The mathematical passage in the Epinomis

Lacey, A. R., THE MATHEMATICAL PASSAGE IN THE EPINOMIS, Phronesis, 1:2 (1956:May) p.81 bona Leer, ASR | \ASe The Mathematical Passage in the Epinomis *) A. R. LACEY HE general theme of the Epinomis is set by its opening question, how can mortal man be wise? The answer is that wisdom and happiness are only for the few, and these will need an intensive intellectual training, whose objects are to be the nature of the universe, its elements and inhabitants, and the basic laws by which it is held together and governed. Nowhere are these laws better exhibited than in the complex but regular and eternal movements of the celestial phenomena, and so, absurd though this may seem to the layman, the path to wisdom lies through astronomy, studied mathematically and not merely empirically (990a2-b4). At this point we have a short but notoriously difficult passage (990c 5-1 b 4) which starts by saying that the study of astronomy involves that of mathematics, and eventually leads up to the panegyric, which ends the dialogue, of the structural unity of the universe, which must be studied and appreciated by those who are to form the Nocturnal Council of Laws 12, Various writers, notably Stenzel, Taylor, des Places, Toeplitz, and Van der Waerden, have tried to interpret this passage, but it seems to me that despite many ingenious and fruitful ideas on individual sentences none of them has succeeded in giving a satisfactory interpretation that takes account of both philosophical probability and the exact Greek text of the entire passage. In this paper I shall try to take the problem one step further by a detailed examination of the grammar and syntax of each of the five sentences into which I have divided the passage, together with an attempt (tentative enough, to be sure) to link them together into a coherent philosophical whole. It need hardly be added that any success I may have will be built on the foundations laid down by previous writers, especially those mentioned above. I have not attempted to say anything on the vexed question of the authorship of the Epinomis. I have in fact assumed that the author is Plato, which the balance of evidence seems to support (see Hans Raeder: Platons Epinomis, Raeder disposes effectively of the view that Philip of Opus wrote it as a deliberate forgery to be passed off under Plato’s name; he says regrettably little about the mathematical passage, to which he gives only two and a half pages, pp. 56-8). This question becomes most pressing in connexion with conflicting doctrines, such as the ether and the demonology; with our present passage the chief task is that of giving

The Mathematical Passage in the Epinomis ?) LACEY HE general theme of the Epinomis is set by its opening question, how T can mortal man be wise? The answer is that wisdom and happiness are only for the few, and these will need an intensive intellectual training, whose objects are to be the nature of the universe, its elements and inhabitants, and the basic laws by which it is held together and governed. Nowhere are these laws better exhibited than in the complex but regular and eternal movements of the celestial phenomena, and so, absurd though this may seem to the layman, the path to wisdom lies through astronomy, studied mathematically and not merely empirically (990a2-b4). At this point we have a short but notoriously difficult passage (990c 5-1 b 4) which starts by saying that the study of astronomy involves that of mathematics, and eventually leads up to the panegyric, which ends the dialogue, of the structural unity of the universe, which must be studied and appreciated by those who are to form the Nocturnal Council of Laws 12. Various writers, notably Stenzel, Taylor, des Places, Toeplitz, and Van der Waerden, have tried to interpret this passage, but it seems to me that despite many ingenious and fruitful ideas on individual sentences none of them has succeeded in giving a satisfactory interpretation that takes account of both philosophical probability and the exact Greek text of the entire passage. In this paper I shall try to take the problem one step further by a detailed examination of the grammar and syntax of each of the five sentences into which I have divided the passage, together with an attempt (tentative enough, to be sure) to link them together into a coherent philosophical whole. It need hardly be added that any success I may have will be built on the foundations laid down by previous writers, especially those mentioned above. I have not attempted to say anything on the vexed question of the authorship of the Epinomis. I have in fact assumed that the author is Plato, which the balance of evidence seems to support (see Hans Raeder: Platons Epinomis. Raeder disposes effectively of the view that Philip of Opus wrote it as a deliberate forgery to be passed off under Plato’s name; he says regrettably little about the mathematical passage, to which he gives only two and a half pages, pp. 56-8). This question becomes most pressing in connexion with conflicting doctrines, such as the ether and the demonology; with our present passage the chief task is that of giving

it a coherent meaning at all, not that of reconciling it with conflicting passages. It does not appear to clash with any other Platonic passages, and its very brevity and obscurity would lead us to conclude that, even if the Epinomis is a forgery, its author had no interest in proclaiming a new doctrine in this sphere, and so would keep as near the Platonic as possible in order to make his work the more plausible.? In fact, with the exception of Laws 894a and 817e-22d, our passage seems to stand in relative isolation among the dialogues. I cannot see that it has any particular connexion with the Nuptial Number (Rep 546ab) or the Tyrant’s Number (Rep 587). These passages, which occur in a political context, have been recently reinterpreted by R. S. Brumbaugh in Plato’s Mathematical Imagination. Brumbaugh does not treat the Epinomis passage in detail, though he connects it with Laws 894a and de Anima 4044 and discusses the general significance of the three mathematical means for Plato (chapter 4). For purposes of exposition I shall divide the passage into five sentences, and devote most of my attention to the last three, though one must bear in mind all through that the sentences form a single passage with (presumably) a consistent development from beginning to end. Epinomis 990c 5-1b4 1. 910 uaÜnuaTov déov dv ely To de ueytotév te Hal rpoitov xal dpidudiv $ e > AUTOY GAA > > 1 > Ff > vo eu ~ Pr x OÙ CWUATA EXOVTOV, GAMA OARS TNG TOD TEPITTOD TE xa dpriov yevéoeac te xai Suvauemc, donv Tapéyerar mods THY TeV SvtHY quan. 2. tadta de uadovir toutots Epic ¿ori è xadovar pÈv apddea yehotoy Gvoua yempetplav, Tv oùx bvrwv Sì ópolov dAANACIS pioer Apr uv Golwarg Tpòs THY TÜV Erimedwv potoav yeyovuta ¿ori Srapavyg è dh Badua oùx dvBporivov dAAL yeyovös Betov pavepóv dv ylyvorro tO duvapéve cuvvoriv. pera de Tabrnv tobs tpic MÉnuévouc xal TH ateped puoer Guotouc’ tous de dvouotoug ad yeyovótas Etéog téyvyn Sporci, TAUTY HY Sy atepcouetpiav Exdiecav ol moootuyetc Kom yeyovótec" 3. 0 dé Oetóv 7’ goriv xal Oavpacrov tots éyxaBopdiat te xal Stavooupévors a A nt 3 3 x \ \ ~ > ~ [4 x Ed 6 Tepi TO dinAdorov del otpepouévys THs Suvauews xai tHe EE évavetac rauen nad’ ExacoTyy dvadoyiav eldoc xai yévos dnotUMOUTAL TEN Odor. 4. N pev dy npam tod Jimiaciov xar’ Apıduöv Ev mpdc dbo xark Adyov [4 A x € x > x 14 x pEpouévn, drrAdotov de Y Kara Sivautv odoa* 7 8° > eis td otepedy TE xal € 4 f Y # > 9 € F AX x 3 > PEAS x ~ ATETOV TAAL ad OLTTALOLOV. AP’ Evög elc ÓXTO Statopevdeica.

5. 4 dE dumAactov piv els uéoov, tows SÌ tod éAattovoc mAéov ¿hartóv ToD pelGovoc, TO 8° Erepov TH adro pépel TÜV &xpwv adtav Ùrepéyov TE xal UTrepexdpevov - Ev pto Sè rod EE npdc tà Ibdexa ouveßn TO Te TpLóMov xal émirpiroy - tobtwy abrav dv tO uéom En’ dupérepa otpepouevy Toîc avOpmmoig albupwvov ypelav nai OÙLUETPOV dreveluat o radis fubuod te xal dpuoviag ydpuw, eddaiuovt xopeta Movoüy dedonevn. 1. And so there will be a need of studies. Now the most important and primary study is that of numbers themselves, not of embodied numbers but of the whole coming-into-being and potentiality crease [by squaring and cubing] of the odd and even, and of of inthe effect which this has towards the nature of things. 2. The man who has learnt this will next study what is called by the ridiculous name of earth-measurement, but has manifes tly become the assimilation to each other, by means of their share in planes, of numbers which are not naturally similar to each other, a wonder which would clearly be of divine, not human, origin, to one who could understand it. After this he will study numbers that have been increased to the third degree, and resemble solids; and those which have become dissimilar again one assimilates by another art, this art which those who first came upon it called solid-measurement indeed — 3. which is also a divine and wonderful thing to those who perceive reflect on how, while the power and the term which and consists in the correlative root to this ever revolve about the double in each proportion, the whole of Nature is stamped out, genus and species. 4. The first term is of the double, being taken numerically from 1 to 2 in ratio. The term which is a power is also a double, and that which goes to the solid and tangible, having gone from 1 to 8. so again is right through 5. Now the term which goes into the middle of a double, but is equally greater than the less and less than the greater, while the other middle position exceeds and is exceeded by the same fractio n of the extremes themselves (the ratios 11 and 14 can be found in the middle of the ratio 6 : 12), in the middle of these, turning towards each of them, this term provides for men a harmonious and balanced practice for the sake of disportation, rhythm and concord, being given to the blessed choir of the Muses. 83

I. 990C5-d1 Stò... TOY ÓVTOV puo. Des Places (Revue des Etudes Grecques, 1935, pp. 540-50; see p. 547) follows Bekker and Z in omitting xaì before dpi8uéiv, and Toeplitz (Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abteilung B! Vol. 2, p. 334ff. (“Die mathematische Epinomisstelle”)3 favours this course. In this case &p@uüv will be predicate to tó Sì péyiotóv te xal Toro (udOnua). Otherwise we must take +d Sì péytoróv te ai moatov as adverbial, and &pıdu.üv as parallel to undnuarwv, and depending on déov. The excision seems favoured by tadra de padóve: in the next sentence. It has been suggested to me that the first a’ implies there are two kinds of &p@uot adrot, the owuarat Eyovres and the oopara oùx Éyovrec, the former being the Forms (which “have bodies” in the sense that they appear through the medium of physical objects), and the latter the Ideal Numbers, which are superior to the Forms. It seems easier to me to follow the usual view, which takes “bodiless numbers” to mean simply numbers, considered in the abstract, and “embodied numbers” to mean sensible numbered groups (see G. Martin, Zeitschrift ftir philosophische Forschung, 1953, p. 191ff, and Tht 196a; cf. also des Places, loc. cit.), the first 4X’ being taken in connexion with the second; the study is not to be of sensible numbered groups, but of “the whole coming-into-being and potentiality of increase of the odd and even”. It seems best to understand Sbvapıv after 6onv, but in a somewhat different sense from dvvanews, because it would be grammatically awkward to make Guvauewc the antecedent of Scyv without making yevésews so too, which would not make sense; but we must assume that it was the proximity of Suvauews that made Plato omit Suvautv after 6onv. yevecewg is not easy to interpret. Aristotle tells us of the generating of numbers from the One and the Indefinite Dyad, and perhaps something of the sort is referred to here. What seems certain from the rest of the passage is that Plato is thinking mainly of the Sbvauc (in whichever sense) of the numbers when already generated, rather than of their generation. 2. 990d1-e1 tabra de pabdvete... yeyovdtec. This must be treated as a single unit, because there are disagreements over its punctuation. O. Becker (QS 4 p. 191), in an article which brings in part of our passage, suppresses the fullstop after ouvvoeiv, and the Sì between puetà and tabrmy, and inserts an ¿ori after Oadux

(deleting the ¿ori before d:apavyjg in the previous clause). He takes è 87 dabya 5 to have a forward reference, his general interpretation being that four studies are in question, arithmetic, geometry, stereometry and harmony, of which the first two have been known for a long time, while the last two are new and therefore of special significance. While this view no doubt has its advantages it would be better to translate the received text if we can, and to give è 3) Vaya a forward reference destroys the grammatical connexion of the sentence with the one before. It would also seem more natural for this interpretation that the next clause should start robe yap &vonolous rather than tobe Sì dvouotoue. As a second and less satisfactory course Becker suggests putting a stop after yeouerpia and enclosing uetà de rabrnv in brackets or commas; this would get rid of the è 87 Bava difficulty, and avoid excising dè, but the bracketing is awkward, and it is hard to see why the geometrical assimilation of numbers should become any clearer to a person who knows stereometry. Keeping to Burnet’s text we shall have to supply some such word as pabytéov (Müller) with tobe reis ndEnuévoue. 6 The more well-known textual difficulties towards the end of this passage do not seem to disturb the meaning so much. I have translated Burnet’s text, and Toeplitz does so too. Des Places, who dislikes tabty as referring to étépa téyvy, and the absence of a subject for ôuotot, reads with Z robs dè dvopoloug ad yeyovóras Eripa téxvn duotoî duota TAUTY, NV ON otepcouetplav..., TabTy referring to geometry. Stallbaum and Müller follow A and O (against L? and Theo of Smyrna) in reading yewpetptav for otepsoperpiav, and it has been suggested to me that this is supported by 87; this reading will cause no trouble so long as we keep 6puota (or éuoix, according as we read érépa téyvn or &repa Téxvn) instead of, or as well as, 6uotoî. The general doctrine of the passage is wellknown. Two numbers are 6yotot when each of them can be represented by a rectangle (if it is split into two integral factors) or a rectangular solid (if it is split into three integral factors) whose dimensions correspond to these factors, and these rectangles or rectangular solids are geometrically similar to each other (Tht 148a, H. Raeder: Platons Epinomis, pp. 56-7, des Places, p. 548, Toeplitz: QS 2 p. 334ff.). (Another sense of &uouoc, in which a number is óuotos if it has symmetrical factors, i.e. if it is a perfect square or cube, is mentioned by Becker (QS 4 p. 185ff.), who supposes it to underlie Rep 5465.) But obviously only a very few numbers are Goto. in this (the first) sense (dvteg Spotor KAANAoıs oboe, as Plato puts it), and the purpose of geometry and stereometry is to assimilate

the others: Toeplitz points out that it is not at once obvious that a number which can be factorised in different ways (e.g. 30 = 6 x sor3 x 10 15) will be similar to the same numbers in each case, and, drawing attention to Euclid’s proof that two numbers are similar when and only when they are related to each other as two squares, he thinks the “assimilation” of two numbers means the constructing of squares (or cubes) which bear the same ratio to each other in area (or volume) as the two numbers do, and that 1 and 2 are assimilated by squares in this sense in the Meno, while the Delian problem is to assimilate them by cubes. In general this seems to be undoubtedly the right interpretation, but it is not necessary to limit it to squares. The general principle of finding the geometric mean between two lines by laying them end to end as the diameter of a circle and dropping the perpendicular from the circle to their meeting-point (see Toeplitz: QS 1 p. 3ff. ($ 1)) can be used equally well to enable one to construct a rectangle which will be similar to a given rectangle and which will stand to it in area as two given lines stand to each other.” Thus if we have the number 6 expressed as a rectangle whose sides are 2 and 3, we can take the number 13, which cannot be expressed as a similar rectangle so long as it is factorised integrally, and express it as a similar rectangle, whose dimensions, however, are surds. Stereometry will do the same thing for numbers which when viewed as solids have become dissimilar again. Why this word “again” (xò)? Presumably what is meant is that if after constructing our two rectangles of areas (say) 6 and 13 we try and turn them into volumes of 6 and 13, we can in the first instance only do so by giving to each of them. unit height, which will of course make them dissimilar again, because the larger one will be too flat. To remedy this we must raise its height and lessen its other two dimensions, while keeping them in the same mutual proportion. This can be done be stereometry.8 (That it can be done, whether or not this solution “counts”, is shown by Archytas’ solution of the Delian problem (T. L. Heath: Greek Mathematics, Vol. 1), so that it seems better to take ôuotot as indicative and not, with Becker, as optative (“môge er durch eine andere Kunst ähnlich gestalten”). A further minor difficulty in translation is to know whether to take épotovc after Tf orepe& pbceı as technical or non-technical; the frequent use of the word in the context, and the presumably technical use of &vopotoug in the next clause, might suggest that it was technical here too; but the fact that the numbers mainly to be studied are those which are not similar until they are made so, coupled with the absence of d¿ANhor and the awkwardness of the dative, 1% otepeà quo, perhaps makes it best

to take it as non-technical. With the technical meaning the phrase would mean: “study numbers with respect to their similarity (sc. or lack of it) in the third dimension”). 3. 990€ 1-4 è dE Delov... rica N odore. It is not easy to know whether to give è dè 0etov a forward or a backward reference. Becker gives it a forward one, as he does to è 5h daöu«, and so does Toeplitz. This makes the sentence rather awkward grammatically, since the &ç clause has to serve as the antecedent of è, although one would naturally take it with totic éyxaBopéat te wat Suavoouuévors, and I have preferred to give it a backward reference, as this helps to unify the passage, and relates the study of stereometry to what is coming.? The next difficulty concerns Suvéuews. Reference to Liddell and Scott, and Ast, shows that the mathematical meaning of this is normally “root”10 or “power”, though Philo uses it for “product”; it seems to mean “that which is potentially something” or “that which something potentially is” (cf. Becker: QS 4 p. 185ff., who quotes Proclus: in Remp. 51.9ff. in interpreting Suvéueva and duvactevdueva of Rep 5465). The very fact that it covered both these meanings suggests that the use of it can hardly have been very strict11, and it would seem difficult if not impossible to take it strictly in either of these senses in our present passage. We only seem to have one instance of a square (4 = 2?) and one of a cube (8 = 23), but we have mention of the number one (&v rpòg duo, 99142), and of the step from 4 to 8 (or else from 1 to 8; 4 $” sig +d ctepeòv, 991 a 3). But the geometrical use of Sbvapuc in connexion with the process of squaring seems to depend on taking a line as being potentially a square, and as becoming a square when drawn out into the second dimension (cf. Leg 894a), and it seems to be part of this view that in just the same sense the point is potentially a line, and the plane potentially a solid; otherwise the symmetry which gives the view its attractiveness disappears. (Cf. Stenzel: ZG 9412: “Das, Können’ besteht gerade in dem Produzieren der nächsten Dimension”. Stenzel refers to Tht 148 b). It seems reasonable enough to add that a point is potentially (though perhaps only indirectly so) a solid, and so with the other cases. This view would enable us to say that any one of the numbers 1, 2, 4, 8 is a düvautç with respect to any of the others, though primarily only with respect to the terms standing next to it in this series. The natural interpretation of hg è évavtiag tabry will on this view be the number with respect to which the Süvauc is a Sbvautc, or the correlative of the Sövauıc. Des Places seems to incline tentatively to this

view (“Ce pourrait être la racine”, p. tempting step of deleting ¿£. 549), and Müller takes the (The rather radical alteration of Theiler (Gnomon 1931 p. 345n3), who reads 10 2& évavrias for tho && Evavriac, seems to be without authority, and it will be better to interpret a safer reading if we can.) Van der Waerden (Hermes, 1943, pp. 185-7) translates: “die Kraft und die aus der Gegensetzung zu ihr... resultierende Kraft”, which he does not comment on in detail, but which if taken literally would suggest a noun évavtta, like ¿vavriórnc, which does not seem to exist. If we are to keep 2& the only alternatives seem to be to supply some feminine noun (like y&pa) meaning “region” or “direction”, or else to understand Suv&pewc again after évavtiac and to take 2& in a vague sort of way as meaning “formed out of” or “consisting in” (“that Sévautc which consists in (or is) the opposite Sbvaus to this”). Taking is Suvápews and Tis È évavtiag as correlatives in this way makes it unnecessary to consider which is root and which power (or which is lower term and which higher term).13 Toeplitz takes Süvauiç in quite a different sense, as “Wirkungsweise (Art zu functionieren)”, and this is followed by Van der Waerden, who takes the Sivauic to be the power (in the ordinary, non-mathematical sense) of the double in creating the 3urAdorov series, and the converse to it as the power of inserting means; in other words both these writers take Suvautc in the non-mathematical sense.14 In interpreting BinAdotov Van der Waerden recalls the Euclidean use of 3urAaciwv A6Yoc to signify the squaring of a ratio (Mathematische Annalen, 118, (1942), p. 287; Euclid 8.8, 18), and says that sentence 4 of our passage has to do with squaring, and sentence 5 with the converse process of inserting means, and in particular the arithmetic and harmonic means, the geometric mean underlying sentence 4 (Hermes, 1943, p. 186, n. 1). But there are difficulties in this. Euclid uses StrAaotwy to signify the square of a ratio, but he uses rpurhacio to signify the cube (Euclid 8.9, 19), and there is no reference to a tpimAdotov here, though there is a reference to a process “from 1 to 8”. The theory seems to be that Sivaytg refers to the multiplication of ratios, of which the particular case where they are multiplied by themselves (8inAccrov) is being used here. In this case we should expect the converse process to be the division of ratios; but this is not altogether satisfactory as a term for mean-building, for though the arithmetic mean is obtained by dividing the sum of the extremes by two, and the harmonic mean by dividing the product of the extremes by the arithmetic mean, the geometric mean is not obtained by dividing at all, but by taking the square root of the product of the extremes, and taking

a square root does not seem to be a special case of division in, the sense that squaring is a special case of multiplication. In any case is it not very vague to talk about the faculty of multiplication and the faculty of division “revolving about the double” when what is meant is that the particular multiplication in question is multiplication by itself, or by two (according to which view one takes)? (It is true that in the first sentence of our passage Suvéuems is used in a sense related to, but not identica with, the technical sense. This merely seems to support Souilhé’s view of the vagueness of the term.) &vaAoyta means primarily mathematical proportion (Tim 31c, 32c). It is also used of the three progressions, and of proportion generally and analogy, and later was used for relation, correspondence, or resemblance (Liddell and Scott; cf. also Stenzel: QS 1 p. 34ff.). It seems to be generally agreed that in our present passage xaf” Éxaornv dvaroyiav refers to the three types of progression, arithmetic, harmonic and geometric, though it is worth nothing that it probably could be used to refer to the three instances of the dixAcctov ratio mentioned in sentence 4, in which case the subject of y in sentence 4 (and presumably therefore in sentence 5) would be dvadoyta rather than (as I shall suggest) Súvayts though in this case we should perhaps rather expect xa0’ Exaotov Adyov instead of 400” Extornv avaroytav. What is not so generally agreed is whether x«0’ ¿xdornv dvaroyiav is to be taken with Tic è Zvavrias cavi (Toeplitz, Van der Waerden), or with orpepouévne (des Places), or with arorunoüraı ráca + pbcıs (Müller, Boulliau (apud Stallbaum)). anoturottat is taken as passive by Boulliau and Toeplitz (though Toeplitz thinks it might have an active sense, with eldog xal yévos as object, the sense being the same), and as middle by Van der Waerden, des Places, Müller, Taylor (apud Harward), and Stenzel. Stenzel (ZG 100) argues that Plato uses the termin the middle on the other occasions when he uses it (Leg 681b, Tht 191d, Tim 39e), and that the last of these is especially significant. But because Plato uses a verb in the middle on three occasions he need not necessarily be using it in the middle on the fourth, and a striking difference between Tim 39e and the present passage is that there the subject is the Demiurge, and here nica 7) oo; it is one thing for the Demiurge to do the stamping, but another thing for Nature herself to do it. Also why ráca à gior? Surely there is no reason to emphasise that the whole of Nature does the stamping — but it would be much more relevant to emphasise that the whole of Nature was stamped, and not just one part of it (such as numbers, as Plato’s readers might have expected from the talk about Sivauc and

Sırıdarov). If axotumottat is then to be taken as passive we are left with the questions: Is the subject nica % quote, or elSoc xal yévos, m&oa à pborc being an explanatory apposition (or even spurious; Reuther, quoted by Stenzel: ZG 99n)? If the former, is elSoc xat yévoc an apposition, or an internal accusative (“stamped into species and genus”)? Stenzel, as is well-known, lays considerable stress on this passage for his triple diaeresis theory, and des Places says one can go further, and that in each number there are two constitutive elements, the genus (the Great and Small) and the specific difference, or species (the One). Whether these or similar interpretations can be accepted cannot be decided on the basis of this passage alone, and all we can do at the moment is to point out that, whereas it is not necessary to take &xotumodrat as middle in order to support Stenzel’s view, it is also possible that the phrase etdog xai yévos is an innocuous apposition (possibly underling räox: “all Nature, both species and genera”). A minor point is that on the technical interpretation of Stenzel, and others, one might perhaps have expected yévos xai eldog instead of eldog xaì yévos, the things which are being stamped being written in the order in which they are being stamped. Toeplitz however takes d&xoturotta: to mean not “stamped out” but “mirrored” or “typified” (abgespiegelt), and considers that the double is used as an example in the various spheres that are mentioned in our passage, since the “assimilation” of 1 and 2 in the Meno gives an example of plane assimilation, and the Delian problem of doubling a cube gives an example of solid assimilation. But to this there are two objections. It is surely not in accordance with the usual meaning of &roruroöode:, which at its other Platonic occurrences (Leg 681b, Tht 191d, Tim 39e) seems to mean “typify” or “give an “stamp out” or “impress” rather than example of”. Secondly, in the final sentence of our passage, which, whatever its detailed meaning, obviously refers in some way to music and harmony, the double has a significance in its own right, and not merely as an instance of a wider set of ratios, for the octave, when one listens to it, at any rate seems to be a fundamental datum of harmony, and not merely something imposed on it for the convenience of theorists. We still have not considered the meaning of otpepouévys, and this brings us to the general interpretation of the whole sentence. What can it possibly mean to say that the root and the power, or the lower and the higher term in a series, “turn about the double” ?15 One thing that suggests itself (especially in view of det. This need not have anything to do with infinite series, but certainly implies that whatever is happening happens more than once. It may be no more than a reinforcement of x0’

£xdormv &vadoytav, if this is taken with otpepopévys) is that the situation is not just a static one, with two Suvéuers and a SimAdorov between them, but that there are either several duvduerc and GirAtoux, or anyway a Suva and its correlate and a SirAdotoy appearing in several positions. This can best be accounted for if we are considering a series. xa0” £x&orny (rather than éxatépav) &vadoyiav suggests that not less than three types of proportion are being referred to. If we construct a scale where each term is half its successor (i.e. its successor is StrA&otov), and then put in the three means, we can write down three scales as follows: Arithmetic: I 14 2 3 4 6 8 4 4/2 Geometric: I [2 2 2/2 8 Harmonic: 1 1$ 2 2% 4 53 8 Now it is obvious here that not only are the basic terms in each of these series (i.e. 1, 2, 4, 8,) ina StrAdotoy relation, but the mean terms in each series are so too (3 is twice 14, 2/2 is twice 2, 23 is twice 14), and so if the Súvapic and its correlate can be treated as general terms for any terms in the basic series (1, 2, 4, 8) then it might be said that the Süvauic which occupies each position in this series in turn, “revolves round” the intervening means, which also form pairs in a SrAdotoy relation. The same could be said of course about the correlate to the Suvaptc, and similarly the means can be regarded as the Sbvauc and its correlate, revolving round the terms of the original dimdcouov series. This is not a very exact account, and it would only apply to the series looked at generally, and not to the first step (14, /2, and 14 are not doubles, though they are, in a looser sense, terms in the StrAdotov relation, in that they are halves), but it does pay some attention to the meaning of the terms involved, and seems no more metaphorical than such a passage as Leg 894a. It will now be seen that it is best to take xa exaotyy dvaroyiav with otpepouévys. To have taken it with the succeeding phrase would have been awkward because it seems most obviously to refer to something mathematical rather than to the physical or metaphysical world that is somehow being derived from or compared with the mathematical one, while to take it with ig ¿£ évavriac tadry would suggest that the Súvapus and its correlate were the basic term (x) and the mean next following (=, jzx, or 5), but though the pro91

portions AuröAtov and Erirpirov shortly to be mentioned there seems no way in which the geometric mean (/2x) could be referred to without mentioning both its extremes (x and 2x), and not merely the lower one (i.e. to mention the geometric mean one must mention three terms, the mean itself and its two extremes, but (in the case of the double) the other means can be named by reference to the lower term only — one-and-a-half, one-and-a-third; 2 is not one-and-an-anything). Also this interpretation would weaken the force of Sdvautc (if my interpretation of that is right), since there is no particular sense in calling a mean the correlate of the lower extreme; and anyway the Greeks had a perfectly good word for “mean” (yécov), but did not have a fixed terminology for distinguishing between roots and powers, or lower and higher terms. 4. 99121-4 N Lv dh rem... Ötaropeudeioe. The first difficulty here is to decide whether the antecedent of 4 (which presumably has the same antecedent in its three occurrences here, and also, unless there is an extreme stylistic barbarity, in sentence 5) is Sovautc (Toeplitz, Van der Waerden) or ¿vadoyia (Müller, Stallbaum, des Places, Taylor (apud Harward; cf. also Harward’s note ad loc.) Stenzel translates the subject of Suaropeudeioa as “Kraft” at ZG 99, but writes avadoyta for it at the top of p. 93). It presumably is not gior. If it is &væAoyix, and the sentence is meant to carry on from the last one, it would seem that we should have to assume that rp@Trn was answered by n de Surkactov in sentence 5, in order to keep to our decision to refer exacta to the three types of proportion, for arithmetic and harmonic proportion are obviously mentioned only in sentence 5. This would involve saying the SirAdotov dè... and % 3” eig td orepeóv ... clauses were parenthetical — but it seems much more natural to take them as answering por, and this in turn would suggest it is these three clauses that are explaining éxdornv; but we have already seen that, though this is not perhaps impossible, we should really expect xa0” éxaotov Adyov rather than xa” ¿xdorny dvaroyiav. Toeplitz and Van der Waerden, who take the antecedent to be Súvapic, both take Sbvauc in the nonmathematical sense of a “Wirkung” (Toeplitz) or “Kraft der Verdopplung” (Van der Waerden) an interpretation whose difficulties I have already discussed. On any interpretation this sentence is probably the most difficult to interpret in detail, but perhaps taking Súvapuc and its converse to mean higher and lower term in a series (and in fact, in the present context,

just “term”)!6 we can, despite the awkwardness of the phrase 4 xarà Súvapiv otca Stvaytc, find an interpretation that is open to less objections than those so far considered. Both Toeplitz and Van der Waerden take tod dumAactov as forming part of the subject with 7; grammatically this is perfectly possible, but the appearance of dirAdcotov as predicate in the two following clauses suggests that it should be taken as predicate here too.17 Van der Waerden explains the general sense of the sentence as being that if one doubles a line one has a geometrical image of the numerical ratio 1 : 2, while if one doubles the side of a square one has the surface-ratio 1 : 4, and if one doubles a solid in each of its dimensions one has the ratio 1 : 8. This seems to be very near the truth, but how does it work out in detail, on the present interpretation of Sovautc? The words qepouévn and Staropevdetoa (and also otpepouévng in sentence 3) suggest that this Süvaurc is something which moves. Now we know that the notion of a point “moving” so as to generate a line, and then this line moving so as to generate a plane, was a common one among the Pythagoreans, and a similar conception applies at Leg 894a, where the initial ¿py remains the subject throughout the whole process. There are really only the static terms, which are considered successively in thought, but this thought is as it were hypostatised and considered as though it were itself a term moving among the others and coming to rest on each of them successively. In this case the occurrences of divaute or its equivalent in the text will refer primarily to the term in the Sırrıkaorov series which has been reached, but also secondarily to this hypostatised moving term which has reached it. The sentence then means that the first term is of the series of the double, being moved numerically from 1 to 2 in ratio, while the term which is a power 18 (4 = 2?) is a double, and so again is the term which takes us to the solid and tangible, having gone right through from 1 to 8. At first sight it would seem that what was being referred to was just this generation of the dimensions that is described at Leg 894a. But there are two reasons whey such an interpretation would by itself be inadequate. Firstly, if xa” Excotny avadroyiav does in fact refer to the three types of proportion, of which sentence 5 seems to mention the arithmetic and harmonic, then we are left with the task of fitting in somewhere some reference to geometric proportion; and secondly, the earlier part of our passage dealt with geometry and stereometry, with high praise of their usefulness, and whether è dè Betov is given a forward or a backward reference, it is desirable to find some connexion between the earlier part of our passage

and this later part. It is also necessary to suppose that in writing this sentence Plato was doing something more useful than merely writin down part of the two-times table. The words xatà Aöyov suggest that the first change has something to do with a ratio, and xar’ «prov suggests that this first clause refers to ordinary numbers (the rpérov uéfnux of sentence 1, leaving the other two clauses to refer to geometry and stereometry respectively, all of which leads us to Van der Waerden’s conception of a line which is doubled, only perhaps it will work out better if we say there are two lines, one of unit length and one of length 2, which are to be compared. If we make squares on both these lines, these squares will bear to each other the ratio 4 : 1, which is double the previous ratio of 2 : 1, and similarly if we make cubes on these Squares we get the ratio 8 : 1, which is again double 4 : 1 (this seems a slightly better way of putting it than Van der Waerden’s, who says that if one doubles a cube in each dimension one gets the 8 : 1 ratio, because if the last clause meant this it would hardly be a step cic! rd orepeóv, since the whole process would be ¿y té oteped the whole time; therefore it seems better to make the third clause refer to a process which arrives at, but does not start from, a cube — though of course the underlying point is the same). This last process one might strictly expect to be described as &nd tettapav eic xt rather than ap’ évéc, but here we must notice the difference in tense betwee n SianopevOetox and the preceding gepouévy; our hypothetical moving term arrives at this last term “having gone (during the whole process, and not only this last stage of it) right through (ux-) from 1 to 8”%; the phrase is a sort of summing-up of the whole process. The significant point is that a numerical ratio which is a double will become doubled again when squares are formed on the lines which are its terms, and yet again when cubes are formed ; this implies that in order to find Squares and cubes which are double other squares and cubes in area or volume one must seek the geometric mean or means, and it is just that that geometry and stereometry help us to do, geometry by the process referr ed to by Toeplitz: QS 1 p. 3f (8 1), and stereometry by Archytas’ solution of the Delian problem. Hence geometry and stereometry are Osióv te xal Davyaoróv to those who understand this later part of the passage . In so far as these geometrical and stereometrical constructions enable one to construct squares and cubes in any ratio, and not merely in that of double, the selection of &v mpd¢ duo for the first clause is arbitrary, and to that extent Toeplitz’s “Abspiegeln” theory is right. The selection of the double for this purpose no doubt depends partly on the symbol ising of the dimensions

by 1, 2, 4, 8, and partly on the real part which the double plays when we come to harmony in the next sentence. 5. 991a5-b4 N SE SixAactov ... SeSouévn. Comparison with Tim 36a 2-5 and Archytas: Fr 2 (Diels) seems to make it clear that the words from tows to bnepeyéuevov are a description of the arithmetic and harmonic means (Toeplitz, QS 2 p. 286ff; Müller’s rejection of tò È’ Étepov ... Smepeyduevov as a gloss, repeating what has gone before, in which he follows Ast, seems quite unnecessary, but it has been suggested to me that tows Si... tod ueltovoc refers to the geometric rather than the arithmetic mean. This might seem to be supported by the definition of the three uéoa in Archytas: Fr 2 (Diels), to which Reuther refers in interpreting the passage (see Stenzel: ZG 99). Archytas says that in arithmetic proportion the interval between the greater terms is less than that between the lesser terms, and vice-versa in harmonic proportion, while in geometric proportion the intervals are equal. But his basic definitions of arithmetic and geometric means are respectively as tpitov Ümepéyet follows: & mp&tog Seurépou Srepéyer, toute DEUTEPOG and è olos npäroc moti tov Sebrepov, xal 6 DEÚTEPOG moti tov tpitov. Now it is possible that Plato was bent on describing the ways in which each of these means was equally removed fromits extremes, the geometric mean being so in the sense described in Archytas’ riders, and the harmonic in the sense described in our text; but on the other hand the use of Sepéyerv (which occurs in the definitions but not the riders of Archytas) in the description of the harmonic mean at least suggests that the same concept is to be understood in the description of the other mean, and this suggestion is strongly supported by the use of ruóMov in the following parenthesis, if that parenthesis is to have any relevance at all). Plato now turns explicitly to dealing with the means, and with the arithmetic and harmonic ones because he has already implicitly discussed the geometric mean (cf. Van der Waerden, p. 186 n. 1). We have already seen that the series of the means, like the series of the extremes, is itself a SirAdotoy series, so that it would be possible to take SirAxotou with % as part of the subject, as is usually done, but it seems much easier nevertheless to take duxAactov with uécov, since in the harmony that we are about to discuss the arithmetic and harmonic means are in fact between extremes which are of the form x : 2x. Moreover if we take dirhactov with y it is difficult to see what answers the uév — one would expect a dè clause with some contrasted subject —, while if we take it

with pécov then the uév is answered quite naturally by the 32 after toux. Theiler (Gnomon, 1931, p. 345) sees the difficulty when he says that what we should expect is: 7 38 SimAuotov [pév] eis péoov, tows [SE] <td uèv (sc. T@v péawv)> Tod EAdrrovog TAgov... On the present interpretation, however, there is no need to alter the text. Possibly my interpretation is also favoured by the lack of an article before Simdactov, since it is easier to talk of a term going into the middle of “a” double rather than to talk of the power of “a” double. We should understand some word like fatvovoa (cf. ouvéfn below) after pécov; the aorist, ouvéBn, presumably signifies merely that a particular example is being given. Van der Waerden’s interpretation of this sentence follows a suggestion by Tannery, which is that, after dividing the octave at the harmonic and arithmetic means to get the fourth and fifth (Mese and Paramese), one should then take a fifth down and a fourth up from the Mese, and divide these, which will give the Enharmonic Lichanos and the Trite respectively, for all the ëriéptov intervals (i.e. those where the numerator exceeds the denominator by 1) up to the tone (9 : 8) are found in the three scales (Enharmonic, Chromatic, Diatonic) mentioned by Archytas (apud Ptolemaeum), 5 : 4 and 8 : 7 between neighbouring terms in the Enharmonic and Diatonic scales respectively, 7 : 6 from the Mese to the Trite common to all the scales, and 6 : 5 from the Mese to the Enharmonic Paranete. Van der Waerden’s article is on Pythagorean music in general, and his purpose in bringing in the Epinomis at all is just to give textual evidence for this otherwise textually unsupported hypothesis of Tannery. Our sentence means, he says, that “dieselbe mittelbildende Kraft, die bereits die Verhältnisse 3 : 2 und 4 : 3 hervorgebracht hat, nun noch einmal von der Mitte aus auf diese beiden angewandt wird” (p. 187). Unfortunately Van der Waerden’s commentary, in the bare two pages which he gives to the whole of sentences 3, 4 and 5, including text and translation (pp. 185-7), is not at all clear in detail. It is hard to see how he can translate SirAaciou uèv ele uécov as “Was nun endlich die (Kraft) der Verdopplung anlangt, die sich nach der Mitte wendet”. If 3urAaotov is to be taken with % the phrase must surely for Van der Waerden refer to the squaring of ratios, which hardly seems to come into this process at all, and it would seem much easier for him to take SrAxotov with u£oov (in fact it was his interpretation that suggested to me that this should be done). It is also not clear how his interpretation quoted above fits with his interpretation of tobtwy abtév év té peo: “von eben diesen (Verhältnissen) in der Mitte (stehend)”; he explains that the “Verhältnissen” are the fuéAtov and ¿xbrprrov just

mentioned, and not, as Toeplitz thought, 8 and 9, and he presumably means that the power, which has just created the jysdAcov and ¿xtrorroy, now stands in the middle of each of them and faces both ways. But the fyutóMov and Ertrprrov in which the power stands are not the jurdrtov and Ertrprrov just created, but a new fifth and fourth created by going a fifth down and a fourth up from the Mese. To divide in this fashion the Autékov and éritpirov just created will not give any of Archytas’ scales at all. The sense in which the ¿xuuóptov intervals up to 9 : 8 are called the fundamental intervals of Archytas’ three scales also seems to be rather a vague sense; it is true that they all occur somewhere in the scales, and that each of the scales contains some of them, but a lot of other intervals occur too, and they do not seem to occur in a very systematic fashion. The three scales of Archytas which Van der Waerden quotes are built with tetrachords of the following intervals (going from Mese to Hypate21) ; E: 5:4, C: 32:27, Dj: 9:8, 36 : 35, 243 : 224, 28 : 27 28 : 27 8 : 7, 28 : 27 Later, after he has finished his discussion of the Epinomis, he mentions three other types of Diatonic scale: Di: 9 : 8, 9 : 8, 256 : 243 Dm: 8:7, 10 : 9, 21 :20 Ds: 10:49, 9 : 8, 16 : ı5 Of Archytas’ three scales 22, as quoted by Van der Waerden, only E contains a harmonic progression (M, Pn, N), except of course for the progression (H, M, N) common to all the scales; both E and D, contain arithmetic progressions (L, M, Pn in E; Ph, L, M and M, T, Pn, and T, Pn, N in D,), and if one takes the cumppévov scale (i.e. the one where the tetrachords have a note in common, as opposed to the Bieteuyuévov scale, where they are separated by a tone) from Pm to M of the octave above, whose H is the original N, D, contains a harmonic progression (T, Pn, Ph) and another arithmetic progression (Pn, Ph, L), while E contains another harmonic progression (Pn, N, M) and another arithmetic progression (Pn, L. M). The C scale however contains no arithmetic or harmonic progressions at all (except those common to all the scales), in either its dveCevypévov or its cuvyuuévov form, though, like D, and the ovvnuuevov form of all the scales, it does contain geometric progressions. We can if we like consider D, as defined by the continued arithmetic progression (M, T, Pn, N), E as a variant which includes the pure major third, but keeps the T of D,, and Cas defined by the geometric progressions (M, Pm, Pn) and (L, Pm, N). It istrue that we can get

EPn by dividing M-N harmonically at Pn, but this does not give us the other member of the tetrachord, T (and hence Ph), and does not itself account for the words en’ augétepa. It is interesting, however, to note that the scale which comes nearest to being generated by a harmonic or arithmetic division of the utéAov and éxitertov is Ds, which Van der Waerden says (p. 189) does not appear until Didymus, writing about 50 A.D. In this scale L is the harmonic mean between H and Pm, and T is the harmonic mean between M and N. What then shall we say that the Epinomis sentence does mean? Van der Waerden takes robrav aùréiv to refer to the proportions 1} and 14, because these have just been mentioned. But they have been mentioned ina parenthetical clause giving an example, and it seems just as permissible to refer robrwv adróv to té&v &xpwv avr 23, also called rd ZAattov and To uettov, in which case robrwv adrév év 76 ptc will perform the natural function of taking up again what was said at the beginning of this rather long and complex sentence, that the Sbvaut¢ goes into the middle of the double (i.e. the octave). If now we want to interpret the sentence as referring to the generation of a whole scale, would it not be better to take that scale as being D,, which appears in the Timaeus, and which Van der Waerden himself says Plato favoured (p. 190), adding that this depends on a conservatism which is especially apparent in the Laws (to which the Epinomis is after all closely connected)? D, is constructed by going a fourth up from M and thena fifth down, and then another fourth up and fifth down (Van der Waerden, p. 189). This interpretation would fit better with ¿vadoyia rather than S3úvayr (in either sense) as the subject of the sentence, though it would not be impossible with $övagız as subject. But need the sentence in fact refer to the whole scale at all? Need it refer to any more than the four basic notes, H, M, Pm, N? Granting that a theory which can bring in all the notes has much to be said for it, yet all we really need is something that will account for the final words of the sentence, from toig dvdpanoıs abupwvov ypelav to Movoüv Sedopéwn?4 — in fact the key words are obupwvov xal obupetpov xpelav?®, since these express what is actually said to be provided (&revetparo); and since the octave, fifth and fourth are undoubtedly the most fundamental intervals and harmonies of Greek musical theory, will not these be sufficient to provide a “symphonious (= harmonious) and balanced practice”? They in any case provide the basis for the rest of the scale, but Plato need not be referring explicitly to the whole scale here. If we are looking for a safe interpretation, that does not go beyond the evidence, it seems quite plausible to take otpepopévy émi, 98 “turning

towards”, as meaning “considered with respect to”, since it is when considered with respect to one or other of the extremes that the means create the harmonies (fifth and fourth) for which they are responsible. That the point of the whole passage is to emphasize the mathematical unity behind widely diverse phenomena is made clear by 991e: 6 de tpdmog Óde - Avayım yap TO ye togoltov ppdler € gi , e 9 7 N , e ! räv Srey pate co ? apuduod te obotyua xal dpuovias obotacw dmacav Tic te TOV dotpwy mepıpopäs Thy duoroyiav olcav ulav drévrov dvapavivar Set tH xard f LA f # EA a A > ~ > à # rporov wavOavovtt, pavhostar de, dv, è Aéyouev, dpOdc tig eis Ev BAéTOV pavdavy - Seopös yap nepuxas révrov téutev ele évapavhosrat Mavooupévols - ei 8° Ks Ts tata petayerprettal tic, tiynv Set xaheiv, donep Hat Ayopev. “And the method is this (for this much we must state): Every geometrical figure, and scale of numbers, and the whole system of harmony and of the revolution of the stars — the oneness of the agreement of all these must become clear to him who learns by the method, and it will do so if, as we say, one learns rightly by looking to unity; for to those who consider the matter there will appear a single bond naturally linking all these things — and if anyone practises these things in any other way one should attribute (his results) to chance, as we maintain.” What our original passage says, in brief, is that we must study the workings of the universe as exhibited in astronomy; this will involve arithmetic, geometry and stereometry, which will appear wonderful sciences to those who understand the theory of proportions, which is so important in music. Three connexions are asserted here, between astronomy and mathematics (arithmetic, geometry and stereometry), between mathematics in this sense and the theory of proportions, and between the theory of proportions and music. The first and the last of these connexio ns are tolerably obvious. Harmony depends on the theory of proportions, and astronomy involves mathematics, and especially arithmetic, which is TO péyiorov as well as tò mpéitov (990c 5); geometry and stereometry are brought in primarily because of what is coming. It is the connexion between these latter and the theory of proportions that is the most difficult to see the full point of. We have seen that probably the best solution is to follow up Van der Waerden’s suggestion that sentence 4 deals with the geometric mean and sentence 5 with the arithmetic and

harmonic means, and to suppose that sentence 2 supplies the method by which the geometric mean serves the purpose it does serve, that namely of providing a mathematical link between the dimensions. Thus the whole of Nature, in all its parts, is built upon the structural laws of mathematics, where connexions are made by the geometric mean or means (for the importance of means cf. Tim 31b-2c), and the formal laws of harmony, where the connexions are made by the other two means, 26 Various methods for connecting the third dimension with the others were thought of, but the earliest achievement was that of Hippocrates of Chios, which was in effect as follows (see T. L. Heath: Greek Mathematics, Vol. 1): A D c € Given a solid of side AB, to construct a similar solid the ratio of whose volume to that of the first solid will be as AC is to AB. Draw BD perpendicular to AB to meet AC in D. Draw DE perpendicular to AC to meet AB produced in E. Draw EC’ perpendicular to AE to meet AC (or AC produced) in C’. By choosing a suitable angle at A, C’ can be made to coincide with C, whereupon AB, AD, AE, AC will be in continuous proportion (by similar triangles), and AD will be the side of the required solid. (A method for getting the correct angle at A was first discovered by Archytas, who used a complicated construction, involving the definition of a point by the intersection of three surfaces, which does not concern us here.) It is tempting to try and relate this to the triangles in the Timaeus, whose grades of size increase in the same way (see Cornford: Plato’s Cosmology). Unfortunately however when AC = 2AB in our diagram the angle at A is not 30°, as it is in Plato’s triangles, and the ratio of the volumes of pyramids built out of his triangles of successive grades of size 8 would be not 2 : 1 but Da to I. Somehow or other the double has come to play a large part in Plato’s

philosophy — probably because its important position in harmony and in the symbolisation of the dimensions led Plato to choose it also as an example when he was mentioning the basis of the relations between the dimensions. Whether or not this double has anything to do with the Indefinite Dyad, or whether Aristotle thought it had anything to do with it, cannot be discussed here. But it is perhaps possible to suggest a reason for the choice of the curious phrase “revolve about the double”, which we explained above as referring to the SumAdctov series and its means. If we take the means in question to be geometric means, and remember the method by which Plato doubled a square in the Meno, by constructing a square on its diagonal (which incidentally does not prove that he did not know any more general method; his purpose in the Meno was limited), we can draw a series of squares, each of which is double its predecessor, by means of a series of lines which represent the SırrAkorov series with the geometric means inserted: S u T W V R Q O P X If OPQR is a square of side and area 1, then OPQR, OQST, OSUV, OUWX, forma series of squares of areas 1, 2, 4, 8, while OP, OQ, OS, OU, OW form a series of lines of lengths 1, /2, 2, 2/2, 4. Any of these lines, let us say OS, revolves round another line (OU), which is itself a double (of OQ), to reach its own double (OW). This notion would at any rate give some sense to the word “revolve”, which in nost commentaries is left in some obscurity. It can hardly be claimed that these notes solve all the problems of the passage satisfactorily, or that everything I have said will meet with general acceptance. What I do hope to have done is to have brought out

the main problems into the light of day bya detailed examination of the grammar and syntax of the passage, and to have made explicit some of the assumptions that are implicit in many of the commentaries on it. London. 1] wish to acknowledge considerable help in discussions, references, etc., from my former supervisor Mr. G. B. Kerferd of Manchester University, who also read the paper at one stage. I have also had the benefit of several very useful comments and suggestions from Professor J. B. Skemp and Mr. D. J. Allan. Finally I am grateful to my friend Mr. I. P. V. Carter of Ferranti’s, Manchester, for mathematical help. 2 It is, I suppose, just possible that the Epinomis is a forgery and the mathematical passage a deliberate piece of meaningless mystification. If so, the unity and coherence which I think the passage contains would seem to show that the forger was more inspired than he thought! 3 Hereafter called QS. 4 sauara L, Theo Smyrnaeus, marginal scribe on O: odyatog A O Stephanus. 5 The minor point whether Qxbya« is to be included with the subject or the predicate does not seem to have any material effect. 6 I accept Bekker’s tele for the rpeïc of the mss. 7 Cf. P. H. Michel: De Pythagore à Euclide, pp. 505-8. 8 Cf. Bed del yewuetpet (Plut.: Symp. Bk 8 Qun 2). Plutarch ends the Question by using the above theorem (which he attributes to Pythagoras) for a symbolic interpretation of the Timaeus. ® The Loeb translation gives it a backward reference. Professor Skemp suggests è 5h Beiov, which would certainly give an easier sense. 19 “Root” occurs only at Tht 147c-8d, but it occurs 6 times there. 1 J. Souilhé: Etude sur le Terme Sôvautc dans les Dialogues de Platon, pp. 105-6 calls its usage in Plato’s time “un peu flottante”, though Tannery (quoted ibid.) would emend the Theaetetus text. Souilhé does not mention our passage. 13 References to Zahl und Gestalt are to the first edition (1924). 13 Stenzel (ZG 101) follows Reuther in referring 2& évavtiag to tmevavtia, Archytas’ word for harmonic proportion (since if x, y, z are in arithmetic proportion, 1/x, 1/y, 1/z will be in (descending) harmonic proportion, and vice-versa). But Stenzel then seems to assume that brrevavtla can apply to the reciprocals of a geometric progression (i.e. to $, 4, }); but these are themselves in geometric progression, and in fr. 2 (Diels) Archytas defines Úrrevavria in terms of harmonic proportion. This, Stenzel says, is because, though no doubt 2& &vavrias has reference to úrevavria, yet “zugleich soll hier auch die eigentliche Bedeutung, die Entstehung dieses Terminus angedeutet werden”. 14 Stenzel (ZG 101) refuses to identify 3évaytc exclusively with either the mathematical or the non-mathematical senses. 15 In common with I think all commentators I prefer 6 mept (02) for boneget (AO). With @onepel we seem to have three alternatives: (1) td SirAdotov is the object of eyxadopúol te xal Guavoouuévoic, donepel meaning “as it were” (in which case è Se Getov must have a forward reference, as there is no connective after rd SirAdauov). (2) +d dimidarov is the object of dmotumotrat, éyxabopdat re xal Stavoouuévorc being absolute (“a wonderful thing to those who reflect on it, as though nature were

stamping the double, while...”). (3) tó BirAdotov is the subject of dnorunodren, ráca $ pbcıs being in apposition, the sentence otherwise being as in (2). None of these seems to give a satisfactory sense. Perhaps bonepel arose through an error from dictation, if it and © mepl were pronounced the same; cf. pls : tpetg above and Sìc : 8’ els below. 18 This does not imply a general usage of Sbvaytg for “term”; the meaning is partly shown by the context here. 17 Ficinus reads SirAdotov xatà divauiv odoa (i.e. omitting Se 4), and translates: “duplum potentiam possidens”. This is not the usual text, and the fact that the first clause has something to do with 2, and the last clause with 8, itself suggests a middle clause having something to do with 4. The words méAtv ad in the last clause might suggest too that something was being said for at least the third time, though Robinson (Plato’s Earlier Dialectic, p. 172) denies any emphasis to this phrase, and Plato’s later style is in any case well known to be prone to hyperbole. 18 At Tim 54b and Pol 266b xatà Sivaptv means “potentially”, being followed in each case by a predicate (“potentially something”). Here, unless we adopt Ficinus’ reading, it has no predicate. At Rep 587 d it might be taken as “potentially” with Sony ärécraoiv as predicate, but it is perhaps better to take it absolutely, meaning “by squaring”. Perhaps the present usage is derived from this: “That which exists (comes into existence) by squaring”, or “which is a power”. Stylistically this would be hardly more barbaric than many things which are to be found in the Laws and Epinomis. 19 8° gig margin of A2: Sig ALO. 20 Cf. the Loeb translation. 21 Abbreviations: E = Enharmonic. C = Chromatic. D! = Diatonic tonaion (Archytas’ Diatonic). D? = Diatonic ditonaion. Dm = Diatonic malakon. Ds = Diatonic syntonon. H = Hypate. Ph = Parhypate. L = Lichanos. Pn = Paranete. M = Mese. Pm = Paramese. T = Trite. N = Nete. 22 The following and similar facts will become clear from tables giving the intervals between each pair of notes in each of the various scales. I give here, as an example, the Synhemmenon form of the Enharmonic scale; the rest (excluded for want of space) can easily be constructed. E: Pm T Pn N=H Ph L M I 28 16 4 112 64 16 27 15 3 81 45 9 I 36 9 4 48 12 35 7 3 35 7 5 35 4 5 4 27 3 3 Each fraction here represents the ratio between the note in whose column it occurs and the note in whose column the figure 1 in the same row occurs. E.g. the ratio of L to T is 48/35. 23 Meibom (apud Stallbaum) and Müller take the antecedent to be 6 and 12. 24 Van der Waerden incidentally makes a good point against the Taylor-Toeplitz interpretation, of an infinite series approaching 6/7 from above and below alternately,

when he says (p. 187): “Es ist aber auch sachlich nicht einzusehen, wie man durch eine solche fortschreitende Approximation jemals zu anmutigem Spiel, Rhythmus und Melodie kommen sollte”. 25 Toeplitz and Van der Waerden take y&pıv as a noun governing the preceding genitives and qualified by cduuetpov. Ido not think this would make very much difference to the point at issue. 26 This passage therefore links together the three studies (arithmetic, the study of length, breadth, and depth, and astronomy) which are simply mentioned seriatim as necessary parts of education at Leg 817e-22d.