The pythagorean Theory of the Derivation of Magnitudes

Pricie, 48: THE “PYTHAGOREAN” THEORY OF THE DERIVATION OF MAGNITUDES ARSETiRNeEAS Mar OF THE GREAT DEBATES in the history of thought have had their origin in, or turned on, what to later times seem minor problems—whether the earth moves, whether transubstantiation in fact occurs, whether we are descended from apes. Early in the fourth century B.c. we begin to hear echoes of what appears to have been such a debate. References then first occur to point, line, plane surface, solid, as the constituent elements in ideal and physical magnitudes. These elements play a major role in the mathematical and metamathematical theories of Plato and the Early Academy, and against such theories Aristotle directs much of the polemic of the Metaphysics. Plato, Speusippus, and Xenocrates, each in differing ways, derive magnitudes from the sequence point, line, surface, solid, and their speculations gave rise to a lively controversy concerning indivisible lines—a controversy that influenced Aristotle’s notions of space, indivisibility, continuum. At the end of the fourth century we find Euclid defining the four terms as basic concepts in his Elements. Our earlier tradition is silent on the origin of these theories of derivation. I propose to discuss here the time when we may reasonably suppose that they arose, and the milieu in which their origin is most probable. The nature of the several theories, and their mutual relations, will be discussed only incidentally, in so far as is relevant to the theme.’ It will be suggested that: (1) We must distinguish between cosmogonical number theories, such as that attributed by Aristotle to the Pythagoreans, and theories of the derivation of solids. (2) A theory of the derivation of solids—point, line, plane sur-' face, solid, or some modification of this sequence—was expounded Metaph. n = Opal o W.D. Ross, Aristotle's Metaphysics (Oxford 1924) quoted by page numbers only. Phys. W. D. Ross, Aristotle's Physics (Oxford 1936). De. An. W.D. Ross, Aristotle: De Anima (Oxford 1961). Theory Robin W. D. Ross, Plato's Theory of Ideas (Oxford 1951). L. Robin, La Théorie Platonicienne des Idées et des Nombres d'après Eudemus Aristote (Paris 1908). F. Wehrli, Die Schule des Aristoteles VIII: Eudemos von Rhodos (Basle Cherniss H. Cherniss, Aristotle's Criticism of Plato and the Academy (Baltimore Heath T. L. Heath, The Thirteen Books of Euclid's Elements? (Cambridge 1926) Vors. H. Diels-W. Kranz, Die Fragmente der Vorsokratiker® (Berlin 1951-1952) 1955), 3 30 SE = en 23 oÈ o 2, +» = © 1944). 3 vols. 3 vols. ES ne Du vi = 2® = -2°¢ Snmnse<dg a= 932% <4 one [A = LEE METEO E aan. =5255 3337 32 (3) In the Early Academy a Pythagorean origin was attributed to such theories, not on historical grounds but because there was in the Academy a vogue for attributing the beginnings of such speculation to Pythagoras. In the first century 8.c. and in succeeding centuries modified forms of the theory reappear as Neopythagorean. (4) Apy such theories must be post-Zenonian, and are unlikely to have been evolved before the beginning of the fourth century. Rymdtoaeh:rgainhegviatfreuirdaony”s1|X3ltoTsd2e9ihr-d6vnes5fa—tp0oei,rntsy, pAbtarihtsortTiEsybaUutelANCg-RdH,hEAtPbpymoshtau-goZesrnytanisu,n, 13] a od References are made as follows: argues. In the first place a distinction must be made between cosmogonical number theories and theories of the derivation of physical magnitudes: that is, between a primary derivation of the universe gua One from numerically determined first principles and a secondary derivation of magnitudes or physical bodies from numbers, whether by the agglomeration of atomic magnitudes—which we might call arithmetical derivation —or by some variety of geometrical derivation based on point, line, plane, surface, solid. The primary or cosmogonical theories are said by Aristotle to have early precursors. For the secondary or physical theories re qebxupeonuycned—d ‚BES 33 by Plato, Speusippus, Xenocrates. Against them, and against such theories as a prevailing trend of contemporary philosophy, Aristotle 1466 J. A. Puiztp THE DERIVATION OF MAGNITUDES DibsmtodseihuantercihwlsdvinrefaoitedinosncnotsauAhmbtromeuahigorsircnteyisuorhtael,d we have no authoritative statement of the time or place of origin beyond the fact that such theories were propounded in the Early Academy. And whereas the secondary theories clearly have a background of mathematical inquiry we need assume no such background for the cosmogonical theories. Any inquiry into these problems must begin from the /ocus classicus of the Metaphysics (985b 23-988a 17). This passage, and the elaborations on it of the commentators, bristles with difficulties; but for our present purposes we need consider only such parts as are immediately relevant to our theme. “Our purpose” Aristotle remarks (986a 13) “is” (not to outline Pythagorean theories in general but) “‘to find out in their case too what first principles they posit and how these coincide with the causes we have mentioned. Now they notoriously consider number to be a first principle, both as material cause of existents and as their modifications and states.”? The elements of Pythagorean number are the odd and the even, the former being limited, the latter unlimited. The It is easy to see how, for the Pythagoreans, numbers as physical existents could be the matcrial cause. By the following phrase Aristotle should allude to some cause other than the material. Ross ad loc. suggests that he is alluding here, as he alludes elsewhere, to a Pythagorean formal cause. It is equally probable that he is alluding to an efficient lythagorcans. cause, or cause of movement, numbers somehow also accounting for qualitative change. Some colour is lent to this suggestion by 987a 5-9 where Aristotle says that the Presocratics anticipated the doctrine of causes in respect of only two of them, the material and the efficient. It seems more likely however that he is simply saying that number is the material substrate end also accounts for the characteristics of existents.

One proceeds from both these elements, being both odd and even. Number proceeds from the One. “And numbers, as I explained earlier, are the whole universe.” Here we have a generation of the universe and then a their existence. Plato on the other hand believed that numbers existed independently of sensible things, and this xwpiouôs marks the difference brief allusion to a post-cosmogonical episode in which number-particulars It is obvious that if Plato's theory of ideas has the characteristics Aristotle describes, and if Plato distinguishes between intelligibles and sensibles, then sensibles cannot proceed directly from number first principles, as they do in Pythagorean theory. For Plato sensibles have the geometrical structure of their intelligible counterparts, and these are the | products of his first principles. Sensibles have such being as they have 34 35 between Platonism and Pythagoreanism.? proceed from the One. Aristotle goes on to complain (989b 29-990a 8) that, though their first principles are abstract and would lend themselves to investigations of another character, nevertheless the Pythagoreans are to be ranged with the physical philosophers in that their whole approach is physical. For they generate the universe, discuss its constitution and phenomena, and to these ends expend their first principles and causes, by their participation in intelligible mathematica, and their elements or constituents are the elements of the mathematica, these imposing order on the “receptacle.” For our purposes we need not consider how these elements are derived from higher principles, nor how Plato, Speusippus, and Xenocrates differed regarding the stages of their derivation. They were all agreed as to the number and identity of the elements except in one particular. Speusippus held that the elements were point, line, plane, and solid. Plato and Xenocrates held that the first element was a minimum indivisible line rather than a point. “Plato,” says Aristotle (992a 20-22), “used to oppose the notion that the point is more than a convention of geometry. He used to assert that the indivisible line was “as agreeing with the rest of the physical philosophers that being is just the perceivable comprised within what we call the universe.” On a later occasion he insists again on this interpretation of Pythagorean theory (1091a 12-20). “It is absurd, and indeed impossible, to have eternal existents generated. Yet there can be no doubt that the Pythagoreans do indeed generate them. For they state explicitly that when the one has been constituted (whether from physical surface or geometrical plane surface or some yet more mysterious constituent) immediately the nearest part of the unlimited is drawn in and delimited by limit.” In these passages we observe Aristotle’s insistence on the fact that Pythagorean theories are cosmogonical. He does however make it clear the apxq or first principle of the line, and frequently maintained this that Pythagorean cosmogony went beyond the creation of a Parmenidean position.’ One to describe, however briefly and unsatisfactorily, the generation of That a theory of the derivation of solids such as Aristotle describes discrete quantity within the One. It apparently “breathes in” a part of the surrounding void (PAys. 213b 22-27; Frs. [Ross] p. 137), this void separating things, and apparently also the number constituents of things, from one another. It is significant that, whereas Aristotle dues tell us what are the elements constituting the Pythagorean One or universe, he JAristotle may seem to deny the validity of the distinguishing mark he has recog. nized, xwptapds, when he says (987b 11-13) chat Plato’s pédetis (participation) and Pythagorean mimesis are equivalent and interchangeable terms. This statement is surprising and isolated. Nowhere else do we hear of Pythagorean mimesis; and if this mimesis implies xwptapds, as we would assume, the statement is a contradiction of what Aristotle always elsewhere assures us is the Pythagorean position, that “things are numbers.” The simplest solution is to regard this as a piece of carcless terminology on the part of Aristotle. He tells us (985b 27) that the Pythagoreans thought they saw duormuora (likenesses) of numbers in existing things, i.c., that they thought they could observe the underlying number structure which things might (loosely) be said to imitate or better represent. Plato likewise says that their real nature is to be explained by is at a loss to explain the origin of particular substances in their theory. The One breathes in the surrounding void, and somehow out of a One substance the many particulars of the physical world come to be. He docs however tell us in general terms how the Pythagoreans conceived these number-particulars. For them “things were numbers” (Metaph. 987b 28) and they thought they saw in number, rather than in elements such as fire, water, air, earth, many similarities or resemblances to existents and things in the process of becoming (985b 27-29). That is, they believed in a numerical structure of the physical universe, and fancied they could observe this structure in physical existents. Aristotle’s insistence that Pythagorean theory had as its primary object the explanation of the origin of the physical world has its counterpart in the distinction he draws between Pythagorean and Platonic theories (987b 22-32), By and large both held that the One was a substance, and that underlying all existents was number. But for the Pythagoreans things were numbers, and there were no mathematicals to mediate can IPUL TER E ten their participation in the ideas. As Aristotle is drawing a parallel between Plato and the Pythagoreans, he may have considered thar there was a real simi'arity in their approach to physical existents, For this similarity see also 109Na 4-7. For discussions of the passage sce Ross ad loc. and Theory 217; Robin 74; Cherniss 180, ‘See Ross ad. loc. and his further discussion in Theory 206-212. Sc also Robin 286-293, PRISE e veanSUOR 472-474 and his Rapports de l'Etre (Paris 1957) 71-73. We are not here concerned with the ramifications of Plato’s Theory of Ideas, nor with the modifications of that theory effected by Speusippus and Xenocrates. Nor are we concerned with the Timacus theory of how ideal magnitudes materialize. Our interest is confined to the general features of the doctrines to which Aristotle testifies. The existence of these doctrines in the Early Academy is questioned only by those who completely reject the testimony of Aristotle on agrapha dogmate. When Aristotle is not referring to the doctrines of a particular thinker—Plato, Speusippus, ar Xenocrates-—he refers to the derivation theory as point-line-plane, as, c.g., Metaph. 1090b 5, An. Pos. 734 35-38.

magnitudes and physical magnitudes. First he shows that mathematica] objects cannot be in sensibles because we then have a meaningless duplication (1076a 38; cf. 998a 7-19), and because, if mathematical and physical magnitudes are derived from point, line, plane, then they must be divisible into these elements, and if the solid is to be divisible the point must be divisible. Next he points out (1076b 11) that if mathewas propounded within the Early Academy we cannot reasonably doubt. We may, however, ask ourselves why they constructed ideal magnitudes, as “things after the numbers,” to mediate the coming-to-be of physical body. There was no reason why the ideal numbers should not produce physical body without the intervention of intermediates. A theory of number-atomism would have resulted. Though geometrical derivation may better account for order and structure, that is not in itself sufficient reason for so surprising an innovation. We can only suppose that in the Academy thinking in geometrical terms either had the charm of novelty or was induced by the mathematical discipline that prevailed. In any case it is a striking fact that Plato and his immediate successors in the headship of the Academy, whatever their other differences, were with one slight exception in agreement as to the derivation of solids. It was to such theories, and to the mathematical tendencies of the Academy, that Aristotle took exception when he wrote (992a 32; cf. Robin 318): “Mathematics has become for our contemporaries the whole of philosophy, though they allege that it is to be pursued (not for its own sake but) for other ends.” That the derivation theory was for him an important aspect of these tendencies is obvious from the arguments he directs against it. In Books M and N of the Metaphysics we have two versions of his polemic against the theory of ideas in its various forms. As Jaeger (Aristotle? [Oxford 1948] 176-193) has shown, Book M to 1086a 12 is the later of the two versions that together constitute these two books; and it is significant that in the later version Aristotle’s arguments against the derivation theory assume a larger place. In Book N, apart from occasional allusions, there is only one major attack (1090a 2 ff.) and that attack is directed against the doctrine of point, line, plane as separate substances. At the time of the writing of Book M it may be that the emphasis on mathematical aspects of the theory of ideas had increased, or it may be that Aristotle had more clearly appreciated their importance. In any case the theory of ideas in its earlier, non-mathematical form, though it is introduced to make the treatment complete, is touched on only cursorily. Elsewhere in the book the theory is discussed as one having mathematical aspects, and as presenting two major problems; (1) Can mathematical objects be said to exist? and (2) Are numbers separate substances and first principles? The mathematical objects Aristotle envisages are either mathematical, i.e., numbers as separate substances, or geometrical, i.e., line, plane, and solid as separate substances. Both are derived, by differing methods, from the same first principles. Aristotle is concerned to show that they are to be considered not as separate substances but as mathematical abstractions from sensibles. Much of Aristotle’s criticism, however, is applicable both to ideal 37 matical magnitudes are separate substances there results a preposterous pullulation of points, lines, and planes. For there must be points prior to the points of the line, lines prior to the lines of the plane, and so forth. And surely points, lines, and planes cannot be prior in substance to body, which possesses unity and completeness and can be ensouled. We need not here discuss the merits of Aristotle’s arguments against the theory of derivation (Robin 427-435), to which we have reason to believe that he once subscribed (Walzer, Frs. [Florence 1934] 28 and n. 3). It is sufficient for us to note the importance of the role it plays. When, after a rather perfunctory discussion of the theory of ideas in its earlier, non-mathematical form, Aristotle reverts to ideal numbers and idea numbers, he turns to the second question he propounded: Are numbers separate substances and first principles? In this discussion he is concerned with the comparability of the units constituting idea numbers and with the stratification of substances from the one down to sensibles. Lines, planes, and solids are referred to only incidentally, to ask whether we must distinguish between mathematical lines and those that come after the ideas, i.e., between an idea of line and mathematical line (1080b 23-30). (That such a distinction was recognized is suggested also by 1084a 37-b 2.) It is only towards the end of his survey that he turns to the special problems of the generation of point, line, plane as the constituents of magnitudes (1085a 7-b 33; Robin 370-371). He argues that Xenocrates, for whom the ideas were numbers, must generate numbers and geometrical objects from identical first principles, and that this involves a weréBacrs els Ado yévos; that Speusippus does not explain how the many points of geometrical objects are to be generated from the one unique point that must be assumed; and finally (this applies also to Plato) that, if there exist indivisible lines, then no line and no continuum can be infinitely divisible. With these and other arguments Aristotle shows the difficulties that arise in generating the points, lines, and planes wtaeeD EUANES 4 v s A». pKtTiymarLeyc,e si that are said to constitute magnitudes. ‘That Aristotle in an early period accepted some geometrical theory of derivation it also suggested by his remark in the Protrepticus (1. Daring, Aristotle's Protrepticus [Göteborg 1961! fr. 33, p. 61 = lambi. Protreg. [Festa] 38.14-22P.) that “things that are prior are causes. .., for if the former ar- removed (ävarpeiraı) the things that have their being from them are removed; fines if numbers are removed, planes if lines, solids if planes, the so-called syllables if letters are removed.”

In the Metaphysics Aristotle shows little or no sympathy for the efforts of Platonists to derive physical magnitudes from ideal magnitudes having are numbers”—a tenet Aristotle repeatedly imputes to the Pythagoreans —then there is no need of a derivation theory, whether arithmetical! or geometrical, to account for the derivation of number-things. We must suppose that someone within the Academy, and probably Plato himself, was inspired by the cosmogonical number theories of the Pythagoreans, but at the same time was impressed (like Aristotle) by the fact that they did not account for the procession of number-things from their cosmogonical first principles. So he devised a theory of derivation, but not on Pythagorean lines. For a Platonist sensibles, as being in continual flux, 38 the three dimensions of the solids of geometry. He regards such attempts as a deplorable aberration of contemporary philosophy. For him sensible particulars are substances, and it is absurd to attempt to account for PSeTRATE them by suggesting that they owe what reality they possess to their participation in a non-sensible geometrical paradigm, itself derived from higher principles. That he was not, however, unaware of the problems that were being discussed in the Academy while he was a member is shown in his other writings, in particular in the Physics. In Book 5 of that treatise he argues for the infinite divisibility of all continua, “the infinite divisibility of movement forming the middle term by which that of extension and that of time are seen to imply one another” (Physics 39 could not be derived from intelligibles. The derivation theory propounded was not an arithmetical one, such as Pythagorean first principles would have suggested, but a geometrical one in which the three spatial dimensions were as it were hypostasized to mediate the derivation of physical 70, n. 1). As a consequence, when we speak of a point in space, a point objects. in a path of movement, or an instant in time, we are not speaking of It may, however, be argued that we cannot exclude the possibility of a Pythagorean derivation theory simply because the testimony of Aristotle appears to exclude it. The Pythagoreans may have developed a derivation theory known to Plato but not to Aristotle, a theory not involving the separation of intelligibles and sensibles. That the Pythagoreans had evolved any such theory does not seem probable, but we can reject the possibility only if we can determine when the geometrical parts of, or entities constituting, space, movement, or time (Physics 212b 24; 220a 4; 222a 15). In the course ofhis discussion Aristotle alludes casually and with some contempt to the notion of indivisible lines (206a 16-18) and he touches briefly on the problems of three-dimensional entities (200a 2-13; 227a 27-32; 231a 20-231b 1). It is obvious from these allusions, and also from his whole reformulation of physical thought in notions and procedures which the theory assumes were developed. But before we attempt to consider this difficult problem let us first turn briefly to the post-Aristotelian history of the derivation theory, and ask terms of matter, form, and privation that he has faced the problems that were agitating the Academy and is offering solutions for them in his own terms. Aristotle’s account in the Afetaphysics, implying as it does lively discussion and important differences within the Academy, and also his own physical theory, both suggest that the problem of the derivation of ideal and physical magnitudes was a live contemporary problem. It arose, or at least its solutions found their application, within the theory of ideas. Point (or indivisible line), line, plane, solid—the three spatial dimensions of geometry-—were borrowed from that discipline to serve in explaining how physical magnitudes were derived from first principles; that is, how the intelligibles are to be related to sensibles. The fact that this derivation theory was a theme of lively discussion, and a theme on which Plato, Speusippus, Xenocrates differed radically, suggests that it had been evolved within the Academy. It is usually claimed, however, that it is a borrowing from Pythagoreanism and represents the pythagorizing tendencies of the Early Academy. Aristotle tells us that the Platonic philosophy was based on Pythagorean theories (987a 29-31). The ideas as “concepts” were a Socratic legacy , but their mathematizaa m, a oerg SEN BESapnetEe,: ourselves why it then appears as Pythagorean rather than Platonic. The pseudo-Aristotelian treatise De Jineis insecabilibus is generally thought to have been written by a member of the Peripatos of the first generation, probably under the headship of Theophrastus. It begins by recapitulating the case for indivisible lines, in part deriving its arguments from the Aristotelian treatises, but in part adducing new arguments of its own.® Once the case has been presented it is answered without preamble, point by point. The answer has the character and develops its arguments after the manner of a formal précis used in instruction. Some Parts are by nature prior to their wholes” (986a 10). S. Pines in “4 New Fragment of Xenocrates and its Implications" (TA Philos 51.2 [Philadelphia April 1961]) discusses this doctrine as one originating with Xenocrates, to be connected with his theories of genus and species (Cherniss 13-17) and indivisible lines and solids. Aristotle (Merapa. 1017b 19) discusses the priority of genus and species without alluding to Xenocrates. The argument from commensurate lines (De din. intre. 986 4: is not an argument deriving from Aristotle, because of the obvious fallacy. Throughout the De lineis inserexistence of “numbers apart from sensibles, whereas the Pythagoreans abilibus the geometrical references are more immediate and technical than in Aristotle; see the references to Euclid in the translation of 1H. H. Joachim, The Works of Aristotle (Oxford 1908). See also G. Vlastos, “Minimal Parts in Epicurean Atomism,” Zsis 56/2 assert that things themselves are numbers” (987b 27-29). Now if “things (1965) 125-129, tion was Pythagorean. Plato, however, is said to have recognized the

of its arguments are as valid against the points of Speusippus as they are against the indivisible lines of Xenocrates (971a 5-7). Reference is made throughout to geometry as an organized discipline, and the meaning can frequently be elucidated by a reference to Euclid, whose Elements must have been published at about this time. No allusion is made to a Pythagorean origin for the derivation theory, one basic aspect of which is being discussed, nor to any Pythagoreans holding it. There is indeed no mention of persons, but the immediate reference to the Aristotelian analysable into elements. In this caricature of Platonic doctrines the most significant feature is that while the doctrines remain recognizably Platonic they are now reported as Pythagorean. Why have thev been rebaptized? Reinhardt (RE, s.v. “ Poseidonios” 647) has pointed out that Stoic pantheism has important similarities with Pythagorean cosmology, the structure of the physical world being for both schools mathematical and so intelligible. But the doctrine that the physical world is understandable gua mathematical is Platonic, and no earlier source suggests that it was also Pythagorean. It seems, however, probable that they were treated as Pythagorean in the school of Posidonius, and we may find some confirmation of this in the Stoic terminology that colours the Hypomnemata quoted by Alexander Polyhistor. Why were doctrines that were 40 yaTRiOrIO Ch Cai treatises makes it certain that the indivisible lines being discussed are the lines of Xenocrates (and Plato), The De /ineis insecabilibus, probably written in the third century, is the last work making reference to the derivation theory as Platonic or Academic. When first we encounter it again it reappears as Pythagorean. Alexander Polyhistor (first century 8.c.) claims to have discovered Pythagorean writings—many such writings appear from the third century on’—from which he quotes as follows (D.L. 8. 25): “The first principle of all things is the monad. From the monad proceeds the indefinite dyad, constituting the material substrate for the monad, the monad being cause. From the monad and the indefinite dyad proceed the numbers, from numbers points, from points plane figures, from planes threedimensional figures, from these sensibles. The elements of sensibles are four: fire, water, air, earth.” i This is obviously a pastiche of doctrines of the Early Academy. It grossly over-simplifies, and it deforms the theory of ideas from which it springs. In the place of the One we have the monad. The indefinite dyad has become material substrate (n). Point, line, plane, and solid now proceed from one another. Sensibles, once constituted, are further *Sce H. Thesleff, An Introduction to the Pythagorean Writings of the Hellenistic Period (Abo 1961). *The procession of line from point, plane from line, solid from plane implies a kinetic nature or principle that is absent from Aristotle's account of derivation in the MetaaFEctLVEPa EN EUNT ST:a CAaRE =Aea oxw.gyLe . 41 phrase xıyndetsav ypéuunr, which I have translated by “a line being set in motion,” is usually rendered by “the movement of a line.” But the whole purport of the passage is not that sensibles are being created by a method of fluxion (so Corniord, Plato and Parmenides [London 1939] 12). Here they are being “ensouled”-—tha: is, an aspect of the relation between soul and mathematicals is being explained, a relation to which Aristotle alludes again in arguing against mathematics as separate substances (Afesaph. 1077a 20-36). Here he argues that both souls {as being in physical bodiss' and monads (as points) have position. If both have position how can we have soul as a self-moving number? Cherniss (396-397) argues that this is a fluxion theory held by Speusippus. For such a theory we have no other evidence, and in view of the “episodic” character of Speusippus’ theories it seems unlikely. It seems more probable that ir is a loose way of speaking of extension, that you advanced a line as you advanced a military forination in line; and that in this rather captious argument Aristotle is hinting that a point when it moves along a line to another position produces a line, not a soul. Similarly, in a much disputed passage (De An. 404b 19-24), we have the form of the one, and primary length, breadth, and depth referred to the autozoon; the one, the two, and the numbers of surface and solid referred to cognitive and perceptive faculties. Though the themes may be germane, they are not directly relevant to the derivation of solids from mathematicals. For souls and mathematicals sce P, Merlan, From Platonism to Neoplatonism? (The Hague 1960) 222 and passim; Robin 487-491. physics and Physics. He himself defines the unit or monad as indivisible quantity not The first explicit mention of a line being described by the motion of a point is by having position, the point as indivisible quantity having position. Line, plane, and solid Proclus (185.6-25) ascribed to Geminus (Heath 1.40}. But Geminus may have referred are the divisible in one, two, and three dimensions. These definitions are coloured by geometrical thinking and were probably developed within the Academy, Book D of simply to the describing of a line by a geometer in the drawing of diagrams and not to a principle of fluxion in a point by reason of which a point in a spatial continuum tends to flow into a line. In a passage in the Laws (894a) where he classifies motions Plato says: “In the case of all things genesis occurs when an ¿px? increases and makes the transition to a second dimension, and from that to the next; and having achieved three dimensions acquires sensible properties for percipients.” A. E. Taylor (The Laws the Metaphysics where they occur being an early book. Aristotle’s account of Platonic derivation theory is conceived in the same terms and is essentially static. So are the definitions of Euclid. There the unit or monad is defined arithmetically. (Elements 7.1; cf. Heath 1.279), and we find the same static notion of plane and solid numbers (Elements 7, def. 16, 17; cf. Heath 287-291). In the De Anima, where the theme of discussion is the soul, we discover a radically different point of view or, perhaps better, a new aspect of the derivation theory. There, at 4091 4, we read: “It is said that a line being set in motion creates a plane, a point 2 line; but then the movements of monads are lines.” This passage occurs in the context of a discussion of Xenocrates’ definition of the sou! as a sel moving number. The un. si;S m mt . AN an {Everyman 1960] 285) interprets this as implying the fluxion theory. But Plato has been talking of collisions of bodies, moving with moving and moving with stationary. As a consequence of these collisions bodics disintegrate and reintegrate in new masses which come to be. This genesis occurs when we have “something to start with" and it acquires three dimensions in the process of integration, thus becoming sensible. Though elliptical chis account of genesis is such as to exclude Auxion of point, line, plane.

THE DERIVATION OF MAGNITUDES clearly Platonic in the fourth century said to be Pythagorean in the ?? second goras was the leader of those who conceive physical body as “what is capable of affecting anything else or being affected” (Pyrr. Hyp. 3.38; Adv. Phys. 1. 366). Plato in the Sophistes (247E) in almost the same words offers this as the mark of the moderate materialist. Again Sextus tells us (Adv. Phys. 2. 261) that for Pythagoras the One “when compounded with itself in respect of difference creates what is called the indefinite dyad”—a curious garbling of Platonic doctrine. What is approximately Aristotle’s definition of time is attributed to Aristotle “or, as some say, to Plato” (Pyrr. Hyp. 3. 136) and a distinction in kinds is made among opposites that derives from the Aristorelian Categories (6b 15) rather than from any Pythagorean source. In all these cases we must suspect a Neopythagorean source that sweeps everything even distantly congruous into the Pythagorean net, and we must expect the same tendency to be at work in the account of Pythagorean doctrines. In Sextus’ discussions of Pythagorean theory in general we are given a sketchy account ofits metaphysical presuppositions and a not altogether consistent account of the theory itself. The dvad, of Platunic origin, ts generated from the monad directly (Pyrr. Hyp. 3. 153), and morad and dyad are the first principles from which numbers are generated. Number are invisible and incorporeal. Physical bodies are a composite of corporeal and incorporeal. They are numerable, the point having the logo» of the monad, the line of the dyad, plane surface os the triad, solid of the tetrad. (The priority of the numbers to the corresponding geometrical dimensions in 154 is Platonic.) The universe is made up of these number bodies, and is ordered after the ratios of the musical chord, these being 42 Burkert (Weisheit und Wissenschaft [Nuremberg 1962] 82-85) has suggested that when with Arcesilaus the Middle Academy turned to scepticism, alleging Socratic origins for the tendency, they could no longer interpret as Platonic the mathematical metaphysics of Plato's latest period. They therefore took literally the pythagorizing of the Early Academy, and considered as Pythagorean the doctrines that were no longer compatible with what they chose to teach as Plaronism. The Early Academy undoubtedly was influenced by, and to a greater or lesser extent borrowed from, a preceding Pythagoreanism. But what the post-Aristotelian tradition presents as Pythagoreanism is largely the pythagorizing Platonism of the Early Academy. This reappears, greatly simplified and sometimes contaminated, in the doxographical account as Pythagoreanism. This thesis of Burkert's is the only plausible explanation of a very curious change. NE ErRMEé pe nmereosna.ir The derivation theory reappears in Neopythagorean writings as characteristically Pythagorean, and as such is implied in the Zivroduction to Arithmetic 2.6-7 (tr. M. L. D'Ooge {New York 1926] 236-241) of Nicomachus of Gerasa. It is no part of our purpose to trace the post-Aristotelian tradition of the theory. A brief outline of the manner in which it is presented by Sextus Empiricus, our most important source for its later form, will suffice to show how the features of the theory were changed in transmission, and will suggest why his account cannot be used for the reconstruction of early doctrines, as was suggested by Cornford. expressible in the first four numbers. These numbers (those of the tetractys) not only constitute the cosmos, Sextus, as a Sceptic, aims at showing the shortcomings and fallacies of all non-Sceptical doctrines, but his discussion shows an interest in the history of thought and considerable philosophical insight. He is not, however, a careful reporter. Of Pythagoras, Empedocles, and “the rest of the Italiots” he tells us (4de. Phys. 1. 127) that they believe there exists a community between men and animals because there is “one pneuma pervading the whole universe, like a soul,” and that this (Stoic) persuasion led them to abstain from meat. Again he tells us that Pytha8G. Borghorst, De Anatolii Fontibus (diss. Berlin 1904) 65 and passim, argues totam de Pythagorico criterio disputationem depromptam esse e Posidonio in Timaeum commentario, and that as a consequence our accounts of the derivation of solids in the mathematical tradition and in commentators on Plato like Chalcidius and Macrobius derive from Posidonius. This thesis perhaps oversimplifies (Pohlenz, Die Stoa? [Góttingen 19591 386-388). Pohlenz points out (255-256; that the tendency to present Posidonius as the first Neoplatonist “verkennt freilich die tiefe Kluft die diesen von Transzendentalismus trennt." Neopythagorean derivation theories, however, usually deform or neglect the transcendental implications of their Plasonic original, or bypass them bv reference to the world soul. they also render it intelligible. For “it is by the ratios of these tour ‘4 2% co(hOripos RET A Pky y ETme lOye<>a. numbers that we conceive both corporeals and incorporeals” (440. Log. 1. 99). The Pythagorean #riterion of truth is the /ogos deriving from mathematics (dv. Log. 1. 92). It is number because everything is measured by number (Adv. Log. 1. 105) and numbers are the first principles and elements of all things (Adv. Phys. 2. 248). Aristotle in the Metaphysics complained that the Pvthagoreans generate their universe of number-things from the One, but do not explain how it is generated. In Sextus we have an account of its generation in which elements of Platonic, Stoic, and Epicurean theory (Pyrr. Hyp. 3. 152) are borrowed. The most characteristic feature of this account is a derivation theory (which for Aristotle is Academic rather than Pythagorean) and that theory is given a peculiar twist. Parlier Pythagorcans, Sextus tells us (Ade. Phys. 2. 270-284), generated the numbers from the One and the indefinite dvad. They then generated from numbers point, line,

It may, however, be objected that the arguments we have adduced so plane, solid (282). But /ater Pythagoreans assert (281) that physical body is generated from the point which ‘flows’ to produce the line, line flowing to produce plane, and plane flowing to produce solid. far are not decisive. We have argued (1) that the problem of the derivation of magnitudes was a critical one for the Early Academy, (2) that the theories evolved in the Academy, and by the Academy loosely called This fluxion theory is novel, but there can be no doubt that it was “Pythagorean” were by succeeding generations considered to be of Pythagorean and not of Academic origin, and (3) that the Neopythagoreans ascribed the origins of the theory to their Founder. Even if all this is conceded it may still be argued that the theory, or some form of the theory, originated long before Plato. And this may be argued even if we are unwilling (as indeed criticism is increasingly unwilling) to believe in a Pythagoras who is the source from which flows, in the secrecy of a verbal tradition through a century and a half, most mathematical knowledge. Let us consider how probable it is that such a theory should have evolved in the fifth century and have been merely adopted by Plato. If it did so evolve it is unlikely that the development occurred in current. Sextus alludes to it several times (Pyrr. Hyp. 3. 154; Adv. Log. 1. 99; Ado. Phys. 1. 375; Adv. Math. 3. 20). Once, not in the course of discussing Pythagoreanism (Adv. Math. 3. 28) he reports a doctrine of the school of Eratosthenes. They hold that the point does not occupy place nor does it measure the breadth of a line: by fowing it creates the line. The line is “a flux of the point.” This flux variation of the derivation theory may be an attempt to escape the consequences of atomic points and indivisible lines. It may however derive, as Sextus suggests, from the theories of later Pythagoreans. “The Pythagoreans specializing in mathematics” (Ado. Math. 4.2)! held that “the monad as first principle underlies and produces all other numbers in their constituted form. The dyad produces length. The monad imposes the logos of the point, the dyad that of the line. . . . For in conceiving the line the mind moved from one point to another, and this is length” (dv. Math. 4. 4). The other dimensions have a similar origin. Like the logos of the soul, that 45 ayer Athens and among Athenians. In the fifth century Athens was a centre of the political and artistic but not of the scientific life of Greece. Though all the great sophists visited Athens and some few of them spent a longer time there, none was Athenian. A Callias or a Pythodorus could of the body is comprised in the number four; and living creatures are, —and at great expense to themselves did—entertain eminent visitors like the cosmos, ensouled in accordance with musical ratios. Here it is the movement of the mind, rather than the flowing of the point, that such as Parmenides and Protagoras. Anaxagoras was an intimate of Pericles, and each contributed to the other’s unpopularity. But the status determines the four dimensions. We are reminded of Aristorle’s account of these visitors was something like that of the Greek philosophers who of the autozoon (De An. 404b 16-27) where it is said that knowledge is later visited or settled in Rome. They were courted, flattered, well-paid. the two because it moves in one direction towards a single object, and But in one essential respect they were inferiors. From politics, the only life in which arete could really be displayed, the sophist was excluded. He of Xenocrates’ definition of the soul as a self-moving number (De /n. 408b 30-409a 10). At a time when the derivation theory and the definition of the four dimensions was a critical problem (Adv. Math. 3. 19) the flux theory may have been suggested by earlier Platonic theories. It appears to have arisen in connection with the diagrams of geometers. We have no grounds for believing that the flux theory, as it is reported by Sextus, had much earlier origins. Neopythagorean doctrine, as it is reflected in Sextus’ account, is a was a foreigner, and a teacher for pay. The contempt for their profession that Plato expresses must have been shared bv others of his class. It was a paradoxical development that Plato himself first created a milieu in which native Athenians could pursue scientific studies; and the first Athenian mathematician of any stature was Theaetetus, himself a pupil of Theodorus of Cyrene but later a member of the Academy. It does not therefore seem probable that a theory of derivation should conflation of Stoic, Platonic, and Aristotelian theories. The extent of its have developed in the Athens of the fifth century and simply have been earlier Pythagorean content seems questionable. In particular the derivation theory has been modified by Neopythagorean mathematical speculation. The distinction between intelligibles and sensibles has been blurred. The dyad no longer has its Platonic role. The function of the theory is to explain how physical body is generated from first principles. adopted by Plato. There are two other possibilities; the first, that it Here we at once think of the mathematici/acusmatici division of Pythagoreans of which Jamblichus speaks. But it seems more probable here that the reference is to a contemporary difference. Mathematicians like Moderatus of Gades or Nicomachus must have been separated by a great gulf from Apollonius of Tyana and his followers. evolved in Ionia, then the centre of mathematical studies (W. A. Heidel, AFP 71 [1940] 1-33); the second, that it was a product of Western Greek speculation. We know that Oinipodes and Hippocrates, both of Chios, visited Athens and that the numerous mathematicians subsequently attracted to the Platonic Academy were predominantly Jonians. It is only Archytas, and a Pythagorean tradition that we have seen reason to regard as suspect, that induce us to think of the West. Archytas, despite his own achievements as a geometer, regarded arithmetic as a more

exact science. None of his writings suggest an interest in physical theory such as might inspire a doctrine of the derivation of magnitudes. Howscene of his principal scientific activity. He had enunciated a theory of the comets before, but probably not long before, 427 8.c. (W. Burkert, Weisheit und Wissenschaft [Nuremberg 1962] 291, n. 76). But his importance for our present purposes lies in the fact that he is said to have written the first Elements. “After these (Anaxagoras and Oinipodes), Hippocrates of Chios, the discoverer of the quadrature of the lune, and Theodorus of Cyrene distinguished themselves in geometry . . . Hippocrates wrote a book of Elements” (Vors. 42.1; Wehrli, fr. 133). This information comes to us through Proclus (Zn Eucl. El, 66.4) from Eudemus’ History of Geometry, and so from the first generation of the Peripatos. Eudemus of Rhodes, himself a scientist and an historian of the sciences, was a candidate for the succession to Aristotle in the headship of the Peripatos when Aristotle died in 322/1 B.c. The choice fell on Theophrastus, probably a contemporary of Eudemus, who lived until 288/7 8.c. The date of Eudemus’ death is not known, but he may well have lived to see the publication of Euclid’s Elements at about the turn of the century. If so, his mention of Hippocrates’ Elements acquires added significance, as does the mention of another of Euclid’s predecessors, Theudius of Magnesia, who is said to have written “a good book on the Elements” and to have “given general application to particular inquiries” (Eudemus fr. 133)." But even if Eudemus did not see Euclid’s 46 ever, no investigation of the complex and controversial history of Greek mathematics is likely to provide us with a satisfactory answer to our problem. We must therefore attempt a secondary line of inquiry—a devrepos mots. In a discussion of the principles of the sciences Aristotle writes (Anal. Post. 76a 37-b 5): “The premisses used in demonstrative sciences are some of them common, some of them peculiar to the particular science. Common premisses are common analogically, since their use is restricted to the field of the particular science. An instance of the premisses of a particular science is that the line and the straight line have such and such characteristics. An instance of common premisses is that if equals are subtracted from equals the remainders are equals. It is enough that each of these premisses shall be valid within the field of the particular science, even if it is not assumed to be true of all sciences, but only in geometry of magnitudes, in arithmetic of numbers. Peculiar to a science, and of necessity assumed to exist, are those entities whose essential attributes the science investigates—in the case of arithmetic these are the units, in the case of geometry point and line.” The first two definitions of the first book of Euclid’s Elements define respectively the point and the line, and it is obvious that even if in theory the assumption of their existence was not made, without point and line there could be no geometry. Jf magnitudes were said to derive from point and line (or a fortiori from indivisible lines) such a theory could have arisen only in connection with geometrical thinking. At what period is it credible that geometry ceased to be empirical—as when Thales calculated the height of a pyramid—and became a body of knowledge with its characteristic assumptions, methods, proofs? Obviously it did not spring fully armed from the head of any one geometer. Oinipodes of Chios, a younger contemporary of Anaxagoras (Vors. 41.1) appears to have been interested primarily in astronomy. He is accused of having stolen from Pythagoras the notion of the ecliptic (Vors. 41.7), the typical device by which the Neopythagoreans annexed for the Master a scientific discovery. As Diels suggests (Yors. 41.2 note), it is probably to the ecliptic that allusion is made in the Erastes (132A), and it may be that Oinipodes first introduced that notion to Greece. (But cf. O. Neugebauer, The Exact Sciences in Antiquity? [Brown UP 1957] 102.) At all events his ather geometrical achievements are said by Proclus (Vors. 41.13) to have had their application in astronomy. Hippocrates of Chios is a more serious contender for the title of “Father of Geometry.” He belonged to the generation after Anaxagoras and Oinipodes (/’ors. 42.1) and it seems probable that Athens was the Di SES TeWseA et CEBRENTEAUSS ies 47 UThe generally accepted foruit of Euclid is 300 s.c. (T. L. Heath, The Thirteen Books of Euclid’s Elements? (Cambridge 1926] 1.2). It is assumed that Eudemus cannot have known his work because the Proclus lemma, deriving however indirectly from Eudemus, is made to end at the words translated by Heath (1.37) as follows: "Those who compiled histories bring the development of this science up to this point.” This is understood to refer to Eudemus. Proclus then continues: “Not much younger than these is Euclid.” On the basis of these two sentences it is argued that Eudemus is chief among those who compiled histories, and that what follows derives from some source other than Eudemus, though Heath (37) concedes that “the style of the summary after this TESÓNRAT “TERDE NODES 4 point does not show any such change from that of the former passage as to suggest different authorship.” The sentence translated by Heath reads in the Greek: of pév obv ras ioroplas avaypayarres péxpe robrov mpo&yovaı TV Tis Eriornuns Tabrys rehewoev. It can also be translated: “Those who made written record of mathematical investigations up to this time advance the perfecting of their science.” (And then Euclid... .) ioropia can mean, and in this context ts more likely to mean (LS), some kind of scientific inquiry. Proclus has been listing geometers and their achievements, with special emphasis on publication. Our sentence does not render it impossible that Eudemus should have said: “Thus far... and then Euclid.” It is, however, generally conceded that Proclus has rearranged the material he found in Eudemus (and has probably added the incongruous tribute to Pythagoras), His lemma does not permit us to maintain that Fudemus did in fact know Euclid and looked to Hippocrates as his earliest predecessor, but it docs not exclude the possibility E that he did. In any case the explicit mention of Hippocrates’ Elements suggests that they were a landmark in the development of geometry, and probably its first general formulation.

predecessors in the perspective of Euclid the fact that the name Stoicheia or Elements, with its implications of system, was used to refer to the treatises of Hippocrates of Chios, Leon, Theudius, and Euclid suggests that they were roughly comparable in theme and intention, if not in investigates first principles. He is unlikely to have accepted or to have devised a theory of the derivation of magnitudes having geometrical characteristics. 48 scope. Hippocrates, in the Athens of about 430/410 8.c., is the first mathematician known to us who attempted a systematic exposition of the LiowvREe+?DSRywE0 elements of geometry. But even if we had no tradition of Hippocrates’ treatise we would not expect that before about the last quarter of the fifth century an attempt could be made to present geometry as a science capable of formulating its first principles and presenting its deductions systematically. Any such development must have been preceded by a period in which geometrical problems, both practica] and theoretical, were not necessarily related to one another nor to first principles. It must also have been preceded, or at least accompanied, by an evolution of logical methods such as we find in the fifth century particularly in the Eleatic school. Geometry as a science having some organization must have preceded the derivation theory. For we can perhaps imagine the crude number-thing theory of the Pythagoreans of which Aristotle speaks (Aferaph. 985b 23-986a 21) evolving into a number atomism earlier than the time of Hippocrates, even though our first account of such a theory having scientific pretences ascribes it to Leucippus and Democritus, who were his contemporaries. But a geometrical theory of derivation assumes a pre-existing geometry. If Hippocrates gives us a terminus post guem for a geometrical theory of derivation, at what time and in what place is the theory most likely to have arisen? Those who favour a relatively late date often suggest that the theory is a product of the Pythagorean tradition and elaborated in the Tarentum of Archytas. But even if we assume that our accounts of a Pythagorean tradition, though largely deriving from Neoplatonist sources, are in substance true, we must ask ourselves what is the character of early Pythagorean mathematics. Aristotle tells us (MetapA. 987b 28) that for the Pythagoreans things were numbers, these number-things proum OBRTPEAN 49 If there are no good grounds for looking to Magna Graecia as the milieu in which such a theory could develop, we must then consider whether it could have arisen with Plato and the Early Academy. We know: (1) that the geometrical presuppositions for a geometrical theory of derivation were present only at about the time of Plato's youth; (2) that geometrical speculations, and in general an intense mathematical activity, characterized the Early Academy; (3) that geometrical theories of derivation were in fact advanced in the Early Academy; (4) that these theories took various forms and occasioned lively su:>dtAr debate, suggesting that the theme was a new one; AA as might be expected to give rise to a geometrical theory of deriva- + (5) that the One and the indefinite dyad were first principles such tion. The early history of Greek mathematics is a tissue of fable and conjecture. Facts are few and far between. An inquiry into origins must remain at best a plausible account. It is, I would suggest, a reasonable belief and one that casts light on the evolution of mathematical thinking within the Early Academy, that the point-line-plane-solid theory was devised in that milieu as an essential part of their mathematical and physical doctrines. But what of Zeno and his paradoxes? Must we not see implied in them a theory of the derivation, or at least of the constitution, of solids such as the theory we are discussing? To this objection we must answer that Zeno is indeed a forerunner, but not therefore a progenitor. He was the first to explore logically problems of space, time, and movement. As such he was “the father ofdialectic.” His method naturally led to geometrical inquiry, the precondition of any derivation theory. But though for us his problems are mathematical problems, Zeno’s approach is logical. In none of the fragments need we assume that he possessed mathematical ceeding in some unexplained fashion from the One. This number was not xwpiorés, but the constituent of sensibles (1086b 18). “For they knowledge, or that he is using a mathematical method. When he uses construct the whole universe out of numbers.” The constructs Aristotle the word uéeyeos (ors. 29B 1, 2), it cannot be given its later and technical describes (985b 26-986a 13) are altogether unscientific, bur they are such sense “magnitude” as it cannot when used by Anaxagoras (ors. 59B 1), as could evolve into an arithmetical theory of derivation, the discrete quantities being atomic magnitudes. If there was in fact some such or by Gorgias (Vrs. 82B 3, 73). H. l’raenkel (Vege und Farmen [Munich tradition it enables us better to comprehend the saving of Archytas (Pors. 47 B4) that arithmetic is the queen of the sciences. His own dimensions. There is no justification for this. The words, if they are principal achievements were apparently in the fields of geometry and of harmonics, and yet he maintains that arithmetic rather than geometry 1955] 220-221) treats péyebos, raxos, öyros (B2) as alluding to the three Mdeul'io ‘a LEA cited from Zeno, are practically synonymous. But in cur fragment they are used by Simplicius, in the passage introducing the lemma, and not by Zeno. If in the “race-course” (Aristotle PAys. 239b 34) Zeno used

the word öyxos, he can only have used it as a synonym for uéyeos in the sense of cûua. But this is Aristotle’s own usage (Bonitz, Index s.v. capa). And so he is probably paraphrasing. In all the other paradoxes Zeno uses visual objects to illustrate. However the paradoxes may be interpreted, it is clear that Zeno employed, and employed effectively, the notion of infinite divisibility. But he is not concerned with surfaces and solids, nor with geometrical space. He applies divisibility to lines, and envisages points. His arrow SOPHIA AND SOPHROSYNE IN EURIPIDES’ Patricia NEILS BOULTER Le so MANY Euripidean plays, the Andromache lacks a central unifying character, and for this reason seems to fall into two parts. Criticism has centred upon this disunity, either attempting to explain it away! or simply condemning it as a major flaw in construction.? A frequent explanation of the two-part structure is that it reflects the joining of two separate legends, one concerning Andromache, the other, Neoptolemos is at a point and a “now.” Achilles and the tortoise progress from point to point. In the “dichotomy” lines are traversed at points. But there is no suggestion that the points are atomic points having magnitude; and the implication of infinite divisibility is that the line divided is a continuum. Let us, however, concede that in the fourth paradox or “racecourse” he may have envisaged indivisible lines and times (PAys. 81-82). If in fact he did so, there is no suggestion that he regarded them as the ultimate constituents of magnitude, though the paradox may have suggested this development to later mathematicians. The notion of indivisible lines, however, implies that all lines are commensurable in terms of these lines. 1f Zeno had propounded any such revolutionary theory, we should expect our tradition to allude to it; or at least we should expect to hear some echo of it, perhaps in connection with Theaetetus’ surds, in discussions of irrationals, or in the tenth book of Euclid where problems of incommensurability are treated. But there are no such echoes. When first we hear of indivisible lines we hear of them in connection with the derivation theory of the Early Academy, as an answer to difficulties arising within that theory when the problems of points having magnitude were realized. It seems therefore reasonable to assume that the derivation theory was posterior to the geometrical developments it implies; and that it arose in the milieu in which for some half century it caused lively debate —in the Early Academy.” If, however, we accept that the derivation theory was developed within the Academy (or in allied mathematical circles of the same time) we are not thereby committing ourselves to an esoteric metaphysic such as is constructed by H. J. Kramer in his article “Die Platonische Akademie und das Problem einer systematischen Interpretation der Philosophie Platons” (Kant-Studien 55/1 [1964] 69-101) and in his preceding book ANDROM ACHE and Hermione.? But in fact, the legendary material concerning Andromache before Euripides’ time seems to have been very slight. In the Iliou Persis Andromache was given to Neoptolemos after the fall of wnHieroCyoEst sr . eRyiPAtC Paty em. e tEen ovsie..” ws -. .-: |$" ’ Arete bei Platon und Aristoteles (Heidelberg 1959). For a criticism of such tendencies, Troy. Euripides undoubtedly used local material for Thetis’ prophecy that Andromache would marry Helenos and that her son Molossos would rule over Epirus.* Between these two given points Euripides freely invented the action of the drama. Similarly with Hermione; earlier writers noted that she had been promised by Tyndareus at the same time to both Orestes and Neoptolemos. Euripides altered this detail for his characterization of Menelaos; in the 4udromache, Menelau< had himself made the promise to Orestes and then reneged when it seemed expedient (966-970). According to earlier legend Neoptolemos was killed at Delphi, and after his death Hermione was returned to Orestes. Here again Euripides made a slight change, for Orestes enters the story before the death of Neoptolemos and in fact contrives it. Thus the idea that two separate legends were patched together and resulted in a two-part plot structure does not really explain much. Euripides was clearly free to invent his story in whatever way he chose, and even to alter certain details of the legend between the two given points, beginning with the known fact that Neoptolemos took Andromache after the fall of Troy, and ending with his death at Delphi and Hermione’s subsequent marriage to Orestes. Therefore we must assume that Euripides deliberately constructed the plot as we have it, and that he did so with a purpose. Many attempts have been made to find a unifying element that binds the two parts of the play together, but they have met with little success. and especially of metaphysical structures erected on point - line - surface - solid see K.-H. Ilting's review of K. Gaiser's Platons Ungeschriebene Lehre in Gnomon 37/2 1H. D. F. Kitto, Greek Tragedy (London 1939) 230; G. M. A. Grube, The Drama of (April 1965). The derivation theory is certainly not a foreign body in the thought of the Early Academy. It is not a borrowing from elsewhere that remains unrelated to principal themes. But neither is it a corner-stone having an obvious and necessary place in a metaphysical structure of which we have ground-plar and elevation. Because Plato's later metaphysic is so problematical it has seemed opportune to confine ourselves here to vindicating the derivation theory for the Early Academy. Euripides (New York 1961) 81-82. *4. R. F. Hyslop, The .Indromache of Euripides (London 1900) xiii. IL. Méridier, Exripide 2 (Paris 1927; 90-98, “The Andromache may well have been written for the young king of Molossia, Tharpys, and produced at his court. Cf. D. S. Robertson, CR 37 (1923) 58-60; Schmid-Stählin, Geschichte der griechischen Literatur 3 e! ~ LI (Munich 1940) 408. Puoenrx, Vol. 20 (1964: 1.