Two pythagorean philosophemes

ably more than a mere juxtaposition. Eretria joined hands with Chalcis and Euboean Cyme to found the colony of Cumae in Campania. It is therefore not at all unlikely that the same city 149 colonial movement is largel built up out of stray allusions and äraf H3aNnU9OtN,vOi—Ld Aeyopera: discard these, and our chapters on colonisation will shrink visibly. Under these conditions it is surely risky to draw conclusions from the lack of references should have taken a draft of Carystian emigrants to Corcyra. to our Euboean colony, which bore an (ili.) It may seem strange that the early date, had a short life, and was Corcyraeans did not ae the type obliterated by a famous and lasting of the principal metropolis, Eretria, settlement on or near the same site. but that of the accessory, Carystus. ii.) In the case of so economical a But the case is not without parallel. writer as Thucydides it is not enough A standing iced of Rhegium was .to say that he could have worked in an a lion’s head, which is clearly derived allusion somewhere. We must be prefrom Samos (Head, Historia Numorum, d to show at what point such an p. 108-110). Yet the Samians con- insertion wag: requisite, Where could tributed but a small and belated draft Thucydides have usefully mentioned a to the population of Rhegium, which was mainly derived from Chalcis and Messene. The argument from coin-types is perhaps not Euclidean in its compactness, but neither is it a mere parcel of sticks. 4. Professor Halliday lays stress on the fact that Greek writers in general, and Thucydides in icular, only mention the Corinthian, but not the Euboean foundation on Corcyra. Here is a real crux of historical method: how far is it legitimate to use the argumentum ex silentio? On this point it may suffice to make two remarks : (i.) The entire history of the Greek and bisection ad indefinstum. It is well known that the later writers of antiquity explain the connexion asserted by the Pythagoreans between rò äpriov and the azrecpoy by saying that ing with a theorem of bn geometry. We have to en think of a *aodisu have been relevant. In any case, it is as unsafe to argue from the reticences of Thucydides as to presume on the silences of Colonel Bramble. Conclusion—The evidence in favour of a Euboean settlement on Corcyra is, of course, not conclusive; but it is as upon it cannot fairly be quoted as a assical example of making sunshine M. Cary. out of cucumber. terminated segment AB of a straight line as made up of a finite number of ‘ points,’ or * untis with position * (uovádes Gécw éyouoai); between any two adjacent ‘ points’ there is an empty interval, what Aristotle calls a «evöv 6 tds puces ‘the even’ can be bisectad eis drespov Bcopite (Phys. 213b 24); if there were (see on this point Burnet EGPh? 288-9, with the references given there). This looks like nonsense, since after (n— 1) divisions of 2" by 2, you get 2 as a quotient, and 2 can no longer be divided into ‘even’ parts. The n° quotient will be x, and thus the ‘ halving’ will come to a stop. The real meaning becomes clear if we understand that we are deald3oy4a9sinudq32bsiu1geC"aW:Tan0y-Iid6ovPXo6)nTyd flections on Corcyraean ordes, in his Sicilian ‘Apyasodoyla? In none of these passages would such a reference strong as that on which many generally accepted statements about early Greek history are founded, and the proof based not this ‘gap’ the two ‘points’ would be identical. In other words, on the view which is in question, it is not true that ‘between any two points on a straight line there is always a third point.’ It seems also to be assumed, as is only natural, that the intervale ls adjacent ‘points’ are all equal. . Now apply this to the bisection of a ‘terminated segment of the straight line. - 2 Euboean settlement on Corcyra? In the Kepevpaixd of Book I., in his re- TWO PYTHAGOREAN PHILOSOPHEMES. 1. The connexion between TO dpriov 1219 | |gti p|iva91owdng;wugira | - Zd'YOT4VL tsed

ably more than a mere juxtaposition. Eretria joined hands with Chalcis and Euboean Cyme to found the colony of Cumae in Campania. It is therefore not at all unlikely that the same city should have taken a draft of Carystian emigrants to Corcyra. (i.) It may seem strange that the Corcyraeans did not adopt the type of the principal metropolis, Eretria, but that of the accessory, Carystus. But the case is not without parallel. A standing coin-type of Rhegium was a lion’s head, which is clearly derived from Samos (Head, Historia Numorum, p. 108-110). Yet the Samians contributed but a small and belated draft to the population of Rhegium, which was mainly derived from Chalcis and Messene. The argument from coin-types is perhaps not Euclidean in its compactness, but neither is it a mere parcel of sticks. 4. Professor Halliday lays stress on the fact that Greek writers in general, and Thucydides in particular, only mention the Corinthian, but not the Euboean foundation on Corcyra. Here is a real crux of historical method: how far is it legitimate to use the argumentum ex silentio? On this point it may suffice to make two remarks: (i.) The entire history of the Greek colonial movement is largely built up out of stray allusions and äraË Neyopeva: discard these, and our chapters on colonisation will shrink visibly. Under these conditions it is surely risky to draw conclusions from the lack of references to our Euboean colony, which bore an early date, had a short life, and was obliterated by a famous and lasting settlement on or near the same site. (ii.) In the case of so economical a writer as Thucydides it is not enough to say that he could have worked in an allusion somewhere. We must be prepared to show at what point such an insertion was requisite. Where could Thucydides have usefully mentioned a Euboean settlement on Corcyra? In the Kepxupaixd of Book I., in his reflections on Corcyraean ordovs, in his Sicilian ’Apyatodoyia? In none of these passages would such a reference have been relevant. In any case, it is as unsafe to argue from the reticences of Thucydides as to presume on the silences of Colonel Bramble. Conclusion.—The evidence in favour of a Euboean settlement on Corcyra is, of course, not conclusive; but it is as strong as that on which many generally accepted statements about early Greek history are founded, and the proof based upon it cannot fairly be quoted as a classical example of making sunshine out of cucumber. M. Cary. TWO PYTHAGOREAN PHILOSOPHEMES. I. The connexion between +d äpriov and bisection ad indefinitum. It is well known that the later writers of antiquity explain the connexion asserted by the Pythagoreans between rö Äpruov and the àrreipov by saying that ‘the even’ can be bisected eis äreupov (see on this point Burnet EGP? 288-9, with the references given there). This looks like nonsense, since after ( — 1) divisions of 2* by 2, you get 2 as a quotient, and 2 can no longer be divided into ‘even’ parts. Then” quotient will be x, and thus the ‘halving’ will come to a stop. The real meaning becomes clear if we understand that we are dealing with a theorem of Pythagorean geometry. We have to think of a terminated segment AB of a straight line as made up of a finite number of ‘points,’ or ‘ units with position ’ (wovades Géow &yovoar); between any two adjacent ‘ points’ there is an empty interval, what Aristotle calls a kevòv à tas pices Gopi£e (Phys. 213b 24); if there were not this ‘gap’ the two ‘ points’ would be identical. In other words, on the view which is in question, it is not true that ‘between any two points on a straight line there is always a third point.’ It seems also to be assumed, as is only natural, that the intervals between adjacent ‘points’ are all equal. Now apply this to the bisection of a terminated segment of the straight line.

First, let the number of ‘points’ in the segment be odd, thus: 2. The One and the ‘Gnomons.’ Aristot. Phys. 203a 13 mepuridepévov, yap Tov TTK A B C D E yropóvov mepi TO Ev kal xopis, OTE me If you could bisect AE at all, the bisection would fall on C. You would have to ‘split’ the ‘unit’ C, and the ‘splitting of the unit’ is logically impossible (Plato, Rep. 525d 8, oicôa yap Milhaud and Burnet seem to have wholly misunderstood this passage, though the meaning is rightly indicated by Themistius in his paraphrase. As to the words, (a) the ‘gnomons’ meant, are clearly, as both Milhaud and Burnet say, the successive series of odd numbers which have to be ‘ put round’ I to produce the series of squares. [1+3=22, 1+3+5=3%, and generally 1+3+5+... (2n-ı)=n2]; (0) kai TOU Tous rep. Tadra detvods aù ús, éav TLS aùTò TO &v ETIXELEN TO Ayo Teuveıv, karayeNdot Te Kal oùk amodexovrau). Such a line cannot be bisected at all. This is why 76 mepırröv and mepas are associated. But now consider the segment AD— A BEE D which contains an even number of ‘units.’ Here bisection will divide AD at E, in the middle of the ‘empty’ interval BC, and is therefore possible. It is then easy to show that ED can in turn be bisected, because the division will not fall on the ‘unit C,’ the only ‘unit’ in the interval ED, but somewhere between C and D, and that, in like manner, the same process can be repeated endlessly. Every ‘bisection’ after the first will fall within the ‘empty’ interval between C and D, and D itself will never be actually reached. Hence the ‘even’ can, as the commentators say, always be bisected ad indefinitum. This is strictly equivalent to the arithmetical proposition that, if » be any natural integer, the sum of # terms of the series = +,5+ a+ oe + x steadily 2 2? 2 approximates to the value ı as # is taken greater and greater, but never reaches it. Thus the proposition that the ‘even’ can always be bisected in indefinitum is strictly true, when we understand what it means. But since, on the given assumptions, not all segments, but only those with an even number of ‘ points’ can be bisected, we see at once that the Pythagorean conception of the point as povds Oéow éxovoa is actually incompatible with the most elementary constructions of the geometry which the Pythagoreans themselves had created. Gro dei yiryverOat To eldos, Ste Òë € (see EGP 103 and ib. n. 2). xopis means ‘and in the other case,’ e contrario, much as at Aeschylus, Agam. 637 xyopls 4 rıun Oedv, where xwpis is apparently adverbial, and the sense is that telling bad news and rejoicing over good fortune are ‘clear contrary’ duties, which should not be blended; (c) as the emphasis given to mepırideuevov TÔv yvwuóvov by its position shows, the two contrasted procedures are not ‘putting the gnomons round the one’ and ‘dispensing with the one ’—this is the mistake committed by Milhaud and Burnet—but ‘putting the gnomons round the one’ and ‘putting something else round the one’; (d) elöos has its standing meaning when it occurs in connexion with geometry, ‘regular polygon.’ (The rectangles Milhaud and Burnet produce by putting successive even numbers round one another are not ‘regular polygons,’ and are never spoken of as elòn.) Aristotle means, then, that if you take the series of sums, I, 1+3, I+3+5..., you getI,4,9..., the second powers of the successive integers. The ‘pattern’ (elòos) remains the same throughout the series; it is a square. But if you add to r successive sums of even numbers you get the series I, I+2,1+2+4,1+2+4+6..., that is I, 3, 7, 13 . . . Now here the ‘pattern’ changes at every step; 3 isa ‘triangle,’ 7 a ‘regular heptagon,’ and so on. Burnet has misunderstood the words he quotes from [Plutarch] Stromat. at EGPH n. 2. In them trav de aprımv opovws mrepurudegévov can only mean, as the antithesis with the preceding clause shows, mepıridenevov TH

povası, ‘if we put the even numbers round I. The writer understands Aristotle, not as Burnet does, but exactly in the way I have just explained. His statement that the numbers which result from the proceeding are érepowúkeus Kal Ävicou mavres must not be pressed to mean that they are one and all ‘oblongs” (products of two unequal factors), still less must ‘oblong’ be taken in the strict sense in which it means a number of the form » (n +1). Such numbers are necessarily even, and none of them could be produced by the method Aristotle and [Plutarch] are describing. Milhaud and Burnet want to find the ‘oblongs’ in the texts, but they have to mistranslate in order to do so. If we write down the first few terms of the series really meant, 1, 1+2,1+2+4... we get I, 3, 7, 13, 21, 31, 43, 57 -.. Out of these eight terms all are prime except 21 and 57, which are products of two unequal factors, 3x7 and 3x19. It is these products of two unequal factors, both odd, which [Plutarch] means by érepowikeis. At any rate this is how Themistius also understood the passage, and I feel sure he is right (In Phys T. Spengel, Pp. 222, oi de äprıoı à mpoorıdewevon TH yovadı Kata Tous épeË£ñs del Tu kauvov eldos mrotoûot, kal N drapo a mpóeuow eis amreıpov, rphyovor, ira artérrrdrywvov €10’ ôT «ai TÚYor). It is interesting that St. Thomas understood the passage correctly. Cf. Comment. in de Anima I. 7, © st enim unitati addatur binarius qui est primus par, consurgit ternarius, qui est numerus triangularis; quibus si vursus addatur quaternarius, qui est secundus par, consurgit septennarius, qui est septangulae figurae, et sic in infinitum. The true sense must have been lost in quite modern times, since Pacius, in his commentary ad loc., only goes wrong on one point; he supposes that to & means 4, because 4 is the ‘ first square’! But he correctly points out that if you add 6 to 4, and get ro, the ‘pattern changes,’ since 10 isa ‘triangle.’ Thus he seems to have understood that an elôos means a ‘regular polygon,’ not the sort of rectangle some modern interpreters bring into the discussion. A. E. TAYLOR. SOME NOTES ON AESCHYLUS, EUMENIDES. IO. ké\oas Em’ GKTÈS VAUTOPOUS TIS Tlavrdées. This description of Apollo’s journey from Delos to Delphi is connected in the Scholia and by modern editors with the route of the Sacred Way through Attica to Delphi. The particular landing-place, however, is in dispute. Is the allusion, perhaps, to Prasiae? Mr. Seltman (A thens, its History and Coinage, pp- 12, 30) argues that Prasiae, on the east coast of Attica, was the original harbour of Athens, whence the Theoria of the Theseus-ship sailed to Delos. The evidence for the view that it sailed from Prasiae in the fifth century is not conclusive, but the traditional connexion between Delos and Prasiae would support Aeschylus’ assumption of a landing by Apollo ‘on the shores of Pallas,’ and Prasiae lies in the direct line between Delos and Delphi. . 285. xpövos xabarped mavra ynpacKkev omov. An interpolation, as most editors agree, yet surely not ‘spurious,’ but an Aeschylean line quoted by some commentator which has strayed into the text. Cp. Prom. Vinci. 981: am ékdddore rave” 6 ynpac Kav xpövos. 328-9.) mi de TÔ Teduuevo 341-2.) Tode uéXos Lae This is generally translated (sc. écri) ‘ Over the victim thisis the song.” For émi I suggest ému—‘ This is the song appointed for the victim.’ Cp. 543: Towa, yap éméorat, and 393-4 (codd.): Emi dé mor yépas mrakatóv; also Agam. 547: 632. Is not Verrall’s translation of evdpoot, ‘for loyal hearts, supported not merely by the general sense of the passage (788-798) in the Agamemnon to which he refers, but also by Choeph. 109: phéyyou xeovoa Kedva Tolaıv edppoour? Cp. also Eumen. 992. M. E. Hirst. UNIVERSITY OF BIRMINGHAM.