Peras and Apeiron in the Pythagorean Philosophy

| Archiv für/Geschichte der Philosophie 14 vol. 1901, 3 &r-399 | VAE GY Bo live. Tipos et | .w.A. Ilépas and *Aretgov in the Pythagorean Philosophy, of intelligent discussion, it requires to be more carefor purposes. fully defined. “Ancipov dans la philosophie pythagoricienne [W. A. Heidel]. Discute les ¡opinions de Preller plus connues que celles des autres historiens de la philosophie sur la base ou la racine (root) de la doctrine pythagoricienne. Y 385 granted that Without detriment to this contention, it may be the ethical ideas which a philosopher accepts and announces may or may not be in accord with his metaphysics: for, like any other two sets of opinion entertained by one aud the same man, these also may be flagrantly inconsistent. But it is not of such opinions that we speak when we say that ethical ideas determine metaphysical thought. much more deeply rooted. There are other ideas that are While inferences and derivative notions are often diametrically opposed one to another, there is a certain XII. core of life and interest too central to be thus rent by contradictions, Merges and “Anevor for round it revolves the entire circle of immediate feeling and untutored emotion. How famous the [lobayégerns soûmes 705 Hiv in the Pythagorean Philosophy. was even in Platos day is evident from Rep. 600A foll. It is within this inner circle of feeling and emotion that we Von W. A. Heidel, Iowa College, Grinnel l. The question as to the root of Pythagorea nism has occupied scholars for a century, but does not yet seem to have reached a deiinite conclusion. The divergencies of opinion are not such. however. as to make an understanding appear hopeless. Among those who have addressed themselves to this problem we may especially mention Schleiermacher, Ritter, Brandis. Heyder. and vier; aud if in the following discussion account is taken chiefly di the views of the last mentioned, it is both because they are more generally known, aud because they deser ve more consideration. We may begin by granting two propositions rightly insisted upon by Zeller: first, Pythagoreanism owed jis vrigin to an ethicoreligious interest: second. the Pytha worcan philosophy was, in its first intention, addressed primarily to the explanation of physical phenomena, and only secondarily concerned itself with problems of theoretical ethics. We have in these propos itions a statement of what appears to me to be the actual truth, but not in sufficient detail to afford a satisfactory account of the facts. | It is not a new observation in the histor y of philosophy that ethical ideas determine physical and metaphysical thought. There Is much truth contained in this view, but to be rendered available must institute our quest for the ethical ideas which effectively influence speculation. And we shall look for them not so much in the views expressly stated as in the latent presuppositions and preconceptions. „AI things io the universe arrange themselves to each person anew, according to his ruling love. Man is such as his affection and thought are.... As he is, so he sees.“ Jn these words of Emerson we have the suggestion that there are levels of being too deep to be influenced by considerations of self-interest or by superficial currents of thought. On these levels are built the foundations of the character that distinguishes au individual or an age. Here, oftentimes, is to be found the cummon ground of systems of thought which in their superstructure stand very far apart. And it is precisely this common ground that must constitute the scope of our especial study if we would learn to seize the trend of the history of thought in its entirety. We need not now pause to enquire why this should be: suffice it for the present to say that the direction of life defines itself in a given set of interests whose organizing influence upon though: may be roughly compared to the lines of force in a magnetic field along which the stray iron-filings leap into place. Now it is just such a practical ethical postulate that lies at

Ileoas and “4neigov Von W. A. Heidel, Iowa College, Grinnell. The question as to the root of Pythagoreanism has occupied scholars for a century, but does not yet seem to have reached a definite conclusion. The divergencies of opinion are not such, however, as to make an understanding appear hopeless. Among those who have addressed themselves to this problem we may especially mention Schleiermacher, Ritter, Brandis, Heyder, and Zeller; and if in the following discussion account is taken chiefly of the views of the last mentioned, it is both because they are more generally known, and because they deserve more consideration. We may begin by granting two propositions rightly insisted upon by Zeller: first, Pythagoreanism owed its origin to an ethicoreligious interest; second, the Pythagorean philosophy was, in its first intention, addressed primarily to the explanation of physical phenomena, and only secondarily concerned itself with problems of theoretical ethics. We have in these propositions a statement of what appears to me to be the actual truth, but not in sufficient detail to afford a satisfactory account of the facts. It is not a new observation in the history of philosophy that ethical ideas determine physical and metaphysical thought. There is much truth contained in this view, but to be rendered available

for purposes of intelligent discussion, it requires to be more carefully defined. Without detriment to this contention, it may be granted that tho ethical ideas which a philosopher accepts and announces may or may not be in accord with his metaphysics; for, like any other two sets of opinion entertained by one and the same man, these also may be flagrantly inconsistent. But it is not of such opinions that we speak when we say that ethical ideas determine metaphysical thought. There are other ideas that are much more deeply rooted. While inferences and derivative notions are often diametrically opposed one to another, there is a certain core of life and interest too central to be thus rent by contradictions; for round it revolves the entire circle of immediate feeling and untutored emotion. How famous the [luôayópetos tpóras tod Blou was even in Plato's day is evident from Rep. 600 A foll. It is within this inner circle of feeling and emotion that we must institute our quest for the ethical ideas which effectively influence speculation. And we shall look for them not so much in the views expressly stated as in the latent presuppositions and preconceptions. „All things in the universe arrange themselves to each person anew, according to his ruling love. Man is such as his affection and thought are.... As he is, so he sees.“ In these words.of Emerson we have the suggestion that there are levels of being too deep to be influenced by considerations of self-interest or by superficial currents of thought. On these levels are built the foundations of the character that distinguishes an individual or an age. Here, oftentimes, is to be found the common ground of systems of thought which in their superstructure stand very far apart. And it is precisely this common ground that must constitute the scope of our especial study if we would learn to seize the trend of the history of thought in its entirety. We need not now pause to enquire why this should be; suffice it for the present to say that the direction of life defines itself in a given set of interests whose organizing influence upon thought may be roughly compared to the lines of force in a magnetic field along which the stray iron-filings leap into place. Now it is just such a practical ethical postulate that lies at

with them are other conceptions, like that of appovia, which are quite as much esthetic as ethical. A particular phase of this general condition has long engaged the attention of historians of philosophy. Aristotle twice refers to a single instance, that of the counter-earth. Once’) he complains of the Pythagoreans as où mpos td patvópeva tabs Adyous wal tas altiac Entoövres, GMA mpôç tivas Adyous xat Òókas abrwv ta pawógeva mpocdaxovtes xal metpedpevor ouyxoopeiv. Again he blames them for allowing esthetic prepossessions to bias their conclusions"). Diels likewise remarks upon the effect of similar considerations among the Eleatics in shaping their doctrine of the sphericity of the world’). The fact that a conclusion so reached maintained itself for so long a time sufficiently proves that the preconceptions ou which it rested were basicin Greek life and civilization. It may be well at this point to guard against a misconception ' °). It is quite in the nature of things that the minds of the Pythagoreans, however predisposed to apply ethical standards in passing judgment on the world, should turn first to that which we regard as the objective, — the problems of cosmology. Only at a later time would they naturally return, as it were, from the outer world to matters of ethics and politics, making it evident that their speculations in these fields were wholly secondary and derivative. 7) Arist. de Caelo II. 13. 293a 25. 5) Arist. Met. [.5. 986a 3: zat dsa elyov ôpoloyobpeva Berxvvar Ev te vois dpıdpois xal Tais dppoviats zpòs ta tod obpavoo nadn zal pépn xal mpdc thy Any kaxdopysev, tata cuvdyovres épfpportov. Kav ef ri nou 6téhetne, mposeylyovto tod cuvetpouévny räsav ‘abtoic elvar chy rpayparelav. déyw 8° olov, ererdh Te)erov à dexäs elvar Boxel xal näaav nepterdynpévar thy Tav dotÔpdy poatv, wal ta gepdpeva xatd tov odpavdv déxa pev elval paatv, Övrwv 82 évvéa _kóvov av pavepd@y Bid todto Bexdrnv thy dytiyBova motodcıv. %) Diels, Parmenides, p. 56: „Zu diesen Resten der überwundenen Weltanschauung des Eleaten gehört nun auch die wunderliche Vorstellung, sich das gesammte Sein unter dem Bilde einer allseitig wohlgerundeten Kugel vorzustellen. Die Vollkommenbeit seines ‘Eév sollte-sich in dem vollkommensten. Körper abspiegeln, ein Gedanke, der pythagoreischer Grübelei entsprungen, wunderlich lange nachgewirkt hat. Die Kosmologie des Platon wie des Aristoteles ruht darauf.“ 10) This in view of the polemic of. Zeller, ibid., 468, foll.

speaks of the dre:pov in a way to disclose its emotional connotations: &v 1 xal to dretpov mai Th draxtov ai ndoa ws elmeiy duoppla xad’ avtyv. But we need not content ourselves with matters so inconclusive. The fundamental conception of the list of contraries was evidently ethical, intended to explain the physical world. Thus Simplicius'®) says: td oöv debtdv xal dvw zat Éprpoodey dyaBdv exdhovv, td ÖL dprotepèv xal xdtw xal dmodev xaxdv Ekeyov, ws abtds 'Aptstoréhns fatépnoev Ev ti ray [udayopsiors dpeoxóvrwv suvaywyÿ. Aristotle himself'?) declares the same even more unequivocally: xDavedtepov dE aofxacw of Tludayépernr Adyew mept adtod, twOévtes dv tH thy dyadöv austoryia td Ev. It was to have been expected that zépas, which to the Pythagorean mind denotes so essential a characteristic of the good, should be employed typically by Aristotle to represent the entire ouatntyia, just as dyadóv was used above’). Aristotle, in fact, readily perceived the affinity between the Pythagorean nemspaouévov and his péoov (for, as we have just seen, td u£onv népus), and acknowledged it?'): En Tb uèv apaptavery nohhayös gory (TO yap xaxdv tod dreipov, cs ot [lodayóperor etxaZov, td GE ayabdv end nenepaapévon), td 88 xatopihodv uovayüs. Had he desired to display his wit instead of giving his sober interpretation of their meaning, he would have adjusted his statement to their contrast gv >< rAndos rather than to répas X Arzıpnv. From the account of Aristotle **) it appears to be unquestionable that there was some difference between the Pythagoreans in regard to the list of contraries, some setting up only the two pairs, odd and even, and limit and unlimited, while others constituted the table of ten pairs. I agree with Zeller**) in regarding the former 18) de C'aelo, 173a 11. 1) Arist. Eth. Nich A. 4. 1096b 5foll. By Plutarch's day the ethical interpretation was fully established, cf. de Iside 48: où pèv IloBayopexot 81a mAetévwy Gvopdtwy xarnyopodcı tod pèv dyaod To Ev td renepacpévov, x. tA. 30) Arist. Met. 987a 13foll. Aristotle here clearly employs menepacpévoy and dretpov instead of their respective series. 21) Arist. Eth. Nic. B. 5. 1106b 28 foll. 27) Arist. Met. A. 5. 985a 15 foll. 33) Zeller, ibid., 355, notes 1 and 2. Archiv f. Geschichte d. Philosophie. XIV. 3.

W. A. Heidel, as presenting the more original views of the Pythagoreans. It will be noted that in the briefer table the distinction of good and evil is not specifically mentioned; but probably nobody would maintain that it was not present to the thought. We may therefore assume that it is implicit in one or both of the pairs enumerated, and few, I fancy, would hesitate to affirm that, of the two, limit and the unlimited most adequately represent it. In the longer table, which Zeller, perhaps rightly, refers to Philolaus, népas Xametpov heads the list, while dyaBóvXxaxóv stands near the end. It is as manifestly unfair to conclude from the latter fact that ethical ideas were introduced as an afterthought, as to infer from the former, without further evidence, that xépasXdnetpov contained the groundwork of the entire system. No conclusion whatever can be properly drawn from the order in which the contraries are enumerated. Without resorting to means so untrustworthy, we may be able to lend a high degree of probability to the thesis that in a certain sense the contrast zépasXdretpov is the basis of all the contraries. We have seen that the earlier Pythagoreans probably contented themselves with setting up two pairs of contraries, mepttt6v>< dott and zerspasuivovXdrerpov. But what is the relation of these two pairs? The key to this question seems to be given by Aristotle**) when he says: 105 62 doubdpnd stuysta vo TE aot wal th meprrzév. For nothing is more evident to even the casual reader of the documents of Pythagoreanism, than the fact that there are here two streams of interest, the ethico-religious and the mathematicoscientific, which never entirely blend, though thev tend ever more and more to merge into one another. These interests are represented in the original table, consisting of two pairs of contraries, respectively by rerspasusvov‘ Aneısov and xepittov’. got. Now since the ethico-religious interest is manifestly the root of Pythagoreanism we might readily be led to affirm that zexsousugvev'y: Äneipov constitutes the basis of the system and that repuróvx Goztov must somehow be deduced directly from it. But I fear we #) Arist. Met. A. 5 986a 1%.

should merely be yielding to the insidious temptation, always strong upon us, to simplify matters overmuch. Philosophies do not grow in that abstract manner. Man lives and gathers experience and acquires convictions and preconceptions before he proceeds to synthesize his thought. Hence it is likely that neither of these pairs of contraries was derived from the other, but that each represented, as we have said, one of the two paramount and parallel interests of the brotherhood. For all that, though recognized as parallel, the two pairs were not regarded as equally basic, as may be seen from the questionable fragments of Philolaus, where r&pas Xärsıpav has quite crowded out its rival. This parallelism finds expression in the identification®®) of the pairs nepettév<apztov and répas X ämetpov. Just here we meet one of the most interesting problems connected with the Pythagorean doctrine of zépas and dretpov. Aristotle*®), in speaking of the relation of the terms drsıpov and dpttov, endeavors to illustrate it by the case of the gnomon. Unfortunately the illustration throws very little light on the problem, being itself scarcely intelligible?’). But even if Aristotle’s account is rightly interpreted and accepted, it must at once be evident, considering the complicated nature of the mathematical relations assumed, that the explanation is at best a late refinement rather than the original conception **). Now the commentaries here offer a different explanation which has been set aside by modern scholars as of a value %) Arist. Phys. T 4. 203a 10: xat ot p&v tO dmetpov elvar to dptiov. Elsewhere, as, c. g. Met. A. 5. 986a 18, Aristotle subsumes äptıov under dretpov, which is logically the more natural as well as the more in keeping with the historical order pointed out above. 26) Arist. Phys. T 4. 203a 10foll. ; 27) Cf. Zeller, ibid., p. 351, n. 2, and Prantl, Arist. Physik, p. 489; see also Burnet, Early Greek Philosophy, p. 310foll. 28) Cf. Burnet, ibid., p. 311, foll.: „The artificial character of all this shows that it does not belong to the groundwork of the system“. Compare Windelband, Gesch. der alten Phil, p. 173, n. 1: „Die Begründung dieser Identification ist so künstlich, dass man deutlich sieht, sie ist ad hoc gemacht, kein natürliches Produkt der Zahlentheorie.“ It does not follow, however, as Burnet seems to think, that the identification has no siguificance for Pytbagoreanism.

W. A. Heidel, inferior to that of Aristotle’s. Simplicius, Phys. 455, 20D, who gives the fullest account, in commenting on the above mentioned passage, says: obtor d& 76 Gmetpov tov Aprıov dptÔpòv Eheyow Sra To Ray wey aptrov, ws pact of BEnynral, els toa Örarpetodatr, to 68 els ica Ötatpoópevov dnetpov xata thy Styotoptay’ A yap els toa val juicy Öraipeors én’ dnetpov™’). To DE meptrtdv rpootsdev repalver adté xwäAdeı yap adtod thy els ta toa dralpectv. odtws pèv oöv of êEnynral te dpriw th dretpov dvardéast xara thy els ta toa Statpectv, xai Önhovórt oùx in dptOpa@y GAN ent peyedöv hapBdvooor thy Em dnerpov topryy... ws os odd 6 ’AptototéAns patverat thy els toa dtaipeayy altiacdpeves tod dmelpnu. gúnote obv év dog tui to apttov aittdv got ndans Statpécems. The attentive reader will perceive at a glance that we have here presented by the ééyyytai a point of view not quite comprehended by Simplicius. He thinks the reasoning erroneous, and assures us that he finds nothing in Aristotle to countenance it. Finally, he clearly associates it with the conception of the divisibility of space or matter ad infinitum popularly current since the days of Zeno*). Philoponus (Phys. 391, 25) goes farther, quite unconsciously confounding the two notions: 76 uèv yap nepırröy reparni wat óptset, th 68 dpruv tHs Em anetoov Tops attidv èstv, dei chy öryaromdv deyouévnv. It is quite clear that he owes his explanation to the same ééryrtai, although he reproduces their statements more freely, according to his own mistaken understanding of them. I hope to make it appear very probable that this explanation, rejected or disregarded by modern scholars, is the reason originally assigned for the identification of the dpttov and the arsıpov, and that it can be traced back in substance to the time of Aristotle. If it should prove to rest upon a conception of number which Aristotle has elsewhere shown his inability to understand, this > 29) Tregard this clause (4, yap .... Er’ dretpov) as an attempted elucidation added by Simplicius. The reasons for this belief will shurtly appear. 3) Windelband (Tufts) p. 46, n. 2. betrays a lurking doubt as to the correctness of the interpretation given by Simplicius, while it likewise becomes evident that he does not understand the é£myntal: „The reason presented for this, viz. that even numbers permit of bisection to infinity (?), is indeed very questionable and artificial.*

would sufficiently account for his adopting the later and more refined explanation. It would seems plain enough from Aristotele’s own account what the Pythagorean theory of number really was. Let us then glance at a few of the passages*'). Met. A, 5. 985b 25: tac toótwv (sc. tüv padnudtwy) dpyàs av Òvrwv doyas AOnsav elvar navımv ... 986a1: ta av apıdpwv otoryeia thy ovtwy otmyeia ndvtuv Oréhafov elva, xat tov Ghov odpavov apuoviav etvar val apıdunv. ibid. M. 6, 1080b 16: xat of [ludayépernr 8 Eva chy padypacexdy (se. dpıdudv elval pact) hiv où xeywptopévoy (Aristotle had just defined dprdudsy padjuanxdy as Tv Gvtwy xeywprouévoy tüv alobrtiv). M. 8. 1083b 12: tiv dprdudy todtoyv eivar pabyuatixdv, döuvarov gow ... N. 3. 1090a 32: «ara pévioe To moreiv € dordudy ta guard smuata, Ex py sydvtwy Bapos py Se xougdtyta Eyovra xovgdrtyta xal Bapos, &oixası nepl aAAov obpavod Afyeıw xal owpdtwv AAN où tidy elsdrtav. A. 8. 990a 21: dprdudv 8’ addov pnôéva eivan rapid tov apıdudv todtov € ob cuvéstyxev 6 xiouos. Two things become very evident as one reads these quotations: first, Aristotle is carrying his own conceptions into his discussion of the Pythagorean doctrine, as he is wont to do*’); second, he is very much perplexed by the theory he is considering and is endeavoring to reproduce it in part in his own terminology. The Pythagoreans assume to be explaining physical phenomena by physical entities, known as numbers; in his desperation, Aristotle pronounces these numbers mathematical but not abstracted from things of sense, though just what that could mean from his point of view is past comprehension. He then says that it is impossible for these numbers to be mathematical, but, assuming that they must be, he berates these philosophers for attempting to explain ponderable objects by elements not pon- 31) These passages have been very diligently collected by Zeller, ibid., p. 343, foll., although it must be said, in justice to him, that he does not draw the same inferences from them. I may add that I arrived at my interpretation independently of Burnet, ibid., p. 306, foll. Our general agreement I regard as a strong corroboration of the view here set forth. %) Compare Zeller, ibid., p. 381, n. 3,

[lépas in the Pythagorean Philosophy. p aud "Areıpov p £ pay 39 5 which Aristotle is as much perplexed as is Simplicius by the explanation which the ëtryrtai offered for the identification of the äptisy and the areıpov. It is of such numbers as these that the repttzsy and the dprnv are the elements. And Simplicius tells us that the dretpov was identified with the äpriov „because every thing that is even is divided into equal parts, and that which is divided into equal parts is unlimited in respect of bipartition; whereas the odd, when added, limits it, since it prevents its division into equal parts.“ Now, bearing in mind the Pythagorean conception of number, we may illustrate the subject of the odd and the even with the following tigures. a B Ta" Let us first take ten, an even number (A). The process of halving, represented by the arrow, goes on without let or hindrance, there being no limit set to it by a solid unit®). But if we take eleven, an odd number, we find that the unit added sets a limit, preventing the indefinite continuance of the process (B). That this illustration is really more than a plausible guess at the meaning of the 2éy;7472!, which Simplicius did not quite understand, is rendered exceedingly probable by a number of passages to which we may next address ourselves. Plutarch**) in an excerpt preserved in Stobaeus, after reciting the facts relative to the gnomon, adduced by Aristotle in his Physics, proceeds as follows: zat uv sis ôóo Ötatpovusvmv tsa wd uiv nen:39nd povas Ev néow nepleott, nù 38 aovton svt heinzrar yoga zat aidsrows zal avapıduns, ws av Bvdends zal drehnùs ovtos. Again Plutarch says*’): zäv yap zaxbv èn Òuaotdoews zat Sagopas ytvecaee Giev wat thy reits dodudy » thv uèv dprinv dvôex wal den, toy G& meptogdv nAnpy T te zal 35) Compare for the solid unit note 33 above and Arist. Phys. T. 7. 207b 5: altıov 8° Or zò Ev dorus adtalpetov, Ört nep av Ev 7, olov dvOpwrog cic ävOgwros zat ch rollt" 4 6° aptdpds Eorıv Eva rielw zal nia’ drra: doze dvdyen otyvat Erl To döralperwv. 36) ('p. Stobaeus, vol.I (Wachsmuth) p. 21foll.; Diels, Dox. p. 96, foll. 37) Plutarch, de Vita et Poesi Homeri, 145.

W. A. Heidel, téhetov dnégnvev. The same author elsewhere writes as follows ®®: tats yap eis tom topats thy dptÔuôv, 6 wiv dpttos ndvrn ÖLtstagévos Onohetret teva Gextixyy dpyúv olov év Éauté xal ZOpay, Ev BE tw mEpittm twadtd raéva uigov del mepleate ths veunosws yóvuov. I trust that I have made it evident that the explanation of the identification of the pairs népasanstpov and neptttov< aptiov offered by the 2éy;7tat had nothing to do with infinite divisibility of space, as seems to have been supposed by Simplicius and as Philoponus clearly believed”). We do not know certainly who these aéy;772¢ were, but we may readily conjecture that Alexander was one of them. Although his sources were of the best, this fact, even if guaranteed, would not enable us to determine the age of the conception we have been trying to establish. Fortunately we possess a bit of evidence, which, though standing alone, yet suffices to date it back to the time of Aristotle. Aristoxenus‘), in his work on the theory of number, writes as follows: tv 62 dpduay äprısı usv etaw ol els toa Statpodusvnt, reptacot ÖL of els Avon zat péoov Éxovtes ... 6 nentoods zal apyyv val Teleuryv xat wéoov ëxst‘). The mention of odd numbers as péonv Èyovres 38) Plutarch, de E apud Delphos, c. 8, 388A. He is speaking of the identification of the 8%) with the dpztov and the dééev with the reprssév. Compare also Plut., Aetia Rom. 102: zal Gtatpovpévwy eis Tas povddas, 6 piv dotuos, zadarep To YA, ywpav petasd aeviy evdidwor, tod Ge TEPLITOD pöptov del wt rhñpes brodeinerau. If we bear in mind that dev) di)» is one of the ten pairs of contraries enumerated by Aristotle and assigned by Zeller to Philolaus, it seems not improbable that the general conception underlying these illustrations was a part of the groundwork of the theory. Certain it is that adpev>< OF Av had to be associated in some way with xépasX aretpov = reputév x dortov, and this conception is undoubtedly more primitive than the physiological and astrological theories current among the Neo-Pythagoreans. For the latter cp. Sext. Empir. adv. Math. V. 6, foll. 39) Compare note 30 above. 40) Aristoxenus, apud Stobaeus (Wachsmuth) vol. I. p. 20. #1) This clause, 6 mept355s ... Eyet, may be merely a paraphrase of the adjective, téhetos, repeatedly used above; cp. Arist. Poetica, 6, 1450b 24, foll. On the other hand it may be a development of the afore-mentioned view reflected perchance in the enigmatic words of Aristotle, de Caelo I. 1, 268a 10, foll.: teAeuch, yap xal pécov zaì dpyh cov dpubuòv Ever tov tod mavtds.

shows clearly that the view above set forth goes back to Aristoxenus, from whom Aristotle himself may have had much of his own knowledge of Pythagoreanism. I believe that | have now shown that Aristotle did not give the only explanation for the identification of népasy<anetpov with Teptttév Xapttov that could have been known to him. Anybody who carefully considers the matter will agree, I fancy, that he chose the more artificial account instead of a more primitive one, which to him was not readily intelligible. The reasons that would determine his choice were doubtless such as vexed the mind of Simplicius as he grappled with the statements of the eyyytat *). If the above argument holds, then it cannot be strictly true of the oldest Pythagorean doctrine that drsıpov = xevév‘*), although the astpov was later made to account for the xevöv, since 6 uèv äptıns [apıdpös] ... ywnav uetaëd xevhv évdiôwstv. The foregoing account of the original conception of the relation between zépasXanstpov and nepırisvXaptıov is readily seen to be in harmony with the view to which we were led by the consideration of the historical root of Pythagoreanism. The odd and the even are characterized with reference to their perfection or imperfection as determined by the inherence or the absence of the limit. Thus the supreme worth of the limit is vindicated, and the ethico-religious interest comes to dominate the mathematico-scientific. Having thus brought to bear upon my main thesis all the evidence and the considerations that support it, I shall now briefly 4%) Even Burnet, ibid., p. 310, n. 37, seems to share the same views: »The commentators usually say that even numbers were called unlimited because they could be halved indefinitely, which, as Simplicius points out in Phys. p. 455, 20D., is not the case.“ 43) For this view, see Ritter, History of Ancient Philosophy, I. p. 374, and Burnet, ibid., pp. 108, 201, 310. Ritter's use of the above mentioned passage of Philoponus is peculiarly unfortunate, as I have shown that it is based upon a thorough misunderstanding of his authorities. Zeller, ibid., p- 386, foll., criticizes this view somewhat, at length. Compare also Shorey’s review of Patin’s Parmenides im Kampfe gegen Heraklit, in the Amer. Journ, of Philol. vol. XXI. 2, p. 208,

remember what Aristotle himself reports of Eurytus**), one of the earlier Pythagoreans, it would seem reasonable to suppose that their method was far simpler. Aristotle says**): oddév 68 òtwpuozat 006: érotépws ot Apıdunt aitior Tv odgröv zal nö etvar, nôtepoy bs Spot, ofov at atiypat av ueyeD@v, zal bs Eüpurns Étarre zis apttuss tives, olov öl uiv Avdpwrou, 6Ôt 68 Inxov, Gonep ot tovs dptDuobs dvovtes els ta oyúparta Tplywvov xal tetpdywvoy, oötws ADOPOLY rals búbors tas moppäastavpurav. Aristotle here speaks of a method of ‘limits’ represented by points. Eurytus used pebbles in lieu of points to mark the outlines of things, then, forming a rude likeness, took the number of pebbles required as the representative or typical number of the thing. We are told**) that Eurytus lived in Tarentum and was a contemporary of Philolaus. Surely, it is altogether more probable that his method was a survival from the earlier procedure of the Pythagoreans than an invention of a contemporary of Plato. We should then have to think of the early Pythagoreans as employing this method of ‘limits’ **) in accordance with which they represented the line by 2, the number of points or limits necessary to define it, the surface by 3, the body by 4. Therefore they could also use 3 to denote a triangle, and 4 to represent a quadrangle, though each had but two dimensions. One can readily see how, with the advance of mathematics, the procedure in vogue in Plato’s day could come to supplant the earlier and cruder method. 51) On Eurytus see Zeller, ibid., 338, n. 5. 52) Arist. Met. N. 5. 1092b 7, foll. 53) Laert. Diog. VIII. 46. Zeller admits that we cannot rely on later accounts regarding the date of Pythagoreans. Eurytus may bave been considerably older than Philolaus, however, an‘! yet have been the instructor of the Pythagoreans whom Aristoxenus knew. Burnet, ibid., p. 314, says of the method of Eurytus: „This was simply a graphic way of showing how many dimensions a thing had, taking a simple pebble as one dimension.“ I confess that I am utterly unable to understand his meaning. How many dimensions should we have to conceive of Eurytus’ man as having, if he used pebbles to delineate bis form and took „a single pebble as one dimension?“ '4) These ‘limits’ are the öpot of Aristotle's account, not the ‘limits’ technically known as such in modern mathematics.