f.; cf. Tarrant H. Scepticism or Platonism? The Philosophy of the Fourth Academy. Cambridge, 1985, 129 n. 9. 63 Galen. De plac. Hipp. et Plat. 4.7.40 = F 165, l. 166f E-K. Under the view common to Pythagoras and Plato obviously the tripartition of the soul is meant (F 142-143, 146 E-K). 64 D. L. 8.29. Here it looks rather peculiar, see Long A. Op. cit., 155f.; Laks. Op. cit., 375. 65 Examples: Burkert. Op. cit., 74; Vander Waerdt P. A. Peripatetic soul-division, Posidonius, and Middle Platonic moral psychology, GRBS 26 (1985), 373–394, at 392. See also: ȦǛǡǏǓ ǖʌǗ ǍʊǛ Ǟʒ ǕʓǍǙǗ ȶǡǙǗ Ǟˍǜ Ǣǟǡˍǜ, ȦǛǡǏǞNjǓ ǎʌ Ǟʒ ȦǕǙǍǙǗ, ǔǛNjǞǙ˹ǗǞǓ ǎʌ Ǟ̆Ǘ ǚNjǒʍǣǗ ȢǖǠʓǞǏǛNj (Ps.-Archyt., 33.15-16); Stob. 1.49.34. 66 Cic. Resp. 1.15–16; Tusc. 1.39: Platonem ferunt… didicisse Pythagorea omnia; De fin. 5.86–87 (= Baltes M., Dörrie H. Der Platonismus in der Antike. Vol. 2. Stuttgart, 1996, 250–256, 526–536). See Tsouni G. Antiochus on Contemplation and the Happy Life, in: Sedley. The Philosophy of Antiochus, 133–150, at 136f. Cf. ɩǞǓ ǞʎǗ ǖʌǗ ǒǏǣǛǑǞǓǔʎǗ ǔNjʐ ǠǟǝǓǔʎǗ ƻǕʋǞǣǗʋ ǠNjǝǓ ǚNjǛʊ Ǟ̆Ǘ ȲǗ ɖǞNjǕʑˋ ƻǟǒNjǍǙǛǏʑǣǗ ȲǔǖNjǒǏ˪Ǘ, ǞʎǗ ǎʌ ȾǒǓǔʎǗ ǖʋǕǓǝǞNj ǚNjǛʊ ƽǣǔǛʋǞǙǟǜ (Anon. Phot. 238.17–19). 67 Alex. In Met. 55.20 = Arist. De bono, fr. 2 Ross; Theophr. Met. 11a27ff. See above, 13 n. 61.

Indefinite Dyad is attested also in the Anonymus Photii and Sextus Empiricus; its formulation leaves no doubt that it derives from the common Stoically coloured source. 68 What is further peculiar to this group is that Plato’s dualistic theory of the opposite principles is subjected to the monistic interpretation that conceives the Monad as the principal arche (active cause) producing the Indefinite Dyad. 69 The Pythagorean Notes puts it as follows: The principle of all things is the Monad. Arising from the Monad the Indefinite Dyad serves as matter for the Monad, which is its cause (NjɎǞʑNj). From the Monad and the Indefinite Dyad arise numbers, from numbers points, from points lines, from lines planes figures, from plane figures, solids, from solids, sensible bodies, the elements of which are fire, air, earth, and water (D. L. 8.25). The basis of this theory is the Platonic derivation of ǗǙǑǞʋ (lines–planes–solids) and NjɎǝǒǑǞʋ from the Monad and the Indefinite Dyad (the four elements at the end are Stoic). There were different ways to explain this, one of them, according to Aristotle, was that the point is a unit with position, a line is generated by the moving point and a plane by the moving line. 70 Under influence of Stoic metaphysics that identified two principles, Ǟʒ ǚǙǓǙ˹Ǘ and Ǟʒ ǚʋǝǡǙǗ, with God (or divine logos) and matter 71 – note that this was Antiochus’ doctrine as well 72 – Platonic derivation was transformed into a doctrine of the active and divine Monad generating the passive material Dyad, which in differing versions became a cornerstone of Middle Platonism. The arithmologist expounds this doctrine as follows: the Monad is likened to god and nous (4a-c, h); the Dyad is generated by ‘flow’ (˸ʕǝǓǜ) of the Monad (8); the Dyad is associated with matter (11). The same theory of the Monad as ȢǛǡʎ ǚʋǗǞǣǗ from which points, lines, planes and solids arise is stated in the Anonymus Photii (238.8–11); the Dyad, though not directly generated from the Monad, is pushed into the background. In Sextus’ source the Monad, active cause, added to itself produces the Indefinite Dyad, passive matter. 73 A corollary of this 68 Anon. Phot. 237.17–23, 238.8–11; Sext. Emp. PH 3.153–154; Adv. math. 10.261–262, 270-278. ɣǞǓ ȿ ǖʌǗ ǖǙǗʊǜ ǔNjǞʊ ǞʎǗ ɎǝʓǞǑǞNj ǔNjʐ Ǟʒ ǖʍǞǛǙǗ ǕNjǖnjʋǗǏǞNjǓ, ȿ ǎʌ ǎǟʊǜ ǔNjǒ’ ɫǚǏǛnjǙǕʎǗ ǔNjʐ ȶǕǕǏǓǢǓǗ (Anon. Phot. 237.19–23). ǙɪǔǙ˹Ǘ ȿ ǖʌǗ ɎǝʓǞǑǜ Ǟ̇ ȳǗʐ ɫǚʋǍǏǞNjǓ…, ȿ ǎʌ ȢǗǓǝʓǞǑǜ ȲǗ ɫǚǏǛǙǡ˝ ǞǏ ǔNjʐ ȲǕǕǏʑǢǏǓ njǕʍǚǏǞNjǓ… ȢǕǕʊ ǔNjʐ ȿ ɫǚǏǛǙǡʎ ǔNjʐ ȿ ȶǕǕǏǓǢǓǜ ǔNjǞʊ ǞʒǗ Ǟ˜ǜ ȢǙǛʑǝǞǙǟ ǎǟʋǎǙǜ ǕʓǍǙǗ ǞʍǞNjǔǞNjǓ (Adv. Math. 10.275–276). For Stoicism in the Anonymus Photii and Sextus’ source, see Reinhardt K. Posidonius, RE 22 (1953) 763–768; Theiler. Philo von Alexandria, 207ff. 69 On this monistic tendency see Isnardi Parente. Sesto, 150f.; Zhmud. Arithmology, 320 with n. 33. 70 De an. 409a3–7. It remains unsettled whom this dynamic theory belongs to. Cf. Speus. fr. 52 Taran with comm.; Xenocr. fr. 195 I. P with comm. Cf. Pl. Leg. 894a2-4. 71 Cf. Stoic doxography in Diogenes Laertius: ƯǙǔǏ˪ ǎ’ NjɪǞǙ˪ǜ ȢǛǡʊǜ ǏɔǗNjǓ Ǟ̆Ǘ ɣǕǣǗ ǎʕǙ, Ǟʒ ǚǙǓǙ˹Ǘ ǔNjʐ Ǟʒ ǚʋǝǡǙǗ. Ǟʒ ǖʌǗ ǙɰǗ ǚʋǝǡǙǗ ǏɔǗNjǓ ǞʎǗ ȦǚǙǓǙǗ ǙɪǝʑNjǗ ǞʎǗ ɯǕǑǗ, Ǟʒ ǎʌ ǚǙǓǙ˹Ǘ ǞʒǗ ȲǗ NjɪǞ˝ ǕʓǍǙǗ ǞʒǗ ǒʍǙǗ (7.134 = SVF 2.300). 72 Cic. Acad. 1.24; Görler. Op. cit., 950. 73 PH 3.153; Adv. math. 10.261. 277. Cf. ƻǟǒNjǍʓǛNjǜ ǞǙʑǗǟǗ ȢǛǡʎǗ Ǟ̆Ǘ ɣǕǣǗ ȢǍʍǗǗǑǞǙǗ ȢǚǏǠʏǗNjǞǙ ǞʎǗ ǖǙǗʋǎNj, ǍǏǗǗǑǞʎǗ ǎʌ ǞʎǗ ǎǟʋǎNj ǔNjʐ ǚʋǗǞNjǜ ǞǙʔǜ ȦǕǕǙǟǜ ȢǛǓǒǖǙʕǜ (Hippol. Philos. 6.23.1, cf. 1.2.6. 9, 4.43.5, 4.51.4). See Theiler W. Einheit und unbegrenzte Zweiheit von Platon bis Plotin, in: J. Mau, ed. Isonomia: Studien zur Gleichheitsvorstellung im griechischen Denken. Berlin, 1971, 103f.; Isnardi Parente. Sesto, 147ff., 150.

theory is the statement that the Monad differs from the numerical one, which is to be found in the Anonymus, the Anonymus Photii and in Sextus’ exposé. 74 All evidence suggests that in the first quarter of the first century BC this system must have been already formed, 75 for it precedes the Anonymus and is attested in one of the earliest Neopythagorean pseudepigrapha, the Pythagorean Notes, which is not dependent on the Anonymus. Another Middle Platonic/Neopythagorean doctrine of principles, known from Eudorus of Alexandria (fl. ca 25 BC), posited a third, supreme principle, above the One and the Indefinite Dyad and therefore was a refinement of this system. 76 If these considerations are correct, then the Anonymus can be seen as an early offshoot of the newly developed Middle Platonic and Neopythagorean number metaphysics, with a more narrow focus on speculations about the first ten numbers. In view of kinship of these two currents of thought, one more general and one more specific, they have to be carefully distinguished. The second always implies the first, but not vice versa. 77 6. The Anonymus and the Vetusta placita The importance of the Anonymus for Neopythagorean doxography is confirmed by the fact that the section on Pythagoras’ principles in the Vetusta placita (Aët. 1.3.8, cf. 1.7.18 on what is god) is consistently arithmological and seems to be almost entirely taken from this work. Pythagoras understandably lacked in Theophrastus’ ǀǟǝǓǔ̆Ǘ ǎǦǘNjǓ, for he was not viewed as a physikos. As nothing certain was known about his physical teaching, sporadic attempts were made to invent it for him. The third-century BC apocryph known as tripartitum included ƻNjǓǎǏǟǞǓǔʓǗ, ƻǙǕǓǞǓǔʓǗ, ǀǟǝǓǔʓǗ; 78 to be sure, no physical views are preserved from this text. The second-century BC compiler of the Stoic theological doxography attributed to Pythagoras an idea that our soul is a part of the divine world-soul, 79 which is echoed in the Pythagorean Notes and in Sextus Empiricus. 80 It is only in the first century BC, when a role of Plato’s teacher was 74 Lydus. De mens 6WDHKOHʋ); Anon. Phot. 237.17–19; Sext. Emp. Adv. Math. 10.262. Cf. Zeller. Op. cit. I, 464ff.; III.2, 108. 76 Eudorus ap. Simpl. In Phys., 181.10–30. For a similar system, see Ps.-Archytas’ On Principles (19.5– 20.17) and Ps.-Timaeus (206.5–17). Merlan Ph. The Pythagoreans, in: A. H. Armstrong, ed. The Cambridge History of Later Greek and Early Medieval Philosophy. Cambridge, 1967, 84ff.; Mansfeld J. Compatible Alternatives: Middle Platonist Theology and the Xenophanes Reception, in: R. van der Broek, T. Baarda, J. Mansfeld, eds. Knowledge of God in the Graeco-Roman world. Leiden, 1988, 92– 117, at 96–100; Bonazzi M. Eudorus of Alexandria and the ‘Pythagorean’ Pseudepigrapha, in: On Pythagoreanism, 385–404; Centrone B. The Pseudo-Pythagorean writings, in: C. Huffman, ed. The History of Pythagoreanism. Cambridge, 2014, 315–340, at 321ff. 77 Cf. Centrone B. Medioplatonismo e neopitagorismo: un confronto difficile, Rivista di storia della filosofia 2 (2015) 399–423 78 D. L. 8.6. 9. 15 = 170.17–172.7 Thesleff; Zhmud. What is Pythagorean, 79f. 79 Cic. ND 1.27–28. See Zhmud. The papyrological tradition, 135f. 80 Soul is “a detachment (ȢǚʓǝǚNjǝǖNj) of aether, both the hot and the cold (…) it is immortal since that from which it is detached is immortal” (D. L. 8.28, cf. 7.143). Sext. Emp. Adv. math. 9.127.

assigned to Pythagoras, and the Pythagoreans started to be seriously regarded as physical philosophers (this is what one of Sextus Empiricus’ sources repeatedly stresses) 81 that a recently created system, Neopythagoreanism, has been successfully and inextricably linked to his name. The decision of the compiler of the Vetusta placita to use the Anonymus as an authoritative source of Pythagoras’ principles does not look self-evident, for our text did not aim to expose Pythagoras’ philosophy. Yet no other authoritative source seemed to be available at this time, which would have better corresponded to the compiler's idea of what Pythagoras’ philosophy should look like; other arithmological texts were either ascribed to his followers 82 or too concise, as the section on principles in the Pythagorean Notes (D. L. 8.25). On the contrary, Pythagoras’ section in the chapter ƻǏǛʐ ȢǛǡ̆Ǘ is much longer than any other and contains, besides two sets of principles, much more arithmology than one would expect. Indeed, one set of principles that goes back ultimately to Aristotle comprises numbers and proportions, which Pythagoras also calls ‘harmonies’. The elements, called ‘geometricals’, are composed out of both of them. 83 Another set consists of the Monad and the Indefinite Dyad, which tend, respectively, to Ǟʒ ǚǙǓǑǞǓǔʒǗ NjɒǞǓǙǗ ǔNjʐ ǏɎǎǓǔʓǗ, ɣǚǏǛ ȲǝǞʐ ǗǙ˹ǜ ɟ ǒǏʓǜ, and to Ǟʒ ǚNjǒǑǞǓǔʓǗ ǞǏ ǔNjʐ ɫǕǓǔʓǗ, ɣǚǏǛ ȲǝǞʐǗ ɟ ɟǛNjǞʒǜ ǔʓǝǖǙǜ. 84 This kind of Stoicized Platonism is familiar to us from other Neopythagorean sources. But unlike them, no attempt is made here to relate numbers to the Monad and the Indefinite Dyad or to derive numbers from them. This is because in the chapter ƻǏǛʐ ȢǛǡ̆Ǘ the compiler did not feel obliged to go beyond the principles, so he omitted both the generation of the Indefinite Dyad from the Monad and further derivation point–line–plane–solid, characteristic for the texts under discussion. 85 What follows and comprises the bulk of the section is arithmology, every element of which is attested in the writings dependent on the Anonymus: The nature of number is the decad. 86 For all the Greeks, all barbarians count up to ten and when they have reached that they revert to the monad. 87 And the power of ten in turn, he says, resides in the four and in the tetrad <…> For example, if one posits one and adds two and three and four to these, then one will complete the number ten. So that the number is in 81 Adv. Math. 1.303, 9.64, 10.45. 248. 250. 255. See above, 5 n. 22. The Hieros Logos in Doric prose (p. 164–166 Thesleff) bearing Pythagoras’ name appeared much later and depends on the Anonymus. See Delatte. Op. cit., 191ff.; Hadot I. Studies in the Neoplatonist Hierocles. Philadelphia, 2004, 69ff. 83 Dox. 281a2–6, cf. Aët. 1.10.2. See Burkert. Op. cit., 58 n. 28. 84 Dox., 281a6-12. Cf. Cic. Acad. 2.118: Pythagorei ex numeris et mathematicorum initiis proficisci volunt omnia. 85 Cf. Philo. De opif. mundi 49; Lydus. De mens. 4.64. 86 Cf. ȿ ǖʍǗǞǙǓ ǎǏǔʊǜ ǚʋǗǞNj ǚǏǛNjʑǗǏǓ ǞʒǗ ȢǛǓǒǖʓǗ, ȲǖǚǏǛǓʍǡǙǟǝNj ǚˍǝNjǗ ǠʕǝǓǗ ȲǗǞʒǜ NjɫǞ˜ǜ (Theon. 106.89); ǔNjǕǏ˪ǞNjǓ <ǎʌ> ȿ ǎǏǔʊǜ ǔǛʋǞǙǜ ǔNjʐ ǚNjǗǞʍǕǏǓNj, ȲǚǏʐ ǚʋǗǞNj ǚǏǛNjʑǗǏǓ ǞʒǗ ȢǛǓǒǖʒǗ ǚǏǛǓʍǡǙǟǝNj ǚˍǝNjǗ ǠʕǝǓǗ ȲǗǞʒǜ ȳNjǟǞ˜ǜ… (Anat. 15.13–14). 87 6HH6WDHKOHʋ86–87; ǚʋǗǞNj ǖʌǗ ǍʊǛ ǞʒǗ ȢǛǓǒǖʒǗ ǏɎǜ ǎǏǔʋǎNj ɂǍNjǍǙǗ, ȲǚǏǓǎʎ ɫǚʌǛ ǎǏǔʋǎNj ǙɪǎǏʑǜ ȲǝǞǓǗ ȢǛǓǒǖʓǜ, ȲǗ Ǟ˝ NjɪǘʏǝǏǓ ǚʋǕǓǗ ȿǖ̆Ǘ ɫǚǙǝǞǛǏǠʓǗǞǣǗ Ȳǚʐ ǖǙǗʋǎNj ǔNjʐ ǎǟʋǎNj ǔNjʐ ǞǙʔǜ ȳǘ˜ǜ (Theon. 99.17f.).

ten by the monad [i.e. the unit], but in four by its power. 88 And that is why the Pythagoreans proclaimed, thinking that the tetrad is the greatest oath, No, by the man who bequeathed the tetractys to our soul which has the fount and root of everlasting nature. And our soul, he says, is composed out of tetrad. 89 For it is from intellect, knowledge, opinion, perception, that every art and every science comes and that we ourselves are rational.90 The rest (Dox. 282a16–283a9) explains, in which way the first four number are related to our cognitive faculties that make up the soul: ǗǙ˹ǜ is one, ȲǚǓǝǞʏǖǑ (knowledge) is two, ǎʓǘNj is three, and NjɒǝǒǑǝǓǜ is four (this part has fallen out). This Early Academic idea is known to us from Aristotle’s On the Soul; it was also discussed in his On Philosophy (see above, 10 n. 44). As the parallels show, this likening was also employed in the Anonymus, 91 but the following explanations of how our critical faculties correspond to the first four numbers do not come from the same source. They are too academic for a popular genre and not similar to the usual arithmological explanations. 7. The pseudo-Pythagorean oath and tetractys The Pythagorean oath containing the tetractys, this kernel of Neopythagorean wisdom, 92 was probably the most popular and thus influential piece of the Anonymus, which left traces in dozens of writings of the Imperial period: 93 ƺɮ, ǖʊ ǞʒǗ ȣǖǏǞʍǛˋ Ǣǟǡˎ ǚNjǛNjǎʓǗǞNj ǞǏǞǛNjǔǞʕǗ ǚNjǍʊǗ ȢǏǗʋǙǟ ǠʕǝǏǣǜ ˸ʑǐǣǖʋ Ǟ’ ȶǡǙǟǝNjǗ. No, by the him who bequeathed the tetractys to our soul, which has the fount and root of everlasting nature. 94 The Pythagoreans swear by Pythagoras because he forbade them to swear by the gods (D. L. 8.22; Iamb. VP 150). This motif comes from the earlier pseudo-Pythagorean literature: at the beginning of his treatise ǀǟǝǓǔʓǗ (cf. above, n. 78) Pythagoras swears not by the gods but by 88 6HH'HODWWH2SFLWI6WDHKOHʋHJɡ ǍʊǛ ȲǗǞǏǕǏǡǏʑˋ ǎǏǔʋǜ, ǞǙ˹ǞǙ ǞǏǞǛʋǜ, ɻǜ ȶǙǓǔǏ, ǎǟǗʋǖǏǓƤ ǏɎ ǍǙ˹Ǘ Ǚɏ Ȣǚʒ ǖǙǗʋǎǙǜ ȦǡǛǓ ǞǏǞǛʋǎǙǜ ȳǘ˜ǜ ǝǟǗǞǏǒǏ˪ǏǗ ȢǛǓǒǖǙʑ, ǎǏǔʋǎNj ǍǏǗǗʏǝǙǟǝǓǗ (Philo. De opif. mundi 47); Theon. 99.20f.; Hierocl. In aur. Carm. 20.14; Lyd. De mens. 2.9 ad fin. 89 Ǚɪ ǖʓǗǙǗ ǎʌ ǞʒǗ ǞǙ˹ ǝʗǖNjǞǙǜ ȲǚʍǡǏǓ ǕʓǍǙǗ ȲǗ ȢǛǓǒǖǙ˪ǜ ǞǏǞǛʋǜ, ȢǕǕʊ ǔNjʐ ǞʒǗ Ǟ˜ǜ Ǣǟǡ˜ǜ (Anat. 8.15, cf. Sext. Emp. Adv. math. 4.3. 5. 8); Ǣǟǡʊ ȢǗǒǛʗǚǙǟ, ɻǜ ƻǟǒNjǍʓǛNjǜ ȶǠǑ, ȲǝǞʐ ǞǏǞǛʋǍǣǗǙǗ ǏɪǒǟǍʗǗǓǙǗ (Lyd. De mens. 2.9). 90 Dox. 281a12–282a16, tr. Laks-Most, slightly modified. 91 ǗǙ˹ǜ ȲǚǓǝǞʏǖǑ ǎʓǘNj NjɒǝǒǑǝǓǜ. ǗǙ˹ǜ ǖʌǗ ɻǜ ǖǙǗʊǜ ȲǗ Ǚɪǝʑˋ, ǔǞǕ. (Theon. 98.4f.); Lyd. De mens. 2.9; Hierocl. In aur. carm. 20.18. Cf. Ps.-Archyt. De intell. 38.19–24. 92 On the Neopythagorean origin of the tetractys, see Zhmud. Pythagoras, 301ff. Cf. Burkert. Op. cit., 72. 93 The most important evidence is collected in Nauck A. Iamblichi De vita Pythagorica liber. Petropoli, Lipsiae, 1884, 216f., 229f.; Delatte. Op. cit., 249ff. They discuss also variations of the text. 94 Ps.-Plut. 877A = Aët. 1.3.8a. This was the original form of the oath (Nauck. Op. cit., 216, 229; Delatte. Op. cit., 249f.).

air and water and incidentally in the same negative form as in this oath. 95 Pythagoras’ name also does not appear in the oath, because the Pythagoreans were not allowed to call him by name and referred to him ‘that man’ (ȲǔǏ˪ǗǙǜ ȢǗLjǛ), a motif known from Apollonius of Tyana (first century AD), the first Neopythagorean biographer of Pythagoras (Iamb. VP 88, 150, 255). The proverbial ƬɪǞʒǜ ȶǠNj (D. L. 8.46), first attested in Cicero (ND 1.10), belongs to the same Neopythagorean milieu as the oath and reflects a widespread since the first century BC belief that the ancient Pythagoreans spoke and wrote Doric.96 Though it is possible that the arithmologist borrowed the oath from a Doricized apocryph, it would be difficult to say what kind of source it was and why has not it left any other traces. Doric arithmological treatises we know of, such as Megillos’ ƻǏǛʐ ȢǛǓǒǖ̆Ǘ (115.15f.) or Pythagoras’ ƶǦǍǙǜ ǚǏǛʐ ǒǏ̆Ǘ (164.1ff.), are much later than the Anonymus and dependent on it. More plausible, therefore, is that the oath appeared first in the Anonymus as a ‘quotation’, which dialect had to underline its authenticity. Delatte hypothesized Timaeus of Tauromenium as the earliest source of the oath,97 but this is related to his tendency to consider many apocrypha as authentic Pythagorean texts. In justice to his acumen it should be said that the next source he indicated was our arithmological treatise. Indeed, the content of the oath is clearly arithmological and its vocabulary belongs to the first century BC. 98 Long after its first appearance the oath occurred predominantly if not exclusively in the arithmological writings or passages directly or indirectly related to the Anonymus, so that the latter can be justifiably regarded as its ultimate source. Robbins suggested that the oath with its doctrinal environment formed an introduction to the Anonymus. 99 It is much more likely, however, that it belonged to the chapter on the number four, for this was its usual place in the arithmological writings, for example, in Philo, Nicomachus, Anatolius, Lydus, etc. The idea that the first four numbers make up ten was inspired by Speusippus (see above, 9), yet the doctrine of the tetractys has been formed in the early first century BC. After the Vetusta placita, it occurs in a condensed and slightly confused form in the Anonymus Photii: ǔNjʐ Ǟʊ ɢǗǞNj ǚʋǗǞNj ȢǛǓǒǖǙʔǜ ǚǛǙǝǑǍʓǛǏǟǙǗ (sc. the Pythagoreans), ɟ ǎʌ ȢǛǓǒǖʒǜ ǝǟǖǚǕǑǛǙ˹ǞNjǓ ǞǙ˪ǜ ǎʍǔNj, ɟ ǎʌ ǎʍǔNj ǝʕǗǒǏǝǓǜ Ǟ̆Ǘ ǞǏǝǝʋǛǣǗ ǔNjǞʊ Ǟʒ ȳǘ˜ǜ ȢǛǓǒǖǙʕǗǞǣǗ ȿǖ̆Ǘ, ǔNjʐ ǎǓʊ ǞǙ˹ǞǙ 95 Ǚɪ ǖʊ ǞʒǗ ȢʍǛNj ǞʒǗ ȢǗNjǚǗʍǣ, Ǚɪ ǖʊ Ǟʒ ɯǎǣǛ Ǟʒ ǚʑǗǣ, Ǚɮ ǔǙǞ' Ǚɒǝǣ ǢʓǍǙǗ ǚǏǛʐ ǞǙ˹ ǕʓǍǙǟ ǞǙ˹ǎǏ (D. L. 8.6). Cf. Diod. Sic. 10.9.1, also from tripartitum, see Schorn S. Die Pythagoreer im zehnten Buch der Bibliothek Diodors, in: M. Berti, V. Costa, eds. Ritorno ad Alessandria. Storiografia greca e cultura bibliotecaria. Tivoli, 179–259, at 228f. 96 See Zhmud. What is Pythagorean, 83f. 97 Delatte. Op. cit., 253. 98 ǞǏǞǛNjǔǞʕǜ occurs first in the oath, further at Anon. Phot. 238.1 and Sext. Emp. Adv. Math. 4.2-3, 7.94. 98. 100; ǠʕǝǓǜ ȢǏǗʋǙǜ is first attested in Posidonius, in a context different from that of the oath (fr. 239 E-K.); ǚǑǍʎǔNjʐ˸ʑǐNj figure once in plural in the Hippocratic corpus (De flat. 7 ad fin.: ɣǚǙǟ ĮੂʌȘȖĮ੿țĮ੿ Įੂ૧઀ȗĮȚIJȠ૨Į੆ȝĮIJંȢİੁıȚ), but all the subsequent occurrences begin with Philo (De congr. erud. 120; Heres 116). See e.g. Ps.-Plut. De lib. educ. 4A; Theon. 18.2; Julian the Methodist (ca 150 AD) ap. Galen. Adv. Julian. 18, 273.1 K. Robbins. Posidonius, 314.

ǞʒǗ ȢǛǓǒǖʒǗ ǚʋǗǞNj ǞǏǞǛNjǔǞʔǗ ȶǕǏǍǙǗ (237.25–238.1). The original sense of the last words was not that any number was called ‘tetractys’, but that the tetrad was called the number of ‘all’, as the parallel in Philo shows: ǔNjǕǏ˪ǞNjǓ ǎ’ ȿ ǞǏǞǛʊǜ ǔNjʐ ‘ǚˍǜ’, ɣǞǓ ǞǙʔǜ ȦǡǛǓ ǎǏǔʋǎǙǜ ǔNjʐ NjɪǞʎǗ ǎǏǔʋǎNj ǚǏǛǓʍǡǏǓ ǎǟǗʋǖǏǓ (De plant. 123). 100 Further Philo specifies that the decad is actually ‘all’, whereas the tetrad is potentially (ibid., 125). At De opif. mundi, 47–52 he sets forth in detail the doctrine of the ǞʍǕǏǓǙǜ ǞǏǞǛʋǜ, beginning as follows: But the heaven in its turn was ordered with a perfect number, the four. You would not go astray in affirming that it is the principle and source (ȢǠǙǛǖʎǗ ǔNjʐ ǚǑǍʏǗ) of the all-perfect number ten; for what the ten is in actuality, the four, it would seem, is potentially. If the numbers from the unit to the four are added up, they will produce the ten. It forms the boundary for the infinitude of numbers, which wind around it like a turning post and turn back. 101 Philo does not use the word ǞǏǞǛNjǔǞʕǜ, preferring to it ǞǏǞǛʋǜ, which creates confusion, because ǞǏǞǛʋǜ denotes the number four, and ǞǏǞǛNjǔǞʕǜ a set of four numbers or items. 102 Other authors also do not distinguish between ǞǏǞǛʋǜ and ǞǏǞǛNjǔǞʕǜ or use them interchangeably. 103 Since this feature is already observed in the Vetusta placita passage, 104 one can suppose that the arithmologist himself used these words interchangeably. Interpretation of the first line of the oath does not cause much difficulty: it is Pythagoras who is meant here, and the variant Ǣǟǡˎ is clearly preferable to ǔǏǠNjǕˎ (first in Sext. Emp. Adv. math. 7.94, though Ǣǟǡˎ in 4.2) and ȖİȞİˎ (first in Nicom. ap. Theol. arith. 22.21), for it is our soul that is related to the tetrad in the arithmological texts (see above, n. 89). ǠʕǝǓǜ ȢǏǗʋǙǜ in the second line has to be understood as the decad, for the latter was considered to be the ‘nature of number’ or to comprise the whole nature of numbers, as was thought already in the Academy. 105 “For under the ‘everlasting nature’”, reports Nicomachus, “they meant the decad, since it is, as it were, the eternal and ageless nature of all things and kinds of thing”. 106 According to Theon, the ratios of all the concords are in “tetractys of the decad” and the decad 100 See Delatte. Op. cit., 254. De opif. mundi, 47, tr. D. Runia. See also ibid. 97–98; De plant. 123-125; De vita Mosi II, 115; In gen,,,6WDHKOHʋ 102 ǞǏǞǛNjǔǞʔǗ ǎʌ ǕʍǍǙǗǞǏǜ ȢǛǓǒǖʓǗ ǞǓǗNj, ɡǜ Ȳǔ ǞǏǝǝʋǛǣǗ Ǟ̆Ǘ ǚǛʗǞǣǗ ȢǛǓǒǖ̆Ǘ ǝǟǍǔǏʑǖǏǗǙǜ ǞʒǗ ǞǏǕǏǓʓǞNjǞǙǗ ȢǚʏǛǞǓǐǏǗ, ɿǝǚǏǛ ǞʒǗ ǎʍǔNj (Sext. Emp. Adv. math. 7.94). ȿ ǖʌǗ ǙɰǗ ǚǛǙǏǓǛǑǖʍǗǑ ǞǏǞǛNjǔǞʔǜ <NjɯǞǑ>, ǔNjǞ’ ȲǚǓǝʕǗǒǏǝǓǗ Ǟ̆Ǘ ǚǛʗǞǣǗ ȢǚǙǞǏǕǙǟǖʍǗǑ ȢǛǓǒǖ̆Ǘ (Theon. 94.10–11). 103 As Delatte. Op. cit., 256 noted: “Chose curieuse, le mot ǞǏǞǛʋǜ, qui devrait être réservé au nombre 4…, est fréquemment employé pour représenter l’ensemble des 4 premiers nombres”. See also Hadot. Op. cit., 64f.; Schibli H. Hierocles of Alexandria. Oxford, 2002, 277 n. 18. 104 DoxI੪ȢȝİȖ઀ıIJȠȣ੖ȡțȠȣ੕ȞIJȠȢIJોȢIJİIJȡ੺įȠȢEXWǞǏǞǛNjǔǞʕǜ in the oath itself. 105 ǏɔǗNjǓ ǎʌ ǞʎǗ ǠʕǝǓǗ ǞǙ˹ ȢǛǓǒǖǙ˹ ǎǏǔʋǎNj (Dox. 281a12-13), see also above, n. 86. ȲǚǏǓǎʎǞʍǕǏǓǙǗȿ ǎǏǔʊǜǏɔǗNjǓǎǙǔǏ˪ǔNjʐǚˍǝNjǗǚǏǛǓǏǓǕǑǠʍǗNjǓǞʎǗǞ̆ǗȢǛǓǒǖ̆ǗǠʕǝǓǗ (Arist. Met. 986a8–9). Speusippus called the decad ǠǟǝǓǔǣǞʋǞǑǗ (fr. 28.10 Tarán). 106 ȢʍǗNjǙǗ ǍʊǛ ǠʕǝǓǗ ǞʎǗ ǎǏǔʋǎNj ʨǗʑǞǞǙǗǞǙ ǞʎǗ ǙɏǙǗǏʐ Ȣ˩ǎǓǙǗ ǔNjʐ NjɎʗǗǓǙǗ Ǟ̆Ǘ ɣǕǣǗ ǠʕǝǓǗ ǔNjʐ ǏɎǎ̆Ǘ ɫǚʋǛǡǙǟǝNjǗ (Theol. arith. 23.1–2). Cf. ǔNjǕǏ˪ǞNjǓ <ǎʌ> ȿ ǎǏǔʊǜ ǔǛʋǞǙǜ ǔNjʐ ǚNjǗǞʍǕǏǓNj, ȲǚǏʐ ǚʋǗǞNj ǚǏǛNjʑǗǏǓ ǞʒǗ ȢǛǓǒǖʒǗ ǚǏǛǓʍǡǙǟǝNj ǚˍǝNjǗ ǠʕǝǓǗ ȲǗǞʒǜ ȳNjǟǞ˜ǜ (Anat. 15.13–14).

constitutes the tetractys (ǞʎǗ ǖʌǗ ǍʊǛ ǞǏǞǛNjǔǞʔǗ ǝǟǗʍǝǞǑǝǏǗ ȿ ǎǏǔʋǜ, 93.17–19) that was venerated by the Pythagoreans, for it seems to embrace the nature of all things (țĮ੿įȠțİ૙IJ੽ȞIJ૵Ȟ ੖ȜȦȞij઄ıȚȞıȣȞ੼ȤİȚȞ94.4). Thus, the decad as ǠʕǝǓǜ ȢǏǗʋǙǜ belongs to the realm of numbers and is the eternal numerical pattern of the universe. A variation in the second line of the oath, ˸ǓǐʗǖNjǞ’ instead of ˸ʑǐǣǖʋ Ǟ’, changed its meaning to “fount containing the roots of everlasting nature”, which prompted more ‘physical’ interpretation of ǠʕǝǓǜ ȢǏǗʋǙǜ: ˸ǓǐʗǖNjǞNj were understood not only as the four numbers, but also as the four physical elements. The link between the tetractys and the four elements (ǝǞǙǓǡǏ˪Nj, not ˸ǓǐʗǖNjǞNj!) was presented already in the Anonymus, as follows, for example, from Philo: “The four elements, out of which this universe was constructed, flowed forth, as from a source, from the four in the realm of numbers”. 107 The oath, however, was not yet directly involved here, as it becomes in Hippolytus’ Refutatio, who viewed Pythagoras, i.e. Neopythagoreanism, as a source of many heretical doctrines he fought against and thus preserved a lot of arithmological material. 108 Hippolytus’ source in the 6th book identifies the oath with the “harmony of the four elements”, for the tetractys is the principle of physical and solid bodies, just as the monad of intelligible ones. 109 Mansfeld suggested that Hippolytus “interprets the ˸ǓǐʗǖNjǞNj of the Pythagorean oath as pertaining to the four elements of (Empedocles’) physics”. 110 Indeed, in the next book Hippolytus quotes Empedocles’ verse ǞʍǝǝNjǛNj Ǟ̆Ǘ ǚʋǗǞǣǗ ˸ǓǐʗǖNjǞNj ǚǛ̆ǞǙǗ ȦǔǙǟǏ (Ref. 7.29.4 = 31 B 6), but three lines earlier he says that, according to Empedocles, there were six elements, not four, so that this parallel does not seem convincing. 111 To be sure, in the excerpt from Nicomachus in Theology of Arithmetic the oath itself is seemingly attributed to Empedocles: ǞǙǓNjʕǞǑǜ ǎʌ ǙɮǝǑǜ ȲǚʗǖǗǟǙǗ ǎǓ’ NjɪǞ˜ǜ ǞʒǗ ƻǟǒNjǍʓǛNjǗ Ǚɏ ȦǗǎǛǏǜ, ǒNjǟǖʋǐǙǗǞǏǜ ǎǑǕǙǗʓǞǓ ǔNjʐ ȢǗǏǟǠǑǖǙ˹ǗǞǏǜ Ȳǚʐ Ǟ˝ ǏɫǛʍǝǏǓ, ǔNjǒʋ ǚǙǟ ǔNjʐ ȸǖǚǏǎǙǔǕ˜ǜƤ ‘Ǚɮ, ǖʊ ǞʒǗ ȣǖǏǞʍǛˋ ǍǏǗǏˎ ǚNjǛNjǎʓǗǞNj ǞǏǞǛNjǔǞʕǗ,/ ǚNjǍʊǗ ȢǏǗʋǙǟ ǠʕǝǏǣǜ ˸ǓǐʗǖNjǞ’ ȶǡǙǟǝNjǗ.’ ȢʍǗNjǙǗ ǍʊǛ ǠʕǝǓǗ ǞʎǗ ǎǏǔʋǎNj ʨǗʑǞǞǙǗǞǙ ǞʎǗ ǙɏǙǗǏʐ Ȣ˩ǎǓǙǗ ǔNjʐ NjɎʗǗǓǙǗ Ǟ̆Ǘ ɣǕǣǗ ǠʕǝǓǗ ǔNjʐ ǏɎǎ̆Ǘ ɫǚʋǛǡǙǟǝNjǗ (Theol. arith. 22.18–23.2). As Delatte correctly explained, however, ǔNjǒʋ ǚǙǟ ǔNjʐ ȸǖǚǏǎǙǔǕ˜ǜ refers not to the following oath – for it is written in Doric and attributed to the Pythagoreans (ȲǚʗǖǗǟǙǗ, ʨǗʑǞǞǙǗǞǙ) 107 De opif. mundi 52, tr. D. Runia, cf. Lyd. De mens. 4.64. For further parallels, see Staehle ʋ27. Mansfeld J. Heresiography in Context: Hippolytus’ Elenchos as a Source for Greek Philosophy. Leiden, 1992. 109 Ref. 6.23.4. Elsewhere Hippolytus, quoting the oath, does not adduce this interpretation (1.2.9–10, 4.51.7–8, 6.34.1), but in the account of ‘Egyptian’ number philosophy he derives the four elements from the number four (4.43.8). Cf. IJİIJȡĮțIJ઄Ȣǜ ȆȣșĮȖȠȡȚțઁȢ ੖ȡțȠȢ, ਵȖȠȣȞ IJ૵Ȟ IJİıı੺ȡȦȞ ıIJȠȚȤİ઀ȦȞ ıȘȝĮ઀ȞȦȞ (Hesych.). 110 Mansfeld. Heresiography, 180. 111 There is even less ground for assuming that ˸ǓǐʗǖNjǞNj belongs to the original version of the oath, which alludes thereby to Empedoclean physics, as in Primavesi O. Empedocles’ Cosmic Cycle and the Pythagorean Tetractys, Rhizomata 4 (2016) 5–29, at 13f.

– but to the preceding ǒNjǟǖʋǐǙǗǞǏǜ ǔNjʐ ȢǗǏǟǠǑǖǙ˹ǗǞǏǜ: similarly to the Pythagoreans, Empedocles admired Pythagoras in his famous verses ɄǗ ǎʍ ǞǓǜ ȲǗ ǔǏʑǗǙǓǝǓǗ ȢǗʎǛ ǚǏǛǓʗǝǓNj ǏɎǎʗǜ… (31 B 129), to which Nicomachus alludes. 112 ˸ǓǐʗǖNjǞNj twice occurring in Nicomachus’ text (Theol. arith. 21.2–3, 23.4–6) refers to the first four numbers, not to the elements. 8. The most generative six The Anonymus attached to the number six an important role in the period of human gestation. The six is “the most generative number” (ǍǏǗǗǑǞǓǔʗǞNjǞǙǜ), 113 because, explains Philo in the remaining fragments of the Questions on Genesis, it is the first number that is both male and female, being a product of even (2) and odd (3). This is why among the ancients some called it ‘marriage’ and others ‘harmony’. 114 The Armenian translation of the Questions preserved a fuller picture of Philo’s arithmological speculations. At 3.38 he presents two other important numbers, 35 and 45, the first consisting of the proportion 6, 8, 9, 12 (in sum they make 35), the second of the proportions 6, 9, 12, 18 (in sum they make 45). Relation of 35 and 45 to gestation is partly clarified later in the same work: 45 is a productive number, for it contains all three basic proportions, the arithmetic, the geometric, and the harmonic, and in the same number of days the embryo of the nine-month babies is formed in the womb, whereas in case of the seven-month babies it takes, as they say, 35 days. 115 Indeed, if we multiply 35 by 6 we make 210, the number of days in seven months, and multiplying 45 by 6 we make 270, the number of days in nine months. This should explain, why the six is ǍǏǗǗǑǞǓǔʗǞNjǞǙǜ. A passage from Varro in Censorinus helps to complete the picture of this generative arithmetic. 116 Varro describes two types of pregnancy “according to Pythagoras”: the seven-month, or 210 days, and the ten-month, or 274 days. The first is based on the number six, the second on the number seven. During the seven-month pregnancy the foetus proceeds through four stages (milky humor, blood, flesh, formed body), which correspond to 6, 8, 9 and 12 days and to the three basic concords, the octave (12:6), the fifth (9:6), and the fourth (8:6). (These four numbers form the Pythagorean ‘musical proportion’, 6:8=9:12, which combines the arithmetic and harmonic means). 117 When added, 6, 8, 9 and 12 produce 35 days, which multiplied by 6 112 Delatte. Op. cit., 252. Phil. De opif. mundi, 13; Lyd. De mens. 2.11. 114 In Gen. 3.38a; Lyd. De mens. 2.11. Cf. Ȳǘ ȢǛǞʑǙǟ ǔNjʐ ǚǏǛǓǞǞǙ˹ Ǟ̆Ǘ ǚǛʗǞǣǗ, ȦǛǛǏǗǙǜ ǔNjʐ ǒʏǕǏǙǜ…, ǎǓʒ ǔNjʐ ȢǛǛǏǗʓǒǑǕǟǜ ǔNjʐ ǍʋǖǙǜ ǔNjʐ ȢǛǞǓǙǚʍǛǓǝǝǙǜ ǔNjǕǏ˪ǞNjǓ (Anat. 10.13–16). For further parallels, see 6WDHKOHʋD 115 In Gen. 4.27, p. 301–302, tr. R. Marcus. 116 Cens. DN 9 and 11, cf. Aul. Gell. 3.10.7-8. See Parker H. N. Greek Embryological Calendars and a Fragment from the Lost Work of Damastes, CQ 49 (1999) 515–534. 117 The numbers 6, 8, 9, 12 first occur in ps.-Platonic Epinomis (990d–991b), see Barker A. Pythagoreans and Medical Writers on Periods of Human Gestation, in Pythagorean Knowledge, 263–276, at 271, but the corresponding ratios were known already to Hippasus (Aristox. fr. 90 = 18 A 12).

makes 210. “And so not undeservedly six is the basis of conception” (11.4, tr. H. Parker). In the second pregnancy the body is fully formed in circa 40 days, which multiplied by 7 makes 280 days, i.e. nine months and ten days, but since the baby is born on the first day of the last week, the exact number of days is 274. The second scheme, unlike the first, is based on the number seven and does not match with that described by Philo. It is unclear, why Varro changed the usual pattern of two pregnancies in 35×6=210 and 45×6=270 days, which is preserved in the Theology of Arithmetic (51.4.–25, most probably, from Nicomachus), 118 Aristides Quintilianus, Proclus and other sources. 119 The Pythagorean Notes, which are not dependent on the Anonymus, briefly refer to a similar theory, where the fetus is formed in 40 days, an arithmetic mean between 35 and 45: “Solidifying first in forty days, the foetus has form, then according to the ratios of harmony, it is completed in seven, nine, or ten months at most, and is born”. 120 The whole topic obviously is too special and too developed to be invented by the arithmologist. Basic ideas of generative arithmetic were attested already in the fifth-century BC philosophy, harmonics, and medicine, partly going back to still earlier number symbolism of the number seven. 121 Indeed, the early embryological calendars, i.e. calculations of the development of the foetus, were based on the number seven, not six, as for example that of Empedocles. The embryo begins the articulation of the limbs from the 36th day (after the fifth hebdomad) and completes it by the 49th day (at the end of the seventh hebdomad); a woman can give birth to a viable child on the 7th or the 10th month (i.e. at the end of a full nine month), 122 which complies with the traditional medical lore. Similar calculations are to be found in the Pythagorean Hippo: “The foetus, he said, was already mature in the seventh month, since the number seven has the greatest power over everything”.123 The Hippocratic treatise On Fleshes (late fifth – early fourth century BCE) offers a more developed scheme: children born at seven months and at nine months and ten days are both viable and have “a precise numerical relationship to sevenday periods”, the first counts exactly thirty seven-day periods (3×10×7= 210) and the second 118 Robbins F. E. Works of Nicomachus, in: Nicomachus of Gerasa. Introduction to Arithmetic. Tr. M. L. D’Ooge. London, 1926, 85, 87. Cf. Bucking S. On Measuring the Range of Anatolius’ Text in the [Iamblichean] Theologoumena Arithmeticae, Grazer Beiträge 18 (1992) 127–148, at 132ff., who ascribes this passage to Anatoly. 119 Theol. arith. 51.4.–25, cf. 63.7–18; Aristid. Quint. 3.18; Procl. In Plat. Rem Publ. II, 34.2–36.2 (Proclus appends his calculations to Empedocles 31 B 69). Plut. De an. in Tim. 1018A gives only the first formula; Macrob. In somn. Scip. I.6.14-17 gives the first formula and alludes to the second. See Delatte A. La Vie de Pythagore de Diogène Laërce. Brussels, 1922, 216f.; idem. Les harmonies dans l’embryologie hippocratique, in: Mélanges Paul Thomas. Bruges, 1930, 160–171, at 166f.; Waterfield R. A. H. Emendations of [Iamblichus], Theologoumena Arithmeticae (De Falco), CQ 38 (1988) 215–227, at 222f., Parker. Op. cit. 120 D. L. 8.29, tr. H. Parker; Long. Op. cit., 152f. 121 Roscher. Die Hebdomadenlehre. 122 31 A 75, 83, B 153a; Hanson A. E. The Eight Months’ Child and the Etiquette of Birth: “Obsit Omen”!”, Bulletin of the History of Medicine 61 (1987) 589–602; Parker. Op. cit., 522f. Cens. DN 7.2 = 38 A 16, tr. H. Parker.

forty seven-day periods (4×10×7=280). A child born at eight months never survives (19, tr. P. Porter). The Hippocratic Regimen relates the life and growth of the foetus to finding the correct attunement, which has concordant intervals the fourth, the fifth, and the octave. 124 Thus, both the combination of harmonics and embryology and the formulas for the seven-month and ninemonths babies, though based on the hebdomad, are presented in the Hippocratic corpus. Still, they does not fully match with two exact formulas of 35 and 45 days multiplied by generative 6, which are given by the Anonymus. 125 Luckily for us, not so long ago Holt Parker published a Hellenistic embryological calendar of a certain Damastes, a medical writer on pediatrics, who can be dated in the second century BC. 126 It is in Damastes that we find for the first time the exact match to both formulas, and their best explanation, which makes him the most probable source of the Anonymus. Here is a passage on the seven-month babies: The seven-month child becomes foam in 6 days, becomes blood in <another> 8, becomes flesh in another 9, takes shape in another 12. Women who are brought to this point complete the number 35. It moves in twice the number, 70, and when this number of days is done, it is born in three times the number, 210 (tr. H. Parker). Thus, the numbers of the ‘musical proportion’, 6, 8, 9 and 12, correspond to four basic periods of foetus formation, then the sum of these numbers is multiplied by two and by three to make 210. The second scheme is 6, 9, 12 and 18, which sum, 45, is multiplied by two and by three to make 270. The only thing that the author of the Anonymus had to do was to replace two and three by six and present this subject in chapter on the most generative number. But this was a decisive step that transformed embryological calculations, partly empirically based but mostly fanciful, into a discourse on the power of the perfect number six. 9. The number five and Aristotle’s Against the Pythagoreans fr. 13 Ross Six was not the only number of marriage, the arithmological tradition also attributed this function to five. If two is the first female number and three the first male number, then five is 124 De victu 1.8. Delatte. Les harmonies, 171 wrongly projects the late theories onto ancient Pythagoreanism; Burkert. Op. cit., 262f.; Huffman. Philolaus, 152. 125 According to an extract from Nicomachus in the Theology of Arithmetic, Diocles of Carystus, the famous doctor of the late fourth century BC, said that the period of 210 days, i.e. seven months of thirty days, equals to 35×6 (Theol. ar. 64.4–15 = Diocles fr. 46 van der Eijk). There is good ground to believe that Diocles, who counted stages of fetus formation in seven-day weeks (fr. 45a-b, from Nicomachus and Macrobius), mentioned both the fifth week (35 days) and the period of 210 days, as the author On Fleshes, quoted next, did (Theol. ar. 64.13–15), but hardly attached any importance to the number six. See Van der Eijk Ph. Diocles of Carystus. 2 vols. Leiden, 2001, fr. 45–46 with comm., and Mansfeld. The Pseudo–Hippocratic Tract, 163ff. Parker. Op. cit.

the nuptial number by addition (ǔNjǞʊ ǝʕǗǒǏǝǓǗ), whereas six is by multiplication (ǔNjǞʊ ǚǙǕǟǚǕNjǝǓNjǝǖʒǗ). Usually ancient writers stuck to one of two versions, 127 but in the Anonymus they the seemed to coexist, as they did, for example, in Plutarch, 128 Nicomachus, 129 Anatolius, 130 the Theology of Arithmetic, 131 Martianus Capella, 132 etc. Plutarch, the first to present the version with the number five, clearly preferred it to the version with the number six that occurs in his writing only once. To be sure, a Pythagorean number of marriage was mentioned already in Aristotle’s Metaphysics, 133 though he nowhere specified what number this was. In a passage from Alexander’s commentary on the Metaphysics, however, which has been identified by Paul Wilpert as the Aristotelian fragment and included by W. D. Ross into his Aristotelis fragmenta selecta, 134 this number appears as five: ǍʋǖǙǗ ǎʌ ȶǕǏǍǙǗ ǞʒǗ ǚʍǗǞǏ, ɣǞǓ ɟ ǖʌǗ ǍʋǖǙǜ ǝʕǗǙǎǙǜ ȦǛǛǏǗʓǜ ȲǝǞǓ ǔNjʐ ǒʏǕǏǙǜ, ȶǝǞǓ ǎʌ ǔNjǞ’ NjɪǞǙʔǜ ȦǛǛǏǗ ǖʌǗ Ǟʒ ǚǏǛǓǞǞʒǗ ǒ˜Ǖǟ ǎʌ Ǟʒ ȦǛǞǓǙǗ, ǚǛ̆ǞǙǜ ǎʌ ǙɱǞǙǜ Ȳǘ ȢǛǞʑǙǟ ǞǙ˹ ǎʕǙ ǚǛʗǞǙǟ ǔNjʐ ǚǛʗǞǙǟ ǞǙ˹ ǞǛʑNj ǚǏǛǓǞǞǙ˹ ǞʎǗ ǍʍǗǏǝǓǗ ȶǡǏǓ (In Met., 39.8–12). Accepting that this passage really derives from Aristotle’s work Against the Pythagoreans, we do not just give preference to the five as the ancient Pythagorean number of marriage, 135 though the identification of even numbers with the female principle, and of odd with male, is first attested in Xenocrates. 136 What is more important, we also need to radically change our entire historical perspective of arithmology. Indeed, what we have in the lines quoted above and on the whole in Alexander’s treatment of the first ten numbers (38.10–39.17), is arithmology, and if this arithmology is the ancient Pythagorean one, then its theoretical foundations were not laid down in the Early Academy and it did not originate as a literary genre in the first century BC. In my previous paper on Greek arithmology I have expressed serious doubts in regard of the authenticity of this fragment of Aristotle, 137 here I can adduce further arguments for my view. ƻǛʒǜ ǞǙʔǜ ƻǟǒNjǍǙǛǏʑǙǟǜ NjƘ figures in the Hellenistic catalogue of Aristotle (D. L. 5.25), and had it contained fully fledged arithmology, it would have been the most important 127 Five: Alex. Aphrod. In Met., 39.8-13; Asclep. In Met. 36.16-20. Six: Philo. In Gen. 3.38a; Clem. Strom. 5.14.93.5, 6.16.139.4; Arist. Quint. 3.6; Theon. 102.4-6; Lydus. De mens. 2.11; Syrian. In Met., 104.25–27; Philop. In phys. comm. 389.1f. 128 Five: Aet. Rom. 264A, 288c-d; De def. orac. 429A; De Is. et Osir. 374A; six: De an. procr. 1018A. 129 Five: ਝijȡȠį઀IJȘ țĮ੿ īĮȝȘȜ઀Į țĮ੿ ਝȞįȡȠȖȣȞ઀Į (Phot. Bibl. 144a36); six: țĮ੿ țȣȡ઀ȦȢ Į੢IJȘ ȝ઼ȜȜȠȞ ਝijȡȠį઀IJȘ ȗȣȖ઀Į IJİ țĮ੿ ȖĮȝȘȜ઀Į țĮ੿ ਝȞįȡȠȖȣȞ઀Į șİȠȜȠȖİ૙IJĮȚ (ibid. 144b6). 130 Five: 9.22-23; six: 10.13-18. 131 Three: 19.20; five: 30.19; six: 43.5. 132 Five: 7.735, six: 7.736. 133 Ǚɏ ǎʌ ƻǟǒNjǍʓǛǏǓǙǓ ǚǛʓǞǏǛǙǗ ǚǏǛʑ ǞǓǗǣǗ ɞǕʑǍǣǗ, ʁǗ ǞǙʔǜ ǕʓǍǙǟǜ ǏɎǜ ǞǙʔǜ ȢǛǓǒǖǙʔǜ ȢǗ˜ǚǞǙǗ, ǙɕǙǗ Ǟʑ ȲǝǞǓ ǔNjǓǛʒǜ ɀ Ǟʒ ǎʑǔNjǓǙǗ ɀ ǍʋǖǙǜ (Met. 1078b22-23). 134 Wilpert P. Reste Verlorener Aristotelesschriften bei Alexander von Aphrodisias, Hermes 75 (1940) 369–396, at 369–376; fr. 203 Rose3 = fr. 13 Ross = fr. 162 Gigon. It should be noted that while fr. 203 Rose3 takes 8 lines (40.26–41.2), fr. 13 Ross takes 3,5 pages (38.8–41.15). 135 Thus e.g. Burkert. Op. cit., 467 and n. 8. 136 See above, 10. Zhmud. Greek Arithmology, 343f.

source for the author of the Anonymus and other writers. As a matter of fact, we see that practically every arithmological item in Alexander’s passage, including that on the number five, has strong parallels with the writings dependent on the Anonymus, 138 whereas nothing in the Anonymus’ remains suggests that its author, or for that matter anybody else in his times, was familiar with Aristotle’s description of ancient Pythagorean arithmology. Wilpert treated very cursorily, if at all, the content of the suggested Aristotelian fragment, especially its arithmological part, and what he said on this account does not always fits the facts. 139 Thus, in Aristotle’s entire criticism of the Pythagoreans he never mentions the generation of numbers or magnitudes, for the obvious reason that this is typically Platonic idea. 140 Aristotle regularly uses language of generative arithmetic and geometry in respect to the Platonists, especially in the Metaphysics Ʒ and Ƹ. 141 In the alleged Aristotelian fragment though the term ȖİȞȞ઼Ȟ is as manifest as in later arithmology: five ǞʎǗ ǍʍǗǏǝǓǗ ȶǡǏǓ from two and three, two generates four, six and nine are generated by three, eight by four, ten by five. According to Wilpert, this term is originally Pythagorean, but the only example from Aristotle he adduces, concerns not the Pythagoreans, but number speculations of Plato. 142 The parallel passage from Theon, adduced by Wilpert, is no less revealing: whereas in Theon Aristotle reports that the Pythagoreans considered the one both even and odd (ȢǛǞǓǙǚʍǛǓǞǞǙǗ), in Alexander the same idea is expressed through language of generation. 143 Further, Wilpert did not mention that Asclepius’ commentary on the Metaphysics contains a passage almost identical to that of Alexander, only with several better readings used by M. Hayduck to improve Alexander’s text. 144 Nor was Wilpert aware that Delatte had published from two Byzantine manuscripts a short anonymous arithmological extract, for the most part coinciding with the relevant commentaries of 138 See e.g.: ੪Ȣ Ƞ੣Ȟ ਙȡȡİȞંȢ IJİ IJȠ૨ ʌȡઆIJȠȣ țĮ੿ ș੾ȜİȠȢ ੒ȝȚȜ઀઺ IJ੹ ʌ੼ȞIJİ ȖȚȖȞંȝİȞĮ Ȗ੺ȝȠȞ Ƞੂ ȆȣșĮȖંȡİȚȠȚ ʌȡȠıİ૙ʌȠȞ (Plut. De E ap. Delph. 387f–388c); IJȠ૨ į੻ ʌİȡȚIJIJȠ૨ ȝ੺ȜȚıIJĮ ȖĮȝ੾ȜȚȠȢ ਲ ʌİȞIJ੺Ȣ ਥıIJȚǜ IJ੹ Ȗ੹ȡ IJȡ઀Į ʌȡ૵IJȠȢ ʌİȡȚIJIJઁȢ țĮ੿ IJ੹ į઄Ƞ ʌȡ૵IJȠȢ ਙȡIJȚȠȢǜ ਥț į੻ IJȠ઄IJȦȞ ੮ıʌİȡ ਙȡȡİȞȠȢ țĮ੿ ș੾ȜİȠȢ ਲ ʌİȞIJ੹Ȣ ȝ੼ȝȚțIJĮȚ (Plut. Aet. Rom. 264A). 139 For reactions to various arguments of Wilpert, see also the useful notes in: Dooley W. E., tr. Alexander of Aphrodisias, On Aristotle Metaphysics 1. Ithaca (NY), 1989, 63ff. 140 The only place where Aristotle speaks of generation of the ȢǤǎʑǣǗ ɢǗǞǣǗ in relation to the Pythagoreans clearly shows that he had in mind a physical process: ɻǜ ǞǙ˹ ȳǗʒǜ ǝǟǝǞNjǒʍǗǞǙǜ, ǏɒǞ' Ȳǘ ȲǚǓǚʍǎǣǗ ǏɒǞ’ Ȳǔ ǡǛǙǓˍǜ ǏɒǞ’ Ȳǔ ǝǚʍǛǖNjǞǙǜ ǏɒǞ’ Ȳǘ ʁǗ ȢǚǙǛǙ˹ǝǓǗ ǏɎǚǏ˪Ǘ, Ǐɪǒʔǜ Ǟʒ ȶǍǍǓǝǞNj ǞǙ˹ ȢǚǏʑǛǙǟ ɣǞǓ ǏɓǕǔǏǞǙ ǔNjʐ ȲǚǏǛNjʑǗǏǞǙ ɫǚʒ ǞǙ˹ ǚʍǛNjǞǙǜ (Met. 1091a13–18), as he himself concedes (1091a18–20). See Philip J. The “Pythagorean” Theory of the Derivation of Magnitudes, Phoenix 20 (1966) 32–50. 141 E.g. Met. 1077a23-31, 1081a22-27, 1081b10-26, 1082b28-33, 1083a32-35, b4-11, 1084a2-7, 1085a 7-b 33, 1090b5-8, 1091a12, 1092a23-24, etc. See Annas J. Aristotle’s Metaphysics: Books Ȃ DQGȃ. Oxford, 1976. 142 Wilpert. Op. cit., 375. Ǟʒ ǎʌ ǎǟʋǎNj ǚǙǓ˜ǝNjǓ ǞʎǗ ȳǞʍǛNjǗ ǠʕǝǓǗ ǎǓʊ Ǟʒ ǞǙʔǜ ȢǛǓǒǖǙʔǜ ȶǘǣ Ǟ̆Ǘ ǚǛʗǞǣǗ ǏɪǠǟ̆ǜ Ȳǘ NjɪǞ˜ǜ ǍǏǗǗˍǝǒNjǓ ɿǝǚǏǛ ȶǔ ǞǓǗǙǜ ȲǔǖNjǍǏʑǙǟ (Met. 997b33–988a1). 143 ǏɔǗNjǓ ǍʊǛ ǞʎǗ ǖǙǗʋǎNj ȧǖNj ȢǛǞǓǙǚʍǛǓǞǞǙǗ, ɡ ȲǎǏʑǔǗǟǏ ǎǓʊ ǞǙ˹ ǍǏǗǗǑǞǓǔʎǗ NjɪǞʎǗ ǏɔǗNjǓ ǔNjʐ ǞǙ˹ ǚǏǛǓǞǞǙ˹ ǔNjʐ ǞǙ˹ ȢǛǞʑǙǟ ȢǛǓǒǖǙ˹Ƥ ȢǛǞʑ̄ ǖʌǗ ǍʊǛ ǚǛǙǝǞǓǒǏǖʍǗǑ ǚǏǛǓǞǞʒǗ ǍǏǗǗˎ, ǚǏǛǓǞǞ̇ ǎʌ ȦǛǞǓǙǗ (In Met., 40.1820). Cf. Theo, 22.5–9 = fr. 9 Ross. 144 Wilpert notes (Op. cit., 375) that according to Asclepius (34.16f.) justice is five, not four, but this is the only difference between two passages; from 34.21ff. Asclepius follows the same source as Alexander.

Alexander and Asclepius, and argued that these three texts derive from an unknown arithmological archetype, perhaps an earlier commentary on Aristotle. 145 Interestingly, right after the pertinent passage Alexander quotes the Peripatetic Aspasius (c. 100–150 AD), who discussed the Pythagorean number philosophy in his commentary on the Metaphysics (In Met. 41.26–28). Could arithmology also come from Aspasius? Authenticity of fr. 13 Ross, the astronomical part of which is no less problematic than arithmological, 146 deserves a special treatment, in the framework of the present paper it suffices to conclude that arithmology presented in Alexander cannot belong to the pre-Aristotelian Pythagoreans. What Alexander says on the number seven only confirms this view. 10. The ungenerated seven By far the longest chapter in the Anonymus is assigned to the number seven: in Philo’s De opificio mundi it takes forty paragraphs (89–128), fifteen of which dealing with mathematicals (91–100, 106–110) and the rest with the corporeal world. Nature takes delight in the number seven, says Philo, listing seven phases of the moon, seven heavenly circles, seven ages of man, seven external and seven internal parts of the body, etc. 147 Symbolism of the number seven has a very rich tradition, 148 that was actively used by our arithmologist. Since from the purely arithmetical point of view the number seven was not a very rewarding topic, the arithmologist divided it into the one and the six, the two and the five and the four and the three and plaid with these pairs of numbers. One specific feature of the seven noted by Speusippus, namely, that it is neither a factor nor a product (fr. 28, l. 30 Tarán), becomes very prominent in arithmological literature. For this reason, says Philo, the Pythagoreans liken seven to the motherless and ever-virgin Maiden (Leg. alleg. 1.15), i.e. Athena, who neither begets nor is begotten. Elsewhere Philo mistakenly ascribes this identification to the other philosophers, saying that the Pythagoreans liken the seven to the ruler of all (De opif. mundi, 100). The entire arithmological literature, however, including Philo’s ‘cousin’ Lydus, 149 contradicts this and relates the connection of the seven with Athena to the Pythagoreans. 150 The passages of Lydus and Philo on the seven as Athena are very close textually and both are full of confusions. 151 Thus, while Lydus ascribes this idea to Philolaus: “Rightly, therefore, did Philolaus call the number seven ‘motherless’; for by nature it alone 145 Delatte. Op. cit., 167–171, at 170. Zhmud. Pythagoras, 343f. 147 Leg. alleg. I, 8-15; De opif. mundi, 101–126. For more details, see Moehring. Op. cit., 200-205; Runia. Op. cit., 301f. 148 See above, 22 and Zhmud. From number symbolism, 26ff. 149 ɣǒǏǗ ǔNjʐ Ǚɏ ƻǟǒNjǍʓǛǏǓǙǓ ȪǒǑǗˎ ǞʎǗ ȳǚǞʋǎNj ȢǗNjǞʑǒǏǗǞNjǓ (De mens. 3.9). 150 Staehle ʋ 43a–43k. Philo’s mistake is best explained by Huffman. Philolaus, 337f. 151 Boyancé. Op. cit., 91ff.; Hooker M., tr. Lydus. On the Months. 20172. Introduction, xxxix ff.

neither begets nor is begotten”, 152 Philo gives a different quote from Philolaus: “There is a ruler and leader of all, god, one, eternal, abiding, without motion, himself like to himself, different from all others”, which Lydus in the next sentence cites under the name of Onetor of Tarentum. 153 This Onetor has been identified by Thesleff with Onetas (Onatas) of Croton, the pseudoPythagorean author of Ȇİȡ੿ șİȠ૨ țĮ੿ șİȓȠȣ, where he discussed and refuted monotheistic ideas. 154 Huffman prefers another Onetor, also suggested by Thesleff, the author of ƻǏǛʐ ȢǛǓǒǖǑǞǓǔ˜ǜ ȢǗNjǕǙǍljNjǜ mentioned in a scholium to Proclus, the fifth book of which considered seven-, eight- and nine-month babies. 155 Existence of such a treatise before the Anonymus is highly unlikely, so that Onetas remains better option, for other attested Onetors also do not suit. 156 Lydus’ quotation from Philolaus that number seven is motherless, placed by Diels-Kranz among spurious fragments (44 B 20), Burkert and Huffman considered as genuine for it has a very close parallel in Aristotle fr. 13 Ross on Pythagorean arithmology: “Since seven neither generates any of the numbers in the decad nor is generated by any of them, they [the Pythagoreans] called it Athena… who is motherless and always virgin”. 157 Does this Lydus’ quote from Philolaus come from the Anonymus, and does Aristotle’s fragment adds to its authenticity? First, we have to bear in mind that Philolaus’ genuine fragments contain no word of any number being generated or assigned to god. On the other hand, there is ample evidence that under his name a pseudo-Pythagorean arithmological treatise circulated, in which angles of the triangle and the square were dedicated to different gods and various numbers were associated with gods. As a matter of fact, Lydus adduces two other such references to Philolaus: the dyad is “a consort of Kronos”, 158 secondly, that the decad is “a receptive of the unlimited”. 159 It seems then reasonable to assume that the quote on the hebdomad in Lydus also comes from the pseudo-Philolaic treatise and not from the Anonymus, for such treatise could not have been written before the Anonymus and cited by the latter. A study of the arithmological work under Philolaus’ 152 ɞǛǒ̆ǜ ǙɰǗ ȢǖʏǞǙǛNj ǞʒǗ ȳǚǞʊ ȢǛǓǒǖʒǗ ɟ ǀǓǕʓǕNjǙǜ ǚǛǙǝǑǍʓǛǏǟǝǏ, ǖʓǗǙǜ ǍʊǛ ǙɮǞǏ ǍǏǗǗˍǗ ǙɮǞǏ ǍǏǗǗˍǝǒNjǓ ǚʍǠǟǔǏ (2.12, cf. 3.19, tr. M. Hooker). 153 ȶǝǞǓ ǍʊǛ ȿǍǏǖʖǗ ǔNjʐ ȦǛǡǣǗ ȣǚʋǗǞǣǗ Ǐɕǜ ȢǏʐ ɼǗ ǒǏʓǜ, ǖʓǗǓǖǙǜ, ȢǔʑǗǑǞǙǜ, NjɪǞʒǜ ȳNjǟǞ̇ ɣǖǙǓǙǜ, ȷǞǏǛǙǜ Ǟ̆Ǘ ȦǕǕǣǗ (Philo. De opif. mundi, 100 = Lydus. De mens. 2.12, tr. C. Huffman). One of Lydus’ manuscripts reads ɞǗLjǞǣǛ instead of ɟ ˸LjǞǙǛ accepted by his editor Wünsch. 154 Thesleff. Op. cit., 138–140. 155 Procl. In Rem publ. II, 378.23 = Onetor (FGrHist 1113 F 4). 156 See FGrHist 1113 F 1–3; Lakmann M.-L. Platonici minores: 1. Jh.v.Chr. – 2. Jh.n.Chr. Leiden, 2017, 212f. 157 Alex. In Met comm. 39.3ff. = Arist. fr. 13 Ross, tr. C. Huffman; Burkert. Op. cit., 249 n. 52; Huffman. Philolaus, 337f. 158 ɞǛǒ̆ǜ ǙɰǗ ɟ ǀǓǕʓǕNjǙǜ ǞʎǗ ǎǟʋǎNj ƵǛʓǗǙǟ ǝʕǗǏǟǗǙǗ ǏɔǗNjǓ ǕʍǍǏǓ, ɡǗ ǔNjǞʊ Ǟʒ ǚǛǙǠNjǗʌǜ ǡǛʓǗǙǗ ȦǗ ǞǓǜ ǏɒǚǙǓ (4.64). 159 ɞǛǒ̆ǜ ǙɰǗ NjɪǞʎǗ ɟ ǀǓǕʓǕNjǙǜ ǎǏǔʋǎNj ǚǛǙǝǑǍʓǛǏǟǝǏǗ, ɻǜ ǎǏǔǞǓǔʎǗ ǞǙ˹ ȢǚǏʑǛǙǟ, ɤǛǠǏʔǜ ǎʌ ǔǕNjǎǙ˹ǡǙǗ, Ȳǘ Ʌǜ ɻǝǏʐ ǔǕʋǎǙǓ ǞǓǗʌǜ ǚʋǗǞǏǜ Ǚɏ ȢǛǓǒǖǙʐ ǠʕǙǗǞNjǓ (1.15);

name is a desideratum. Thesleff wrongly decided that Philolaus’ doxography derives from the early Academic and Peripatetic sources, while his fragments even if they are inauthentic can be dated at least to the mid-fourth century BC. 160 Due to this he did not distinguish a specific arithmological treatise of Philolaus, although there is plenty of reason to assign to it such fragments as B 8, 11, 20a-c and 23, and testimonia A 10, 11, 12, 13 ad fin., 14, as well as other material. Therefore, there is sufficient ground to believe that the idea of the seven that ǙɮǞǏ ǍǏǗǗˎ ǙɮǞǏ ǍǏǗǗˍǞNjǓ has been ascribed to the Pythagoreans only after Speusippus and Aristotle and thus, that Aristotle’s fr. 13 Ross is not genuine. 160 Thesleff. Op. cit., 149. To be sure, he considered as spurious Lyd. De mens. 2.12 (on the seven) and Lucian. Pro laps. in salut. 5 (on the tetractys).