"Apeiros" [Greek] and Circularity Kaplan, Michael Greek, Roman and Byzantine Studies; Summer 1975; 16, 2; ProQuest pg. 125

"A1tEtPOC and Circularity 
Michael Kaplan 

THE ATTEMPT to connect cX'TT'EtpOC 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 cX'TT'ELPOV, 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.1 
Porphyry, whose investigation of a'TT'Etpoc and circularity I shall con­
sider 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.!, 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 demon­
strate here that cX'TT'EtpOC may indeed be related to a notion of circularity 
in itself, but that this is a latent meaning and therefore seldom ex­
pressed with absolute clarity, and that this meaning of a'TT'€LpOC by 
itself was obscured after Pythagorean doctrine spread and gained 
notice. Furthermore, I submit that''oKEavbc, 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 cX'TT'EtpOC which Porphyry presents 
has its origin. 

I want to approach cX'TT'ELPOC first of all by considering its etymology. 
I d · I he h' '" • I n so Olng must stress t e lact t at a'TT'ELpWV, a'TT'EtpECtOC, a'TT'EpEtCWC 
and &'TT'ELPL'TOC are all epic variants of a'TT'ELpOC, which dominates later 
prose usage. Moreover, these epic variants tend to have their own 
restricted formulaic usages, as &'TT'EP€LCWC does, for instance, in the 

1 Ph. 3.4-8 contains a general discussion of a'1Tf:lpoc. 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) Iff. 

125 



126 JI AnEIPOC AND CIRCULARITY 

Homeric phrase a7TEpEtn' a7ToLva.3 Considering the narrow range of 
these individual words and their eventual telescoping into a7TELpoc, we 
may consider their etymologies together and not make serious 
distinctions among them.4 

The root of all these words is Indo-European *per, which had an 
< end-directed' signification. Kahn has well argued that the alpha­
privative in a7T€LpOC negates not the noun 7Tlpac but the verbal root 
*per-, which may be seen in 7TEtpw, 7TEpaw, 7TEpatvw, as well as in 
numerous preverbs, such as 7Tp6, 7Tapa and 7TEpt.5 Schwyzer goes so far 
as to say that "7Tapa und die Nebenform 7Tapat gehoren etymologisch 
zunachst mit 7Tapoc 'froher' zusammen, weiter auch mit WEpt, 7Tlpa, 
wp6, 7Tp6c, USW."6 When one further considers that 7TEpt may appear 
as 7TEP and that 7TEpa is often joined in compounds in the form 7TEP-, it 
is easy to see that confusion could arise between different, developed 
denotations of *per-. Frisk, for example, glosses 7TEpa as "darober 
hinaus, weiter, langer, mehr, jenseits," while he glosses 7TEpt as 
Hringsum, tiberaus, durchaus." Contrast these developments to the 
original *per-, which Schwyzer says meant "im Hinausgehen, Hintiber­
gehen tiber, im Durchdringen."7 

Among modem philologists Schulze was the first to stress the 7TEpl 
aspect of aWELpoc or, more accurately, of a7TEtpL'TOC. He analyzed 
a7TEtpL'TOC as *a-peri-itos and explained the suffix -itos as drawn from 
lEvaL, for which he compared aJLagL'T6c and the Latin orbita; he trans­
lated it as that which 'drcumiri nequit'. He allowed, however, that it 
was possible that it might mean 'transire', with the -peri- equivalent 
to Latin per.s Nevertheless, it has been his first explanation which 
later philologists have accepted. Bechtel agreed with Schulze in his 
identification of -itos with lEvaL, and he gave an equivalent translation 

a In the Iliad 11 tirnes with a1ToLva; once with lSva in both the Iliad and the Odyssey . 
.. Cf. Ch. Kahn, Anaximander and the Origins of Greek Cosmology (New York 1960) 231 

[hereafter KAHN, AOGC]. 
5 Kahn, AOGC 232. Ann L. Bergren, The Poetics of a Formulaic Process: Etymology and Usage 

of 1TEipap in Homer and Archaic Poetry (Diss. Harvard 1973), stresses the *per significance of 
1rEipap as 'goal-oriented'. 

• Ed. Schwyzer, Griechische Grammatik II (Munich 1939-53) 491f; cf. also H. Frisk, Griech­
ische etymologisches WOrter'buch (Heidelberg 1960-72) S.Y. 1TEpl. 

7 Schwyzer, op.cit. (supra n.6) II. 499f: "Diese Bedeutungen kennt auch noch das Griech­
ische; doch ist hier wie irn Indisch-Iranischen 'rings urn, urn' die Hauptsbedeutung 
geworden. Ursprlinglich war von 1rEpi in dieser Bedeutung aMi 'zu beiden Seiten' ver­
schieden; doch verblasste der Unterschied, bes. bei aMi." 

8 w. Schulze, Quaestiones epicae (Giitersloh 1892) 116 n.3. 



MICHAEL KAPLAN 127 

of cmEtpL'TOC as that "urn den man nicht herum gehen kann."9 More 
recently Chantraine has said that it "pourrait ... signifier' dont on ne 
peut faire Ie tour' de a-TT€pt-t'Toc." Thus he too relies upon Schulze's 
comparison with all-cxgL'TCJc and assumes that the base of the word is a 
negated *per(i).l0 Frisk, however, is troubled by the -i- in -itos; 
Schwyzer offers a qualitative interpretation which satisfies neither 
Chantraine nor Frisk.l1 

The significance of aTT€tpoc, 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.12 The Heraclitean concept of circu­
larity and the applicability of aTT€LpOC 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 aTT€lpL'TOC 
and hence aTT€tpoc (from *aTTEpLoc)aTT€LpOC by metathesis, as aTT€p€{cLOC 
= aTT€LpEctOC) with TT€pl. aTT€tpoc, moreover, is often associated with 
TT€pL-compounds, especially TT€PtEXW, in philosophic speculation. 
Aristotle informs us that Anaximander (as is likely, to judge from the 
context) stated that his aTT€LpOV surrounded (TT€ptEX€LV) the world. 
Anaximenes replaced Anaximander's 'TO aTT€LpOV as apx~ with an 
aTT€tpoc a~p; he still allowed it to surround the world.13 Elsewhere 

8 F. Bechtel, Lexilogus ZU Homer (Halle 1914) 49. 
10 P. Chantraine, Dictionnaire etymologique de la langue grecque (Paris 1968-) S.V. c1.7mp€ctOc. 
11 See Frisk, op.cit. (supra n.6) s.v. cY.rmp€ctOc; Schwyzer, op.cit. (supra n.6) I.l06 n.3. 
12 Kahn, AOGC 232f. Porphyry, Quaestionum Homericarum ad I1iadem pertinentium reliquiae 

14.200 (fasc. II pp.189ff ed. H. Schrader, Leipzig 1882), had already hinted at the relative 
quality of cX7mpoc, when he wrote that C7]p.atv€t 8~ TO a77€tpOV Ka~ TO 77€77€paCJ·dvov /.L~ TV 
€avrov .pvw, ~/.Liv 8' c1.77€po..TJ77TOV (regarding Schrader's text, see infra n.30). G. J. M. Bartelink, 
who wrote the articles on a77fdptTOC and a77€{pwv in Lexicon des fruhgriechischen Epos fase. VI 
(Gottingen 1969), stresses that the endlessness is relative to the viewer. See also P. J. Bick­
nell, "TO cX77€tpOV, cX77€tpOC chip and TO 77€pt€XOV," Acta Classica 9 (1966) 39. 

13 Arist. Ph. 203b12; Aetius 1.3.4 (=H. Diels and W. Kranz, Die Fragmenta der Vorso­
kratiker16 [Dublin-Zurich 1972] 13 B 2 [hereafter Diels-Kranz]). Cf. also Arist. Cael. 303b12, 
o 77€pt€X€LV .pad 77aVTaC TOlk ovpavovc cX77EtpOV DV, and Pi. Ti. 31A4, 31A8, and 33B1. On the 
whole question of 77Ep.lXEtv as a reminiscence of Anaximander, see A. E. Taylor, A Commen­
tary on Plato's Timaeus (Oxford 1928) ad 31A4; 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 fruhgriechischen 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 Philoso­
phers (Cambridge 1969) 108-10, argue on the basis of early usage that the spatial sense of 



128 "AllEIPOC AND CIRCULARITY 

Aristotle says that certain philosophers granted the a7TEtpOV the right 
and prerogative of TO 7TaVTa 7TEp"XELV Ka~ TO 7Tav ~V EaVTCjJ lXEW.14 
Aristotle replaced it, however, in his scheme with the oupavoe. In this 
scheme the oupavOc encloses a complete system. It is the function of 
TO 7Tav to surround, not the a7TE£pov, which Aristotle defines as a 
potential but not realized whole; the a7TE£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 a7TE£pov as material cause: ou 

I , \ \ , I :t " " r C'., , , 7TEP£EXEL al\l\a 7TEp£EXETat. U a7TE£pov .•. 7TEptEXETa£ yap we 7J VI\7J EVToe 

Ka~ TO a7TELpov, 7TEp"XEL 8E TO El8oe.15 Aristotle's discussion in these 
sections fairly bristles with 7TEP"XW in its many forms, with active and 
passive forms opposed to one another. He is upbraiding those philoso­
phers who have granted to the a7TELpOV (= material cause) the pre­
rogative of the formal cause, that of defining and outlining the whole, 
in this case, the world. IS 

Aristotle is here changing the a7TEtpOV 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 im­
possible within Aristotelian terminology, since "body is defined as 
that which is limited by a surface."l? To the end of Physics 3.6 he is 
occupied with exposing the fallacies involved in equating a7TELpov and 
TO 7Tav (= oAov). 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. 

a1TE'poc predominates (though not necessarily in the sense of 'infinite'), but they consider it 
uncertain that Anaximander intended precisely this. Certainly, however, Aristotle under­
stood the word as 'infinite', and in his discussion 'qualitatively indeterminate' (i.e:. amor­
phous) 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; if. 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. 

u Ph. 207a19. 
15 ibid. 207a25-bl. Cf also Cael. 312aI2-13, .paJLEv 8~ T~ JLEv 1TEpdxov TOU EZ80vc ElvlU. ~ 8~ 

1TEP'EXOJLEVOV rijc V)"7JC. 
16 Cf the language ofPl. Ti. 3IA4 and 3IA8 in a similar context. Also note LSJ 1TEpLlxw I. Lb. 
11 W. D. Ross, Aristotle: Physics. Revised Text with Introduction and Ccmtmentary (Oxford 

1955) 364. Cf Sweeney, op.cit. (supra n.2) 92, 170f. 



MICHAEL KAPLAN 129 

We obviously have to deal with two senses of the a7TEtpov: in the 
first, that of the earlier physicist-philosophers, it is external and active 
(7TEpdXEL), while in the second view, that of Aristotle, it has become 
an internalized phenomenon and is now a passive factor (7TEp£EXE'TCU). 

At the same time as he is involved in changing the orientation of 
the a1TE£pov, Aristotle alters its definition to keep it in agreement 
with his concept of the infinite as always undefined and incomplete: 

• \ -~ ~\ "I: .\\. ~" "I: • , - " ,. 18 Th OU yap ov P.T}OEV E!:.W. a/\/\ ov aEL TL E!:.W ECTL. TOUTO a7TELpov ECTtV. e 
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 d.7TE£POV surrounded all things, he 
probably assumed that all of space was occupied with his a7Topov, and 
therefore that it was a type of OAoV. This, however, was unacceptable 
to later philosophers.19 In the Timaeus, for instance, Plato speaks of a 
cosmos which contains ~V OAOV EKaC7'OV (of its four constituent elements) 
and which leaves P.EpOC ov8€v OV8EVOC ••• ;gW(}EV, but on the contrary 
is "whole and wholly complete."2o Furthermore, he describes this 
sphere as lK P.ECOV 7Tav7!} 7TpOC 7'aC 7'EAEV7'aC ZCOV cX7TEXOV, which is, more­
over, reminiscent of the explanation generally accepted for Parmen­
ides' <sphere', where 7TEtpa7'a are imposed upon it (fr.8) to serve as 
the confines of an unvarying reality.21 Guthrie notes that the argu­
ment 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 indude."22 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 Parmen­
ides; following the lead of Plato, Aristotle is making a fundamental 
return to a position taken (but with serious objections concerning the 
existence of anything d.7TELpOV) by Parmenides. Yet it is not until 
Aristotle that we can see an explicit definition of the status of the 
d.7TEtPOV, and it is in the course of his definition that he manifestly 
diverges from Parmenides, both because Parmenides absolutely 

18 Ph. 207al. 
111 Solmsen, op.cit. (supra n.13) 120-22. 
20 Ti. 32c5-33A7. 
21 See Taylor, op.cit. (supra n.13) ad 33B4-5, 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) 1I.43ff. 

23 Guthrie, op.cit. (supra n.13) 11.38. 



130 "AIlEIPOC AND CIRCULARITY 

denied the existence of an (}:TrELPOV and also because he did not posit a 
physically existent sphere. 

In attempting to define further the nature of the (}:TrELPOV, Aristotle 
indicates that it is that which is &8LEglT'T]TOV, '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 &8LEglT'T]TOV 

I11TELpoV to Zeno's paradoxes of motion, particularly the first one 
(Ph. 239bU-14), which may be complemented by an infinite regress 
to disallow the possibility of motion entirely or (as here) the possi­
bility of reaching a terminus. Elsewhere in the Physics Aristotle uses 
similar language, once in defining the sense in which something is 
'intraversable' (TO &8VvCtTOV 8LEA8E'iV) and again in his disquisition on 
circular motion at the end of the Physics (8LEABEtV 8£ T~V a1TELpov [sc. 
cpopd.v] &8VVCtTOV).23 Solmsen has opined that the source of the concept 
of an a8LEglT'T]TOV for Anaximander's thought is Hesiod, Theogony 
736ff. 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 a1TELpov, which they find confirmed in Homer. 
It is for this reason that Homer, while he describes both the earth 
and sea as a1TElpwv, nonetheless imposes 1TElpCtTCt upon them-their 
1TElpCtTCt 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 '{2KECtv6c, of course, which provides the 1TElpCtTCt for the earth, 
and, conversely, the earth's shores are the inner 1TEtpCtTCt for the River 
Okeanos. According to Bergren, "the 1TeLpCtTCt yCtt'T]c is the earth's 
physical extremity ... it is the line between opposite elements." It is 
thus coextensive with the 1TEtpCtTCt '{2KECtVO'iO. Bergren maintains that 
the most archaic signification of 1TEtpCtp in Greek is the concrete 
designation of the earth's extremity and that every time 1TElpCtTCt 

23 Ph. 204aI4, 204a4, 265al9-20. Cf also the comment on Zeno in Ph. 233a22, 'T~ p.~ lv8£XIE­
cOw. 'TO: a7TlEtpa 8,&\8IEty. Porphyry recalls Aristotle's terminology in the phrase &8'IE~'-n7'TOV 
lr.7T£lpov (p.192.24 ed. Schrader [supra n.12]). Also, Simp!. in Phys. 470-71 opposes a7TIE'poc to 
8'IE~o81EV'T(k and 8,a7TOplEV'TOC. 

USee Solmsen, op.cit. (supra n.13) 122f, esp. 123 n.58, for the Hesiodic origin of lr.&£~l'"1'TOY. 
Cf Pind. Pyth. 10.63. 



MICHAEL KAPLAN 131 

yaLTJC 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.25 Even as the earth, described as &7Tfdpwv, is delimited at its 
boundaries by Okeanos, so Okeanos, which provides the 7TelpaTa to 
the earth, is itself &'7Tdpwv, since it lies even beyond the earth's im­
measurable 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 TO a7TELpov (and so 
itself becomes a7T€Lpoc) because it encircles (7TEpdXEt) the earth, which 
is itself &7TEtpWV, 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 a7TELpoc : Ka~ yap TOVC SaKTVALOVC 

Ct7TELpOVC AfYOVC£ TOVC JL~ EXOV'TaC ccfEvSovTJv, OTt al€l Tt ;gw ECTt AaJL{3a­

VEtV. To be sure, he goes on to reproach this (colloquiaI1) usage for a 
lack of precision: Ka()' OJLOto'TTJTa JL€V Ttva A€yOV'TEC, ou JL€V'TOt KVPLWc" 

O€, yap 'TOV'TO T€ lmapx€LV Kat JLTJO€7TO'T€ 'TO av'TO AaJL{3aV€c(Jat· €V SE 'TcfJ 
I \ " ., '\ \. " "'/" {;~ '" Hid KVKlIlp ov ytYV€TaL OV'TWC allll atEt TO €'f'€S TJC JLOVOV E'TEPOV. e cone u es 

by associating this with his opposition of 'TO a7TEtpOV and 'TO 7Tav and 
·hhi f'" {;' " ,.,.. l' " , WIt s concept 0 aOL€r;t'TTJ'TOV: a7T€LpOV JL€V ovv €CT£V ov Ka'Ta TO 7TOCOV 

\ Q I "\ Q I " "i:. l' "" "" "I:. A , " lIaf.LfJavovCLV aLEL Tt lIaf.LfJaVELV €CTLV Er;W. OV DE f.L'fJDEV Er;W, TOVT ECTL 

T€A€£OV Kat. OAOV.26 

That a circle may be called a7T€£pOC is evidently a developed geo­
metric concept,27 which the early philosophers seized upon as a con­
venient and intriguing method to express that continuity which 
remains unbroken, temporally or otherwise. Heraclitus, fr.103 

2S Bergren, op.cit. (supra n.5), goes on to say that '1T£{paTa 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 KAHN, 
"Anaximander"]. This idea may well be indebted to medical concepts; see Kahn, "Anaxi­
mander" 25-27, and G. S. Kirk, Heraclitus: The Cosmic Fragments (Cambridge 1954) 113-15. 

Hesychius picks up the geometrical possibilities when he glosses a'1T£'pov as '1TOAO, rX.y£VCTOV 
(confusing what are actually two different words), '1T£p",p£P(C, CTP0rYVAOV, a,eX TO P..qT£ apxIJv 
1L~T£ '1T(pac ~X(LV. Latte, in his ed. ofHesychius (Copenhagen 1953), notes that this explanation 
was borrowed from Diogenianus ofHeraklea, a Greek grammarian of Hadrian's time; this 
shows the continuity and persistence of this explanation. 



132 "AIlE/poe AND CIRCULARITY 

~VVOV (yap) cipX~ Ka2 'TTlpac br2 KVKAOV, quoted by Porphyry in his 
discussion of the circularity implied in eX.7T€LPOC, 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 quanti­
ties; but it very early came to be associated with the idea of con­
tinuous motion, which can only be found on a circle. Thus Aristotle 
echoes this idea more than once, as when he says TOU O€ KVKAcp C6JJLaTOC 
• " 1 <to ., l:. ".. '\ ~ d h h h o aVTOC T07TOC 0 €V 7]p",aTO Kat €tC OV T€I\€VTq., an w en e says t at 
continuous motion is possible only on a circle, since elsewhere OU yap 
CVVa7TT€' -rfj cipxfj TO 7Tlpac.28 In addition Alcmaeon is quoted in the 
Problemata on human mortality as follows: TOVC yap civ8pcimovc cp7]dv 
'A,\ 1 ~ \ ~ ",\ '\ II <t , ~ 1 ",... 1,\ I\KJLa,wv o,a TOVTO a7TOI\I\VClJa,. on ov ovvaVTa, 7"'l}V apX7]v TqJ T€I\€' 
7Tpocaapa,.29 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 eX.'TT'E:'POC evidently 
intrigued Porphyry, for when he was compiling his Quaestiones 
Homericae he devoted several pages to an exegesis of the various senses 
of eX.'TT'€'poc.30 His lemma was Iliad 14.200£: €lfl-' yap dI/JOJLIV7J 'TT'oAvcp6p{1ov 

, I I.n 1 ll'" 1 \ 1 IT' 0 1 I hi 7T€'paTa ya"7]c. ,)4K€aVOV T€. IJ€WV Y€V€Cl.V. Kat fl-'YJTEpa J. 'YJ VV. n s 
subsequent discussion Porphyry indicates various senses of eX.'TT'E:'POC : 
(1) 7j KaTa JLly€Ooc 7j KaTa 'TT'AfjOoc,31 (2) the relativistic eX.'TT'€tPOC 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 

18 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.ll). 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 Yen. 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.) 

31 The division of Ci:Tr€'poc into 1) KaT<l p.Ey€9oc 1) KaT<l 71'Mj8oc is at least as old as Zeno; if. 
Diels-Kranz 29 B 1, B 3 and Simp!' in Phys. 22.9 (=Diels-Kranz 13 A 5). 



MICHAEL KAPLAN 133 

references in poets which associate a7TEtpOC 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 1l7T€LPOC by poets 
in a few instances need not imply that all unlimited objects are con­
sidered round, but rather that the notion of circularity is contained 
in the nouns to which the adjective is attached.33 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 

th ' ., I ". ~ " f} ~" 1\ ear . WCT€ cvvaY€Tat, EL7T€P 1J Y1J 7T€7T€pa 'l.L€V1J P1J €LCa a7TEtpOC 7Ta/uv 

, 'f} \ ~ \ \ \ 'l:. I ,\ l' \ 'f}' A f} " 
€PP1J 1J, IL1J OLa TO IL1J €f;LT1JTOV aVT1JV ELVaL KaTa JL€Y€ OC €Lp1JC a£ a7T€tpOV, 
~ \ ~ \ \ J.. ~ A t \ I ,\ \ A • \ A J..f} 
OLa DE TO C<paLpOEto1J ELVa£ KaL TOLaVT1JV aVT1JV KaTa cX1JILa V7TELII1J<p aL. 

Porphyry likely had Heraclitus, the composer of the Homeric alle­
gories, 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 TO TOV KOCILOV ccpaLpO€LSEC. Later he writes that Il7TELPOV S' Ilv 

• I \ , I Y ~ I , ~ I " " ~ ~l:. I , o KVKIIOC oVOILa<:,OLTO OLKaLWC, €7TEL01J7TEp aIL1Jxavov ECTL OELf;aL 7T€paC €V 

avTCp TL. Lastly he quotes as the <clearest' proof of the spherical world 
the symbol of Achilles' shield.34 

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 (Nauck2); Eur. Or. 25 and fr.941 
(Nauck2). Cornford, Principium Sapientiae (supra n.2) 173, also quotes Empedocles fr.28 , 
where he takes a'7Tdpwv E.paipoc: ICVK>'OT£P1}C 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." Phrone­
sis 10 (1965) 51-53; and Bicknell. op.dt. (supra n.12) 41. Bicknell maintains that TO aTr£'pov 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"). 

34 Heraclitus, Allegories d'Homere, chs. 47-48 (ed. F. Buffiere [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 aTr£lpova yaiav (as well as Hera­
clitus' comments) demonstrate the anteriority of the argument to the Porphyrian state. 



134 II AIlE/poe AND CIRCULARITY 

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 C'TpOyyvAoc, 
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.36 Heidel further asserts that Posidonius is likely to have 
credited Parmenides with positing a spherical world, and he also says 
that Posidonius was <Cat least the proximate source for the statements 
of Aetius (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."37 
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. lOSE £f) 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.3S 

For Heraclitus a1T€tpoc 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 to show the world 
surrounded at its edges by the River Okeanos. The Hesiodic Scutum 
presents a similar picture of Okeanos, flowing round the rim and 

36 Eustathius,: ad II. 7.446: '!crloJl s~ J'n rfiJl :>..ly££ &1T£lpoJla &vr~ 'TOV 1f.1T££POJl, <be EVP£1TlSTJc 
b ~alSp~ 4nJcl. Kcd £'lPTJTat p.& Ka~ aMaxov 1T£Pl. 'Tom-ov' Kal vW S~ PTJ'TloJl In 'T£ ffl£v8£J1 
1TapwPpacac EVp£1TlSTJe &'Tlpp.oJla l.c01T'Tpa £t1T£ 'Ta KUKAO'T£pfj, &rr£p 'TaUT6J1 Ecn 'Tip &TTdpova, 
E1T£~ Kal 'TO 'Tlpp.a Kal. 'TO 1Tlpac 'TO aUTO S.qAOVC£. Kal. In Ka8' • Op.TJpoJl p.~JI &1T£lpwv if JATJ rfi a 
Ecn c.patpo££S~c Ka~ C'Tpcryyt1ATJ' 

36 W. A. Heidel, The Frame of the Ancient Greek Maps (New York 1937) 68-74. 
37 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.2l) 95ff, for a convincing denial that Parmen­
ides 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. 

88 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 loe. 



MICHAEL KAPLAN 135 

enclosing the other scenes of the shield.39 The belief in an outer river 
which surrounds the fringes of the inhabited world is ancient, com­
mon to Mesopotamian legend and Egyptian lore long before it be­
came 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.4o Herodotus shows the extent of 
this belief when he ridicules Homer and the contemporary map­
makers because they represented the circular earth as surrounded by 
the River Okeanos.41 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.42 

Some Homeric usages aid in the attempt to associate Okeanos and 
a7TELpoc in the sense of an unbroken circular river. First of all, there is 
&ifi6ppooe, '(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 ad. 20.65) glosses the word as follows: 'Aifi6p­

pOOC oe 'QKWVOC 0 KVK'\o/ rije yfjc 7TEptVOCTWV Ka~ aifi 7TCX'\W €7T~ TO aVTO 
., " \\" ~ • <:." \ \,J.. 'Y tKVOVfLEVOC KCt.TCt. TO, 7TEptTEIIIIOfLEVWV EVLaVTWV. Ot OE 7TCt.IICt.tOt 'f'pa."ovct 

\., , .f. ' .,. \ , \' , ~ , \ ~ 8 '\ \ 
Kat OVTWC a'f'0ppooc, ° ELC ECt.VTOV avallvwv EV To/ ELIIE'C Ct.t KVKllo/ 7TEpt 

T~V yfjv.43 The thought behind Iliad 18.402f ( ... 7TEP~ oe pooe 'QKEaVOtO I 
39 II. 18.607f; Scut. 314f. 
40 BM no.92687. This tablet is reproduced in Cune~form Texts in the British Museum pt.XXII 

(1906) pI. 48. It is also reproduced in Kahn. AOGC pI. I. 'Okeanos' is probably a non-Indo­
European word; see Frisk. op.cit. (supra n.6) s.v. 'QKw\l6c; Kirk and Raven. op.cit. (supra 
n.13) 14 n.3; and P. Weizsacker in W. H. Roscher, Ausfuhrliches Lexicon d. griechischen u. 
romischen Mythologie III. 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 AntCl24 (1955) 27f. bur see E. Vermeule, Greece in the Bronze Age (Chicago 1964) 18f and 
6Off, for a critical appraisal of our knowledge ofPelasgian. Lastly. the situation of other sea­
related words in Greek (e.g. aAc. Tr£Aayoc, Tr6VTOC; the origin of 8&Aacca 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 Tr6VTOC 
see E. Benveniste in Word 10 (1954) 256f. 

u 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.lO 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 II. 18.399 and Od. 20.65. In the latter passage, it is curious 
that Homer should speak of the Trpoxoalof Okeanos, since it is normally considered an 
unbroken circular stream (and hence without a mouth). The explanation of this apparent 



136 "AilE/POC AND CIRCULARITY 

acppljJ p.opp.vpwv P€EV UC7TETOC) is very similar, where 7TEP~" • P€EV 
provides the idea of circularity and UC7TETOC 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 U7TELpOC as that which cannot be 
circumnavigated. 

Homer never applies U7TUPOC directly to Okeanos; he uses the *aper­
family with 'Yata and 7TOVTOC. From his usage of it with the latter we 
may conclude that he associates it with the notion of annular circu­
larity. For instance, in Odyssey 10. 194f Odysseus climbs a rocky look­
out and observes an island Ti]v 7T€PL 7TOVTOC a7TElp£TOC €CTEcpavwTa£, 
Hwhich the sea encircles in an unbroken ring.".t4 The sense in Hymn. 
Hom. Yen. 120 (7Tal'op.Ev, ap.cp!' S' op.£Aoc a7TElp£TOC €CTEcpavwTo) may well 
be similar, though here a7TElp£TOC could mean 'uncounted'. These 
usages, associated with ap.cf>l or 7TEpt, recall the associations with 
7TEPL€XW, Homer adds two more words to our list, ap.cplaAoc (used five 
times in the Odyssey, always in the phrase ap.cp£aAcp 'J(JaKTJ) and 
7TEplpptJ'TOC (used once in the Odyssey, of Crete). Eustathius connects 
ap.cplaAoc with the line of the Odyssey quoted above when he writes 
,~, "\..... " , I ,~, " T7}V oE KaT aVT7}v V'1]COV, 7TEp£ 7TOVTOC a7TELpLTOC ECTE't'aVWTaL, 'T}'YOVV 

KVKAcp 7TEP£€XE£ WC ap.cplaAov. Thus the idea that Okeanos binds to­
gether 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 , 
7TOVTOC. 

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 under­
ground sources of Okeanos (or Okeanos as the source of other rivers) or perhaps of Acheron­
like 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 
'It£lpcrra yalTJc found in ad. 4.563 and the Nekyia of Book 11. (In formulaic terms, the phrase 
must be related to 'ltpoxovc 'ltOTap.ofJ, in ad. 11.242, etc., but this does not demean the im­
portance of the transfer.) 

" See R. Mondolfo, El Infinite en el pensamiente de fa antigUedad cldsica, transl. F. Gonzalez 
Rios (Buenos Aires 1952) ch. 5. It is wrong to translate a'lt£{p'TOC here as 'impossible to 
traverse'. Odysseus has in fact just crossed this strait. The circularity implied in it is made 
emphatic by (CT£t/>&'vWTa,. To be sure, the phrase could mean merely 'surrounded by a huge 
expanse of sea', and 'ltOVTOCin 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 loco But the use of 
'ltlp' • •. EC'T£t/>&.vClYNU seems to me against this; cf. LSJ S.w. cT£t/>av6w and 'It£p"T£t/>avow, esp. 
the reference to [Arist.] Mund. 393b17. 



MICHAEL KAPLAN 137 

An intriguing possibility is afforded by the connection of a:1TElpwv 
with 1T6vTOC. 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 a,1TelpWV 1T()VTOC in the sense of that which 
cannot be circumnavigated. The use of a,1TEtpwv in the phrase 'EAA7Jc-
1TOVTOC a,1Telpwv (II. 24.545) and the uses with ya:ia: then are secondary 
and generalized, and they mean more simply 'huge, immense', the 
transference of meaning which Kahn favors and which I have men­
tioned above.45 

Seligman considers Okeanos to be a highly developed antecedent 
of Anaximander's a1TEtpov. He is particularly impressed by the 
iconographic significance of Okeanos as a source for the development 
of the metaphysical a'TTHpOV, 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 representa­
tion regulated the conceptual, abstract product of mythology. The 
concrete myth of Okeanos, on this theory, preceded the metaphysical 
symbol of the a'TTEtpov.46 

We must take account of the astral and temporal qualities of 
a'TTEtpOc. 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 a,PX7J and 'TTEpaC, but the source of the temporal concept is the phases 
of the heavenly bodies and the seasons.47 This is the source of Alc­
maeon'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 'EAA~C'710V'TOC a'71£lpwv 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'lmagerie phbticienne et la mythologie 
iconologique ehez les Grees (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. 



138 "AIlE/POC AND CIRCULARITY 

destruction occur £t a:TrElpov alwvoc avaKvKAovfLlvwv 1TavTwv aVTWv, a 
variation on the Homeric phrase which employs 1TEpLTEAA6fLEVOC 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 apx~ and 1Tlpac of mortals, is 
the origin of the eternal motions of the Milesians and of 'eternity' in 
general.4S 

One problem in analyzing the early significance of /X.1TELPOC is to 
assess correctly its relationship, very noticeable later, with 1Tlpac. 
Certainly Homer calls the earth a1TElpwv but still places 1TElpaTa 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 
/X.1TELPOV as an apx~ in Physics 203b7-8 connects it with TlAoc and 1Tlpac; 
this entire discussion is commonly considered to be directed pri­
marily at Anaximander,49 yet we have no certain grounds for positing 
that he specifically associated /X.1TELPOV and 1Tlpac. Moreover, regarding 
the suggestion that he may have argued for the infinitude of his 
a1TELpov on the basis of its having neither an apx~ nor a 1Tlpac, we are 
faced with the use of Aristotelian-Peripatetic terminology, where 
&:Px~ 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).00 On the other hand, Anaximander 
may well have described the a1Topov as &:t8wv ••• Ka~ a'Y~pw (Hippol. 
Haer. 1.6.1= Diels-Kranz 12 B 2) and &:OavaTov •.• Ka, &:VWAEOpOV 
(Arist. Ph. 203b13), usages for which there is prior warrant;51 the 

U To Kahn's references in "Anaximander" 27 (Philo, de Opif.Mund. 13.44 and Arise. 
Cael. 284a3-13) may be added Arise. Metaph. 1074a37-38. 

" 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 ofehe apX11 is summarized in Kirk and Raven, op.cit. (supra n.13) 10~8. 
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 i'&~c,c and 9&'VC%TOC or </>9op&., he is 
unlikely to have expressed his argument in eerms of the abstracts i"V~C'C and </>9op& them-



MICHAEL KAPLAN 139 

(1:.7dpaJlTOJl reported in Aetius (1.3.3= Diels-Kranz 12 A 14) is possibly 
the phraseology of Anaximander, but far more likely it is applied to 
him by Aetius (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 (a71'€tpov o~ TO a71'€paVTWC OL€CT7JKOC). That Aristotle connected 
a71'€£pov with 71'lpac as its negative partner admits of little doubt, as we 
may gather from Physics 203b7-852 and 207al-15, as well as from his 
discussion of TO odtooov in Physics 204a3-6. 

Aristotle's collocation of a71'€tpov-'1Tlpac 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 (ef 990a8-9). It is certainly not until after the Pythagoreans posited 
the opposition that the association of a71'€tpov with 71'lpac became so 
important and dominated subsequent thought insofar as a71'€£pov was 
thenceforth considered the negative of 71'lpac.53 

Though mention of a71'€tpoJl would have been anathema to Parmen­
ides, whose use of 71'EpaC 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 a71'ELpov.54 The fragments of a later Pythagorean, Philolaus, 
show the same opposition (Diels-Kranz 44 B 1, B 2; ef A 9).55 

selves (found in Simpl. in Phys. 24.17ff[ = Diels-Kranz 12 B 1] in the context of his quotation 
of Anaximander; if. [Plut.] Strom. 2 [=Diels-Kranz 12 A 10], Hippol. Haer. 1.6.1 [=Diels­
Kranz 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 q,Oopa more than of Y€VWc; Y€VWc is found in 
Homer, though in a somewhat concrete signification. and y€v£Ctc Ka~ {))o.£Opoc in Parm. 
fr.8.27 (cf 8.21) should be noted. As a doublet, however, what Kirk postulates of y€vmc Ka~ 
q,Oopa is true: they are apparently not found together earlier than Plato (if. LS] S.w. 
Y€VWc, q,Oopa). On the adjectives mentioned here,cf.Pl. Ti.33AZ and 33A7 (&yr]pwv Ka~ ~vocov) 
and the parallels adduced by Taylor, op.cit. (supra n.13) ad loco 

ss Kahn, AOGC 233 n.1, assumes on the basis of this passage that Anaximander "probably 
defined 'nl ~1TOpOV by opposition to 1Tlpac." 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. 

53 Thus Solmsen, op.cit. (supra n.13) 116, and Bicknell, op.cit. (supra n.12) 39. 
1>4 Cf. Owen, op.cit. (supra n.21) 97-101. 
55 The fragments ascribed to Philolaus may not be his, but may rather have been 



140 "AnEIPOC AND CIRCULARITY 

Otherwise among the Presocratics &'7TELPOC 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 signifi­
cance, as in [plut.] Stromateis 2 and 7 (= Diels-Kranz 12 A 10 and 68 A 39), 
Jg Cx7TELPOV cxlwvoc (or xp6vov), but probably its most frequent context 
is that of 'TO KCX'TCt. j.t€YE{}OC or 'TO KCX'TCt. 7TAfj{}Oc.56 

What can we conclude about &'7TELPOC 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 &'7TELPOC is often used in conjunction 
with words compounded of 7TEpt. Okeanos certainly represents a pre­
Greek, non-Indo-European forerunner of the Cx7TEtpWV 7T6v'Toc; 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 &.7TELPOV 
and 7Tlpcxc, 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.57 

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, IZ7TfiLPOC could after­
wards 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 I); but above all Zeno (Diels-Kranz 29 B I, B 3). The reason for its 
appearance is obvious. Zeno was denying motion (and plurality) by establisWng 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.