Two kinds of equality

abs x Closer Er : Mini evrüuk lags TWO KINDS OF EQUALITY by F. D. HARVEY All animals are equal but some animals are more equal than others, George Orwell, Animal Farm. Equality was a fundamental principle of Greek democracy. In this paper ] propose, first, to trace the history uf one form of opposition to that principle, the theory of geometric proportion; and, secondly, to examine some recent speculation on geometric and harmonic proportion*), LARITHMETICAL AND GEOMETRIC PROPORTION: THE HISTORY OF AN IDEA (1) Lhe background Greek democrats believed that all citizens should be cqual!). The claim is casily misunderstood; modern theorists prefer to say that no-one should receive preferential treatment unless there is a relevant justification for it?). But Greek democrats, with their simpler formulation of the principle, were open to the criticism that in their passion for equality they neglected the claims of merit. The idea that an exceptional man deserves exceptional rights is *) Fam greatly indebted to Mr. P. A. Brunt, Professor G. E. 1. Owen and Mr. G. E. M. de Ste. Croix for help and criticism at an early stage; they are not, however, responsible for my views as they now stand. 1 regret that at the time of writing two dissertations which deal with this subject were not available to me: A. Meper Der athenische Demos zur Zeit des peloponnesischen Krieges im Lichte zeitgenössischer Quellen, Diss. Munich, 1938 (p. 37 with n. 112); and G. Grossmann Politische Schlagwörter aus der Zeit des Peloponnesischen Krieges Diss. Basel, 1950 (p. 44 IL). 4) Hor. 3,80,6; Eur. Suppl. 438-413; Isocr. 20 (Lochites) 20; Ps.-Anpoc. 4 (Alcib.) 13, 16, 27; PL. RP 557 A; An. Pol. 1317 B 3-7; etc. 3) eg. S. 1. Been and R. S. Perers Social Principles and the Democratic State (Lone don, 1959), 110, 114.

first found in a military context, in Homer, Achilles in the dad for example, in the Republic Socrates criticizes democracy for loérrsé siva dpolws tous te ai dvicotg Stavépousat). Plato claims no novelty for grumbles that a man gets an cqual share of booty whether he just sits around, or whether he fights his hardest: dy 62 1} tj Mute noxds ii noi eobAdc*), This is a legitimate grumble, for whether a man fights or not is relevant to the amount of booty he deserves. Aristotle, 103 this: it is a well-known complaint, says Adcimantus?). But Plato has in mind a more detailed argument which by this had attached itself Lo the old ideas. The new stream of thought has its source in mathematics, however, cites this line in a political context: of yeplevtec, he says, taaızlouotw nepl Thy tidy, Edy teat’ lev nal “Ev BEIN ru judy xands 482 vat &00%07’.*) In Homer, the zazég was the coward, and the ¿00%ós the brave man; but the words are used, like of yaplevres, to denote social classes. teo is now political power; and the justification is altogether different’). Jt is in the pages of ‘Thucydides that we first encounter this line of reasoning transferred to the political sphere. It is attributed by Athenagoras of Syracuse to his opponents; “Someone will say that democracy is not intelligent or tou, but that those who have money are also those who arc best at ruling”°). ‘he criticism now takes the (ii) The three proportions of Archylas Archytas of Tarentum, a Pythagorean who flourished in the carly fourth century, wrote in his work on music): “There are three proportions in music: (i) the arithmetical, (ii) the geometric, (iii) the subcontrary which they call harmonic. The arithmetical is when there are three terms which stand in tbe following relation to one another in proportion: the first exceeds the second by the same amount as the second exceeds the third (e.g. 6, 4 and 2, where 6 - 4 = 4 - 2). And in this proportion it is the case that paradoxical form that democratic isérns is in fact not tov, But it is not only the critics of democracy who make use of this idea, for it certainly lies behind the well-known passage in Pericles’ funeral speech: péreori dì vata piv tobe vépous meds tx ida Stapopa ritor Tú lou, nave SE thy dilmow, 05 Exaatos Ev tw ebSoxmet, 00% dh uépous tó rhéov ig To xuvà Han’ aperto porter: “but while che law secures cquality to all alike in their private disputes, the claim of excellence is also recognized; and when a citizen is in any way distinguished, he is gencrally preferred to che public service, not in +) PL. RP 558 G. 9) yvebprea Meyers, ib. Cf. n. 195-196. w) Dieıs-Kranız Die Fragmente der Vorsohratiker (Berlin, 1951) (hencelorward DK) 47 Ba: méane Sé èvti thts 7% provai, pix nev dprlgenrind, Sevtépa di à jenperpizz, pére 8 Orevavrla, dv xx £ovre dppovixav, dpOpytuck ps, Onxa Eaves Tpeïc Spor zack Tv solav dnepoydy dvà Aúyow" © rpüros beuräpun brepéyer, tosti Sebtepos teitou bnepeye:. ual dy Taûre <TH Gvaloyle ouurinre fiuev sò sv petévenv Gewv Ätdorqun ctor, ch 88 tv pevivov peitor. è yemperpieà BE, dune teaver olos 6 pates mori thy Seórepav. ral è Seórepos rock tev zpleov. Tubrav S ol uelQoves laov zowüvsen To Barna unì vi uetnus.& 8” y tla, Ev xadoti; , Gna Evil TOUL > à pros öpvs Orepéyer sub Bevrépou AUTADTOU dec, “ét E uéons tod Tplrou imepéyet 700 teltou rotation, but for merit”?). Pericles is claiming, in fact, that the oppouéper. yiverar 8’ dv sabre rà Gvadoyle zu TOY partéven 6hp Ötdornuz peión, 20 dè mó nents of democracy were wrong in their assertion that democracy was perovery jtetov. Passagesin bracketsin the translation are not of course in the text of so obsessed with equality that it ignored all other considerations. Archytas. J take the examples from Diels, Yet these same crilicisms cuntinued to be made by non-democrats; It would be as well to define the quasi-technica) terms used in this theory: avadoyla: propurtion (sce M.L. d’Ooge’s ts. of Nicomachus of Gerasa, Arithn. (New York, 1926), n. On 2,21,1 (p. 264)). Setotgpa: ratio (an anual meaning, but the only one that will make sense ju the 9) Hit 9,319. Line 320, which interprets the “equal share” as death, is usually, and rightly, regarded as an interpolation. 1) A 1-2. Pol, 1267 >) cl. Eur. Hecuba 306 8; Ken. Gyrop. 2,2,18 - 2,3,16; the contexts here are still military, as in the /liad, For a non-military context, cf, Soru. Antig. 519-20. 9) Tu. 6,39,1. Y Tu, 2,37,1, Us. Jowett revised Brant. Archytas passage. LS] s.v. dséatqux 4 cite only this and Ar. Phys. 202 A 18 (where, again, no other sense is possible) for this meaning). ladeng: equality; equality of ratios; proportion plan: mean; proportion geoóras: mean; proportion (this second sense is not given by LSJ; but the wurd is certainly so used, c.g. ps.-Archytas ap. Stob. 4,1,137. See d'Ooge, loc. cit.) ópoc: term of a ratio (but not in this sense in Pl. RP 443 Az sce pp. 142-144) Onepoyh: excess.

the ratio between the larger terms is smaller, and that between the smaller greater (i.e. 6 is 12/, times 4; but 4 is twice 2, a greater ratio). The geometric is when the first stands in the same relation to the constitution, cach man does not get equal rights. But it docs take 104 second as the second to the third (c.g. 8, 4 and 2, where 8 : 4 :: 4 : 2). And of these the greater are in the same ratio as the smaller (i.e. 8 is twice 43 and 4 is twice 2). 105 account of the value of cach number, and of cach man. Here there is perfect equality; all the way up the scale, the ratio remains the same (4 is twice 2; 8 is twice 4; 16 is twice 8; and so on); in political Lerms, what a man receives is always equal to his worth!*). This, it must be confessed, is a scheme of great ingenuity, and even The subcontrary, which we call harmonic, is when they are as elegance. The moral is clear. The equality of which the democrats follows: the first term exceeds the sccond by the same fraction of boast is really only a superficial equality; whereas the so-called itself as the fraction of the third by which the second term exceeds the third (c.g. 6, 4 and 3, where 6 - 4 = 2, i. c. 1/, of 6; and 4-3 = 1, ic. */, of 3). And in this proportion the ratio between the greater terms is greater, and that between the lesser less (i.c. 6 is 1*/, limes 4; but 4 inequality of the non-democrats another fragment concerned with fogismos!?). “Logismos when discovered is 14/4 Limes 3, a lesser ratio)”"). stops stasis”, says Archytas, “and increases homonoza; when it occurs, is in fact a true and satisfying equality. Who was the first man to apply this scheme to politics? I believe that a very strong case can be made out for Archytas, on the basis of there is no péeonexia, but there is isotés; for by this we settle our disputes. By this, then, the poor take from the powerful, and the rich give Lo the (iii) The political application of the proportions of Archytas Unlikely as it might scem, the first two of these proportions were needy, both sides trusting that through this they will gettò tou. IL is a rule and it prevents men from doing wrong: it stops those who know how applied to politics. The arithmetical proportion, it was claimed, was a to calculate before they do wrong, persuading them that they will not paradigm of a democracy; the geometric, of a “better” form of conbe able to escape notice, when they come to it; and it prevents those stitution. ‘he analogy works out as follows. Democrats praise equality; but their equality is arithinctical proportion. This at first sight would appear to be perfect equality, since cach number stands at an equal 3) This is a difficult and occasionally obscure topic, and it is regrettable that the latest discussion of it, by G. R. Morrow Plato's Cretan City (Princeton, 1960), distance from its neighbours (12, 10, 8, 6, 4, 2). So, in a democracy, 524 5, is confused. cach man gets equal rights. But it takes no account of the value of cach proportion “may be the source of the idea that the mean in politics is a mixture number, or of cach man. Here there is flagrant inequality: the higher up the scale you go, the smaller the ratio (4 is twice 2; Gis 11/, times 4; of ils two extremes” (525). He then cites Thucydides’ reference (8,97,2) to the Theramencan constitution as a “measured fusion” to the benefit of the few and the many. But the two ideas are totally distinet. In the theory which we are dis- 8 is 11/, times 6 — and so on); in political terms, the better the man, cussing, cach different kind of proportion symbolizes a different constitution; the the less his worth is rewarded. But now look at geometric proportion. ‘This is not democratic, but (the argument goes) it is much fairer. At first sight, equality would seem to have vanished, for cach number no longer stands at an equal distance from its neighbours (64, 32, 16, 8, 4, 2). So, in this kind of 12) I have translated the passage on harmonic proportion for the sake of completeness; it was never applied to politics by Archytas or any writer in the classical period; but see pp. 123-126 and part 1. The mathematical formulas, he writes (524), “suggest that the mean carries in itself proportionate elements of the two extremes”. ‘The theory of terms stand for individuals, for peuple. ‘Chere is no question whatever of a mean between constitutions, where the terms stand for constitutions, nor of a mixture of constitutions (whatever that may be: see R. Robinson, trs. of and commentary on Awusrorit’s Politics INI and IV (Oxford, 1962), go-1, 93-4, 100-1). It so happens that ps.-Archytas discusses both the proportion idea (Stob. 4,1,137) and the mixed constitution (Stob. 4,1,138); but he docs not assimilate them. 8) DK 47 B 3: oraaw piv Exavacy, dudvoray di aufnaev Aoniogc ebpeOels” nacoveZle ce yap wx dar tobrou vevoévou val lobrac Zativ’ touren yap negli Tv cada ro SoMaccónela. dtd tobtov obv of nevntes Zapfdvuvi mapa civ Suvapivov, of te Robot Sibóvre tots Senutvote, meateduvres Appótepos Sta tubto Tb laov Eeiv. raw Sì zal xadlutip tv adtnodvtev <dav> robo py Entovageévoos Anyiteollar velo dduety Exauac, neloas Ste 0% Suvacoivrat AgMetv, Gray En’ adróv Eevee’ cobs Sè yeh emearayegyous, Ev até orar Köizulvrac, Exa daa.

who do not know from doing wrong, showing by that very fact (ie. (hat they do not know how tu calculate) that they are doing TWO KINDS OF EQUALITY 107 all, if an cducated Greek could swallow the Platonic number!*), then he could easily swallow this. wrong”). The significance of this passage depends, of course, entirely on the meaning given to the word ogismos. Its usual meaning is “calculation” in the most general sense; and that would make some sort of sense here. But can Archytas perhaps be alluding to geometric proportion? This is a very templing suggestion: let us sec what this interpretation would entail’), Let us imagine an Archytan state, in which there is a first-rate man, A; we may give him the value 16. There is also an incapable buffoon, B, value 2!9). They both realise their own worth. (This extra premiss is essential.) So there is no stasis or pleonexia, because B has the rights and dutics and the wealth he deserves, and so has A, this means homoneia, and true equality in the sense already explained. But suppose that B, for some reason, is short of money, and docs not have the amount which, as a 2-man, he deserves, while A is (00 prosperous for a 16-man, The remedy is simple: “both sides trust that through the logismos they will get cò teov”: the rich man gives the required amount to the poor, This is exactly what did happen in Archytas’ Tarentum, as Aristotle informs us'?). The same would also apply to political power and honour. Crime, too, is prevented. Our 2-man, B, thinks that he will rob A; and then it strikes him that it will be obvious that he will, if the enterprise is successful, have more than other 2-men, and that A will have less than other 16-men; everyone will see this, and he will be caught. But C, another 2-man, fails Lo work all this out; he goes ahead and robs A, and his very ignorance of the application of the logismos shows that he is a criminal. Alf this, it might be suggested, is more than a trifle ludicrous. But it is, 1 suggest, what the fragment means. After 1) ‘The last clause is somewhat obscure: I follow the suggestion made in the foutnote in DK (1 438). Perhaps the text is not sound. 1) ‘his passage is discussed more fully on pp. 33-7, where further reasons are given for the suggestion that there is a reference to geometrie proportion. J. S. MORRISON CAL ms. viii (1958), 213 If., implies that fogismos is harmonic proportion; but this was not applied to politics until late antiquity: see part II. 16) If this seems very strange, cl. PL. Plt, 257 A -B; sce p. 107. 1) Ar. Pol. 1320 B 9 1(.; sec p. 199 140. (iv) Plato From Archytas, this picture of the state was passed on, directly or indirectly, to Plato. There is no clear evidence!) that the two philosophers met before its first occurrence in Plato, in the Gorgias®) ; but certainly in that dialogue it is said to be a belicf of the cogot, in other words, of the Pythagorcans*). “It seems to have escaped you”, says Socrates tu Callicles, “that geometric proportion has immense power among both gods and men; you think that you should cultivate pleonexia instead?2): this must be duc to your ignorance of gcometry”*) We have already noticed the occurrence of the idea in the Republic”). We see how geometric proportion was applied to the individual at the opening of Plato’s Politieus®), a passage which is a striking confirmation of our picture of the Archytan state. Socrates: “Theodorus, I am really very much indebted to you for my introduction to ‘l'heactetus and to our guest from Elea”. ‘Theodorus: “Good, but you are likely to be three times as much in my debt, Socrates, when they have done their task and defined the Statesman and the Philosopher as well as the Sophist for you”. Socr.: “Three times as much? Really, my dear ‘Theodorus, must it go on record that we heard our greatest mathematician say that?” Uhco.: “What do you mean, Socrates?” Socr.: “Are we to say that we heard you reckoning all these three as of equal value when their real values differ to an extent that defies all your geome- ) Pr. RP 546 A ff. 19) See n. 190. 2) Pr. Gorg. 508 A, My rejection of Morrison’s arguments (part 11} removes objections to dating the Gorgias alter 388; sce E, R. Dodds’ edition of the Gorgias (Oxford, 1959) 26-7, esp. n. 3 to p. 26. 22) Dons, Gorgius, 337-8. n) We have seen (pp. 104-105) how Archytan theory dealt with pleonexia, ) Dodds note on this passage (Gorgias, 939-40) is the best treatment of this pe il mt rae pane sia to me. But I do not believe that Republic: see sce pp. p . 142-144. 142-414 «public: ren E : proportion is to be found in the ur 25) Pr. Plt, 257 AB. ‘The translation is that of J. B. Skesse slighdy changed. Skemp has a good note ou the passage (p. ly (London chest

god of rei trical expressions of proportion?” ‘T'hco.: “By Ainmon, blun srl well said, Socrates, and a fair hit! Your dropping on myremem berec calculation (logismoi) like this shows that you have really évexo: most people would not put up with it; it is to be judiciously 108 . . hematics!” ned to the idea, and set it out ing. ar‘ the end of his life, Platouslyretur he had only alluded to it in passin full in the Laws®*), for previo na “For servants and masters never can be friends”, says the Athe pe ce decla are they se becau y merel stranger; “nor can good and bad, h ER e becom s cqual have equal privileges. For to unequals in they are not harmonized by measure; and both by reason ° aqua o and by reason of inequality, cities are filled with seditions?®). The truc; happy and alsovor ®), isis happy friendshship”**), i makes; friend zur; g, (hat “equality sayin is ty but there is obscurity and confusion as to what sort of equali trie geome and l metica meant. For there are two equalities (i.c. aritlu in ma u proportion), which are called by the same name, but areen may Pe many ways almost the opposite of one another; one of the introduced without difficulty by any state or any legislator in r, distribution of honours; namely, that of measure, weight,ty,andolnumbe r bette a equali which he ensures by the lot. But there is another judge the is This and higher kind, which is not so easily recognized. is er, howev little, ment of Zeus; among men it avails but little39); that ”). I or it Brees the source of the greatest good to individuals and states to the to the greater more and to the inferior less, and in proportion r nature of each; and above all, greater honour always to the greate w their re virtue, and to the less less; and to either in proportion is spective measure of virlue and education. And this is penne, and ever the true principle of states, at which we ought to aim ‘ Despite this extravagant praise, however, Plato thinks that geod, 2) Pr. Laws 756 8-757 E. The translation, which is that of Jowett* (Oxfor 1953), IV 324, covers only 757 A-C. | 7446 37-395 do ni el (ll DK 478 3, lineïa 7;(loc.Ar.cit.),Pol.and1301theA saying s of Solon quote 13) cf. RP 658 C. 29) Pasti; ch. Archytas’ homono n d experiment of Archytastheal Tan is thinking of the isolate ay Perhaps Platoy ff; ans Atheni that ing admitt is he s perhap or ; sce p.139) (Ar Pol 1320 b did pay some attention to this principle (ln. 2,37,1, quoted on p. 102). 32) of. Gorg. 508 A. 109 metric proportion is not to be used undiluted, duoxoAlas cv rol mixed with ordinary equality?2). Why did he call it Avs xptous? The answer surely lics in the contrast with the democratic equality of the lot. This is &,Ogarwv xpiots, a choice involving man’s idea of cquality, where cach man is to be regarded as having the same value as the next. Atos zplors, on the other hand, is the God's-cye view: it is the choice which Zeus would make if he were appointing people to office; and he, of course, would consider candidates on their merits. There is also another passage in the Laws which shows the practical effects of geometric proportion, and with its aid we can sce whether our account of the Archytas fragment is plausible, “It would be well that every man should come to the colony having all things equal; but seeing that this is not possible, and one man will have greater possessions than another, for many reasons and in particular to preserve cquality in special crises of the state, qualifications of property must be unequal, in order that offices and contributions and distributions may be proportioned to the value of cach person’s wealth, and not solely to the virtue of his ancestors or himself, nor yet to the strength and beauty of his person, but also tu the measure of his wealth or poverty; and so by a law of inequality, which will also be one of proportion, he will receive honours and offices as equally as possible (65 laatrara ra dvicw cuuuérpe Sè), and there will be no quarrels and disputes. To which end there should be four different standards appointed according to the amount of property: there should be a first and a second and a third and a fourth class... »33) Here, then, arc four property-classes, determined on the principle of geometric proportion. Perhaps geometric proportion also lies behind the three classes of the Republic). 32) Laws 757 li. 02) Pr. Laws 744 BC. The translation is again (hat of Jowett, IV 313. 91) Wealth, of course, is arranged on different principles in the Republic; but political rights are distributed in accordance with the general requirements of geometrie proportion, Was this intentional? Plato says nothing about it; but it would square nicely with the immense importance given to geometric proportion in the Gorgias (508) A), and would provide a clear link between that work and the Laws, As for harmonie proportion in the Republic, itisa mirage: sce pp. 142-144. Ghd M.

Plato’s high opinion of geometric proportion (iséras) may provide the answer to two related problems. “Pamphila*5) in the 25th book of her Memorabilia says that when the Arcadians and Thebans were founding Megalopolis, they invited Plato to be their legislator; but when he found that they were not willing to have equality, he refused to go”). Plato was no egalitarian; and as Morrow points out?”), this must be an allusion to geometric proportion — that is, if there is any value in the story at all. Secondly there is a reference in Plato’s seventh Letter to the Sxatos rai lodvopog modreta%). The same question ariscs: why is Plato praising equality? And the answer is surely the same: he is thinking of geometric proportion??). 115 would not prefer to share in a form of government under which his own worth shall not pass unnoticed, rather than be lost in the hurlyburly of the mob and not be recognized for what he is”)? It is often suggested") that there is an allusion to the doctrine of the two equalilies here. This is perhaps misleading. We might divide that doctrine into three parts: (i) the assertion that democratic equality is not the best type of equality; (ii) the assertion that geometric proportion is true equality; and (iii) the demonstration of these propositions by geometric “proof”. Of these, not one is found in this passage: what Isocrates says is that equality itself is wrong - and this is not the same as (i) — and that a monarchy where every man is honoured according to his worth is an ideal state of affairs. No attempt is made to claim that this too is a kind of equality, and a better kind (ii), or to “prove” (v) fsocrates “] imagine”, wrote Isocrates, “that we all believe that it is altogether monstrous that the good and the bad should be thought worthy of the same privileges, and that it is of the very essence of justice that distinctions should be made between them, and that those who arc unlike should not be treated alike but should fare and be rewarded in each case according to their deserts. Now oligarchies and democracies seck equality for those who share in the administration of them; and the doctrine is in high favour in those governments that one man should not have the power to get more than another ~ a principle which works in the interest of the worthless! Monarchies, on the other hand, make the highest award to the best man, the next highest to the next best, and in the same proportion to the third and the fourth and so on. ¿ven if this practice does not obtain everywhere, such at Icast is the intention of the polity. And, mark you, monarchics more than other governments keep an appraising cye upon the characters and actions of men, as everyone will admit. Who, then, that is of sound mind it geometrically (ii). We might surmise that Isocrates knows the theory, but thinks it too complicated to put forward in a general work. (As we have seen, even Plato never gives the full mathematical working-out, although the Georgias and the Politicus passages show that he is familiar with it). Isocrates’ “philosophy” was, after all, one of practical results, not of theory. There is little difference between his remarks and the general assertions made in the previous century"), What makes it clear that he knows the new theory is the fact that he borrows its language: wh Tous dvopotous mov éuolos tbyy&vew!) is not far from Plato’s “distributing a kind of equality to equals and uncquals alike”). The theory, never wholly convincing, is particularly ill-adapted to the absolute monarchy advocated in the Nicocles, “one of the strangest documents produced under the Athenian democracy by anyone who claimed to be a leader of political opinion”), Isocrates himself seems 4) Isocr, 3 (Mic). 14-16, The translation is that of G, Norlin, Loeb Isecrates 185. 39) Ffareit during the reign of Nero. 36) Poy. LAERT. 3,23. 3) G. R. Morrow, Plato's Cretan City, 524 n. 10. a) Ep. 7, 326 D. #) For the sake of completeness it should be added that two means also vecur in the Timaens, this time the arithmetical and the harmonic (7 tm. 36 A (ef. Ar. fr. 47 Rose); cf. Taylor ad loc.). But here the context is cosmological, not political. 10) Eg. by Norlin in his note on Isocr. 7 (4reop,) 21, Locb Isocrates 1, 116, note b, 12) Sec pp. 101-102. 4) Isocr, 3 (Nic.) 14. 4) PL. RP 558 C. 45) Norman Baynes “lsucrates", in Brzantine Studies and other Essays (London,

embarrassed by the fact that what he says so glibly is clearly and demonstrably false: “Monarchics make the highest award to the best man, the next highest to the next best... and u on. Even if this practice does not obtain everywhere, such at Jenst is the intention of the polity”. Or in other words, “What } have just said is nol truc, bul never mind, monarchics mean well.” Not only is the idea a borrowed one; it is ineptly borrowed, and its quasi-intellectual backbone 3s | removed for popular consumption’). "The theory also makes a brief but less shadowy appearance in the Areopagiticus. Here the two equalities arc explicitly menBuneil; the “proof” is as usual suppressed, or taken for granted. But what contributed most to their good government of the state (i.c. the carly Athenian state) was that of the two recognized kinds of equality - that which makes the same award to all alike and that which gives to cach man his due — they did not fail to grasp which was the more serviceable; but, rejecting as unjust that which holds that the good and the bad are worthy of the same honours, and preferring rather that which rewards and punishes every man according to his deserts, they governed the city on Uhis principle, not filling the offices by lot from all the citizens, but selecting the best and the ablest for cach sed . function of the state”47). Two points are worth noticing here. First, the theory is retrojecte into the remote past*). Secondly, arithmetical equality dar specifically with the lot, an association made also in Pair Laws }; 113 voting, which entails the counting of heads, it is, of course, the most obvious example of the practical application of arithmetical equality. The Areopagiticus is generally dated between 360 and 354%); the Laws presumably appeared soon after 347, the ycar of Plato’s death. Yet I find it hard to believe that the new detail is to be attributed to Isocrates - not on the grounds that Isocrates was incapable of an original thought, but because when a writer takes over a theory and states it briefly and generally, it is not likely that he will make a small but important innovation of this nature. This observation, if correct, might provide support for those who believe in an carlier version of the Laws used by Aristotle), Alternatively, if the Laws is simply a record of what Plato had been urging in lectures and conversation for many years, Isocrates may perhaps have heard of it in that way. A third possibility, of course, is a common source, now lost‘), (vi) Aristotle It is in Aristotle’s Politics that the theory is discussed at greatest length. It is much to Aristotle’s credit that he saw some of its inadequacies, but, in my view, he did not go nearly far enough: he failed to sec the force of his own criticisms. It is perhaps best to take cach passage as il occurs in the Politics. “The first thing to note is what marks men give of oligarchy and but not in any earlier occurrence of the theory. The idea is growing, democracy, and what are oligarchic and democratic justice. introduced the lot into this context. Together with (he practice of Way, and they do not state the whole of the absolutely just. For cxample, justice is thought to be equality; and so it is, but for equals, 4) The same idea is said to be expressed in an carlier work, To Micucles (a), * ; not for everybody, Inequality is also thought tu be just; and so it is, and new concrete ideas are being added. It is difficult to say who first All men grasp justice to some extent; but they only go part of the “Ht js monstrous that the worse should rule the better, and pas the ne but for unequals, not for everybody. ‘They omit the “for whom” and is the whole point of the nern forimi — ha It is merely a comu 357, W. Jarcer, 14.5.C.P. Suppl. Vol. 1 (Athenian Studies presented to W. S, Fershould give orders to men of greater wisdom”. This tou is cited as wi; en pa the two equalities by Norlin in his note (Loeb Isocrates II, on pore de È. niet vague in the extreme — more so than the Nicocles: the very wor cqua - 4 h br SA A0 (Areop.) 21-423 the translation is again that of Narlin, Loch Isoratic gr +, akin to those of the crates, II 117. in) ef. pp. 120 123. 4) Pr. Lattes 757 B 5; p. 108. Old Oligarch. . $) Usually 356 or 355; carly 354, Mathicu, Bude Isocrates 11}, 56; perhaps late guson), 439 (sce pp. 09-39). *t) G. R. Morrow Aristotle's Comments on Plato's Lares, in Avistotle and Plato in the Mid-fourth Century, cd I. Düring and G. E. L. Owen (Göteborg, 1960), 145-62, esp. (60-2, Morrow docs not attempt to date this earlier version. 5) Archytas, even?

judge badly. That is because they are judging about themselves. Most men arc bad judges in their own cause. | o Justice is relative to persons, and requires the saine ratio fui the persons as for the things, as was said carlier in the Ethics. ven wom the cquality of the things but dispute that of the Er : ris n mainly because, as just mentioned, they Judge badly in their ow E affairs, but also because each side is really saying are Moins about justice and hence thinks it is saying the whole u ver sa as, party, bcing unequal in some respect, as posesion, in Ì net wholly unequal. The other party, being equal in some respect, è eather ke freedom, thinks itself wholly equal”®®). Here Aristotle views the theory as a theory about justice, rat ict than about the narrower issue of constitutional rights; in this he is at one with Archytas®'), But constitutional rights are of course paco “Justice is thought to be cquality”: this is the ene parti o view, as Aristotle says later®*), “And so it is”, he adds, “but [os e : not for everybody”. ‘There may be a certain amount of truth bc rine what Aristotle is trying to say; but his terminology is highly renee Justice involves equality: that is, everybody is to be ir cated a “+ unless good grounds can be adduced in a particular <a ue n ss justify an inequality - a privilege or a curtailment of rights. sit, of course, imply that some men are naturally better than ot 7 on the contrary, it denies it; but that view is inextricably boune ap with Aristotle’s terminology, “Inequality is also thought to Re Just”: this is the oligarchic point of vicw, as Aristotle says laterTM). And a perhaps we should say “differences ”, are taken into account in justice; but only those relevant to the particular matter in hand. Aristotle docs however realise the uscles sness of claims to equality and inequality which are made in sort of grading of human beings. “Equality (or inequality) o Ml persons” is distressingly off the point: for this suggests that the n di a question is an empirical matter, whercas it Is a decision about ca dure which rests on a moral belief about human rights. And this flaw 53) Ar, Pol. 1280 A 7 25, Ws. R. Robinson (Oxford, 1962), 28. RE) 5 A 29; 1317 B 2 10; 1318 À 3-8, LT These . rn e a, Teor passages areà quoted ul on pp. 117 and 119. 56) Ar. Pol. 1301 A gt. the air: the important question, as he sees it, is to discover what kinds of inequa lity (or differences) entitle a man to what kinds of inequality (or differences) Aristotle’s answer is that it is those who “contr of treatment. ibute most to political society” (door avußdidovrar mÀziatov ele thy totabryy (i.c. rohrixhv) xotvwyiav) who are entitled to a larger share in the city®?). But this reply suffers from two vaguenesse s. (i) It is not at first clear which qualitics are to count as contributing to political society; so that Aristotle’s answer would not settle the rival claims of, for instance, a rich man and a wise man and a brave man (not that such men can be su neatly distinguished execpt in theory). This is a question which Aristotle later attempts to answer), (ii) It is not clear how this entitlement to a larger share in the city is to work out in practi ce: what are these privileges? 11, as seems likcly, they are political offices, then on what principles are they to be measured vut, divided, and shared round ? Aristotle scems to be tentatively feclin g his way towards formulating the principle of relevance; but he does not reach it in this discussion. A Jater account which he gives, however, does go as far as to point out ications which were alleged in the inadinissibility of some of the qualif his day to be relevant to political it is, but for unequals, not for everybody”, Here again Di in complain that to speak of “uncquals” is to mislead ; for it imp ics in 115 is fundamental to Aristotle’s reasoning. Certainly, inequalities, or *) ib. 1281 A 4 ff. 28) pp. 115-116, In the course of an BINSON (Aristotle's “Politics”, books distinctions. interesting discussion of this passage, HH and I V, 31-3) answers this “In 1280 B 5 political goodness looks R. Roquestion as follows: like good character and good behavi our towards the city and its laws and citizens . In 1283 A 7 it looks like the power to produce good character and good behaviour in the other citizens, The latter is probably Aristotle’s idea: political goodness is being good at making men good by ruling them” (p. 31). 1f this interpr etation is right, then this is the qualit y which is lo count as “contributing to politic al society”; but we still do not know how it is to be measured or assessed; and my second objection (p. 13) still holds gootl. And there is the further objection that one cannot avoid having doubts about men who are “good at making men guod by ruling them”: what sort of people would they be? Aristotle's fundamental political principle is involved, and this is misguided (Robinson, op. cit. xvii-xx viii, esp. xxii-xxv},

“All men hold that justice is some kind of equality; and up to a certain point they agree with what bas been determined in our philosophical discussions on ethical matters. That is, they say that justice is a certain distribution to certain persons, and must be equal for equals. What we have to discover is equality and inequality of what sort of persons. ‘That is difficult, and calls for political philosophy. Perhaps someone would say that superiority in any good whatever deserves inequality in the assignment of offices, provided that in all other respects the men do not differ but are alike, on the ground that whoever differs has a different right and worth, But if this is true, complexion and height and any good whatever entitle those who excel in it to an excess of political rights. But is this not obviously false”5%)? Here then Aristode uses the relevance principle; but it should have been used in the previous argument 100%), He even gocs some way towards formulating it: “What we have to discover is equality and inequality of what sort of persons”. Aristotle answers his question with an example from flute-playing®) which is very much to the point; but his conclusion is disappointing: “It is clear that in politics also it is reasonable not to claim office on the ground of any and every inequality. ... The claim must be based on a difference in something that helps to constitute a city. Hence it is reasonable for the noble and free and rich to claim the honour, because the citizens must be free and have taxable property; TWO KINDS OF EQUALITY 117 abandoning it half way. What relevance has wealth? ‘The rich pay taxes, says Aristotle; but why is this a good reason for giving them more political power than, say, a poor sailor, who can also claim to serve the state by fighting for it? Aristotle has grasped the point that diflerences between men, and inequalities in the state, must both be taken into consideration; he realises that these must be related to each other in a reasonable and relevant way. Why then this blindness, this lack of rigour in Aristotle’s argument? Because, no doubt, his mind was so accustomed to thinking of birth and wealth as entitling a man to power that, despite his own arguments, he did not stop to question them. In many Greek states this was an accepted fact of political life. Aristolle’s practical conclusions were at variance with his abstract arguments; if we want to pass a harsher judgement, we might say that his prejudice obscured the force of his own logic. Little remains to be said about Aristotle’s further remarks on the attitudes of democrats and oligarchs towards justice; but the relevant passages should be quoted. “We must first assume the starting point, that many forms of constitution have come into existence with everybody agrecing as to what is Just, that is proportionate equality, but failing to attain it... ‘Thus democracy arose from men’s thinking that if they are equal in any respect they are equal absolutely (for they suppose that because they are all alike free they are equal absolutely), oligarchy arose from a city could not consist entirely of needy persons, any morc than of slaves. And, if those attributes are necessary, evidently justice and their assuming that if they are uncqualas regards some one thing they are go on. Or, rather, without the former, it cannot exist, and without to participate in all things in cqual shares, while the oligarchs as political goodness are necessary too. Without these, also, a city cannot these it cannot go on well”#?). ‘This conclusion contradicts Aristolle’s previous arguments. If + here means political power, as it clearly docs, what relevance has good birth? None whatever, as Aristotle would have seen if he had followed his reasoning to the conclusion which it demands, instead of uncqual wholly (for being unequal in property they assume that they are uncqual absolutely); and then the democrats claim as being equal being uncqual seek to have a larger share, for a larger share is uncqual. All these forms of constitution then have some clement of justice, but from an absolute point of view they are erroncous; and owing to this cause, when each of the two parties has not got the share in the constitution which accords with the fundamental assumption that they happen to entertain, class war ensues, And of all men those 29) Ar. Pal, 1282 B 18 30, ds. R. Robinson (p. 41). 5) ib. 1280 À 7 25.. 61) ib, 1282 B 3o- 1283 À y. 2) ib. 1283 À 10-22, us. R. Robinson (p. 42). who excel in virtue would most justifiably stir up faction, though they are least given to doing so; for they alone can with the fullest reason be deemed absolutely unequal. And there are some men who

119 being superior in birth claim uncqual rights because of this inbirth and virtue are found in few men, but the qualifications specified cquality; for persons who have ancestral virtue and wealth behind them are thought to be noble”s3) in more: nowhere are there a hundred men nobly born and good, but ‘The importance of this passage is that it tells us that equality was the there are rich men in many places, But for the constitution to be framed absolutely and entirely according to either kind of cquality watchword of democrats, and inequality of oligarchs. Not, of course, is bad. And this is proved by experience, for not one of the constitutions that oligarchs would have said openly “Incquality is a splendid thing” formed on such lines is permanent. And the cause of this is that it is - the whole theory of geometric proportion is a subtle attempt to impossible for some evil not to occur ultimately from the first and iuiavoid doing just that, an attempt Lo call incquality “truc equality” - tial crror that has been made. Hence the proper course is to employ but rather that their practice presupposes such an attitude, No-one could defend inequality, because the word loov meant “fair” as well as numerical equality in some things and equality according to worth in equal; and no-one would boast of an unfair system). So far, Aristotle has not explicitly mentioned geometric proportion, others”66), It is interesting that Aristotle says that it is bad for the constitution to be framed entirely according to either kind of equality; Plato had still less its mathematical basis. But it is clear that (his has been in the forescen practical dilficultics®?), but Aristotle’s position is a long way background all the time: it lies behind the whole notion of equality from his exalted view of geometric proportion as “God's choice”, and as proportionate tu the worth of the person. It is made explicit in a later having “the greatest power among gods and men”) Why has Aripassage, which arises from a discussion of stasis*): stotle stepped down? Most probably it is a natural reaction from “For party strife is everywhere duc to inequality, where classes that Plato in the light of experience®). Or perhaps the theory of geometric are unequal do not receive a share of power in proportion ...; for proportion had been subjected to such cogent attack from the demogenerally the motive for factious strife is the desire for cquality. But crats that Aristotle could no longer put it forward with the complete equality is of two kinds, numerical equality and equality according to confidence of his master. worth — by numerically equal I mean that which is the same and equal in number or dimension, by cqual according to worth that which is ‘Lhe last references to the theory in the Politics occur in book 6, where Aristotle shows that the principle of equality, or arithmetic equal by proportion; for instance numerically 3 exceeds 2 and 2 equality, as he calls it, is the basis of democracy: exceeds 1 by an equal amount, but by proportion 4 exceeds 2 and 2 “But one factor of liberty is to govern and be governed in turn; for exceeds 1 equally, since 2 and 1 are equal parts of 4 and 2, both being the democratic principle of justice is to have equality according to halves. But although men agree that the absolutcly just is what is number (zo toov Eyew zar dowdy), not worth, and if this is the prinaccording to worth, they disagree (as was said before) in that some ciple of justice prevailing, the multitude must of necessity be sovercign think that if they are equal in something they are wholly cqual, and and the decision of the majority must be final and must constitute others claim that if they are unequal in something they deserve an justice, for they say that cach of the citizens ought to have an equal unequal share of all things. Owing to this two principal varieties of share; so that it results that in democracies the poor are more powerful constitution come into existence, democracy and oligarchy; for noble 6) ib. 1301 A 46 -B 4, ax. I. Rackham (Loeb), pp. 371-3. #1) Cf. the remark of K, J. Dover, in The Political Aspect of Aeschylus’ “Lumenides”, J. HS, Ixxvn (1957), 233: “Democrats do not boast of their dvapzia, nor do uligarchs claim to impose Boudela”. 65) CE Arcuvras, DK 47 B 3; Pi. Laws 757 A, 744 C. $) Ar. Pol, 1401 B 27-1302 A 8, tes. HI, Rackham (pp. 375-7). Compare the working-oul with that of Archytas, DK 47 B 2. 9) Po. Laws 757 E. ©) ib. 757 passim; Gorg. 508 A. 2%) Ar. Pal, 1302 A 4.

than the rich, because there are more of them and whatever is decided n” by the majority is sovercig7°), He makes the same point again later: “But what is thought to be the extreme form of democracy and of popular government comes about as a result of the principle of justice that is admitted to be democratic, and this is for all to have equality according to number (+4 taov Éyeuv dravrag var’ apilubv). For it is [true] equality?!) for the poor to have no larger share of power than the rich, and not for the poor alone to be supreme but for all to 121 days of Solon’); and it is interesting to sce how this notion was elaborated in later antiquity. Jt need hardly be added that this retrojection of the theory cannot possibly have any basis in historical fact; it must represent later propaganda. “It is said”, wrote Plutarch, “that a remark which Solon had previously made to the effect that *equality docs not create war’ was in circulation; and this pleased both those with property and those without; for the former expected to get equality based on worth and excellence (&Efo and &petà), the latter equality based on measure and number”#), Perhaps Solon did declare 74 toov rörzpov ob moreè?s); govern cqually”?2). but the suggestion that the poor expected arithmetical equality, and the upper classes geometrie proportion, is of course a fiction, Plutarch (vii) Plutarch and the retrojection of Ihe theory repeats the same story elsewhere??): “Solon’s remark about the state, We have seen how Isocrates attributed geometric proportion to the that equality does not make stasis, was thought to be too much in “sood old days” of early Athens’). These “good old days” were the 8 favour of the rabble (6,265) in introducing arithmetical and democratic proportion instead of the excellent geometric proportion”). 19 ib. 1337 B 2-10, us. H. Rackham (slightly changed), pp. 489 91. 71) In other words, geometric proportion, 2) Ar, Pal, 1318 A 3-8, trs. 1. Rackham, p, 493. 427" dpUjesy at the end of (his sentence (line 8) is rightly excised by Rackham; it has come in from line 5, and makes nonsense of the passage. Ross however prints it, without comment; so does Newman, It would be interesting to know what Plutarch’s source was. For this is not simply Isocratean propaganda; it is its very opposite. For Isocrates, the Solonian constitution was an example of geometric proportion; for Plutarch’s source it was arithmetical equality: those of 1 do not wish to embark here on a discussion of the difficult passage 1318 A 3-B 5 (rejected by some as spurious), But E suspect that it is in fact an attempt to put geometric proportion into practice, and that the puzzling phrase tiv xatà volte disappointed, It is not difficult to sce the origin of the discrepancy, loérgta (1318 A 14), translated by Newman (IV, 504) as “equality in respect of property-qualification”, comes to very much the same as “geometrie proportion”. It is certainly not “cquality in proportion to number”, i.e. arithmetical proportion Isocrates believed that Solon instituted, not the radical democracy of the fourth century, but the “good old democracy” — in other words, Solon's supporters who wanted geometric proportion were greatly (Rackham, 494, note a). not There is also a brief reference to this topic at Pol. 1308 A 11-13: “For that equality which men of democratic spirit seek for in the case of the multitude is not only just but also expedient in the case of those who are alike”. For the sake of completeness, it should be added that the whole apparatus, mathematical examples and all, turns up also in the ethical works of Aristotle. 1 refrain, however, from discussing these passages: they present difficulties of interpretation of their own, and they take us too far afield from our subject. ‘The referensource, on the other hand, believed that Solon was the founder of the ces arc: EN, 1191 Aw Beg EN. 1158 B 27-33 EK, 1238 BR 19 21 SE 1242 B 11-31 EJE. 1241 B 32-40 ALAL 1193 B 19-1194 A 25, of which only 1193 B 36-1194 A 18 is peculiar to the ALM. ‘The great flaw of regarding (he cquality of men as some measurable empirical quality (cf. p. 12) makes the claborate discussion of justice in the EN. practically worthless. 29) suor. 7 (dreop.) 235 p. 111 ite. a democracy at all: hence geometric proportion. Plutarch’s democracy; so arithmetical equality is appropriate. IL throws a bright light on the practical value of these theories that the (wo opposite and 4) ib. 16, #5) Puur. Sul, 14.4. **) Solon did of course believe in a “balanced” state (e.g. frs. 5, 25 Dichl); + ion here might mean something like a “balance” between the claims of the two sides, 7) Puur, Mor, 484 B (de frat. amore). 79) Note the variant: equality does not make rózeyos (Plut. Sol. 14.4); it does not make stasis (here; cl, Archytas DK 47 B 3; PI Laws 757 A, 744 €; Ar. Pol, 1301 A 39, B 26 ff). ‘This is an indication that Platarch quotes from memory, rather than that he had two sources.

incompatible equalities could both be thought of as symbolizing the with the God’s-cye vicw#!); and that arithmetical equality, as always, 122 123 is explicitly connected with democracy, and geometric proportion same constitution”). Equally unhistorical is Plinarch’s attribution of the theory ka Lycurgus, “Plato combined Lycurgus, no less than Pythagoras, with Socrates, as Dicacarchus believed. For, you know, Lycurgus expelled arithmetical proportion from Sparta, as being democratic and favourable to the rabble (6y%xb05), and introduced geometric proportion, which is suited to sober oligarchy and law-abiding kingship. For the former distributes equality by number, while the latter distributes what a man deserves by proportion (4 pty yap éeOnd sò Lau, $ SE Ayo To at’ ¿Elo &rovéper); it does not mix everything up together <unlike arithmetical equality», but it makes a clear distinction between the good and the bad; it is not by means of an even balance, nor even by means of the lot that they receive their share, but they get what befits (hem in accordance with how much they differ in virtue and vice. God applics this proportion to things, my dear Tyndares; it is called diké and nemesis, and it teaches us that we should make justice too» (fair?), but not make equality just; for God destroys as far as possible the equality which the majority pursue, with a non-democratic constitution. (viii) The later history of the idea At the tail-end of antiquity, we find the idea expounded by Bocthius, in a section of his work on arithmetic called “Quac medictates quibus rerum publicarum statibus comparentur”. But here something remarkable has happened. “And so arithmetical proportion is compared to a state which is governed by a few, because there is a greater ratio between its lesser terms (i.c. 6 is 11/, times 4; but 4 is twice 2, a greater ratio). But they say that harmonic proportion is the state governed by the optimates, because the greater ratio is found in its greater terms (ie. 6 is 114, times 4; but 4 is 1'/, times 3, a lesser ratio). Geometric proportion is the proportion of a somehow democratic and equalized state. For it is composed of an cqual ratio of all, of both greater and lesser terms, and there is a parity between all which consists of a proportion which which is the greatest of all injustices, but he preserves cquality accordretains an equal justice in its ratios (i.c. 8 is twice 4; and 4 is twice 2)”#?). ing to worth, distinguishing geometrically the lawful by the proportio- This is extraordinary: the whole meaning of the theory has been | changed. But a little reflection will show that it is not, as might be nate”#0), | This, with its allusion to Lycurgus, is no more than fanciful antisupposed, totally bogus: the new interpretation leaves the mathematiquarianism; but it has not been forgotten that the theory is in origin cal respectability of the theory unimpaired. According to Bocthius, in that the political application of the proportions had anything to do with Pythagoras himself: we can quite safcly ignore both Pythagoras and Lycurgus in this context. Chronology, we observe, has gonc for equated by the writer with an oligarchy, get most power, In the Pythagorean. There is, of course, not a scrap of cvidence elsewhere the arithmetical proportion, the men of the least worth, arbitrarily harmonic proportion, the men of the greatest worth get most power And in the geometric proportion, all, whatever their worth, get nothing: if it is Pythagorean, how docs Lycurgus come to know all about it? We may note that geometric proportion is again associated #1). As at Pi. Laws 757 B. 2%) Bowrmus Avithm, 2.45: Atque ideo arithmetica quidem rei publicac compa- 29) This is because the two authors held opposite and incompatible views about the Solonian state. But neither Isocrates nor Plutarch's source approved of demoeracy; so much is clear from their use of the theory of geometric proportion; is it is a built-in characteristic of that theory that whenever it makes an appearance, geometric proportion is bound to win. 80) Pros. Mer. 719 À € (Quest. ennwir. 8,2,2). ralur, quac paucis regitur, idcirco quod in minoribus cius terminis maior proportiv sit. musicam vero medictatem optimatium dicunt esse rem publicam ideo, quod in maioribus terminis maior proportionalitas invenitur. geometrica medietas popularis quodammodo et cxacquatae civitatis est. namque vel in maioribus vel in minoribus acquali omnium proportionalitate componitur, et est inter omnes paritas quacdam medietatis acquum ius in proportionibus conservantis, (Passages in brackets in the translation are not, uf course, in the text).

equa! power. For Boethius, then, arithmetical proportion is oligarchy, harmonic proportion is the rule of the optimales; and geometric pro- 125 through superiority in arelé or riches or power. ‘That then which distributes equally is democratic, that which distributes uncqually, portion is democracy*), But for the classical age, arithmetical proportion was democratic, and geometric proportion was undemocratic, a graded society of increasing and decreasing privilege. Bocthius is able to turn this mathematical somersault by means of ignoring the Are we to suppose that this is pure Archytan doctrine, that Plato. theory took into account, and using only the values of the numbers, the only distance between cach number and its neighbour, which the classical and the ratios between them. aristocratic or oligarchic”., Who is this author? If we take Stobacus’ word for it, he is none other than Archytas himself. If this is truc, it would be most startling, Isocrates, Aristotle and Plutarch all got it wrong, and that Boethius is trustworthy exponent of an uncorrupted philosophical tradition which comes to him straight from the carly fourth century But Bocthius docs not stand alone: Stobacus quotes an author who gives both the same heterodox interpretation of the proportions, and B.C, 285) Surely not. It is @ priori implausible; and the whole weight of the evidence gocs against it. ‘The fragment cannot be the work of the same mathematical explanation*): Archytas**), “Wherefore some make justice aristocratic, some democratic, and How can we account for the Bocthius and pseudo-Archytas passages? some oligarchie: aristocratic justice is according Lo the subcontrary Someone has obviously tried to revive the old theory; but, clearly, mean: for this proportion distributes greater ratios to greater, and less Lo less. Democratic justice is according Lo the geometric proportion: for in that the ratios of greatcr and smaller amounts are equal. Oligarchic and tyrannic justice is according to the arithmetical proportion: for this is the opposite of the subcontrary: it distributes greater ratios to less, and less to greater. ‘These, then, are the forms (iSéar) of the distribution, but the imitations (elxéves) are scen in constitutions and houscholds: for rewards and punishments and offices are either distributed equally to greater and less, or unequally, cither he is ignorant of its classical application, and has worked it out for himself, with startling new results; or else it is a deliberate reinterpretation. The latter seems more likely, But who was responsible for it? Not Bocthius himself: the use of dicunt*?) shows this. Nor pseudo-Archytas: the use of zotodvz") shows this. Bocthius docs not translate straight from pscudo-Archytas, nor vice-versa: this is shown by the fact that they discuss the proportions in a different order. There must be a common source. Dicunt and xotodv=:, both plurals, suggest that more than one author had developed the new interpretation. ‘The authors political views can be detected from his theory. Harmonic proportion was thought to be better than geometric, as e) What, it might be asked, is the difference between uligarchy and the rule of the optimales? ‘The mathematical working-out shows that the distinction is between geometric was better than arithmetical. The originator of the new application must therefore have believed that the rulc of the best was rule, the optimates. "This interpretation is borne out by the pseudu-Archytas fragmost desirable; second on his list came democracy; and worst of all, the rule of any few men — any old oligarchy — and that of the few who deserve to zi dv dprotoxparinoy zul de Bapoupariniv mM) De rates — Seb TuTbBixatuv apiotonparuzds uud TEV Umevavılav peste Taig ment (p. 124-125). zul SÌ Oeyapymby moriva: nal tev ap pétoot péfovas tag Abyos, Tots HE pont pñovas Sravejeee & Srey, mire To Sì Saporparizòv uae rev yeomerpeciv” dy yap tara tol hóyot Tout av uetiveay sem góvos peyédecov: th Sè uyapymóv val supavindy ALTA TRY Apr rirdy avreater yxp ara TH brevavela’ tots yap poor pálovac tes Adyeng, tots Se pétoat phovas. ral piv Gy Wer rig Buavopäs tuaabrar, tal dì eleñves tu tats molte nai zeig alas Vewpgovrat’ teunl re yap vat xurdtoreg val dpyal <p> e kam tots péfoae val uo Seavépovrat, 4 El Ayla À 1H dperd Snepézer Y tH noize) Fj wat Buvdpasi. T piv CV E Lau deporgarizóv, Tu Sè 8% vico dptatonpatiniy Y, Oeyapyicey, 85) If dis fragment represents Archytas’ views, then Archytas himself must have got the theory wrong when he wrote DK 47 B 3, if our interpretation (pp. 105--106) is correct. **) It has often been condemned as spurious on other grounds. This passage, with others also attributed to Archytas by Stobacus, is discussed in more detail in part IT, pp. 131-138, where the grounds for rejecting them are stated in full. #) Boethius, line 4 of the quotation in n, 82. **) ps.-Archytas, line 2 of the quotation in u, 84. CI. k M,

127 the rule of a few. A strange outlook. I must confess that I am unable to notion of equality in its simple form. Once this distortio n had been go any further than this“). ‘The theory of geometric proportion had a long life. One late echo, allowed to creep in, then it was an casy step to claborate, and to introduce the “superior” geometric proportion. “Democr atic arithmetical proportion” is, then, a concept introduced by anti-dem ocrats; and we will have finished with it: some lines from Johannes "Tzetzes, the twelfth-century grammarian, on the words inscribed over Plato's it is what they said that democrats believed in; but it is not what demodoor?) : erats did believe in, This is clear enough from the fact that by itself the concept of “democratic arithmetical proportion” is pointless; it only ph thay rpwWipuv tov abro ypatac brnpye llore makes sense in the context of the comparison with the non-democ ratic geometric proportion. The theory is based on putting into the mouths of democrats views which they never held. Mudela dycopérensos lotto, por thy otdyyy' rouréoruw, &8tuus aqSels maperoepytole ride" loérne yep vai Binary gate yanpzrpla'). Furthermore, the principle that those with special abilities should be given jobs where they could make use of them - which is one of (ix) Criticisms In dealing with this topic, I have made certain criticisms of the theory of geometric proportion. ‘To conclude the first part of this the ideas behind geometric proportion - was in fact at all times recognized at Athens, as Pericles claimed). At no time were the most paper I would like to recapitulate these, and to add a few more. In the first place, no Greek democrat would have said that he of believed in arithmetical proportion; he just believed in equality pure and simple, equal rights, equality before the law, and so forth. It is the opponents of democracy who interpreted these values as avithmetical proportion: in place of the democratic ideal they substituted a scale - something entirely different, and indeed opposed to the important offices of state entrusted to the lot, that example arithmetical proportion. ‘The opponents of par excellence democracy insisting on something which no democrat at Athens ever were denied. But democrats never clothed their ideas in mathematical trappings. They insisted on certain qualifications for the more specialized jobs (as opposed to mere attendance at the council and assembly ): generals were elected, so that experienced men led the Athenian army; there was a property-qualification for the post of tamias, so that il he took 8°) ‘Two negative points. The theory was known tu Plutarch in its original im But it is not safe to infer a date between the first and sixth centuries A.D. for the moncy, it could be recovered; and so forth. These were particular qualifications for particular offices with particular justificat ions: no new version, It may be that the new interpretation was current in ] lutarch s day, and that Plutarch either had not heard about it, or preferred to ignore it Dr de purposes of his ancedote about Solon. 1 suggest (part II, p. 28, n ia € sa in the first century A.D. for Diotogenes, Sthenidas and Ecphantus: shou d per aps general theory was framed, but ability and merit a similar date be given to pseudo-Archytas? ‘There are certain close similarities in to suit two radically different evaluations of constitutions, thought between all these authors (part II p. 133, n. 119). | o Nor can anything be deduced from the surprising order in which Bock u Be sents the proportions: arithmetical, harmonic, geometrie, *] hey are not in the + a of their merit, as can be seen by a comparison with ps.-Archytas account. Perhaps, having stated the basic proportion, the arithmetical, Roethius then juups to the harmonic because it is the vue that interests him most as a musician. 2°) 'Yzwırzes Hist. var. chiliades 8, 972-5. ka 98) [ refrain from pursuing the topic through the pages of the mathematicians and musical theorists, where the arithmetical, geometrie and harmonic propor Hana are often fully treated, but not given a political meaning. See Nicomacnus ov Cirasa dritlon. 2,21 25: 2,28,6; Borrittus Afus, 1,10. 2,12. 2,14. Of these, Boethius 2,62 succeeds in making the somewhat complex theory extremely clear. were not overlooked. Again, the translation of the mathematical basis of the theory political terms is valucless, since it can be adapted into with no difficulty the “classial” application, and that of Bocthius and pseudo- Archytas, The gencral idea behind the theory, that men should have power in proportion to their worth, is also valueless, It is too vague, Plato makes Socrates use geometrie proportion against Callicles*3); yet Callicles himself uses the idea on which geometric proportion rests: “it is Just”, he says, 2%) “Tu. 2,37,1, cited on p. 102; ef. Ps.-Xun. Ath. Pol. 1.3 (distortcd). 99) Pr. Gore, 508 A,

“that the greater should have more than the less, and the ge crats' principle of equality by the introduction of a “superior” form of 128 than the weaker”), The phrase links in one Callicles’ “immoralism and the very words most characteristic of geometric propartion. The theory is thus too adaptable: it can be used on one side or the su with equal effect; it can even be called in to justily absolute monarchy? À To represent the equality of the democrats in numerical terms is 10 misunderstand a claim about the way in which people ought to be treated as an empirical statement about what pcople are; and the doctrine of geometric proportion also ignores the criterion of relevance. Briefly, this is: every difference in treatment must be justified, not by any difference between the men treated, but by a relevant difference. Most of the supporters of the principle of geometric proportion failed to take this into account. Aristotle did take it into account; but we may have serious doubts about his “political goodness”. go pushes the problem back one stage further: what coustitutes a goodness” ? “political a Finally: it is all very well to say that one man 15 better Less another; but who is to decide this? Not the men themselves: o7ed6v ol msiora galzor xpırai mept civ olzelav"®). Or, as Popper says: “Against all this anti-equalitarianism, | hold, with Kant, that it must 129 equality is thus a complete failure), 11 GEOMETRIC AND HARMONIC PROPORTION IN ARCHYTAS AND PLATO In two recent articles”), Mr. J. S. Morrison has maintained that Plato abandons geometric proportion in the Republic and adopts instead a third kind, harmonic proportion. He argues as follows): “I noticed, when the theory of justice or social order put forward in the Republic was being considered, that it differed from the theory mentioned bricfly in the Gorgias. In the latter dialogue the formula which is the basis of social and physical order is geometrie proportion . . . The bond of unity in the Republic is the harmonic fsotes ... We are told that Pythagoras discovered the theory of means (Lamb. in Nie. p. 100,19 If. (Pistelli) | — DK 18.15]), and knew of three, the arithmetic, geometric, and sub-contrary, but that the third was named harmonic by Archytas and Hippasus. In a fragment of Archytas’ work on music the three proportions are defined and the third is described as “the subcontrary which we call harmonic”. These facts suggest very strongly that Plato’s acquaintance with Archytas was the cause of his adoption of the harmonic proportion as the bond of unity in soul and state in the Republic, be the principle of all morality that no man should consider himself more valuable than any other person. And ] assert that this principle is the only one acceptable, considering the notorious impussibility of judging oneself impartially”®*). ‘The individual cannot judge himself, then; nor is anyone else, or any group, in a position to give a verdict on such a question. No group of men “known to be good” can decide; for an infinite regress is set up: who is to decide who these men are? So the evaluation and grading of citizens which the theory of geometrie proportion requires is an impossible task. | | The attempt of the opponents of democracy to discredit the demo- 99) ‘Chere are some striking resemblances between geometric proportion and what is known as “dominance hierarchy” among baboons (for a popular account, sce S. L. Wasumurs and I. DeVore The Social Life af Baboons, Scientific American, June 1961, 62-71, esp. 69-70, from which the quotations that follow are taken). “The most dominant males will be more frequently groomed and they occupy fecding and resting positions of their choice”: the best receive honours and privileges xx" ééiav; equality for equals, incquality for uncquals. “When a dominant animal approaches a subordinate one, the lesser animal moves out of the way”: contrast the state of affairs in a democracy (where arithmetical proportion prevails) people and cven animals are always bumping into you in the street (Ps.-XEN. Ath, Pol. 1,10, PL RP 563 C). “The hierarchy has considerable stability ... The usual effect of the hicrarchy ... is to decrease disruptions in the troop”: geometric proportion prevents stasis (Archytas DK 47 B 3; Pr. Laws 757 A, 744 C; Ar. Pol. 1301 À 39, B 26 (f.; PLuT. Afor. 484 R (de frat. amore).) “Although dominance depends ultimately on force, it leads to peace, order and popularity": homoncia (Archyras DK 47 B 3) and q0érs (Pr. Laws 757 A). 93) ib. 583 E. . %) Isocr. 3 (Ne). 14 10. 98) Ar. Pol. 1280 A 16. ") K. R. Porrer The Open Society and its Enemies? (London, 1957), 1 256, tl, 20. 92) J. S. Morrison Pythagoras of Samos, C.Q. ns. vi (1956), 135-565 “The Origins of Plato's Philosopher-Statesman”, C.Q. n.s. viii (1958), 198-218. 10) (2. ns. viii (1958), 213-5. The same arguments may be found in the carlicr article, C.Q. ns. vi (1956), 154-6, where the political fragments of Archytas are also used as evidence (sce pp. 131 -138).

the Fons Not ouly did he [i.e. Archytas] employ the name harmonic among of the social but in a work on mathematics he gives an interesting account application of such logismoi (DK 47 B3)”. are The passages in the Republic which are alleged to reflect these ideas was 132 AMD and 443 D. This theory also entails that the Gorgias | written before Plato visited Sicily. I do not find Morrison’s hypothesis at all persuasive, for the 131 2. Archytas did not apply the harmonic proportion to politics. So far my disagreement with Morrison has been trivial. Even if Archytas did not invent the term “harmonic proportion”, it is still possible that it was he who first applied the concept to politics, I believe, however, that there is no clear indication that he did. ‘To prove this it will be necessary to discuss at some length (i) some fragments preserved in the anthology of Stobacus, which are alleged to be by following reasons: Archytas, and in which the harmonic proportion is applied to politics: 1. Archytas did not invent the term »harmonic proportion« fragment of Archytas (DK 47 B 3): this, I shall argue, docs not refer these, I shall argue, are spurious; and (ii) an undisputedly genuine was not only known before Arda); ‘The third kind of proportion it was also, according to lamblichus, called “harmonic beiure Archytas by Hippasus, an early Pythagorcau!®). fF urthermote, un is said by NicomachusTM) to have given a reason for this mame, I hile an was a contemporary of Socrates!); if Nicomachus is right, this implies that the term was in use before Philolaus’ time, and thus well before Archytas. And although Archytas speaks in one place") of “the subfragse 1 the> same refe to itt in ay®, he refers AA 5 l07) appovin contrary, &v ahobpev 15 no there Thus zav. ment) as “the subcontrary, dv xoA€ovtt Gppovi a person a from it need to suppose that Plato must have learnt about + ut ete TE meeting with Archytas in Sicily'®*). . \ be perdi Pythagoras+4“+Arima lamblichus shows, n quotaiion from , As Mosto pe = 118.23 Nic. (ie e elsewher us Lamblich it. with 191) Morrison's reference, 431 D, appears to be a slip. been familiar that it was a Babylonian discovery and that Pythagoras was Min Im o nnS it into Greece. He then gives a list of Am + mes who used it: Aristacus, . and later Plato in the chytas, hi “among the first” hedges over this. Alter er a ! piper‘phrase to different centuries. On the date of appa À salati cl Funaeus. Archytas belong see K. von Frrrz The discovery of incommensurability by Hipprasus of Metapontion, of Mathematics xivi (1945), 245- d 104) Nico. Arifhm. 26.2 = DK 44 A 24. ws) CH, PL. Pad. Gr E. to) DK 47 B 2, p. 436 line 9. to?) “We, the Paloma a not “I, Archytas”. sii n. R 2, p. 435 line 20. {47 i i in a case, that the tradition is confused. pag ga tie ne the | sel ive itd la nb av nepl "Apydrav vat “Innzoov suggests that he, writing is mista ra iena> re A.D., regards these philosophers as contemporaries. If he may also be mistaken in attributing to Archytas the alteration of punir. th to the harmonic proportion, (1) The Authenticity of the Stobacus fragments Stobacus!!”) quotes an elaborate application of the three kinds of proportion to politics, which he attributes to Archytas. The text and a translation will be found on p. 124, Many modern writers have wanted to reject this, together with the other political fragments of Archytas!!!) as spurious, But there has been great diversity of opinion on the subject"), Morrison, in his earlier article, regards them as genuine, concurring with carlier scholars who defended them against the charge 119) STOBAEUS 4.5,137 (Wachsmuth-Hense: all future references will be to this edition). 111) Sron. 4,1,132, 135 8, 5,61. 112) It may be of interest to list the views of previous scholars, Hartenstein (1833) regarded them as authentic, with later interpolations. Gruppe (1840) attributed them to an Alexandrian Jew of the first century A.D. Zeller (1844 cto) thought that they were the work of nco-Pythagorcans of the first century B.C. Beckmann (1844), according to Morrison, believed them to be genuine. Diets {1903 etc.) does not print them in his Fragmente der Vorsokratiker, and lists them as spurious. Arsann Denarre Essai sur da Politique Pythagoricienne, (Liège, 1921), where the political fragments of Archytas are most thoroughly discussed, and where the views of Hartenstein, Gruppe and Zeller are reported and criticized) finds no indication that they are late or apocryphal. W. TneiLer (Gnomon ii (1926), 147-56), reviewing Delatte, regards them as Hellenistic, influenced by Plato and Aristotle, but earlier than Posidonius. He is followed by E. Gounenousn (“Che Political Philosophy of J lellenistic Kingship”, Yale Classical Studies i (1928) 55-102). E. L. Minar (Early Pythagorean Politics (Baltimore, 1942), 111) speaks of them as genuine; so, with some doubt, dues Louis DeLarre (Les Traités de la Royanté, cic., (Liège, 1942), 160). T. A. Sıncr.am (History of Greek Political Thought (Loudon, 1951), 293; 301) regards them as late productions. Thus the general opinion is that they are spurivus; but there is no unanimity.

133 of containing Platonic and Aristotelian matter. I will first consider the political fragments as a whole, and then the “proportion fragment” in dialect, and the attribution to Ecphantus, probably a fourth century particular. figure!!?), show that their author or authors intended them to be taken o All the fragments are written in Doric, A careful examination of the dialect by Armand Delatte!!) leads him to the conclusion that il resembles literary Dorie of the fifth and fourth centuries B.C., and in particular that of the authentic Pythagorean fragments, But the content might make us suspicious, for Stobacus also quotes from writings attributed to Diotogenes, Ecphantus and Sthenidas!!4), which express ideas similar to those of the Archytas fragments") ; and these too are in Doric. Although no agreement has been rcached on the date of the Diotogenes, Ecphantus and Sthenidas fragments!**), one thing is certain: they can hardly be carlier than the Hellenistic period; yet their as fourth-century writings. ‘The political fragments of Archytas thus appear in disreputable company; but then, the same might be said of many of our pre-Socratic fragments whose authenticity is undisputed; and the genuine fragments of Archytas come from the same context. We must look next at their contents"*), The parallels in thought between the political fragments of Archytas and the admittedly late fragments of Diotogenes, Ecphantus and Sthenidas arc remarkable!!®), and they led Goodenough!?) to regard the former as roughly contemporary with the latter. Louis Delatte!®?), too, although he dues not believe in a late date for the Archytas fragments, continually cites them o | Ure: Oxford Classical Dictionary is rightus) the sion of da gentlemen from (1949), 194. nf. cit. 113) (iu n. 112), 78. ly castigated by El. Lasr, 7.2.3. xxxix | followed by M. R. Crtartesworru (CR. oa an Due (op. cit. in n. 112),they belong to the second century A.D. 135) See n. 119. Ixiii (1949), 22-3), maintained that “Alexander the Great and the Unity of Mankind’ , Proc. Brit. Acad. xix (1933), 32, n. 43, thinks that Diotogenes has Demetrius Poliorcetes in mind in his éme of the ideal king. This view is supported by some very remarkable parallels, nes Goodenough and Sinclair (opp, citt. in n. ı 12) regard them as Hellenistic. WWITARN, in sense and in phrasing, with Plutarch’s life: ol Demetrius. This shows, Tarn a Jieves, that Diotogenes was contemporary with Demetrius, or very nearly so. ‘larn is right, Sthenidas and Ecphantus should also belong to tes period, for they can hardly be separated. Or is Plutarch the source of Diotogenes: There are also striking parallels between the ideas of these three authors and those of Dio Chrysostom, tu which Mr. Oswyn Murray has kindly drawn my attention. The ideas of Dio on kingship are put in this general context by L. Delatte (np. cit. in n. 112, 151-2); but the parallels are perhaps closer than ıs generally realised, and may possibly shed some fight on the dating of Divtogenes, prg and Ecphantus. ‘Thus, the king should be a shepherd of his people, not one whe destroys the sheep (Ecphantus ap. Sron. 4,7,54 (vol. 5 P- 276), “Archytas Hus Sron. 4,5,61; Dio CHRYSOSTOM 1,13. 3,41); he derives his power as a stewarc up from Zeus, and governs his peuple according to Zeus” ordinances, for he is his representative on carth (Diotogenes, Stos. 4,7,62 ad fin. (pp. 269-70); Sthenidas, he is com- Srov. 4.7.63 (pp. 270-1); Ecphantus, Srow. 4,7,64 (p. 275); Pio 1,45); pared with the sun (Ecphantus, Sron. 4,7,04 (p. 272 ff1.); Dio 3,73B1). A more careful comparison than 1 have made would probably reveal further correspon- Dio took his ideas from Diotogenes a Pr is, however, difficult to determine whetherfrom him, or whether there is some ces. and the others, or whether they took theirs | common source. On general grounds, it seems unlikely that the ideas originated with Dio. If we could safely assume that Diotogenes, Sthenidas and Ecphantus were his source, then Louis Delatte's date (second century A.D.) would have to be in comparison with the thought of Diotogenes, Ecphantus and Sthenidas. On the other hand, Armand Delatte has pointed out a surprising number of parallels with fourth-century thought (the parallels with later writers are, however, more numerous and impressive); these can, of course, be interpreted cither as the influence of fourth-century writers (to support a late date), or as a reflection of the common atmosphere of fourth-century thought (to support an carly date). But the former seems more probable. Archytas’ allusiont#), in Aristotelrejected in favour of something earlier. Why not the first century A.D.? This is suggested by the parallels with Dio and (despite Tarn) Plutarch, (CE part I, n. 89). I would not, however, press this conjecture: there may have been a common source; and an carlier date cannot be excluded. The “Archytas” fragments have much in common with those of Diotogenes, Sthenidas and Ecphantus (see n. 119): should they be given the same date? 11?) On his date, sce G. S. Kirk and J. E. Raven The Pre-Socratic Philosophers (Cambridge, 1957), 247 n 1. 1") Most conveniently sununarized in Sinclair, op. cit. (in n. 112), 293 4. 119) To take an example: Archytas (STOR, 4,1,135, p.82 lines 20-1), like Divtogencs (Sros. 4,7,61, p. 263 line 19), says that the king is vónos Endozos, and compares vipog with the sun (Stos. 4,1,138, p. 87 lines 17-20). Ecphantus speaks of the dazzling light of the Godlike king (Srov. 4,7,46, p. 2721.). CH. also n. 116. The most useful discussions of the fragments of Diotogenes, Ecphantus and Sthenidas are by Louis Delatte and (with translations) E. Goodenough (opp. citt. in n. 14). 120) Op. cit. (in n, 112). 131) Op, cit. (in n. 112). 122) Sros. 4,1,135 (p. 83, lines 7-9): 25 26704 Evo, th Groyos and và rin.

135 ian terminology, to the Platonic three-part soul, is unlikely to belong to an independent fourth-century philosophical tradition. And_ his remark poi 3% yo nioav nowenlay & dpyovtes zul dpyupévio auvéorzpsy val tetroy vouey!?%) is almost certainly a correction of the Aristotelian view of a world of natural rulers and natural subjects!??), It is surely impossible to maintain that such a sentence could be of Aristotle, tended to be purely theoretical!29); the more practical tone of the Archytas fragments, the working-out of the relationship ol written before Aristotlc!#). things were a reality!?). here is a man with much higher powers, the faoi2d¢ who is an Euyuzas vópos!53), A fourth-century discussion of kingship, such as that the king to the law, and the discussion of his nature and dutics and powers, point to a date later than the fourth century, a date when such The political fragments of Archytas, then, whatever their date, can So much for the fragments as a whole. But we must cxamine the hardly be considered to be the genuine work of Archytas the carly fourth-century philosopher. ‘They may be purely literary exercises, or they may be forgerics with a political purpose'®*); whatever their “proportion fragment”TM!) in greater detail, for it is on this fragment that origin, they are spurious. ‘True, there are references to oligarchies and democracies; but the polis did not die with Alexander; and in any case these allusions serve only as a background to the main theme, kingship. ‘This emphasis on kingship tells strongly against a fourth-century date. Stobacus 4,5,61 is entirely concerned with the nature and powers ul the true ruler, the ¿AaDivós Zpywv; and there are continual references to kingship in Stob. 4,1,135-8. While the word (Suc. is sometimes associated by the author with Sparta'*), the type of ruler envisaged my argument depends. It contains three features which might arouse our doubts. First, the author calls the third kind of proportion breve» siz; but Archytas, as we know! *), called it dopnovxd. But it might be argued that Archytas may have used both terms interchangeably (in fragment 2 he speaks of “the Srevavita which they (or we) call Sppoviz2”); and a forger would perhaps have taken pains not to make such a slip. Secondly, there is the phrase tat piv Ov Wea tao Jtavepas tocabrar, tat Se eluóves du vaig xodttelars ual tots oïxotc Dempfovitai, One's immediate reaction is to suspect the influence of the Platonic theory of forms, but Armand Delatte!) most aptly cites Aristotle's Metaphy- 123) ib, p. 82 lines 19 20. 121) Pol. 1 passim. 125) "his suspicion is strengthened by Archytas’ use of the Aristotelian phrase è roue xowevia (Stop. 4,136, 137 (p. 83 line 16, p. 84 line 9)). 120) Charlesworth (of. cit. in n. 116) asks what audience the writings of Diotogenes, Sthenidas and Ecphantus, in their archaic Doric, were intended to influence, if they are in fact late productions. "The same question could be asked of the Archytan fragments, but in both cases it would miss the point, In the ancient world, the respectability of a political doctrine was guaranteed by its antiquity (lo make revolution in Greck is vewtepi%ewv), and thus, if one could claim the support of an archaic document, one’s case was thereby strengthened. ‘The prestige of Archytas was great; Doric was needed for plausibility. stes 51): ni per yap lludayóperor uipqow 72 iva paalv siva: tov apri, and this phrase might be taken as a very striking example of that doctrine!35), Yet although iSéa. and elxóves can have the meanings required here even as carly as the filth century, the combinations of 128) Srou, 4,1,135 (p. 82 line 21). ‘Phe implications of this exalted view of kingship are discussed by Goodenough, of. cit. in n. 112, 61 ff. i 15) See J. P. V. D. Barsvon The ‘Divinity’ of Alexander, Historia à (1950), 36871. Xenophon's Gyropaideia and Isocrates’ Nicocles and To Micaces do, however, attesta growing interest in monarchy in the fourth century. y IL is not a great step from “Archytas” on kingship to Euschius on Christian On the other hand, the fragments might be purcly literary productions in the style of Archytas (see Armand Delatte, op. cit. in n. 112), which perhaps intenkingship: sce N. Baynes Eusebius and the Christian Empire, Mélanges Bidez 1 (Paris, tionally, perhaps not, were circulated as genuine. 1934), 13-18, esp. 17. 127) Sron. 4,1,198 (p. 85 lines 13, 21). The reference to the classical Spartan constitution, and in particular to the dual kingship (lines 13-14), would suggest a date before the end of the third century B.C.; but this may be only plausible antiquarianism. IU is certainly very odd to regard the ephors as an oligarchic 131) Son, 4,1,137. 132) Sec p. 130, 33) Op, cit. in n 112. 195) 987 B 11. clement in the Spartan state (line 15). Nor is there anything very democratic, as Archytas alleges, about the irraypére, the cavalry commanders. 1 would suggest 1%) However, it might be maintained that Aristotle distorts the teaching of both Plato and the Pythagoreans in this problematic passage. See II. Cuerniss ‘The that the author docs not know what he is talking about. Riddle of the Early Academy (Berkeley, 1945).

137 the two in the same sentence docs scem very Platonic; but admittedly But perhaps Plato scizcd on the fact that in geometric proportion the Metaphysics passage might account for this too. remarkable men get remarkable privileges, forgetting that it was associated with democracy? Yet it remains odd, if this fragment really Finally, there is the fact that this Archytas fragment proposes an entirely new interpretation of the political significance of the propordocs tions. In the classical theory, the equation was: proportion not only in the Gorgias, but also in the Laws, where he calls it OnDsorarn xal dptarg”), and where he clearly implies") that Arithmetical proportion = democracy Gcometric proportion = a superior constitulion, which Geometric proportion = democracy that Plato praises geometric known to have thought very highly of arithmetic — more so than of de ddeyapyizov rai tupavvixdy (sc. Sixarov) was zur tay éplunztzx" (sc. keodrarz). All these arguments tend to suggest that Archytas was not the author of Stobacus 4,1,137; and there is another piece of evidence Subcontrary proportion {or harmonic) teaching, geometry'#2) - makes it most improbable that he would write that 7d But Archytas asserts that: = oligarchy and tyranny Archytan arithmetical proportion is democratic. Again, the fact that Archytas is recognizes merit. Arithmetical proportion represent = aristocracy. ‘The most extraordinary point here is that democratic justice is said which will increase our suspicions, There is just one parallel for the association of geometric proportion with democracy. But this parallel, to be in accordance with geometric proportion, zé SÌ Sauorpatixòv for which we search classical writers in vain, comes from the sixth (sc. Sixatov) uate tay yempetpindy (sc. peoòtara). This is unparalleled in the whole of classical Greck political thought. But that does not mean century A.D. 1 medielates quibus rerum publicarum statibus comparentur«, which not only that it is spurious. We know what the constitution of Tarentum was associates geometric proportion with democracy, but also associates like during Archytas’ lifetime!®), but we do not know how it was arithmetical proportion with the rule of the “best” men, just as the “Archytas” fragment docs. Morcover, Bocthius gives the same mathedescribed. It is not impossible that it was called a democracy, and in that case Archytas would be claiming that in a democracy, remarkable men (such as himself?) ought to be given special privileges; the mathcrefer to the passage in BocthiusTM*), headed »Quae matical reasons. ‘The passages are so exactly similarTM*) that there can be no doubt that both authors reflect a single source’®). matics of this will work out, though not in the usual way!*?). On the other hand, it might simply be the statement of a bungling forger unfamiliar with the orthodox application of proportions to politics, or, more probably"), of a late innovator trying to put new meaning into an old theory. It would certainly be very curious, if Archytas had praised geometric proportion as democratic proportion, that Plato, hardly a supporter of democracy, should take it over uncritically and, without any reservations, speak so cnthusiastically of it in the Gorgias"*?). 138) Sce pp. 138 139. . . 13?) See pp. 123- 124. Butitisimprobable, if Archytas were recommending geometric proportion, that he would also mention a superior type of proportion, the harmonic, 49) 757 B; cf. p. 141. 5 ib. Mt) DK 47 B 4. 142) Arithm. 2,45, quoted and translated on p. 123. 154) But neither is a translation of the other: theorder in which the three proportions are discussed is different. 118) See p. 125. Neither Boethius nor “Archytas” is the inventor of this new ape plication of the theory: they both report (dicunt, rninüvet). Why, it might be asked, docs the Archytan version of the proportions make no reference to kingship? Docs not this suggest an carly date, before the Hellenistic period, when kings had not yet begun to dominate the political scene? For no-one writing later than Alexander could ignore monarchy. But kingship might well be regarded as the extreme and most praiseworthy form of aristocracy (= harmonic 138) Ser p. 125. proportion); in fact, there is no other way of accommodating it within the mathematical framework. The political theory of proportions was not designed to take monarchy into account; it fits badly, and the author ignores it. There are, in any

The cumulative force of the arguments against the authenticity of the political fragments of Archytas is thus very strong. Admittedly, some of the objections can he met; there is no evidence which is absolutely decisive; but there is enough, I believe, to conclude that in all probability Archytas did not write the fragments of the repl vépou 200 dixazocóvas with which he is credited. (ii) The dopropós fragment This leaves us with Archytas’ praise of kvyuoués in an undoubtedly genuine fragment, DK 47 B 3: oréoiv pév Exavaey, opdvoray SE uE nos» huyiapig shpelelc" masovella re Yip odx dom subrou yevopévou nai iootag gate. The question arises, what is this 70y10p65? Morrison #6), in my opinion wrongly, that it is harmonic proportion. It implies will be necessary to consider the question in some detail. Archytas has been saying that one must become émezépov, and one can do this by discovering a 2uywopso!!"). What this Aoywoués is, is not stated; nor, indeed, is it clear that Archytas has a particular Aoyiopée in mind atall he might mean any formula or compulationTM*). But it seems at first sight very probable that Archytas was thinking, not of the harmonie proportion, but of the geometric proportion; for this is, after all, the only proportion apart from the arithmetical (which is, of course, out of the question) to which fourth-century writers attached any political significance. The suggestion that Archytas Doytoués is geometric proportion is strengthened by the following considerations. Archytas’ Aoy:0uós stops atéots; atéots, says Aristotle, arises because of 76 Xvsoev, but not when there is geometric proportion"), IL increases ópóvota: laorng pubraro axepyatera, says Plato"), and proceeds to discuss the virtues TWO KINDS OF EQUALITY 139 of geometric proportion. It prevents zreovetla: riceverla is the opposite of geometric proportion, says Plato elsewhereTM), The identification is moreover supported by what we know about Tarentum during the lifetime of Archytast32). We have two apparentl y conflicting pieces of evidence. (a) Strabo 6.280 = DK 47 A 4: toyvoay SÉ mute ul T'apavrivor zal!’ brepiudpy TOMTENÓUEVO! Sqoxpatexin g . .. aredtzavto Se val thy Un darróperos prdusoplay, diagepdvtes 8’ "Apyizay, It has been held that this shows that Tarentum was an extreme?) democracy under Archytas, but as Minar!5!) shows, the first sentence very probably refers to the period after the Persian wars, when Aristotle says that democracy was established after a revolution ! §), The compression of Strabo’s account is misleading. (b) Aristotle, Politics 20 B y - 14: zane 8 Eyer urpetotias!*) vat ta "l'apavetvene txcives Yap “ows newiviss TÁ ATA LAT tuts dndpurs ext thy yprorv sbvouv mapaozentZavar to res En SE the après sita Enoigoms Sirs, tag tv alperas vis dì virpwris. Tag iv rhnprric dense 6 dins avtáv psttyy tag Sè alpetas iva xodtredoverat FICA The present tense shows that this refers to the political organization during Aristotle’s lifetime-a mixed constitution. As Armand Delatte15? ) and Minar'**) have observed, there is a striking correspondence surely alludes to the king (Ejujoyos vógeos,, Stob. 4,1,135, p. 82 line 21). 146) See p. 129. 147) "This appears to be the connection of thought between the first and second halves of this fragment; Diels, however, thinks that the two parts arc not connected. 1) "The context will not allow the very general meaning of “consideration”. 159) Pel. 1301 B 26-27, reading 65 py et with Newman and Ross, 160) Eanes 757 À. between this and Archytas fragment 3, in particular between Archytas’ 312 Tobrov ulv ot névates AapAdvevTr xape Tüv Suvapévey, ul te TAuboLr Mi. Gorg. 508 A. ha fact, of course, it is the opposite of any sort of labzys. 58) In this 1 follow E. L. Minar, op. cit. (in n. 112). 86 1£ The particular situation at 1 arentum is more relevant than Pythagorean political beliels in general, on which see ib. 15-35. case, continual references to kingship in the fragments as a whole (sec p. 1:34); and within this fragment, the phrase póvxpzos vópos (Stub. 4,1,137, p- 84 lines y -10) de nat rpcéorr, Tis ROdemg rudo» yedvov. 2 143) IL is not easy to decide whether x20" órepindy belongs with toyvaav or Inpoxparizios. Minar takes it with dqusxpatwéic, but earlier editors placed acomma alter énepBodjy. 1) Loe. cit. in n. 152. 23) Pal. 1303 A 3. me) “Savonarola introduced a system like the Tarentine at Florence probabl being influenced by the teaching of the Politics” (Newman ad loc.) 87) Op. cit. in n. 112. 169) Op. cit. in n. 112.

SwWdvet vois deouévous and Aristotle’s éxetvor yap row nouwbvres tà uthpaca tots anépurc!59), Archytas, then, was a leading politician in a state which combined democratic and non-democratic features. In such a position he might well fcel the need for a theory to show that exceptional men deserve exceptional privileges, and the analogy of geometric proportion would fulfil this function admirably. It is an attractive suggestion — but perhaps an implausibly neat one — that it was devised to justily his highly successful but illegal tenure of the arparnyia for seven successive years?®), So it looks as if Auyıapög might very well mean geometric proportion. However, another fragment'*) shows (i) that & Asyıorıza for Archytas meant arithmetic, and (ii) that he regarded arithmetic as superior to gcometry, on the interesting grounds that unlike the latter it could furnish proofs!#2). This might suggest that Jovy:epés in fragment 3 should be connected with arithmetic rather than with geometry; and that Archytas is perhaps unlikely to have preferred geometric proportion to arithmetical. Certainly it must make us hesitate before assuming too confidently that Aoyioués is geometric proportion; perhaps, after all, we should translate it in the most general way possible, as “formula”, or, better, “calculation” — which is what we would expect the unquali16) "The fact that Archytas was elected otpxtqyég seven times in succession orb TWO KINDS OF EQUALITY 141 fied word Auyroués Lo mean!9), But there is a strong case, as we have scen, for saying that Archytas’ Xuytopég is geometr ic proportion, and these objections, [ believe, are not sufficiently weighty to refute it. There is, moreover, one argument against taking hoyioués as “cal. culation”: while it makes excellent sense to talk of the “discovery” of geometric proportion (Aoyiouds eúpesic), it is hardly as natural to speak of calculation in general as eópedeiz. If we accept the identification of %oyıspög with geometric proportion, as 1 suggest we should, we could then draw the satisfying conclusion that we had found the origin of the application of the proportion to politics, and the exact reference of the allusion in the Gorgias'*), We may now summarize the conclusions of this section. doyisuós in Archytas fragment B 3 could be simply “calcula tion”; the evidence which suggests that it is geometric proportion is very strong; but we have found nothing whatever to make us suppose that it might be harmonic proportion. i 3. Geometric proportion is always recommended as the modelfor political justice. I have already!%) touched on another argument against Morrison’s theory, an argument which is, I think, both simple and conclusive. The whole tradition of Platonic and Aristotelian thought recommends geometrie proportion as 74 Ölxarov té rorirév is), but we hear nothing thy rodırav (the Suda, DK 47 A 2 - perhaps a misleading variant on roderóv of this harmonic proportion in the whole of classical political &arparhynas, Diog. Laert. 8.79 = DK A 1, the source of other material in this sophy!*), For Plato in the Laws, geometric proporti context) dees imply a democratic constitution. 160) See n. 159. is axqeardry za &plorn!°®), and is even more highly praised 161) DK 47B 4; authenticity undisputed, been in the Gorgias!®); for Aristotle, too, the oligarch 162) On the implications of this, sce O, Nevcenauer The Exact Sciences in Antiquity? (Providence, 1957), 147-8. Archytas is, of course, writing before the discoveries of Eudoxus, whose great advances in geometry “put him in the very first rank of mathematicians of all philoon, the Ards xptouc, than it has ic opposite of the democratic arithmetical proportion is the gcometric!*), and this is the model of one form of justice in the Ethics"), Nor do Plato and Aristotle times” (JT. L. Hears, Greek Mathematics 1 (Oxford, 1921), 323. That Plato should regard gcometry as superior to arithmetic (as his application of the proportions implies) indicates that he was not influenced on this point by Archytas, but rather 163) "Ia + “ ES î sd wa of doyGeoae in the same fragment would Support this conclusio n. by the achievements of Archytas’ pupil in geometry, Eudoxus, and by his own 185) p. 136, friend and pupil Thcactetus, (The fact that Eudoxus was a pupil of Archytas is 166) Pr. Laws 757 €. taken by Diogenes Laertius (8,86) from the Pinakes of Callimachus, a literary history in 120 volumes, and is more likely to be true than the statement that he was a pupil of Plato, which Diogenes takes from Sotion; for Callimachus wrote se ari pans is made by Douds in his edition of the 199) 508 A. before Sotion, who in his Audoyat was the first to employ the artificial system of master-pupil relationships (sce Kirk and Raven, The Pre-Socratie Philosophers, *)). 9 Pol, 1301 B 26 fl. m) EN 1131 Borg Of Gorgias, 20 n. 3.

stand alone: Isocrates!?2) and, later, Plutarch!??) give exactly the same version of the theory!?!), 143 Morrison, following Cornford, translates the opening words of (i) “bringing into tune those three parts like the terms in the proportion of a musical scale”. I do not find all this in the Greek. The octave (Stamasév) was composed of two tetrachords which might be conjunct 4. The »Republic« does not allude to the harmonic proportion Furthermore, do the passages in the Republic to which Morrison refers allude to the harmonic proportion? In the first, Plato is describing the harmony within the soul of the truly just man: (Grm), lowest (vetta) and fourth (péon) notes were thus the three fixed points in the scale. A modern writer making the same points might perhaps say “bringing them into harmony, exactly like tonic, mediant and dominant” - Plato intends (i) vat owapuócavra spia Gvia Gansp Opus tests Gppuviag ATELVÓS, wink te sal Ómáros vol péogg, val el Wr dera peral svyuiver Eva, mivra tabra amdicavta ual mavrémaaw Eva yevijevov En TODO, Gappave vai ipuuopivor. .. (RP 443 D-E) In the second, he is comparing opposés with a Gppovio (431 E): (ii) Ana DU Gres ares tetatai, Où naci nopezopevy awedurtas TUNG TE hollevsarkruve rasrbv yal tog lopnporároos zul tobs ptoous, el piv Bob, ppovine, el dì Budact, lait, el 33, nab de 4 xphuaow À Dis bramüv TO rounbreov Gare éplérar dv paipev tadtTyy The Guévoay supernova clyat, yeigovig te uxt spetvoves vata piow suppeviay, inétepov Set &pyew, nal Ev Casa AB} moher ai dv tut État. or successive!??); the highest!?®) a metaphor of concord, no more, There is no direct reference to proportion here, unless we lay great stress on the word Spot. But this is quite unnecessary: ópot may be terms in a proportion, but they can also be the fixed notes of the scale, the “boundaries” of the two tetrachords. This is what Plato has uppermost in his mind; I do not deny that in his allusive style he may intend a glancing mathematical suggestion. But this cannot be the primary meaning: for (a) if it were, what could we make of wat ct dda ¿rta peratò tuyydve: ivrea? Other notes in the scale would produce all sorts of proportions, not simply the favoured harmonic proportion — so that the phrase would be inept, according to Morrison’s interpretation; but they would also produce further harmonious chords, which is surely the point of Plato’s It is important to take into account the contexts of these passages. Neither deals with the ideas to which the metaphor of proportion 1s applicd elsewhere. (i) is concerned with justice in the individual, whereas the various kinds of proportion, when used in political philosophy, are concerned with justice in the state. Although a moment later!75) Socrates claims to have discovered tiv pèv dtzatov na &vSpa xal mé, the discovery of the just city must be a reference to carlier passages, for Socrates has explicitly said that he is now talking not about thy ¿do npäfıv riv abroü, but about chy ¿vrós!*9). (ii) is a description of cuppocóva in the state, not dixaoodvy, and is thus not phrase, as Shorey saw'*). (b) Again, why #22 and not £201? Plato is no longer speaking of the chief notes of the scale; the others cannot strictly be called spor. If he had meant Spor as “proportions”, then he would have written &%or. (c) Finally, the verbs cuvappécavra ... vai... cuvèfgavia are more applicable to musical notes than to terms of a proportion. The earlier editions of Liddell and Scott gave “proportions” as the meaning both here and at Philebus 17 D (Spor véiv dtactaustey); but this has been corrected in more recent editions to »rotes which limit the intervals in the musical scale”!%), A glance at Aristoxenus’ work on the relevant as an indirect parallel. er) Li + Mor. RAT (de frat, amare); 719 A © (Qnaest. convtw. $,2,2). a2 Areop, 21; cf. 3 (Nie) 14. 171) See pp. 110 113, 120 122. 79) 444 A. 158} 443 D. . 123) ARISTOXENUS Harn. 59. 123) Highest, that is, for the hand in playing the Aithara, but lowest in pitch. Sec M. 1, Henderson in the New Oxford History of Music 1 (Oxford, 1957), 345. 1) Loeb translation, note ad doc.; his Shakespearian quotation is to the point. 1%) Similarly, the recent edition of Aristoxenus by Rios (Rome, 1954) gives intervalli limes as an equivalent for Sans.

elements of music is sufficient to settle the question. There we read that an interval (dikorape) is something bounded wer, by two notes!8!), And sods riv dior, uate Spuug)**) are clearly the posts aries of the intervals”, as Macran translates, just as agai elsewhere ) pus are “bounding-notes”, a synonym of roi jpiléovte@y in the same possibly have been Archytas!®) from whom he heard about the different kinds of proportion, and perhaps about the relation of Aoytouéc to politics. ‘This was claborated by some of Plato’s Pythagor ean friends as follows: Democracy employs arithmetical proportion; this is the He was born at Tarentum, and reccived instruction from his father, Plato has been so disgusted'?!}. A good constitution has a chaptert#t). We could hardly ask for better evidence than Aristoxenus. Here he met various Pythagorean philosophers (among whom may igórac which is the basis of the Athenian constitution, with which | who was a friend both of Archytas and of Socrates'#). that claim the that so (ii); in Similarly, there is no such reference better basis, namely geometric proportion. That this doctrine was Pythagore an is virtually certain from the allusion to it in the Gorgias!®2) in connection maintained '**), Plato’s regular name for the Pythagorcans!#). The possibility Plato here abandons geometric proportion for harmonic cannot be with another undoubtedly Pythagorean belief, attribute d to of Gugol, might have been the invention of Archytas is not ruled that it out by our discussion of the doyopes fragment!*); it is very tempting Conclusions ‘Thus Morrison’s theory can be shown to go far beyond the evidence. Its rejection allows us to place the composition of the Gorgtas ee hesitation alter Plato’s visit to Sicily in 388. Dodds in his edition ‘) inclines towards this later date; Morrison’s argument would be evito say that it was, for this would enable us to give a simple and straightf orward account of the parentage of Plato’s idea. This theory appealed to Plato's love uf mathematics, and on his return he incorporated it (and other Pythagorean teaching) in the Gorgias, where the term i, dence for a date before the Sicilian journey. But the frequent references to Pythagorcanism in the Gorgias constitute a strong reason for dating it alter the visit); and we can now do so confidently. of philosophy. In of the history This enables us to rewrite this fragment 388 Plato visited [taly and Sicily “ut Pythagorac inventa perdiscerct”**). 191) ARISTOXENUS Harm, 15. 12) ib, 49, cl. Pi. Phib, 17 D. 182) "The same meaning is required in ib. 59, rn cited in n. 177. . 86. | a tg Pist.) tells us that the harmonic proportion si Tamanicnus (in Mic. 118.23Babylonians) was used by various en ar ry 15) See Macran’s edition (Oxford, 1902), introd., he thinks was invented by the “Plato in the Tiaueus” (sce n. 102). This is quite true (Tim. 36 A-B) ; and if pis ten implies “in the Timeeus but not elsewhere” - as it surely does it would contra our conclusions. 187} 26-7. In Fubcro, ee ern16: sed audisse te credo,Ivaliam mt Cic. de Re Publica 1, (10, discendi et in Sici a con causa, post in we) Dodds, doc. cit. | \ mortuo primum in Acgyptum tendisse, ut Pythagorac inventa perdisceret, eumque et cum Archyta + in / es multum fuisse; no doubt an inference from Plato’s 7th Letter, 326 \-B, bd Plato gives dissatisfaction with contemporary politics and interest in politic: philosophy as motives for visiting Italy and Sicily. Demosthenic, is quite possibly fourth century (N.W. and N.J. de Witt, Loeb Demosthenes VII, 41). ‘There is, however, no clear evidence for a meeting in 388; yet such a mecting is taken for granted by many modern scholars (c.g. FieLv Plato and his Contemporaries (London, 1930), 14; Minar Early Pythagorean Politics, 92: SouiLué, translation of Plato’s Letters (Paris, 1949), n. on 338 C; SincLarr History of Greek Political Thought, 124; SARTON History of Science 1 (London, 1953), 397; Dodds, edition of Gorgias, introd. 26.) Morrison, while granting o 192) jb. 56. 124) À later meeting with Archytas is certain (Pl. Ep. 7. 338 C, referring to Platos second visit in 367); on the good effect which Archytas’ friendship with Plato had on Archytas’ reputation, sec ps.-Demosth. 61 (Erot.), 46; this work, even if not this (C.Q. ns. viii (1958), 202 n. 1), argues as though a meeting in 388 were more or less certain, relying on Cicero (cited in n. 92), whose account of sufficiently precise to furnish any chronological data. Plato’s western travels is not (‘Two letters survive purporting to be from Plato to Archytas (PI Ep. 9; and 12 = Diog. Lacrt. 8,79), and one from Archytas to Plato (Diog. Laert. 8,80). Of these, Plato’s 12th letter and that of Archytas are certainly spurious, and Plato's gth letter almost certainly so, despite Cicero's citations, de Fin. 2,14, de Offic. 1,7. Vhe reasons for rejecting them are clearly stated by Souilhé in the Budé edition of the Letters, introd. XCIV-XCVIII.) 191) Ep. 7, 326 AB. 192) 508 A. 192) Sce Dodds’ long note un pp. 337-8 of his edition. Cf. Plut. Mor. 719 AC (Quaest. convir. 8,2,2), where Dicacarchus {fr. 42) is quoted as saying that Plato “mixed Lycurgus and Pythagoras with Socrates” by introduci ng geometric proportion, 191) See p. 141.

loben à yeoperpzi “is introduced without a word of explanation, as if it were already familiar”!9), This is curious; either the Gorgias was intended in the first place fur Plato’s immediate circle, to whom he had explained the idea, or else Plato is making a deliberately casual reference to a newly-acquired piece of knowledge — not an uncommon human habit, An echo of the idea recurs for a moment inthe Republici9o), ladrate. two ópolos touts ve xal Avlaoıg Stavipovoa; and in his last work, the Laws'**), Plato expounded it at length. Plato never swerved from his loyalty to geometric proportion into a flirtation with harmonic proportion, and Aristotle after him espoused the same principle. Geometric proportion was discovered by the Pythagoreans, and it was they who first applied it to political philosophy; but there is no good evidence that either they, or Plato, or Aristotle attempted to do the same with harmonic proportion), Harvey F. D., Two kinds of equalily [chez les Pythagoriciens] ; cf. Plato Philosgphus. | ; . | ARVEY F. D., Two kinds of equality : C&M XXVI 1965 101-146. | L’his- | toire de la théorie de la proportion géométrique, opposte à la proportion | arithmétique en tant que fondement de l’égalité démocratique, et la critique de l'hypothèse de J. S. Morrison (cf. APh XXVII 144, sous Pythagorica, | & XXIX 148, sous Plato) portent à placer la composition du Gorgias après le voyage en Sicile de 388, voyage au cours duquel Platon aurait rencontré des philosophes pythagoriciens qui lui auraient communiqué leurs idées sur les différents genres de proportions et la relation du Aoytouée avec la politique. 12) Dodds on 508 A. We cannot be sure that the theory of geometrie proportion was not fairly widely known in intellectual circles in Athens in the fourth century. In that case, Plato might have heard about it there just as easily as in the West; but there is no evidence for such an assumption. 195) 558 €, 7) 757 A if. 122) My conclusions, if correct, are surprising: given the Pythagorean preoccupation with music and dpyevlar and their interest in politics, one would cxpeet them to have applied harmonic proportion to politics, But what might be expected is not necessarily the truth.