Apeiros and Circularity

GREEK » ROMAN » AND + BYZANTINE + KAPLAN, HA. Summer 1975 VOLUME 16 Po NuMBER 2 Annual Subscription $10.00 Issued Quarterly GREEK’"ROMAN’AND BYZANTINE’ STUDIES Single Number $3.00 Founded by Jorn J. Brurrz Editorial Board Senior Editor Editor for Greek Studies Editor for Roman Studies Editor for Byzantine Studies Assistant Editor Business Manager For abbreviations in footnotes, this journal follows the usage described in the American Journal of Archaeology 74 (1970) 1-8, and for ancient and Byzantine authors and titles, the practice of the Oxford Classical Dictionary? Assistant Business Manager (Oxford 1970) ix-xix and A Patristic Greek Lexicon, ed. G. W. H. Lampe (Oxford 1961) xi-xlv. Contributors Circulation Manager are requested to observe these usages in preparing their manuscripts. Manuscripts should be typed in = a WituiaM H. Wis, Duke University Wituam M. Carver ID, Columbia University Prmup A. STADTER, University of North Carolina Dero J. GEANAKOPLOS, Yale University KENT J. Ricssr, Duke University DeCourser Fates jr, Emerson College Iten Fares, Cambridge, Massachusetts Dorotuy Rounns, Cambridge, Massachusetts Advisory Board Price H. De Lacy, University of Pennsylvania KAPLAN M., "Anetpos and circularity [Vassociation du mot avec mécag A l’époque de Pythagore] ; cf. Gree ct dialectes helléniques. Kaptan M., "Argtsoc and ciscularity : GRBS XVI 1975 125-140. | The notion of circularity is inherent in the word, although it is duubtful the word ever had this meaning by itself. The etymological association of the word with répæc dates only from Pythageresn times. WZ 475 STERLING Dow, Boston College Henry R. IMMERWAHR, University of North Carolina James H. OLıver, The Johns Hopkins University Eric G. Turner, University College London George H. Wuriams, Harvard Divinity School vj We Volume 16 Summer 1975 Number 2 DUKE UNIVERSITY 7 DURHAM, NORTH CAROLINA

HE ATTEMPT to connect &reipoc with the notion of circularity is not a novel concept. The way was indicated as far back as Aristotle, who includes in his Physics under the discussion of the theoretical possibility of the existence of the &reipov, the ‘infinite’, a mention of the application of the adjective to objects such as rings which are uniform and characterized by the absence of a bezel. Porphyry, whose investigation of &reipoc and circularity I shall consider at length in the body of this essay, collected several examples of import similar to that of Aristotle’s ring. More recently Cornford concluded that it actively has the meaning ‘circular’.2 The latter two discussions, of which Porphyry’s is dependent upon Aristotle and Cornford’s practically a restatement of Porphyry, have both gone awry and have convinced no one who has considered the matter carefully. I, too, believe that their position is substantially untenable, but I am, however, prepared to grant that this was a result more of their method than of what they intuitively sensed. I intend to demonstrate here that &reıpoc may indeed be related to a notion of circularity in itself, but that this is a latent meaning and therefore seldom expressed with absolute clarity, and that this meaning of &reıpoc by itself was obscured after Pythagorean doctrine spread and gained notice. Furthermore, I submit that ’Qxeavdc, the River Okeanos, is the primal concept behind the idea of circularity in it and that it is from here that the picture of the circular ärepoc which Porphyry presents has its origin. I want to approach ärepoc first of all by considering its etymology. In so doing I must stress the fact that areipwv, éreupécioc, amepeicioc and areipiroc are all epic variants of @reıpoc, which dominates later prose usage. Moreover, these epic variants tend to have their own restricted formulaic usages, as @repeicioc does, for instance, in the 1 Ph. 3.4-8 contains a general discussion of &reıpoc. The example of the ring is in Ph. 207a2-7. 2 F. M. Cornford, “The Invention of Space,” in Essays in Honour of Gilbert Murray (London 1936) 226; and Principium Sapientiae (Cambridge 1952) 171-77. For a general review of all the arguments see L. Sweeney, Infinity in the Presocratics (The Hague 1972) 1ff.

Homeric phrase &repeicı” &moıwa.3 Considering the narrow range of these individual words and their eventual telescoping into @reıpoc, we may consider their etymologies together and not make serious distinctions among them. The root of all these words is Indo-European «per, which had an ‘end-directed’ signification. Kahn has well argued that the alphaprivative in &reıpoc negates not the noun répac but the verbal root «per-, which may be seen in zreipw, mepdw, mepaivw, as well as in numerous preverbs, such as mpé, rap and epi.5 Schwyzer goes so far as to say that “rap« und die Nebenform rapai gehören etymologisch zunächst mit répoc ‘friiher’ zusammen, weiter auch mit repi, répa, mp6, mpéc, usw.”® When one further considers that repi may appear as rép and that répa is often joined in compounds in the form zep-, it is easy to see that confusion could arise between different, developed denotations of «per-. Frisk, for example, glosses répa as “darüber hinaus, weiter, länger, mehr, jenseits,” while he glosses wepi as “ringsum, überaus, durchaus.” Contrast these developments to the original «per-, which Schwyzer says meant “im Hinausgehen, Hinübergehen über, im Durchdringen.”7 Among modern philologists Schulze was the first to stress the rept aspect of &reipoc or, more accurately, of @reipıroc. He analyzed drreípuroc as «a-peri-itos and explained the suffix -itos as drawn from ievaı, for which he compared dapatiréc and the Latin orbita; he translated it as that which ‘circumiri nequit’. He allowed, however, that it was possible that it might mean ‘transire’, with the -peri- equivalent to Latin per.8 Nevertheless, it has been his first explanation which later philologists have accepted. Bechtel agreed with Schulze in his identification of -itos with ievaı, and he gave an equivalent translation 3 In the Iliad 11 times with ¿rowa; once with édva in both the Iliad and the Odyssey. 4 Cf. Ch. Kahn, Anaximander and the Origins of Greek Cosmology (New York 1960) 231 (hereafter Kann, AOGC]. 5 Kahn, AOGC 232. Ann L. Bergren, The Poetics of a Formulaic Process : Etymology and Usage of reîpap in Homer and Archaic Poetry (Diss. Harvard 1973), stresses the «per significance of meipap as ‘goal-oriented’. 6 Ed. Schwyzer, Griechische Grammatik II (Munich 1939-53) 491f; cf. also H. Frisk, Griechische etymologisches Wörterbuch (Heidelberg 1960-72) s.v. repi. ? Schwyzer, op.cit. (supra n.6) II. 499f: “Diese Bedeutungen kennt auch noch das Griechische; doch ist hier wie im Indisch-Iranischen ‘rings um, um’ die Hauptsbedeutung geworden. Ursprünglich war von zepi in dieser Bedeutung audi ‘zu beiden Seiten’ verschieden; doch verblasste der Unterschied, bes. bei audi. 8 W. Schulze, Quaestiones epicae (Gütersloh 1892) 116 n.3.

of @reipıroc as that “um den man nicht herum gehen kann.” More recently Chantraine has said that it “pourrait . . . signifier “dont on ne peut faire le tour’ de &-7epu-itoc.”” Thus he too relies upon Schulze’s comparison with &oéirdc and assumes that the base of the word is a negated xper(i).1° Frisk, however, is troubled by the -i- in -itos; Schwyzer offers a qualitative interpretation which satisfies neither Chantraine nor Frisk. The significance of &reıpoc, keeping in mind its «per root, is ‘what cannot be passed over from end to end’ with a connotation of circular movement; Kahn maintains that this easily passes into the sense of ‘immense, enormous’ in relation to human perspective, a sense associated with Homeric usage.!? The Heraclitean concept of circularity and the applicability of &reipoc to a circle I shall consider below when I examine Porphyry’s arguments. On the whole, then, it is best to posit the connection of areipıroc and hence &reıpoc (from «@répioc) ämeupoc by metathesis, as àrepeicioc = @rretpécioc) With epi. &reipoc, moreover, is often associated with rrepi-compounds, especially wepıexw, in philosophic speculation. Aristotle informs us that Anaximander (as is likely, to judge from the context) stated that his &reıpov surrounded (repréyeuw) the world. Anaximenes replaced Anaximander's ro &mewpov as «px% with an äreipoc anp; he still allowed it to surround the world.!3 Elsewhere ® E, Bechtel, Lexilogus zu Homer (Halle 1914) 49. 10 P, Chantraine, Dictionnaire étymologique de la langue grecque (Paris 1968-) s.v. ameıpecuoc. 11 See Frisk, op.cit. (supra n.6) s.v. drreupécioc; Schwyzer, op.cit. (supra n.6) 1.106 n.3. 12 Kahn, AOGC 232f. Porphyry, Quaestionum Homericarum ad Iliadem pertinentium reliquiae 14.200 (fasc. II pp.189ff ed. H. Schrader, Leipzig 1882), had already hinted at the relative quality of ¿meipoc, when he wrote that cnuaiveı Sì TO dmeıpov Kal TO merepacuevov pev TH éauro duce, iuiv 8° drepiAnmrov (regarding Schrader’s text, see infra n.30). G. J.M. Bartelink, who wrote the articles on dreipıroc and &reipwv in Lexicon des frühgriechischen Epos fasc. VI (Göttingen 1969), stresses that the endlessness is relative to the viewer. See also P. J. Bicknell, “ré &rreipov, ärepoc anp and ro rrepiéxov,” Acta Classica 9 (1966) 39. 13 Arist. Ph. 203b12; Aétius 1.3.4 (=H. Diels and W. Kranz, Die Fragmenta der Vorsokratiker!® [Dublin-Zürich 1972] 13 8 2 [hereafter Diels-Kranz]). Cf. also Arist. Cael. 303b12, $ mepiéxew dacì mévrac roùc oùpavodc üreıpov dv, and Pl. Ti. 3144, 3148, and 3381. On the whole question of repuéxew as a reminiscence of Anaximander, see A. E. Taylor, A Commentary on Plato’s Timaeus (Oxford 1928) ad 3144; and F. Solmsen, “Anaximander's Infinite: Traces and Influences,” AGPh 44 (1962) 109-31. It is beyond the scope of this paper to discuss the problem of ‘qualitatively indefinite’ versus ‘quantitatively infinite’. H. Frankel, Wege u. Formen frühgriechischen Denkens (Munich 1955) 189ff, declares for “qualitatively indefinite’, and W. K. C. Guthrie, A History of Greek Philosophy I (Cambridge 1962- ) 83ff, prefers this meaning (without excluding the other). G. S. Kirk and J. E. Raven, The Presocratic Philosophers (Cambridge 1969) 108-10, argue on the basis of early usage that the spatial sense of

Aristotle says that certain philosophers granted the &7espov the right and prerogative of ro wavra mepiéyeiv Kal TO av Ev éaur@ éyew.4 Aristotle replaced it, however, in his scheme with the oöpavcc. In this scheme the oöpavöc encloses a complete system. It is the function of To mév to surround, not the @reıpov, which Aristotle defines as a potential but not realized whole; the éze:pov is a mere part, and so it is impossible that it should embrace and define anything. This follows, according to Aristotle’s logical system, for two reasons. First, in his division of causes he defines the &reıpov as material cause: où TEPLEXEL GAAG TEPLÉXETOL, D ETTELPOV ... TrepiéxETOL yap we Y VAN Evröc Kal TO Ameıpov, mepiéyer de To eldoc.l5 Aristotle’s discussion in these sections fairly bristles with repiéxw in its many forms, with active and passive forms opposed to one another. He is upbraiding those philosophers who have granted to the &reıpov (= material cause) the prerogative of the formal cause, that of defining and outlining the whole, in this case, the world.16 Aristotle is here changing the &reıpov from the external factor that it was in Anaximander, Anaximenes and others into an internal factor. Beyond a matter of the four causes, Aristotle is also faced with the problem of a body infinite in extension. Such a thing appears impossible within Aristotelian terminology, since “body is defined as that which is limited by a surface.”17 To the end of Physics 3.6 he is occupied with exposing the fallacies involved in equating &respov and To rráv (=6Aov). This is in keeping with the overall tenor of Physics 3.4-8, which is a general discussion on the possibility of the existence of infinity. &repoc predominates (though not necessarily in the sense of ‘infinite’), but they consider it uncertain that Anaximander intended precisely this. Certainly, however, Aristotle understood the word as ‘infinite’, and in his discussion ‘qualitatively indeterminate’ (i.e. amorphous) expectedly gives way to an overriding emphasis on Form. One should imbibe the salutary warning of Guthrie, however, that with Anaximander we are not at a stage where “distinctions between different uses of the same word are possible” (op.cit. 1.86; cf. 109). It is likewise not the aim of this paper to consider the question of innumerable worlds in Anaximander. Recent discussions of this problem (with references to earlier work) may be found in Kirk and Raven, op.cit. 121-23; Kahn, AOGC 46-53; and Guthrie, op.cit. 1.106-15. 14 Ph. 207a19. 15 ibid. 207a25-b1. Cf. also Cael. 312a12-13, papev de TO pèv rrepiéxov Tob eidouc elvas, TO de mepieyxopevov THC VANc. 16 Cf. the language of PI. Ti. 3144 and 3148 in a similar context. Also note LSJ repiéxw 1.1.b. 17 W. D. Ross, Aristotle: Physics, Revised Text with Introduction and Commentary (Oxford 1955) 364. Cf. Sweeney, op.cit. (supra n.2) 92, 170f.

We obviously have to deal with two senses of the &reipov: in the first, that of the earlier physicist-philosophers, it is external and active (repiéyei), while in the second view, that of Aristotle, it has become an internalized phenomenon and is nowa passive factor (rrepiéxeras). At the same time as he is involved in changing the orientation of the &rrespov, Aristotle alters its definition to keep it in agreement with his concept of the infinite as always undefined and incomplete: où yap od umdev ¿Ew, GAN od dei Ti ¿Ew Ecri, Toûro Grreupóv ecriv.18 The basis for his definition of it is the principle of infinite division and addition which he enunciates just prior to this. To be sure, when Anaximander stated that the äreıpov surrounded all things, he probably assumed that all of space was occupied with his äreipov, and therefore that it was a type of öXov. This, however, was unacceptable to later philosophers.!? In the Timaeus, for instance, Plato speaks of a cosmos which contains dv 6Aov ¿xacrov (of its four constituent elements) and which leaves pépoc oùdèv oddevdc ... éÉw0ev, but on the contrary is “whole and wholly complete”? Furthermore, he describes this sphere as ex pécou mavrn mpoc Tac TeNevtàc icov améxov, which is, moreover, reminiscent of the explanation generally accepted for Parmenides’ ‘sphere’, where reipara are imposed upon it (fr.8) to serve as the confines of an unvarying reality?! Guthrie notes that the argument for confining all reality within bounds seems to be that “what is apeiron is essentially unfinished, incomplete, never a perfect whole however much of it one may include.”?? Parmenides, of course, did not say this in so many words; it is, however, a valid extrapolation of his doctrine from the viewpoint of Aristotelian terminology. The trail leads irrevocably back from Aristotle, to the Timaeus, to Parmenides; following the lead of Plato, Aristotle is making a fundamental return to a position taken (but with serious objections concerning the existence of anything &reipov) by Parmenides. Yet it is not until Aristotle that we can see an explicit definition of the status of the drreipov, and it is in the course of his definition that he manifestly diverges from Parmenides, both because Parmenides absolutely 18 Ph. 207al. 19 Solmsen, op.cit. (supra n.13) 120-22. 20 Ti. 3205-3347. 21 See Taylor, op.cit. (supra n.13) ad 3384-5, for a reference to Parmenides. On the question of the ‘sphere’, see now G. E. L. Owen, “Eleatic Questions,” CQ n.s. 10 (1960) 95-101; and Guthrie, op.cit. (supra n.13) II.43ff. 22 Guthrie, op.cit. (supra n.13) 11.38.

denied the existence of an &reıpov and also because he did not posit a physically existent sphere. In attempting to define further the nature of the éze:pov, Aristotle indicates that it is that which is advegiryrov, ‘incapable of being crossed from side to side’. In part his reason for saying this is because the infinite always has something further to be negotiated. Almost certainly he owes something for this conception of the adıe&irnrov &meıpov to Zeno’s paradoxes of motion, particularly the first one (Ph. 239b11-14), which may be complemented by an infinite regress to disallow the possibility of motion entirely or (as here) the possibility of reaching a terminus. Elsewhere in the Physics Aristotle uses similar language, once in defining the sense in which something is ‘intraversable’ (ro &dvvarov &eA0etr) and again in his disquisition on circular motion at the end of the Physics (SveAOeiv Se riv &meupov [sc. popav] aduvarov).23 Solmsen has opined that the source of the concept of an adveEirnrov for Anaximander’s thought is Hesiod, Theogony 736. The poets, he thinks, had not yet discovered the possibility of ‘absolute’ infinity; a space demanding more than a year to negotiate boggles the simple mind and is felt to be (practically) infinite.24 This is very close to the manner in which Porphyry and Kahn arrive at a relativistic concept of &reipov, which they find confirmed in Homer. It is for this reason that Homer, while he describes both the earth and sea as dreipwv, nonetheless imposes reipar« upon them—their meipara are so distant relative to the capacity of the Homeric man for travel that they are, for all intents, beyond the grasp of the mortal imagination. It is *Skeavóc, of course, which provides the reípara for the earth, and, conversely, the earth’s shores are the inner reipara for the River Okeanos. According to Bergren, “the reípara yainc is the earth’s physical extremity ... it is the line between opposite elements.” It is thus coextensive with the reipara ’Qkeavoto. Bergren maintains that the most archaic signification of retpap in Greek is the concrete designation of the earth’s extremity and that every time zeipara 23 Ph. 204a14, 204a4, 265a19-20. Cf. also the comment on Zeno in Ph. 233a22, 76 ph évôéyecdo Ta arretpa ÖreAdeiv. Porphyry recalls Aristotle’s terminology in the phrase ddvetirirov ametpov (p.192.24 ed. Schrader [supra n.12]). Also, Simpl. in Phys. 470-71 opposes ärepoc to dcetodevroc and duarropeuroc. 24 See Solmsen, op.cit. (supra n.13) 122f, esp. 123 n.58, for the Hesiodic origin of dde&irnrov. Cf. Pind. Pyth. 10.63.

yainc denotes the end of the earth in Homer (which it does in all but one instance of the phrase), the context includes the streams of Okeanos.2 Even as the earth, described as @reipwv, is delimited at its boundaries by Okeanos, so Okeanos, which provides the reipara to the earth, is itself &reipwv, since it lies even beyond the earth’s immeasurable magnitude and therefore surpasses it in distance from our hypothetical observer, as well as because it has a circumference obviously larger than the orbis terrarum and is an unbroken circle, according to one implication of the later Heraclitean fragment. Extrapolating from Anaximander’s point of view, then, Okeanos would be a physical, geometrical representation of ró &rreipov (and so itself becomes aretpoc) because it encircles (wepıexeı) the earth, which is itself areipwv, according to the relativistic Homeric interpretation. Aristotle is the first to remark that a uniform ring which has no socket for a gemstone may be called &reipoc: Kat yap rode SaxrvAtouc ameipouc A€éyouct Toùc un Exovrac cHevddvyny, Ott ariel Te CEw ¿cri AapBEvew. To be sure, he goes on to reproach this (colloquial?) usage for a lack of precision: kab” duovóryTa uév riva Aéyovrec, où pevror kuplwcdet yap TOUTÓ TE Ündpyeuw Kal underrore TO adTO AapBavecBau Ev de TO KUKAw od yiyveTat oùrwc GAN atei TO ébeËñc povov Erepov. He concludes by associating this with his opposition of ro éretpov and 76 r&v and with his concept of adveEirnrov: &meupov uev odv écriv 08 kara TO rrocóv NauBavovew atei te Auußaveıv écriv é£w. od de undev 2éw, Tobr’ Ecrı tédetov Kal dAov.?6 That a circle may be called &reıpoc is evidently a developed geometric concept,?” which the early philosophers seized upon as a convenient and intriguing method to express that continuity which remains unbroken, temporally or otherwise. Heraclitus, fr.103 25 Bergren, op.cit. (supra n.5), goes on to say that reipara denotes not a physical material as such but the function of anything that binds or defines, and which forms the limit of anything’s outward extension. 26 Ph. 207a2-9. 27 This is Kahn’s argument in “Anaximander and the Arguments Concerning the Apeiron at Physics 203b4-15,” in Festschrift Ernst Kapp (Hamburg 1958) 28f [hereafter Kann, “Anaximander”]. This idea may well be indebted to medical concepts; see Kahn, “Anaximander” 25-27, and G. S. Kirk, Heraclitus: The Cosmic Fragments (Cambridge 1954) 113-15. Hesychius picks up the geometrical possibilities when he glosses detpov as mo, &yevcrov (confusing what are actually two different words), repupepéc, crpoyytov, did TO rre &pyv pre mepac éxew. Latte, in his ed. of Hesychius (Copenhagen 1953), notes that this explanation was borrowed from Diogenianus of Heraklea, a Greek grammarian of Hadrian’s time; this shows the continuity and persistence of this explanation.

Euvov (yap) apyt Kal mepac émi kikdov, quoted by Porphyry in his discussion of the circularity implied in @reıpoc, does not mean to imply more than the coincidence of the beginning and end in a circle, for here Heraclitus is concerned with the coupling of opposite quantities; but it very early came to be associated with the idea of continuous motion, which can only be found on a circle. Thus Aristotle echoes this idea more than once, as when he says rod de KdKAw coparoc 6 aùròc réroc 60ev ipéaro Kai eic Gv releur&, and when he says that continuous motion is possible only on a circle, since elsewhere où yep cuvdrrei TH «px TO mépac.28 In addition Alcmaeon is quoted in the Problemata on human mortality as follows: rove yap av9pwrrove dyciv AÂkualwv Sia roûro AmoAAucdaı, Ori où Övvarraı THY Apxıv TO Tédel mpocdha.2 It is at once obvious, particularly if one considers the possibility that this may have been a common saying, that Aristotle has in the former instance paraphrased Alcmaeon. Porphyry’s quotation from Heraclitus was undoubtedly influenced by Aristotle, who was himself influenced by Heraclitus. Aristotle’s remarks on the annular possibilities of &reipoc evidently intrigued Porphyry, for when he was compiling his Quaestiones Homericae he devoted several pages to an exegesis of the various senses of &retpoc.2° His lemma was Iliad 14.200f: elus yap obouévn moAvbopßov meipara yainc, | "Qxeavdv re, Hewv yéveciv, Kai unrepa Tn0yv. In his subsequent discussion Porphyry indicates various senses of &reipoc: (1) } xarà péyeboc 7 Kare mAndoc,31 (2) the relativistic éretpoc already noted, (3) that associated with objects of exceeding beauty, and (4) that connected with circular or spherical objects. It is in the context of the last meaning that Porphyry quotes the Heraclitus fragment to which I have already referred. He continues by quoting several 38 Cael. 279b2, Ph. 264b27. To Aristotle’s quotations we may add the similar sentiments of [Arist.] MXG 977b4 (= Diels-Kranz 21 A 28) on Xenophanes (apropos a sphere, however) and 974a9-11 (=Diels-Kranz 30 A 5) on Melissus (of temporal continuity). 29 [Arist.] Pr. 916a33-35. 30 My discussion of Porphyry is based on pp.189ff of Schrader’s ed. (supra n.12). Schrader’s bold reconstruction of the text of Porphyry can no longer be accepted; see H. Erbse in Zetemata 24 (1960) 17-77. The long comment on II. 14.200 here under discussion comes from Codex Ven. B (manus secunda) and so (following Erbse) its authenticity is beyond doubt. The philosophical nature of the argument also is an indication of its Porphyrian origin. (I am indebted to Professor Henrichs for his help in resolving my questions on the text of Porphyry.) 81 The division of ärepoc into % Kara uéyedoc 7 kara rAgdoc is at least as old as Zeno; cf. Diels-Kranz 29 8 1, B 3 and Simpl. in Phys. 22.9 (= Diels-Kranz 13 A 5).

references in poets which associate &reıpoc and unbroken circularity, on the basis of which Cornford concluded that in classical Greek it may actually and actively entail circularity.32 As other scholars have shown, however, the fact that round objects are spoken of as ézretpoc by poets in a few instances need not imply that all unlimited objects are considered round, but rather that the notion of circularity is contained in the nouns to which the adjective is attached.83 Porphyry, moreover, not only leads astray those who refer to him, but he is led astray by his own sources when he says that Homer believed in a spherical earth: cre cuvayerai, eimep Y y merepacpévn pndeica Gmeupoc maAWw eppyOn, un dia TO py e&irnrov aurnv elvas kara méyeloc eipñcôœ &reupov, dia de TO chœpoedÿ elvar Kai ToLradTny aurnv Kara cyua dredndBae. Porphyry likely had Heraclitus, the composer of the Homeric allegories, in mind when he wrote this; the passage of Homer cited as the lemma and the general tone of the disquisition confirm it. Heraclitus first of all cites the movements of the winds as a proof that Homer believed in ro ro5 kócuov chatpoerdéc. Later he writes that &reıpov 8’ av 6 KUKAoc ovou&loiro Sikaiwc, ETELÖNTTEP aunyavov ecri Setar mépac ev aùT@® Ti. Lastly he quotes as the ‘clearest’ proof of the spherical world the symbol of Achilles’ shield.54 Heraclitus and Porphyry were not the only ones who thought that Homer had believed in a spherical world; Eustathius also makes this same mistake, perhaps misled by the Porphyry passage, but, if not, at 32 The references are Ar. fr.250 (Edmonds); Aesch. fr.379 (Nauck?); Eur. Or. 25 and fr.941 (Nauck?). Cornford, Principium Sapientiae (supra n.2) 173, also quotes Empedocles fr.28, where he takes aweipwv Sdhaipoc xvrdorepijc as one extended phrase meaning ‘spherical’. 33 See, among others, G. Vlastos’ review-article on Cornford, Principium Sapientiae (supra n.2) in Gnomon 27 (1955) 74 n.2; H. B. Gottschalk, “Anaximander’s Apeiron,” Phronesis 10 (1965) 51-53; and Bicknell, op.cit. (supra n.12) 41. Bicknell maintains that 70 dzecpov in Anaximander is spherical, arguing thus: Cornford will have been correct in regarding this apeiron as a spherical thing, but not because the word bears of itself any such sense. The apeiron is spherical because in its original state it was coterminous with the present cosmos, which appears spherical to the observer (or rather hemispherical, for the other half “follows from the observation of the movements of the heavenly bodies and is demanded by the dictates of symmetry’). 84 Heraclitus, Allégories d’Homere, chs. 47-48 (ed. F. Buffiére [Paris 1962]). Such comments as Heraclitus and Porphyry present are in large part from the common stock of allegorical interpretation in existence concerning Homer. A neoplatonist such as Porphyry would be aware of these interpretations, and it is difficult to believe that this particular lemma and disquisition are unrelated to Heraclitus. On Heraclitus’ own predilections, cf. infra n.38. Lastly, as Professor Henrichs advises me, earlier glosses on areipova yaiav (as well as Heraclitus’ comments) demonstrate the anteriority of the argument to the Porphyrian state.

least ultimately misled by the same body of allegorical scholia behind all these interpretations; the old error persists.35 According to Heidel the confusion in our sources between the circle and the sphere is common. This is due in part to the ambiguity of the term czpoyyvAoc, which may mean either ‘round’ or ‘spherical’. In the fifth century B.c. the term was not used exclusively—or even generally—with reference to a sphere.3® Heidel further asserts that Posidonius is likely to have credited Parmenides with positing a spherical world, and he also says that Posidonius was “at least the proximate source for the statements of Aétius (3.10.1) that Thales and the Stoics and their respective adherents taught the sphericity of the earth and for the assertion of Diogenes Laertius (2.1) that Anaximander held that doctrine.” Certainly no one now believes that Homer or Hesiod (or Thales and Anaximander) conceived of the earth as spherical (not even Cornford said that). Hippolytus (Haer. 1.6.3= Diels-Kranz 12 a 11), moreover, informs us that Anaximander believed in a circular but flat earth. To the best of our knowledge Plato (Phd. 1088 ff) was the first to conceive of a spherical earth in the center of a cosmic sphere. The key to this discussion of the spherical earth lies, I submit, in the train of thought which Heraclitus the Allegorizer presents to us.** For Heraclitus &reıpoc and sphericity< circularity are inextricably entwined with the description of Achilles’ shield in the Iliad. Nor is this surprising, for the oblong body-shield purports toshow the world surrounded at its edges by the River Okeanos. The Hesiodic Scutum presents a similar picture of Okeanos, flowing round the rim and 35 Eustathius,"ad Il. 7.446: "Icréov Sè Sri yv Ayer ameipova avri rob ametpov, we Evpımiönc dv Paldpa Incl. kai elpyrar pev Kai ¿Aaxoú mepi Toûrou Kal viv dé pnréov dru Te évrebber napadpacac Evpunidnc drépuova ¿comrpa ele TG kukdoTEp%, Ömep Tabrôv Ecrı TH dmeipova, érrel Kal TO Tépua Kai TO mépac TO aùrò Sidova. Kal örı Kad” “Opnpov pev areipwv 7 An yi 6 Ecrı cóopociónc Kai crpoyyvAn. 36 W. A. Heidel, The Frame of the Ancient Greek Maps (New York 1937) 68-74. 87 ibid. 67. This obviously is an easy error to commit, judging by the number of scholars who have so erred. See Owen, op.cit. (supra n.21) 95ff, for a convincing denial that Parmenides meant us to understand his system as establishing a spherical world. Diogenes Laertius elsewhere (8.48) has a rather vague and confusing statement assigning a ‘round’ earth to various philosophers; see Heidel, op.cit. (supra n.36) 73, and Sweeney, op.cit. (supra n.2) passim. 38 Heraclitus’ interpretation accords well with his attitude toward classical writers. Reinhardt, “Herakleitos 12” in RE 15 (1912) 508-10, comments that “Dieser Traktat verfolgt den Zweck, die Dogmen der anerkannten Philosophen, besonders Platons, Aristoteles, und der Stoiker, in systematischer Folge aus den Homerischen Epen abzuleiten.” Cf. Cic. Nat.D. 1.41 and A. S. Pease’s commentary (Cambridge [Mass.] 1955) ad loc.

enclosing the other scenes of the shield.?? The belief in an outer river which surrounds the fringes of the inhabited world is ancient, common to Mesopotamian legend and Egyptian lore long before it became a fixture in Greek civilization. Indeed, a Babylonian world-map on a cuneiform tablet now located in the British Museum shows the earth encircled by this outer river. Herodotus shows the extent of this belief when he ridicules Homer and the contemporary mapmakers because they represented the circular earth as surrounded by the River Okeanos.*! The Greeks also shared with the Egyptians the belief that the sun, after setting in the west, journeys back to the east along Okeanos; the Greeks had it sail in a golden bowl (representing the sun itself), the Egyptians in a ship.4? Some Homeric usages aid in the attempt to associate Okeanos and ameipoc in the sense of an unbroken circular river. First of all, there is abóppooc, ‘(Okeanos) flowing back into itself (as it encircles the earth)’. Homer uses the word twice, both times of Okeanos, once in the Iliad apropos the structure of Achilles’ shield and once in the Odyssey. Eustathius (ad Od. 20.65) glosses the word as follows: ’Aéppooc dé "S2keavoc 6 KUKAW Tic yc Tepivoct@v Kat ab radıv érri TO adTO LKVOUJLEVOC KATH TO, MEPLTEAAouEVWV éviaur@v. ol de TaAQLOL hpdbouct Kal oÙTwc: abdppooc, 6 eic éavtov avahiwy Ev Tm eldetcBar KvKrAW Tepi Tv yñv.48 The thought behind Iliad 18.402f (. . . wept de póoc "Qkeavoto | 39 Il. 18.607f; Scut. 314f. 4° BM no.92687. This tablet is reproduced in Cuneiform Texts in the British Museum pt.XXII (1906) pl. 48. It is also reproduced in Kahn, AOGC pl. 1. ‘Okeanos’ is probably a non-IndoEuropean word; see Frisk, op.cit. (supra n.6) s.v. *Qxeavoc; Kirk and Raven, op.cit. (supra n.13) 14 n.3; and P. Weizsäcker in W. H. Roscher, Ausführliches Lexicon d. griechischen u. römischen Mythologie M.1 (Leipzig 1908) 816. Cf. R. B. Onians, The Origins of European Thought (Cambridge 1951) 249; P. Seligman, The Apeiron of Anaximander (London 1962) 142; and F. Gisinger in RE 17 (1937) 2309f. More recently A. Carnoy has proposed a Pelasgian origin in AntCl 24 (1955) 27f, but see E. Vermeule, Greece in the Bronze Age (Chicago 1964) 18f and 60ff, for a critical appraisal of our knowledge of Pelasgian. Lastly, the situation of other searelated words in Greek (e.g. @Ac, mélæyoc, movroc; the origin of daAacca is unknown) should be noted, since they are often non-Indo-European words or words with a new signification. See Frisk, op.cit. (supra n.6), and Chantraine, op.cit. (supra n.10), on these words; on zróvroc see E. Benveniste in Word 10 (1954) 256f. 41 Hdt. 2.23 and 4.36. For a convenient summary of the matter see Kirk and Raven, op.cit. (supra n.13) 11-14. 42 The Greek belief is presented in Mimnermus fr.10 and Stesichorus fr.6 (Diehl). See Kirk and Raven, op.cit. (supra n.13) 14f, and Seligman, op.cit. (supra n.40) 134. 43 The Homeric passages are Il. 18.399 and Od. 20.65. In the latter passage, it is curious that Homer should speak of the apoyoai of Okeanos, since it is normally considered an unbroken circular stream (and hence without a mouth). The explanation of this apparent

adp@ popuúpwv péev äcreroc) is very similar, where epi... péev provides the idea of circularity and &creroc the notion of continuity. Okeanos is the circling stream which joins its end to its beginning and, as such, is a primary model for the evolution of the Heraclitean circle, and so it fulfills the definition of dzeipoc as that which cannot be circumnavigated. Homer never applies &reıpoc directly to Okeanos; he uses the xaperfamily with yata and rôvroc. From his usage of it with the latter we may conclude that he associates it with the notion of annular circularity. For instance, in Odyssey 10.194f Odysseus climbs a rocky lookout and observes an island rv mepı môvroc ameipitoc écrepävwrau, “which the sea encircles in an unbroken ring.”## The sense in Hymn. Hom. Ven. 120 (mailouev, audi 8° Guidoc Arreipıroc ecrepavwro) may well be similar, though here «reipıroc could mean ‘uncounted’. These usages, associated with audi or zepi, recall the associations with mepiéyw. Homer adds two more words to our list, éupia)oc (used five times in the Odyssey, always in the phrase audıcAw ’Iddkn) and mepippvroc (used once in the Odyssey, of Crete). Eustathius connects ddiadoc with the line of the Odyssey quoted above when he writes Tv de Kar’ avr vijcov, mepi mévToc Ameipıroc ECTEPÁVOTOL, Nyovv KUKA@ mepıexeı we audiadov. Thus the idea that Okeanos binds together and encircles the earth is transferred to passages in which the sea encircles an island—the River Okeanos is to the earth as the sea is to an island. Thus circular continuity advances from Okeanos to TÓVTOC. inconsistency lies in Penelope's wish: she wants to be carried off to the end of the world (=Okeanos) and go down to Hades to see Odysseus. She is thinking either of the underground sources of Okeanos (or Okeanos as the source of other rivers) or perhaps of Acheronlike appearances of rivers from below ground, since such places were commonly considered to afford descent to the underworld. This is not merely Hades as the land beyond the meipara yalnc found in Od. 4.563 and the Nekyia of Book 11. (In formulaic terms, the phrase must be related to mpoyoÿñc rorauoô, in Od. 11.242, etc., but this does not demean the importance of the transfer.) 44 See R. Mondolfo, El Infinito en el pensamiento de la antigiiedad clásica, transl. F. González Ríos (Buenos Aires 1952) ch. 5. It is wrong to translate &meipiroc here as ‘impossible to traverse’. Odysseus has in fact just crossed this strait. The circularity implied in it is made emphatic by ecredavwrau. To be sure, the phrase could mean merely ‘surrounded by a huge expanse of sea’, and zróvroc in the sense of ‘a path over dangerous terrain’ would not hinder this; cf. W. B. Stanford in his edition of the Odyssey (London 1967) ad loc. But the use of mépr . . . ecrepavwwros seems to me against this; cf. LSJ s.vv. crebavow and epıcredavow, esp. the reference to [Arist.] Mund. 393b17.

An intriguing possibility is afforded by the connection of «reípwv with wévroc. Accustomed as we are because of our world overview to our capability of sailing the open seas without worry regarding the status of our ultimate destination, we often forget that the Greeks historically tried to avoid such voyages and instead preferred to sail along the coastline, occasionally island-hopping as was possible through the Cyclades. Consider, then, a northern voyage around the Aegean: during the winter months when sailing would be prohibited this would properly be an &reipwv mévroc in the sense of that which cannot be circumnavigated. The use of &reipwv in the phrase ‘EAAijcmovroc ametpwr (Il. 24.545) and the uses with yata then are secondary and generalized, and they mean more simply ‘huge, immense’, the transference of meaning which Kahn favors and which I have mentioned above.*® Seligman considers Okeanos to be a highly developed antecedent of Anaximander's &reıpov. He is particularly impressed by the iconographic significance of Okeanos as a source for the development of the metaphysical &reipov, and for this he rightly refers to the Babylonian cuneiform tablet. In addition he mentions the French orientalist Clermont-Ganneau, who posited that an optic mythology has preceded every aural mythology and that a pictorial representation regulated the conceptual, abstract product of mythology. The concrete myth of Okeanos, on this theory, preceded the metaphysical symbol of the &rreıpov.*® We must take account of the astral and temporal qualities of @meipoc. The Greek notion of time was not strictly linear, but circular, stretching infinitely into the past and future with some remote junction. As such it was always connected with the astral phases. Eternity as a philosophic concept first appears in a dialectical analysis of epy and répac, but the source of the temporal concept is the phases of the heavenly bodies and the seasons.*? This is the source of Alcmaeon’s saying in Problemata 916a33-35 (supra p.132; cf. Ph. 264b27) and may also be the origin of Anaximander’s belief, reported in [Plut.] Stromateis 2 (=Diels-Kranz 12 a 10), that generation and 45 It is interesting that the phrase ‘EAAjcrovroc ameipwv attracted Gibbon; see Decline and Fall 11.145 in Bury’s 6th ed. (London 1913). 46 The reference to Ch. Clermont-Ganneau is to his L’Imagerie phénicienne et la mythologie iconologique chez les Grecs (Paris 1880) p.xvii. 47 See Kahn, “Anaximander” 28, and the sources quoted by him there; cf. also Guthrie, op.cit. (supra n.13) 1.351-53.

destruction occur éé ameipou aidvoc avakukAoupevwr mavrwv avTOV, a variation on the Homeric phrase which employs wepıreAAduevoc with the particular time period in question. Indeed, temporal infinity is one of the basic types of infinity which Aristotle allows (Ph. 206a9-b3). Kahn goes so far as to say that the idea of incessant recurrence in the eternal life of nature, as opposed to the &py and répac of mortals, is the origin of the eternal motions of the Milesians and of ‘eternity’ in general.18 One problem in analyzing the early significance of &reipoc is to assess correctly its relationship, very noticeable later, with répac. Certainly Homer calls the earth areipwv but still places reipar« upon it. Yet Homer is far from opposing the two terms in a figura etymologica; they do not occur next to one another there. At the very earliest it may be Anaximander who opposes the two, but we cannot be certain since we have so little of his actual wording and since later information about him and explanations of his doctrine are often expressed in terminology developed after his lifetime. Aristotle’s explanation of &rreupov as an epy in Physics 203b7-8 connects it with réloc and répac; this entire discussion is commonly considered to be directed primarily at Anaximander,*® yet we have no certain grounds for positing that he specifically associated &reıpov and répac. Moreover, regarding the suggestion that he may have argued for the infinitude of his &rreıpov on the basis of its having neither an &py# nor a répac, we are faced with the use of Aristotelian-Peripatetic terminology, where ápxí is the material principle, the substratum, and in the Aristotle passage it carries this significance in addition to its sense of ‘beginning’. It certainly bears this signification in the problematic passages of Simplicius (in Phys. 24.13= Diels-Kranz 12 a 9.5) and Hippolytus (Haer. 1.6.2= Diels-Kranz 12 a 11).5° On the other hand, Anaximander may well have described the &reıpov as atdiov .. . Kat aynpw (Hippol. Haer. 1.6.1= Diels-Kranz 12 B 2) and ad@avarov... Kat avwAebpor (Arist. Ph. 203b13), usages for which there is prior warrant;5 the 48 To Kahn’s references in “Anaximander” 27 (Philo, de Opif.Mund. 13.44 and Arist. Cael. 284a3-13) may be added Arist. Metaph. 1074a37-38. 49 As Solmsen, op.cit. (supra n.13) 109-14. This is also the basis of Kahn’s argument in “Anaximander.” Cf. Sweeney, op.cit. (supra n.2) 74ff, esp. 87-92. 50 The problem of the &py1) is summarized in Kirk and Raven, op.cit. (supra n.13) 104-08. 51 See Kahn, AOGC 43, and Solmsen, op.cit. (supra n.13) 114 n.19. Though Anaximander may have used these adjectives to express privation of yévecic and Odvaroc or ¿0opa, he is unlikely to have expressed his argument in terms of the abstracts yevecıc and ¢8opa¢ them-

amepavrov reported in Aétius (1.3.3= Diels-Kranz 12 A 14) is possibly the phraseology of Anaximander, but far more likely it is applied to him by Aétius (or his source) on the basis of its similarity to the other qualifying phrases, its common use (in prose and poetry) from the fifth century onwards, and particularly on the basis of its use in Aristotle in describing infinity in Physics 204b21 and Metaphysics 1066b33 (&reıpov de TO Arrepvrwc StecrnKoc). That Aristotle connected &rreupoy With mépac as its negative partner admits of little doubt, as we may gather from Physics 203b7-852 and 207a1-15, as well as from his discussion of ro diétodov in Physics 204a3-6. Aristotle’s collocation of dreipov-médpac is an opposition which we encounter first of all in the Pythagoreans. We know for certain that Aristotle was cognizant of the Pythagorean association of these terms since he reproduces it in the Table of Opposites in Metaphysics 986a23- 26 (cf. 990a8-9). It is certainly not until after the Pythagoreans posited the opposition that the association of @reıpov with répac became so important and dominated subsequent thought insofar as éretpov was thenceforth considered the negative of répac.53 Though mention of @reıpov would have been anathema to Parmenides, whose use of répac serves to mark not a limit in time but rather the invariancy of the subject, Melissos dissents (fr.2) from his stand on time (past, present, future) by granting the existence of these states and of &reipov.54 The fragments of a later Pythagorean, Philolaus, show the same opposition (Diels-Kranz 44 B 1, B 2; cf. A 9).55 selves (found in Simpl. in Phys. 24.17ff [= Diels-Kranz 12 8 1] in the context of his quotation of Anaximander; cf. [Plut.] Strom. 2 [=Diels-Kranz 12 a 10], Hippol. Haer. 1.6.1 [= DielsKranz 12 A 11], and Arist. Ph. 203b8), which were well established in Peripatetic terminology but do not belong to early Presocratic vocabulary, at least according to Kirk and Raven, op.cit. (supra n.13) 117f. I find this to be true of #0op& more than of yévecic; yevecıc is found in Homer, though in a somewhat concrete signification, and yevecıc Kai öAedpoc in Parm. fr.8.27 (cf. 8.21) should be noted. As a doublet, however, what Kirk postulates of yevecıc kai ¢Oopaé is true: they are apparently not found together earlier than Plato (cf. LSJ s.vv. yévecic, bBopa). On the adjectives mentioned here, cf. Pl. Ti. 3342 and 3347 (aynpwv kai &vocov) and the parallels adduced by Taylor, op.cit. (supra n.13) ad loc. 52 Kahn, AOGC 233 n.1, assumes on the basis of this passage that Anaximander “probably defined 76 &reıpov by opposition to rrépac.” Though possible, it is improbable, as I hope to have shown; it is rather a collocation Aristotle has taken over from the Pythagoreans and assimilated into his own teaching (the discussion of this point falls next in this article). Cf. Sweeney, op.cit. (supra n.2) 87-92. 58 Thus Solmsen, op.cit. (supra n.13) 116, and Bicknell, op.cit. (supra n.12) 39. 54 Cf. Owen, op.cit. (supra n.21) 97-101. 55 The fragments ascribed to Philolaus may not be his, but may rather have been

Otherwise among the Presocratics &reipoc tends generally to be utilized in several contexts, all of which parallel our own sense of ‘countless’, ‘boundless’, or ‘infinite’. It may have a temporal significance, as in [Plut.] Stromateis 2 and 7 (= Diels-Kranz 12 a 10 and 68 A 39), ¿E ameipov aidvoc (or ypôvou), but probably its most frequent context is that of 70 kara ueyedoc Or ro kara mAïñboc.56 What can we conclude about &zeipoc then? I believe that we may assert that the notion of circularity is indeed inherent in it. That it ever actively in itself had this sense is doubtful on the basis of our present evidence; no incontrovertible example of it can be produced. It is significant, however, that &reipoc is often used in conjunction with words compounded of wepi. Okeanos certainly represents a preGreek, non-Indo-European forerunner of the areipwv rróvroc; Homer presents indications of this. Although only a conjecture, I submit that it may represent the Greek abhorrence of sailing the open seas, especially during the wintry season. Finally, the collocation of &reıpov and répac, based on an etymological association which is seemingly obvious and therefore plausible, dates only from Pythagorean times, after which it has been generally accepted as valid.5? HARVARD UNIVERSITY October, 1974 written by someone dependent upon Aristotle’s account of the Pythagoreans (see Kirk and Raven, op.cit. [supra n.13] 308-11, for a summary of the problem). For my purposes it is of little consequence since I am positing that the doublet, important to the Pythagorean school, caused the two words to be associated thereafter. Nonetheless, &reipoc could afterwards still signify not only ‘boundless’, but a Homeric ‘immense’ as well, as Bicknell, op.cit. (supra n.12) 40, observes. 56 For example, Simpl. in Phys. 22.9 (=Diels-Kranz 13 a 5); Diog.Laert. 9.44; Anaxag. fr.1 (= Diels-Kranz 59 B 1); but above all Zeno (Diels-Kranz 29 8 1, B 3). The reason for its appearance is obvious. Zeno was denying motion (and plurality) by establishing limits within a regression (and progression) in an infinite, geometrical series. Cf. Porphyry’s analysis, supra. 57 I am grateful to Professors G. E. L. Owen, T. Irwin and A. Henrichs, and to Dr Martha C. Nussbaum for help with this essay.