The Unity of Mathematics in Plato's 'Republic' 3515110763, 9783515110761

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Theokritos Kouremenos

The Unity of Mathematics in Plato’s Republic Klassische Philologie Franz Steiner Verlag

Palingenesia 102

The new theory was not derived from experience. Viktor could see this quite clearly. It had arisen in absolute freedom; it had sprung from his own head. The logic of this theory, its chain of reasoning, was quite unconnected to the experiments conducted by Markov in the laboratory. The theory had sprung from the free play of thought. It was this free play of thought–which seemed quite detached from the world of experience–that had made it possible to explain the wealth of experimental data, both old and new. The experiments had been merely a jolt that had forced him to start thinking. They had not determined the content of his thoughts. All this was quite extraordinary… His head was full of mathematical relationships, differential equations, the laws of higher algebra, number and probability theory. These mathematical relationships had an existence of their own in some void quite outside the world of atomic nuclei, stars and electromagnetic or gravitational fields, outside space and time, outside the history of man and the geological history of the earth. And yet these relationships existed inside his own head. Vasily Grossman, Life and Fate

CONTENTS Preface ..............................................................................................

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1. ASTRONOMY IN THE REPUBLIC 1.1. 1.2. 1.3. 1.3.1. 1.3.2. 1.3.3. 1.4. 1.5. 1.5.1. 1.5.2. 1.5.3. 1.6. 1.6.1. 1.6.2. 1.7. 1.7.1. 1.7.2. 1.7.3. 1.7.4. 1.7.5.

Introduction ........................................................................................ The two astronomies in R. 7 ............................................................. The contemporary astronomy familiar to Glaucon .......................... Introduction ...................................................................................... The fixed-star phases ......................................................................... The theory of the fixed-star phases .................................................... The hypotheses of contemporary astronomy ................................... The criticism of contemporary astronomy ........................................ Why is contemporary astronomy mired in the sensible world? ......... Arithmetical interlude ....................................................................... Archytas of Tarentum ....................................................................... The future astronomy envisioned by Socrates ................................. The hypotheses of the future astronomy .......................................... Celestial motions in the hypotheses of the future astronomy ............ The future astronomy and solid geometry ........................................ The problem of cube-duplication ...................................................... Archytas’ solution ............................................................................ Menaechmus’ solution ....................................................................... Eudoxus’ solution and his theory of homocentric spheres .............. Future astronomy, stereometry and the theory of homocentric spheres .

11 19 21 21 23 26 30 34 34 36 44 46 46 51 56 56 60 61 62 66

2. THE UNITY OF MATHEMATICS IN THE REPUBLIC 2.1. 2.2. 2.2.1. 2.2.2. 2.2.3. 2.2.4. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9.

Introduction ........................................................................................ Arithmetic as unifier of mathematics ............................................... The superiority of arithmetic and incommensurability ..................... Incommensurability and the future astronomy in R. 7 ........................ Incommensurability in the divided-line simile ................................. Philip of Opus ................................................................................... Proportion-theory as unifier of mathematics ................................... Mutual benefit as unifier of mathematics ........................................... The unity of soul and Callipolis as mutual benefit of their parts ..... The unity of mathematics and the Meno .......................................... Two modern mathematicians on the unity of mathematics .............. Albert Lautman on the unity of mathematics ................................... Mathematics and philosophy ...........................................................

71 74 74 77 79 80 82 83 86 90 94 95 98

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Contents

3. THE MYTH ABOUT PLATO’S ROLE IN THE DEVELOPMENT OF MATHEMATICS 3.1. 3.2. 3.2.1. 3.2.2. 3.2.3. 3.2.4. 3.2.4.a. 3.2.4.b. 3.2.4.c. 3.2.4.d. 3.2.4.e. 3.3. 3.3.1. 3.3.2.

Plato’s role in the development of astronomy ............................ The tradition of the Delian problem and its sources .......................... The main testimonies ................................................................... The generative principle of the tradition ....................................... Evidence for the application of this principle in Theon .................. The sources of the tradition in the Platonic corpus ..................... Meno ........................................................................................... Laws 7 ......................................................................................... Epinomis ........................................................................................ Republic 7 ..................................................................................... Timaeus ........................................................................................ The sources of the myth about Plato’s impact on astronomy ........ Laws 7 .......................................................................................... Laws 1 ..........................................................................................

103 109 109 111 113 115 115 116 117 119 120 120 120 124

Bibliography ................................................................................ Index of passages .........................................................................

132 136

PREFACE In the seventh book of his Republic Plato says that, before the future philosopherrulers begin their study of philosophy, they must engage in an intense and prolonged study of mathematics, ultimately in order to grasp the community and kinship of all its branches, its deep unity. He does not explain how the unity of mathematics is supposed to be understood, however, despite the paramount importance he attaches to this feature of mathematics. The first chapter of this monograph, which develops further Kouremenos (2004), attempts to throw some light on Plato’s conception of astronomy in the seventh book of the Republic as a propedeutic to philosophy by taking into account a possible connection between fourth-century-BC astronomy and solid geometry that could have shaped Plato’s view on the unity of mathematics: the solution to the problem of cube-duplication by Eudoxus of Cnidus has not come down to us, but he could have solved this problem with his famous astronomical theory of homocentric spheres. The second chapter argues that Plato conceives of the unity of mathematics exactly in terms of the mutually benefiting links between its branches, not as imparted by one of them to the rest, over which it is somehow privileged and through which it thus runs, just as he conceives of the unity of the state outlined in the Republic in terms of the common benefit for all citizens, not in the light of the privileged role accorded to its philosopher-rulers. The third chapter expands Kouremenos (2011) and concerns two well-known stories: that the solutions to the problem of cube-duplication put forth by Greek mathematicians in the fourth century BC had been motivated by Plato’s interpretation of a Delphic oracle given to the inhabitants of the island of Delos, and that the philosopher Plato spurred the mathematician and astronomer Eudoxus to come up with his theory of homocentric spheres. All components of these stories, however, including Apollo’s relation with mathematics and the contribution of his oracles in the progress of mathematics in Greece, can be easily traced back to passages in the Platonic corpus. We must thus conclude that both stories are nothing but biographical anecdotes (re)constructing episodes in Plato’s life from the Platonic corpus. Fig. 4 is reproduced from Knorr (1993), fig. 5 from Yavetz (1998) and figs. 6– 8 from Riddell (1979). The passage from Grossman’s Life and Fate is quoted in the epigraph from the translation by Robert Chandler. I would like to thank the editor of the Palingenesia series Prof. Dr. Christoph Schubert for accepting this monograph and for his helpful comments, the staff at Franz Steiner Verlag, my friend Alexandros Kampakoglou, who always responds promptly to my requests for bibliography unavailable here, and my wife Poulheria Kyriakou for her unstinting help and, especially, her kind support. I dedicate this monograph to the memory of my colleague Paraskevi Kotzia. Theokritos Kouremenos Aristotle University of Thessaloniki

1. ASTRONOMY IN THE REPUBLIC 1.1. INTRODUCTION Arithmetic is one of the five branches of mathematics which the future philosopherrulers of the city outlined in the Republic will study for a decade before they move on to dialectic, i.e. philosophy, according to book 7, 537b7–c3. It is introduced in book 6 together with another branch, geometry, in the context of the simile of the divided line. Socrates is presented as asking Glaucon, his codiscussant and Plato’s brother, to imagine a line divided into two unequal parts, liken one to sensibles and the other to intelligibles and then divide each part in the same proportion. The first section of the “sensible” part contains shadows, images and reflections on all kinds of surfaces; the second contains the objects that cast shadows and are pictured or reflected (509d6–510b1). Arithmetic and geometry are introduced in the description of the contents of the “intelligible” part of the divided line (510b2–511c2): Σκόπει δὴ αὖ καὶ τὴν τοῦ νοητοῦ τοµὴν ᾗ τµητέον. Πῇ; Ἧι τὸ µὲν αὐτοῦ τοῖς τότε µιµηθεῖσιν ὡς εἰκόσιν χρωµένη ψυχὴ ζητεῖν ἀναγκάζεται ἐξ ὑποθέσεων, οὐκ ἐπ’ ἀρχὴν πορευοµένη ἀλλ’ ἐπὶ τελευτήν, τὸ δ’ αὖ ἕτερον [τὸ] ἐπ’ ἀρχὴν ἀνυπόθετον ἐξ ὑποθέσεως ἰοῦσα καὶ ἄνευ τῶν περὶ ἐκεῖνο εἰκόνων, αὐτοῖς εἴδεσι δι’ αὐτῶν τὴν µέθοδον ποιουµένη. Ταῦτ’, ἔφη, ἃ λέγεις, οὐχ ἱκανῶς ἔµαθον, ἀλλ’ αὖθις ἦν δ’ ἐγώ· ῥᾷον γὰρ τούτων προειρηµένων µαθήσῃ. οἶµαι γάρ σε εἰδέναι ὅτι οἱ περὶ τὰς γεωµετρίας τε καὶ λογισµοὺς καὶ τὰ τοιαῦτα πραγµατευόµενοι, ὑποθέµενοι τό τε περιττὸν καὶ τὸ ἄρτιον καὶ τὰ σχήµατα καὶ γωνιῶν τριττὰ εἴδη καὶ ἄλλα τούτων ἀδελφὰ καθ’ ἑκάστην µέθοδον, ταῦτα µὲν ὡς εἰδότες, ποιησάµενοι ὑποθέσεις αὐτά, οὐδένα λόγον οὔτε αὑτοῖς οὔτε ἄλλοις ἔτι ἀξιοῦσι περὶ αὐτῶν διδόναι ὡς παντὶ φανερῶν, ἐκ τούτων δ’ ἀρχόµενοι τὰ λοιπὰ ἤδη διεξιόντες τελευτῶσιν ὁµολογουµένως ἐπὶ τοῦτο οὗ ἂν ἐπὶ σκέψιν ὁρµήσωσι. Πάνυ µὲν οὖν, ἔφη, τοῦτό γε οἶδα. Οὐκοῦν καὶ ὅτι τοῖς ὁρωµένοις εἴδεσι προσχρῶνται καὶ τοὺς λόγους περὶ αὐτῶν ποιοῦνται, οὐ περὶ τούτων διανοούµενοι, ἀλλ’ ἐκείνων πέρι οἷς ταῦτα ἔοικε, τοῦ τετραγώνου αὐτοῦ ἕνεκα τοὺς λόγους ποιούµενοι καὶ διαµέτρου αὐτῆς, ἀλλ’ οὐ ταύτης ἣν γράφουσιν, καὶ τἆλλα οὕτως, αὐτὰ µὲν ταῦτα ἃ πλάττουσίν τε καὶ γράφουσιν, ὧν καὶ σκιαὶ καὶ ἐν ὕδασιν εἰκόνες εἰσίν, τούτοις µὲν ὡς εἰκόσιν αὖ χρώµενοι, ζητοῦντες δὲ αὐτὰ ἐκεῖνα ἰδεῖν ἃ οὐκ ἂν ἄλλως ἴδοι τις ἢ τῇ διανοίᾳ. Ἀληθῆ, ἔφη, λέγεις. Τοῦτο τοίνυν νοητὸν µὲν τὸ εἶδος ἔλεγον, ὑποθέσεσι δ’ ἀναγκαζοµένην ψυχὴν χρῆσθαι περὶ τὴν ζήτησιν αὐτοῦ, οὐκ ἐπ’ ἀρχὴν ἰοῦσαν, ὡς οὐ δυναµένην τῶν ὑποθέσεων ἀνωτέρω ἐκβαίνειν, εἰκόσι δὲ χρωµένην αὐτοῖς τοῖς ὑπὸ τῶν κάτω ἀπεικασθεῖσιν καὶ ἐκείνοις πρὸς ἐκεῖνα ὡς ἐναργέσι δεδοξασµένοις τε καὶ τετιµηµένοις. Μανθάνω, ἔφη, ὅτι τὸ ὑπὸ ταῖς γεωµετρίαις τε καὶ ταῖς ταύτης ἀδελφαῖς τέχναις λέγεις. Τὸ τοίνυν ἕτερον µάνθανε τµῆµα τοῦ νοητοῦ λέγοντά µε τοῦτο οὗ αὐτὸς ὁ λόγος ἅπτεται τῇ τοῦ διαλέγεσθαι δυνάµει, τὰς ὑποθέσεις ποιούµενος οὐκ ἀρχὰς ἀλλὰ τῷ ὄντι ὑποθέσεις, οἷον ἐπιβάσεις τε καὶ ὁρµάς, ἵνα µέχρι τοῦ ἀνυποθέτου ἐπὶ τὴν τοῦ παντὸς ἀρχὴν ἰών, ἁψάµενος αὐτῆς, πάλιν αὖ ἐχόµενος τῶν ἐκείνης ἐχοµένων, οὕτως ἐπὶ

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1. Astronomy in the Republic τελευτὴν καταβαίνῃ, αἰσθητῷ παντάπασιν οὐδενὶ προσχρώµενος, ἀλλ’ εἴδεσιν αὐτοῖς δι’ αὐτῶν εἰς αὐτά, καὶ τελευτᾷ εἰς εἴδη. “Look now at how the intelligible part must be divided.” “How?” “In this manner: the soul is forced to study one part of it from hypotheses, using things that were imitated earlier on as images, not ascending to a starting point but descending to an endpoint, but with regard to the other part, it ascends from a hypothesis to an unhypothetical starting point and approaches it without its images, with and through the forms themselves.” “I did not get what you just said,” he replied. “But again ,” I said. “You will understand my point more easily after the following. As you know, I am sure, the students of geometry, arithmetic and the like lay down odd and even, figures, three kinds of angle and other things akin to these in each field, and as if they knew these things, turning them into hypotheses, they do not deign to give either to themselves or to others an account of what is hypothesized, assuming that it is clear to everybody, but start from their hypotheses and go through the subsequent stages to arrive consistently at what they set out to investigate.” “I certainly know this,” he said. “So you also know that they use visible shapes and argue about them, but actually do not think about them but about those things that the visible shapes resemble, their proofs concerning the square itself and the diagonal itself, not that diagonal they draw, and so on–that is, they use as images the shapes they make up and draw, of which there are also shadows and reflections in water, in their attempt to see those things themselves that one can see with no other means than thought.” “It is true,” he said. “These were the intelligibles I was talking about in whose study the soul is forced to rely on hypotheses without ascending to a starting point, since it cannot transcend its hypotheses, but using visible images that are considered to be clearer than the originals and thus prized.” “I see,” he said, “that you are talking about geometry and its kindred fields.” “So you can see that the other section of the intelligible part I was talking about is what reason itself grasps with the power of dialectic, employing hypotheses not as starting-points but as genuine hypotheses, let us say as footholds and launchers, so as to reach what is unhypothetical, the principle of all, and then, having gotten hold of it, turn back and, grasping what depends on it, descend in this manner to the end-point, using no sensibles whatsoever but the forms themselves through themselves to themselves, and end up with forms.”

The first section of the “intelligible” part of the line contains the objects studied in mathematics via their visible images and problematic definitions, “hypotheses”;1 the second contains the forms studied in philosophy without such aids. Plato seems to view what is studied in mathematics as forms approached in a particular way. Below in R. 7, 533a10–c6, he has Socrates say that mathematics sees beings in a dream via unclear hypotheses for which not accounts are given, not in the state of wakefulness, as dialectic does. Here he has Socrates give the square itself with the diagonal itself as example of an object studied in geometry. Forms have been introduced as the only beings at the end of R. 5, in the description of the philosophers (473e5–480a13) after the claim that, unless philosophers rule or rulers philosophize, humankind’s troubles will not end (473c11–e4). Philosophers want to learn about forms such as the beautiful itself, the intelligible and unchanging 1

For hypotheses in the divided-line simile as definitions see Bostock (2009) 13.

1.1. Introduction

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objects of knowledge, each of which is unique but, since it is associated with the changeable sensibles, appears everywhere as many, e.g. beautiful things: the latter resemble their form but are subject to change, and thus cannot possibly be objects of knowledge but only of opinion, though according to non-philosophers they are the only existents.2 Forms are not sensible because that they are immaterial.3 They seem to be conceived as eternal or atemporal entities not existing in space.4 In terms of the traditional ontological categories, they are usually thought to be abstract properties, not definable in observational terms.5 As mathematical objects, forms are best regarded as abstract particulars since in mathematics what does not look like a thing, e.g. a function, is regularly treated as such.6 If mathematical objects are forms, the sections of the “intelligible” part of the divided line do not answer to two different kinds of intelligibles, one studied by mathematics, the other by philosophy, each discipline approaching its objects in its own way, but to the distinct ways in which philosophy and mathematics approach intelligibles of a single type, forms; if so, the sections of the “sensible” part of the divided line similarly do not correspond to two kinds of sensibles but to two distinct ways in which sensibles are approached, and forms can be objects of belief and sensibles of knowledge insofar as they are related to forms.7 We can restore to mathematics its own objects, intelligible ones distinct from forms but similar to them in two crucial respects that explain the use of the same terminology for the description of both kinds of entities, if we rely on the testimony of Aristotle. According to Aristotle, between forms and sensibles Plato wedged mathematical objects as a third kind of existents. These are similar to forms in two respects, hence intelligible, and to sensibles in another: the so-called intermediates are similar to forms, and differ from sensibles, in that they are eternal and cannot move or suffer any change, but also resemble sensibles, and differ from forms, in that for each of them there are many alike (Metaph. Α 6, 987b14–18). Just as there is a single form of beauty over the many beautiful sensible things, there is a single form over the many intermediates that are alike. Aristotle contrasts mathematical numbers, each of which contains its predecessor plus one unit, from those numbers that do not each contain their predecessors: mathematical numbers consist of undifferentiated and combinable units, but each number of the other type has its own units, not combinable with those of any other number (Metaph. M 6, 1080a12–35). The units in the numbers of either type lack magnitude, are partless and indivisible (cf. Metaph. M 6, 1080b16–20, and 8, 1083b8–17). Aristotle calls numbers which are sets of undifferentiated and indivisible units “monadic” (from µονάς, “unit”). Numbers with combinable units are intermediates since Aristotle 2

3 4 5 6 7

The discussion of the Good at R. 6 contrasts the oneness of an intelligible form with the many sensibles associated with, or “participating” in, it and thus also named after it (507b1–9); for the contrast see also Phd. 78c10–79a5. For their immateriality see Sph. 246a7–c3. See Ti. 48e2–52d1 and the description of beauty itself in Smp. 210e2–211b5. On whether forms are timeless or eternal see Sorabji (1983) 108–112. See e.g. Fine (1999) 215 n. 1. See Gowers (2008) 10. For a précis of Platonism in mathematics see Brown (2005) 59–60. See Fine (1999). All forms can thus be only those of mathematical objects; see ch. 2.9.

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1. Astronomy in the Republic

says that for each one of them there exist infinitely many alike (Metaph. M 7, 1081a5–12); numbers consisting of non-combinable units, on the other hand, are said to be forms since a form is unique (Metaph. M 7, 1082b24–28). There is no hint in Plato’s works that he introduced the distinction between intermediates and forms, just as nothing in R. 6, 510b2–511c2, hints that his example of an object studied in mathematics, the square itself with the diagonal itself, is not a form but an intermediate.8 Not unreasonably, scholars have doubted that Plato had put forth this distinction even in his discussions with members of the Academy.9 It is implausible that Aristotle simply foisted it on him, however. An Academic argument for the existence of forms discussed in his On forms was based on the objects of the sciences: the objects of a science exist; they are not particulars, for these are infinitely many and undetermined but each object of a science is single and determined; thus there are things that are different from particulars, and these things are forms (Alex. Aphr. in Metaph. 79.8–11 Hayduck). Aristotle would agree with Plato that the objects of mathematics are determined in the sense that each of them is what it is since e.g. lines are just lines, without breadth and depth, and straight ones lack any curvature (Euc. El. 1 Def. 2 and 4). But he might object that each one of them is not unique: no number of lines etc. is assumed in geometry, so if the argument shows that there must exist some things different from sensible particulars in that they are determined, these things will not be forms, each of which is unique, but form-like in that each of them must be eternal and not subject to change or motion if it is not a sensible particular and what is not a sensible particular is eternal and does not change or move. Assuming that there are other eternal things that do not change or move, the forms, each of which is unique, Aristotle could argue that Plato is committed to intermediates, thereby trying to answer a question raised by the passage from the Republic translated above. In it Plato talks about the visible shapes used in geometrical proofs and about which the geometers seem to argue, such as a square drawn with its diagonal, one of a great many such shapes that can be or are drawn or exist in the physical world, and he also distinguishes them from the intelligible objects that are truly studied in geometry, such as the square itself with the diagonal itself. These are described by him in the same way that he describes forms: the square itself with the diagonal itself seems to answer to the intelligible form of beauty, the beautiful itself, a single being that is associated with many sensibles and appears everywhere as square things, such as the figures drawn in the context of geometrical proofs, 8 9

E.g. Yang (1999) argues that it is an intermediate, Franklin (2012) 494–497 that it is a form. For references see Arsen (2012) 201, who argues in favor of mathematical intermediates. For a survey of older literature against intermediates in Plato’s ontology see Brentlinger (1963). He attempts to strike a middle position suggesting that as intermediates, in a weaker sense than that in which the term is employed by Aristotle, Plato must have regarded the objects of the definitions of arithmetic and geometry: definitions are said in Ep. 7, 342a7–344d2, to be one of the four means by which everything is knowable, so their objects, which are different from both sensibles and forms, whose representations they are, are indispensible to mathematical knowledge, actually of forms, a crucial fact mathematicians fail to grasp, ending up treating erroneously as objects of mathematical knowledge what are only means to it. Brentlinger does not explain, however, why Aristotle speaks of Plato’s intermediates as eternal, like forms.

1.1. Introduction

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which are like it but subject to change. Now, how do the things defined in the hypotheses of mathematics fit into the apparently exhaustive division of existents into sensibles and their forms, which sensibles resemble? As defined in Euclid’s Elements, which can be reasonably assumed to reflect the state of mathematics in Plato’s time, the square of our example is not a sensible object (El. 1 Def. 19 and 22); nor can it be the square itself, for the latter is unique but geometry does not limit the number of objects defined as square. As will be argued next, Plato probably viewed the things defined in the hypotheses of mathematics as neither eternal nor lacking change and motion but as rarefied mental images of sensibles in the simplest cases (e.g. visible squares). Since these sensibles share in a form (e.g. the square itself) and the mental images of them cannot but participate in the relevant form themselves, these images and their (mentally) visualized motions and changes can be used to approach the form. But Aristotle could insist polemically that Plato’s own principles force him to introduce intermediates as a new category of existents and go on to charge him with ontological profligacy. That the objects studied in geometry such as lines and circles are not like any sensible objects was pointed out by Protagoras. Aristotle’s testimony, embedded in his critique of Plato’s supposed theory of intermediates, presupposes the immobility of mathematical objects as conceived by Plato (Metaph. B 2, 997b12–998a6): ἔτι δὲ εἴ τις παρὰ τὰ εἴδη καὶ τὰ αἰσθητὰ τὰ µεταξὺ θήσεται, πολλὰς ἀπορίας ἕξει· δῆλον γὰρ ὡς ὁµοίως γραµµαί τε παρά τ’ αὐτὰς καὶ τὰς αἰσθητὰς ἔσονται καὶ ἕκαστον τῶν ἄλλων γενῶν· ὥστ’ ἐπείπερ ἡ ἀστρολογία µία τούτων ἐστίν, ἔσται τις καὶ οὐρανὸς παρὰ τὸν αἰσθητὸν οὐρανὸν καὶ ἥλιός τε καὶ σελήνη καὶ τἆλλα ὁµοίως τὰ κατὰ τὸν οὐρανόν. καίτοι πῶς δεῖ πιστεῦσαι τούτοις; οὐδὲ γὰρ ἀκίνητον εὔλογον εἶναι, κινούµενον δὲ καὶ παντελῶς ἀδύνατον· ὁµοίως δὲ καὶ περὶ ὧν ἡ ὀπτικὴ πραγµατεύεται καὶ ἡ ἐν τοῖς µαθήµασιν ἁρµονική· καὶ γὰρ ταῦτα ἀδύνατον εἶναι παρὰ τὰ αἰσθητὰ διὰ τὰς αὐτὰς αἰτίας· εἰ γὰρ ἔστιν αἰσθητὰ µεταξὺ καὶ αἰσθήσεις, δῆλον ὅτι καὶ ζῷα ἔσονται µεταξὺ αὐτῶν τε καὶ τῶν φθαρτῶν. ἀπορήσειε δ’ ἄν τις καὶ περὶ ποῖα τῶν ὄντων δεῖ ζητεῖν ταύτας τὰς ἐπιστήµας. εἰ γὰρ τούτῳ διοίσει τῆς γεωδαισίας ἡ γεωµετρία µόνον, ὅτι ἡ µὲν τούτων ἐστὶν ὧν αἰσθανόµεθα ἡ δ’ οὐκ αἰσθητῶν, δῆλον ὅτι καὶ παρ’ ἰατρικὴν ἔσται τις ἐπιστήµη καὶ παρ’ ἑκάστην τῶν ἄλλων µεταξὺ αὐτῆς τε ἰατρικῆς καὶ τῆσδε τῆς ἰατρικῆς· καίτοι πῶς τοῦτο δυνατόν;…ἀλλὰ µὴν οὐδὲ τῶν αἰσθητῶν ἂν εἴη µεγεθῶν οὐδὲ περὶ τὸν οὐρανὸν ἡ ἀστρολογία τόνδε. οὔτε γὰρ αἱ αἰσθηταὶ γραµµαὶ τοιαῦταί εἰσιν οἵας λέγει ὁ γεωµέτρης (οὐθὲν γὰρ εὐθὺ τῶν αἰσθητῶν οὕτως οὐδὲ στρογγύλον· ἅπτεται γὰρ τοῦ κανόνος οὐ κατὰ στιγµὴν ὁ κύκλος ἀλλ’ ὥσπερ Πρωταγόρας ἔλεγεν ἐλέγχων τοὺς γεωµέτρας), οὔθ’ αἱ κινήσεις καὶ ἕλικες τοῦ οὐρανοῦ ὅµοιαι περὶ ὧν ἡ 10 ἀστρολογία ποιεῖται τοὺς λόγους, οὔτε τὰ σηµεῖα τοῖς ἄστροις τὴν αὐτὴν ἔχει φύσιν. Again, if one posits intermediates besides forms and sensibles, he will face many difficulties. For it is clear that there will be lines besides the lines themselves and sensible lines, and similarly with the objects studied in each of the other sciences. Thus, since astronomy is one of them, there will be a cosmos besides the sensible cosmos and a Sun and a Moon and all the other celestial objects. How can one believe these things, however? It is not plausible that the cosmos is immobile and that it moves is completely impossible. The same, moreover, applies to the objects studied in optics and mathematical harmonics. These, too, cannot exist apart from sensibles for the same reasons. For, if there are intermediate sensibles and sensations, it 10

Metaph. B 2, 997b32–998a4 = Protag. DK 80 B 7.

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1. Astronomy in the Republic is clear that there will also be animals between the animals themselves and the perishable animals. A further difficulty is what beings must be studied in these sciences. For, if geodesy differs from geometry only in this respect, i.e. in that the former is concerned with sensibles and the latter with non-sensibles, then it is clear that besides medicine there will be another science between the medicine itself and the medicine of sensibles, and similarly with each of the other sciences. But how is this possible?...On the other hand, astronomy would be neither about sensible lines nor about this cosmos. For there are no sensible lines such as those a geometer’s proofs are concerned with (since nothing sensible is as straight or circular, given that a ruler touches a circle not at a point but as Protagoras used to say taking exception to the geometers), nor are the motions and helixes of the cosmos like those about which astronomy gives proofs, nor do celestial objects have the same nature as points.

Fig. 1

Protagoras pointed out the obvious clash between what geometry shows, i.e. that a straight line touches a circle at a point, in Euclid’s Elements the porism of theorem 3.16, and what we can see right away when we use a ruler to draw a straight line touching a circle drawn with a compass, i.e. that the ruler, and the line we draw with it, touches the circle over a region of perceptible magnitude. The theorem, whence it follows that a straight line touches a circle at a point, asserts that (a) a straight line drawn at right angles to a circle’s diameter falls outside the circle, (b) between this straight line and the circumference no other straight line can be interposed and (c) the angle of the semicircle, the mixed angle of the circle’s diameter and its circumference, is greater, and the similarly mixed angle of the circumference and the straight line at right angles to the diameter less, than any acute rectilinear angle.11 In fig. 1 we assume a straight line ΓΑ at right angles to a circle’s diameter falling within the circle ΑΓΒ and also draw ΓΔ, but since ΓΔ = ΔΑ as radii of a circle and the angle ΔΑΓ is right, the angle ΔΓΑ of triangle ΔΑΓ is also right, hence a triangle’s two internal angles are absurdly both right, and it is thus clear that a straight line perpendicular to the diameter BA at A cannot share with the circle another point: this line must be AE, which is seen, though, to share with the circle not a point but a stretch of its circumference, as Protagoras pointed out. (b) is a further example of this clash between geometry and sensibles: it 11

On the angle of the circumference and its tangent see Heath (1956) vol. 2, 39–43.

1.1. Introduction

17

asserts that AZ does not exist, though it can be easily drawn. For, if ΔΗ is drawn from Δ perpendicular to ΑΖ, then since the angle ΑΗΔ will be right, the angle ΔΑΗ will be less than a right angle and thus the straight line ΔΑ will be greater than the straight line ΔΗ, in the triangle ΔΑΗ the greater side subtending the greater angle (Euc. El. 1.19): but ΔA is a radius of the circle and cannot be greater than ΔΗ which has been drawn from Δ perpendicular to AZ outside the circle, unless the part ΔΘ of ΔΗ, Θ being the point where ΔΗ cuts the circumference, is absurdly greater than the whole. If AZ does not exist, no acute angle BAZ exists and (c) follows immediately: the mixed angle formed by the diameter BA and the circumference AΓ is greater, and the mixed angle formed by the straight line AE at right angles to the diameter BA and the circumference AΓ less, than any acute rectilinear angle. We do not know what lesson Protagoras drew from the fact that in geometry a straight line is shown to touch a circle at a point, whereas a line drawn with a ruler is seen to touch a circle drawn with a compass over a region of some length. But it is implausible that in this self-evident fact Plato could have seen evidence that the intelligible and mind-independent objects geometry really studies are unlimitedly many, eternal, non-sensible and “perfect” lines and circles, like those defined in the hypotheses of contemporary geometry: lines and circles that not only lack even the slightest deviations from true straightness and circularity but also cannot move and change, unlike all sensibles that resemble them and via them unique forms, i.e. the circle itself and the line itself, a single circle and a single line somehow different from, though as “perfect” and motionless as, the countlessly many intermediate or mathematical circles and lines that are like them and over which they each preside. In the geometry of Plato’s time basic geometrical objects were most probably defined through their kinematically visualized production, just as in the Euclidean Elements, where the sphere is generated by a rotating semicircle (El. 11, Def. 14): Σφαῖρά ἐστιν, ὅταν ἡµικυκλίου µενούσης τῆς διαµέτρου περιενεχθὲν τὸ ἡµικύκλιον εἰς τὸ αὐτὸ πάλιν ἀποκατασταθῇ, ὅθεν ἤρξατο φέρεσθαι, τὸ περιληφθὲν σχῆµα. A sphere is the figure comprehended when, the diameter remaining fixed, the semicircle is carried around and is restored to the same position from which it began to be moved.

The sphere is one of the three solid figures defined by Euclid in a kinematically visualized manner. The cone, too, is defined as the figure generated when a right triangle turns about one of the sides of its right angle (El. 11, Def. 18): Κῶνός ἐστιν, ὅταν ὀρθογωνίου τριγώνου µενούσης µιᾶς πλευρᾶς τῶν περὶ τὴν ὀρθὴν γωνίαν περιενεχθὲν τὸ τρίγωνον εἰς τὸ αὐτὸ πάλιν ἀποκατασταθῇ, ὅθεν ἤρξατο φέρεσθαι, τὸ περιληφθὲν σχῆµα. A cone is the figure comprehended when, one of the sides around the right angle of a rightangled triangle remaining fixed, the triangle is carried around and is restored to the same position from which it began to be moved.

Similar also is the Euclidean definition of the cylinder (El. 11, Def. 21):

18

1. Astronomy in the Republic Κύλινδρός ἐστιν, ὅταν ὀρθογωνίου παραλληλογράµµου µενούσης µιᾶς πλευρᾶς τῶν περὶ τὴν ὀρθὴν γωνίαν περιενεχθὲν τὸ παραλληλόγραµµον εἰς τὸ αὐτὸ πάλιν ἀποκατασταθῇ, ὅθεν ἤρξατο φέρεσθαι, τὸ περιληφθὲν σχῆµα. A cylinder is the figure comprehended when, one of the sides about the right angle of a rightangled parallelogram remaining fixed, the parallelogram is carried around and is restored to the same position from which it began to be moved.

Would Plato have thought that the kinematic definitions of the cone etc. describe each a class of infinitely many geometrical objects that exist independently of our mind and are actually unsusceptible to motion and change, hence eternal, or rarefied mental images of sensibles participating in forms, like the sensibles themselves, and also capable of undergoing mentally visualized motions and changes? The first alternative is unlikely given Plato’s view on squaring, addition, and application of areas (R. 7, 527a1–b11): the geometers’ talk of performing these operations on the objects of their study presupposes that these objects do come to be and also pass away, though they are in fact eternal beings, which suggests that it cannot be they themselves that are treated by the geometers as if they were not eternal but some other, non-eternal things similar to, and thus participating in, them. These things can only be sensibles and mentally formed images thereof.12 That an intelligible object, be it a supposed intermediate or a form, instantiates perfectly a hypothesis of mathematics is ruled out by the non-spatiality of beings, which means that they do not have shape (see Phdr. 247c4–d1), and has been plausibly denied by Proclus: he points out that in the realm of the immaterial causes extended things exist without extension, divided things without division, magnitudes without magnitude, figures without shape (in Euc. 54.1–8 Friedlein). The forms of geometrical objects cannot be geometrical, nor can form-numbers be monadic, which is how numbers would be defined in the hypotheses of Plato’s contemporary arithmetic (El. 7, Def. 1–2).13 Examples of hypotheses are given in R. 6, 510b2–511c2, from geometry and arithmetic, and are followed by a cursory reference to their kin in each of the other branches of mathematics. These are astronomy and harmonics if the distinction in the following book between plane and solid geometry is not presupposed here. If it is, then solid angles and solid figures will be defined in the hypotheses of solid geometry; this must hold as much for the immature solid geometry in the dramatic time of the Republic as for its advanced descendant that will be of propedeutical use to philosophy (R. 7, 528a10–c7). Plato, moreover, has Socrates regard not only solid geometry but also astronomy and harmonics as not sufficiently developed to be propedeutically useful to dialectic. This service will again be rendered by an advanced astronomy and harmonics of the future (R. 7, 528e1–531c8). 12

13

Cf. above n. 9 on Brentlinger’s suggestion. Franklin (2012) argues in a similar manner that Plato viewed intermediates as “theoretical fictions”. It should be noted that Plato does not despise visualization in mathematics. In this he is remarkably modern; see Mancosu (2005) 13 –30. Wedberg (1955) 66, 80–84 and 120 denies that form-numbers can be monadic since forms are simple; he also cites van der Wielen (1941) ch. 7, esp. 87–89, according to whom Aristotle misrepresented Plato’s form-numbers (van der Wielen [1941] is extensively reviewed in Cherniss [1947] 235–251). See also Findlay (1974) 56–57, Tarán (1981) 13–29 and (1991) 206–224; cf. below 1.5.2.

1.2. The two astronomies in R. 7

19

Numbers will feature in the hypotheses of the future harmonics that will serve philosophy propedeutically, just as they are implied to have featured in those of harmonics in the dramatic time of the Republic (R. 7, 531b2–c5). Is the advanced astronomy of the future that will aid philosophy propedeutically also assumed to share at least some of its hypotheses with its primitive ancestor in the dramatic time of the dialogue? Since in R. 7, 530b6–c4, the future astronomy that will propedeutically serve philosophy is reduced to (plane) geometry (see 1.7.5), its hypotheses will be about some forms of “geometrical” objects. These are static and intelligible, i.e. existing independently of both their sensible manifestations, whether in the sky or in figures drawn for proofs, and all our mental visualizations and definitions of them. The astronomy in the dramatic time of the Republic, however, is assumed to have a single hypothesis concerning a geometrical object that is a mental visualization of something sensible and also undergoes motion.14 1.2. THE TWO ASTRONOMIES IN R. 7 In R. 7 Plato has Socrates suggest that he, Socrates, and his codiscussant, Glaucon, consider astronomy to be the third subject after arithmetic and geometry to which philosopher-rulers in the making must apply themselves before taking up the study of philosophy. Glaucon readily agrees, pointing out the importance that better awareness of the seasons, months and years has not only for agriculture and sailing but also for generalship (527d1–4). Glaucon’s answer is in line with the keen interest he has shown earlier on in the military applications first of arithmetic, much to Socrates’ amusement, who, though, plays along for the sake of the discussion (522c1–e2), and then of geometry. In this context Socrates points out that a smattering of geometry and arithmetic suffices for military purposes (526c7–d7). It is not that Plato portrays Socrates as indifferent about the important practical applications, in general, and the military ones in particular, of arithmetic and geometry. He has Socrates try to hint to Glaucon that arithmetic is a subject that the future philosopher-rulers must study before they embark on their philosophical studies by pointing out to his codiscussant that the discipline he has in mind is useful to military men like the future philosopher-rulers (521d8–12); offering another hint, Socrates says that it is common to all arts and crafts, no matter how humble, and to all knowledge, and that it is one of the first things we learn (522b1–c9). By this dialogic device Plato 14

According to Mueller (1992) 192–194, the hypotheses of the astronomy that will serve as a true propedeutic to philosophy will posit uniform and circular motions; see also the conception of the subject-matter of this astronomy in Mourelatos (1981) 27–30, Robins (1995) 373–378 and Burnyeat (2000) 56–63 (this is a view that seems to go back to Simplicius; see ch. 3.1). But motion cannot have anything essential to do with the mathematical objects that these hypotheses will be about; nor is there any hint that the mathematical objects, whose supposedly rotational motions will form the subject-matter of the philosophically useful future astronomy, will be circles and spheres (Mourelatos, Burnyeat and Mueller, as it turns out from Mueller [1981] 103–111) or the regular solids inscribed in spheres (Robins [1995] 373).

20

1. Astronomy in the Republic

hints that the propedeutics of knowledge, and thus knowledge itself, which falls within the purview of philosophy and has forms as its objects, are humbly rooted in the world of becoming, which falls within the purview of the other crafts and arts. That much has been made clear in the myth of the cave at the beginning of R. 7, where the unity of the worlds of becoming and being is further emphasized by the striking return to the cave of the initially unwilling escapees, who will come to stand for the accomplished philosopher-rulers. What matters next, however, when Plato has Socrates turn his and Glaucon’s attention to the propedeutic studies of the future philosopher-rulers, is the escape from the cave, which stands for the revolving of the soul away from the nocturnal day of becoming to the true day of being (521c1–d7). Socrates has to remind Glaucon of this after Glaucon agrees that astronomy is to be on the propedeutic curriculum of the future philosopherrulers in view of this discipline’s practical utility (527d5–528a3). After Glaucon’s fumbling answer (528a4–5), Socrates asks his codiscussant to leave astronomy aside, for not astronomy but solid geometry must be the third propedeutic of philosophy after arithmetic and, as it now emerges, plane geometry (528a6–d11). Socrates goes on to reintroduce astronomy as the fourth propedeutic, and an amazed Glaucon realizes that the astronomy Socrates considers the fourth propedeutic is a future astronomy, not contemporary astronomy (528e1–530c4). It is the latter that Glaucon must have in mind when he agrees to the inclusion of astronomy in the propedeutic studies of the future philosopher-rulers on account of the discipline’s practical utility. Socrates, on the other hand, suggests the inclusion of astronomy with the future astronomy in mind. There are thus two astronomies in R. 7. When in R. 6 Plato has Glaucon agree with Socrates that all branches of mathematics start from hypotheses, in view of R. 7 he portrays Glaucon as implicitly agreeing that Socrates gives an accurate description of the situation in contemporary astronomy. However, it is unlikely that Plato intends his readers to also assume retrospectively that actually Socrates refers implicitly to the hypotheses of astronomy thinking already of the future astronomy he envisions in the next book. In R. 6 nothing hinges on the distinction between contemporary and future astronomy. Astronomy, whether contemporary or future, is a branch of mathematics: as such, it must start from hypotheses if all branches of mathematics start from hypotheses, which is all that matters in the section of R. 6 under discussion. Another reason why the distinction Plato draws in R. 7 between contemporary and future astronomy might not be of relevance to the discussion in R. 6 about the hypotheses of mathematics is that the two astronomies do not differ as far as hypotheses are concerned: their hypotheses might be implicitly assumed in R. 7 to be the same, either wholly or in part, those of future astronomy augmenting those of contemporary astronomy. Let us see first which are the hypotheses of contemporary astronomy, the discipline as it was at the dramatic date of the Republic, the early 410s BC at the latest (cf. n. 76). There is no reason to assume that, by portraying Glaucon as being amazed at the realization that the astronomy Socrates considers an appropriate propedeutic of philosophy is not the astronomy of their time, Plato intends his audience to impute anachronistically to Glaucon knowledge of the astronomy at

1.3. The contemporary astronomy familiar to Glaucon

21

the time the dialogue was composed, too, in the mid- to late 370s BC on a generally accepted dating. As will be seen below in 1.7, it is Socrates who, with his vision of future astronomy, is in all likelihood presented as prescient of the developments in the field after the dramatic date of the Republic. This does not imply, though, that the astronomy of the first half of the fourth century BC is the future astronomy Plato has Socrates dream about in R. 7. 1.3. THE CONTEMPORARY ASTRONOMY FAMILIAR TO GLAUCON 1.3.1. Introduction The contemporary astronomy Glaucon has in mind when he agrees with Socrates’ remark that all branches of mathematics arrive at their conclusions starting from hypotheses must be linked with better awareness of the seasons, months and years. Enhancing this awareness seems to have been a major concern for Greek and, in particular, Athenian astronomers at the time the discussion in the Republic is supposed to have taken place. The oldest known Greek astronomical observation is of the summer solstice of 432 BC by Euctemon and Meton, and observations of solstices are related to the determination of the length of the year defined as the time between two successive summer or winter solstices, or that between two successive vernal or autumnal equinoxes (tropical year).15 They are also related to the determination of the lengths of the seasons defined astronomically as the time between vernal equinox and summer solstice (spring), summer solstice and autumnal equinox (summer), autumnal equinox and winter solstice (autumn), and winter solstice and vernal equinox (winter); the discovery of the inequality of the seasons is attributed to Euctemon and Meton.16 Finally, the length of the year is related to the construction of lunisolar cycles. The problem is to find a number of years which contains almost a whole number of synodic months: the solution allows one to determine how many of these years must have a thirteenth month added so that a lunar calendar, such as those used by Greek cities, does not get noticeably out of step with the seasons. In our sources Euctemon or Meton or both are credited with the introduction of the 19-year cycle in Athens, beginning at the summer solstice of 432 BC.17 19 tropical years are almost 235 synodic months; thus in a period of 19 years 12 years must have each 12 months (144 months in total), whereas each of the remaining 7 years must have 13 month each (91 months in total). The 19-year cycle is called the Metonic cycle. Around the dramatic date of the Republic astronomers in Athens were thus occupied with a set of problems whose solutions could be said to have aimed at better awareness of the seasons, months and years. Glaucon, however, cannot 15 16 17

See Ptol. Alm. 1.205.15–21 and 207.7–10 (Heiberg); cf. Lehoux (2007) 88–90. See Simp. in Cael. 497.17–22 (Heiberg) = Eudem. fr. 149 Wehrli, and PPar 1 (the so-called Ars Eudoxi) col. 23; cf. Neugebauer (1975) 627–629 summarized in Evans (1998) 458 n. 10. See Diod. Sic. 12.36.2–3, Ptol. Alm. 1.207.7–10 (Heiberg), Gem. Elem. Astr. 8.50, Theodos. De diebus 2.18, Censor. De die natali 18.5.

22

1. Astronomy in the Republic

speak having in mind the regulation of the civil calendar per se by the Metonic cycle when he says in R. 7 that astronomy enhances the useful awareness of the seasons, months and years in order to explain why he agrees with Socrates that the field must be on the propedeutic course of the future philosopher-rulers. This awareness, he says, comes in handy for the sailor, the farmer and, not least, the general. As it is, we expect that whatever might be on his mind as he voices his agreement must have been widely known to be useful for sailors, farmers and military leaders. No matter why they needed to be aware of the seasons, months and years, however, Greek sailors, farmers and generals could not have been aided by lunar calendars regulated by the Metonic cycle. The reason is that, even in Plato’s time, the Metonic cycle was not used by the Greeks, Athenians included, to regulate their notoriously chaotic civil calendars.18 What role the Metonic cycle played is obscure. A plausible possibility is that the cycle was introduced not to regulate the civil calendars per se but rather the position in the year of certain phenomena called phases of the fixed stars.19 Already in Hesiod’s Works and Days the farmer judges the changing of the seasons and the right time for his various tasks by observing these phenomena, which also help the sailor avoid sailing in stormy seas. By the end of the fifth century BC epigraphical lists of fixed-star phases in order of occurrence in the course of the year might have already appeared in Greek cities (extant fragmentary specimens are later). The fixed-star phases could be accompanied by weather predictions in these calendars, which let the public know the occurrence of the fixed-star phases without having to observe the actual phenomena. Into a hole bored in the stone next to the entry for each day was inserted a movable peg to indicate the current day of the year, hence the Greek word for this calendar, παράπηγµα. A Greek civil calendar without a regular pattern of intercalation of months, and occasionally even with days freely intercalated or deleted for various reasons, could not have been used for constructing παραπήγµατα, a situation which might have led Meton, Euctemon or both to either come up themselves with the 19-year cycle for this purpose or borrow it from Babylonia, where it had been used to regulate the calendar from the beginning of Darius’ reign (522 BC).20 Both astronomers feature prominently in the literary parapegmatic tradition from which modern scholars learned about the calendars of the fixed-star phases before the discovery of the first epigraphical specimens at the beginning of the last century.21 Even if Meton and Euctemon simply argued in their works for the systematic presentation of the body of accumulated lore surrounding the phases of the fixed stars and inscriptional tables based on their ideas began to be constructed only after Plato’s time, certainly the fixed-star phases were widely known to be useful for sailors and farmers much earlier than the dramatic time of the Republic.22 We can be reasonably certain that 18 19 20 21 22

See Neugebauer (1975) 616–617 and Stern (2012) 51–53. See again Neugebauer (1975) 616–617 and Lehoux (2007) 90–93. On the 19-year cycle and the Babylonian calendar see Stern (2012) 105–124. The history of παραπήγµατα is exhaustively treated in Lehoux (2007). Whether Euctemon and/or Meton had composed literary or inscriptional παραπήγµατα cannot be asserted with any degree of certainty; see Lehoux (2007) 20–26 and 212–213.

1.3. The contemporary astronomy familiar to Glaucon

23

Glaucon’s first agreement in R. 7 to put astronomy on the propedeutic curriculum of the future philosopher-rulers is meant to remind the reader mainly of the phases of the fixed stars;23 secondarily, perhaps, of the technical work by the astronomers Euctemon and Meton. Meteorological predictions based on the fixed-star phases would presumably have been of interest to military commanders, too, during operations. At any rate, since military operations were conducted in summer, they might be associated by Glaucon with agricultural works, such as sowing etc., and sailing, in that they, too, were seasonal activities, hence his claim that astronomical knowledge of the seasons based on the phases of the fixed stars matters not only to the farmer and the sailor but to the general as well. 1.3.2. The fixed-star phases The phenomena called phases of the fixed stars are consequences of the annual motion of the Earth around the Sun, which, to adopt the geocentric point of view appropriate here, we perceive as the annual motion of the Sun around the Earth in the circle called ecliptic, against the background of the constellations of the zodiac. Parts of this stellar backdrop are hidden by the Sun as it moves slowly along the ecliptic from west to east, at the rate of about 1° per day. Stars hidden by the Sun cannot be seen from the Earth to rise and set. But, when the Sun will have moved sufficiently eastwards along the ecliptic, or a hitherto hidden star will have slipped sufficiently westwards, then (I) just before sunrise (a) the star will be seen rising in the east for the first time after a period of invisibility and (b) a star will be seen setting in the west for the first time (before this event its setting occurred during the day and was thus invisible); (II) soon after sunset (c) a star will be seen setting in the west for the last time and (d) a star will be seen rising the east for the last time (after these events the setting of the one star and the rising of the other will take place during the day and will be invisible). (a) – (d) are the four phenomena called fixed-star phases: (a) is the visible morning rising and (b) the visible morning setting; (c) is the visible evening setting and (d) the visible evening rising. They are called visible to be distinguished from the true fixed-star phases: the true morning rising and setting, and the true evening setting and rising. They occur when a star crosses the horizon at the same time as the Sun, hence they cannot be seen. As such, they were not of interest to farmers and sailors. As said above, it was by means of the fixed-star phases that already in Hesiod’s time the farmer determined the right time for his agricultural tasks and the sailor the best time for sailing. In the Works and Days the farmer is advised to harvest the wheat when the Pleiades make their morning rising, in May, and to plow the land and sow the wheat when the Pleiades make their morning setting, in November (383–387):

23

Cf. Robins (1995) 373 and Burnyeat (2000) 11–12; see also next n.

24

1. Astronomy in the Republic Πληιάδων Ἀτλαγενέων ἐπιτελλοµενάων ἄρχεσθ’ ἀµήτου, ἀρότοιο δὲ δυσοµενάων. αἳ δή τοι νύκτας τε καὶ ἤµατα τεσσαράκοντα κεκρύφαται, αὖτις δὲ περιπλοµένου ἐνιαυτοῦ φαίνονται τὰ πρῶτα χαρασσοµένοιο σιδήρου. When the Pleiades, daughters of Atlas, are rising, start harvesting, plowing when they are setting; for an interval of forty nights and days these stars remain hidden, and, as the year turns round, they reappear when the iron sickle is first sharpened.

The evening rising of Arcturus signals the coming of spring, the return of the swallow and the time to prune the vines (564–570): Εὖτ’ ἂν δ’ ἑξήκοντα µετὰ τροπὰς ἠελίοιο χειµέρι’ ἐκτελέσῃ Ζεὺς ἤµατα, δή ῥα τότ’ ἀστὴρ Ἀρκτοῦρος προλιπὼν ἱερὸν ῥόον Ὠκεανοῖο πρῶτον παµφαίνων ἐπιτέλλεται ἀκροκνέφαιος. τὸν δὲ µέτ’ ὀρθογόη Πανδιονὶς ὦρτο χελιδὼν ἐς φάος ἀνθρώποις ἔαρος νέον ἱσταµένοιο· τὴν φθάµενος οἴνας περιταµνέµεν· ὣς γὰρ ἄµεινον. When after the turnings of the Sun sixty wintry days Zeus has completed, then the star Arcturus, leaving the Ocean’s holy stream, rises shiny for the first time at dusk; next Pandion’s mournful daughter, the swallow, appears to men, when spring is beginning anew. Prune the vines before her arrival, as is best to do.

The time for threshing is signaled by the morning rising of Orion (597–599): Δµωσὶ δ’ ἐποτρύνειν Δηµήτερος ἱερὸν ἀκτὴν δινέµεν, εὖτ’ ἂν πρῶτα φανῇ σθένος Ὠρίωνος, χώρῳ ἐν εὐαεῖ καὶ ἐυτροχάλῳ ἐν ἀλωῇ. Have your slaves winnow Demeter’s holy grain when strong Orion first appears, in an airy place and a level threshing floor.

As the evening rising of Arcturus signals the time to prune the vines, its morning rising is the astronomical sign that the time of vintage has come, and is followed by the morning setting of the Pleiades, with which the agricultural year ends. Stormy weather follows the morning setting of the Pleiades, after which sailing must be avoided (609–623): Εὖτ’ ἂν δ’ Ὠρίων καὶ Σείριος ἐς µέσον ἔλθῃ οὐρανόν, Ἀρκτοῦρον δὲ ἴδῃ ῥοδοδάκτυλος Ἠώς, ὦ Πέρση, τότε πάντας ἀποδρέπεν οἴκαδε βότρυς, δεῖξαι δ’ ἠελίῳ δέκα τ’ ἤµατα καὶ δέκα νύκτας,

1.3. The contemporary astronomy familiar to Glaucon

25

πέντε δὲ συσκιάσαι, ἕκτῳ δ’ εἰς ἄγγε’ ἀφύσσαι δῶρα Διωνύσου πολυγηθέος. αὐτὰρ ἐπὴν δὴ Πληιάδες θ’ Ὑάδες τε τό τε σθένος Ὠρίωνος δύνωσιν, τότ’ ἔπειτ’ ἀρότου µεµνηµένος εἶναι ὡραίου· πλειὼν δὲ κατὰ χθονὸς ἄρµενος εἴη. Εἰ δέ σε ναυτιλίης δυσπεµφέλου ἵµερος αἱρεῖ· εὖτ’ ἂν Πληιάδες σθένος ὄβριµον Ὠρίωνος φεύγουσαι πίπτωσιν ἐς ἠεροειδέα πόντον, δὴ τότε παντοίων ἀνέµων θυίουσιν ἀῆται· καὶ τότε µηκέτι νῆα ἔχειν ἐνὶ οἴνοπι πόντῳ, γῆν δ’ ἐργάζεσθαι µεµνηµένος ὥς σε κελεύω· When Orion and Sirius come to the middle of the sky and rosy-fingered Dawn looks upon Arcturus, it is time, Perses, to cut off all grapes and bring them home; expose them to the sun for ten days and ten nights, then cover them for five days and on the sixth pour into vessels the gifts of joyful Dionysus. But when the Pleiades, the Hyades and strong Orion set, it is time to be mindful of plowing in season; may the seed be sowed firmly in the earth. Now, if desire for dangerous seafaring grips you, when the Pleiades, from the strong Orion fleeing, fall into the misty sea, then unpredictable winds blow strong; do not keep your ship at sea any longer, mindful of working the land, as I instruct you.

If Glaucon is indeed portrayed by Plato as speaking with the fixed-star phases paramount in his mind when he remarks in R. 7 that astronomy must be on the propedeutic curriculum of the future philosopher-rulers, given its importance not only to agriculture and sailing but also to generalship, then it is the theory of the fixed-star phases that must be identified with the contemporary astronomy he is familiar with. It is with this theory that Glaucon contrasts in amazement the future astronomy envisioned next by Socrates as appropriate propedeutic for the study of philosophy. He cannot contrast it with the fixed-star phases themselves or in their association with the coming of the seasons, the timing of agricultural works and the weather.24 The fixed-star phases are phenomena that can be easily observed, and their observation does not require any theory, as is clear from their prominent place in Hesiod’s Works and Days. The same goes for their function as signs. Associating the morning setting of the Pleiades with the time for plowing and the evening rising of Arcturus with the coming of spring requires as much theory as does the realization that the return of the swallows heralds the coming of spring. But, as said above in 1.2, the contemporary astronomy, whose inclusion in the propedeutic 24

Robins and Burnyeat, as pointed out in the previous n., associate Glaucon’s conception of astronomy with the fixed-star phases, to which they refer as risings and settings without further specification, but do not identify the astronomy Glaucon has in mind with the theory of the fixed-star phases.

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curriculum of the future philosopher-rulers Glaucon justifies in R. 7 on account of its relevance to the needs of farming and sailing, is identical with the astronomy implicitly assumed in R. 6 by Glaucon himself to proceed from hypotheses. It has things similar to the odd and even numbers in arithmetic and to the three kinds of angle and the various figures in geometry. It is like arithmetic and geometry in that it proves various conclusions about the objects defined in its hypotheses, as e.g. arithmetic proves theorems about the odd and even numbers. Factoring in its link with better awareness of the seasons, months and years, with the phenomena called phases of the fixed stars, it can only be the theory of these phenomena. 1.3.3. The theory of the fixed-star phases What is called here theory of the fixed-star phases is not predictive. It does not aim at foretelling the occurrence of the phenomena with which it is concerned but rather at providing a theoretical understanding of them in geometrical terms. It distinguishes the true from the visible phases of the fixed-stars, and then proceeds to prove theorems about the order of the true and the visible phases; the length of the period at which a star is seen rising each night between its visible morning and evening risings, as well of the period at which a star is seen setting each night between its visible morning and evening settings; the length of the period between its visible morning rising and setting, and of the period between its visible evening rising and setting, depending on the position of the star in the sky; the length of the period between the true morning rising and evening rising of a star, and between its true evening setting and morning setting; the length of the period at which a star cannot be seen between its visible evening setting and morning rising, depending on the position of the star in the sky; the order of the visible phases of a star, depending on its position in the sky (the position of a star in the sky is not given in any system of celestial coordinates but in terms of the ecliptic: a star is in or near the ecliptic, north or south of it). These are some of the theorems proven by Autolycus of Pitane in his short treatise On Risings and Settings.25 Dated to the last decades of the fourth century BC, it is one of the earliest surviving Greek mathematical works; the other is On the Moving Sphere, another treatise by Autolycus.26 Assuming that all its theorems, or the most important of them, had been known to, or proven by, Greek astronomers already in the dramatic time of the Republic does not involve any anachronism. To get an idea of the simple geometrical style of the theory of the fixed-star phases as developed by Autolycus, let us see the demonstration of the first proposition in the first book of On Risings 25

26

The two books of On Risings and Settings are actually different versions of the same treatise, the second better than the first; see Schmidt (1952) and Neugebauer (1975) 751. There is no reason to assume that Autolycus depends on a lost work by Eudoxus; see Neugebauer (1975) 750. For a concise presentation of the theory of fixed-star phases see Evans (1998) 190–198. These works are often regarded as older than Euclid’s Elements and Phenomena and thus as the oldest surviving Greek mathematical and astronomical treatises. But there is no good reason for this assumption; see Neugebauer (1975) 750.

1.3. The contemporary astronomy familiar to Glaucon

27

and Settings: the visible morning risings and settings of all fixed stars follow the true ones, whereas the visible evening risings and settings precede the true ones. In fig. 2 the circle αβγδ is the horizon and αεγζ the circle of the Sun, the ecliptic. α marks the east, γ the west and the semicircle αεγ is below the Earth. When the Sun is rising at α, a star rising simultaneously at δ is making its true morning rising, which is invisible, for the light of the star is swamped by the light of the Sun. After some days the Sun will have moved eastwards to ε, and the star at δ will be far enough from it to be seen for the first time rising before sunrise. This is the visible morning rising of the star, which thus occurs after the true one. Again, as the Sun is rising at α, another star is simultaneously setting at β, thus making its true morning setting, which is invisible. After some days, when the Sun will have moved to ε, the star at β will set while the Sun is below the horizon and will be observed setting for the first time soon before sunrise, i.e. making its visible morning setting, which thus follows the true one. Now, when the Sun is setting at γ, a star is simultaneously rising at δ, thus making its true evening rising, which is invisible. A few days earlier the Sun was at η, when the star at δ rose while the Sun was below the horizon: it could be observed rising for the last time soon after sunset, i.e. making its visible evening rising, which thus precedes the true one. Again, as the Sun is setting at γ, another star is simultaneously setting at β, thus making its true evening setting, which is invisible. A few days earlier the Sun was at η, when the star at β set while the Sun was below the horizon: it could be observed setting for the last time soon after sunset, i.e. making its visible evening setting, which precedes the true one.

Fig. 2

The two circles αβγδ and αεγζ are introduced by Autolycus as great circles on a sphere, the cosmos: ἔστω ἐν κόσµῳ ὁρίζων κύκλος ὁ αβγδʹ, ὁ δὲ τοῦ ἡλίου κύκλος θέσιν ἐχέτω ὡς τὴν αεγζʹ (“let the circle αβγδ be the circle of the horizon on the cosmos and the circle of the Sun be positioned as the line αεγζ”). What Autolycus calls cosmos is the so-called celestial sphere. The celestial sphere is thought to be concentric with the comparatively insignificant Earth, and the stars visible at night to an Earth-based observer are assumed to be its bright points. The horizon of any Earth-based observer bisects this sphere if extended in thought arbitrarily far, which shows that the Earth is a mere point by comparison, and the

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celestial equator, the equator of the celestial sphere, is coplanar with the equator of the Earth. It can be assumed that it is not the Earth but the enormous celestial sphere that rotates once a day, about an axis which is an extension of the Earth’s own axis of rotation, but in the opposite direction to that of the Earth’s true rotation: all basic phenomena caused by the true rotation of the Earth, the diurnal risings and settings of the stars and the Sun, will be observed. Although a fiction, the celestial sphere is considered real in ancient cosmology, the physical surface that is the extremity of the cosmos, below which are the stars, the most remote objects from the Earth. It is a notion still useful in astronomy. The zodiac is a zone on the celestial sphere defined by the constellations through which we see the Sun pass in a year as it apparently orbits the Earth, a reflection of the Earth’s true orbiting of the Sun. The plane of the Earth’s orbit is not perpendicular to our planet’s rotational axis, and forms an angle with its equator: thus the ecliptic, the projection of the Sun’s apparent annual orbit around the Earth on the celestial sphere, another great circle on that sphere like the celestial equator and the horizon, is seen to be at an angle to the celestial equator, as is also the belt of the zodiac.27 Though not mentioned in the proof from the first book of On Risings and Settings presented above, or in the other proofs in this book, the celestial equator plays a role in several proofs in the second book of the treatise. The other surviving work of Autolycus, On the Moving Sphere, is named after the diurnally rotating celestial sphere. Many of its simple propositions concern the diurnal risings and settings of points on this sphere. At the beginning of all proofs in the first book of On Risings and Settings the horizon and the ecliptic are introduced as great circles on the cosmos, the celestial sphere.28 There are no other spheres in the simple theory of the phases of the fixed stars, unlike in the famous theory of the homocentric spheres developed in the fourth century BC by Eudoxus of Cnidus.29 Now, a single solid is mentioned by Socrates in R. 7, 528a6–10, as what astronomy studies. He asks Glaucon, who has just agreed with him that astronomy, given its practical applications, must be the third propedeutic for the future philosopher-rulers, to leave this field aside for a moment. For not astronomy but stereometry must be regarded as third propedeutic of philosophy after arithmetic and plane geometry, as becomes evident in retrospect: Ἄναγε τοίνυν, ἦν δ’ ἐγώ, εἰς τοὐπίσω· νῦν [δὴ] γὰρ οὐκ ὀρθῶς τὸ ἑξῆς ἐλάβοµεν τῇ γεωµετρίᾳ. Πῶς λαβόντες; ἔφη. Μετὰ ἐπίπεδον, ἦν δ’ ἐγώ, ἐν περιφορᾷ ὂν ἤδη στερεὸν λαβόντες, πρὶν αὐτὸ καθ’ αὑτὸ λαβεῖν. “Let us then backtrack,” I said, “for we chose to put after geometry the wrong subject.” 27 28

29

The plane of the Earth’s orbit, too, is confusingly called ecliptic, in a heliocentric sense of the term. The horizon and the ecliptic are also introduced at the beginning of all proofs in the second book. In it the tropics and the celestial equator are also introduced twice (4–5) and the meridian once (9). The theory will be discussed below in 1.7.4–5.

1.3. The contemporary astronomy familiar to Glaucon

29

“How,” he asked, “did we err?” “By considering a rotating solid,” I said, “after the plane before the solids themselves.”

Since Glaucon agrees in 528b3, Socrates is portrayed as mentioning to him a prominent characteristic of the astronomy that he, Glaucon, is familiar with, the contemporary astronomy. But if this astronomy is, as argued above, the theory of the fixed-star phases, the single rotating solid identified by Socrates as its subject must be the diurnally rotating celestial sphere.30 It might be objected that in the lines just cited the subject of astronomy cannot be identified with a single rotating solid, for Socrates contrasts astronomy with solid geometry in terms of what they each study, and solid geometry cannot but be assumed to study the solids themselves, not a single solid: στερεὸν αὐτὸ καθ’ αὑτό must thus be used collectively, and so must be ἐν περιφορᾷ ὂν στερεόν, which means that astronomy studies many rotating solids, not a single one, as argued above, and if so, the identification of Glaucon’s contemporary astronomy with the theory of the fixed-star phases is doubtful. Although στερεόν must be used collectively for solid geometry, it is very unlikely that in the lines under discussion Plato identifies the subject-matter of solid geometry with that of astronomy minus rotational motion, as he must do if he uses στερεόν collectively for astronomy, too, and the objection is followed to its logical end. In this case, he wrote the lines under discussion having in mind an astronomy studying as many solids undergoing rotational motion as were studied in solid geometry at the dramatic time of the Republic. What we know about the solid geometry of that time, however, shows that it studied a range of solids that never played any role in astronomy.31 If Plato does use στερεόν collectively for solid geometry, too, but without implying the identification of the subject of solid geometry tout court with that of astronomy minus rotational motion, he can only think of the subject of astronomy as a multitude of spheres: a set of spheres is the only multitude of solids which Plato could plausibly consider as playing a role in astronomy, assuming that he would implicitly identify the fourth-century-BC theory of homocentric spheres with the astronomy which Glaucon is portrayed as familiar with at the dramatic time of the Republic. An anachronistic picture of astronomy at the dramatic time of the dialogue would not in itself be a problem in a literary work such as the Republic. If the astronomy of the late fifth century BC familiar to Glaucon is viewed anachronistically as the fourth-century-BC theory of homocentric spheres, however, then Glaucon must be intentionally or accidentally portrayed by Plato as mistaken when he speaks excitedly about the relevance of contemporary astronomy to farmers, sailors and generals, for the theory of homocentric spheres did not have any such relevance. Nothing in the text hints at this, though. The problem with what Glaucon says when he agrees that astronomy is to be the third propedeutic for the future philosopher-rulers given its practical applications is by no means the non-existence of such applications but rather Glaucon’s emphasis on 30 31

Cf. κόσµος στρεφέσθω and στρεφοµένου τοῦ κόσµου in Autol. de Ort. 2.1 and 2. Democritus stated theorems on cones and cylinders having the same base and on pyramids and prisms having the same base (Euc. El. 12.7 and 10); see Archim. Eratosth. 2.430.1–9 Heiberg. Cf. section 1.7.2 below on Archytas’ solution to the cube-duplication problem.

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1. Astronomy in the Republic

them. It is thus preferable to assume that the astronomy familiar to Glaucon does not study a number of rotating solids but a single rotating solid, which must be the celestial sphere if this astronomy is the theory of the fixed-star phases in view of its relevance to the concerns of farmers, sailors and generals. Let us now turn to the hypotheses of this astronomy. 1.4. THE HYPOTHESES OF CONTEMPORARY ASTRONOMY Various plane figures and the three kinds of plane angle are defined at the beginning of the first book of Euclid’s Elements, a treatise usually dated not much later than the two surviving works of Autolycus. A different set of definitions at the beginning of the arithmetical section of the Elements, books 7–9, contain those of the odd and even numbers. Similarly, various solid figures and the solid angle are defined at the beginning of the final section of the treatise, books 11–13, which is devoted to solid geometry. We do not know anything about the structure of the precursors of Euclid’s Elements in Plato’s time, but it is not far-fetched to assume that they already prefaced the treatment of arithmetic and geometry with some kind of definitions of the objects studied in each subject and of some of their parts: Plato might have had such definitions in mind when he had Socrates mention the various hypotheses of arithmetic and geometry in R. 6, 510c2–d3. Alternatively, in Plato’s time definitions might have not yet been formally set out as in Euclid’s Elements: perhaps contemporary mathematicians did not yet feel any need to include definitions in their mathematical treatises, though they operated with some definitions, perhaps often those in Euclid’s Elements, and would have readily provided them if asked to, in which case Plato would have had in mind such formally unstated definitions when he had Socrates talk about the hypotheses of arithmetic and geometry. This alternative has in its favor the virtual absence of definitions such as those in the Euclidean Elements from the surviving works of Autolycus, which perhaps allow us a glimpse of the situation between the dramatic date of the Republic, as well as the date of its composition, and Euclid’s time, if they are indeed the oldest surviving Greek mathematical works.32 On the Moving Sphere begins with a single definition, at most three if the second and third are not obelized as additions.33 At the beginning of On Risings and Settings there are only two definitions, the first of the true and the second of the visible phases of the fixed stars; as said in 1.3.3, at the beginning of all proofs in both books of On Risings and Settings the horizon and the ecliptic are introduced as great circles on the celestial sphere, but neither this sphere nor its important great circles are defined (Autolycus, however, as we will see in a moment, does define the celestial sphere at the beginning of On the Moving Sphere, not formally but implicitly in his first theorem). Now, when Plato 32 33

Cf. above n. 26. The first definition is of uniform motion, the second and third of a sphere’s axis and poles. The second and third definitions appear in only one manuscript. The first definition is suspect, too: it is equivalent to the second theorem.

1.4. The hypotheses of contemporary astronomy

31

presents Glaucon in R. 6 as readily agreeing with Socrates that all branches of mathematics, hence the contemporary astronomy familiar to Glaucon, too, the theory of the fixed-star phases, as argued in 1.3.2–3, start each from some hypotheses, the objects implicitly assumed to be defined in the hypotheses of this astronomy must be the celestial sphere and its important great circles. This conclusion follows from the contrast in R. 7, 528a6–10, between the subject-matters of stereometry and the astronomy familiar to Glaucon. Solid geometry studies solid figures, which must be defined in its hypotheses according to R. 6, 510c2–d3. Glaucon agrees that a single rotating solid is studied in astronomy, i.e. the celestial sphere, as argued in 1.3.3; by analogy with solid geometry, therefore, this solid must be defined in a hypothesis of the astronomy familiar to Glaucon, its most fundamental definition presupposed by all the rest, of circles such as the horizon, the celestial equator and the ecliptic. Although the celestial sphere is not formally defined at the beginning of On the Moving Sphere, as already noted, the first proposition of the treatise is in effect a definition of the celestial sphere. Autolycus demonstrates that, if a sphere rotates uniformly about its axis, all points on the surface of the sphere except those on the axis, i.e. except the poles, will trace parallel circles that have the same poles as the sphere and are perpendicular to its axis: Ἐὰν σφαῖρα στρέφηται ὁµαλῶς περὶ τὸν ἑαυτῆς ἄξονα, πάντα τὰ ἐπὶ τῆς ἐπιφανείας τῆς σφαίρας σηµεῖα ὅσα µὴ ἔστιν ἐπὶ τοῦ ἄξονος κύκλους γράψει παραλλήλους τοὺς αὐτοὺς πόλους ἔχοντας τῇ σφαίρᾳ, καὶ ἔτι ὀρθοὺς πρὸς τὸν ἄξονα. If a sphere rotates uniformly about its axis, all points of the sphere’s surface not on its axis will trace out parallel circles whose poles are the same as the sphere’s and which are perpendicular to the axis.

The theorem seems to lack any astronomical relevance, as do all theorems proven in On the Moving Sphere. Its astronomical relevance is obvious, however, since we see point-like stars tracing uniformly at night the same parts of unequal parallel circles whose centers are on the same line as if they were traced by the points of a uniformly rotating sphere, and we can easily imagine that in the period of a day they all complete their circles: hence the stars are fixed, bright points of a uniformly rotating sphere, the celestial sphere, which completes a rotation in the course of a day. As it is, the first theorem in On the Moving Sphere turns out to provide an implicit definition of the celestial sphere when we adjoin to it an astronomically basic observational fact: the celestial sphere is a sphere that rotates uniformly and completes a rotation in a day, and whose luminous points are identified with the stars because in a day they each complete parallel circles, just like those traced by the points of a uniformly rotating sphere. Thus soon after Plato’s time we find the celestial sphere defined, albeit implicitly, at the beginning of an astronomical work related closely to the theory of the phases of the fixed stars, just as the odd and even numbers and various geometric figures, which Plato regards as starting points of arithmetic and geometry in R. 6, 510c2–d3, are defined at the beginning of the treatment of these subjects in Euclid’s Elements, perhaps a little later than Autolycus.

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The Phaenomena, an elementary astronomical treatise attributed to Euclid, begins with the statement that the stars must be assumed (θετέον) to be carried in circles and to be set into a single body, the observer being equidistant from the circumferences of those circles, for the stars are seen not only always rising from the same place and setting in the same place, those rising together always rising together and those setting together always setting together, but also always moving from rising to setting having the same angular separations from one another: with things that move circularly, as observedly do the stars from rising to setting, this happens only when the observer is equidistant from the circular paths. The single body into which the stars must be assumed to be set, and by which they must also be assumed to be carried in circles as it rotates, is the celestial sphere conceived of as a solid, the diurnally rotating cosmos, near the surface of which the stars are set: Ἐπειδὴ ὁρᾶται τὰ ἀπλανῆ ἄστρα ἀεὶ ἐκ τοῦ αὐτοῦ τόπου ἀνατέλλοντα καὶ εἰς τὸν αὐτὸν τόπον δυόµενα καὶ τὰ ἅµα ἀνατέλλοντα ἀεὶ ἅµα ἀνατέλλοντα καὶ τὰ ἅµα δυόµενα ἀεὶ ἅµα δυόµενα καὶ ἐν τῇ ἀπ’ ἀνατολῆς ἐπὶ δύσιν φορᾷ τὰ πρὸς ἄλληλα διαστήµατα τὰ αὐτὰ ἔχοντα, τοῦτο δὲ γίνεται ἐπὶ τῶν ἐγκύκλιον φορὰν φεροµένων µόνον, ἐπὰν ἡ ὄψις πάντῃ τῆς περιφερείας ἴσον ἀπέχῃ, ὡς ἐν τοῖς ὀπτικοῖς δείκνυται, θετέον, τὰ ἄστρα ἐγκυκλίως φέρεσθαι καὶ ἐνδεδέσθαι ἐν ἑνὶ σώµατι καὶ τὴν ὄψιν ἴσον ἀπέχειν τῶν περιφερειῶν. The fixed stars are always seen rising from the same place and setting in the same place, those that rise together always rising together and those that set together always setting together, and they are also seen to keep the same distances from one another as they move from rising to setting, which happens only with things that move in circles if the observer’s eye is in every direction equidistant from the circumference, as is shown in Optics,34 so it must be supposed that the stars are carried in circles and are fixed into a single body, the observer’s eye being equidistant from the circumferences.

This is a definition of the celestial sphere that also explains, as does its implicit analogue in On the Moving Sphere, why this important notion arises naturally. Next comes a description of the diurnal circles of the stars, which shows that these circles are best imagined as parallel circles on a sphere, and a definition of the celestial equator and of the ecliptic as a great circle on the celestial sphere oblique to the celestial equator; this section of the introduction to the treatise ends by concluding that the sphericity of the cosmos, i.e. the existence of the celestial sphere conceived of as the boundary surface of the cosmos, must be assumed from all the above (διὰ δὴ τὰ προειρηµένα πάντα ὁ κόσµος ὑποκείσθω σφαιροειδής).35 Scholars have raised doubts about the authenticity of the introduction to the Phaenomena.36 But it is interesting to note that the author of this introduction thought it fit for a treatise on astronomy to begin with a definition of the celestial sphere, just as the treatment of arithmetic and geometry in the Euclidean Elements begins with e.g. the definitions of the odd and even numbers and various figures, which Plato considers in R. 6, 510c2–d3, to be starting points of these two fields of 34 35 36

Nothing in Euclid’s Optics corresponds to this reference; cf. Neugebauer (1975) 756 n. 13. For an analysis of the introduction see Berggren & Thomas (2006) 48–52. See Neugebauer (1975) 756 and Berggren & Thomas (2006) 8–13.

1.4. The hypotheses of contemporary astronomy

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mathematics. What is more, the author brings in the celestial sphere by using a verbal adjective (θετέον) deriving from the same verb as the noun Plato uses for the starting points of a branch of mathematics (ὑποθέσεις). Even if the treatise did not originally include the introduction, the concept of the celestial sphere is presupposed in the proof of its first proposition, which states that the Earth is in the middle of the cosmos, the celestial sphere, occupying the position of the center with respect to the cosmos. The proof rests on the simple fact that, when one of two diametrically opposite points of the zodiac, tacitly assumed to be a great circle on the celestial sphere, is seen from the Earth rising at a point in the eastern part of the circle of the horizon, then the second point can be seen setting at a diametrically opposite point in the western part of the horizon: this means that the observer is in the middle of a diameter of the celestial sphere.37 As it is, when much later than Autolycus and Euclid, in the second century AD, Claudius Ptolemy begins in the introduction to his Almagest the discussion of the preliminary notions to the study of astronomy with the notion of the celestial sphere, he seems to follow a long tradition that can be traced as far back as the dramatic date of the Republic (Alm. 1.10.4–11.13 Heiberg): Τὰς µὲν οὖν πρώτας ἐννοίας περὶ τούτων ἀπὸ τοιαύτης τινὸς παρατηρήσεως τοῖς παλαιοῖς εὔλογον παραγεγονέναι· ἑώρων γὰρ τόν τε ἥλιον καὶ τὴν σελήνην καὶ τοὺς ἄλλους ἀστέρας φεροµένους ἀπὸ ἀνατολῶν ἐπὶ δυσµὰς αἰεὶ κατὰ παραλλήλων κύκλων ἀλλήλοις καὶ ἀρχοµένους µὲν ἀναφέρεσθαι κάτωθεν ἀπὸ τοῦ ταπεινοῦ καὶ ὥσπερ ἐξ αὐτῆς τῆς γῆς, µετεωριζοµένους δὲ κατὰ µικρὸν εἰς ὕψος, ἔπειτα πάλιν κατὰ τὸ ἀνάλογον περιερχοµένους τε καὶ ἐν ταπεινώσει γιγνοµένους, ἕως ἂν τέλεον ὥσπερ ἐµπεσόντες εἰς τὴν γῆν ἀφανισθῶσιν, εἶτ’ αὖ πάλιν χρόνον τινὰ µείναντας ἐν τῷ ἀφανισµῷ ὥσπερ ἀπ ἄλλης ἀρχῆς ἀνατέλλοντάς τε καὶ δύνοντας, τοὺς δὲ χρόνους τούτους καὶ ἔτι τοὺς τῶν ἀνατολῶν καὶ δύσεων τόπους τεταγµένως τε καὶ ὁµοίως ὡς ἐπίπαν ἀνταποδιδοµένους. µάλιστα δὲ αὐτοὺς ἦγεν εἰς τὴν σφαιρικὴν ἔννοιαν ἡ τῶν αἰεὶ φανερῶν ἀστέρων περιστροφὴ κυκλοτερὴς θεωρουµένη καὶ περὶ κέντρον ἓν καὶ τὸ αὐτὸ περιπολουµένη· πόλος γὰρ ἀναγκαίως ἐκεῖνο τὸ σηµεῖον ἐγίνετο τῆς οὐρανίου σφαίρας τῶν µὲν µᾶλλον αὐτῷ πλησιαζόντων κατὰ µικροτέρων κύκλων ἑλισσοµένων, τῶν δ’ ἀπωτέρω πρὸς τὴν τῆς διαστάσεως ἀναλογίαν µείζονας κύκλους ἐν τῇ περιγραφῇ ποιούντων, ἕως ἂν ἡ ἀπόστασις καὶ µέχρι τῶν ἀφανιζοµένων φθάσῃ, καὶ τούτων δὲ τὰ µὲν ἐγγὺς τῶν αἰεὶ φανερῶν ἄστρων ἑώρων ἐπ’ ὀλίγον χρόνον ἐν τῷ ἀφανισµῷ µένοντα, τὰ δ’ ἄπωθεν ἀναλόγως πάλιν ἐπὶ πλείονα· ὡς τὴν µὲν ἀρχὴν διὰ µόνα τὰ τοιαῦτα τὴν προειρηµένην ἔννοιαν αὐτοὺς λαβεῖν, ἤδη δὲ κατὰ τὴν ἐφεξῆς θεωρίαν καὶ τὰ λοιπὰ τούτοις ἀκόλουθα κατανοῆσαι πάντων ἁπλῶς τῶν φαινοµένων ταῖς ἑτεροδόξοις ἐννοίαις ἀντιµαρτυρούντων. It is a reasonable assumption that the ancients got their first notions on the matters under consideration from the following kind of observations. They saw that the Sun, the Moon and the other celestial objects were carried from east to west always along parallel circles, and that they began to rise up from below, as if from the Earth itself, gradually getting up high and then in a similar fashion turning and getting lower, until they were gone from sight as if they had fallen to the Earth, and that after they had stayed invisible for some time, they rose and set once again, these times, as well as the places of risings and settings, being fixed and, for the most part, the same.

37

See Berggren & Thomas (2006) 52–55 and Evans (1998) 88–89.

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1. Astronomy in the Republic What chiefly led the ancients to the concept of the celestial sphere was the revolution of the ever visible stars, which was observed to be circular and to take place about a single center, the same [for all]. This point necessarily became [for the ancients] a pole of the celestial sphere. The stars that were closer to it revolved on smaller circles, while those farther away described ever larger circles in proportion to their distance, until the distance to the stars that became invisible was reached, of which those near the ever visible stars were observed by the ancients to stay invisible for a short time, while those farther away stayed invisible for longer, again proportionately [to their distance from the pole]. Originally, therefore, it was only this kind of observations that led the ancients to the aforementioned concept, but as their researches went on, they grasped that everything else was in accord with it, for all phenomena without exception contradict alternative notions.

It is worth noting that a few lines below (Alm. 1.13.10–12 Heiberg) Ptolemy calls the concept of the celestial sphere a hypothesis (ὑπόθεσις). As said in 1.3, we can be reasonably certain that Glaucon’s first agreement in R. 7 to put astronomy on the propedeutic curriculum of the future philosopherrulers is meant to remind the reader mainly of the theory of the fixed-star phases, whose most basic definition must be of the celestial sphere, and secondarily, perhaps, of the work by Euctemon and Meton (see 1.3.1). The definition of the celestial sphere can be regarded as the most fundamental hypothesis in their work, too. The equinoxes and the solstices, through which the lengths of the four seasons and the length of the year are defined, are points on the ecliptic, one of the most important great circles on the celestial sphere, whereas the synodic month is the time the Moon needs to orbit the Earth moving against the background of the zodiacal belt on the celestial sphere, the point of reference for a complete orbit being the Sun. 1.5. THE CRITICISM OF CONTEMPORARY ASTRONOMY 1.5.1. Why is contemporary astronomy mired in the sensible world? We saw in 1.2 that in R. 7, 527d5–528a3, after Glaucon agrees that astronomy must be on the propedeutic curriculum of the future philosopher-rulers in view of this discipline’s practical utility, Socrates has to remind his codiscussant that the propedeutic studies aim at making the mind turn away from the nocturnal day of becoming to the true day of being, as he puts it when he and Glaucon start discussing in detail the propedeutic program (521c1–d7). Next Socrates asks his codiscussant to put astronomy aside for the moment: it is not astronomy but solid geometry that must be regarded as the third propedeutic of philosophy after arithmetic and plane geometry, as is evident now (528a6–d11). Socrates then reintroduces astronomy as the fourth propedeutic of philosophy, and Glaucon agrees that his codiscussant rightly chided him when he praised the inclusion of astronomy in the propedeutic studies of the future philosopher-rulers on account of its practical usefulness: it is evident to all that astronomy does force the mind to turn upwards from what is down here (528e1–529a2). Glaucon intends this statement to be understood as true, literally and metaphorically. Astronomy forces the mind to turn upwards literally, to the sky from everything around us, and metaphorically, to

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intelligible being from sensible becoming. Glaucon was reminded of this when Socrates related what is studied by the astronomy familiar to Glaucon and what is studied in solid geometry (528a6–10).38 Earlier on, moreover, Glaucon had no problem admitting that plane geometry draws the mind to being, though geometers talk misleadingly about the objects they study as if they were part of the world of becoming, hence subject to change, generation and decay (527a1–b11). The reason for this must be that change etc. are undergone by the visible or mentally visualized representations of the objects under study, not by the latter themselves, which are not only eternal and unchangeable but also free of all motion, as is evident from Aristotle’s Metaph. B 2, 997b12–998a6 (see section 1.1; as will be argued in this and the following section, though, Aristotle caricatures Plato’s position when he refers ironically to a non-sensible, immobile celestial sphere [οὐρανός] beyond the visible one, and to a host of non-sensible, immobile luminaries beyond those in the sky). The mathematical objects themselves can be grasped only by the mind, not the senses; the mind, though, must necessarily be aided by visible or mentally visualized representations of the objects that it actually studies. Glaucon agreed easily to this in R. 6, 510d5–511a3, after Socrates explained how all branches of mathematics reach their conclusions: through visible images of the objects under study and definitions of mental visualizations of these objects (see 1.1). But, if plane geometry draws the mind to being despite the fact that the intelligible objects of its study must be regarded as subject to generation, change and motion, contemporary astronomy, which studies a rotating solid, can indeed draw the mind to being no less effectively than does solid geometry, which cannot but consider the various objects it studies, the solids themselves, as subject to motion etc., just as its plane counterpart regards its own objects. To Socrates it is by no means clear that contemporary astronomy forces the mind to turn upwards to intelligible being from sensible becoming (529a3–c2): Ἴσως, ἦν δ’ ἐγώ, παντὶ δῆλον πλὴν ἐµοί· ἐµοὶ γὰρ οὐ δοκεῖ οὕτως. Ἀλλὰ πῶς; ἔφη. Ὡς µὲν νῦν αὐτὴν µεταχειρίζονται οἱ εἰς φιλοσοφίαν ἀνάγοντες, πάνυ ποιεῖν κάτω βλέπειν. Πῶς, ἔφη, λέγεις; Οὐκ ἀγεννῶς µοι δοκεῖς, ἦν δ’ ἐγώ, τὴν περὶ τὰ ἄνω µάθησιν λαµβάνειν παρὰ σαυτῷ ἥ ἐστι· κινδυνεύεις γὰρ καὶ εἴ τις ἐν ὀροφῇ ποικίλµατα θεώµενος ἀνακύπτων καταµανθάνοι τι, ἡγεῖσθαι ἂν αὐτὸν νοήσει ἀλλ’ οὐκ ὄµµασι θεωρεῖν. ἴσως οὖν καλῶς ἡγῇ, ἐγὼ δ’ εὐηθικῶς. ἐγὼ γὰρ αὖ οὐ δύναµαι ἄλλο τι νοµίσαι ἄνω ποιοῦν ψυχὴν βλέπειν µάθηµα ἢ ἐκεῖνο ὃ ἂν περὶ τὸ ὄν τε ᾖ καὶ τὸ ἀόρατον, ἐάν τέ τις ἄνω κεχηνὼς ἢ κάτω συµµεµυκὼς τῶν αἰσθητῶν τι ἐπιχειρῇ µανθάνειν, οὔτε µαθεῖν ἄν ποτέ φηµι αὐτόν, ἐπιστήµην γὰρ οὐδὲν ἔχειν τῶν τοιούτων, οὔτε ἄνω ἀλλὰ κάτω αὐτοῦ βλέπειν τὴν ψυχήν, κἂν ἐξ ὑπτίας νέων ἐν γῇ ἢ ἐν θαλάττῃ µανθάνῃ. 38

Robins (1995) 373 thinks that Glaucon “takes the solids” studied in solid geometry “to be the bodies visible in the skies, and he detaches observation of them from any correlation with times and seasons”. What is studied by solid geometry, however, cannot be plausibly related to what is studied by astronomy in this manner; see above section 1.3.3. Nor is there any hint in the text that Glaucon now lets the connection between astronomy and seasonal changes fall by the wayside.

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1. Astronomy in the Republic “Maybe,” I said, “it is clear to everybody except me, because I don’t think so.” “But,” he asked, “why?” “I think that astronomy, as pursued by those who relate it to philosophy, does nothing but make us look downwards.” “What,” he said, “do you mean?” “You seem to me,” I answered, “to have a nice view of the study of the things up there! You give me the impression that you think that one sees not with his eyes but with his mind even if he tries to learn something by bending his head backwards to look at decorations on a ceiling! You might be right and I naïve. But I cannot help thinking that only the study of what is and what the eyes cannot see can make the soul look upwards and, if someone attempts to study something sensible looking upwards open-mounted or squinting downwards, I say that he will never learn anything, for there is no knowledge about any of these things, and that his soul looks not upwards but downwards even if he tries to learn lying on his back on the earth or in the sea.”

What Plato portrays Socrates as suggesting to Glaucon is that astronomers do not study their rotating solid as a visible or mentally visualized representation of an intelligible and static object, which they actually study. The rotating solid in the proofs of the theorems constituting the simple theory of the fixed-star phases, or the theory in Autolycus’ On the Moving Sphere, is studied as a visible or mentally visualized representation of something visible, the celestial sphere, not of what the mind alone can grasp aided by visualization, mental or not (cf. 1.4). Geometrical objects undergoing motion in definitions or proofs can be easily considered mental visualizations of static and abstract forms, whose nature we can grasp not through our sight but only with our mind, aided by visible instances of them, our mental visualizations thereof and the motions our visualizations can undergo: the rotating solid that is the celestial sphere cannot be so considered, however. 1.5.2. Arithmetical interlude The R. 7 discussion of the preparatory studies of the future philosopher-rulers begins with a statement of the goal that must be served by these studies and will thus be the criterion for the selection of their contents: the prisoners’ forced exit from the cave of the famous simile at the beginning of the book (514a1–521b11) and the soul’s revolving away from the nocturnal day of coming to be to the true day of being (521c1–d7). That mathematics will be the sole subject of these studies can be easily deduced from the similes of the cave and the divided line. As Socrates explains to Glaucon in 517a8–c4, a prisoner’s forced exit from the cave and her ascent to the surface is the ascent from the sensibles to the intelligibles, i.e. to forms, and the Sun is the Good. Hence, in the light of the divided-line simile, the shadows cast by the things outside the cave as the Sun illuminates them and their reflections on water, which is all that the former prisoner can look at directly at first, stand for forms as studied in mathematics; the things themselves, which she will eventually be able to behold directly before bearing to view the Sun itself at the end, stand for forms as studied in dialectic. Mathematics is thus implicitly the sole propedeutic to philosophy, for only what is studied in arithmetic, geometry and the other branches of mathematics is contained in the “lower” section of the

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“intelligible” part of the divided line. But Plato does not have Socrates introduce mathematics as sole propedeutic to philosophy by simply reminding Glaucon of the divided line and the cave. He has Socrates try to hint to Glaucon that arithmetic is the first subject the future philosopher-rulers must study before they embark on their philosophical studies by pointing out that he has in mind something useful to soldiers, like the future philosopher-rulers (521d8–12): this is not only common to all arts and crafts, no matter how humble, and all knowledge but also one of the very first things we learn (522b1–c9). To an uncomprehending Glaucon Socrates reveals that he is thinking of “that trivial thing, to distinguish the one and the two and the three; I mean in general all number and calculation” (522c5–7: τὸ φαῦλον τοῦτο, ἦν δ’ ἐγώ, τὸ ἕν τε καὶ τὰ δύο καὶ τὰ τρία διαγιγνώσκειν· λέγω δὲ αὐτὸ ἐν κεφαλαίῳ ἀριθµόν τε καὶ λογισµόν).39 It is not yet clear, however, why arithmetic can aid the soul’s revolving away from the nocturnal day of becoming to the true day of being. Before passing on to that, Plato has Socrates poke fun at the importance laid on the military uses of arithmetic by the presentation of the hero Palamedes in tragedy, the mythological inventor of numbers, thanks to whom it had been made possible for the Greeks to array their troops and count their ships at Troy and for Agamemnon to count his feet (522c7–d9)!40 A little geometry, thus a little arithmetic, too, suffices for military needs, Socrates observes below, but the mind’s turning away from the sensibles to the intelligibles needs much more than that (526c11–e5).41 As Plato has Glaucon put it memorably, being human needs counting and calculating much more than troop-deployment does (522e1–4). Addressing now the question why arithmetic can help in drawing the soul away from the sensibles towards being, Socrates first distinguishes between those sensibles that force the intellect to inquire and those that do not. Unlike the second, the first always present themselves to the senses together with their opposites, so the senses, although they do operate under normal conditions, always perceive no more each one of them than its opposite. A finger e.g. presents itself to sight as what it is, a finger, without being accompanied by anything opposite to it, and the intellect is not forced to ask what a finger is. But its softness presents itself to touch along with its hardness, and its smallness, relative to another finger, presents itself to sight along with its largeness, relative to a third finger. In these cases the reports of the senses to the intellect that a soft thing is hard and a small thing large need investigation, so the intellect asks whether in each case it is dealing with a single property or two, which the relevant sense perceives tangled into one but it, the intellect, is forced to recognize as separate (522e5–524c9). This is how we first wonder what largeness is and what smallness is, Socrates concludes, and he also reminds his codiscussant of what they said in R. 5, 478e7–479c5, about largeness itself and smallness itself: they are intelligibles, forms, unlike the mix of largeness and smallness perceived in a finger by the sense of touch (cf. 475e6–476b11). This passage helps us understand the problem with a finger’s softness and hardness or 39 40 41

Plato calls the first propedeutic ἀριθµητική and/or λογιστική. λογιστική is another name for ἀριθµητική: its results often involve addition etc., ratio, λόγος, and proportion, ἀναλογία. Cf. A. frr. 181a–182 Radt. The reorientation of the mind is a battle, though, “the mother of all battles”; see R. 7, 534b3–d2.

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smallness and largeness. Any sensible property of its soft parts that touch might identify with softness is present in its hard parts and might also be identified with hardness, which is absurd; its seen length makes it smaller than a second finger and larger than a third, so if sight identifies this length with smallness, then it absurdly identifies the same length with largeness. It is now clear in what sense some of the sensibles incite the intellect to inquire according to Socrates (524c10–d5): Οὐκοῦν ἐντεῦθέν ποθεν πρῶτον ἐπέρχεται ἐρέσθαι ἡµῖν τί οὖν ποτ’ ἐστὶ τὸ µέγα αὖ καὶ τὸ σµικρόν; Παντάπασι µὲν οὖν. Καὶ οὕτω δὴ τὸ µὲν νοητόν, τὸ δ’ ὁρατὸν ἐκαλέσαµεν. Ὀρθότατ’, ἔφη. Ταῦτα τοίνυν καὶ ἄρτι ἐπεχείρουν λέγειν, ὡς τὰ µὲν παρακλητικὰ τῆς διανοίας ἐστί, τὰ δὲ οὔ, ἃ µὲν εἰς τὴν αἴσθησιν ἅµα τοῖς ἐναντίοις ἑαυτοῖς ἐµπίπτει, παρακλητικὰ ὁριζόµενος, ὅσα δὲ µή, οὐκ ἐγερτικὰ τῆς νοήσεως. Μανθάνω τοίνυν ἤδη, ἔφη, καὶ δοκεῖ µοι οὕτω. “So this is how it first occurs to us to ask what actually is the large and the small?” “Certainly.” “And we called the latter intelligible, the former visible.” “Absolutely,” he said. “These are then the things I was having in mind a moment ago when I said that some of them incite the intellect but others do not, defining those that incite the intellect as those affecting the sense together with their opposites and those that do not do so as not inciting the intellect.” “Now I understand,” he said, “and I agree.”

Back to astronomy, now, sight grasps as unambiguously the shape of what, as a result, is called the celestial sphere, as it does the shape of the path traced out by each star (cf. section 1.4), and thus does not provoke the intellect to examination, just as it does not do so when it perceives a finger: had sight failed to grasp that shape, the intellect would have been roused to seek an intelligible, immobile mathematical object that could be linked as paradigm to the indeterminate sensible shape presented to sight, a distorted image (cf. section 1.6.2). Contemporary astronomy forces the intellect to turn literally upwards since it studies the sky and the phenomena it exhibits but not also in the crucial metaphorical sense of interest here, for it does not draw the soul toward something intelligible beyond the relevant visible things and their phenomena. Although its students look literally up, metaphorically they are bogged down here, in the sensible world of becoming. It is worth taking a close look at the rest of the discussion of arithmetic, for it makes clear that, as noted in the introduction to this chapter, form-numbers are not monadic. Socrates goes on to ask Glaucon the following question: are what is one and the numbers presented to any sense as what they each are, i.e. one and a given number, without their being accompanied by their opposites, so that the soul is not forced to wonder what the one itself is or what a given number itself is, or are they always presented to the senses together with their opposites and thus prompt the soul to ask these questions (524d6–525a3)? To paraphrase Glaucon’s reply, sight sees at the same time one finger and three phalanges, three being by implication as incompatible with one as smallness is with largeness in the example in the previous

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discussion, though Plato has Glaucon say exaggeratingly “we see the same thing as one and at the same time as an indefinitely large multitude” (525a4–6). Socrates next observes, and Glaucon easily agrees, that this applies to all numbers (525a7–9): the sensible things that are called e.g. three fingers are said at the same time to be nine phalanges. Thus, although a finger, insofar as it is a finger, unlike its simultaneously present smallness and largeness, does not force the soul to ask what a finger itself is, three fingers do force the soul to ask what the number three itself is and what the number nine itself is. Socrates and Glaucon conclude then that arithmetic, the study of numbers, can indeed help the mind move towards truth, emerging from sensibles, which always change, to come in contact with eternal and unchangeable being (525a10–c7). It is easy to see how this conclusion is reached. Given R. 5, 478e7–479c5, the passage alluded in 524c10–d5, the answer to the question what smallness itself or largeness itself is does not refer to something sensible but to a form: similarly, the answer to the question what a number itself is refers to an intelligible form-number which is only, and thus truly, e.g. three, not nine or any other number. A form-number seems to be implicitly regarded here as monadic, a group of things each of which is truly one, unlike sensible things which are e.g. three fingers as much as they are nine phalanges, each of them being not truly one thing but divided into many. This cannot be so, however. In the above account of the discussion of arithmetic up to 524c10–d5, the last passage translated, “intellect” translates νόησις; in this passage διάνοια appears for the first time in the discussion of arithmetic as an alternative to νόησις. These two terms are familiar from the divided-line simile in R. 6, where Plato declares for the first time that mathematics studies intelligible objects (see 1.1). Mathematicians employ visible shapes about which they construct their arguments but actually they think not about these sensibles, such as a drawn square with its diagonal, but about objects which the sensibles only resemble, forms such as the square itself with the diagonal itself, and rely on sensibles only as means in their quest to see the objects of their interest, which can be understood only through thought; other means the mathematicians employ in their quest are hypotheses, problematic definitions of the objects under study in need of elucidation. The intelligibility of mathematical objects is clearly hinted by the mention of intellect or thought, διάνοια, as the sole available means of approaching them in contradistinction to the senses. This noun is cognate to the verb διανοεῖσθαι used by Plato to remark that mathematicians are not actually interested in the various visible shapes about which they construct their proofs. This is said in the description of the first section of the “intelligible” part of the divided line, where contains forms as studied in mathematics. After Socrates finishes the description of the second section, which contains forms as studied by dialectic, in themselves without sensibles and hypotheses, Glaucon summarizes the crucial difference Socrates has explained between the ways in which dialectic and mathematics approach forms. Here διάνοια appears for the first time as terminus technicus for the approach to intelligible objects peculiar to mathematics and the resulting cognitive condition. For the dialectical approach, which relies neither on sensibles nor on hypotheses but only on the forms themselves, and the resulting

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cognitive condition Glaucon uses νοῦς, which Socrates replaces with the cognate νόησις when immediately next he agrees with Glaucon and also associates similar terms with the two sections of the line’s “sensible” part (511c3–e5). For one’s being in the cognitive conditions called διάνοια and νοῦς or νόησις, towards which lead the hypothetical and unhypothetical approaches respectively these two nouns also denote, Plato has Glaucon use periphrases containing the nouns at issue, διανοίᾳ θεᾶσθαι, which is an equivalent to διανοεῖσθαι, and νοῦν ἴσχειν. In R. 7, 533c8–534a8, Plato employs again the noun διάνοια as a terminus technicus for the approach to forms which is peculiar to mathematics and the cognitive condition it results in, but here he contrasts διάνοια not with νοῦς or νόησις but with ἐπιστήµη, using νόησις as a general term for both διάνοια and ἐπιστήµη. He has Socrates refer back to the introduction of διάνοια in R. 6 and signal the terminological shift with the comment that he and Glaucon will not quarrel about words when they have so important matters to discuss (533d6–9); not many lines after this comment the original term νοῦς is put in Socrates’ mouth for the cognitive state of a dialectician, one who can give an account to himself and to others of what each thing is (534b3–6). In the discussion of arithmetic in R. 7 it is clear that up to 524d5 the two terms διάνοια and νόησις are not used in the technical sense. When Socrates says that a finger is not presented with an opposite, he specifies that he talks about the souls of the majority (523d3–4: τῶν πολλῶν ἡ ψυχή). There is no signal that, as he passes on to the sensibles that are presented each with its opposite, he suddenly turns to the souls of mathematicians and philosophers in the cognitive states resulting from the mathematical and dialectical approach respectively to the intelligible objects studied in mathematics and dialectic. διάνοια can be certainly said to be summoned up by a mathematically trained soul when it is confronted with sensibles in order to approach intelligibles but νόησις certainly cannot, for it approaches intelligibles only in and through themselves, without having to rely on any sensibles. Thus in the discussion of arithmetic up to 524d5 διάνοια and νόησις are employed interchangeably for the pre-theoretic seeds of what they denote below in 533c8–534a8 and above in the divided-line simile at the end of the previous book: they mean “intellect”, the reasoning faculty that is innate to humans and called διάνοια or νόησις if studying intelligibles in the manner appropriate to mathematics or philosophy respectively. νόησις, however, is surely used by Socrates in the technical sense when Plato has him conclude in 525b9–c6 for the first time that arithmetic is a propedeutic to dialectic: Socrates says that the future philosopher-rulers must study arithmetic until νόησις itself allows them to view the nature of numbers. Since this comes after an argument that numbers are not sensible but intelligible, forms, to view the nature of numbers here means to grasp them as form-numbers without reliance on sensibles or hypotheses, so νόησις itself is the cognitive condition of someone who has managed to see form-numbers in themselves, and Socrates here talks about not the end of the mathematical studies of the future philosopher-rulers but the end of their dialectical studies. If so, what will have been achieved after such a long and arduous journey can hardly be assumed to have already been revealed by Socrates in the preceding brief argument that numbers are not sensible but forms: this assumption, however, cannot

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be avoided if this argument does commit Plato to the view that form-numbers are monadic. The description of the form of beauty in Smp. 210e2–211b5 leaves no doubt that a form is conceived by Plato as completely unlike anything sensible or thinkable associated with and named after it. However, assuming that the concept of monadic numbers makes sense, they can hardly be called completely unlike the groups of sensible things which participate in them and are described by them. Socrates is not yet done with arithmetic. He says to Glaucon that now he understands that arithmetic is subtle subject conducive in many ways towards the realization of their goal, provided that its study is pursued not for retailing but for obtaining knowledge. Here Socrates appears to be about to introduce one or more new points: this seems to be that arithmetic sharpens one’s mind even if one is not mathematically gifted, and since it is a most demanding subject, it is a propedeutic to dialectic and must be studied by the future philosopher-rulers (526b5–c6). But first he surprisingly restates their conclusion that arithmetic launches powerfully the soul into an upward journey, forcing it to discuss the numbers themselves and take exception to the view of a codiscussant that numbers have visible or tangible bodies. Why this is so has been shown in the preceding discussion, where a number is implicitly assumed to be a group of things each of which is one, whereas all collections of sensible things are made up of objects each of which are divided into many. In the preceding discussion this situation has been argued to prompt an inquiring soul to wonder what numbers really are and, as is easily inferable from the context though not explicitly stated, to conclude that, if numbers consist of things each of which is indivisible, unlike the members of collections of sensible things, they must be not sensible but intelligible. This conclusion is now explicitly stated together with the assumption about numbers from which it follows, and is presented as a conclusion arrived at quite easily not by an inquiring soul but by mathematicians (525c8–526a7): Καὶ µήν, ἦν δ’ ἐγώ, νῦν καὶ ἐννοῶ, ῥηθέντος τοῦ περὶ τοὺς λογισµοὺς µαθήµατος, ὡς κοµψόν ἐστι καὶ πολλαχῇ χρήσιµον ἡµῖν πρὸς ὃ βουλόµεθα, ἐὰν τοῦ γνωρίζειν ἕνεκά τις αὐτὸ ἐπιτηδεύῃ ἀλλὰ µὴ τοῦ καπηλεύειν. Πῇ δή; ἔφη. Τοῦτό γε, ὃ νυνδὴ ἐλέγοµεν, ὡς σφόδρα ἄνω ποι ἄγει τὴν ψυχὴν καὶ περὶ αὐτῶν τῶν ἀριθµῶν ἀναγκάζει διαλέγεσθαι, οὐδαµῇ ἀποδεχόµενον ἐάν τις αὐτῇ ὁρατὰ ἢ ἁπτὰ σώµατα ἔχοντας ἀριθµοὺς προτεινόµενος διαλέγηται. οἶσθα γάρ που τοὺς περὶ ταῦτα δεινοὺς αὖ ὡς, ἐάν τις αὐτὸ τὸ ἓν ἐπιχειρῇ τῷ λόγῳ τέµνειν, καταγελῶσί τε καὶ οὐκ ἀποδέχονται, ἀλλ’ ἐὰν σὺ κερµατίζῃς αὐτό, ἐκεῖνοι πολλαπλασιοῦσιν, εὐλαβούµενοι µή ποτε φανῇ τὸ ἓν µὴ ἓν ἀλλὰ πολλὰ µόρια. Ἀληθέστατα, ἔφη, λέγεις. Τί οὖν οἴει, ὦ Γλαύκων, εἴ τις ἔροιτο αὐτούς· “Ὦ θαυµάσιοι, περὶ ποίων ἀριθµῶν διαλέγεσθε, ἐν οἷς τὸ ἓν οἷον ὑµεῖς ἀξιοῦτέ ἐστιν, ἴσον τε ἕκαστον πᾶν παντὶ καὶ οὐδὲ σµικρὸν διαφέρον, µόριόν τε ἔχον ἐν ἑαυτῷ οὐδέν;” τί ἂν οἴει αὐτοὺς ἀποκρίνασθαι; Τοῦτο ἔγωγε, ὅτι περὶ τούτων λέγουσιν ὧν διανοηθῆναι µόνον ἐγχωρεῖ, ἄλλως δ’ οὐδαµῶς µεταχειρίζεσθαι δυνατόν. “Now that we have talked about the study of arithmetic,” I said, “I realize that it is a subject subtle and useful in many ways for fulfilling our purpose, provided that one studies it not for retailing but in order to obtain knowledge.”

42

1. Astronomy in the Republic “In what ways?” he asked. “I’m thinking of what we just said, that it leads the soul somewhere very high and forces someone to discuss the numbers themselves, refusing without the slightest doubt to agree with anyone if they mention in a discussion numbers with visible or tangible bodies. As you certainly know, if someone in a discussion attempts to divide what is one, those that are experts in arithmetic laugh at him and refuse to agree multiplying it if you break it out of fear that what is one might turn out not to be one but to have many parts.” “This is absolutely true,” he agreed. “Then what do you think would happen, Glaucon, if someone asked them: ‘what are those numbers you’re talking about in which what is one is equal to any other and without even the slightest difference and has no parts?’. What do you think they would answer?” “They would say, I believe, that they’re talking about those numbers that only thought can approach and cannot be grasped in any other way.”

In the immediately following discussion of geometry, Plato will present Socrates introducing a view of what geometry does which even those who know only basic geometry, hence geometers, too, would easily adopt (527a1–b11). Similarly, here he has Socrates point out that mathematicians would not find it especially hard to concede the intelligibility of numbers. At the same time, Plato makes clear that this is not a view on the nature of numbers mathematicians would adopt by default but one they would easily accept if someone made them see in a discussion that the numbers they consider to be collections of indivisible units, indistinguishable from one another, cannot be sets of sensible things, which are divisible, unless each of them is ridiculously assumed to be an indivisible unit and its parts to be other indivisible units. The imaginary debate between mathematicians and a dialectical opponent, whose crucial question is tellingly put in Socrates’ mouth, would offer to mathematicians a chance to engage in dialectical reasoning. At the end of their discussion of all branches of mathematics Socrates says, before he passes on to dialectic, that mathematicians are not dialecticians, capable of giving and receiving an account, and Glaucon comments that he has met very few dialectically capable mathematicians (531d8–e5). As Socrates says in the simile of the divided line, “the students of geometry, arithmetic and the like lay down odd and even, figures, three kinds of angle and other things akin to these in each field, and as if they knew these things, turning them into hypotheses, they do not deign to give either to themselves or to others an account of what is hypothesized, assuming that it is clear to everybody, but start from their hypotheses and go through the subsequent stages to arrive consistently at what they set out to investigate”. Mathematics is forced to rely on hypotheses and, unlike dialectic, cannot transcend them to reach the forms, whose shadows and reflections are the objects defined in the hypotheses of mathematics according to the similes of the cave and the divided line: dialectic is responsible for giving accounts of these hypotheses, for “destroying” them and approaching the forms that are studied by mathematics in a dreamlike state as they are in themselves, i.e. as forms (R. 7, 533a10–534a8). In the imaginary dialogue of mathematicians and a dialectical opponent, the mathematicians are unable to give an account of what the numbers studied in arithmetic really are, and their dialectical opponent gently nudges them towards the realization that these numbers cannot be collections of sensible objects: if the

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numbers studied in arithmetic are made up of indivisible and undifferentiated units, they cannot be collections of sensible things, so these numbers must be intelligible. In view, however, of what is said in R. 6 and 7 about mathematics and dialectic, it is unlikely that these monadic numbers are form-numbers. Would Plato present even those very few mathematicians known to Glaucon to be dialectically minded as capable of seeing form-numbers in themselves by responding to a quite simple question by a dialectician? Form-numbers must be monadic insofar as they are still approached by mathematics, hypothetically, when its relevant hypotheses have been subjected to a slight dialectical correction. “I think they would say that they’re talking about those numbers that only thought can approach and cannot be grasped in any other way” is Glaucon’s report of the mathematicians’ answer to the dialectician’s crucial question. Glaucon uses the aorist infinitive of διανοεῖσθαι for being approached by thought. In Glaucon’s report of a reply offered by mathematicians to a dialectician, Plato’s choice of the verb is significant, given the usage of διάνοια in the passages from R. 6 and 7 discussed above: it hints that form-numbers are monadic insofar as they are viewed not in themselves, by dialectic, but hypothetically, as by arithmetic, via definitions dimly capturing them. Presupposed here by the mathematicians must be a definition of number similar to that in Euclid’s Elements: “a number is a multitude composed of units” (El. 7, Def. 2: ἀριθµὸς δὲ τὸ ἐκ µονάδων συγκείµενον πλῆθος), where the unit is defined as “that with respect to which each thing is called one” (El. 7, Def. 1: µονάς ἐστιν, καθ᾽ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται). “Each thing” cannot be anything other than each thing around us, a sensible. In his imaginary conversation with some mathematicians, the dialectician forces the mathematicians to face the fact that, if numbers are defined, as in arithmetic, to be collections of things each of which is one, numbers cannot be collections of sensible things, for no sensible thing is really a unit but many multitudes, unless whatever a given sensible thing set out as unit might be divided into is not absurdly a part of it but itself another unit. The conclusion that arithmetic studies not sensible but intelligible and monadic numbers to which the mathematicians jump results from a dialectical assault on arithmetical hypotheses, but not from a full-blown one that could “destroy” these hypotheses to reach the intelligible objects at issue in themselves and see them clearly as forms, for it does not transcend the hypotheses of arithmetic but only clarifies them. Its objective is restricted, to demonstrate that the numbers studied in arithmetic are intelligible. It removes the explicit or implicit assumption that the numbers at issue are sensible but leaves the rest intact, whence follows that they are monadic. Transcending the hypotheses of arithmetic to provide a dialectical account of numbers as forms does not mean to merely understand that arithmetic studies intelligible and not sensible numbers: in the case of the Good, grasping its intelligibility is not an account of it, as is evident from R. 6, 504e3–509d5. This passage shows that the Republic showcases not an advanced dialectic approaching forms in themselves and through themselves but rather one using hypotheses about its objects and images, as mathematics does.42 The imaginary conversation between 42

See Fine (1999) 236 with further references in n. 37.

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mathematicians and a dialectical opponent similarly exemplifies a dialectic unable to reveal form-numbers more satisfactorily than the Good, independently from the hypotheses of arithmetic which formalize the common-sense conception of number presupposed in the first part of the discussion of arithmetic. Plato could very easily have presented Glaucon as using νοῦν ἴσχειν instead of διανοεῖσθαι in the report of the mathematicians’ reply to the dialectician’s crucial question if he had wanted to signal to his alert audience that form-numbers are the intelligible and monadic numbers which the mathematicians would easily consider to be the objects really studied in arithmetic. After Glaucon’s report Socrates concludes once again that arithmetic must be included in the propedeutic studies of the future philosopher-rulers because arithmetic forces the soul to rely on νόησις itself and ascend towards the truth itself (526b1–3: προσαναγκάζον αὐτῇ τῇ νοήσει χρῆσθαι τὴν ψυχὴν ἐπ’ αὐτὴν τὴν ἀλήθειαν). In the light of 533c8–534a8, one of the passages discussed above, the pronoun qualifying νόησις and truth hints that the monadic numbers, whose intelligibility would be admitted without much fuss by mathematicians, are not forms but their shadows, forms approached using, or in the epistemic state of, διάνοια, from hypotheses: this is a form of νόησις inferior, in part on account of its hypothetical character, to the non-hypothetical approach to forms by dialectic, the superior form of νόησις, νόησις itself, which is free from hypotheses, as is the truth about intelligibles approached in this way. The inferior form of νόησις cannot fail to guide the soul towards νόησις itself, hence arithmetic is a propedeutic to dialectic. As Socrates says at 525b9–c6, when Plato has him conclude for the first time that arithmetic is a propedeutic to dialectic, the future philosopher-rulers must study arithmetic until νόησις itself allows them to view the nature of numbers. διάνοια, the lower-grade νόησις, cannot grasp it: only νόησις itself will do so when it will have finally succeeded in “destroying” the hypotheses of arithmetic, apparently starting from them, from within arithmetic. The arithmetic whose numbers are collections of units entirely indistinguishable from one another is called in Phlb. 56d4–e6 “arithmetic of the philosophers” not because these numbers are forms but faute de mieux, “something adequate”, as Plato describes it in hazy terms at Phd. 101c9–e3, which we will reach in our search for forms from hypotheses by trying to give full accounts of these hypotheses through concatenations of progressively higher, i.e. explanatorily better, hypotheses in each case. In his works Plato does not talk about form-numbers in definitive terms any more than he does about the form of beauty and that of the Good. 43 1.5.3. Archytas of Tarentum Back to R. 7, 529a3–c2 (see 1.5.1), what Socrates actually says in this passage is that astronomy, as is conceived of at his and Glaucon’s time by those who connect it with philosophy, makes the mind look very steeply down. The anonymous target of Plato’s ironic criticism here can be quite plausibly identified with the Tarentine 43

Plato is interested only in showing that numbers are not sensibles but forms. But the passages discussed here could have suggested to Aristotle that form-numbers are monadic; see 1.1.

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mathematician and Pythagorean Archytas. If so, Plato might have allowed himself to commit a slight anachronism here, for the best estimate of the date of Archytas’ birth is between 435 and 410 BC. In R. 7, 530d6–10, Plato has Socrates say that astronomy and harmonics form a pair of “kindred sciences” (ἀδελφαί τινες αἱ ἐπιστῆµαι), an echo of the description of arithmetic, geometry, astronomy and harmonics by Archytas at the beginning of his Harmonics (Ἁρµονικός) or On Mathematics (Περὶ µαθηµατικῆς; DK 47 B 1).44 In view of the emphasis Archytas placed in his Discourses (Διατριβαί; DK 47 B 4) on the superiority of arithmetic to all other arts “in regard to wisdom” (σοφία), to which astronomy also contributes given its kinship with arithmetic (see ch. 2.2.1), Plato seems to have had the Tarentine in mind when in 529a3–c2 he chose to portray Socrates as criticizing those who connect contemporary astronomy with philosophy, despite the fact that this astronomy makes the mind look in a steeply downward direction.45 If the only thing we knew about the beginning of Archytas’ Harmonics or On Mathematics was the description of arithmetic etc. as kindred sciences, Plato’s criticism would lead us to surmise that, for the Tarentine, astronomy dealt (a) with risings and settings, either diurnal, theorems about which belong to the elementary theory of the celestial sphere set out in On the Moving Sphere by Autolycus, or those called fixed-star phases, whose theory is presented by Autolycus in his other surviving work, On Risings and Settings, with the help of the rotating celestial sphere; (b) with the time it takes the Sun to circuit the zodiac, and the Moon to do the same and catch up with the Sun (cf. below n. 90). This is indeed how Archytas conceives of the science of astronomy in the introduction to his Harmonics or On Mathematics: καλῶς µοι δοκοῦντι τοὶ περὶ τὰ µαθήµατα διαγνώµεν καὶ οὐθὲν ἄτοπον ὀρθῶς αὐτούς, οἷά ἐντι, περὶ ἑκάστων φρονέν· περὶ γὰρ τᾶς τῶν ὅλων φύσιος καλῶς διαγνόντες ἔµελλον καὶ περὶ τῶν κατὰ µέρος, οἷά ἐντι, καλῶς ὀψεῖσθαι. περί τε δὴ τᾶς τῶν ἄστρων ταχυτᾶτος καὶ ἐπιτολᾶν καὶ δυσίων παρέδωκαν ἁµῖν σαφῆ διάγνωσιν καὶ περὶ γαµετρίας καὶ ἀριθµῶν καὶ οὐχ ἥκιστα περὶ µωσικᾶς. ταῦτα γὰρ τὰ µαθήµατα δοκοῦντι εἶµεν ἀδελφεά. I think that the students of mathematics make good distinctions, and it is not at all surprising that they understand correctly how individual things are. For, having made good distinctions about the nature of wholes, they were bound to also see well how individual things are. Thus they handed down to us clear distinctions concerning the speed of celestial bodies and risings and settings, concerning geometry and numbers and, not least, concerning music. For these sciences seem to be akin.46

ἁ τῶν ἄστρων ταχυτάς refers most probably not only to the time it takes the fixed stars to describe their parallel circles, to the tropical year and the synodic month but also to the tropical period of each planet known in antiquity, Mercury, Venus, Mars, Jupiter and Saturn, i.e. to the time it takes a planet to complete a circuit of the zodiac (cf. below n. 90). ἐπιτολαὶ καὶ δύσιες, moreover, must refer not only to the daily rising and setting of the fixed stars but also to their phases, phenomena of great 44 45 46

For the title see Huffman (2005) 126 and cf. 187–188. Cf. Huffman (2005) 234–235 and 244–245 on what Archytas calls “wisdom”. The fragment is considerably longer; for full discussion see Huffman (2005) 103–161.

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importance to early Greek astronomy. It is conceivable that the terms also pick out the phenomena of planetary phases observed in the east or west in the morning before sunrise or in the evening after sunset.47 In the case of the Sun and the Moon, διάγνωσις περὶ ἐπιτολᾶν καὶ δυσίων might refer to the palindromic displacement north and south of the rising and setting positions of the Sun and the Moon along the horizon in a year and a month respectively.48 The reference in R. 7, 530d6–10, to astronomy and harmonics as kindred sciences occurs in a criticism of Pythagorean harmonics on the ground that it, too, fails to free the mind from the sensible world of becoming (530e1–531c8). This is a second instance of Platonic critique of Archytas in R. 7.49 R. 7, however, also contains an acknowledgement of a significant achievement of the Tarentine in the field of solid geometry, as will be seen below in 1.7. We will now pass on to the hypotheses of the future astronomy that Plato has Socrates dream of in R. 7 as suitable propedeutic for the study of philosophy and Glaucon contrast in amazement with the contemporary astronomy familiar to him. 1.6. THE FUTURE ASTRONOMY ENVISIONED BY SOCRATES 1.6.1. The hypotheses of the future astronomy In R. 7, 527a1–b11, Glaucon agrees readily with Socrates that geometry draws the mind to being, as everybody who knows a smattering of it would admit, though geometers do talk about changes which the objects of their study undergo. Similarly, in 529c3–5 he does not hesitate to agree with Socrates that astronomy, as viewed currently by those who link it with philosophy, makes the mind look very steeply down. He now sees that Socrates thinks of an astronomy that does not yet exist as another propedeutic for the future philosopher-rulers: Δίκην, ἔφη, ἔχω· ὀρθῶς γάρ µοι ἐπέπληξας. ἀλλὰ πῶς δὴ ἔλεγες δεῖν ἀστρονοµίαν µανθάνειν παρὰ ἃ νῦν µανθάνουσιν, εἰ µέλλοιεν ὠφελίµως πρὸς ἃ λέγοµεν µαθήσεσθαι; “I got what I deserved,” he said. “Your rebuke is correct. But in what manner different from ours you were saying that astronomy ought to be pursued if its study is going to be conducive to our purpose?”

He ought to have realized that Socrates considers contemporary astronomy unsuitable as a propedeutic for the future philosopher-rulers when Socrates asked him in 528a6–10 to leave astronomy aside for the moment, for not it but solid geometry must be considered the third propedeutic of philosophy after arithmetic 47 48

49

These phenomena were very important to Babylonian astronomy, but it seems that early Greek astronomers paid no attention to them. They have no connection with seasonal changes. Huffman (2005) 53 suggests that ἁ τῶν ἄστρων ταχυτάς refers to the periods of the five planets, the Sun and Moon, ἐπιτολαὶ καὶ δύσιες to the phenomena of fixed-star phases. But this is unnecessarily restrictive. Cf. Huffman (2005) 64.

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and, as it now emerges, plane geometry, on account of the logical priority of solid geometry, the study of solids themselves, to contemporary astronomy, the study of a rotating solid. After Socrates gives a brief description of solid geometry (528b1– 2), to which we will turn below, Glaucon immediately comments with amazement that solid geometry has yet to develop (528b3–4). By reminding Glaucon of the logical priority of solid geometry over astronomy and then having him recall the undeveloped state of solid geometry in their day, Socrates tries to suggest to his codiscussant that he has misidentified a subject in the propedeutic curriculum of the future philosopher-rulers as contemporary astronomy. Astronomy will be on this curriculum, but Socrates is having in mind an astronomy that does not yet exist, not the astronomy of their time. Compared to it, contemporary astronomy is primitive, just as contemporary solid geometry has yet to make progress. This subtle point is lost on Glaucon, who forces Socrates to make it explicit that the astronomy which will be a propedeutic to philosophy has yet to appear; after Socrates has tried to clarify his vision, Glaucon exclaims that this astronomy will far exceed contemporary astronomy in difficulty (530c3–4). To Glaucon’s question in the passage last quoted Socrates gives the following answer (529c6–d6): Ὧδε, ἦν δ’ ἐγώ. ταῦτα µὲν τὰ ἐν τῷ οὐρανῷ ποικίλµατα, ἐπείπερ ἐν ὁρατῷ πεποίκιλται, κάλλιστα µὲν ἡγεῖσθαι καὶ ἀκριβέστατα τῶν τοιούτων ἔχειν, τῶν δὲ ἀληθινῶν πολὺ ἐνδεῖν, ἃς τὸ ὂν τάχος καὶ ἡ οὖσα βραδυτὴς ἐν τῷ ἀληθινῷ ἀριθµῷ καὶ πᾶσι τοῖς ἀληθέσι σχήµασι φοράς τε πρὸς ἄλληλα φέρεται καὶ τὰ ἐνόντα φέρει, ἃ δὴ λόγῳ µὲν καὶ διανοίᾳ ληπτά, ὄψει δ’ οὔ· ἢ σὺ οἴει; Οὐδαµῶς γε, ἔφη. “In this manner,” I said. “Since these decorative patterns in the sky have been made in what is visible, they will be considered the most beautiful and exact of their kind but much inferior to true motions performed relative to one another and carrying along what is inside them with invariant fastness and invariant slowness,50 in true number and all true figures, all of which can be grasped by thought, not sight.51 Or you think otherwise?” “By no means,” he said.

The first relative clause explains τῶν ἀληθινῶν, sc. ποικιλµάτων, with which τὰ ἐν τῷ οὐρανῷ ποικίλµατα are unfavorably compared. It makes clear that τὰ ἐν τῷ οὐρανῷ ποικίλµατα, which are visible, are some celestial motions:52 for they are compared with the motions, φοράς, mentioned in the first relative clause, τὰ ἀληθινὰ ποικίλµατα. These motions, their fastness and slowness and the numbers associated with them, i.e. the numerical measures of their fastness and slowness, and their figures, which can only be the shapes of the paths of the motions at issue, will 50 51

52

On the meaning of τὸ ὂν τάχος καὶ ἡ οὖσα βραδυτής cf. Burnyeat (2000) 59 n. 86. τὸ ὂν τάχος καὶ ἡ οὖσα βραδυτὴς φορὰς φέρεται καὶ τὰ ἐνόντα φέρει is strange, but cf. φέρεται γὰρ καὶ ἐν φορᾷ αὐτῶν ἡ κίνησις πέφυκεν in Tht. 156d2–3, from which it is a small step to τὸ ὂν τάχος καὶ ἡ οὖσα βραδυτὴς φορὰς φέρεται καὶ τὰ ἐνόντα φέρει in view of τάχος δὲ καὶ βραδυτὴς ἔνι τῇ κινήσει αὐτῶν in Tht. 156c8. Which ones they are will be seen below in this section. τὰ ἐν τῷ οὐρανῷ ποικίλµατα cannot be constellations, as proposed by Bulmer-Thomas (1984) 108–109; cf. Burnyeat (2000) 60 n. 88.

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be διανοίᾳ ληπτά, ὄψει δ’ οὔ: they will not be visible but graspable by our mind in the hypotheses-dependent epistemic state called διάνοια in the context of the simile of the divided line at the end of R. 6 (see above section 1.5.2).53 The astronomy of the future envisaged by Socrates as propedeutic to philosophy will break free of the sensible world of becoming, in which contemporary astronomy is trapped, and will ascend to the intelligible world of being as this is approached by mathematics, thereby erasing the gap between mathematics and contemporary astronomy. Its assimilation to (plane) geometry is stated in R. 7, 530b6–c4.54 Its hypotheses will thus be about the abstract mathematical objects, τὰ ἀληθῆ σχήµατα, that are implicitly contrasted with the paths of certain celestial motions, their visible representations (cf. 529d7–530a3), just as the objects defined in the hypotheses of geometry are contrasted with their sensible representations.55 As suggested above in section 1.1, it is likely, in view of Euclid’s Elements, that in Plato’s time important geometrical objects would have been defined via their kinematic generation; we can thus plausibly think that in the astronomy of the future, too, the geometrical objects in its hypotheses will be defined via their generation by the motions whose paths they are implied to be. If so, other geometrical objects than those generated will undergo these motions.56 These motions cannot be assumed to have counterparts in 53

54 55

56

For τὰ ἀληθῆ σχήµατα as shapes of paths of motions cf. Burnyeat (2000) 59. He does not explain what traces them out, however, and his identification in n. 84 of the shapes in question with spheres of various diameters has no basis in the text. Cf. n. 56 below. It will be discussed below in section 1.7.5. See R. 6, 510c2–511a3, quoted in section 1.1. On ἀληθής, -ές see below in this section. ἐνδεῖν in the passage discussed here suggests that the subject, τὰ ἐν τῷ οὐρανῷ ποικίλµατα, is sensible, whereas the object, τῶν ἀληθινῶν, sc. ποικιλµάτων, is abstract and represented visibly by the subject; cf. Phd. 74d4–75a4. In view of the parallelism between the future astronomy and geometry, Plato does not hint that abstract motions are made visible as celestial motions. His point is rather that the shapes of the paths of celestial motions visibly represent abstract geometrical objects; see the following discussion. Cf. Post (1944) 299. τὰ ἐνόντα, sc. φοραῖς, in our passage can be identified as the generating objects; cf. ἡ γραµµὴ ἡ τοῦ κύκλου ἐν φορᾷ ἐστίν in [Arist.] Mech. 851b34–35. They are not the heavenly bodies visible in the sky, as Burnyeat (2000) 59 has it: τὰ ἐνόντα must be invisible, like the motions that carry them (we do not have to saddle Plato with intelligible copies of the visible heavenly bodies, as Aristotle does in Metaph. B 2, 997b12–998a6, quoted in section 1.1, and identify them with τὰ ἐνόντα). Mourelatos (1981) 2–5 and 27–30 seems to think that τὰ ἐνόντα φοραῖς are other φοραί, all of these motions being the revolutions of concentric circles and rotations of concentric spheres. There is no reason to see circles and spheres in R. 7, 529c6–d6, however; nothing in this passage or its context hints at these figures (on motion see next n.). If τὰ ἐνόντα are geometrical figures that move so as to generate other such objects, which objects could be generators and which generated? In R. 7, 529c6–d6, or in its context there is no hint that Plato wanted to answer this question. Post (1944) 299 mentions lines produced by the movement of points, lines and planes (he also suggests that Plato’s future astronomy will include mechanics, which can be safely ruled out as completely unlikely, however). τὰ ἀληθῆ σχήµατα are identified by Burnyeat with spherical shapes of paths of motion (see above n. 53) but he does not explain what it is that moves. Robins (1995) 373–378 thinks that τὰ ἀληθῆ σχήµατα are circular paths traced out by the rotating regular, or Platonic, polyhedra inscribed in spheres. But circles or semicircles, spheres or other solids are not hinted at in R. 7, 529c6–d6, or in its context. As for the regular polyhedra, in his Timaeus Plato analyzes the four elements into the true elements, two types of right triangles that are planar

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the intelligible mathematical universe, just as neither coming into being nor decay nor any other change of geometrical objects might occur in it.57 It is only the mentally visualized representations of geometrical objects, not the latter themselves, that can come to be, suffer changes and undergo motion in the formal context of definitions or proofs; similarly, the geometrical objects that undergo motion so as to produce other such objects when the latter are defined kinematically must be mere mental visualizations of the respective geometrical objects themselves, which again are not subject to motion, generation, decay or change.58 In R. 7, 527b2–11, a few Stephanus pages before the passage just quoted, τὸ ἀεὶ ὄν is used as a collective term referring to the intelligible geometrical objects themselves as eternal entities, ungenerated and indestructible, which must also be completely immobile, as can be safely inferred from Aristotle’s Metaph. B 2, 997b12–998a6, quoted in section 1.1; ἀλήθεια is what geometry draws the mind to, insofar as the objects of its study are ungenerated and indestructible. One could thus hardly expect not much below the use of the adjective ἀληθινός, -ή,!-όν as a qualification of motions and the numerical measures of their swiftness and slowness, or the use of the participle ὤν, οὖσα, ὄν as modifier of this swiftness and slowness independently of numerical measures. For, in view of R. 7, 527b2– 11, this adjective and the participle seem to leave no doubt that eternal motion exists in the intelligible mathematical universe: in it, however, no motion can possibly exist since the mathematical objects themselves, the only things existing therein, are immobile, and motion cannot but be motion of something. As it is, we can only assume that in R. 7, 529c6–d6, the adjective ἀληθής, -ές does denote the ungeneratedness and indestructibility of the mathematical objects themselves that will be defined in the hypotheses of the future astronomy kinematically, i.e. through the envisaged generation of their mental representations when mental representations of other geometrical objects undergo visualized motions. ἀληθινός,

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elements of space: one makes up the faces of regular tetrahedra, octahedra and icosahedra, the particles of fire, air and water, the other makes up the faces of cubes that are the corpuscles of earth (Ti. 53c4–56c7). As for the dodecahedron, the fifth regular solid in three-dimensional Euclidean space, it is associated by Plato with the cosmos (Ti. 55c4–6). Whatever this might mean, there is no reason to assume that he accorded astronomical significance to the other four regular solids, too, presaging Kepler’s Mysterium Cosmographicum. To flesh out the relation between future astronomy and stereometry we do not have to bring in the Platonic solids; see section 1.7. Given (a) the assimilation of the future astronomy to (plane) geometry in R. 7, 530b6–c2, (b) the emphasis Plato lays in 527a1–b11 on the basic fact that the objects studied in geometry are not subject to change as well as generation and decay, and (c) Aristotle’s description in Metaph. B 2, 997b12–998a6, of Platonic astronomy as studying what is immobile, it is unlikely that the hypotheses of the future astronomy will define uniform, circular motions (Mueller [1992] 192– 194); or that the future astronomy is to be identified with the elementary spherical astronomy presented e.g. in Autolycus’ On the Moving Sphere (“spherics”; Mueller [1981] 103–111); or with a science of pure kinematics that will focus on the revolution of concentric circles and spheres (Mourelatos [1981] 2–5 and 27–30); or, in general, that the geometrical objects it is concerned with, be they the five Platonic polyhedra or not (cf. previous n.), are actually rotating. It seems that for Plato the “geometrical” objects themselves not only do not move, change, come to be and perish, as their visualized representations necessarily do, but also lack extension and shape; see above section 1.1.

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-ή,!-όν, however, does not apply to these motions in the same sense. Nor does the epithet apply in the sense at issue to the numerical measures of their fastness and slowness, or the participle ὤν, οὖσα, ὄν to this fastness and slowness themselves. Otherwise, i.e. if the second adjective and the participle were interpreted in the light of R. 7, 527b2–11, the Platonic mathematical universe would contain not only intelligible objects but also motions reifying the visualizable motions undergone by mental representations of geometrical objects. When in R. 7, 529c6–d6, Plato employs the adjective ἀληθινός, -ή,!-όν in describing some motions as well as the numbers which express or measure the fastness and slowness of these motions and the participle ὤν, οὖσα, ὄν in describing this fastness and slowness per se, no ontological import should be read into these qualifications. They can only signal the uniformity of the visualizable motions in question, the invariability of their fastness and slowness relative to one another and of the numbers that would, in theory, measure their relative fastness and slowness. They do not hint at the mindindependently objective existence of eternal motions in the abstract mathematical universe that are as close to eternal rest as is possible, being periodic over the same path, and, having constant fastness or slowness relative to one another, recall the Heraclitean rest by motion (DK 22 B 84a: µεταβάλλον ἀναπαύεται).59 As seen above in section 1.1, the sphere is defined in Euclid’s Elements as the solid figure produced by a semicircle as it rotates about its fixed diameter. This rotation does not occur anywhere in physical space but is mentally visualized. It is supposed to stop after a single turn, but can also be thought of as being periodic over the same path. Although the Euclidean definition contains no assumption as to whether the speed of this intuited motion is constant or not, the semicircle can be plausibly considered to rotate uniformly, for its rotational motion must be either uniform or non-uniform and there is no reason to assume that it speeds up or slows down since this would not affect the outcome of its rotation.60 Well after Plato, in the fifth century AD, the Neoplatonist Proclus regards as uniform the visualizable motions whose possibility is taken for granted in the first two Euclidean postulates, the drawing of a straight line from any point to any point and the production of a finite straight line continuously in a straight line. In these postulates Euclid asserts, according to Proclus, a consequence of the kinematic conception of a line as the flowing of a point, and of a straight line as the uniform and undeviating flowing of a point. In this context, the point flows undeviatingly in the sense that it flows in a straight line, and it flows uniformly in the sense that it moves at constant speed, unless uniform flowing means the same as undeviating flowing in a pointless repetition (in Euc. 185.8–19 Friedlein): τὸ µὲν γὰρ ἀπὸ παντὸς σηµείου ἐπὶ πᾶν σηµεῖον εὐθεῖαν γραµµὴν ἀγαγεῖν ἑπόµενόν ἐστι τῷ ῥύσιν εἶναι τοῦ σηµείου τὴν γραµµὴν καὶ τὴν εὐθεῖαν ὁµαλὴν καὶ ἀπαρέγκλιτον 59 60

On the participle ὤν, οὖσα, ὄν as indicating invariability cf. Burnyeat (2000) 59 n. 86. In the kinematic definition of the Archimedean spiral (Archim. Spir. Def. 1) the rotation of a line and the motion of a point along the line are both characterized as uniform; if the speed of the motion of the point along the line is not constant but grows exponentially, then the resulting spiral is not Archimedean but logarithmic. Whether motion is at constant speed or not makes no difference in the examples of kinematic generation of geometrical figures considered here.

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ῥύσιν. νοήσαντες οὖν τὸ σηµεῖον κινούµενον τὴν ὁµαλὴν καὶ ἐλαχίστην κίνησιν ἐπὶ θάτερον σηµεῖον καταντήσοµεν, καὶ τὸ πρῶτον αἴτηµα γέγονεν οὐδὲν ποικίλον ἡµῶν ἐπινενοηκότων. εἰ δὲ δὴ τῆς εὐθείας σηµείῳ περατουµένης ὡσαύτως νοήσαιµεν τὸ πέρας αὐτῆς κινούµενον τὴν ἐλαχίστην καὶ ὁµαλὴν κίνησιν, ἔσται τὸ δεύτερον αἴτηµα πορισθὲν ἀπὸ εὐµηχάνου καὶ ἁπλῆς ἐπιβολῆς. The drawing of a straight line from any point to any point follows from the conception of the line as the flowing of a point and of the straight line as its uniform and undeviating flowing. Thus, if we think of a point as moving uniformly over the shortest path, we will arrive at another point and we have obtained the first postulate without any complicated thought. And since a straight line ends at a point, if we think of its end-point as moving uniformly over the shortest path, then the second postulate will have been established by an easy and simple thought.

Proclus does not describe as uniform the mentally visualizable motion whose possibility is taken for granted in the third Euclidean postulate (in Euc. 185.19–25 Friedlein), the description of a circle with any point as center and any distance as radius. Its uniformity, however, is certainly implicit. Nothing, of course, compels us to assume that, well before Proclus, already Plato had regarded such mentally visualizable motions as uniform. This could back up the view that the geometrical figures implied in R. 7, 529c6–d6, to be shapes of paths of uniform motions are produced, and thereby defined, by the uniform motions in question, hence they are mental representations of mathematical objects and the motions generating them are mere visualizations. But this view comes easily to mind in the light of the nature of Greek geometry to which the difficult astronomy of the future sketched out by Socrates in R. 7, 529c6–d6, is reduced. And, as Proclus makes clear, the assumption this reading entails, that the intuited motion in the definition of geometrical objects via their kinematic generation is uniform, can also be held easily. Euclid does not hint at the uniformity of motion, which is mathematically useless. Thus one naturally wonders why Plato has Socrates in R. 7, 529c6–d6, refer to the uniformity of motions in a potentially confusing way if the motions at issue are assumed to be visualized motions of some geometrical figures resulting in the production of other figures that will be kinematically defined in the hypotheses of the future astronomy. We can glean a possible answer from the unfavorable contrast of visible celestial motions with these visualized motions. Though mathematically useless, the uniformity of the latter might be invoked as a hint to the identity of the celestial motions of interest here: compared with the motions that will appear in the kinematic definitions of the future astronomy, these celestial motions are far inferior because they are not uniform but slow down and speed up. 1.6.2. Celestial motions in the hypotheses of the future astronomy The only available candidates are the zodiacal motions of the Moon, the Sun and the five planets known in antiquity: Mercury, Venus, Mars, Jupiter, Saturn.61 The hypotheses of the future astronomy will define mathematical objects whose visible 61

Aristotle states their non-uniformity in Cael. B 6, 288a13–17; see Kouremenos (2010) 86–93.

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representations are thus paths of zodiacal motions decorating the sky. Plato seems to hint at the complex zodiacal motions of the planets as the particularly striking ones.62 In R. 7, 529c6–d6, the phrase τὰ ἐν τῷ οὐρανῷ ποικίλµατα describes the celestial motions that will be far inferior to the motions through which geometrical objects will be kinematically defined in the hypotheses of the advanced future astronomy; their inferiority is explained by the causal clause ἐπείπερ ἐν ὁρατῷ πεποίκιλται. The perfect participle of the same verb is used in the Timaeus as a qualification of the wanderings of the five planets. These wanderings are their zodiacal motions, whose periods are contrasted with the aforementioned zodiacal period of the Sun and the synodic period of the Moon, the tropical year and the synodic month, and with the period of the diurnal rotation, the day (39c5–d2): τῶν δ’ ἄλλων τὰς περιόδους οὐκ ἐννενοηκότες ἄνθρωποι, πλὴν ὀλίγοι τῶν πολλῶν, οὔτε ὀνοµάζουσιν οὔτε πρὸς ἄλληλα συµµετροῦνται σκοποῦντες ἀριθµοῖς, ὥστε ὡς ἔπος εἰπεῖν οὐκ ἴσασιν χρόνον ὄντα τὰς τούτων πλάνας, πλήθει µὲν ἀµηχάνῳ χρωµένας, πεποικιλµένας δὲ θαυµαστῶς. Men, with very few exceptions, have not understood the periods of the other celestial objects, have no names for them, do not compare their numerical measures and thus in effect do not know that their wanderings, bewildering many and marvelously variegated, constitute time.63

If we take our cue from the use of the verb ποικίλλεσθαι in the Timaeus, then the celestial motions that will turn out to be far inferior to the motions that will appear in the kinematic definitions of the hypotheses of the future astronomy seem to be the zodiacal motions of the five planets: in view of the above, therefore, it is only the paths of these zodiacal motions that seem to be considered visible images of the geometrical objects that will be defined in the hypotheses of the future astronomy, not the paths of the zodiacal motions of the Sun and the Moon and the diurnal paths of the fixed stars as well. The Moon orbits the Earth in an eastward direction, as the Sun appears to do. A further similarity with the Sun is that it appears to move in a circle on a plane slightly inclined to the ecliptic, thus against the background of the zodiacal constellations, though it circuits the belt of the zodiac in about a month, whereas the Sun needs a year. The other planets of our solar system also orbit the Sun in an eastward direction on planes inclined a few degrees to the ecliptic and passing through the zodiacal belt, within whose boundaries the planets are always to be found. To the Earth-based observer the Moon and the five planets that are visible to the naked eye and were known in antiquity appear thus to move against the backdrop of the distant stars from west to east, traveling near the ecliptic. Each of these celestial objects needs a characteristically different time to circuit the zodiac, with the exception of Venus and Mercury, which, like the Sun, they need a year. 62 63

Cf. the discussion of R. 7, 529c6–d6, in Burnyeat (2000) 56–63. Plato has mentioned the tropical periods of Mercury and Venus in 38d2–4, but this does not mean that they are excluded here. The synodic periods of the planets might also be referred to in this context (on the distinction between the tropical and the synodic period of a planet see the discussion in Evans [1998] 289–295).

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The planets are thus naturally grouped together with the Sun and Moon on account of their traveling eastwards within the zodiac. These seven celestial objects are distinguished from the stars in that they are seen to constantly shift positions, relative to both the stars and each other, because of their easterly motions, which combine with the simultaneous participation in the diurnal rotation producing spiral motions.64 This is why the Greeks used to call the celestial objects at issue wandering, in contrast to the fixed ones, the stars, or wanderers, πλανητὰ ἄστρα (Ti. 38c6), from whence is derived the modern term “planet”. The planets, moreover, appear from the Earth to wander also in a manner totally peculiar to them. At regular intervals, their eastward motion is seen to be interrupted. A planet appears to be stationary in the sky (first station), then begins to move again, though in the opposite direction, but after a while stops for a second time (second station), and when its motion resumes, the direction is once again to the east. As a result of this reversal in the direction of zodiacal motion (retrogradation), the planet is seen to trace out a looping or zigzag path; the shape of a retrograde path is not the same from one retrogradation to the next, a striking deviation from circularity in the path of the planet’s mainly eastward motion against the stellar backdrop. Retrogradations are as spectacular as they are puzzling within a geocentric worldview. The phenomenon is explained easily from a heliocentric point of view, however, given the differently sized orbits of the planets around the Sun, and their unequal orbital speeds. A planet’s threading its way along the background of the zodiac, with variably shaped retrograde bends and loops distributed at irregular intervals, can be statically described in metaphorical terms as a ποίκιλµα, i.e. as an embroidered design or a tracery on a vault, i.e. the celestial sphere, against the background of which the zodiacally moving planet is observed to trace out its variable path. Scholars have doubted that fourth-century-BC astronomers had knowledge of planetary retrogradations.65 In the Timaeus passage quoted above, however, τὰς τούτων πλάνας can only be the zodiacal motions of the five planets, for their periods are contrasted with the year and the month: why would Plato describe these motions as πεποικιλµένας θαυµαστῶς if he did not want to hint at the phenomenon of planetary retrogradations? His description suggests that the planets, unlike the Moon and the Sun, do not really move around the zodiac in concentric circles, as he assumes in Ti. 38b6–39b2.66 The helixes, moreover, Aristotle mentions at the end of the Metaph. B passage quoted in 1.1 are curves “drawn” by celestial motions, sensible lines such as those mentioned in his account of Protagoras’ objection to geometry. To show that astronomy can be thought to deal not with what we see in the sky but 64

65

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Cf. Ti. 39a5–b2, the first mention of the phenomenon and its correct explanation. We need not follow Dicks (1970) 129 and Vlastos (1975) 54–55 in attributing to Plato the discovery of the phenomenon and/or its explanation. See Goldstein (1997) 4 and cf. Bowen (2013) 230–248, who argues that passages which scholars have taken to indicate knowledge of the phenomenon prior to the second century BC cannot be read to entail such knowledge. Cf. Mendell (1998) 264–265 n. 5. On the circular paths of zodiacal motions in Ti. 38b6–39b2 see ch. 3.3.

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with non-sensibles such as Plato’s intermediates, Aristotle argues first that, since this science brings geometry to bear on the study of observed celestial motions, it does not actually deal with the curves traced out by celestial motions, for these curves are sensible, since they are generated by some sensible objects, but, as Protagoras had argued, the lines studied in geometry are not the same as sensible lines. Aristotle next points out that the curves we see celestial motions trace out are not similar to the curves studied in astronomy. This is clearly a second point, so the dissimilarity in question can only be the discrepancy between some geometrical models of celestial motions studied in astronomy and some features of the actual celestial motions, a further reason, besides the Protagorean argument, for the view that astronomy does not really deal with the sky we see. However, the celestial motions whose paths are described as helixes cannot be the spiral motions of the planets, the Sun and the Moon mentioned above, for their paths could be plausibly said to differ from their counterparts in astronomy only in the Protagorean sense. Aristotle calls the twists of the intestines “helixes” (PA Γ 14, 675b22–27), a term also used by him for the epididymis including the vas deferens (GA A 3, 717a23–29). In our context, this term is most probably used for the lines traced out by the planets in the episodes of retrogradation:67 as will be seen below in 1.7.4–5, in Aristotle’s time there probably existed geometrical models of these lines markedly clashing with observation. ποίκιλµα means “tracery” in R. 7, 529a3–c2, quoted in section 1.5.1. There Socrates objects to Glaucon that it is by no means clear to him that contemporary astronomy forces the mind to turn upwards to intelligible being from sensible becoming: contemporary astronomers are likened to people looking at traceries on a vault, ἐν ὀροφῇ ποικίλµατα, and learning whatever it is that they learn by the sense of sight, not by the mind. If, as argued above, Plato here portrays Socrates as reminding Glaucon that contemporary astronomers do not study the rotating celestial sphere as a visible representation of an intelligible object with which they are actually concerned, the geometrical designs on the vault of the sky at which they look are traced out by the stars as the celestial sphere rotates. They are the parallel diurnal circles of the stars, to which we can add the ecliptic, one of the few other circles apart from the diurnal circles of the stars that were of interest to contemporary astronomy. When ποικίλµατα recurs a few lines below, in 529c6– d6, it is at first construed in the same sense. As soon as it is made clear in this passage, however, that at issue now are geometrical designs traced out on the vault of the sky by non-uniform motions, no doubt is left that these motions are zodiacal motions, for the diurnal motion of the stars is unwaveringly uniform. And since the circular paths traced out by the zodiacal motions of the Sun and the Moon do not compare with the irregular paths described by the zodiacal motions of the planets in variability, a significant semantic component of the noun ποίκιλµα, this term seems to be used in 529c6–d6 only for the complex path of a planet’s zodiacal motion. Now, this variability helps explain why the future astronomy, unlike contemporary astronomy, will force the mind to turn upwards to intelligible being from sensible becoming. In view of R. 7, 522e5–525c7, where the same question is 67

Cf. Kouremenos (2010) 43–46 on Ph. Θ 10, 267a21–b6, and 48 n. 113. Aristotle knew the basic planetary phenomena; see fr. 211 Rose, needlessly dismissed by Bowen (2013) 167–168 n. 283.

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answered for arithmetic, sight grasps accurately the shape of the path traced out by a diurnally moving star and does not rouse the mind into activity (cf. section 1.5.2). But what is the shape of the path traced out by a zodiacally moving planet? To this question sight does not give us a satisfactory answer. For it reports that this shape is not uniform everywhere around the zodiac, and that the deviations from the overall shape, which do not themselves have constant shapes, do not always occur in the same places of the zodiac. Thus a shape that, after a period of observation, sight might identify as that of the path traced out by a zodiacally moving planet is as much as is not the sought after shape, hence it does not qualify as an answer to the question. The failure of sight to give an answer by singling out a sensible shape forces the mind to face the question, perhaps to also get beyond the evidence of the senses, though at first incontrovertible, that the zodiacal paths of the Sun and the Moon are circular. The mind will thus try to give an acceptable answer by finding an intelligible object, most probably at the end of a series of other such objects more and more akin to that sought after, to which the shape of the zodiacal path grasped by the senses in each case is linked as its sensible representation (cf. R. 7, 529d7–530a3). A hypothesis of the future astronomy, as argued above, will concern such an intelligible object, a form that exists mind-independently and will be defined in the future astronomy via the kinematic generation of its mental representations, just as e.g. the sphere and the cylinder are defined in the Euclidean Elements through their generation by a rotating semicircle and a rotating rectangle respectively. Subject to generation and motion will be the mental representations of forms, not these mindindependent objects themselves since neither generation nor motion occurs in the Platonic universe of forms. As seen above in section 1.1, in Metaph. B 2, 997b12– 30, Aristotle says that for Plato the true subject-matter of astronomy is an absurdly static celestial sphere (oὐρανός), which exists beyond its sensible diurnally rotating counterpart and is complete with a static Sun, Moon and all the other things in the sky (τἆλλα ὁµοίως τὰ κατὰ τὸν οὐρανόν). We also saw that there is reason to seriously doubt Plato’s belief in the existence of an abstract counterpart to the celestial sphere as such, and there is thus no reason to attribute to him a belief in the existence of abstract counterparts to the seven celestial objects within the celestial sphere, i.e. the planets, the Sun and the Moon. However, it seems that he did believe in the existence of intrinsically interesting abstract counterparts to some of the things that are within the celestial sphere, i.e. the visible shapes of the zodiacal paths of at least the five planets: thanks to these sensibles, the overarching abstract objects can be mathematically approached.68 The kinematic construction of mathematical objects as a feature of the future astronomy is suggested by the interlude where Socrates has Glaucon recall the undeveloped state of solid geometry in their day. As pointed out in 1.6.1, in so doing Socrates tries unsuccessfully to suggest that suitable for the propedeutic curriculum of the trainee philosopher-rulers can only be a future astronomy, by comparison to which the astronomy of their day, the study of a rotating solid, is as primitive as contemporary solid geometry. 68

A possible connection with the astronomy of his time is discussed in the next section. On the Platonic unity of distinct branches of mathematics such as astronomy and geometry see ch. 2.

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1.7. THE FUTURE ASTRONOMY AND SOLID GEOMETRY69 1.7.1. The problem of cube-duplication Socrates asks Glaucon to put astronomy aside for the moment, for not it but solid geometry must be considered the third propedeutic of philosophy after arithmetic and plane geometry in view of the logical priority of solid geometry to astronomy. Then he gives a description of solid geometry as the study of “the growth of cubes and what partakes of depth”, and Glaucon retorts that results concerning these things have yet to be obtained. Next, Socrates explains that the lack of progress in solid geometry is due not only to the difficulty of this field but also to the need for a superintendent of those who work in it and to the lack of support for it from all cities in Greece, let alone to its being lightly esteemed (R. 7, 528a6–c7): Ἄναγε τοίνυν, ἦν δ’ ἐγώ, εἰς τοὐπίσω· νῦν [δὴ] γὰρ οὐκ ὀρθῶς τὸ ἑξῆς ἐλάβοµεν τῇ γεωµετρίᾳ. Πῶς λαβόντες; ἔφη. Μετὰ ἐπίπεδον, ἦν δ’ ἐγώ, ἐν περιφορᾷ ὂν ἤδη στερεὸν λαβόντες, πρὶν αὐτὸ καθ’ αὑτὸ λαβεῖν· ὀρθῶς δὲ ἔχει ἑξῆς µετὰ δευτέραν αὔξην τρίτην λαµβάνειν. ἔστι δέ που τοῦτο περὶ τὴν τῶν κύβων αὔξην καὶ τὸ βάθους µετέχον. Ἔστι γάρ, ἔφη· ἀλλὰ ταῦτά γε, ὦ Σώκρατες, δοκεῖ οὔπω ηὑρῆσθαι. Διττὰ γάρ, ἦν δ’ ἐγώ, τὰ αἴτια· ὅτι τε οὐδεµία πόλις ἐντίµως αὐτὰ ἔχει, ἀσθενῶς ζητεῖται χαλεπὰ ὄντα, ἐπιστάτου τε δέονται οἱ ζητοῦντες, ἄνευ οὗ οὐκ ἂν εὕροιεν, ὃν πρῶτον µὲν γενέσθαι χαλεπόν, ἔπειτα καὶ γενοµένου, ὡς νῦν ἔχει, οὐκ ἂν πείθοιντο οἱ περὶ ταῦτα ζητητικοὶ µεγαλοφρονούµενοι. εἰ δὲ πόλις ὅλη συνεπιστατοῖ ἐντίµως ἄγουσα αὐτά, οὗτοί τε ἂν πείθοιντο καὶ συνεχῶς τε ἂν καὶ ἐντόνως ζητούµενα ἐκφανῆ γένοιτο ὅπῃ ἔχει· ἐπεὶ καὶ νῦν ὑπὸ τῶν πολλῶν ἀτιµαζόµενα καὶ κολουόµενα, ὑπὸ δὲ τῶν ζητούντων λόγον οὐκ ἐχόντων καθ’ ὅτι χρήσιµα, ὅµως πρὸς ἅπαντα ταῦτα βίᾳ ὑπὸ χάριτος αὐξάνεται, καὶ οὐδὲν θαυµαστὸν αὐτὰ φανῆναι. “Let us then backtrack,” I said, “for we chose to put after geometry the wrong subject.” “How,” he asked, “did we err?” “By considering a rotating solid,” I replied, “after the plane before the solids themselves. The right way is to consider the third dimension after the second. This is the subject dealing with the growth of cubes and with what partakes of depth.” “True,” he said. “But it is believed that results in this subject are yet to be obtained.”70 “There are two reasons for this,” I said. “First, since no city esteems it, due to its difficulty it is not studied strenuously and, second, those who work in it need a superintendent, without whom they are not going to get any results. But such a person is hard to emerge in the first place and, even if one did emerge, in the present situation those who seek these results would not listen, self-confident as they are. If the entire city esteems the field and acts as superintendent, however, they will listen and the constant and assiduous pursuit of results will bear fruit. Even now, although most people disrespect this subject and denigrate it while its students cannot explain its usefulness, it grows nonetheless thanks to its beauty, and it will be no surprise if results in it do turn up.”71 69 70 71

This section develops further Kouremenos (2004). On the unity of mathematics see ch. 2. Glaucon most probably reports a mathematician’s view, not his own. Socrates regards not only astronomy but also harmonics as undeveloped; see R. 7, 531b2–c5. It is not unlikely that early fourth-century-BC astronomy and harmonics, as well as arithmetic and

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It has been argued that in the phrase ἡ τῶν κύβων αὔξη the noun αὔξη means “addition of a dimension”, as in the preceding clause, hence ἡ τῶν κύβων αὔξη does not mean “the growth of cubes”, as translated here, but rather “the addition of the third dimension to squares making cubes”, the latter being an example of a kind of solids, τὸ βάθους µετέχον, in a very general description of what is studied in solid geometry.72 This is an unnatural reading of the phrase at issue, however. If we interpret αὔξη in this phrase, too, to mean the addition of an extra dimension to those already existing, we can only be led to the conclusion that the phrase must refer implausibly to four-dimensional cubes: for the genitive τῶν κύβων cannot but pick out the objects which grow by the addition of an extra dimension, not those which result from the addition of an extra dimension to objects of fewer dimensions, just as does the genitive depending on αὔξη in 521e1–2 (γυµναστικὴ µέν που περὶ γιγνόµενον καὶ ἀπολλύµενον τετεύτακεν· σώµατος γὰρ αὔξης καὶ φθίσεως ἐπιστατεῖ).73 It is preferable to assume, therefore, that ἡ τῶν κύβων αὔξη means “the increase of cubes” and take it to refer to a specific stereometric problem, that of increasing the volume of a given cube by any given ratio. The problem has come to be known as the duplication, or doubling, of the cube after the assumed double ratio of increase. Thus Glaucon’s statement that results concerning the increase of cubes and what partakes of depth have yet to be obtained points out the lack of solutions to a particular stereometric problem, that of doubling a cube, in order to illustrate concretely the undeveloped state of contemporary solid geometry, i.e. the dearth of results about three-dimensional objects. Plato’s description of solid geometry as undeveloped does not fit the state of this branch of mathematics in the first decades of the fourth century BC, when the Republic was probably written. The mathematician Theaetetus of Athens died in all probability in 391 BC, and must have had less than a decade of original work before his early death to obtain stereometric results of great significance: the construction of the five regular solids, their inscription in a sphere and their comparison.74 Plato immortalized the short-lived Theaetetus in the important dialogue named after the mathematician, and was so fascinated by the regular solids that he famously brought them to bear on cosmology in his Timaeus, thus earning them the familiar appellation Platonic.75 We cannot determine when the Republic was composed, but, on a generally accepted dating of the dialogue to the

72 73

74 75

geometry, both plane and solid, had not yet developed sufficiently in Plato’s view to truly serve philosophy propedeutically. If so, in R. 7, 528a6–c7, Plato comments on the primitive state in contemporary Greece not only of stereometry but of mathematics in general. See Heath (1981) vol. 1, 297. Cf. Huffman (2005) 386. Fowler (1990) 117–121 argues that ἡ τῶν κύβων αὔξη, which he translates as “the dimension of cubes”, refers not to the cube-duplication problem but to a far more difficult and still only partially understood problem that could have naturally arisen within Greek mathematics in the time of Plato, as Fowler speculatively reconstructs it, from the solution to the problem of doubling the cube: understanding the “anthyphaeretic” expansion of the ratio 3√ 2 : 1. i.e. of the sides of the unit cube and its double, in modern terms the continued fraction expansion of 3√ 2. The probable date of Theaetetus’ death is discussed in Nails (2002) 275–277. On the regular solids in Plato’s Timaeus see above n. 56.

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mid- to late 370s BC, it is unlikely that it would include a reference to the solid geometry after 391 BC as undeveloped, unless at the time Plato had yet to realize the importance of Theaetetus’ achievements in this field. If so, it must be the solid geometry at the dramatic date of the Republic, i.e. around the early 410s BC at the latest, that is assumed to be undeveloped in 528a6–c7.76 This seems to be confirmed by the three consecutive stereometric books of Euclid’s Elements. The two most advanced books reflect the state of solid geometry in the fourth century. Derived in book 11, the first of the stereometric books, are various elementary results about the parallepipedal solids; in book 12 results about volumes are obtained by the method of exhaustion famously developed by Eudoxus and, according to the testimony of Archimedes, two of the theorems which Eudoxus proved by this method had already been stated, and probably somehow proved, by Democritus;77 book 13, finally, contains the investigation of the Platonic solids and depends on the work of Theaetetus. The doubling of the cube seems to have come up first in the work of the mathematician Hippocrates of Chios, sometime before the end of the fifth century BC.78 Hippocrates asserted that if between two given lines A and B are constructed two mean proportional lines X and Y, i.e. such that the proportion A : X = X : Y = Y : B holds, then since it follows that A3 : X3 = AXY: XYB = A : B, if line B is two times line A, it turns out that line X is the side of a cube which is double the cube with line A as side; and since A and B can have to each other any ratio, we can evidently increase a cube in any ratio we please.79 The construction of two mean proportionals was effected by three mathematicians of Plato’s day, each employing a different technique: Archytas, whose birth should be placed sometime between 435 and 410 BC, as already said, and who was most probably the older of the three; Eudoxus, who lived between 408 and 355 BC on a traditional dating but most probably between 390 and 337 BC;80 finally, Menaechmus, whose origin is unknown and who seems to have been a generation younger than Eudoxus.81 Archytas could have published his cube-duplication by the dramatic date of the Republic. In the context of a complaint about the undeveloped state of solid geometry in the early 410s BC, a mention of the problem of doubling a cube, if by that time Archytas had published his solution, would not have been out of place. 76 77 78

79 80 81

On the problems surrounding the dramatic date of Plato’s Republic see Nails (2002) 324–326. See above n. 31. Our source for the problem of doubling a cube in the fifth and fourth century BC is Eutoc. in Sph. Cyl. 3.88.13–90.11 Heiberg. The commentator quotes a letter addressed by Eratosthenes of Cyrene to his royal patron, Ptolemy III Euergetes. In his letter the Cyrenean polymath describes the construction of an instrument of his own design for doubling a given cube. For a convincing defense of the letter’s authenticity against the objections of Wilamowitz (1962) see Knorr (1993) 17–24. For the famous anecdote, as a result of which the problem came to be known as Delian, see ch. 3.2. Cf. Procl. in Euc. 212.24–213.11 Friedlein. On the reduction of the cube-duplication problem to the construction of two mean proportionals see Saito (1995) and Netz (2004) 274–275. On Eudoxus see Zhmud (2006) 95–96. See Zhmud (2006) 99 n. 90. For the solutions of Eudoxus, Menaechmus and Archytas to the problem of cube-duplication see Eutoc. in Sph. Cyl. 3.56.2–10 (a solution attributed to Eudoxus is flawed and cannot be considered genuine, hence it is not set out), 78.14–84.7 and 84.13–88.2 Heiberg.

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Plato might have chosen to mention this problem because the construction of two mean proportionals around the dramatic date of the Republic, the first substantive result obtained in stereometry, had brought into stark relief the dearth of results about solids. On the other hand, the cube-duplication problem might have been still awaiting solution when the discussion in the Republic is supposed to have taken place: in this case, its brief mention in the seventh book, in the context of a complaint about the undeveloped state of stereometry at the dialogue’s dramatic time, highlights directly the situation at issue. Those among the dialogue’s contemporary audience who were knowledgeable about the great advances in mathematics after the dramatic time of the Republic probably would not fail to recall that the progress of solid geometry in their time was due, among other things, to the duplication of the cube by Archytas and others. If they also realized that Plato considered propedeutic to philosophy a future astronomy, whose mathematical sophistication relative to the astronomy of both the dialogue’s dramatic time and their own day will be as that of contemporary solid geometry compared to its ancestor at the dialogue’s dramatic time, then the importance of the kinematic construction of figures to the future astronomy would register with them, to be confirmed by the sketch of this astronomy in 529c6–d6.82 82

According to Plu. Moralia 718E7–F4, Plato criticized Archytas, Eudoxus and Menaechmus for their instrumental and mechanical constructions of the two mean proportionals that are required to solve the cube-duplication problem (cf. Marc. 14.9–11). It is unlikely that Plato’s remarks in R. 7 on the undeveloped state of solid geometry presuppose such a criticism. Eratosthenes, who had himself constructed an instrument for a practical solution to the cubeduplication problem by quickly finding mean proportionals, observes in his letter to king Ptolemy that the solutions found by Archytas, Eudoxus and Menaechmus were impracticably abstract (Eutoc. in Sph. Cyl. 3.90.8–11 Heiberg). The muddle of the tradition about Plato and the cube-duplication problem is evident not only in the anecdote which has been mentioned in n. 78 above but also in the fact that an instrumental solution is ascribed to Plato himself (Eutoc. in Sph. Cyl. 3.56.14–58.14 Heiberg). The author of the Anonymous Prolegomena to Platonic Philosophy (6th century AD) even credits Plato with the proportion-theoretic reduction of the problem of cube-duplication, which Eratosthenes ascribes to Hippocrates, though now the problem is reduced mistakenly into constructing a single mean proportional (Anon. Proleg. 5.13–24 Westerink). According to Huffman (2005) 392–401, Plato had criticized Archytas’ solution to the cube-duplication problem not for its mechanical character, as the tradition has it, but because, though truly abstract and brilliant, it was headed in the wrong direction insofar as, instead of being integral part of a systematically explored discipline, it was an isolated result motivated by problems suggested by the sensible world: doubling the size of an altar according to the anecdote mentioned in n. 78 above. It is not this story, however, that can help us understand Plato’s remarks in R. 7 on the undeveloped state of solid geometry but the other way around. Nor is there anything in these intriguing remarks to suggest that they might have been motivated by a dispute between Plato and some mathematicians, i.e. Archytas and, perhaps, Eudoxus, as Huffman thinks. Plato does speak in this context of stereometers as arrogant types. He does so, however, in emphasizing their presumed unwillingness to be directed by anyone (provided that a potential director would suddenly appear) as things stand in contemporary Greece, where there is no institutional support for mathematics, which thus languishes undeveloped. As it is, what Plato is really saying is that contemporary stereometers have not been habituated to working in the framework of institutions supported by the state under the inspired leadership of one or more outstanding individuals, so even if one willing and able to direct research in stereometry (no matter how that might be envisaged by Plato) would appear, probably one of

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1.7.2. Archytas’ solution There is no doubt about the kinematic character of Archytas’ ingenious solution to the problem of cube-duplication: in it the construction of two mean proportionals depends on the intersection of two curves, one generated by a rotating semicircle and the other by a rotating triangle.

Fig. 3

To construct two mean proportional lines between two given lines ΑΔ 〉 ΑΒ in the manner of Archytas, we assume first that ΑΔ is the diameter of a circle AZΔB and ΑΒ a chord of the same circle (see fig. 3). Next we assume that a right semicylinder is erected on the semicircle ΑΒΔ and consider a semicircle with diameter ΑΔ, which lies on the rectangle of the right semicylinder and is thus perpendicular on the circle AZΔB. We imagine this semicircle rotating around point A, which stays fixed, and thus generating a half horn-torus. As the rotating semicircle cuts the curved surface of the semicylinder, its circumference draws a curve on this surface (in fig. 3 this semicircle is AKΔ, shown in a frozen instant of time; the curve, which at this instant has the common point K with the semicircle, is not shown). Next, the chord AB of the circle AZΔB is produced until it meets at Π the tangent of this circle drawn at Δ and forms the triangle ΑΔΠ. We imagine this triangle rotating about ΑΔ as axis and generating a right cone. AB thus traces out a semicircle BMZ perpendicular to the circle AZΔB, and as the side ΑΠ of the rotating triangle cuts the curved surface of the right semicylinder, it draws on this surface a curve (not shown in fig. 3). When this curve meets at point K the curve traced out by the rotating semicircle AKΔ, point B becomes M. Dropping from point K the perpendicular KI on AΔ and drawing the line MΘ, it is not difficult to prove, first, that MΘ is perpendicular to BZ, then, drawing the line MI, that the angle the workers in the field, the others would laugh at any attempt of this person to influence their work in some way–what else to expect from brilliant individuals used to work in isolation? I can see nothing here that hints at Plato’s having bruisingly encountered at some point the super-sized egos of some stereometers.

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AMI is right and, finally, that the three triangles ΑΚΔ, ΑΙΚ and ΑΜΙ are similar, whence follows the proportion ΑΜ : ΑΙ = ΑΙ : ΑΚ = ΑΚ : ΑΔ. The lines ΑI and ΑK are, therefore, mean proportionals between the lines ΑΒ = ΑΜ and ΑΔ. If AB = ½ AΔ, then AI is the side of a cube whose volume is double the volume of the cube with side AB.83 1.7.3. Menaechmus’ solution Menaechmus, like Archytas, constructed two mean proportional lines through the intersection of two curves, either two parabolas or a parabola and a hyperbola, most probably initiating the study of conic sections; his construction, too, was probably kinematic.

(a)

(b) Fig. 4

Let the two given lines BG and ΒΑ be arranged at right angles to each other, as in fig. 4. In fig. 4a, the proportion BG : BH = BH : BK follows from similar triangles. As point H slides to the right on the horizontal axis and simultaneously point K moves upwards on the vertical axis, the proportion holds for all positions of these two points on their respective axes, and the point N of the rectangle KBHN draws a parabola. In fig. 4b, now, the proportion BH : BK = BK : ΒΑ follows once again from similar triangles, and as point H slides to the right on the horizontal axis and point K moves simultaneously upwards on the vertical axis, it holds for all positions of these two points on their respective axes, while the point X of the rectangle KBHX draws a parabola. When the two curves intersect, the lines BH and BK for both are equal, hence BG : BH = BH : BK = BK : ΒΑ. The lines BH and BK are thus mean proportionals between the given lines BG and ΒΑ. If BG = ½ BA, BH is the side of a cube whose volume is double the volume of the cube with side BG.84 83 84

For the full proof, and for using the same letter Δ to label two points, see Huffman (2005) 349–360. Based on Knorr (1993) 61–66. Menaechmus, in other words, solved the problem of doubling a cube by employing the parabolas y2 = 2x and x2 = y, whence, by substituting the expression of the second parabola in that of the first, we have x4 = 2x, thus x4 – 2x = 0 and x(x3 – 2) = 0,

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1.7.4. Eudoxus’ solution and his theory of homocentric spheres We do not know how Eudoxus constructed the required two mean proportionals and solved the problem of cube-duplication. We know, however, that this construction can be effected with his astronomical theory of homocentric spheres.85 The nested spheres, after which this famous theory takes its name, rotate simultaneously and uniformly. What matters here is that, as a point of the innermost of four homocentric spheres orbits the center of the system, it can trace out a curve vaguely similar to the one which a planet is observed tracing out as it moves zodiacally round the Earth. In fig. 5a the outermost sphere is omitted.

(a)

(b)

(c)

Fig. 5

According to what can be justly called the traditional reconstruction of the theory of homocentric spheres, the point corresponding to the planet is assumed to be on the equator of the innermost sphere.86 It and the next sphere are assumed to spin

85 86

with x = 0 being the intersection of the curves at the beginning of the axes and the cubic x3 – 2 = 0 giving 3√ 2 = x as the length that solves the problem of finding the side of a cube whose volume is double that of a cube with volume a cubic unit. Alternatively, Menaechmus used the hyperbola xy = 2 and the parabola y = x2. See Riddell (1979) 13–19. For two alternative attempts at reconstructing Eudoxus’ lost solution to the problem see Knorr (1993) 52–61. All reconstructions are speculative. The classic reconstruction of Eudoxus’ theory is due to Schiaparelli (1875); an interesting alternative reconstruction has been proposed by Yavetz (1998) and (2001). For an outline of both reconstructions see Kouremenos (2010) 33–40. They are undeniably speculative. The

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oppositely, but with the same period, corresponding to the synodic period of the planet. Due to their combined rotations, the point traces the 8-shaped curve in fig. 5b, which is called hippopede, “horse-fetter”.87 The third and outer sphere carries the hippopede in the direction of its rotation, its period corresponding to the zodiacal period of the planet. The combined rotations of all three spheres cause the point on the equator of the innermost one to describe above and below the vertical equator of the outermost sphere, which stands for the ecliptic, the curve shown in fig. 5c, whereupon its motion is principally to the direction of the rotation of the outermost sphere but occasionally reverses and then resumes its principal direction.88 As seen from fig. 5b, the hippopede is the intersection of the sphere whose equator is horizontal with a cylinder internally tangent to the sphere at the point where the curve crosses over itself. The projection of the curve on the equatorial plane of this sphere is a circle, which touches the sphere at the same point and is described by the projection on this plane of the hippopedally moving “planet”. Now, it is this planar projection that allows the construction of two mean proportionals, and the solution to the problem of doubling a cube, if we assume that the rotational period of the inclined sphere is double that of the next sphere. In this case, the path traced out by the “planet” is a closed curve, which is, though, not a hippopede (see fig. 8; point P is the equatorial “planet” A moving

87

88

theory of homocentric spheres is sketched out by Aristotle in Metaph. Λ 8, one of only two sources on which reconstructions of the theory are based; the other is Simplicius’ extensive commentary on Cael. B 12, on which see now Bowen (2013). The reliability of Aristotle and Simplicius as sources for the early Greek planetary theory, and thus for all reconstructions of a Eudoxean theory of the planets from their testimony, is rejected by Bowen (2002). According to Bowen, Eudoxus spoke only of planetary phenomena, among them those at the horizon but not retrogradations; although retrogradations figure prominently in both reconstructions of the theory of homocentric spheres, Bowen argues that they were unknown in Greece at the time (see 1.6.2, where I argue that Aristotle probably hints at the phenomenon); Aristotle simply tried to imagine how many rotational motions of nested spheres would be required to produce the planetary phenomena at issue. Bowen admits that there is no evidence that the phenomena of the planets at the horizon were known in fifth- or fourth-century-BC Greece, but he argues that knowledge of them could have been imported from Mesopotamia and that there is no need to assume the same for retrogradations given that the transmission of astronomical knowledge from Mesopotamia to Greece was a very fragmentary process prior to the second century BC. I find this argument unconvincing. Representing retrogradations need not have been the primary motivation for the introduction of the hippopede. This is only one possibility, alongside the representation of the distance from the Sun of Mercury and Venus, the two interior planets, and of the invisibility periods of all planets; see Mendell (1998) 228–229. His detailed study of the mathematics of the hippopede shows that the theory of homocentric spheres, as traditionally reconstructed, can represent to some degree a number of planetary phenomena, none of which, though, can be selected as the theory’s empirical foundation in the light of this mathematics alone. This can be an artifact of the dearth of evidence at our disposal; alternatively, it might show the fertility of Eudoxus’ theory, provided that the sources on which our reconstructions of it rest are mainly correct. For the hippopede see e.g. Neugebauer (1953), Riddell (1979) and, especially, the detailed treatment in Mendell (1998). On iconography of horse-fetters see Mendell (2000) 74 n. 21. The hippopede also appears in the alternative reconstruction of the theory (see previous n.); this is not recognized in Yavetz (1998) but see Yavetz (2001) 70–75. The fourth, outermost, sphere omitted in fig. 5a stands for the celestial sphere.

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under the influence of both rotations). Its projection on the equatorial plane of the next sphere is still described by the projection of the “planet” moving in a circle which touches the equator internally, though now this circle moves around the equator epicyclically. In fig. 6 the effect of the slow rotation is to move the equatorial point A to Q through an angle of rotation τ and, at the same time, raise it and B above the plane AOB of the equator until Q comes to P1 and B to B1 (not shown). The projections of P1 and B1 on the plane AOB of the equator are P´1 and B´1. It is the epicyclic motion of the circle GP´1Q that generates the projection on the plane AOB of the path traced out by the equatorial point. The opposite fast rotation carries this circle through an angle of rotation 2τ so that it becomes the circle HP´J. A perpendicular P´T to OA cutting OJ at S (see now fig. 7) can produce three similar triangles JRO, STO and RSO, whence follows a continuous proportion OT : OS = OS : OR = OR : OJ. The lines OS and OR are thus mean proportionals between the lines OT and OJ.

Fig. 6

Fig. 7

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For the three triangles JRO, STO and RSO to be similar, and thus for the continuous proportion OT : OS = OS : OR = OR : OJ to obtain, the angle RSO or, equivalently, the complementary angle JSR, must be right. As seen from fig. 7, the angle JSR = σ + α is a right angle if α = τ.89 But if α = τ, then β = τ, which means that the angle JSR is a right angle if the triangle KSR is isosceles with KS = KR. Since KS = KJ, it follows that the angle JSR is a right angle if KJ = KR = ½ JR. Since JR = RQ and KJ = P´J = P´1Q, however, the condition for the angle JSR to be right becomes P´1Q = P´1R = ½ RQ and, in view of the similar triangles QP´1G and QRO, B´1B = B´1O = ½ OB. But the point B1, to which the point B is raised above the plane AOB of the equator by the slow rotation through an angle τ, is projected on the plane as B´1, so B´1B = B´1O = ½ OB, i.e. the condition for the angle JSR to be right, is the amount by which the slow rotation raises B above the plane AOB, in other words the inclination of the axis of the slowly rotating outer sphere to the vertical axis of the fast rotating inner sphere. If this inclination is appropriately chosen, therefore, the continuous proportion OT : OS = OS : OR = OR : OJ obtains, and to double a cube we only need the equality OT = ½ OJ = ½ OA. If Π is a plane perpendicular to the equatorial plane and cutting halfway between O and A, the equality obtains when P´ is the projection of the “planet” at the moment it pierces this plane as it traces out its curvilinear path under the influence of the two combined rotations (see fig. 8). OS is, therefore, the side of a cube whose volume is double the volume of the cube with side OT.

Fig. 8 89

If τ + σ = 90º in triangle QP1´G, then angle OST = σ in triangle STO. Thus angle P´SJ = σ, and angle P´JΗ = σ since triangle JP´Η is triangle QP1´G transposed; angle JP´S = 2τ since τ + σ = 90º and the sum of the angles P´SJ, P´JΗ και JP´S is two right angles since these three angles are interior angles of a triangle. In the quadrilateral KJP´S, which we construct by dropping line SK parallel to line P´J, we have the equalities angle JKS = angle JP´S = 2τ and angle KJS = angle JSK = σ.

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1.7.5. Future astronomy, solid geometry and the theory of homocentric spheres We can see now that the kinematic construction of figures as a characteristic of the future astronomy envisaged by Socrates in R. 7 might well be hinted at in the interlude where he has Glaucon recall the undeveloped state of solid geometry in their day as illustrated by the lack of solutions to the cube-duplication problem. We assumed above, in 1.6.1, that, in so doing, Socrates attempts in vain to make his codiscussant realize that suitable for the propedeutic curriculum of the trainee philosopher-rulers can only be a future astronomy, by comparison to which the astronomy of their day is as primitive as contemporary solid geometry. Perhaps only Archytas had published his solution to the cube-duplication problem by the time Plato commented briefly in the Republic on the undeveloped state of solid geometry at the dramatic time of the dialogue. Those among Plato’s contemporary audience who were abreast of the advances in mathematics after the dramatic time of the Republic, and also realized that Plato considered propedeutic to philosophy an astronomy of mathematical sophistication comparable to that of contemporary solid geometry, as illustrated by Archytas’ approach to the problem of doubling the cube, would have guessed without any difficulty the implicit centrality of the kinematic construction of figures to the future astronomy dreamt of by Socrates. As noted in 1.7.4, we do not know how Eudoxus solved the problem of cubeduplication. The fact, however, that the construction of two mean proportionals can be effected with his astronomical theory of homocentric spheres makes it quite likely, in view of the connection drawn by Plato between the problem of cubeduplication and astronomy in R. 7, that this is indeed how Eudoxus managed to double the cube, that his stereometric and astronomical work had been done by the time Plato commented in R. 7 on the undeveloped state of solid geometry at the dramatic date of the dialogue, and also that it is hinted at by Plato’s remarks. If so, the theory of homocentric spheres probably underlies the outline of the future astronomy in 529c6–d6. It is difficult not to see in this description hints to features of Eudoxus’ theory. The hypotheses of this astronomy, as seen above, will define the geometrical objects themselves that are visibly represented in the sky as paths of the zodiacal motions of the planets: mental visualizations of these figures will be defined via their kinematic generation by other geometrical objects, also mental visualizations, as they move uniformly relative to one another. These motions can be plausibly understood as alluding to Eudoxus’ theory and its uniform rotations of the nested spheres in fig. 5a, whose end-product is the curve in fig. 5c that the zodiacal motions of the planets resemble. Plato’s contemporaries, if they knew the basics of Eudoxus’ theory, could not have made anything else of what Plato has Socrates say about the advanced future astronomy that can propedeutically support philosophy. The absence of any allusion to spheres from the description of this astronomy can very well be an implicit warning to his contemporary audience against the hasty identification of the future astronomy with the Eudoxean theory. The latter is to be brought in mind only to illustrate the character of the astronomy Plato has Socrates envision: it will investigate complex geometrical configurations generating kinematically representations of the mathematical objects that are also

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approximated by the observed paths of planetary zodiacal motions. The lack of good fit between the observed paths of the zodiacal motions of the five planets and those produced by the theory of homocentric spheres (see below) presumably would not allow Plato to identify the theory with the astronomy he has Socrates envision at the dramatic time of the Republic. Plato has Socrates conclude his brief discussion of this future astronomy with Glaucon by emphasizing a specific feature of this astronomy. This feature brings the future astronomy even closer to the Eudoxean theory of homocentric spheres, as we reconstruct it independently of any considerations about R. 7. As already said, Plato has Socrates state in 530b6–c2 the assimilation of the advanced future astronomy to geometry as he and Glaucon are wrapping up their tortuous discussion of the fourth propedeutic to philosophy: Προβλήµασιν ἄρα, ἦν δ’ ἐγώ, χρώµενοι ὥσπερ γεωµετρίαν οὕτω καὶ ἀστρονοµίαν µέτιµεν, τὰ δ’ ἐν τῷ οὐρανῷ ἐάσοµεν, εἰ µέλλοµεν ὄντως ἀστρονοµίας µεταλαµβάνοντες χρήσιµον τὸ φύσει φρόνιµον ἐν τῇ ψυχῇ ἐξ ἀχρήστου ποιήσειν. “It is thus through problems,” I said, “that we will pursue astronomy, just like we do with plane geometry, and we will leave the things in the heavens alone if we are to partake truly of astronomy and turn the naturally intelligent part of the soul from uselessness to usefulness.”90

90

“Leaving the things in the heavens alone” means, in view of the above, that the observed zodiacal paths of the planets will concern the future astronomy not in themselves but as the starting points for the investigations of mathematical structures, the true subject-matter of this astronomy. It does not imply that observation will have to be abolished; see also the following discussion. Nothing suggests that the observational component of the future astronomy will need not be taught to the philosopher-rulers in the making (Gregory [2000] 48–60) or that the mathematical/philosophical astronomers who will study astronomical problems arising from the results of observation will not have to do observational work (Vlastos [1981] 16). A call for the cessation of observation cannot be plausibly read in R. 7, 530a4–b5, where Plato has Socrates deny that the relation (συµµετρία) between night and day, the relation of night and day to the month and the year as well as the relations of the other celestial objects, i.e. of the periods of the planets, to the year and the month, can be accurately determined because all of these relations concern the variable motions of sensible and material bodies. In the reference to the relation between night and day and the year and the month, night and day are most probably to be understood as the 24-hours period, νυχθήµερον, and the reference itself as an allusion to the work of Meton and Euctemon, on which see above section 1.3.1. However, it is implausible that Plato did not appreciate the practical importance of such efforts and called for the cessation of the relevant empirical work: this does not follow from his warning that the motions of material bodies cannot have stable periods, which means that there is no point in trying to determine the relations between these periods with absolute precision–this is inherently unattainable, unlike the knowledge of abstract mathematical structures sensibly manifested in celestial motions. The reference to the relations of the periods of the planets to the year and the month might betray awareness of the existence of the Babylonian relations of the numbers of tropical and synodic cycles of a planet and a number of years; see Evans (1998) 304–305. The relation between night and day can be plausibly understood as the varying relation between the lengths of night and day in the course of the year. It is unlikely that in Plato’s time there was any attempt to calculate the length of day and night at any given time of the year for a given latitude; the first known attempt at a solution to the problem is Hypsicles’ Anaphorikos from the first half of the second century BC, but the problem could have been posed quite earlier.

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Now, the stereometrical interlude in 528a6–c7 after Glaucon’s abortive initial introduction of astronomy as the third propedeutic of philosophy would probably suggest, as argued above, to the contemporary audience of the Republic, if they knew the advances in mathematics after the dramatic time of the dialogue, that the future astronomy envisioned by Socrates as the fourth propedeutic is modeled on contemporary solid geometry. Plato, however, makes it clear at the beginning of the stereometrical interlude that next the term γεωµετρία will be used for plane geometry alone: as noted above in section 1.6.1, he has Socrates ask Glaucon to backtrack from astronomy as the propedeutic to philosophy after γεωµετρία, for logically not astronomy but the study of solids follows γεωµετρία, i.e. the study of the plane (528a6–10: Ἄναγε τοίνυν, ἦν δ’ ἐγώ, εἰς τοὐπίσω· νῦν [δὴ] γὰρ οὐκ ὀρθῶς τὸ ἑξῆς ἐλάβοµεν τῇ γεωµετρίᾳ. Πῶς λαβόντες; ἔφη. Μετὰ ἐπίπεδον, ἦν δ’ ἐγώ, ἐν περιφορᾷ ὂν ἤδη στερεὸν λαβόντες, πρὶν αὐτὸ καθ’ αὑτὸ λαβεῖν· ὀρθῶς δὲ ἔχει ἑξῆς µετὰ δευτέραν αὔξην τρίτην λαµβάνειν. ἔστι δέ που τοῦτο περὶ τὴν τῶν κύβων αὔξην καὶ τὸ βάθους µετέχον). As it is, with the stereometrical interlude in 528a6–c7 Plato seems to suggest that the future astronomy that will serve philosophy propedeutically is to be viewed in the light of the advanced fourth-century BC solid geometry, but at the end of his discussion of this astronomy he leaves no doubt that the future astronomy will be pursued through προβλήµατα in the manner of plane, not solid, geometry. προβλήµατα in this context are best understood as construction-problems, though the noun is also used not much below (531b8–c4) without connoting constructions.91 If the future astronomy will seek to effect constructions in the plane, the cube-duplication, which exemplifies the fourth-century BC solid geometry as the background against which the future astronomy is to be understood, illustrates nicely this feature of the future astronomy. The intersecting curves in Archytas’ solution to the problem results in the construction of three triangles in a plane and thus to the two sought after mean proportionals; Menaechmus constructed planar curves to find the two mean proportionals; to the same end Eudoxus followed in the footsteps of Archytas if he doubled the cube within his theory of homocentric spheres. But if the future astronomy recalls to the mind this theory since it can be plausibly assumed to study geometrical figures that are kinematically generated by other figures and are resembled by the observed zodiacal paths of the planets, then it does so even more when we take into account that it will also aim at effecting certain constructions in the plane. The latter will probably be related to the shapes of planetary zodiacal paths. The planar projection of a retrograde path of a planet, however, has been shown to be constructible, with compass and straightedge, within the stereometrical framework of Eudoxus’ theory of homocentric spheres, on its classic reconstruction outlined above as well as on a more recent alternative reconstruction, whereby the point-like planet is assumed to be on a circle of latitude near a pole of the innermost of the three spheres; on the alternative reconstruction, the combined rotations of the two inner spheres make the planet trace a long loop that, unlike the hippopede, does not 91

Cf. Procl. in Euc. 77.7–78.13 Friedlein on construction-problems vs. theorems.

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intersect itself.92 How likely is it that Plato’s explicit call for planar constructions in the future astronomy, as well as the rest of his brief outline of this science, mirrors accidentally the theory of homocentric spheres, which cannot throw any light on the future astronomy?93 It is not only at the beginning of the stereometrical interlude that Plato warns us about the subsequent use of the noun γεωµετρία in the sense “plane geometry” but also at its end, just before the second, and successful, introduction of astronomy as a suitable propedeutic to the study of philosophy. After Socrates says that the lack of progress in solid geometry is due not only to the inherent difficulty of this field but also to the need for a superintendent of those mathematicians who work in it and to the lack of support for it from all contemporary cities in Greece, as well as to this beautiful subject’s being lightly esteemed, Glaucon agrees with his codiscussant on the beauty of solid geometry and asks for a clarification (528d1–10): Καὶ µὲν δή, ἔφη, τό γε ἐπίχαρι καὶ διαφερόντως ἔχει. ἀλλά µοι σαφέστερον εἰπὲ ἃ νυνδὴ ἔλεγες. τὴν µὲν γάρ που τοῦ ἐπιπέδου πραγµατείαν γεωµετρίαν ἐτίθεις. Ναί, ἦν δ’ ἐγώ. Εἶτά γ’, ἔφη, τὸ µὲν πρῶτον ἀστρονοµίαν µετὰ ταύτην, ὕστερον δ’ ἀνεχώρησας. Σπεύδων γάρ, ἔφην, ταχὺ πάντα διεξελθεῖν µᾶλλον βραδύνω· ἑξῆς γὰρ οὖσαν τὴν βάθους αὔξης µέθοδον, ὅτι τῇ ζητήσει γελοίως ἔχει, ὑπερβὰς αὐτὴν µετὰ γεωµετρίαν ἀστρονοµίαν ἔλεγον, φορὰν οὖσαν βάθους.94 “It is certainly a very beautiful subject,” he said. “But explain to me what you just said. You defined as geometry the study of plane surfaces.” “Yes,” I answered. “Then,” he said, “at first you placed astronomy after it, but later you backtracked.” “Because I am in a hurry to go through everything at once,” I said, “it takes me more time! The study of the dimension of depth does come next, but it is laughably undeveloped, so I bypassed it and mentioned astronomy, which is concerned with a moving solid, after the geometry of the plane.”

Plato seems to want to make sure that γεωµετρία is construed in the correct sense “plane geometry” when it occurs next after the stereometrical interlude, whereby solid geometry is brought in as both the third propedeutic to philosophy and the backdrop against which the fourth propedeutic, the future astronomy, is to be understood. The next occurrence of the term is in the last remark on the future astronomy to the effect that this astronomy, viewed in the light of solid geometry though it must be, will pursue construction-problems in the manner of γεωµετρία, 92 93

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For the alternative reconstruction of the theory of homocentric spheres see above n. 86; for the construction of the planar projection of a planet’s retrograde path see Yavetz (2001) 77–93. There is no implicit suggestion here that the planar constructions Plato calls for in the future astronomy must be performed by using only straightedge and compass. Insistence on this restriction is attributed to Plato in connection with the cube-duplication problem, which cannot be solved by employing only straightedge and compass, and probably in view of the story of his objections to contemporary solutions mentioned above in n. 82 (see e.g. Stewart [20043] 78). The genitive βάθους in the expression ἡ βάθους αὔξη is not subjective, unlike the genitive τῶν κύβων in the expression ἡ τῶν κύβων αὔξη in R. 7, 528a6–c7, discussed above in section 1.7.1, but appositive.

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i.e. will seek to produce planar constructions that are related to certain astronomical questions. Since within the framework of the theory of homocentric spheres, on either of its available reconstructions to date, one can produce planar constructions related to the astronomical question that will probably be central to Plato’s future astronomy, namely the shape of planetary zodiacal paths, this interesting technical specification in his brief outline of the future astronomy suggests that this rough sketch was most probably influenced by the theory of homocentric spheres.95 In this theory Plato could see what mattered to him: aspects of the constantly variable world of the senses and motion resembling, to various degrees, intelligible and static mathematical entities. On its classic reconstruction, the theory of homocentric spheres gives retrograde paths of the planets with shapes remotely, if at all, similar to those that are observed in the sky; on its more recent reconstruction, however, it is able to postdict the shape of retrograde paths for some planets with surprising accuracy, provided that certain crucial parameters are chosen appropriately. It might well be the case that, instead of two competing reconstructions, we are faced with two versions of the same theory, both of which Eudoxus and his school had investigated. Be that as it may, neither of the two reconstructions can result in the observed variation in the shape of the retrograde path from one retrogradation to the next, nor do they fare any better with the uneven bunching of retrogradations around the zodiac. Irrespective of how it is reconstructed, the theory lacks predictive power. There is no evidence that developing even crudely accurate predictive models of celestial motions had been a goal of Greek astronomers in the time of Eudoxus.96 We do not have to assume that in R. 7 Plato presents Socrates as foreshadowing clairvoyantly the future of astronomy, though this would not be a problem in view of the portrayal of Socrates in Phd. 84c8–85b9 as a servant of Apollo. As said above, Archytas might have obtained and published his solution to the problem of doubling the cube by the dramatic time of the Republic: conceivably, his realization that the kinematic construction of curves might be fruitfully brought to bear on the geometrical study of the complex zodiacal wanderings of the planets influenced the stereometrical and astronomical efforts of Eudoxus, who is said in our sources to have been Archytas’ student.97

95

96 97

On the story that it was Plato who influenced the development of the theory see ch. 3. There is no direct evidence in the dialogues that Plato had knowledge of the theory of homocentric spheres; cf. Zhmud (2006) 98 with n. 87. See Jones (1991) and cf. Yavetz (1998) 246–247. See D.L. 8.86.

2. THE UNITY OF MATHEMATICS IN THE REPUBLIC 2.1. INTRODUCTION In R. 7, 531c9–d5, Plato has Socrates note that the ultimate goal of the study of mathematics the future philosopher-rulers have to go through before they take up the study of philosophy is to grasp the affinity and kinship between arithmetic, plane geometry, stereometry, astronomy and harmonics, the only areas of study which in the preceding discussion have been picked out as capable of serving the study of philosophy propedeutically (521b1–531c8). Next Plato has Glaucon emphasize the difficulty of this task: Οἶµαι δέ γε, ἦν δ’ ἐγώ, καὶ ἡ τούτων πάντων ὧν διεληλύαµεν µέθοδος, ἐὰν µὲν ἐπὶ τὴν ἀλλήλων κοινωνίαν ἀφίκηται καὶ συγγένειαν, καὶ συλλογισθῇ ταῦτα ᾗ ἐστὶν ἀλλήλοις οἰκεῖα, φέρειν τι αὐτῶν εἰς ἃ βουλόµεθα τὴν πραγµατείαν καὶ οὐκ ἀνόνητα πονεῖσθαι, εἰ δὲ µή, ἀνόνητα. Καὶ ἐγώ, ἔφη, οὕτω µαντεύοµαι. ἀλλὰ πάµπολυ ἔργον λέγεις, ὦ Σώκρατες. “In my opinion,” I said, “the study of all these subjects that we have just discussed has also some relevance to our purposes and is not wasted effort if it leads to the affinity and kinship between the subjects in question and to understanding how they are related to one another; if it does not, though, it is wasted.” “This is my hunch, too,” he said. “But you are proposing a very difficult task, Socrates.”

But Plato does not explain how the affinity and kinship between the branches of mathematics is supposed to be understood, despite the importance he obviously attaches to this feature of mathematics. Although in R. 7, 530d6–10, he has Socrates say that astronomy and harmonics are “kindred sciences” (ἀδελφαί τινες αἱ ἐπιστῆµαι), the respect in which astronomy and harmonics are conceived of as kindred here cannot link them with the remaining three branches of mathematics. Astronomy and harmonics are akin in that each concerns itself with a different kind of motion: harmonics deals with audible motion (530d7), astronomy with the visible motion of a solid (see 528a9–10, where the subject-matter of astronomy is described in its contrast to the subject-matter of solid geometry, the solids per se).1 1

Harmonics studies audible motion since pitch is for Plato a function of the speed of sound; see the account in Ti. 67a7–c3 and 79e10–80b8. The astronomy whose subject-matter is a rotating solid is a calendaric astronomy that aids us in determining the changes of seasons and is useful for agriculture and seafaring (R. 7, 527d1–4). The changes of seasons were determined by the phenomena called phases of the fixed stars, and the only rotating solid involved in the theory of these phenomena is the celestial sphere, so Plato must have the latter in mind when he describes astronomy as the study of a rotating solid; see ch. 1.3–4. In having Socrates contrast solid geometry with astronomy as regards the subject that each studies, Plato does not say unambiguously that what is studied in astronomy is a single rotating solid. The phrase he uses for the subject-matter of astronomy is ἐν περιφορᾷ ὂν στερεόν, but he describes in the same

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But motion cannot link astronomy and harmonics with the other branches of mathematics (solid geometry is emphatically said to concern itself with the solids themselves and not, as astronomy does, with a moving solid). It might be thought that Plato hints where the kinship of all branches of mathematics lies in R. 6, in the divided-line simile (see ch. 1.1). There he singles out two features common to all branches of mathematics: deriving conclusions, via chains of consistent steps, from problematic hypotheses of which no account is given, as if they were clear to everyone, i.e. from some definitions of mathematical objects, which are not sensibles but forms; relying on visible representations of forms (510b2–511a9). The examples of hypotheses, two from geometry, those about the three kinds of angle and the various figures, and one from arithmetic, those about odd and even numbers (510c2–5), are followed by a mention of their kin in the other branches of mathematics (510c5: ἄλλα τούτων ἀδελφὰ καθ᾽ ἑκάστην µέθοδον). If the kinship of the hypotheses transfers to the bodies of proofs that are dependent on hypotheses and constitute the branches of mathematics, Plato views all branches of mathematics as akin at least insofar as each of them erects its proofs on definitions of its objects that are presented as self-evident, though they are not. The two features that are common to all branches of mathematics are set out in R. 6 as a description of the lower section of the “intelligible” part of the divided line. Given in unclear terms by Socrates without even an implicit reference to mathematics (510b2–8), this description prompts the confused Glaucon to ask for an explanation (510b9), after which he realizes that Socrates associates this section of the “intelligible” part of the divided line with geometry and its indeterminately many kindred sciences (511a10–b1: Μανθάνω, ἔφη, ὅτι τὸ ὑπὸ ταῖς γεωµετρίαις τε καὶ ταῖς ταύτης ἀδελφαῖς τέχναις λέγεις). “Kindred” in this context can be plausibly understood to mean that the kinship between all branches of mathematics lies in their necessarily relying on problematic starting points as well as in their constructing visible representations of purely intelligible forms. These are the two crucial respects in which all branches of mathematics are akin, and which thus characterize a single section of the divided line corresponding to all branches of mathematics as akin to one another. The higher section of the “intelligible” part of the divided line answers to dialectic, i.e. philosophy. Dialectic is also concerned with forms but approaches them in a completely non-mathematical way: that is, without relying on any visible representations of these intelligible objects and also by using hypotheses not as self-evident starting points, no account of which is given, for reaching conclusions without grasping what these hypotheses are about but as true foundations, on which full accounts of forms are built (511b2–c2). The branches of mathematics seem thus to be akin to dialectic insofar as all of them study intelligible objects, just like philosophy does, though their approach to breath what stereometry studies as στερεὸν αὐτὸ καθ᾽ αὑτό, undoubtedly a collective phrase (solid geometry cannot be plausibly said to study a single solid). But the theory of the phases of the fixed stars is the only astronomy that can be implied here. It involves a single rotating solid, the celestial sphere, so Plato cannot use the noun στερεόν in a collective sense to describe the subject-matter of astronomy; see ch. 1.3.3.

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intelligible objects is not only different from but also inferior to the approach to being that is characteristic of philosophy. If central to Plato’s conception of all branches of mathematics in R. 6 as kindred sciences (ἀδελφαὶ τέχναι) is his conception of what they study as intelligible, it stands to reason that, when in R. 7 he has Socrates attribute to anonymous Pythagoreans the description of astronomy and harmonics as kindred sciences (ἀδελφαὶ ἐπιστῆµαι), a possible echo of the description of all branches of mathematics by Archytas of Tarentum in his lost treatise Harmonics (Ἁρµονικός) or On Mathematics (Περὶ µαθηµατικῆς; DK 47 B 1), he implicitly contrasts two diametrically opposite conceptions of the kinship between the various branches of mathematics: the Platonic, which is grounded in the intelligibility of the objects studied in each branch, and the Pythagorean, whereby the objects studied in each branch are confined to the sensible world, as is suggested by Plato’s critique of Pythagorean harmonics and astronomy in R. 7, 529a3–c2 and 530d6–10.2 It is, however, unlikely that, in Plato’s view, the future philosopher-rulers will have to study only mathematics for a decade (R. 7, 537b7–c3) merely in order to understand that all of its branches are akin to one another insofar as each one of them, first, studies not sensible but intelligible objects, though visibly represented, and, second, relies on starting points assumed to be clear to everyone, which is not the case. Plato thinks that the characterization of what is studied in mathematics as not sensible but intelligible is unproblematic. Concerning the “operations” such as applying an area, squaring and addition that are constantly “performed” in geometry, as if the latter dealt with perceptible, constructible things, he says that even people who have little experience in this subject would not doubt that talk of performing such operations is counter to the true nature of the objects that geometry studies: they would readily agree that geometry is not about performing operations, for it is not concerned with things subject to changes such as coming to be and passing away but instead aims at obtaining knowledge of changeless, eternal and intelligible objects, though in order to do so it has to rely on visible images of these objects, hence the indispensability of the funny talk about operations so prominent in it (R. 7, 527a1–b11). Plato presents the realization of the intelligibility of the objects studied in arithmetic as similarly unproblematic (R. 7, 525c8–526a7; see ch. 1.5.2). As regards astronomy and harmonics, the situation is more complex. Plato has Socrates criticize astronomy and harmonics as mired in the sensible world, which makes both of them incapable of serving philosophy, and dream of a future astronomy and harmonics that will be oriented towards intelligibles, as geometry and arithmetic are (R. 7, 528e1–531c8).3 However, it is clear that, when and if the astronomy and the harmonics that will serve philosophy propedeutically become available, by Plato’s lights even those who will know only a smattering of them will readily agree, before embarking on a rigorous course of study, that these branches of mathematics, too, in fact deal with intelligible objects, not the sensible world. 2 3

For astronomy see ch. 1.5 with section 3 on Archytas. On the future astronomy see section 2.1 below and for full discussion ch. 1.6–7.

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A decade-long demanding course in mathematics would not be plausibly needed for the future philosopher-rulers to grasp what in the case of geometry is considered by Plato as clear to all who know a smattering of this subject. Nor does one need a decade-long immersion in mathematics to see that mathematical proofs rely on starting points such as the three kinds of angle, the various figures in geometry and the odd and even numbers in arithmetic, or that the starting points, despite their being set out as self-evident, raise considerable difficulties. As goal of the decade-long mathematical studies of the future philosopherrulers Plato seems to posit the grasp of a deeper kinship between the branches of mathematics. Realizing and appreciating it requires time and hard practice. A plausible answer to the question how he might have conceived of the kinship at issue must satisfy this requirement. 2.2. ARITHMETIC AS UNIFIER OF MATHEMATICS 2.2.1. The superiority of arithmetic and incommensurability The metaphorical conception of the branches of mathematics as a family can be traced back to the Pythagorean Philolaus of Croton, who considered geometry the mother-city of the other branches of mathematics (DK 44 A 7a). The description of geometry as mother-city (µητρόπολις) of the other branches of mathematics probably suggests that geometry was the first mathematical science to develop.4 There is no evidence, however, that Plato followed Philolaus in singling out, rightly or wrongly, one branch of mathematics as the source from which all the others originated, and which, therefore, makes them akin. As suggested above, Plato’s conception of the branches of mathematics as somehow akin to one another might echo Archytas’ description of all of them as sister-sciences. The kinship of the branches of mathematics seems to reside, according to Archytas, in that each of them starts from some general principles, “wholes”, within which successive distinctions are then made, apparently of less and less generality, until particular objects and phenomena of our experience, “individual things”, are reached, and knowledge about them is thereby obtained.5 This seems to chime in well with Plato’s critique of Pythagorean harmonics and astronomy in R. 7, 530d6–10, as hopelessly mired within the sensible world. On the Platonic conception of these and the other branches of mathematics, the true objects of knowledge are not sensible things but rather the intelligible entities for whose sake sensible things are studied, and which also render sensible things knowable, to whatever extent they are knowable. It thus seems plausible to assume that Plato simply takes over mutatis mutandis Archytas’ conception of the kinship between the branches of mathematics. This view, however, must be rejected for the reason that, as seen in the previous section, it forces us to deny that the Platonic

4 5

See Huffman (2005) 69–70. See Huffman (2005) 64–68.

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unity of all branches of mathematics resides in their two common features set out in the context of the divided-line simile. In his Harmonics or On mathematics Archytas conceived of the branches of mathematics as a happy family, but in another treatise titled Discourses he singled out “logistic” (λογιστική), in all probability arithmetic plus the theory of arithmetic ratios and proportions, as far superior to geometry and all the other branches of mathematics (DK 47 B 4): καὶ δοκεῖ ἁ λογιστικὰ ποτὶ τὰν σοφίαν τῶν µὲν ἀλλᾶν τεχνῶν καὶ πολὺ διαφέρειν, ἀτὰρ καὶ τᾶς γεωµετρικᾶς ἐναργεστέρω πραγµατεύεσθαι ἃ θέλει. καὶ ἃ ἐκλείπει αὖ ἁ γεωµετρία, καὶ ἀποδείξιας ἁ λογιστικὰ ἐπιτελεῖ καὶ ὁµῶς, εἰ µὲν εἰδέων τεὰ πραγµατεία, καὶ τὰ περὶ τοῖς εἴδεσιν. Logistic seems indeed to be far superior to the other arts as regards wisdom and to deal with what it wishes more concretely than geometry. As regards what geometry leaves out, logistic completes geometrical proofs and, if it deals with shapes, similarly with what concerns them.6

It is not improbable that in his work Harmonics or On mathematics Archytas described all branches of mathematics as members of a family also in view of his conception of arithmetic and proportion-theory as the foundation that binds to itself the other branches of the discipline into a unity.7 Knorr has argued that Plato’s conception of the unity of mathematics is to be understood in the light of Archytas’ view that arithmetic is superior to geometry and the rest of mathematics. This view, according to Knorr, reflects “the recognition that certain classes of geometric and stereometric problems–the study of incommensurability being the prime instance–give rise to arithmetic relations which are ultimately examined via the principles of number theory. It is for this reason surely that Archytas says “arithmetic completes the proofs of geometry” (fr. 4); and it is also in this sense that Plato’s conception of the unity of the mathemata should be interpreted.”8 Knorr understands incommensurability as prime instance of the superiority of arithmetic over geometry in view of two theorems that can be plausibly attributed to the mathematician Theaetetus of Athens, on the evidence in Plato’s famous dialogue named after him (147d3–148b2), and a theorem attributed to Archytas by Boethius in his De institutione musica (285.7–286.19 Friedlein = DK 47 A 19). Theaetetus seems to have shown that the side of a square and the unit-line are commensurable if the square contains a square number of unit-areas, and that similarly the side of the cube and the unit-line are commensurable if the cube contains a cubic number of unit-volumes. Archytas proved that a superparticular ratio n+1/n, such as 9/8, 4/3 and 3/2, which correspond to the tone, the fourth and the fifth, the basic intervals out of which the octave and all Greek musical scales are constructed, cannot be divided into equal parts by the interpolation of a mean proportional number. What this means geometrically is that, given a rectangular 6 7 8

On logistic see Huffman (2005) 240–244. Cf. Huffman (2005) 64–65. Knorr (1975) 93. For an alternative interpretation of what Archytas says in DK 47 B 4 about geometrical proofs and arithmetic see Huffman (2005) 232–240.

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parallelogram whose sides have to each other a superparticular ratio, both sides are incommensurable with the side of a square whose area equals the area of the given rectangular parallelogram;9 moreover, given two unequal squares, their sides are incommensurable if the area of one square stands to the area of the other square in a superparticular ratio n+1/n.10 Relevant here is also Aristotle’s reference in APr. A 23, 41a21–32, to an arithmetical reductio ad absurdum proof that a square’s side and its diagonal are incommensurable: the assumption of their commensurability leads to the absurdity that even numbers are odd.11 As it is, arithmetic helps not only geometry, both plane and solid, as is clear from the two results attributed to Theaetetus, but also harmonics derive certain results of fundamental importance that cannot be obtained otherwise. Its indispensability to these other branches of mathematics can thus be thought to bring them together into a unity, where arithmetic dominates. But if Knorr’s view on the Platonic unity of mathematics is to be accepted, it must extend to astronomy as well. There is a trivial sense in which it does, as is shown by Plt. 299e1–2, where Plato refers to “arithmetic in its entirety, whether pure or plane or in depths or in speeds” (σύµπασαν ἀριθµητικὴν ψιλὴν εἴτε ἐπίπεδον εἴτ’ ἐν βάθεσιν εἴτ’ ἐν τάχεσιν οὖσάν που). Arithmetic can be pure, “plane”, with numbers representing areas, “in depths”, whereby numbers are volumes, or “in speeds”, probably the visible or audible motions mentioned in R. 7. As argued above, motion can account for the kinship of astronomy and harmonics but not for their kinship with arithmetic and geometry. Plato, however, as is clear from Plt. 299e1–2, seems to think that, insofar as the visible motions studied in astronomy are numerically expressed, this branch of mathematics, too, not only its related harmonics, is connected with plane and solid geometry via arithmetic, the unifying bond of all other branches of mathematics. This seems to be supported by the mention of “true numbers” in R. 7, 529c6–d6, Socrates’ obscure outline of a future astronomy that will aid philosophy propedeutically (cf. ch. 1.6.1). That arithmetic is manifested in astronomy and all the other branches of mathematics is also clear from Lg. 5, 747a1–5, where there is a reference to the “variations of numbers, both those variations that numbers have in themselves and their variations in lengths and depths, as well as in sounds and motions, both upward and downward in a straight line and circular” (τὰς τῶν ἀριθµῶν…ποικίλσεις, ὅσα τε αὐτοὶ ἐν ἑαυτοῖς ποικίλλονται καὶ ὅσα ἐν µήκεσι καὶ ἐν βάθεσι ποικίλµατα, καὶ δὴ καὶ ἐν φθόγγοις καὶ κινήσεσι ταῖς τε κατὰ τὴν εὐθυπορίαν τῆς ἄνω καὶ κάτω φορᾶς καὶ τῆς κύκλῳ περιφορᾶς). We can do better than that, however, and argue that Plato might conceive of astronomy as related to plane and solid geometry and harmonics by arithmetic in a stronger sense, too, via incommensurability, i.e. in 9 10

11

See Euc. El. 2.14 and 6.13; cf. Arist. de An. B 2, 413a17–20, and Metaph. B 2, 996b20–21. X is the middle proportional between A and B if A2 : X2 = A : B, but if A = n+1 and B = n, there is no number to measure the side of X2. Doubling a square is the related problem set by Socrates to the slave-boy in the Meno: the solution is that the diagonal of a square is the side of another square whose area is double that of the first. See Knorr (1975) 22–24.

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the same manner in which, according to Knorr, he thinks that arithmetic relates plane and solid geometry to each other and to itself. By Theaetetus’ theorem, the side of the unit-cube and that of a cube whose volume is two times the unit-cube are incommensurable: if one square has double the area of the other, their sides are incommensurable, and the analogous result holds of cubes. To construct the side of the cube whose volume is two times the unit-cube, in other words to double or duplicate a given cube, is the famous cubeduplication problem. The discussion of stereometry in R. 7 mentions this problem in connection with the astronomy that will serve philosophy propedeutically.12 This seems to allow us to assume that arithmetic, in Plato’s view, does relate astronomy to the rest of mathematics via the study of incommensurability. 2.2.2. Incommensurability and the future astronomy in R. 713 In R. 7 Plato has Socrates refer to the problem of cube-duplication when he briefly describes stereometry as the study of “the growth of cubes and what partakes of depth” (528b1–2: ἔστι δέ που τοῦτο περὶ τὴν τῶν κύβων αὔξην καὶ τὸ βάθους µετέχον). Glaucon immediately protests that results concerning these things have yet to be obtained (528b3–4); this is a reference as much to the undeveloped state of solid geometry, i.e. to the dearth of results about three-dimensional objects (“what partakes of depth”), as to the ignorance of how cubes can be increased, to the lack of solutions to a particular stereometric problem, that of doubling a cube. Plato’s description of solid geometry as undeveloped does not fit the state of this branch of mathematics in the first decades of the fourth century BC, when the Republic was probably written. Thus it must be the solid geometry of the early 410s BC, the dramatic time of the Republic, which in the seventh book of the dialogue is said to be undeveloped. Now, why would Plato want to remind his contemporary audience of the undeveloped state of solid geometry at the dramatic time of the Republic? He has Socrates stress the logical priority of stereometry over contemporary, practical astronomy and describe stereometry after he has Glaucon identify contemporary, practical astronomy as the third discipline which the future philosopher-rulers must study before taking up philosophy (527d1–528b1). By pointing out to his interlocutor the priority of solid geometry over astronomy and then having him recall the undeveloped state of solid geometry in their day, Socrates tries to gently nudge Glaucon away from the misidentification of a subject in the demanding propedeutic curriculum of the future philosopher-rulers as contemporary, practical astronomy. Astronomy is to be on this curriculum: Socrates, though, is having in mind not the astronomy of their day but a future astronomy, compared to which the astronomy of their day is primitive, just as contemporary solid geometry is undeveloped. Glaucon misses this subtle point. He thus forces Socrates to state 12 13

On the cube-duplication problem see ch. 1.7.1. Like squaring the circle and trisecting an angle, it is unsolvable with straightedge and compass; see Stewart (2008) ch. 8 and cf. below n. 29. This section is based on Kouremenos (2004) and is further developed in ch. 1.6–7.

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explicitly that the astronomy he envisages as a propedeutic to philosophy is not contemporary astronomy; after Socrates has tried to clarify his vision, Glaucon realizes that this astronomy will far exceed contemporary astronomy in difficulty (528d1–530c4). Those among Plato’s contemporary audience who were familiar with the history of mathematics were conceivably supposed to latch onto the reference to the undeveloped state of solid geometry at the dramatic time of the Republic and thus realize, before this point is made explicit in the text, that, unlike Glaucon, Socrates does not consider as propedeutic to philosophy the practically oriented astronomy of his time. Eudoxus of Cnidus was one of the three mathematicians who solved the problem of doubling a cube in Plato’s time. We do not know how he solved it.14 A solution, however, can be obtained from his astronomical theory of homocentric spheres.15 If this is how he doubled the cube, let us assume that Eudoxus had done his stereometric and astronomical work by the time Plato commented in R. 7 on the undeveloped state of solid geometry at the dramatic time of the dialogue. Then, in light of the priority of solid geometry to astronomy and the reference to the cube-duplication problem as an illustration of the undeveloped state of stereometry at the dialogue’s dramatic time, those among the work’s contemporary audience who were aware of the developments in mathematics after the dramatic time of the Republic would have probably recalled that the progress of solid geometry in their time was due, among other things, to the doubling of the cube by Eudoxus. This stereometric feat was connected with his achievement in astronomy, the theory of homocentric spheres, which in its treatment of the planets, where its connection with the solution to cube-duplication problem lies, went beyond the astronomy identified by Glaucon as the third subject the future philosopher-rulers ought to study before philosophy. In his discussion with Glaucon of the first two subjects, arithmetic and plane geometry, Socrates makes it very clear that propedeutic to philosophy are only subjects whose value exceeds their practical utility (522b5– 523a3 and 527b6–c11). This is lost on Glaucon; if it were not lost on those among Plato’s contemporary audience who were familiar with the advances in mathematics since the dramatic date of the Republic, they would probably realize that Plato has Socrates, unlike Glaucon, view as a propedeutic to philosophy not the practically oriented astronomy of his time but an astronomy cast into the mold of the theory of homocentric spheres. Plato has Socrates give a hazy description of the future astronomy he dreams of as a propedeutic to philosophy. It is possible, however, to see in this description allusions to the broad features of Eudoxus’ theory of homocentric spheres. The future astronomy will treat visible motions in the sky as far deficient in exactness compared with true motions that invariable swiftness and invariable slowness in true numbers and all true figures undergo relative to one another, carrying what is inside them (529c6–d6). The motions, wherein invariable swiftness and invariable slowness are expressed in true numbers, can be plausibly understood as an allusion to the Eudoxean theory which takes its name from its nested solids: they spin 14 15

See ch. 1 n. 81. See ch. 1.7.4.

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simultaneously and uniformly, each, in general, with its own period, so that e.g. a point in the innermost one traces out a repeatedly self-intersecting curve, similar to that along which a planet is observed to move zodiacally round the Earth in an eastward direction but regularly reversing direction. It is indeed hard to see what else Plato’s contemporary audience who knew about the theory of homocentric spheres could have made of what Plato has Socrates say about the future astronomy that can propedeutically support philosophy. If this context contains an allusion to the theory of homocentric spheres, Plato has Socrates mention not rotations of spheres but vague motions most probably as an implicit warning to his contemporary audience against the hasty identification of the future astronomy with Eudoxus’ theory. The latter is to be brought in mind as a mere illustration of the nature of the future astronomy that Plato has Socrates envision: it will investigate abstract structures, whatever, and however complicated, they might be, generating kinematically further geometric structures that are approximated by the paths of visible motions in the heavens. The metaphorical description of these motions with both a noun and a cognate verb denoting their beauty and variety (529c6–7: τὰ ἐν τῷ οὐρανῷ ποικίλµατα…ἐν ὁρατῷ πεποίκιλται) seems to suggest that the celestial motions that matter here are the zodiacal motions of the planets, the treatment of whose retrograde paths is the centerpiece of the theory of homocentric spheres. In Ti. 39d1–2 the same verb appears in a description of planetary zodiacal motion and probably hints at the variously shaped, and unevenly spaced, paths of successive retrogradations (τὰς τούτων πλάνας, πλήθει µὲν ἀµηχάνῳ χρωµένας, πεποικιλµένας δὲ θαυµαστῶς). Within Eudoxus’ theory, moreover, a planar projection of an already observed retrograde path is constructible with compass and straightedge. It is probably this feature of the theory of homocentric spheres that Plato wants his contemporary informed audience to recall when he has Socrates declare that, to be of propedeutic service to philosophy, astronomy has to move beyond empirical utility, to solving problems in the manner of plane geometry, and next emphasizes, via Glaucon’s surprised exclamation, that this goes well beyond the mathematical sophistication of the astronomy at the dramatic time of the Republic (530b6–c4). 2.2.3. Incommensurability in the divided-line simile Plato might, therefore, think that arithmetic relates not only plane and solid geometry and harmonics but astronomy, too, via incommensurability. A potential hint to such a crucial role of incommensurability can be seen in the context of the divided-line simile. There, as seen above, all the branches of mathematics are said to rely on kindred starting points for their proofs, examples of which are from geometry the kinds of angle and various figures, from arithmetic the odd and even numbers. As example of the second similarity which is shared by all branches of mathematics, i.e. their need to construct visible representations of the intelligible objects under study, Plato gives a square and a diagonal drawn in a proof about the square and its diagonal: the proof, though, aims at obtaining knowledge of the

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square itself and the diagonal itself, which are abstract objects, and has nothing to do with that particular square and that particular diagonal that have been drawn on a given occasion as visible representations of the intelligible objects actually studied. The proof about the square and its diagonal Plato has Socrates speak of cannot but derive the incommensurability of the side and the diagonal of a square. As said above in 2.2.1, in APr. A 23, 41a21–32, Aristotle refers to a proof in which this is established by showing that the assumption that the side and the diagonal of a square are commensurable with each other leads to the absurdity of even numbers being odd. As it turns out, when in the divided-line simile Plato has Socrates give the three kinds of angle, the different figures and the odd and even numbers as some examples of kindred starting points from which mathematical proofs establish their conclusions via chains of consistent steps, in all probability he does not select starting points at random. He seems to choose a particular arithmetical starting point, the odd and even numbers, and a category of starting points from geometry, the angular plane figures such as the square, both of which feature in a proof of the fundamental geometric proposition alluded to immediately next: the incommensurability of the side and the diagonal of a square. What his choice seems to imply is that the kinship of the arithmetical and geometrical starting points lies exactly in the establishment of incommensurability, the bond by which arithmetic unites itself with geometry. 2.2.4. Philip of Opus The view that all branches of mathematics are kindred according to Plato insofar as arithmetic unites all the others with itself through its role in the study of incommensurability begins to lose its attractiveness as soon as one realizes that it seems also to be the view on the unity of mathematics apart from astronomy in the Epinomis, probably by Philip of Opus.16 The author asserts the superiority of arithmetic over plane and solid geometry and harmonics, astronomy being superior to them all (989e1–991b4). Geometry is defined as the assimilation to planes of numbers that are dissimilar by nature to each other, and solid geometry as the assimilation to solids of numbers that are dissimilar by nature to each other, but not in the same way as in geometry (990d1–4 and d6–e1). The description of geometry refers to the representation of a number as a rectangle, i.e. as the rectangle’s area, and its representation as a square, which is the construction of a square with the same area as the rectangle; the description of stereometry refers to the representation of a number as a rectangular solid, i.e. as the solid’s volume, and its representation as a cube, i.e. the construction of a cube equal in volume to the rectangular solid. The numbers at issue being dissimilar, i.e. non-square and non-cubic, their turning from dissimilar rectangle and rectangular solid into similar square and cube respectively produces incommensurable lines. The author goes on to couch the overarching superiority of arithmetic to both the geometrical study of 16

On Plato and the authorship of the Epinomis see the relevant section in Tarán (1975).

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planes and solids in an alternative manner that allows him to bring harmonics into the fold: the sequence of numbers 1, 2, 4 = 22, 8 = 23 reaches to what is tangible and solid (23) through the plane (22) starting from numbers in double ratio, and the interpolation of the arithmetic and geometric means between numbers in this ratio generates the basis for the organization of musical scales (990e1–991b4). By Philip’s lights, arithmetic not only is common, and hence superior, to plane and solid geometry and harmonics but also unifies all three of them precisely on account of its superiority, just as at a higher level all four of them are welded into a superior unity by astronomy, whose indispensible propedeutics they all are (cf. 990b8–c5): understanding the divine providence immanent in nature, the goal of astronomy (991b5–c1), which encompasses dialectic (991c2–6), is most probably the “single bond” (δεσµὸς εἷς) that imparts unity to all branches of mathematics (991d5–992a3). Also quite probable is that according to Philip arithmetic unifies plane geometry, stereometry and harmonics not only because of its superiority to all of them in the sense explained above: his arithmetical characterization of plane and solid geometry can be easily thought to point to incommensurability as the phenomenon via whose necessarily number-theoretical handling in all of geometry and in harmonics arithmetic binds together these other branches of mathematics. Knorr thinks that the Epinomis, which is strongly influenced by Plato’s thought, offers support to his view that in the Platonic conception of the unity of mathematics arithmetic unites all the others branches of this science with itself via its important role in the study of incommensurability.17 Mueller also sees in the Epinomis support for a reading of Plato’s view on the unity of mathematics similar to Knorr’s: he notes that Plato “is no doubt interested in the whole of mathematics, but his focus is strongly on the numerical component in all of its branches”, and explains that he “seems to be looking for a maximization of the role of number in the description of both mathematical (e.g. geometric) and cosmic (e.g. astronomical and musical) phenomena”.18 In the Epinomis, however, the conception of arithmetic as superior to geometry, both plane and solid, and harmonics, hence as their unifier, too, whether through its role in the study of incommensurability or not, leaves out astronomy: even if we extend this conception to accommodate astronomy, as Mueller suggests, in the Epinomis astronomy, or rather astronomical theology, absorbs dialectic un-Platonically and usurps its role as coping stone of mathematics in the Republic, so the accompanying view on the unity of all geometry and harmonics might be irrelevant to Plato’s own view on the issue in the Republic. Indeed, a plausible answer to the question of Plato’s view must be based, as said above, on the requirement that, no matter exactly where he sees the unity of mathematics, grasping it requires hard practice and time, since it is the goal of the decade-long intense mathematical studies of the future philosopher-rulers. But it is as a trivial fact that arithmetic is said in R. 7, 522c1–11, to be common to all thought, arts and sciences, the other four branches of mathematics in all probability included, so grasping such a trivial fact cannot be considered the ultimate goal of an intense and prolonged engagement with mathematics. As for 17 18

See Knorr (1975) 93–94. See Mueller (1991) 103–104.

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incommensurability, in the discussion of arithmetic and geometry in the Laws ignorance of this important phenomenon is said to be ridiculous and disgraceful, and the study of elementary arithmetic and geometry by all citizens is prescribed as necessary remedy of this ignorance (7, 819d5–820e7). How likely is it that in the Republic Plato views incommensurability, and its inevitable handling with the aid of arithmetic, as binding together the rest of mathematics, whose unity requires a long period of study to be understood? 2.3. PROPORTION-THEORY AS UNIFIER OF MATHEMATICS As said in 2.2.1, in all probability Archytas singles out not only arithmetic but also the theory of arithmetic ratios and proportions as far superior to geometry and the other branches of mathematics. We saw that, even if he described all branches of mathematics as members of a family because arithmetic unifies them, in Plato’s Republic the unity of mathematics does not seem to be conceived in such terms. Nor can proportion-theory be plausibly regarded as the bond that relates the other branches of the discipline to each other in Plato’s Republic. The notion that proportion binds together all branches of mathematics is ascribed to Eratosthenes of Cyrene by Proclus (in Euc. 43.22–44.9 Friedlein). This view on the unity of mathematics was probably put forward in Eratosthenes’ Platonicus as an attempt at accounting for the unity of mathematics in the Republic.19 Although ratios and proportions do appear abundantly in all branches of mathematics, a prolonged immersion in each one of them cannot be plausibly thought to culminate in the appreciation of this fact. Proclus rejects Eratosthenes’ view on the ground that proportion is only one among the features pervading mathematics. According to him, the immediate bond (προσεχὴς σύνδεσµος) of all branches of mathematics is a “single and universal mathematics” (ἡ µία καὶ ὅλη µαθηµατική): it contains the principles of each branch in simpler form, explains their common nature and their differences, which traits are the same in all of them and which belong to more or fewer of them. What exactly Proclus has in mind is unclear. Elsewhere he regards as the highest principles that pervade all mathematical objects the limit and the unlimited, and, after singling out proportion-theory as the body of theorems that apply to all mathematical objects, he goes on to say that beauty and order are also common to all branches of mathematics, as are proof and the methods called analysis and synthesis (in Euc. 7.13–8.20 Friedlein). A reference to the Republic shows that Proclus puts forth these views as Plato’s own, but their relevance to what Plato says in the Republic cannot be seriously entertained. The single bond of all branches of mathematics as Proclus conceives of it contains what in Metaph. E 1, 1026a23–27, and K 7, 1064b6–9, Aristotle describes as “universal mathematics” (ἡ καθόλου µαθηµατική). It belongs in common to all branches of the science, and 19

See Solmsen (1942) 193–195. For evidence on the Platonicus see Hiller (1870). That ratio and proportion unifies mathematics in R. 7 is argued in Fowler (1990) ch. 4, within a speculative reconstruction of Greek mathematics in Plato’s time, and Robins (1995) 387–389 (who does not accept Fowler’s reconstruction). Against Robins (1995) see Kouremenos (2004).

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its propositions apply to what is studied in all of them, be it numbers, i.e. discrete quantities, or magnitudes, i.e. continuous quantities (see Metaph. M 2, 1077a9–10 and 1077b17–22). In view of APo. A 5, 74a17–25, we can plausibly assume that Aristotle’s universal mathematics is a theory of proportion applicable to both basic types of mathematical objects, numbers and magnitudes, commensurable as well as incommensurable. Book 5 of Euclid’s Elements sets out such a proportiontheory, whose ancestor was a similarly general proportion-theory put forth by Eudoxus in the time of Plato and Aristotle.20 The abstractness of this theory would be attractive to Plato, but nothing suggests that the general theory of proportion unifies mathematics in his Republic. For a decade-long immersion in mathematics cannot be plausibly assumed to be required for the future philosopher-rulers to grasp the applicability of the general proportion-theory to each branch of this science. 2.4. MUTUAL BENEFIT AS UNIFIER OF MATHEMATICS In R. 7, 537b7–c8, Plato has Socrates refer to the final goal of the study of mathematics the future philosopher-rulers have to go through after they finish their elementary education as a general or comprehensive view of the kinship between the branches of mathematics and the nature of being (σύνοψις οἰκειότητός τε ἀλλήλων τῶν µαθηµάτων καί τῆς τοῦ ὄντος φύσεως). What is described here is in fact the goal reached by the future philosopher-rulers at the end of their study of both mathematics and philosophy. Even when they have successfully finished only their mathematical studies, they have grasped the kinship between the branches of mathematics and, although they do not yet realize it, the kinship between it and the nature of being, the subject-matter of philosophy whose study will then bring this implicit knowledge out. In all probability Plato here envisages a considerably tighter connection between mathematics and philosophy than the trivial one seen in the first section of this chapter (see also the final section of this chapter). As propedeutic to the study of philosophy, the study of mathematics aims at the comprehensive view of the kinship between the branches of mathematics because philosophers are comprehensive viewers by definition. This capability must, therefore, be tested and promoted during their propedeutic studies: Μετὰ δὴ τοῦτον τὸν χρόνον, ἦν δ’ ἐγώ, ἐκ τῶν εἰκοσιετῶν οἱ προκριθέντες τιµάς τε µείζους τῶν ἄλλων οἴσονται, τά τε χύδην µαθήµατα παισὶν ἐν τῇ παιδείᾳ γενόµενα τούτοις συνακτέον εἰς σύνοψιν οἰκειότητός τε ἀλλήλων τῶν µαθηµάτων καὶ τῆς τοῦ ὄντος φύσεως. Μόνη γοῦν, εἶπεν, ἡ τοιαύτη µάθησις βέβαιος, ἐν οἷς ἂν ἐγγένηται. Καὶ µεγίστη γε, ἦν δ’ ἐγώ, πεῖρα διαλεκτικῆς φύσεως καὶ µή· ὁ µὲν γὰρ συνοπτικὸς διαλεκτικός, ὁ δὲ µὴ οὔ. Συνοίοµαι, ἦ δ’ ὅς. “After this period,” I continued, “those of the twenty-year-olds who have been selected will be promoted over the others and must then bring together the disciplines they were taught 20

See Knorr (1978) and Mendell (2007).

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2. The unity of mathematics in the Republic as children in a disconnected manner so that they may gain a comprehensive view of the ties these disciplines have to one another and to the nature of what is”.21 “Indeed,” he said, “only those who have been taught thus have secure knowledge”. “And it is also the greatest test of whether one is naturally predisposed to philosophy or not,” I added. “It is the ability to form comprehensive views that marks off those who are philosophers from those who are not”. “I agree,” he said.

It has been suggested that the comprehensive view of the kinship between the branches of mathematics is to see them in the exact order in which Plato has Socrates and Glaucon discuss them in R. 7: arithmetic, plane geometry, solid geometry, astronomy and, finally, harmonics.22 This order reflects an increase in complexity: from what is extensionless to extended magnitude, from two to three dimensions and from immobility to motion. To see the kinship between the five branches of mathematics comprehensively means, on this view, to see them as a unified system, in which the prior and simpler form the basis for more elaborate developments. This view recalls in part Philip’s view in the Epinomis on the unity of mathematics, and to see in the prior and simpler branch of mathematics a foundation for those that are posterior and more complex, as this dependence is understood by the view under discussion, cannot be plausibly thought to require ten-year-long studies in mathematics. That Plato views the unity of mathematics in terms of a hierarchy of its branches, one of them, e.g. arithmetic, being somehow superior to the rest or each one being somehow superior to its successor in a sequence, seems to be supported by the fact that in the passage just translated he refers to a general view of the kinship between the branches of mathematics and philosophy. He certainly considers philosophy to be superior to mathematics, and likens it memorably to a cornice of a building or wall in R. 7, 534e2–535a2: Ἆρ’ οὖν δοκεῖ σοι, ἔφην ἐγώ, ὥσπερ θριγκὸς τοῖς µαθήµασιν ἡ διαλεκτικὴ ἡµῖν ἐπάνω κεῖσθαι, καὶ οὐκέτ’ ἄλλο τούτου µάθηµα ἀνωτέρω ὀρθῶς ἂν ἐπιτίθεσθαι, ἀλλ’ ἔχειν ἤδη τέλος τὰ τῶν µαθηµάτων; Ἔµοιγ’, ἔφη. “Do you agree, therefore,” I asked, “that dialectic occupies a higher position than the branches of mathematics, as if it were a cornice, and that we cannot rightly place any other subject still higher but have now completed our discussion of the curriculum?” “I agree,” he replied.

If a comprehensive view of the kinship between mathematics and philosophy focuses on their hierarchy, it is a reasonable assumption that in a comprehensive view of the kinship between the branches of mathematics, too, the focus is on their hierarchy, no matter how Plato might conceive of it: it is this hierarchy that turns them into a unified whole, thereby allowing one to view them comprehensively, just as is the case with mathematics as a unity and dialectic on the next and final 21 22

On the elementary mathematical education of the future philosopher-rulers see 536d4–537a3. See Gaiser (1986) 101 and Burnyeat (2000) 67–70.

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level in the ordering of disciplines which do not concern themselves with human opinions and desires, production and manufacture of goods as well as their care and maintenance (R. 7, 533a10–c6). Moreover, philosophers are comprehensive viewers in the sense that they are able to relate the forms which they study to the superordinate Good, so their study of mathematics can be reasonably assumed to prepare them for this by familiarizing them with a hierarchy between the branches of mathematics and the abstract objects studied in them, an image of the hierarchy existing between forms and the Good in philosophy. This assumption cannot be accepted, however, if we cannot specify a sense in which Plato might conceive of one branch of mathematics as superior to all the rest or of each one of them as superior to its successor in some appropriate order; at issue, moreover, must be a superiority which is similar to that of philosophy over mathematics and of the Good over forms in that it is not trivial but illuminating, which is why it and the unity imparted by it to mathematics need a long time to be grasped. Such a sense is not forthcoming. There is no direct evidence that Plato takes one branch of mathematics to be somehow illuminatingly superior to all the rest, or each one of them to be illuminatingly superior to its successor in a sequence. As for the order in which the fields of mathematics are discussed in R. 7, if it is indeed based on increase in complexity, from extensionless numbers to magnitudes in two and then three dimensions and from immobility to motion, then it does by no means reflect an illuminating hierarchy between the branches of mathematics comparable to that between philosophy and mathematics as a whole and between the Good and forms. Arithmetic, in particular, which has been thought to be superior to all the other branches of mathematics, is discussed by Plato first in all probability because it is common to all thought, arts and sciences, which must include the other four branches of mathematics, and also because one of the first things we learn is counting (522c1–11). This fact clearly does not raise arithmetic to a position of superiority in the requisite sense over the other four branches of mathematics, however. Nor does this sense obtain when one takes into account the central role of arithmetic in the study of incommensurability by other branches of mathematics. What this phenomenon shows is that arithmetic fails strikingly to be universally applicable to them: that is, if arithmetic is assumed to be illuminatingly superior to other branches of mathematics in that it is indispensible to their study of the phenomenon of incommensurability, these other branches, too, are superior to arithmetic in the same sense. Even if Plato views arithmetic as superior to the other branches of mathematics in some sense, incommensurability hints that the branches of mathematics studying it are bound together by their mutually beneficial contacts. That such contacts are exactly what the comprehensive view of the kinship between the branches of mathematics focuses on according to Plato is suggested, as will be seen next, by his conception of another unity, which is also of immense importance in the Republic, that of the soul. The three parts of the soul, the rational (λογιστικόν), the spitited (θυµοειδές) and the appetitive (ἐπιθυµητικόν), answer to the five branches of mathematics. The unity of the soul is, moreover, correlated with that of the city outlined in the Republic. Its parts are classes of its citizens, each corresponding to a part of the soul: the philosopher-rulers, the military class

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(the auxiliaries) and the productive class. As is expected, Plato’s conception of this unity, too, leads us to the same conclusion. 2.5. THE UNITY OF SOUL AND CALLIPOLIS AS MUTUAL BENEFIT OF THEIR PARTS How Plato conceives in the Republic of the unity of the tripartite soul sheds light on how he understands the unity of mathematics because the future philosopherrulers, whose mathematical studies must culminate in a comprehensive view of the kinship between the five branches of mathematics, will be just (7, 520a6–e3). But justice is defined in the Republic in view of the unity of one’s tripartite soul:23 studying mathematics for a decade so that the kinship of all its branches can be viewed at the end comprehensively must play a major part in unifying the tripartite souls of the future philosopher-rulers into just wholes, whose unity is thus most probably conceived of in the same manner as the unity of mathematics. Now, the rational part of the soul is unquestionably dominant. It rules over the spirited and the appetitive, but the unity of the soul does not lie in the forceful subjugation of its spirited and appetitive parts by its rational part but rather in the agreement by all three parts that one of them, the rational, must rule over the other two because it knows what is beneficial both for each one of them separately and for all three of them as a whole. It is exactly the mutual benefit of the parts of the soul that unifies them into a just whole, a bond transcending the polar opposition between dominant and dominated parts. This is clear from the definition of the brave, wise, temperate and just individual in R. 4, 442b10–d7: Καὶ ἀνδρεῖον δὴ, οἶµαι, τούτῳ τῷ µέρει καλοῦµεν ἕνα ἕκαστον, ὅταν αὐτοῦ τὸ θυµοειδὲς διασῴζῃ διά τε λυπῶν καὶ ἡδονῶν τὸ ὑπὸ τῶν λόγων παραγγελθὲν δεινόν τε καὶ µή. Ὀρθῶς γ’, ἔφη. Σοφὸν δέ γε ἐκείνῳ τῷ σµικρῷ µέρει, τῷ ὃ ἦρχέν τ’ ἐν αὐτῷ καὶ ταῦτα παρήγγελλεν, ἔχον αὖ κἀκεῖνο ἐπιστήµην ἐν αὑτῷ τὴν τοῦ συµφέροντος ἑκάστῳ τε καὶ ὅλῳ τῷ κοινῷ σφῶν αὐτῶν τριῶν ὄντων. Πάνυ µὲν οὖν. Τί δέ; σώφρονα οὐ τῇ φιλίᾳ καὶ συµφωνίᾳ τῇ αὐτῶν τούτων, ὅταν τό τε ἄρχον καὶ τὼ ἀρχοµένω τὸ λογιστικὸν ὁµοδοξῶσι δεῖν ἄρχειν καὶ µὴ στασιάζωσιν αὐτῷ; Σωφροσύνη γοῦν, ἦ δ’ ὅς, οὐκ ἄλλο τί ἐστιν ἢ τοῦτο, πόλεώς τε καὶ ἰδιώτου. Ἀλλὰ µὲν δὴ δίκαιός γε, ᾧ πολλάκις λέγοµεν, τούτῳ καὶ οὕτως ἔσται. Πολλὴ ἀνάγκη.

23

Justice is defined as each part’s doing its own job, whereas the unity of the soul, the agreement and concord between its parts, is identified with temperance; see the passage translated next on the virtues of the soul and cf. R. 4, 430d3–433b4, on the virtues for the city. Each part cannot do its own job, however, unless there is agreement and concord between them, and vice versa: the definitions of justice and temperance seem to collapse them into each other. Indeed, in R. 4, 443c9–444a2, quoted and translated below, justice is described first as each part’s doing its own job and then as the unity of all three parts and their agreement, though in 435b4–8 justice seems to be considered distinct from all the other virtues of a city.

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“Moreover, we describe someone as brave by virtue, I believe, of this part–that is, when his spirited part cleaves through pain and pleasure to what reason has ordered to be feared and not feared.” “Correct,” he replied. “Wise, of course, by virtue of that little part, the one that ruled in him and issued the orders in question knowing what is beneficial for each part and for the whole unity of all three of them.” “Absolutely.” “And don’t we describe someone as temperate by virtue of the friendship and agreement between these same parts–that is, when the ruling part and its twin subjects agree that the rational part must rule and there is no rebellion against it?” “Temperance,” he agreed, “is certainly nothing else than this, for both the city and the citizen.” “But someone will, of course, be just because of what we often mention and in this manner.”24 “Necessarily so.”

A few lines below, in 443c9–444a2, the unity of the soul is likened to a harmony, a single whole above the many differently pitched tones that are arranged into it: Τὸ δέ γε ἀληθές, τοιοῦτόν τι ἦν, ὡς ἔοικεν, ἡ δικαιοσύνη, ἀλλ’ οὐ περὶ τὴν ἔξω πρᾶξιν τῶν αὑτοῦ, ἀλλὰ περὶ τὴν ἐντός, ὡς ἀληθῶς περὶ ἑαυτὸν καὶ τὰ ἑαυτοῦ, µὴ ἐάσαντα τἀλλότρια πράττειν ἕκαστον ἐν αὑτῷ µηδὲ πολυπραγµονεῖν πρὸς ἄλληλα τὰ ἐν τῇ ψυχῇ γένη, ἀλλὰ τῷ ὄντι τὰ οἰκεῖα εὖ θέµενον καὶ ἄρξαντα αὐτὸν αὑτοῦ καὶ κοσµήσαντα καὶ φίλον γενόµενον ἑαυτῷ καὶ συναρµόσαντα τρία ὄντα, ὥσπερ ὅρους τρεῖς ἁρµονίας ἀτεχνῶς, νεάτης τε καὶ ὑπάτης καὶ µέσης, καὶ εἰ ἄλλα ἄττα µεταξὺ τυγχάνει ὄντα, πάντα ταῦτα συνδήσαντα καὶ παντάπασιν ἕνα γενόµενον ἐκ πολλῶν, σώφρονα καὶ ἡρµοσµένον, οὕτω δὴ πράττειν ἤδη, ἐάν τι πράττῃ ἢ περὶ χρηµάτων κτῆσιν ἢ περὶ σώµατος θεραπείαν ἢ καὶ πολιτικόν τι ἢ περὶ τὰ ἴδια συµβόλαια, ἐν πᾶσι τούτοις ἡγούµενον καὶ ὀνοµάζοντα δικαίαν µὲν καὶ καλὴν πρᾶξιν, ἣ ἂν ταύτην τὴν ἕξιν σῴζῃ τε καὶ συναπεργάζηται, σοφίαν δὲ τὴν ἐπιστατοῦσαν ταύτῃ τῇ πράξει ἐπιστήµην, ἄδικον δὲ πρᾶξιν, ἣ ἂν ἀεὶ ταύτην λύῃ, ἀµαθίαν δὲ τὴν ταύτῃ αὖ ἐπιστατοῦσαν δόξαν. “Justice is, it seems, really such a thing, though it pertains not to how the parts of a person’s own soul act externally but internally, to one’s own self and to one’s own parts who does not let each part in oneself do another’s job and prohibits the parts of one’s soul from interfering in one another’s job. This person has truly set his house in order and is a master of his own self, having arranged it well into a state of inner peace through the combination of the three parts, as if they really were the main sounds in a musical scale, the highest, the lowest and the middle.25 Having put together all of them, and any other sounds that happen to be between them, and turned from a multiplicity into a perfect unity, temperate and finely tuned, he then acts in this manner if he does something concerning the acquisition of property or the care of his body or in his capacity as an official or as a private citizen, everywhere considering and calling an act just and fine if it preserves and promotes this inner state, the knowledge that 24 25

“Because of what we often mention and in this manner” (ᾧ πολλάκις λέγοµεν, τούτῳ καὶ οὕτως) refers to each part’s doing its own job; cf. R. 4, 433a8–b4. For this harmonic simile cf. R. 4, 430d3–432b2. The relations of the middle to the highest and the lowest are the proportions 3 : 2 and 4 : 3. But the context leaves no doubt that proportion is an example of unity, not unity itself: in Ti. 31c2–32a7 Plato does declare that the best bond is proportion, for it effects in the best way the fullest unification of the things which are bound together, but he probably means that it is the best example of a unifying bond. Cf. next n.

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2. The unity of mathematics in the Republic presides over such an act being in his view wisdom, but unjust if it damages it, the belief that presides over such an act being for him ignorance.”

As the mutual benefit of the parts of the soul unifies them into a just whole, the mutual benefit of all citizens similarly unifies them into the community of a city. There is no worse evil for a city than what breaks it apart according to R. 5, 462a2– b3, and hence no greater good than what binds it together and makes it one. In R. 7, 519e1–520a5, shortly before the discussion of the mathematical curriculum for the future philosopher-rulers, what binds a city together is identified with the mutual benefit of all citizens, which is strongly contrasted with the disruptive advantage of one group of citizens at the expense of the others. This passage explains why the philosopher-rulers will not have any trouble coming back down to the dark cave from which they have escaped to get their hands dirty with ruling. To suspect that they will be unwilling to abandon their bliss is to forget that they cannot lose sight of the mutual benefit of all citizens, for they are just, i.e. have their souls unified by the common benefit of their parts, hence they know and cannot but do what is just, i.e. contribute to the common benefit of all citizens: Ἐπελάθου, ἦν δ’ ἐγώ, πάλιν, ὦ φίλε, ὅτι νόµῳ οὐ τοῦτο µέλει, ὅπως ἕν τι γένος ἐν πόλει διαφερόντως εὖ πράξει, ἀλλ’ ἐν ὅλῃ τῇ πόλει τοῦτο µηχανᾶται ἐγγενέσθαι, συναρµόττων τοὺς πολίτας πειθοῖ τε καὶ ἀνάγκῃ, ποιῶν µεταδιδόναι ἀλλήλοις τῆς ὠφελίας ἣν ἂν ἕκαστοι τὸ κοινὸν δυνατοὶ ὦσιν ὠφελεῖν καὶ αὐτὸς ἐµποιῶν τοιούτους ἄνδρας ἐν τῇ πόλει, οὐχ ἵνα ἀφιῇ τρέπεσθαι ὅπῃ ἕκαστος βούλεται, ἀλλ’ ἵνα καταχρῆται αὐτὸς αὐτοῖς ἐπὶ τὸν σύνδεσµον τῆς πόλεως. Ἀληθῆ, ἔφη· ἐπελαθόµην γάρ. “Once again you forgot, my friend,” I said, “that the law is not interested in how one class in the city will be better off than the others but contrives to impart this to the whole city, uniting the citizens by both persuasion and compulsion and making them share with one another the benefit each one of them might be able to offer to the community. The law produces for the city people who have this ability not in order for each one of them to be free to go in whatever direction they want but in order to use them to bind the city together.” “You’re right,” he replied. “I forgot it.”

It is not the dominant position of the philosopher-rulers itself in the city that binds together all of its classes into a unity, for ruling in a city is not a good desirable in itself, as is made clear in the following lines (520a6–d6), but it is unity that is the greatest good for a city. It resides in the mutual benefit of all citizens, and the rule of the philosopher-rulers is the function of one class of citizens serving the greatest good as much as does the function of each of the other two classes. The position of the passage just translated, close to the beginning of the discussion of the mathematical curriculum for the future philosopher-rulers, leaves no doubt as to why grasping the kinship between the branches of mathematics is thought to be of immense importance for a future philosopher-ruler and thus to constitute the ultimate goal of her or his mathematical preparation for the study of philosophy. Since philosophy is inextricably linked with ruling in a city and the rulers are assumed to be guided by the common benefit of all citizens that turns their city into a unified and thus just whole, a comprehensive view of the kinship

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between the branches of mathematics helps unify the tripartite souls of the future philosopher-rulers into just wholes not only for their own good but also, and much more importantly, for the good of the community over which they will rule. As seen above, what is good in both cases has a common nature, and if it is a reasonable assumption that a comprehensive view of the unity of mathematics offers an abstract paradigm of this nature, this view must necessarily focus on the common benefit of all branches of mathematics even if one of them is somehow superior to the rest. Viewing comprehensively how they are all united by their mutually beneficial links, none of them wronging or being wronged by another, will enable the future philosopher-rulers to first assimilate their own souls as much as possible to this eternally stable order and, when the time comes, to mold the entire life of the city to it. This is Socrates’ justification in R. 6 of his belief that only philosophers must rule, particularized to mathematics as sole propedeutic to philosophy (500b8–d10): Οὐδὲ γάρ που, ὦ Ἀδείµαντε, σχολὴ τῷ γε ὡς ἀληθῶς πρὸς τοῖς οὖσι τὴν διάνοιαν ἔχοντι κάτω βλέπειν εἰς ἀνθρώπων πραγµατείας, καὶ µαχόµενον αὐτοῖς φθόνου τε καὶ δυσµενείας ἐµπίµπλασθαι, ἀλλ’ εἰς τεταγµένα ἄττα καὶ κατὰ ταὐτὰ ἀεὶ ἔχοντα ὁρῶντας καὶ θεωµένους οὔτ’ ἀδικοῦντα οὔτ’ ἀδικούµενα ὑπ’ ἀλλήλων, κόσµῳ δὲ πάντα καὶ κατὰ λόγον ἔχοντα, ταῦτα µιµεῖσθαί τε καὶ ὅτι µάλιστα ἀφοµοιοῦσθαι. ἢ οἴει τινὰ µηχανὴν εἶναι, ὅτῳ τις ὁµιλεῖ ἀγάµενος, µὴ µιµεῖσθαι ἐκεῖνο; Ἀδύνατον, ἔφη. Θείῳ δὴ καὶ κοσµίῳ ὅ γε φιλόσοφος ὁµιλῶν κόσµιός τε καὶ θεῖος εἰς τὸ δυνατὸν ἀνθρώπῳ γίγνεται· διαβολὴ δ’ ἐν πᾶσι πολλή. Παντάπασι µὲν οὖν. Ἂν οὖν τις, εἶπον, αὐτῷ ἀνάγκη γένηται ἃ ἐκεῖ ὁρᾷ µελετῆσαι εἰς ἀνθρώπων ἤθη καὶ ἰδίᾳ καὶ δηµοσίᾳ τιθέναι καὶ µὴ µόνον ἑαυτὸν πλάττειν, ἆρα κακὸν δηµιουργὸν αὐτὸν οἴει γενήσεσθαι σωφροσύνης τε καὶ δικαιοσύνης καὶ συµπάσης τῆς δηµοτικῆς ἀρετῆς; Ἥκιστά γε, ἦ δ’ ὅς. “In fact, Adeimantus, one whose mind is truly fixed on beings has no time to look downwards at the affairs of people and be filled with envy and ill-will in his disputes with them but, always directing his gaze towards some things which are organized and always unchangeable, seeing them neither wronging nor being wronged by one another but 26 unexceptionally forming a rational order, he tries to imitate them and assimilate himself to them as much as possible. Or do you think it is possible for someone not to imitate whatever it is that in whose company he delights?” “It is impossible,” he replied. “As it is, since the philosopher is associated with what is divine and orderly, he becomes orderly and divine himself to the extent this is possible for humans. But all others slander him.” “Absolutely.” “If, therefore,” I continued, “he is compelled for some reason to apply what he sees there to human customs and bring it to both private and public affairs and not only mold himself to it, do you think he will turn out to be a bad craftsman of temperance, justice and all civic virtue?” “Certainly not,” he said.

How closely related to ruling is for Plato the comprehensive view of the unity of all mathematics, a template for harmonious order, is shown by Laws 12, 967d4–968a4: 26

κόσµῳ καὶ κατὰ λόγον ἔχοντα can be translated as “being ordered by proportions”, but the emphasis is on the rationality of a unifying order exemplified by proportion; cf. previous n.

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2. The unity of mathematics in the Republic ΑΘ. Οὐκ ἔστιν ποτὲ γενέσθαι βεβαίως θεοσεβῆ θνητῶν ἀνθρώπων οὐδένα, ὃς ἂν µὴ τὰ λεγόµενα ταῦτα νῦν δύο λάβῃ, ψυχή τε ὡς ἔστιν πρεσβύτατον ἁπάντων ὅσα γονῆς µετείληφεν, ἀθάνατόν τε, ἄρχει τε δὴ σωµάτων πάντων, ἐπὶ δὲ τούτοισι δή, τὸ νῦν εἰρηµένον πολλάκις, τόν τε εἰρηµένον ἐν τοῖς ἄστροις νοῦν τῶν ὄντων τά τε πρὸ τούτων ἀναγκαῖα µαθήµατα λάβῃ, τά τε κατὰ τὴν µοῦσαν τούτοις τῆς κοινωνίας συνθεασάµενος, χρήσηται πρὸς τὰ τῶν ἠθῶν ἐπιτηδεύµατα καὶ νόµιµα συναρµοττόντως, ὅσα τε λόγον ἔχει, τούτων δυνατὸς ᾖ δοῦναι τὸν λόγον· ὁ δὲ µὴ ταῦθ’ οἷός τ’ ὢν πρὸς ταῖς δηµοσίαις ἀρεταῖς κεκτῆσθαι σχεδὸν ἄρχων µὲν οὐκ ἄν ποτε γένοιτο ἱκανὸς ὅλης πόλεως, ὑπηρέτης δ’ ἂν ἄλλοις ἄρχουσιν. ATH. No mortal man can ever truly respect the gods who has not grasped these two truths now stated. First, soul is the most important of all things that come to be, and is also immortal and rules over all bodies. In addition to that, as we have often said now, he must also understand the rationality of the phenomena exhibited by the celestial objects as well as the subjects that necessarily come before their study and, having surveyed their harmonious associations, he must use them to arrange customs and laws in the same way, being also able to give an account of what has an account.27 One who is unable to achieve the above in addition to possessing the civic virtues could never become a competent ruler of the city as a whole but merely a servant for others who rule.28

2.6. THE UNITY OF MATHEMATICS AND THE MENO We can see now in a new light the kinship that according to Plato binds together the starting points of the odd and even numbers and the square, the angular plane figure he seems to have in mind in R. 6 when he presents Socrates illustrating the notion of a starting point in geometry with the three kinds of angle and the various figures. These hypotheses are thought to be akin to each other not only in that they are laid down as self-evident, though they are far from being so, no matter what the reasons for this might be. Of much greater interest is their kinship insofar as one is from arithmetic and the other from geometry, two different branches of mathematics, but appear next to each other in the same proof to the effect that the side and the diagonal of a square cannot but be incommensurable with each other, a proposition of fundamental importance for both geometry and arithmetic: arithmetic helps geometry with the study of a basic object, the square, and its help is returned at the same time by the recipient, for what is established about the square studied in geometry bears crucially on the numbers studied in arithmetic, which by dint of 27

28

The subjects coming before astronomy are arithmetic and geometry; cf. Lg. 7, 817e5–822d1, on the basic education of all citizens of the planned Cretan city whose laws are discussed in the work. The Athenian stranger here talks about the wise members of the nocturnal council, the highest governing body in the planned Cretan colony. Like the philosopher-rulers in the Republic, they will study higher mathematics (Lg. 7, 817e5–818a3). But in Lg. 12, 960e9–967d3, mathematics does not depend on dialectic, “giving an account of what has an account”, in the manner of R. 6– 7. For the inadequacy of virtue alone see Lg. 12, 963a1–964d9. The importance of mathematics for both the soul and society via governing is related to unity by Burnyeat (2000) 74–81, who has a different conception of the unity of mathematics, however, as pointed out in section 4 of this chapter; cf. n. 48 below.

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the geometrical result about the square turn out surprisingly to fail to describe everything. There is, therefore, strong evidence to assume that, as suggested above, incommensurability hints that the unity of mathematics resides not in the superiority of arithmetic over its other branches, for which arithmetic is necessary one way or the other, but rather in the mutually benefitting connections between its branches that can be fruitfully explored in discovering mathematical truths. As we saw in section 2.1, moreover, Plato might have written the section of R. 7 on stereometry intending his contemporary informed audience to recall a possible link between Eudoxus’ astronomical theory of homocentric spheres and his solution to the stereometric problem of doubling a cube. It is probable that here lies another hint that all branches of mathematics are akin to one another insofar as there are mutually benefiting connections between them. If in R. 6 Plato alludes to a case of basic concepts of one branch spearheading an important proof in another branch and, at the same time, revealing their own limited applicability, then in R. 7 he can very well hint at the solution to a problem in one branch that turns out to help tackle a problem in another. Doubling a cube allows us to make some progress towards understanding the phenomenon of retrogradation, a most puzzling aspect of planetary motion, which in turn allows a fuller understanding of the stereometric problem by providing a broader canvass against which it can be studied.29 If the prolonged engagement of the future philosopher-rulers with mathematics aims at their understanding and coming to appreciate the multiple connections between the branches of mathematics, the achievement of this goal understandably needs a long time, during which the future philosopher-rulers live deeply the life of mathematics, experiencing first-hand the often unexpected unifying relationships between its branches and seeing for themselves how one branch can be of benefit to another. In R. 7, 528b5–c7, Plato has Socrates envisage a director of mathematics in the state governed by the philosopher-rulers: s/he will focus attention on important problems awaiting solution, such as the doubling of a cube at the dramatic time of the work. The context is the explanation of the lack of progress in solid geometry: Διττὰ γάρ, ἦν δ’ ἐγώ, τὰ αἴτια· ὅτι τε οὐδεµία πόλις ἐντίµως αὐτὰ ἔχει, ἀσθενῶς ζητεῖται χαλεπὰ ὄντα, ἐπιστάτου τε δέονται οἱ ζητοῦντες, ἄνευ οὗ οὐκ ἂν εὕροιεν, ὃν πρῶτον µὲν γενέσθαι χαλεπόν, ἔπειτα καὶ γενοµένου, ὡς νῦν ἔχει, οὐκ ἂν πείθοιντο οἱ περὶ ταῦτα ζητητικοὶ µεγαλοφρονούµενοι. εἰ δὲ πόλις ὅλη συνεπιστατοῖ ἐντίµως ἄγουσα αὐτά, οὗτοί τε ἂν πείθοιντο καὶ συνεχῶς τε ἂν καὶ ἐντόνως ζητούµενα ἐκφανῆ γένοιτο ὅπῃ ἔχει· ἐπεὶ καὶ νῦν ὑπὸ τῶν πολλῶν ἀτιµαζόµενα καὶ κολουόµενα, ὑπὸ δὲ τῶν ζητούντων λόγον οὐκ ἐχόντων καθ’ ὅτι χρήσιµα, ὅµως πρὸς ἅπαντα ταῦτα βίᾳ ὑπὸ χάριτος αὐξάνεται, καὶ οὐδὲν θαυµαστὸν αὐτὰ φανῆναι. “There are two reasons for this,” I said. “First, since no city esteems it, due to its difficulty it is not studied strenuously and, second, those who work in it need a superintendent, without whom they are not going to get any results. But such a person is hard to emerge in the first place and, even if one did emerge, in the present situation those who seek these results would not listen, self-confident as they are. If the entire city esteems the field and acts as 29

Nothing hinges on whether Eudoxus started from stereometry and went on to astronomy, or the other way around. It should be noted here that, since the problem of cube-duplication is not solvable with straightedge and compass, it showcases the interaction of geometry and algebra.

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2. The unity of mathematics in the Republic superintendent, however, they will listen and the constant and assiduous pursuit of results will bear fruit. Even now, although most people disrespect this subject and denigrate it while its students cannot explain its usefulness, nonetheless it grows thanks to its beauty, and it will be no surprise if results in it do turn up.”

Since we have no evidence to assume that for some reason Plato would like to keep the teaching of mathematics to the future philosopher-rulers separate from the advancement of all branches of the discipline under the aegis of the director, the ten-year-long immersion of the future philosopher-rulers in mathematics seems to be devoted not only to learning but also to doing mathematics, from a certain point onwards, and trying to solve outstanding problems.30 Plato’s example of such a problem, the cube-duplication, suggests that their solutions are often nexuses of different branches of mathematics, whose deep intrinsic unity they thus bring out. This is not just because there exists a possible connection between Eudoxus’ solution to this stereometric problem and his astronomical theory of homocentric spheres. Menaechmus, of unknown origin, was one of the three mathematicians who solved the problem of doubling a cube in Plato’s time.31 He solved the problem through the intersection of two curves, either two parabolas or a parabola and a hyperbola, bringing out a connection between the study of solids such as the cube and of certain plane curves that at first glance could not have anything to do with solids.32 In bringing the intersection of two plane curves to bear on the solution to the cube-duplication problem, Menaechmus followed in the footsteps of Archytas, probably the first of Plato’s contemporary mathematicians to solve this problem, though his curves do not enjoy the fame of Menaechmus’.33 Menaechmus was probably the discoverer of the conic sections and the first to study some of their properties, and it is not unlikely that he did not just bring parabolas and hyperbolas to bear on the solution to the cube-duplication problem but, what is more, discovered them in his investigation of the problem, a discovery that immediately led him to a new solution to it: a link between two branches of mathematics, solid and plane geometry, might have provided both a solution to an outstanding problem in the former and at the same time a key that allowed passage from the former to the latter, from cubes to hitherto unsuspected curved lines, whose study was thus opened up. Once again, two branches of mathematics are kindred by virtue of a mutually benefitting connection: the theory of conic sections helps stereometry tackle a problem and stereometry reciprocates simultaneously by allowing the theory of conic sections to emerge through the mathematicians’ attempt to solve this problem.34 In R. 7, 528b5–c7, mathematics is 30

31 32 33 34

That Plato wants to keep the teaching of mathematics to future philosopher-rulers distinct from doing mathematics has been argued by Gregory (2000) 55–60 with respect to astronomy. On the importance of problems in mathematics see Atiyah (1985) 26–28 and Gray (2006) 3–5. Cf. ch. 1 n. 78 and 81. Cf. ch. 1.7.3. Cf. ch. 1.7.2. Plato’s proposal in R. 7, 531b2–c5, for the future harmonics that will serve as a propedeutic to philosophy entails its mutually benefitting connection with arithmetic; cf. Barker (1978). On the ways in which one branch of mathematics can benefit another see Gowers (2000) 76.

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said to grow through the pursuit of solutions to open problems, despite the absence of state support and despite public hostility, thanks to its beauty. The kinship of its fields is one strand constituting its beauty as internal motor of its growth. This is suggested by the parallel in the Meno between knowledge and recollection. Here the kinship of nature is explicitly assumed to be a precondition for the possibility of knowing, skepticism about which the parallel aims to remove (81c5–d5): Ἅτε οὖν ἡ ψυχὴ ἀθάνατός τε οὖσα καὶ πολλάκις γεγονυῖα, καὶ ἑωρακυῖα καὶ τὰ ἐνθάδε καὶ τὰ ἐν Ἅιδου καὶ πάντα χρήµατα, οὐκ ἔστιν ὅτι οὐ µεµάθηκεν· ὥστε οὐδὲν θαυµαστὸν καὶ περὶ ἀρετῆς καὶ περὶ ἄλλων οἷόν τ’ εἶναι αὐτὴν ἀναµνησθῆναι, ἅ γε καὶ πρότερον ἠπίστατο. ἅτε γὰρ τῆς φύσεως ἁπάσης συγγενοῦς οὔσης, καὶ µεµαθηκυίας τῆς ψυχῆς ἅπαντα, οὐδὲν κωλύει ἓν µόνον ἀναµνησθέντα – ὃ δὴ µάθησιν καλοῦσιν ἄνθρωποι – τἆλλα πάντα αὐτὸν ἀνευρεῖν, ἐάν τις ἀνδρεῖος ᾖ καὶ µὴ ἀποκάµνῃ ζητῶν· τὸ γὰρ ζητεῖν ἄρα καὶ τὸ µανθάνειν ἀνάµνησις ὅλον ἐστίν. Since, therefore, the soul is immortal and has been born many times, it has seen what is here and in Hades and all things, and there is nothing it has not learned. Consequently, it is not surprising if it is able to recall what, of course, it also knew in the past, about virtue and about other things. For, since all nature is akin and the soul has learned everything, nothing prevents a person, if he has recalled a single thing, which people call “learning”, from discovering all other things by himself, provided that he is courageous and does not get tired in his search. All inquiry and learning is thus recollection.

As said above, Plato’s brief comment in R. 7 on the undeveloped state of solid geometry at the dramatic time of the dialogue was probably intended to lead those among his contemporary audience who were aware of the developments in mathematics after the dramatic time of the work to recall the progress of solid geometry in their time as exemplified by the solutions to the cube-duplication problem. In the Meno it is the plane analogue of this problem, doubling a square, towards “remembering” the solution to which a slave-boy is guided by Socrates in an experiment that illustrates the possibility of knowing (82b9–85b7). We have, therefore, two contexts, one in the Meno and one in R. 7, where Plato’s concern is with the possibility of obtaining knowledge, and where the piece of knowledge used as example is the solution to a certain problem in two dimensions and to its three-dimensional analogue respectively. But the kinship of all nature is in the Meno a precondition for the possibility of knowing, and, in view of the parallel in the Meno between acquisition of knowledge and recollection, the kinship of all nature lies in the interconnections between the facets of nature that allow us to pass from knowing one of those facets to knowing others, just as something reminds us of something else, which is related to it and is thus recalled by us. Hence a kinship between some things must be considered to be a precondition for the possibility of the growth of mathematical knowledge in the related context in R. 7, too, and must also be understood as lying in the interconnections between these things. But relevant to Plato’s implicit comment in R. 7 about the advances in stereometry after the dramatic time of the dialogue can only be, in view of the rest of R. 7, the kinship between the branches of mathematics, hence their kinship must lie in the interconnections that tie them together.

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2.7. TWO MODERN MATHEMATICIANS ON THE UNITY OF MATHEMATICS It should be emphasized that the view on the unity of mathematics attributed to Plato here does not arise from a particular philosophical attitude to mathematics, Platonic or not, but from within mathematics itself. The unity of mathematics so conceived was the topic Sir Michael Atiyah discussed in his presidential address to the London Mathematical Society on 19 November 1976: I want…to use this occasion to express my personal attitude to mathematics, but to do this by way of simple example rather than by philosophical generalities. The aspect of mathematics which fascinates me most is the rich interaction between its different branches, the unexpected links, the surprises, and my aim will be to illustrate this by considering some simple problems.35

The example presented in the talk was the interaction of number theory, geometry and analysis as illustrated by the connection between failure of unique factorization in a ring, the Möbius band and integro-differential equations, a connection from which a great wealth of results has sprung in topology, algebraic and differential geometry and functional analysis through the use of analogies: The main theme of my lecture has been to illustrate the unity of mathematics by discussing a few examples that range from Number Theory through Algebra, Geometry, Topology and Analysis. This interaction is, in my view, not simply an occasional interesting accident, but rather it is of the essence of mathematics. Finding analogies between different phenomena and developing techniques to exploit these analogies is the basic mathematical approach to the physical world. It is therefore hardly surprising that it should also figure prominently internally within mathematics itself. I feel that this needs to be emphasized because the axiomatic era has tended to divide mathematics into specialist branches, each restricted to developing the consequences of a given set of axioms. Now I am not entirely against the axiomatic approach so long as it is regarded as a convenient temporary device to concentrate the mind, but it should not be given too high a status.36

Jean Dieudonné has expressed similar thoughts: As for the old classification of mathematics into Arithmetic, Algebra, Geometry and Analysis, it has become as out-of-date as the divisions of the ‘animal kingdom’ by the early naturalists into species grouped according to fortuitous and superficial resemblances. Modern mathematical objects appear as centers where surprising combinations of many diverse structures come to converge. A typical modern mathematical paper will unfold as follows: The aim is to demonstrate that a certain group occurring in number theory (and, more precisely, in the arithmetic theory of roots of unity) is finite. One begins by interpreting this group using homological algebra, which reduces the theorem to be proved to a result concerning the cohomology of certain discrete subgroups of a Lie group: and finally this last result is obtained by calling upon E. Cartan’s theory of symmetric spaces and Hodge’s theory of harmonic forms. It started with ‘Arithmetic’, and then passed via ‘Algebra’ to end up ultimately in ‘Geometry’ and ‘Analysis’! One could give many other analogous examples, showing conclusively that the old conceptions can only be intolerable fetters in the understanding of mathematics today.37 35 36 37

Atiyah (1978) 69. Atiyah (1978) 75–76. See also Atiyah (1985) 31–32 and cf. Atiyah (2006). In Lautman (2011) xlii.

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2.8. ALBERT LAUTMAN ON THE UNITY OF MATHEMATICS The passage just quoted is from Dieudonné’s introduction to the 1977 edition of the works of the short-lived French philosopher Albert Lautman. It comments on Lautman’s views in his essay on the unity of mathematics.38 Lautman starts from a distinction drawn by H. Weyl in the introduction to his Group Theory and Quantum Mechanics between two different kinds of modern mathematics: one which is unlike Greek mathematics because it privileges arithmetic over geometry and culminates in complex analysis, and a more recent one which resembles Greek mathematics in that it is concerned with distinct domains, i.e. abstract algebra with its groups and fields, which resembles Greek mathematics with its two distinct domains of objects under study, numbers and continuous magnitudes. According to Lautman, Weyl misdiagnoses an accident of historical development as an essential split of mathematics into two irreducibly distinct kinds: what is essential to mathematics is its profound unity evident in the penetration of analysis, the domain of the continuous and the infinite, by techniques imported from algebra, the domain of the discontinuous and the finite. (It should be noted that analytic tools are imported into the branch of number theory called for this reason “analytic”, hence to algebra, too, which overlaps with number theory.39 ) The Lautmanian unity of mathematics resides exactly in the relations between the continuous and the discontinuous, the finite and the infinite: Lautman calls these relations abstract ideas and what they relate notions, pairs of polar opposites whose tension is resolved by the ideas as they become concretized in theories throughout mathematics thereby structuring and unifying it. Lautman’s “ideas” do not seem to have more in common with Platonic forms than their name, which is a transliteration of a Greek term for forms, but Lautman himself does not think so, arguing that in its true Platonic sense this term refers to what he calls structural schemes, according to which mathematical theories are organized and unified. This is dubious, however, as is evident from his attempt to relate his philosophy of mathematics to Plato through the work of Becker and Stenzel on the one and the indefinite two as the generators of formnumbers and forms in Plato’s unwritten doctrines (whereby all forms turn out to be what Lautman calls ideas, relations between poles of an opposition that structures the domain of forms).40 Lautman, moreover, elaborates the genesis of mathematics from ideas in Heideggerian terms.41 What is central to this interesting philosophy of mathematics, and can undoubtedly be traced back to the historical Plato, is the emphasis on the unity of mathematics as the essential feature of the discipline that resides in the various interesting points of contact among its diverse and seemingly unconnected fields. Having demonstrated that two distinct types of decomposition, each of which can be thought characteristic of only one of Weyl’s two kinds of modern mathematics, are applicable to the same entity in functional analysis Lautman concludes thus: 38 39 40 41

Lautman (2011) 45ff. On Lautman’s philosophy of mathematics see Petitot (1987). On analytic number theory see Lautman (2011) 207–219. Lautman (2011) 189–191. Lautman (2011) 199–206.

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2. The unity of mathematics in the Republic The examples that have been given thus permit us to understand that if there are different ways of thinking in mathematics, it is very unlikely that these differences in method correspond to differences of domain. The duality of the types of decomposition which have been stressed throughout this chapter is a certain fact that asserts itself for any observer, but this duality of methods does not end up constituting two different mathematics, that which would be a promotion of the arithmetic of whole numbers, and that which would be an extension of algebra. The same entities are able to be studied in both ways and it is this encounter of methods that gives rise to the profound unity of mathematics.42

Atiyah, Dieudonné and Lautman use examples from advanced modern mathematics to illustrate a conception of the essential unity of this discipline that Plato seems to have been the first to form, though the mathematics of his day could provide scant support, divided as it broadly was into arithmetic, which studied discontinuous numbers, and geometry, which dealt with continuous magnitudes. Arithmetic and geometry touched only insofar as magnitudes could be treated like numbers, i.e. in the case of commensurable magnitudes, hence the arithmetical component in the study of incommensurability, based on which Knorr, as said above, sees in the Platonic unity of mathematics the penetration of geometry by arithmetic. On the view proposed in this chapter, the Platonic unity of mathematics, as exemplified by the study of incommensurability, resides not in the penetration of geometry by arithmetic but rather in their mutually profitable interaction. Whatever view on the unity of mathematics one prefers to attribute to Plato, there can be no denying the fact that the mathematics of his day would have offered only insignificant evidence for it: the isolated example of incommensurability does not compare with the extensive range of the examples from modern mathematics that are marshaled by Lautman for the permeation of analysis, the study of the continuous, by tools of algebra, the study of the discontinuous, or those that can be adduced for the interplay of these two and other branches of mathematics. Aristotle, who was the first to put forth the formalist conception of mathematics as split into specialist branches, with which Atiyah contrasts the view of the discipline as a unity, is much closer than Plato to the situation of his time when he denies any contact between arithmetic and geometry except in the case of commensurable magnitudes. Aristotle’s view on the interrelationship between arithmetic and geometry, the two main branches of Greek mathematics, is found in APo. A 7, 75a38–b20: Οὐκ ἄρα ἔστιν ἐξ ἄλλου γένους µεταβάντα δεῖξαι, οἷον τὸ γεωµετρικὸν ἀριθµητικῇ. τρία γάρ ἐστι τὰ ἐν ταῖς ἀποδείξεσιν, ἓν µὲν τὸ ἀποδεικνύµενον, τὸ συµπέρασµα (τοῦτο δ᾽ ἐστὶ τὸ ὑπάρχον γένει τινὶ καθ᾽ αὑτό), ἓν δὲ τὰ ἀξιώµατα (ἀξιώµατα δ᾽ ἐστὶν ἐξ ὧν)· τρίτον τὸ γένος τὸ ὑποκείµενον, οὗ τὰ πάθη καὶ τὰ καθ᾽ αὑτὰ συµβεβηκότα δηλοῖ ἡ ἀπόδειξις. ἐξ ὧν µὲν οὖν ἡ ἀπόδειξις, ἐνδέχεται τὰ αὐτὰ εἶναι· ὧν δὲ τὸ γένος ἕτερον, ὥσπερ ἀριθµητικῆς καὶ γεωµετρίας, οὐκ ἔστι τὴν ἀριθµητικὴν ἀπόδειξιν ἐφαρµόσαι ἐπὶ τὰ τοῖς µεγέθεσι συµβεβηκότα, εἰ µὴ τὰ µεγέθη ἀριθµοί εἰσι· τοῦτο δ᾽ ὡς ἐνδέχεται ἐπί τινων, ὕστερον λεχθήσεται. ἡ δ᾽ ἀριθµητικὴ ἀπόδειξις ἀεὶ ἔχει τὸ γένος περὶ ὃ ἡ ἀπόδειξις, καὶ αἱ ἄλλαι ὁµοίως. ὥστ᾽ ἢ ἁπλῶς ἀνάγκη τὸ αὐτὸ εἶναι γένος ἢ πῇ, εἰ µέλλει ἡ ἀπόδειξις µεταβαίνειν. ἄλλως δ᾽ ὅτι ἀδύνατον, δῆλον· ἐκ γὰρ τοῦ αὐτοῦ γένους ἀνάγκη τὰ ἄκρα καὶ τὰ µέσα εἶναι. εἰ γὰρ µὴ καθ᾽ αὑτά, συµβεβηκότα ἔσται. διὰ τοῦτο τῇ γεωµετρίᾳ οὐκ ἔστι δεῖξαι ὅτι τῶν ἐναντίων µία ἐπιστήµη, ἀλλ᾽ οὐδ᾽ ὅτι οἱ δύο κύβοι κύβος· οὐδ᾽ ἄλλῃ 42

Lautman (2011) 59. On the unity of mathematics as conceived by Lautman see Petitot (1982).

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ἐπιστήµῃ τὸ ἑτέρας, ἀλλ᾽ ἢ ὅσα οὕτως ἔχει πρὸς ἄλληλα ὥστ᾽ εἶναι θάτερον ὑπὸ θάτερον, οἷον τὰ ὀπτικὰ πρὸς γεωµετρίαν καὶ τὰ ἁρµονικὰ πρὸς ἀριθµητικήν. οὐδ᾽ εἴ τι ὑπάρχει ταῖς γραµµαῖς µὴ ᾗ γραµµαὶ καὶ ᾗ ἐκ τῶν ἀρχῶν τῶν ἰδίων, οἷον εἰ καλλίστη τῶν γραµµῶν ἡ εὐθεῖα ἢ εἰ ἐναντίως ἔχει τῇ περιφερεῖ· οὐ γὰρ ᾗ τὸ ἴδιον γένος αὐτῶν, ὑπάρχει, ἀλλ᾽ ᾗ κοινόν τι. It follows that it is impossible to prove something by passing to it from another kind, e.g. to prove a geometrical truth with arithmetic. For there are three elements in demonstration: what is proved, the conclusion (which is an attribute belonging to a kind in itself); the axioms (which are premises of the proof); third, the underlying kind whose attributes and properties that hold of it in itself are revealed by the demonstration. The axioms, which are premises of demonstration, may be identical in two or more sciences: in the case of two different kinds such as arithmetic and geometry, however, you cannot fit arithmetical demonstration to the attributes of magnitudes, unless the magnitudes in question are numbers; how this is possible in certain cases I will explain later.43 Arithmetical proof always has its own kind, and so do the proofs in the other sciences. Thus, if a proof is to cross from one science to another, the kind must be the same absolutely or to some extent. Otherwise crossing is evidently impossible since the extreme and the middle terms must come from the same kind;44 for, if they do not hold in themselves, they hold incidentally. This is the reason why geometry cannot prove that opposites fall under a single science, nor even that two cubes make a cube;45 nor can a theorem of one science be proved by means of another science, unless they are related so that one falls under the other, as optical theorems are related to geometry or harmonic theorems to arithmetic. Nor can geometry prove of lines any property which they do not possess as lines and as depending on the principles proper to them, for example that the straight line is the most beautiful of lines or the contrary of curves, as these things do not belong to lines in virtue of their proper kind but in virtue of something common.

Unlike Plato, Aristotle speaks not of the affinity and kinship between two branches of mathematics but of a proof from one fitting, or being accommodated by, the other or the objects studied in it. A proof about the properties of the kind (γένος) of objects studied in a branch of mathematics does not cross over to, or fit, a different kind of objects studied in another branch, unless the objects in the second kind are in a sense those belonging to the first kind: magnitudes, the kind of objects studied in geometry, are like numbers, the kind of objects studied in arithmetic, insofar as they are commensurable:46 proofs from arithmetic cross over to geometry to the limited extent that magnitudes are commensurable, and, apart from the trivial case of one proof in arithmetic and another in geometry starting off from the same common principles (ἐξ ὧν µὲν οὖν ἡ ἀπόδειξις, ἐνδέχεται τὰ αὐτὰ εἶναι), there can be no other link between arithmetic and geometry. All Aristotle sees in the arithmetical proof he mentions in APr. A 23, 41a21–32, to the effect that the side and diagonal of the square are not commensurable, is thus the failure of arithmetic to fit geometry and not a mutually beneficial interaction of these two branches of mathematics, which is exactly what Plato sees, as argued above. In this proof Aristotle sees the barrier between geometry and arithmetic but Plato a bridge 43 44 45 46

Cf. APo. A 9, 76a4–15, and 13, 78b34–79a16. Cf. APo. A 9, 76a4–9. Euc. El. 9.4. Pace Barnes (19932) 131 Aristotle’s “unless the magnitudes in question are numbers” (εἰ µὴ τὰ µεγέθη ἀριθµοί εἰσι) is not a jocular hypothesis; cf. Heath (1956) vol. 2, 113.

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between the two that is much more interesting than the trivial treatment of geometrical magnitudes as numbers, for it is the breaking down of the application of already established results about numbers to magnitudes that revealingly delimits these kinds of mathematical entities: numbers do not wear their difference from geometrical magnitudes on their sleeves but reveal it with the help of a geometrical object, the square, while elucidating the relationship between its side and diagonal. Aristotle in all probability did appreciate this Platonic point of view, but the formalist conception he pioneered of each branch of mathematics as an axiomatic system on its own is unable to capture the interaction between two or more of them, or e.g. stereometrical spinoffs of astronomy and plane-geometrical spinoffs of solid geometry. Nevertheless he is closer to the situation in the mathematics of his time than Plato, whose conception of the unity of all mathematics as understood here is more appropriate to modern mathematics and was barely noticeable in his time, testifying to the power of his vision if he did hold it. 2.9. MATHEMATICS AND PHILOSOPHY As seen in 2.4, in R. 7, 537b7–c8, Plato has Socrates refer to the final goal of the study of mathematics the future philosopher-rulers have to go through after the end of their elementary education as a comprehensive view of the kinship between the branches of mathematics and the nature of being. Unlike in R. 7, 531c9–d5, what is described here by Plato is the goal reached by the future philosopher-rulers at the end of their study of mathematics and philosophy: it thus seems that at the end of their successful completion of their mathematical studies they have already grasped the kinship between the branches of mathematics and, although they do not yet realize it, that between mathematics and being, the subject of philosophy whose subsequent study will draw out this implicit knowledge. As suggested in 2.4, Plato probably envisages here a tighter kinship between mathematics and dialectic than the one residing solely in the intelligible nature of their objects, whereby the approach of mathematics to the objects under study is different from but also inferior, and only propedeutic, to that which is peculiar to philosophy and also able to somehow make the objects studied in mathematics fully knowable (R. 6, 510b2–511d5). For, if all branches of mathematics are akin to one another in the sense that there are a great many kinds of mutually beneficial and illuminating connections among them, does this not suggest that the kinship between mathematics and philosophy is to be similarly understood? It must lie not in the intelligibility of what they both deal with, important though this is, and in the epistemic dependence of mathematics on philosophy, but in their mutually beneficial interdependence. Philosophy renders the objects studied in mathematics fully knowable by rising above the problematic hypotheses used in mathematics as definitions of its objects and relating the latter to a starting point that must be the Good (R. 6, 511c3–e5). But, no matter how this foundational service dialectic renders to mathematics is to be understood, dialectic seems to duly reciprocate for the service that has already been rendered to it by mathematics. As seen in 2.5, what is good for the city and the soul is unity, the

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interrelations binding together distinct relata for the common benefit, so the most important object of study in philosophy, the Good, is to be identified with a purely intelligible unity. It must be from mathematics, however, that this notion has been bequeathed to philosophy: for what else might Plato mean by having the future philosopher-rulers study for ten long years mathematics in order to see the kinship and affinity between its branches if this abstract unity is not somehow crucially linked not only to ruling, as seen above, but also, through it, to the highest object of dialectic, which in the Republic is the science of governing? Dialectic elucidates the true nature of the objects studied in mathematics just as mathematics itself seems to reveal to dialectic the nature of the highest object it, dialectic, studies, thus enabling it to pay back its foundational debt: a cornice crowns a wall or a building as much as is supported by it. The intimate connection between the Good and the unity of mathematics is suggested by R. 7, 531c9–d5, the passage quoted at the beginning of this chapter. In these lines Plato has Socrates declare to Glaucon that the study of all branches of mathematics has some relevance to their purposes and is not wasted effort if it leads to grasping the affinity and kinship between the subjects in question and to understanding how they are related to one another. This observation comes after Glaucon has agreed with Socrates that, when harmonics will have sufficiently advanced in the future to serve dialectic propedeutically, it will be of use in the attempt to understand the Good and beauty (R. 7, 531b2–c8). Thus the purposes at issue here are the dialectical goal of understanding the Good and beauty. Harmonics might be thought to be more relevant to this goal than the other branches of mathematics, given the aesthetic and moral content of music, whose scales are the empirical subject studied mathematically in harmonics. Socrates is made to warn his codiscussant, however, that all branches of mathematics can aid the dialectical effort to understand the Good and beauty, provided that their intense study aims at bringing out and highlighting their own kinship. Thus the unity of all branches of mathematics must be closely linked with the Good and beauty. When Plato has Socrates declare to Glaucon that the study of all branches of mathematics has some relevance to their purposes, this is a rhetorical understatement: for nothing seems to be more relevant to the dialectical study of the Good and beauty than the unity of mathematics revealed by the study of all its branches. The proposal that in the Republic the Good is somehow intimately related to the unity of mathematics is supported by Aristoxenus’ report of Aristotle’s oral testimony about a public lecture Plato delivered on the Good. To the surprise of the audience, who expected to hear a talk on things commonly thought to be good such as wealth, health and power, Plato talked on mathematics, numbers, geometry and astronomy, and at the end he identified the Good with unity. Presumably this unity is not that of all branches of mathematics, examples of which are geometry and astronomy, but rather the ontological unity that is adumbrated by the unity of mathematics, i.e. the unity of the intelligible entities studied in mathematics. The Good can be plausibly understood as the structure of the abstract universe of the forms, all of them fitting together into a harmonious unity as perfectly good parts of

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it (cf. Grg. 503d6–504a4 and Phd. 98a6–b3).47 Unless this universe contains only forms of mathematical objects, then in his lecture about which Aristoxenus informs us Plato touched upon a part of the Good only and extolled mathematics as the sole tool we have to study it philosophically in its entirety (Harm. 39.8–40.4 da Rios): καθάπερ Ἀριστοτέλης ἀεὶ διηγεῖτο τοὺς πλείστους τῶν ἀκουσάντων παρὰ Πλάτωνος τὴν περὶ τἀγαθοῦ ἀκρόασιν παθεῖν· προσιέναι µὲν γὰρ ἕκαστον ὑπολαµβάνοντα λήψεσθαί τι τῶν νοµιζοµένων τούτων ἀνθρωπίνων ἀγαθῶν οἷον πλοῦτον, ὑγίειαν, ἰσχύν, τὸ ὅλον εὐδαιµονίαν τινὰ θαυµαστήν· ὅτε δὲ φανείησαν οἱ λόγοι περὶ µαθηµάτων καὶ ἀριθµῶν καὶ γεωµετρίας καὶ ἀστρολογίας καὶ τὸ πέρας ὅτι ἀγαθόν ἐστιν ἕν, παντελῶς οἶµαι παράδοξόν τι ἐφαίνετο αὐτοῖς, εἶθ’ οἱ µὲν ὑποκατεφρόνουν τοῦ πράγµατος, οἱ δὲ κατεµέµφοντο. This is what, as Aristotle used to report, happened to most of those who listened to Plato’s lecture on the Good. Everyone came assuming they would then manage to acquire one of those things considered by people to be good such as wealth, health, power, in one word a marvelous happiness. But when the lecture turned out to concern mathematics, numbers, geometry and astronomy and concluded that unity is the Good, it seemed to them, I think, completely absurd, and some laughed at it while others raised objections.48

Reports of Plato’s unwritten doctrines are generally suspect, but here we do not read something that cannot also be read off from the few comments in the Republic about the unity of all mathematics.49 In the Republic Plato is reluctant to have Socrates talk concretely about the nature of the Good (see 6, 504e6–505b3 and 506d5–e1). Since there is no doubt as to its intelligibility, however, in view of the explicit characterization of the good for a soul and a city as a unity, the value Plato accords to the intelligible unity of mathematics for the training of future philosopher-rulers suggests the identification, partially at least, of what this unity is rooted in, i.e. the kinship of the abstract objects studied in mathematics, with the Good. If, moreover, the unity of mathematics lies in the mutually illuminating interactions of its branches and access to the Good is granted to us only through these links, we can see why Plato has Socrates deny sufficient knowledge of the Good (R. 6, 504e6–505b3): the requisite connections, as said above, remained few and far between well after the dramatic time of the Republic. As already said, Plato has Socrates regard not only stereometry and astronomy but also harmonics as not 47

48

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See Fine (1999) 228–229. Plato often speaks of the Good as a form (see e.g. R. 6, 508e3) but it is clearly not a form like the others (see R. 6, 509b6–c2). Not all things called forms seem to be the same; cf. below n. 50. For various views on the Good see Schindler (2008) 107–117. τὸ πέρας can be taken as part of what Plato said, not an adverbial expression. If so, τὸ πέρας ὅτι ἀγαθόν ἐστιν ἕν identifies the Good with a unifying limit: Sedley (2007) 270 translates it “that the Good is a unification of limit” and rejects the view that the Good is unity arguing that it is “an ideal proportionality, intelligible only through the conceptual framework of a highlevel mathematics” (for a similar view see Burnyeat [2000] 74–81 who thinks that the Good is unity generated as well as sustained by concord, attunement and proportion, all of which are fundamental to aesthetics and mathematics). If it is only through such a framework that the Good is intelligible, however, since in the Republic this framework cannot be anything other than the unity of mathematics, the Good must be unity, and there is no reason to think that this unity is restricted to proportionality; see section 3 of this chapter. On the Good as unity see also Arist. EE A 8, 1218a24–30.

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sufficiently developed to be of propedeutical service to philosophy. Most probably he did not think that this was true only of the dramatic time of the dialogue, and we can assume that he did not regard contemporary arithmetic and plane geometry as developed to an adequately high degree either, though we have no evidence for this. But if all or, at least, most branches of mathematics cannot serve dialectic propedeutically because of their undeveloped state, this must be so for no other reason than their insufficient exhibition of the unity of mathematics. Only future mathematics will have progressed sufficiently for a great multitude of mutually beneficial interconnections between its branches to have emerged so as to prepare their students for their first and partial encounter with the Good that will begin to become accessible to the philosophy of their time. Alternatively, Plato might well think that the universe of forms contains only those of mathematical objects, and in his lecture about which Aristoxenus informs us he might have discussed the Good in its entirety. In the Republic he seems to think that dialectic does not have more forms as special objects of study than those corresponding to the objects investigated in mathematics: it is distinguished from mathematics only by its characteristic approach to the same objects as those that are studied in mathematics (see ch. 1.1 and 5.2). If so, the Good is the unity of the forms of mathematical objects, which is shown by the intrinsic unity of mathematics, and thus the connection of mathematics and dialectic becomes even closer. Of the forms presupposed in the central books of the Republic, those of the half, double (and thus triple etc.), large and small need not perhaps be understood as different from the forms of numbers and those of “geometrical” objects, however the latter might be thought to admit of size-comparisons (see ch. 1.1).50 If, moreover, the Good is unity, then Plato’s definition of the virtues in the Republic in terms of unity raises naturally the question why we need forms of justice and all the other virtues apart from the Good: Plato’s tacit answer might well be that there is a single form of all virtues, the Good, in which case there are no other forms than those of mathematical objects if the unity of the latter is the Good. As for the form of beauty, an exquisite beauty, unsurpassed by its sensible manifestations in the sky as well as by anything sensible in general, is attributed by Plato in R. 7, 529c6–530a2, to the intelligible mathematical objects that will be revealed by astronomy in the future, when this science will have advanced enough to aid philosophy propedeutically: the form of beauty can thus easily be identified with the Good if this is the unity of all forms of mathematical objects.51 Finally, his discussion of astronomy and harmonics in R. 7 can 50

51

What is half, double, triple etc. with regard to something else can be assumed to participate in a form-number standing in a certain order-relation to the form-number in which the other thing participates (there can be nothing apart from form-numbers to be considered as the half itself, the double itself and the triple itself, for there can be no form of n/2, 2n, 3n etc. since there is no form of number; see Arist. Metaph. B 3, 999a6–10). The forms of “geometrical” objects are required for size-comparisons, such as that of the diagonal and the side of a square, that cannot be expressed as ratios of whole numbers. There is no need to postulate forms of largeness and smallness distinct from the form-numbers, or forms of “geometrical” objects, participated in by the larger and smaller of any two things under comparison. Cf. R. 7, 531b2–c8, discussed above in this section, and 528b5–c7, discussed in 2.6. Beauty and mathematics seem to have an interesting neurobiological connection; see Zeki et al. (2014).

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be plausibly understood to suggest that all forms corresponding to the fundamental contents and properties of the physical world are of mathematical objects, a position perhaps adopted in the Timaeus, where there is no need for distinct forms of the elements (see Ti. 48e2–52d1) from those of four regular solids, which can also stand for the forms of heaviness and lightness as well as for those of hardness and softness (see Ti. 62b6–63e8).52 Needless to say, the notoriously generous range of forms in R. 10, 596a6–b3, does not square with the view that all of them are forms of mathematical objects, unlike the more limited range of forms that seems to be assumed in Plt. 262a8–263a1 and Phdr. 265e1–3. If Plato does think that the universe of forms is populated only by those of mathematical objects and that the Good is the unity of this universe, this need by no means imply his belief in the future coalescence of advanced mathematics and philosophy, although he might well have envisaged a heavily mathematized branch of philosophy that could be able to render the objects studied in mathematics proper fully knowable in terms of the Good (whether he thought of the latter as wholly identical with the unity of the mathematical universe or only partially). The “likely story” in the Timaeus seems to suggest that he probably envisaged the development of physics into a heavily mathematized branch of philosophy but without collapsing mathematics and natural philosophy even if he took all forms that correspond to properties of the physical world to be those of mathematical objects: philosophy in the Timaeus seems to be superior to mathematics by dint of its characteristically inclusive vision, which, just as it will be able to somehow reveal the nature of the abstract objects studied in mathematics through its comprehensiveness, it will also fully encompass the natural world alongside the deep unity of these intelligible objects in order to see how they are all linked together, thereby concretizing the Good. Plato might have thought, moreover, that this concretization would feed back into mathematics, just as in the Republic he seems to envisage that the application of geometry and arithmetic to celestial motions and musical sound in the astronomy and harmonics of the future will enrich geometry and arithmetic.53 One is tempted to believe that Plato did envisage the complete absorption into mathematics not only of natural philosophy but also of the philosophical study of the foundations of mathematics, whether he wholly identified the Good with the unity of the mathematical universe or not. But if he did, it is doubtful that he would also think of moral and political philosophy, the study of how this abstract unity can be concretized in the human soul and society, as subsumable under mathematics. 52

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Cf. ch. 1 n. 56. The regular dodecahedron in the Timaeus can be understood in this light. It is associated with the cosmos, whereas the other four regular solids are each associated with a particle of an element: since there are no other regular polyhedra in three-dimensional Euclidean space, the related forms of these exquisitely symmetrical mathematical structures determine the kinds of matter existing in the cosmos at its microscopic scales and somehow describe a feature of its largest scale, thereby uniting these two opposite scales. The regular dodecahedron can be associated with the cosmos because its face, the regular pentagon, is used in the construction of a regular pentadecagon, whose side subtends an angle approximately equal to the obliquity of the ecliptic; see Euc. El. 4.16 and Procl. in Euc. 269.8–21 Friedlein. An important feature of the largest scale of the cosmos is thus encoded in the form of a mathematical object. On astronomy and geometry see ch. 1; on harmonics and arithmetic see n. 34 of this chapter.

3. THE MYTH ABOUT PLATO’S ROLE IN THE DEVELOPMENT OF MATHEMATICS1 3.1. PLATO’S ROLE IN THE DEVELOPMENT OF ASTRONOMY According to a famous story in Simplicius, Plato firmly believed that all celestial motion is circular, uniform and orderly, and thus set the following problem before contemporary astronomers: by hypothesizing which circular, uniform and orderly motions of the Moon, the Sun and the planets can we save the phenomena of these celestial objects? The phenomena at issue here are the variable speed with which all these celestial objects travel against the background of the zodiac, and the stations and retrogradations of the planets, due to which the path of the zodiacal motion of these celestial objects deviates from circularity (see ch. 1.6.2). Simplicius distinguishes the hypothesized circular, uniform and orderly motions from both the visible zodiacal motions of the Moon, the Sun and the five planets, which are not true but merely apparent, and the ungraspable truth about their mechanism of production, in lieu of which astronomy substitutes its hypotheses, with whose help it aims only to save, i.e. derive, the relevant phenomena, not to really understand them. He also notes that Eudoxus put forth his theory of homocentric spheres rising to the challenge issued to astronomers by Plato (in Cael. 488.10–24 Heiberg): ὁ δέ γε ἀληθὴς λόγος οὔτε στηριγµοὺς αὐτῶν ἢ ὑποποδισµοὺς αὐτῶν οὔτε προσθέσεις ἢ ἀφαιρέσεις τῶν ἐν ταῖς κινήσεσιν ἀριθµῶν παραδεχόµενος, κἂν οὕτω φαίνωνται κινούµενοι, οὐδὲ τὰς ὑποθέσεις ὡς οὕτως ἐχούσας προσίεται, ἀλλὰ ἁπλᾶς καὶ ἐγκυκλίους καὶ ὁµαλεῖς καὶ τεταγµένας τὰς οὐρανίας κινήσεις ἀπὸ τῆς οὐσίας αὐτῶν τεκµαιρόµενος ἀποδείκνυσι· µὴ δυνάµενοι δὲ δι’ ἀκριβείας ἑλεῖν, πῶς αὐτῶν διακειµένων φαντασία µόνον ἐστὶ καὶ οὐκ ἀλήθεια τὰ συµβαίνοντα, ἠγάπησαν εὑρεῖν, τίνων ὑποτεθέντων δι’ ὁµαλῶν καὶ τεταγµένων καὶ ἐγκυκλίων κινήσεων δυνήσεται διασωθῆναι τὰ περὶ τὰς κινήσεις τῶν πλανᾶσθαι λεγοµένων φαινόµενα. καὶ πρῶτος τῶν Ἑλλήνων Εὔδοξος ὁ Κνίδιος, ὡς Εὔδηµός τε ἐν τῷ δευτέρῳ τῆς Ἀστρολογικῆς ἱστορίας ἀπεµνηµόνευσε καὶ Σωσιγένης παρὰ Εὐδήµου τοῦτο λαβών, ἅψασθαι λέγεται τῶν τοιούτων ὑποθέσεων Πλάτωνος, ὥς φησι Σωσιγένης, πρόβληµα τοῦτο ποιησαµένου τοῖς περὶ ταῦτα ἐσπουδακόσι, τίνων ὑποτεθεισῶν ὁµαλῶν καὶ τεταγµένων κινήσεων διασωθῇ τὰ περὶ τὰς κινήσεις τῶν πλανωµένων φαινόµενα. The true account accepts neither their stations nor their retrogradations nor any additions or subtractions in the numbers of their motions, though they appear as moving in this manner, and does not allow hypotheses of such a kind, but proves from the real nature of celestial motions that they are simple, circular, uniform and orderly. Being unable to understand how, given the truth concerning these motions, the phenomena they exhibit are not real but mere appearances, the astronomers contented themselves with discovering what ought to be assumed if the phenomena presented by the motions of the celestial objects known as wandering were to be saved through uniform, orderly and circular motions. As Eudemus says in the second book of his History of Astronomy, followed by Sosigenes, who relies on the authority of Eudemus, the first 1

This chapter incorporates Kouremenos (2011).

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3. The myth about Plato’s role in the development of mathematics among the Greeks to work on such hypotheses was Eudoxus of Cnidus, after Plato, as is reported by Sosigenes, had set the following problem for people who had studied these things: given what hypothesized orderly and uniform motions, the phenomena exhibited by the motions of the wandering celestial objects could be saved?2

Simplicius thus attributes to Plato the cornerstone of Ptolemaic astronomy, the circularity and uniformity of all celestial motion, be it the diurnal rotation of the celestial sphere, i.e. the diurnal revolution of the fixed stars, or the zodiacal motion of the Moon, the Sun and each of the five planets known in antiquity. Ptolemy lays it down before he presents for the first time in his Almagest the hypotheses of the eccentric and epicyclic motion, through which the observed non-uniformity of all zodiacal motion and its observed deviations from circularity in the case of the planets are to be explained as merely apparent (1.216.3–18 Heiberg): Ἑξῆς δ’ ὄντος καὶ τὴν φαινοµένην ἀνωµαλίαν τοῦ ἡλίου δεῖξαι προληπτέον καθόλου, διότι καὶ αἱ τῶν πλανωµένων εἰς τὰ ἑπόµενα τοῦ οὐρανοῦ µετακινήσεις, ὥσπερ καὶ ἡ εἰς τὰ ἡγούµενα φορὰ τῶν ὅλων, ὁµαλαὶ µέν εἰσιν πᾶσαι καὶ ἐγκύκλιοι τῇ φύσει, τουτέστιν αἱ νοούµεναι περιάγειν εὐθεῖαι τοὺς ἀστέρας ἢ καὶ τοὺς κύκλους αὐτῶν ἐπὶ πάντων ἁπλῶς ἐν τοῖς ἴσοις χρόνοις ἴσας γωνίας ἀπολαµβάνουσιν πρὸς τοῖς κέντροις ἑκάστης τῶν περιφορῶν, αἱ δὲ φαινόµεναι περὶ αὐτὰς ἀνωµαλίαι παρὰ τὰς θέσεις καὶ τάξεις τῶν ἐν ταῖς σφαίραις αὐτῶν κύκλων, δι’ ὧν ποιοῦνται τὰς κινήσεις, ἀποτελοῦνται, καὶ οὐδὲν ἀλλότριον αὐτῶν τῆς ἀιδιότητος περὶ τὴν ὑπονοουµένην τῶν φαινοµένων ἀταξίαν τῷ ὄντι πέφυκε συµβαίνειν. τὸ δ’ αἴτιον τῆς ἀνωµάλου φαντασίας κατὰ δύο µάλιστα τὰς πρώτας καὶ ἁπλᾶς ὑποθέσεις ἐνδέχεται γίνεσθαι. Our next task is to demonstrate the apparent anomaly of the Sun, but first we have to remark that all of the eastward motions of the wandering celestial objects are naturally uniform and circular, just like the westward rotation of the universe, which means that the imaginary straight lines carrying around these celestial objects or their circles describe in every case without exception equal angles at the center of their revolution in equal times. The apparent anomalies of their motions result from the position and order of these circles in the sphere of each by means of which they perform their motions, and there is nothing really alien to their eternity in the supposed disorder that their phenomena exhibit. The cause of their apparent irregularity can be explained by means of two hypotheses that are the most fundamental and simple.

The Platonic challenge to astronomers seems to echo Socrates’ vision of a future astronomy in R. 7, 529c6–d6, which will study intelligible motions and thus be suitable to serve as propedeutic to philosophy. Simplicius, of course, does not describe the circular and uniform motions of the challenge as intelligible; Plato, moreover, does not have Socrates even hint at the circularity of the intelligible and 2

Simplicius here seems to adopt the so-called instrumentalist interpretation of “saving the phenomena” popularized by Duhem (1908) as the ultimate goal of Greek astronomy from Plato onwards down to Ptolemy (against this interpretation see Lloyd [1991] and Musgrave [1991]): the phenomena are said to be saved if they are reproduced in a systematic way, and can also be predicted (assuming that prediction is a desideratum, which could not have been in the case of the theory of homocentric spheres) from models, i.e. hypotheses, void of explanatory content, i.e. not referring to, or describing, the real workings of nature, for the latter is inaccessible to experience. In Ptolemy’s Almagest the phrase “saving the phenomena” occurs only once (Alm. 2.532.19–533.3 Heiberg), but is used in a thoroughly realist sense; cf. Goldstein (1997) 8.

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uniform motions which the future astronomy will study, nor does he seem to think of them as mere devices intended to help us make up for our inability to grasp the truth about the celestial motions at issue. As regards the hypotheses of the future astronomy envisioned by Socrates, for Simplicius they will concern the intelligible and uniform motions this astronomy will study, whereas I argue in ch. 1.6 that they are best viewed as defining geometrical objects kinematically through the uniform motions of other geometrical objects generating those defined: the latter will be the figures approximated by the visible paths of the zodiacal motions. I also argue in ch. 1.7.5 that the brief outline of the future astronomy in R. 7 was inspired by Eudoxus’ theory of homocentric spheres, whereby the curve in fig. 5c, which the zodiacal motions of the planets vaguely resemble, is kinematically generated by the uniform rotations of the nested spheres in fig. 5a. Since Plato models this future astronomy on the paradigm of solid geometry as exemplified by the duplication of the cube (see ch. 1.7.1), the fact that the future astronomy strongly calls to mind the theory of homocentric spheres, with which the cube can be doubled, makes it likely, as argued in ch. 1.7.5, that Eudoxus did double the cube with his theory of homocentric spheres, and that Plato outlined the future astronomy with this theory in mind rather than issued with his outline a challenge which motivated Eudoxus to develop his theory. Indeed, Plato has Socrates conclude his discussion of the future astronomy with Glaucon by stressing a feature of this astronomy that brings it even closer to the Eudoxean theory of homocentric spheres as scholars have reconstructed it independently of any considerations about R. 7 (see ch. 1.7.5). It is thus equally unlikely either that Plato challenged contemporary astronomers to do something so specific or, alternatively, that his brief discussion in R. 7 of the future astronomy mirrors in this regard, too, the theory of homocentric spheres only accidentally, and does by no means betray the influence of Eudoxus’ theory, which thus does not illuminate Plato’s future astronomy, however else the latter might be understood. The Neoplatonist acknowledges his sources for this story about the Platonic influence on the development of the theory of homocentric spheres in the passage quoted above from his extensive commentary on Aristotle’s de Caelo: the History of Astronomy by Eudemus of Rhodes and the treatise On the Unwinding Spheres by Sosigenes (second century AD), teacher of Alexander of Aphrodisias. He repeats the story not much below (in Cael. 492.31–493.11 Heiberg): καὶ εἴρηται καὶ πρότερον, ὅτι ὁ Πλάτων ταῖς οὐρανίαις κινήσεσι τὸ ἐγκύκλιον καὶ ὁµαλὲς καὶ τεταγµένον ἀνενδοιάστως ἀποδιδοὺς πρόβληµα τοῖς µαθηµατικοῖς προὔτεινε, τίνων ὑποτεθέντων δι’ ὁµαλῶν καὶ ἐγκυκλίων καὶ τεταγµένων κινήσεων δυνήσεται διασωθῆναι τὰ περὶ τοὺς πλανωµένους φαινόµενα, καὶ ὅτι πρῶτος Εὔδοξος ὁ Κνίδιος ἐπέβαλε ταῖς διὰ τῶν ἀνελιττουσῶν καλουµένων σφαιρῶν ὑποθέσεσι. Κάλλιππος δὲ ὁ Κυζικηνὸς Πολεµάρχῳ συσχολάσας τῷ Εὐδόξου γνωρίµῳ µετ’ ἐκεῖνον εἰς Ἀθήνας ἐλθὼν τῷ Ἀριστοτέλει συγκατεβίω τὰ ὑπὸ τοῦ Εὐδόξου εὑρεθέντα σὺν τῷ Ἀριστοτέλει διορθούµενός τε καὶ προσαναπληρῶν· τῷ γὰρ Ἀριστοτέλει νοµίζοντι δεῖν τὰ οὐράνια πάντα περὶ τὸ µέσον τοῦ παντὸς κινεῖσθαι ἤρεσκεν ἡ τῶν ἀνελιττουσῶν ὑπόθεσις ὡς ὁµοκέντρους τῷ παντὶ τὰς ἀνελιττούσας ὑποτιθεµένη καὶ οὐκ ἐκκέντρους, ὥσπερ οἱ ὕστερον. It has been said above, too, that Plato, having unhesitatingly assigned the properties of circularity, uniformity and orderliness to celestial motions, set before the mathematicians a problem, i.e. which

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3. The myth about Plato’s role in the development of mathematics things must be supposed if the phenomena the wandering celestial objects present are to be saved via circular, uniform and orderly motions, and that Eudoxus of Cnidus was the first to attack this problem via hypotheses involving the spheres known as unwinding.3 Callippus of Cyzicus, a student of Polemarchus, who was an associate of Eudoxus, went to Athens after him and spent time with Aristotle, correcting and fleshing out with him Eudoxus’ discoveries. Operating on the belief that all celestial objects must move about the center of the universe, Aristotle liked the hypothesis of the unwinding spheres, for in it the spheres are assumed to be concentric with the universe and not eccentric, as posterity believed.

This story has been dismissed as an anecdote, though on the weak ground that it seems to derive not from the fourth-century-BC Eudemus but only from the much later, and far less trustworthy, Sosigenes.4 However, not much can be built on the fact that Simplicius names only Sosigenes as his source for Plato’s role in his account of the development of the theory of homocentric spheres. A few lines earlier Simplicius mentions both Sosigenes and Eudemus as his sources for Eudoxus and the theory of homocentric spheres, and also emphasizes that Sosigenes followed Eudemus; moreover, we cannot but consider quite seriously the possibility that Simplicius knew Eudemus exclusively through the account of Sosigenes.5 Knorr, assuming that most probably Simplicius’ only source for the story of a Platonic influence on the development of the theory of homocentric spheres was Sosigenes, has argued that Socrates’ vision in R. 7 of a future astronomy, which will be geometrically advanced, unlike contemporary astronomy, “casts Socrates in very much the role that Sosigenes seems to have in mind for Plato when he says he “posed the problem to those engaged in astronomy”. It thus emerges as a likely source of the tradition on which Sosigenes relies for this claim”.6 All the more so, we should add, since Plato has Socrates stress in R. 7 that this advanced astronomy will pursue planar problems after having argued in the preceding stereometrical interlude that the lack of progress in contemporary solid geometry is due not least to the need for an overseer of the mathematicians who work in this undeveloped field (see ch. 1.7.5). For a bizarre principle of ancient biography is the assumption of an agreement between the life and the writings or thought of the person biographied. What a philosopher was known to have done or said at some point in his life was accordingly considered to be indispensable to the interpretation of his thought and writings, and, conversely, his thought and writings were hypothesized to be safe guides to his life: one was free to look for biographical “facts” in his writings.7 Hence the pseudo-biographical story that has Plato play a significant role in the development of contemporary astronomy as a superintendent of astronomers who challenges them to tackle a problem and spurs 3 4 5 6 7

The expression “unwinding spheres”, which appears in the title of Sosigenes’ work mentioned above, is a metonymy for a system of homocentric spheres; see Mendell (2000) 92–93. See Zhmud (2006) 86–87. For Sosigenes and his treatise On the Unwinding Spheres (the title is recorded by Procl. Hyp. 4.98.3–4) see Moraux (1984) 335ff. See Bowen (2002) 317–318 and Mendell (2000) 88–89; cf., though, Zhmud (2006) 87 n. 23. Knorr (1990) 325. Cf., though, Burnyeat (2000) 63 and Gregory (2000) 97–100. See Mansfeld (1999) 20. Cf. Chitwood (2004) for the biographical tradition about the life and death of Heraclitus, Empedocles and Democritus, and Lefkowitz (1981) for the lives of Greek poets.

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Eudoxus to develop his theory of homocentric spheres. Proclus credits Plato with greatly advancing mathematics, geometry in particular, because of his passion for mathematical studies, and gives a long roster of the mathematicians “who lived together in the Academy, making their researches communally”, and among whom figure prominently the great Archytas, Eudoxus and Menaechmus (in Euc. 66.8–68.4 Friedlein).8 Plato had a crucial role in these collaborative researches, according to Philodemus’ History of the Academy (PHerc 1021 col. Y.3–17 Dorandi) found among the carbonized remains of a library in the Villa dei Papiri at Herculaneum, on the Bay of Naples, probably Philodemus’ personal library: he functioned as a director of mathematics which progressed markedly in his time, setting out problems which the mathematicians investigated eagerly.9 Nothing suggests, however, that Plato had actually functioned, or just fancied himself, as the director of mathematics whom he has Socrates mention in the stereometrical interlude of R. 7, 528a6–c7.10 The belief that the motion of the Moon, the Sun and the five planets is uniform and, in the case of the five planets, circular, too, is attributed to the Pythagoreans by Geminus, the author of an introductory textbook in astronomy, who seems to have written in the first century BC.11 Geminus considers this belief to be the foundation of all astronomy. The Pythagoreans were, in his view, the first to study astronomy, and it was they who first stated the problem of how the phenomenon of non-uniformity exhibited by the zodiacal motion of the Moon, the Sun and the planets, and that of departures from exact circularity in the case of the planets, can be explained via hypotheses of circular and uniform motions, probably as being the true ones all these celestial objects undergo (1.19–21): Ὑπόκειται γὰρ πρὸς ὅλην τὴν ἀστρολογίαν ἥλιόν τε καὶ σελήνην καὶ τοὺς πλανήτας ἰσοταχῶς καὶ ἐγκυκλίως καὶ ὑπεναντίως τῷ κόσµῳ κινεῖσθαι. Οἱ γὰρ Πυθαγόρειοι πρῶτοι προσελθόντες ταῖς τοιαύταις ζητήσεσιν ὑπέθεντο ἐγκυκλίους καὶ ὁµαλὰς ἡλίου καὶ σελήνης καὶ τῶν πλανητῶν ἀστέρων τὰς κινήσεις. Τὴν γὰρ τοιαύτην ἀταξίαν οὐ προσεδέξαντο πρὸς τὰ θεῖα καὶ αἰώνια, ὡς ποτὲ µὲν τάχιον κινεῖσθαι, ποτὲ δὲ βράδιον, ποτὲ δὲ ἑστηκέναι· οὓς δὴ καὶ καλοῦσι στηριγµοὺς ἐπὶ τῶν πλανητῶν ἀστέρων. Οὐδὲ γὰρ περὶ ἄνθρωπον κόσµιον καὶ τεταγµένον ἐν ταῖς πορείαις τὴν τοιαύτην ἀνωµαλίαν τῆς κινήσεως προσδέξαιτο ἄν τις· αἱ γὰρ τοῦ βίου χρεῖαι τοῖς ἀνθρώποις πολλάκις αἴτιαι γίνονται βραδυτῆτος καὶ ταχυτῆτος. Περὶ δὲ τὴν ἄφθαρτον φύσιν τῶν ἀστέρων οὐδεµίαν δυνατὸν αἰτίαν προσαχθῆναι ταχυτῆτος καὶ βραδυτῆτος. Δι᾽ ἥντινα αἰτίαν προέτειναν οὕτω, πῶς ἂν δι᾽ ἐγκυκλίων καὶ ὁµαλῶν κινήσεων ἀποδοθείη τὰ φαινόµενα. It is a fundamental assumption of all astronomy that the Sun, the Moon and the five planets move without varying their speeds, circularly and oppositely to the rotation of the cosmos. For 8

9 10 11

The passage is part of Eudem. fr. 133 Wehrli = in Euc. 64.16–68.6 Friedlein, an account of the development of geometry from the beginnings to the fourth century BC, which is very likely to incorporate material from Eudemus’ history of geometry, but whose Platonocentric section can be plausibly traced back to a Neoplatonist source that incorporates material from a work by a member of the early Academy; see Zhmud (2006) 179–190. The passage is a quotation from the work of an unknown author, probably a member of the early Academy; see Zhmud (2006) 87–89. Pace Zhmud (2006) 107; cf. ch. 1 n. 82. On Geminus’ date see Evans & Berggren (2006) 15–22.

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3. The myth about Plato’s role in the development of mathematics the Pythagoreans, who were the first to approach inquiries of this kind, hypothesized that the motions of the Sun, the Moon and the five planets are uniform and circular. The reason is that they did not accept in things eternal and divine such disorder as faster motion at one time, slower at another, and stopping at yet another, pauses called “stations” in connection with the five planets. One would not accept such anomaly of movement even in the walking of an orderly and decorous person. The business of life is often the cause of slowness and quickness in humans, whereas in the case of the incorruptible nature of the celestial objects it is impossible to invoke any cause of quickness and slowness. This is why the Pythagoreans put forth the following problem: how could it be possible to explain the phenomena through circular and uniform motions?

This story smacks of the Neopythagorean tendency to connect with Pythagoreanism, and with Pythagoras himself, the rise of astronomy and mathematics in Greece. Neopythagoreanism begins in the first century BC, so its influence on Geminus is possible, and since another tendency in Neopythagoreanism is the attribution to Pythagoreanism of all things Platonic, a trend whose origins can be traced back to the Academy right after the death of Plato, it is likely that Geminus tells the same story as Simplicius, Plato having simply been replaced by the Pythagoreans.12 As will be seen below in section 3, for his testimony on the “Pythagoreans” Geminus or his source seems to draw on Plato’s Lg. 7 and 10. The fanciful notion of Plato’s impact on the exact sciences of his time is also reflected in the well-known anecdote, according to which Archytas, Eudoxus and Menaechmus applied themselves to solving the problem of cube-duplication after Plato had interpreted an oracle given to the Delians. I will discuss the sources of this famous story in Plato’s works first and then turn to those of the related anecdote about his impact on astronomy.

12

We learn from the pseudo-Aristotelian work Problems that Archytas held the view that natural motion traces circles (XVI.9, 915a25–32 = DK 47 A 23a). Now, Archytas was a prominent Pythagorean of Plato’s time, but as far as we know, he invoked the conception of natural motion as circular only in the context of explaining why parts of plants and animals, such as stems, shoots, thighs, arms and trunks, are of curved shape. There is absolutely nothing to suggest that he inferred the circularity of all celestial motions simply from their naturalness, whence we could conjecture that he was the first to pose the astronomical problem whose statement Geminus attributes to the Pythagoreans. Nor is there any evidence that Archytas did any work in astronomy; cf. Huffman (2005) 295. The most probable reason for the Greek belief in the circularity and uniformity of all celestial motions is simple: the wish to bring a few anomalies, the irregular and (in the case of the five planets only) non-circular motions of the seven wandering celestial objects, into line with the rule: the regular and circular motions of the fixed stars, the vast majority of the celestial objects the naked eye can observe. We can confidently suppose that no matter exactly when this belief took shape in Greece, no single individual or group was needed to form it in a stroke of trailblazingly original theorizing. We ought to resist the tendency of the Greeks to seek prōtous heuretas, understandable though it may be, and not put much stock on either the story in Simplicius or its version in Geminus, preferable though the second may seem to the first.

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3.2. THE TRADITION OF THE DELIAN PROBLEM AND ITS SOURCES 3.2.1. The main testimonies Plutarch records in his work On the Divine Sign of Socrates a story on account of which the problem of cube-duplication came to be named after a Greek island, Delos. On the Divine Sign of Socrates is a dialogue in which the Theban Simmias, one of Plato’s associates and Socrates’ interlocutors in Plato’s Phaedo, reports a meeting that occurred on his and Plato’s return trip from Egypt, where they had recently traveled (579B1–D3): κοµιζοµένοις ἡµῖν ἀπ’ Αἰγύπτου περὶ Καρίαν Δηλίων τινὲς ἀπήντησαν δεόµενοι Πλάτωνος ὡς γεωµετρικοῦ λῦσαι χρησµὸν αὐτοῖς ἄτοπον ὑπὸ τοῦ θεοῦ προβεβληµένον. ἦν δ’ ὁ χρησµὸς Δηλίοις καὶ τοῖς ἄλλοις Ἕλλησι παῦλαν τῶν παρόντων κακῶν ἔσεσθαι διπλασιάσασι τὸν ἐν Δήλῳ βωµόν. οὔτε δὲ τὴν διάνοιαν ἐκεῖνοι συµβάλλειν δυνάµενοι καὶ περὶ τὴν τοῦ βωµοῦ κατασκευὴν γελοῖα πάσχοντες (ἑκάστης γὰρ τῶν τεσσάρων πλευρῶν διπλασιαζοµένης ἔλαθον τῇ αὐξήσει τόπον στερεὸν ὀκταπλάσιον ἀπεργασάµενοι δι’ ἀπειρίαν ἀναλογίας ἣν τὸ µήκει διπλάσιον παρέχεται) Πλάτωνα τῆς ἀπορίας ἐπεκαλοῦντο βοηθόν. ὁ δὲ τοῦ Αἰγυπτίου µνησθεὶς προσπαίζειν ἔφη τὸν θεὸν Ἕλλησιν ὀλιγωροῦσι παιδείας οἷον ἐφυβρίζοντα τὴν ἀµαθίαν ἡµῶν καὶ κελεύοντα γεωµετρίας ἅπτεσθαι µὴ παρέργως· οὐ γάρ τοι φαύλης οὐδ’ ἀµβλὺ διανοίας ὁρώσης ἄκρως δὲ τὰς γραµµὰς ἠσκηµένης ἔργον εἶναι [καὶ] δυεῖν µέσων ἀνάλογον λῆψιν, ᾗ µόνῃ διπλασιάζεται σχῆµα κυβικοῦ σώµατος ἐκ πάσης ὁµοίως αὐξόµενον διαστάσεως. τοῦτο µὲν οὖν Εὔδοξον αὐτοῖς τὸν Κνίδιον ἢ τὸν Κυζικηνὸν Ἑλίκωνα συντελέσειν· µὴ τοῦτο δ’ οἴεσθαι χρῆναι ποθεῖν τὸν θεὸν ἀλλὰ προστάσσειν Ἕλλησι πᾶσι πολέµου καὶ κακῶν µεθεµένους Μούσαις ὁµιλεῖν καὶ διὰ λόγων καὶ µαθηµάτων τὰ πάθη καταπραΰνοντας ἀβλαβῶς καὶ ὠφελίµως ἀλλήλοις συµφέρεσθαι. On our return trip from Egypt, some Delians met us near Caria asking Plato as a geometer to solve for them a strange oracle the god had given them. According to the oracle, the present troubles would stop for the Delians and the other Greeks if they doubled the altar at Delos. They could not grasp what was asked, however, and were facing ridiculous difficulties with the construction of the altar (doubling each of the four sides,13 they ended up making the volume eight times larger since they were unfamiliar with the proportional increase effected by the double length), so they called upon Plato to help them get out of their impasse. Recalling the Egyptian,14 Plato answered that the god teases the Greeks for their neglect of intellectual pursuits, insulting us, as it were, for our ignorance and ordering us to engage seriously in geometry, given that finding two mean proportionals, the only way to double the volume of a cubic body by increasing it in all three dimensions,15 takes a mind which is not inferior or dull but supremely trained in geometry. At any rate, Eudoxus of Cnidus or Helicon

13

14 15

The sides at issue here are of each square face of the cubic altar, as is clear by the following specification that the Delians “were unfamiliar with the proportional increase effected by the double length”–they did not know that, by doubling the length of the side of each square face of the cubic altar, they got an altar enlarged not two times, as was required, but eight. This passage has vexed scholars (see Nesselrath [2010] 88 n. 76), but makes good sense if we just read ἣν τὸ µήκει διπλάσιον παρέχεται (following Sieveking) instead of the mss reading ἣ τῷ µήκει διπλάσιον παρέχεται. See 578E9–579B1. For the relation between the cube-duplication problem and two mean proportionals see below.

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3. The myth about Plato’s role in the development of mathematics of Cyzikus would find for them a solution to the problem;16 they must think, though, that this is not what the god wants, but that he orders all Greeks to be in the company of the Muses forsaking war and destruction, and to come to terms with one another harmlessly and beneficially, soothing the passions with the help of philosophy and mathematics.

Plutarch repeats the story briefly in his On the E at Delphi, 386E3–10, so the god who issued the geometrical oracle to the Delians was Apollo, and the oracle was issued at Delphi. Theon of Smyrna recounts this story in the introduction to his Exposition of the Mathematics Useful for Reading Plato (2.3–12 Hiller) and concludes with the remark “following the advice of Apollo, Plato himself has written extensively on the usefulness of mathematics” (2.13–14 Hiller: ἀκολούθως δὲ τῇ τοῦ Πυθίου παραινέσει πολλὰ καὶ αὐτὸς διέξεισιν ὑπὲρ τοῦ ἐν τοῖς µαθήµασιν χρησίµου).17 When Plutarch says “according to the oracle, the present troubles would stop for the Delians and the other Greeks if they doubled the altar at Delos”, he seems to be having in mind the famous horned altar of Apollo at Delos (Κερατὼν βωµός, Plu. Thes. 2.21 = Dicaearch. fr. 85 Wehrli = 74 Mirhady), whose construction Callimachus describes in his Hymn to Apollo, 60–63.18 Theon seems to be speaking of just an altar at Delos, however. Another altar of Apollo at Delos had existed behind the horned altar according to Diogenes Laertius, 8.13, who refers to Aristotle’s Constitution of the Delians (fr. 489 Rose) as source for the association of this other Delian altar with Pythagoras. Theon adds the detail that the Delians had consulted the god while suffering from a plague. He does not mention any mathematician(s) named by Plato as being capable of solving the problem of cube-duplication that puzzled the islanders, or a political lesson the philosopher saw in the oracle, though the educational lesson is stated here, too. In Theon’s account, moreover, the Delians sent envoys to Athens seeking Plato for help. This version, slightly different from Plutarch’s, goes back to Eratosthenes of Cyrene, whose dialogue Platonicus Theon cites as his source.19 The Alexandrian polymath gave a different version of the anecdote in a letter he addressed to his royal patron Ptolemy III Euergetes detailing an instrument of his own design for finding two mean proportional lines, and thus doubling a cube (for the relation between the two problems see below). The letter is quoted by the sixth-century-AD Eutocius of Askalon, in his comments on Archimedes’ On the Sphere and Cylinder (for the section relevant here see in Sph. Cyl. 3.88.13–90.11 Heiberg).20 Eratosthenes says in his letter that, after the Delians sent emissaries to 16

17 18 19

20

According to the thirteenth letter attributed to Plato, Helicon was an associate and student of Eudoxus (360c2–4). Plutarch reports in Dion 19.4 that Helicon accompanied Plato on his third visit to Sicily (361 BC): see, however, Zhmud (2006) 244 n. 69. Theon’s remark is discussed in section 3.2.3. For the identification of the altar in Plutarch’s report with the famous horned altar of Apollo at Delos see Nesselrath (2010) 87 n. 75. For fragments and reconstruction of the Platonicus see Hiller (1870); further literature cited in Frazer (1972) 410 n. 268. After Hirzel (1895) vol. 1 405ff. it is usually taken for granted that this work was a dialogue. Solmsen (1942) 193 n. 3 thought that the issue is undecidable; against the assumption of dialogic form see Geus (2002) 141–194. Cf. ch. 1 n. 78. For further testimonies on the Delian problem see Riginos (1976) 141.

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Athens asking the geometers in Plato’s Academy for help, the geometers Archytas, Eudoxus and Menaechmus applied themselves diligently to the problem of doubling a cube, and were finally able to obtain solutions. In this version, too, Plato is not said to have referred off-handedly to any mathematician(s) as being able to obtain a solution to the problem at issue; here, moreover, the philosopher does not draw an educational moral from the oracle. The Delians are said to have been asked by the oracle to double one of many altars, presumably of Apollo at Delos. The letter makes clear that the problem of doubling a cube did not come to the attention of Greek mathematicians suddenly in Plato’s time, when the Delians received the oracle asking them to double the altar at Delos and, having failed to achieve this, called upon Plato for help. By that time, Greek mathematicians were familiar with it. According to Eratosthenes (in Sph. Cyl. 3.88.18–23 Heiberg), the fifth-century BC mathematician Hippocrates of Chios (DK 42 A 4), was the first who realized that in order to double a cube one has to find two mean proportionals, that is two lines X and Y such that, given two other lines A and B, we have A : X = X : Y = Y : B, whence it follows that A3 : X3 = AXY : XYB = A : B, or that line X is the side of a cube which is double the cube with line A as side if line B is double line A;21 after Hippocrates had effected the reduction of the cube-duplication problem to that of finding two mean proportionals, the Delians received the oracle, went to the geometers in Plato’s Academy for help, and the problem was at last fully solved by Archytas, Eudoxus and Menaechmus, each of whom managed to construct two mean proportionals in a different way.22 3.2.2. The generative principle of the tradition The implicit point of the account relating Plato and the various fourth-century-BC solutions to the problem of doubling a cube, especially the version of this account in Plutarch, is that, as head of the Academy, Plato had presided imperiously over a coterie of highly competent mathematicians, to whom he handed over problems to solve. Centuries after Eratosthenes, who is the oldest source of the Delian anecdote known to us, Proclus comes very close to saying just this when, as seen in 3.1, he credits Plato with greatly advancing mathematics, in particular geometry, because of his passion for these studies, and then proceeds to give the long roster of the mathematicians “who lived together in the Academy, making their inquiries communally”, prominent among whom are Archytas, Eudoxus and Menaechmus. Eutocius, who details in his above mentioned commentary various solutions to the problem of doubling a cube, put forth from the fourth century BC onwards, even ascribes a solution to Plato himself (in Sph. Cyl. 3.56.14–58.14 Heiberg).23 This solution obtains the two straight lines, which are mean proportionals between the two given straight lines, by employing a specially designed instrument. In his Sympotic Questions, however, Plutarch reports that Plato objected to Archytas, 21 22 23

Cf. Procl. in Euc. 212.24–213.11 Friedlein. On the reduction cf. ch. 1 n. 79. On the chronology of the three mathematicians see ch. 1.7.1; on their solutions see ch. 1.7.2–4. On this solution see Knorr (1993) 57–61 and Netz (2003).

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Eudoxus and Menaechmus for their “instrumental and mechanical constructions” of the two mean proportionals required for doubling a given cube (718E7–F4; cf. Marc. 14.9–11); Eratosthenes, moreover, who had himself manufactured a tool for an easy solution to the problem by quickly finding mean proportionals, notes in his letter to king Ptolemy that the solutions developed by Archytas, Eudoxus and Menaechmus could not be applied since they were all purely theoretical (in Sph. Cyl. 3.90.8–11 Heiberg). The author of the Anonymous Prolegomena to Platonic Philosophy (6th century AD) even credits Plato himself with the proportion-theoretic reduction of the cube-duplication problem ascribed by Eratosthenes to Hippocrates: in this “Platonic” reduction the problem translates mistakenly into finding one mean proportional (Anon. Proleg. 5.13–24 Westerink). This confusion makes one suspect that the tradition linking Plato, the problem of doubling a cube and a Delphic oracle lacks historicity.24 We can assume that the story of his objections to Archytas, Eudoxus and Menaechmus is an accretion due to Plutarch;25 that Plato came to be erroneously credited with the mechanical solution to the cube-duplication problem as a result of this method’s having been described by the character Plato in Eratosthenes’ Platonicus just to be criticized;26 and, finally, that the unknown author of the Anonymous Prolegomena to Platonic Philosophy simply made, or repeated, a mistake. However, not a shred of historical evidence can be adduced in support of the broader tradition in which the story of the Delian problem is inscribed and that assigns to Plato a purportedly important role in the development of contemporary mathematics, a myth that goes back to the early Academy and was subsequently utilized by the Neoplatonists to exalt Plato.27 Eratosthenes, as seen above, knew what seems to be a more plausible explanation of how the problem of doubling a cube came to the attention of Greek mathematicians–through the work of Hippocrates of Chios in the fifth century BC, not because of a Delphic oracle the Delians received in the next century and its sagacious interpretation by Plato. In his letter to king Ptolemy Eratosthenes grafts for some reason onto this historically plausible account a toned-down version of the story about Plato’s supposed connection with the history of the solutions to the cube-duplication problem, but the two can be easily unpicked.28 We can seriously doubt even the possibility that an oracle asking for the duplication of a cubic altar could have ever been received by the Delians.29 The 24 25 26 27 28

29

See Knorr (1993) 22 and Zhmud (2006) 84–86. Cf., though, Huffman (2005) 376–378. See Riginos (1976) 146, Huffman (2005) 378–385 and Zhmud (2006) 85 n. 17. See Huffman (2005) 381. See Zhmud (2006) ch. 3 and ch. 5, 179–190. No one seems to have followed Sachs (1917) 150 in also denying the historicity of the reduction of the problem of cube-duplication to that of finding two mean proportionals. There is nothing suspicious about it. It is conceivable that, after the problem of cube-duplication arose in Hippocrates’ work, interest in its solution received new impetus from practical considerations having to do with a Delphic oracle given to the Delians, who asked Plato to help them with it (cf. Huffman [2005] 378). As will be argued next, it is doubtful that an oracle asking for the doubling of an altar, whether the horned altar at Delos or not, could have been received, and, at any rate, Plato’s association with all things Apolline is inherently suspect. Fontenrose (1978) 333 rejects this oracle as non-genuine.

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tradition of the Delian problem aside, however, there is no literary evidence that Apollo’s horned altar at Delos had been cubic, assuming that the tradition concerns that famous altar.30 Its location, moreover, is unknown. If it was located in the socalled Monument of the Apse,31 the structure offers no evidence as to the shape of any altar it might have housed. Since archaeological evidence for cubic altars in Greece is absent, it is doubtful that Apollo’s horned altar, wherever it might have been, or another altar at Delos, could have had the shape of a cube, and thus that an oracle could have called for it to be doubled.32 If the Monument of the Apse did house Apollo’s famous horned altar, the fact that in the 340s BC the north and south flanks of this structure were doubled in thickness with the addition of a wall made of porous limestone to each,33 assuming that here lies the historical basis of the story of the Delian problem, shows that this story must be pure invention: the problem of doubling a cube has nothing to do with doubling the thickness of walls.34 Finally, we should not lose sight of the fact that the close relation of Plato to the god Apollo is emphasized in the extant biographies of the philosopher and in biographical anecdotes about him.35 But the study of all these anecdotes leaves no doubt that they are easily traceable back to passages in the Platonic corpus:36 the same must be the case with the tradition of the Delian problem, according to which Plato had privileged insight into Apollo’s will.37 It can only have sprung from the principle behind the amply documented tendency of ancient biography of mining “facts” out of an author’s works: the purportedly necessary agreement of his life with his writings. 3.2.3. Evidence for the application of this principle in Theon The principle behind this curious practice is evident from Theon’s testimony on the Delian problem in the preface to his compendium Exposition of the Mathematics Useful for Reading Plato.38 After stating the characteristically Platonist position that one cannot live the best possible life without knowledge of mathematics, Theon observes that Plato has made it abundantly clear that one ought not to disregard mathematics (2.1–2 Hiller). Given the aim of Theon’s short compendium, which is expressly stated in the preceding lines and in the title of the work, we expect to find included an anthology, 30 31 32

33 34 35 36 37 38

The testimonies on the horned altar are collected in Bruneau & Fraisse (2002) 61–67. See Bruneau & Fraisse (2002) ch. 4. That there is no archaeological evidence for cubic altars in Greece has been pointed out in connection with the tradition of the Delian problem by Sarton (1993) 279 who refers to Yavis (1949). See Bruneau & Fraisse (2002) 32 and 76. Cf. the discussion in Fontenrose (1978) 333. See Riginos (1976) ch. 1. See Riginos (1976) 29–32. Riginos (1976) 141–146 discusses the tradition of the Delian problem separately from the stories about Plato’s Apolline nature, together with those about Plato and the Academy. On Theon and his only surviving work see Petrucci (2009).

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or summaries, of Platonic passages where the value of mathematics is extolled and the need for its study is emphasized. We do find this (2.13–16.2 Hiller), but first Theon tells the story of the Delian embassy to Plato as Eratosthenes reported it in his Platonicus, its moral being that the god gave to the Delians the oracle asking for doubling the size of the altar as a reproach to the Greeks for their neglect of mathematics and disregard for geometry (2.3–12 Hiller). As illustration of the importance Plato attached to mathematics, this story is given priority over what the philosopher says in his works about the importance of mathematics. It leaves no doubt that, in Theon’s opinion, the value of mathematics has been made clear mainly by Plato’s life itself, to which the views on mathematics the philosopher wrote down–in accordance, of course, with his life–come second. Indeed, Theon goes on to say that, Plato wrote much on the usefulness of mathematics following the advice of none other than Apollo. What is more, Theon clearly thinks that Plato has hinted in writing that failure to study mathematics in the right way entails impious disregard for oracles. After his remark that Plato followed the advice of Apollo and wrote much on the usefulness of mathematics, Theon opens his list of the relevant textual evidence with a few lines from the Epinomis, whose attribution to Plato he evidently accepts (cf. ch. 2, n. 16). These lines, 992a3–6, contain a short exhortation to the appropriate study of mathematics as the only way to eudaimonia, and ends with the phrase “it is not right to show disregard for the gods” (ἀµελῆσαι δὲ οὐ θεµιτόν ἐστιν θεῶν). In this context, to disregard the gods can only mean, by Theon’s lights, to disregard the oracle Apollo supposedly gave to the distraught Delians, and heeding which Plato himself wrote much on the value of mathematics: the particle (γάρ) in the sentence with which Theon introduces the quote from the Epinomis leaves no doubt that this quote backs up his view that Plato praised in writing the importance of mathematics because he heeded Apollo’s advice to the Greeks, which was delivered through the god’s oracle to the Delians. Whether written by Plato or not, the Epinomis does not, of course, say anything about oracles in the lines at issue. “It is not right to show disregard for the gods” is a traditional maxim which, though, takes on a new, Platonist, meaning in the Epinomis: “it is not right to show disregard for mathematics”, whose study in the right way leads to the realization of the truth “all things are full of gods and we are never neglected by our superiors due to forgetfulness or disregard” (991d4–5). The Platonicus of Eratosthenes is the oldest source to which we can trace back the tradition linking the problem of doubling a cube, Plato as a philosopher fervently believing in the immense value of mathematics and an oracle issued by Apollo. Based on which Platonic passages other than R. 7, 528a6–c7 (see 3.1), did Eratosthenes, more probably his source(s), originate this tradition, which cannot have sprung, pace Theon, from Epin. 992a3–6? How deeply this tradition is rooted in the writings of Plato has been underestimated: the view of Wilamowitz that the story because of which the problem of cube-duplication came to be known as the Delian problem can very well be true is quoted approvingly to date.39 Surprising 39

Wilamowitz (1962) 48; cf. Huffman (2005) 377. The story has been considered evidence for the ritual origin of geometry: see Seidenberg (1963) 493–496 and van der Waerden (1980) 37–38.

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though it is, Delphic oracles as drivers of the advancement of mathematics in Greece appear in the Platonic corpus (see section 3.2.4.c), not only in the story of the Delian problem: thus, given the practice of mining biographical “facts” about an author out of his works, it is, on the contrary, very implausible that the story is true. It smacks of anecdotal etiology crassly blended with Platonic hagiography. It purports to “explain”, on the basis of certain passages in the Platonic corpus, an important episode from the history of geometry in Plato’s day, the construction by Archytas, Eudoxus and Menaechmus of two mean proportionals, which is needed to solve the problem of cube-duplication. It does nothing but reflect the proPlatonic tendencies of the source(s) that originated it, and of all the later sources that repeat it, just as other biographical anecdotes about Plato, which are also traceable back to passages in the Platonic corpus, reflect anti-Platonic tendencies.40 3.2.4. The sources of the tradition in the Platonic corpus a. Meno In her exhaustive study of the anecdotes concerning the life and writings of Plato, Riginos suggests that inspiration for the anecdote relating Plato and the fourthcentury-BC solutions to the cube-duplication problem probably came from the Meno and the slave-boy’s famous attempt to “recollect” prenatally obtained knowledge.41 Socrates guides deftly the slave-boy towards “recollecting” the side of a square whose area is double a given square with side two length-units. This is the planar analogue of the problem facing the Delians in the anecdote at issue. The slave-boy initially proposes that the solution is four length-units (Men. 82b9–e3): this is a mistake because the sides of two squares whose areas stand in the ratio 1 : 2 are not commensurable–the side of a square whose area is double that of a given square is the given square’s diagonal, √2 if 1 is the original square’s side.42 The mistake is analogous to that of the Delians in the anecdote. Doubling a square is linked to doubling a cube by more than formulation: the planar problem reduces to finding one mean proportional line between two lines, whereas the solid problem boils down to finding not one but two mean proportional lines between two lines.43 The mistake in the proportion-theoretic reduction of the cube-duplication problem ascribed to Plato in the Anonymous Prolegomena to Platonic Philosophy does not, therefore, merely testify to the mathematical incompetence of the work’s author: it also points to the Meno as a Platonic dialogue that provided some material for the story about Plato’s role in the solution to the cube-duplication problem. The duplication of a square and the duplication of a cube are also connected as regards each problem’s role within its context under consideration here. For, in the 40 41 42 43

Cf. Riginos (1976) 199–201. See Riginos (1976) 145. Cf. Pl. Tht. 147d3–148b2: if the area of a square is n area-units, the side of the square is incommensurable for n which is not a square number, e.g. for n = 2. For the planar problem see Euclid’s Elements 6.13; it is equivalent to 2.14 (squaring an area).

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Meno, Socrates sets before the slave-boy the problem of doubling a square in order to convince his interlocutor that believing in the possibility of knowledge makes us industrious and disposed to inquire, it makes us better and manlier than we would be if we adopted the opposite view that seeking out knowledge is pointless (86b6– c2). This comes quite close to why Apollo set before the Delians the problem to double a cube–in order to induce them, together with all Greeks, to abandon their shameful disregard for education and mathematics and inquire. The two related problems, the duplication of a square and that of a cube, are used in the Meno and the anecdote about the Delian problem respectively to draw out the same lesson: it is through study that we become better, especially through the study of mathematics. b. Laws 7 By challenging with his oracle the Delians to double a cube so as to reproach not only them but all the Greeks for their neglect of mathematics, Apollo essentially rebukes the Greeks for their ignorance of incommensurability. The Delians first tried to build a cubic altar whose side was double that of the original cube. They failed because the sides of two cubes whose volumes stand in the ratio 1 : 2 are not commensurable, just like the sides of two squares whose areas are in this ratio. For, let 1 be the side of a cube (i.e. one length-unit): its volume is 13 = 1 (i.e. one volume-unit), and doubling this cube is expressed by the equation x3 = 2, where x is the side of the cube with volume double that of the original cube, but x = 3√2, which is an irrational number or incommensurable length.44 However, widespread indifference towards, and ignorance of, mathematics in Greece of Plato’s time is emphasized and harshly castigated by the philosopher in his Laws, where the deep ignorance of mathematics among the Greeks is evinced exactly by their ignorance of incommensurability. The relevant passage cannot but have contributed to the shaping of the anecdote we are discussing here. One of the codiscussants, an anonymous old man from Athens known as the Athenian stranger, expresses forcefully his disgust at the shameful ignorance of incommensurability that is current everywhere in Greece, including Athens. A widely traveled man, who has thought long and hard on law-related matters, thus on public education, too, which in his view is of great importance for a state and the preservation of its laws, he admits with shame to his having learned of incommensurability only late in life: as he puts it, not to know that magnitudes do not always have a common measure, and, as a result, do not always stand to each other in the ratio of one whole number to another, befits not human beings but rather a kind of pig-like creatures (Lg. 7, 819d5–820c9). Cleinias, the elderly Cretan codiscussant of the Athenian stranger and member of a commission charged with making laws for a colony that Cnossos and other cities of Crete are about to found on the island, the topic of the dialogue, has never heard of incommensurability.45

44 45

Cf. Pl. Tht. 147d3–148b2: if the volume of a cube is n volume-units, the side of the cube is incommensurable for n which is not a cubic number, e.g. for n = 2. On Lg. 7 see also section 3.3.1.

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c. Epinomis How incommensurability is related to the general, arithmetical formulation of the planar analogue of the cube-duplication problem and the solid problem itself is said by the author of the Epinomis to be a divine marvel for those able to grasp it (990d4–6, e1–2). What is actually said to be a divine marvel is, in plane geometry, the assimilation to planes of numbers that are dissimilar by nature to each other; in stereometry, correspondingly, it is the assimilation, not in the same way as in plane geometry, to solids of numbers that are dissimilar by nature to each other (990d1– 4, d6–e1). The description of plane geometry refers to the representation of a number as a rectangle, i.e. as the area of a rectangle, as well as to its representation as a square, i.e. to the construction of a square of the same area as the rectangle (squaring a given area entails, as pointed out above in 3.2.4.a, the construction of one mean proportional, the planar analogue of constructing two mean proportionals and thus doubling a given cube). The description of stereometry refers similarly to the representation of a number as a rectangular solid, i.e. as the volume of a solid, and to its representation as a cube, i.e. to the construction of a cube that is equal in volume to the rectangular solid, a problem that is equivalent to constructing two mean proportionals and thus duplicating the cube.46 If the numbers represented geometrically are dissimilar, i.e. if they are non-square and non-cubic in the case of plane and solid geometry respectively, their geometric transformation from rectangle and rectangular solid into square and cube respectively produces incommensurable lines.47 As it is, arithmetic unifies plane and solid geometry through the phenomenon of incommensurability.48 What is undoubtedly hinted at in the Epinomis about incommensurability and the unity conferred through it by arithmetic on plane and solid geometry is, therefore, that it is a marvel of divine origin for those able to comprehend it. This, however, is clearly parallel to what we have in the story about the Delian problem: an oracle from Apollo led the Delians to marvel uncomprehendingly at the phenomenon of incommensurability, manifested stereometrically in the problem of doubling a cube, and seek help from Plato, who, marveling comprehendingly at this phenomenon of divine origin, did give it to them. It cannot be accidental that the author of the Epinomis hints at the Apolline origin of the divine marvel that e.g. the cube-duplication is for those able to grasp it, for he couches next the overarching superiority of arithmetic to both the geometrical study of planes and solids in a manner that allows passage to harmonics, a branch of mathematics that he takes to be arithmetically coherent with the plane and solid geometry through the theory of means: in this context he argues that the insertion of two means between terms in the ratio 1:2 has been given to humans by the blessed choir of the Muses (990e1–991b4). He refers to the insertion of the harmonic and arithmetic means between two terms an octave apart

46 47 48

See Knorr (1975) 93–94. For the problems of constructing a cube which is equal in volume to a rectangular solid and doubling a cube see Saito (1995) 121–122. Cf. above n. 42 and 44. On the unity of mathematics in the Epinomis see ch. 2.2.4.

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that produces the basis for the organization of scales:49 if this is a gift of the Muses to humans, however, the same must apply to its stereometric analogue, the insertion of two means between lines in a given ratio, which enlarges, e.g. doubles, a cube in this ratio, a marvel of divine origin for those able to understand it. What originates from the Muses can be plausibly assumed to originate ultimately from Apollo, the leader of their blessed choir.50 The connection of the problem of cubeduplication with Apollo in the anecdote of the Delian problem can thus be traced back to the Epinomis. The important role that, according to the Delian anecdote, a Delphic oracle played in the solution to the problem of cube-duplication can be traced back to another passage from the same work. This story links in effect progress in a branch of mathematics, stereometry, with oracles issued by Apollo. Now, according to the author of the Epinomis, knowledge of the planets had been first obtained by the peoples of the Near East, Egypt and Syria, due to the meteorological conditions prevailing there at summertime that allow easy observations of all celestial objects, and has then spread to Greece and everywhere else (986a8–987a6). In his view, however, the Greeks perfect all their borrowings from non-Greeks, and, although it is difficult to discover indisputably everything related to the planets and the other celestial objects, there are many good hopes that the Greeks, being reliant on their multifaceted education and the Delphic oracles, will be able to pay better and more just respect to all these divinities, the celestial objects, than the rumored service done to them by the pioneering foreigners (987d3–988a5). This bizarre mention of Delphic oracles here can be plausibly explained in light of the fact that in the Epinomis, as in Plato’s Laws 10, 899b3–9, all celestial objects are thought to be divinities.51 The author of the Epinomis seems to insinuate that this is not a Greek conception of the celestial objects but rather a Near Eastern import.52 He has the Athenian stranger urge his codiscussants to enshrine unhesitatingly the divinity of all celestial objects as well as the understanding of the periodicity of their motions, to which he attaches paramount religious importance, in the laws of the future Cretan colony (987a6–7), i.e. in the sacred laws, which in the Cretan city will be kept distinct from the other laws.53 But Plato has the Athenian stranger declare in Lg. 6, 759c6–d1, that laws concerning religious matters must be brought to the Cretan city from Delphi; hence, according to the author of the Epinomis, the admission of the divinity of the celestial objects, a non-Greek concept, into the city’s sacred laws must be vetted by the Delphic oracle; the oracle will also promote the development of astronomy in the Greek world, for it is a discipline of religious significance amounting to the worship of the gods that are the celestial objects, and, unless it advances as much as humanly possible, proper respect to all 49 50 51 52

53

For a succinct explanation see Burnyeat (2000) 49–51. For Apollo as leader of the Muses see Graf (2009) 33–34. Cf. Pl. Ti. 39e3–40d5. On the divine nature of the planets, the Sun and the Moon, the Greek “wandering” stars, in Mesopotamia see Rochberg (2005) 326–327; for the association of the stars with the gods Anu, Ea and Enlil see Evans (1998) 8–9. See Morrow (1993) 407–408.

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these gods will not be paid. At the same time, however, the author of the Epinomis might have had a hazy idea of the role that divination had played in contemporary Near East in the advancement of astronomy, hence he hoped that in Greece, too, divination, i.e. the Delphi, the most important oracular center in Greece, would help astronomy grow.54 One who took the Epinomis, where Apollo’s oracles are said to be factors that will positively affect the progress of astronomy in Greece, as a genuine Platonic work would have been easily led to assume that, according to Plato, other branches of mathematics, too, still undeveloped in his day, would eventually advance in Greece thanks to Delphic oracles, all the more so since the author of the Epinomis emphasizes not only the intrinsic unity conferred by arithmetic on plane and solid geometry and harmonics but also a deeper unity that is bestowed on all these branches of mathematics by astronomy, and as source of which he must also consider the Muses and thus, ultimately, Apollo. d. Republic 7 As seen in ch. 1.7.1, stereometry is presented in R. 7 as an undeveloped branch of mathematics (528a6–c7), and the only example of the problems it deals with Plato gives in this passage is “the increase of cubes” (528b2), the generalization of the cube-duplication problem. Based on the assumption of the “agreement” between the life and writings of Plato, one surmised that the philosopher must have played a role in the solutions to this problem that appeared in his day and that an oracle from Apollo must also have played a role in the same episode: this was the historical basis that allowed “Plato” to speak in the Epinomis of Delphic oracles as drivers of progress in astronomy, and thus in all branches of mathematics, whose unity is implied in the same work to derive from the Muses and thus, ultimately, from Apollo. That the purpose of the oracle must have been to shake the Greeks out of their indifference towards mathematics followed from the reasons Plato gives in R. 7 for the undeveloped state of stereometry: the lack of progress in this field is due to its inherent difficulty as well as to the need for a superintendent of those mathematicians who work in it and the non-existing support for mathematics from all contemporary cities in Greece, let alone to this subject’s being lightly esteemed. The discussion of solid geometry in R. 7 echoes Plato’s complaint in the Laws over the ignorance of incommensurability among the Greeks, and must be one of the sources for the story of Plato as a director of contemporary mathematics who presented the mathematicians with problems to solve, another result of the application of the fruitful principle for uncovering biographical “facts” about an author that assumes an “agreement” between his actual life and his writings. Since mathematics is propedeutic to philosophy, and Plato famously thinks in R. 5 that, until philosophy rules in the states, they will not get rid of their present troubles (473c11–e4), but he also makes it clear in R. 7 that, until mathematics has advanced 54

On the important relation between divination and astronomy in ancient Mesopotamia see the brief but very informative account in Evans (1998) 296–299. On celestial divination in ancient Mesopotamia see Rochberg (2004).

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sufficiently, it will not propedeutically serve philosophy (530b6–c4; 531b2–c5), the political lesson for all Greeks he reportedly descried in Apollo’s cube-doubling oracle was also extracted via applying this principle. e. Timaeus The god’s promoting peace among the Greeks in the anecdote by forcing upon them the study of a proportion-involving geometrical problem also harks back to the Timaeus with its description of proportion as supremely beautiful binder and unifier (31c2–4): in this context, the Demiurge, a god like Apollo, forges friendship and unity on the level of the whole cosmos via a theorem of arithmetic involving two mean proportionals, like the cube-duplication problem in geometry, and cubic numbers (31c4–32c4).55 3.3. THE SOURCES OF THE MYTH ABOUT PLATO’S IMPACT ON ASTRONOMY 3.3.1. Laws 7 Besides the discussion of the propedeutics to philosophy in R. 7, a further likely source for the story in Simplicius is the similar discussion in Lg. 7 of the subjects in which the citizens of a planned Cretan colony must be educated. Two of the three codiscussants have already been introduced: an anonymous old man from Athens, commonly known as the Athenian stranger, and the Cretan Cleinias. The third is the Lacedaemonian Megillus. All three are of the same age group and confer about the laws and institutions suitable for a colony that Cnossos and other Cretan cities plan to found on the island; Cleinias has been appointed to the ten-member commission charged with making laws for the new city. The Athenian stranger argues that the colony must provide to its citizens basic training in the sciences of numbers, of all three dimensions (in geometry) as well as of the paths in which the celestial objects, compared with one another, travel according to nature (817e5–818a1). Plato has the stranger say forcefully that the Greeks are mistaken in believing that the Sun, the Moon and the five planets never travel each along the same celestial path. The reference here is certainly to the spirals whose coils are the diurnal circles that these celestial objects are each observed to describe over a period of time: for they follow the celestial sphere in its diurnal rotation and at the same time also move zodiacally eastwards at an inclination to the celestial equator, on the ecliptic, in the case of the Sun, or near it, in the case of the Moon and the five planets. In our very few sources for the early history of Greek astronomy, the spirals of the planets, the Sun and the Moon are first mentioned by Plato, in his Timaeus, who also gives the correct explanation of the phenomenon (38b6–39b2). However, when Plato has the stranger castigate the Greeks for believing that the Sun, the Moon and the planets 55

Given two cubic numbers a3 and b3, it is a3 : a2b = a2b : ab2 = ab2 : b3 (Euc. El. 8.12).

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never travel each along the same celestial path, it is conceivable that he wants his audience to also see an allusion to the observed deviations of a planet’s zodiacal path from circularity due to retrogradations (821b5–c5): ΑΘ. Ὦ ἀγαθοί, καταψευδόµεθα νῦν ὡς ἔπος εἰπεῖν Ἕλληνες πάντες µεγάλων θεῶν, Ἡλίου τε ἅµα καὶ Σελήνης. ΚΛ. Τὸ ποῖον δὴ ψεῦδος; ΑΘ. Φαµὲν αὐτὰ οὐδέποτε τὴν αὐτὴν ὁδὸν ἰέναι, καὶ ἄλλ’ ἄττα ἄστρα µετὰ τούτων, ἐπονοµάζοντες πλανητὰ αὐτά. ΚΛ. Νὴ τὸν Δία, ὦ ξένε, ἀληθὲς τοῦτο λέγεις· ἐν γὰρ δὴ τῷ βίῳ πολλάκις ἑώρακα καὶ αὐτὸς τόν τε Ἑωσφόρον καὶ τὸν Ἕσπερον καὶ ἄλλους τινὰς οὐδέποτε ἰόντας εἰς τὸν αὐτὸν δρόµον ἀλλὰ πάντῃ πλανωµένους, τὸν δὲ ἥλιόν που καὶ σελήνην δρῶντας ταῦθ’ ἃ ἀεὶ πάντες συνεπιστάµεθα. ATH. Now, my friends, nearly all we Greeks lie about great gods, the Sun and Moon. CL. What lie do we tell? ATH. We say that they and some other celestial objects along with them never travel in the same path, and we call them wanderers. CL. Yes, by Zeus, stranger, what you say is true; in my life I have often watched the morning and the evening stars and some other celestial objects never travel along the same path but wander every which way, and we all know that the Sun and Moon are constantly doing this.

As the Athenian stranger explains next, each of the celestial objects in question here moves zodiacally always along the same circular path and only appears to wander in many, just as the one among them whose zodiacal motion is the fastest (i.e. the Moon) appears, and is thus mistakenly believed, to be the slowest, for its zodiacal motion is not considered independently from the diurnal rotation (822a4–c6): ΑΘ. … οὐ γάρ ἐστι τοῦτο, ὦ ἄριστοι, τὸ δόγµα ὀρθὸν περὶ σελήνης τε καὶ ἡλίου καὶ τῶν ἄλλων ἄστρων, ὡς ἄρα πλανᾶταί ποτε, πᾶν δὲ τοὐναντίον ἔχει τούτου – τὴν αὐτὴν γὰρ αὐτῶν ὁδὸν ἕκαστον καὶ οὐ πολλὰς ἀλλὰ µίαν ἀεὶ κύκλῳ διεξέρχεται, φαίνεται δὲ πολλὰς φερόµενον – τὸ δὲ τάχιστον αὐτῶν ὂν βραδύτατον οὐκ ὀρθῶς αὖ δοξάζεται, τὸ δ’ ἐναντίον ἐναντίως. ταῦτ’ οὖν εἰ πέφυκεν µὲν οὕτως, ἡµεῖς δὲ µὴ ταύτῃ δόξοµεν, εἰ µὲν ἐν Ὀλυµπίᾳ θεόντων ἵππων οὕτως ἢ δολιχοδρόµων ἀνδρῶν διενοούµεθα πέρι, καὶ προσηγορεύοµεν τὸν τάχιστον µὲν ὡς βραδύτατον, τὸν δὲ βραδύτατον ὡς τάχιστον, ἐγκώµιά τε ποιοῦντες ᾔδοµεν τὸν ἡττώµενον νενικηκότα, οὔτε ὀρθῶς ἂν οὔτ’ οἶµαι προσφιλῶς τοῖς δροµεῦσιν ἡµᾶς ἂν τὰ ἐγκώµια προσάπτειν ἀνθρώποις οὖσιν· νῦν δὲ δὴ περὶ θεοὺς τὰ αὐτὰ ταῦτα ἐξαµαρτανόντων ἡµῶν, ἆρ’ οὐκ οἰόµεθα γελοῖόν τε καὶ οὐκ ὀρθὸν ἐκεῖ γιγνόµενον ἦν ἂν τότε, νῦν ἐνταυθοῖ καὶ ἐν τούτοισι γίγνεσθαι γελοῖον µὲν οὐδαµῶς, οὐ µὴν οὐδὲ θεοφιλές γε, ψευδῆ φήµην ἡµῶν κατὰ θεῶν ὑµνούντων. ΚΛ. Ἀληθέστατα, εἴπερ γε οὕτω ταῦτ’ ἐστίν. ATH. …This opinion, my friends, about the Sun and Moon and some other celestial bodies is not correct, i.e. that they wander, but the truth is precisely the opposite—each of them always travels in one and the same path, not in many but always in a single one, circularly, although it appears to move along many paths—and the quickest of them is wrongly thought to be the slowest and vice versa. If these are the facts but we think otherwise, suppose that we held a similar view about horses racing at Olympia or about long-distance runners, and that we thus proclaimed the quickest to be slowest and the slowest quickest, and sang praises to the loser as the winner: the praises for the runners would be neither right nor acceptable to them, though they were mortals. Now, however, when we make the same

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3. The myth about Plato’s role in the development of mathematics mistake about gods, do we not think that what would have been ridiculous and wrong there and then is, here and now and in such a matter, by no means ridiculous but impious, for we sing a false tale about gods? CL. Very true if these are the facts.

The citizens of the future Cretan colony, the Athenian stranger then suggests, ought to study astronomy if he and his codiscussants can provide a demonstration of the above, i.e. that each of the so-called wandering celestial objects actually moves always along the same circular path and that sight errs in perceiving the fastest of them as the slowest. Moreover, the study of astronomy in the Cretan city will proceed as far as the introduction of these two facts (822c7–d1): ΑΘ. Οὐκοῦν ἂν µὲν δείξωµεν οὕτω ταῦτ’ ἔχοντα, µαθητέα µέχρι γε τούτου τὰ τοιαῦτα πάντα, µὴ δειχθέντων δὲ ἐατέον; καὶ ταῦτα ἡµῖν οὕτω συγκείσθω; ΚΛ. Πάνυ µὲν οὖν. ATH. So, if we will manage to show that there are the facts, should astronomy be studied up to this point but left aside if we will not? Do we agree on this? CL. Of course.

If the wanderings of the planets are not only their spiral motions, as is the case with the wanderings of the Sun and the Moon, but also their retrograde motions, then here Plato has the Athenian stranger expect himself and his codiscussants to demonstrate something quite close to what, according to the account in Simplicius, the philosopher himself challenged the astronomers to show. Unlike the sketch of the future astronomy in R. 7, this passage from Lg. 7 does mention emphatically circular motions of the planets, contrasting them with the apparent, and implicitly non-circular, wanderings of these celestial objects; moreover, the derivation of the shape of the paths of the apparent motions from that of the circular ones could be plausibly assumed to furnish the proof at issue. But Plato makes it clear that in this passage he does not throw any challenge to astronomers through the stranger as his mouthpiece: for the Athenian stranger is presented as expecting himself and his codiscussants, not astronomers, to prove that each of the so-called wandering celestial objects in fact does not wander but moves always in the same circular orbit. Nor can we, moreover, plausibly assume that Plato has the Athenian propose that the citizens of the planned Cretan colony ought to study astronomy only if the astronomers, on whom as experts the stranger and his codiscussants can be tacitly supposed to rely, will be able to prove the open conjecture that each of the socalled wandering celestial objects actually moves always in the same circular path: Plato has the stranger know that this is so (why he suggests that the citizens of the future Cretan colony must study astronomy, if he and his codiscussants are able to demonstrate it, will be explained below). Moreover, this will be one of the two lessons in which the study of astronomy by the citizens of the future colony will culminate; but the other, i.e. that sight perceives erroneously the fastest of the wandering celestial objects as the slowest, is elementary, as befits the astronomy that the citizens ought to study: whence follows that the first lesson, too, is elementary. As it is, that each of the wandering celestial objects in fact does not

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wander but moves always in the same circular orbit is not a challenging conjecture in astronomy, posed by Plato through the Athenian stranger as his mouthpiece, but a piece of elementary astronomy. Leaving aside for now the question whether the wanderings of these objects referred to in 821b5–c5 include the retrogradations of the planets, let us turn to a further piece of evidence pointing in the same direction. Between his criticism of the Greeks for believing that the Sun, the Moon and the five planets never travel each along the same celestial path (821b5–c5) and his explaining to Cleinias and Megillus that each of the celestial objects at issue here actually moves always in the same circular path and only appears to move in many (822a4–c6) the Athenian stranger first explains why he attaches such importance to the correction of the false belief prevailing among the Greeks with regard to the motions of the Moon, the Sun and the planets: celestial objects are divinities and, since disorder is completely foreign to the divine, to view them as wanderers, and thus disorderly, is blasphemy; he goes on to reveal that it was only late in life that he finally rid himself of this mistaken belief (821c6–822a3). Earlier on, in the discussion of arithmetic and geometry, Plato has him reveal that it was only late in life that he also rid himself of a ridiculously and disgracefully ignorant belief about magnitudes: arithmetic and geometry will be taught to the citizens of the colony with the aim of remedying this ignorance, i.e. that not all geometrical magnitudes are commensurable (819d5–820e7). The Athenian stranger is thus presented as having learned late in life what the citizens of the planned Cretan colony will have to learn in their mathematical and astronomical studies: first, not all magnitudes are commensurable; second, the socalled wandering celestial objects do not wander but always move each in the same circular path. However, just as the first is undoubtedly a basic result that was well established by the time Plato wrote the above passages, so must be the second. It cannot be plausibly assumed that, although Plato requires from public education in elementary mathematics the wide dissemination of knowledge of a well-established fact at the time, i.e. incommensurability, he poses as precondition for public education in elementary astronomy the proof of a conjecture he himself has made: one whose truth, if shown, ought to be widely disseminated, and will thus form one of the two culminating lessons of public education in elementary astronomy. The Athenian stranger’s suggestion that the citizens of the future Cretan colony ought to study astronomy if he and his codiscussants will be able to demonstrate the circularity of the zodiacal path along which each of the so-called wandering celestial objects actually moves always does not imply that at the time of the discussion this circularity was a still open conjecture; if it were, how could he know it as an indisputable fact?56 It is meant to allay his Dorian codiscussants’ caution toward the subjects he recommends for the basic curriculum in the colony. After he explains why the three sciences he recommends are worth studying (818b7–e2), Cleinias agrees reluctantly (818e3–4). Prompted perhaps in part by the guarded attitude of the Cretan towards his proposals, the stranger remarks next that, since it is difficult to make the educational program he advocates into law, the 56

On the possible origin of the conception of all celestial motion as circular see above n. 12.

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precise details of this are best left for a later occasion (818e5–7). Cleinias then observes that the Athenian appears scared by the habitual neglect in Dorian areas of the subjects just suggested to be most appropriate for the basic schooling of a Cretan colony’s citizens, and the stranger promptly agrees (818e8–819a1). What Plato has the sophist Hippias say in Hp.Ma. 285b5–c5 about the Lacedaemonians does apply to all Dorians: they cannot bear to listen about the stars and celestial phenomena, nor do they enjoy listening about geometry, most of them being unable even to count! Indeed, in 821b5–c5 (quoted above) Plato presents Cleinias as blithely unaware of the fact that the morning and evening stars are one single celestial body, and of the exact number of the so-called wandering celestial bodies. Scientific illiteracy is not an exclusive problem of Dorian lands, however. As the stranger’s “autobiographical” references subtly suggest, it prevails even in such an intellectually vibrant Greek state as Athens. 3.3.2. Laws 10 The Athenian stranger does not proceed immediately to allay his codiscussants’ reservations about the inclusion of astronomy in the curriculum of the colony by convincing them of the circularity of the path in which each of the so-called wandering luminaries always moves zodiacally. In view of the above, we expect an argument presupposing as little knowledge of astronomy as possible and contributing to the eradication of blasphemy from the city and the establishment of foundations for firm belief in the divine (cf. the beginning of the discussion of astronomy’s suitability for the curriculum in 820e7–821b4). We do find such an argument in Lg. 10. The conclusion of this argument comes very close to the requisite one, deviating from it in a very interesting detail. The argument at issue is embedded in a broader argument showing that the heavens with their contents are moved by one or more rational and good souls (896d5–897c3):57 ΑΘ. Ἆρ’ οὖν τὸ µετὰ τοῦτο ὁµολογεῖν ἀναγκαῖον τῶν τε ἀγαθῶν αἰτίαν εἶναι ψυχὴν καὶ τῶν κακῶν καὶ καλῶν καὶ αἰσχρῶν δικαίων τε καὶ ἀδίκων καὶ πάντων τῶν ἐναντίων, εἴπερ τῶν πάντων γε αὐτὴν θήσοµεν αἰτίαν; ΚΛ. Πῶς γὰρ οὔ; ΑΘ. Ψυχὴν δὴ διοικοῦσαν καὶ ἐνοικοῦσαν ἐν ἅπασιν τοῖς πάντῃ κινουµένοις µῶν οὐ καὶ τὸν οὐρανὸν ἀνάγκη διοικεῖν φάναι; ΚΛ. Τί µήν; ΑΘ. Μίαν ἢ πλείους; πλείους· ἐγὼ ὑπὲρ σφῷν ἀποκρινοῦµαι. δυοῖν µέν γέ που ἔλαττον µηδὲν τιθῶµεν, τῆς τε εὐεργέτιδος καὶ τῆς τἀναντία δυναµένης ἐξεργάζεσθαι. ΚΛ. Σφόδρα ὀρθῶς εἴρηκας. ΑΘ. Εἶεν. ἄγει µὲν δὴ ψυχὴ πάντα τὰ κατ’ οὐρανὸν καὶ γῆν καὶ θάλατταν ταῖς αὑτῆς κινήσεσιν, αἷς ὀνόµατά ἐστιν βούλεσθαι, σκοπεῖσθαι, ἐπιµελεῖσθαι, βουλεύεσθαι, δοξάζειν ὀρθῶς ἐψευσµένως, χαίρουσαν λυπουµένην, θαρροῦσαν φοβουµένην, µισοῦσαν στέργουσαν, καὶ πάσαις ὅσαι τούτων συγγενεῖς ἢ πρωτουργοὶ κινήσεις τὰς δευτερουργοὺς αὖ παραλαµβάνουσαι κινήσεις σωµάτων ἄγουσι πάντα εἰς αὔξησιν καὶ φθίσιν καὶ διάκρισιν καὶ σύγκρισιν καὶ τούτοις ἑποµένας θερµότητας ψύξεις, βαρύτητας κουφότητας, σκληρὸν 57

Presupposed is the conception of soul as first cause of all motion (895e10–896b2).

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καὶ µαλακόν, λευκὸν καὶ µέλαν, αὐστηρὸν καὶ γλυκύ, καὶ πᾶσιν οἷς ψυχὴ χρωµένη, νοῦν µὲν προσλαβοῦσα ἀεὶ θεὸν ὀρθῶς θεοῖς, ὀρθὰ καὶ εὐδαίµονα παιδαγωγεῖ πάντα, ἀνοίᾳ δὲ συγγενοµένη πάντα αὖ τἀναντία τούτοις ἀπεργάζεται. τιθῶµεν ταῦτα οὕτως ἔχειν, ἢ ἔτι διστάζοµεν εἰ ἑτέρως πως ἔχει; ΚΛ. Οὐδαµῶς. ΑΘ. Πότερον οὖν δὴ ψυχῆς γένος ἐγκρατὲς οὐρανοῦ καὶ γῆς καὶ πάσης τῆς περιόδου γεγονέναι φῶµεν; τὸ φρόνιµον καὶ ἀρετῆς πλῆρες, ἢ τὸ µηδέτερα κεκτηµένον; βούλεσθε οὖν πρὸς ταῦτα ὧδε ἀποκρινώµεθα; ΚΛ. Πῶς; ATH. Must we, therefore, agree next that soul is the cause of what is good and bad and beautiful and ugly and just and unjust and of all opposites if we are to consider it the cause of all things? CL. How can we not? ATH. But, since soul governs and dwells in all things that move in every which way, must we not say that it also governs the heavens? CL. Yes. ATH. One soul or more? More–let me answer for you. But let us assume no less than two, a beneficial soul and one capable of producing the opposite effects. CL. Most true. ATH. Very well. Soul, therefore, moves all things in the heavens, on Earth and in the sea with its own motions called will, thought, care, planning, true and false belief, joy and grief, confidence and fear, hate and love, and with all other motions that are related to them or primary and assume control over the secondary motions of bodies, driving all of them to increase and decrease and separation and combination, hence to heating and cooling, heaviness and lightness, hardness and softness, whiteness and blackness, bitterness and sweetness and all those that soul employs to guide everything rightly and happily when it operates together with reason, always a veritable god among gods, but brings about the exact opposite when it consorts with unreason. Are we to assume that this is so or are we still under the suspicion that it might be otherwise? CL. By no means. ATH. Which kind of soul are we, therefore, to say controls the heavens and the Earth and the entire rotation? The wise and full of goodness or the other one, which lacks both traits? Do you want us to give an answer in this manner? CL. How?

The first alternative will be established if it is shown that the entire motion of the heavens and their contents is similar to that of reason (897c4–d2): ΑΘ. Εἰ µέν, ὦ θαυµάσιε, φῶµεν, ἡ σύµπασα οὐρανοῦ ὁδὸς ἅµα καὶ φορὰ καὶ τῶν ἐν αὐτῷ ὄντων ἁπάντων νοῦ κινήσει καὶ περιφορᾷ καὶ λογισµοῖς ὁµοίαν φύσιν ἔχει καὶ συγγενῶς ἔρχεται, δῆλον ὡς τὴν ἀρίστην ψυχὴν φατέον ἐπιµελεῖσθαι τοῦ κόσµου παντὸς καὶ ἄγειν αὐτὸν τὴν τοιαύτην ὁδὸν ἐκείνην. ΚΛ. Ὀρθῶς. ΑΘ. Εἰ δὲ µανικῶς τε καὶ ἀτάκτως ἔρχεται, τὴν κακήν. ΚΛ. Καὶ ταῦτα ὀρθῶς. ATH. Let us assume, my friend, that if the entire motion of the heavens and of all the things in them is similar in nature and akin to the motion of reason and its rotation and thoughts, then clearly we must say that the good soul takes care of the whole cosmos and drives it on its aforementioned course. CL. Correct.

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3. The myth about Plato’s role in the development of mathematics ATH. But we should opt for the bad soul if the cosmos moves in a dissordely and irrational way. CL. Correct again.

At this point, however, the stranger warns that, since we do not have a sufficiently good understanding of the nature of reason, it is best to try to answer by means of an image (εἰκόνα) the obvious question which motion is that of reason (897d3–e3): ΑΘ. Τίνα οὖν δὴ νοῦ κίνησις φύσιν ἔχει; τοῦτο ἤδη χαλεπόν, ὦ φίλοι, ἐρώτηµα ἀποκρινόµενον εἰπεῖν ἐµφρόνως· διὸ δὴ καὶ ἐµὲ τῆς ἀποκρίσεως ὑµῖν δίκαιον τὰ νῦν προσλαµβάνειν. ΚΛ. Εὖ λέγεις. ΑΘ. Μὴ τοίνυν ἐξ ἐναντίας οἷον εἰς ἥλιον ἀποβλέποντες, νύκτα ἐν µεσηµβρίᾳ ἐπαγόµενοι, ποιησώµεθα τὴν ἀπόκρισιν, ὡς νοῦν ποτε θνητοῖς ὄµµασιν ὀψόµενοί τε καὶ γνωσόµενοι ἱκανῶς· πρὸς δὲ εἰκόνα τοῦ ἐρωτωµένου βλέποντας ἀσφαλέστερον ὁρᾶν. ΚΛ. Πῶς λέγεις; ATH. Well, what is the nature of the motion of reason? It is difficult, my friends, to give a reasonable answer to this question, so now you should let me help you with the answer. CL. You are right. ATH. Let us not, therefore, answer by looking straight in the eye of the Sun, bringing darkness to the noon, as if we could ever behold and understand reason satisfactorily with mortal eyes. It is safer to try to see it looking at an image of what our question is about. CL. What do you mean?

The image is provided by the first item in an earlier division of ten kinds of motion and change.58 The motion of reason must be similar to motion that is circular, in the same place round a center, and uniform (897e4–898b9): ΑΘ. Ἧι προσέοικεν κινήσει νοῦς τῶν δέκα ἐκείνων κινήσεων, τὴν εἰκόνα λάβωµεν· ἣν συναναµνησθεὶς ὑµῖν ἐγὼ κοινῇ τὴν ἀπόκρισιν ποιήσοµαι. ΚΛ. Κάλλιστα ἂν λέγοις. ΑΘ. Μεµνήµεθα τοίνυν τῶν τότε ἔτι τοῦτό γε, ὅτι τῶν πάντων τὰ µὲν κινεῖσθαι, τὰ δὲ µένειν ἔθεµεν; ΚΛ. Ναί. ΑΘ. Τῶν δ’ αὖ κινουµένων τὰ µὲν ἐν ἑνὶ τόπῳ κινεῖσθαι, τὰ δ’ ἐν πλείοσιν φερόµενα. ΚΛ. Ἔστι ταῦτα. ΑΘ. Τούτοιν δὴ τοῖν κινήσεοιν τὴν ἐν ἑνὶ φεροµένην ἀεὶ περί γέ τι µέσον ἀνάγκη κινεῖσθαι, τῶν ἐντόρνων οὖσαν µίµηµά τι κύκλων, εἶναί τε αὐτὴν τῇ τοῦ νοῦ περιόδῳ πάντως ὡς δυνατὸν οἰκειοτάτην τε καὶ ὁµοίαν. ΚΛ. Πῶς λέγεις; ΑΘ. Τὸ κατὰ ταὐτὰ δήπου καὶ ὡσαύτως καὶ ἐν τῷ αὐτῷ καὶ περὶ τὰ αὐτὰ καὶ πρὸς τὰ αὐτὰ καὶ ἕνα λόγον καὶ τάξιν µίαν ἄµφω κινεῖσθαι λέγοντες, νοῦν τήν τε ἐν ἑνὶ φεροµένην κίνησιν, σφαίρας ἐντόρνου ἀπεικασµένα φοραῖς, οὐκ ἄν ποτε φανεῖµεν φαῦλοι δηµιουργοὶ λόγῳ καλῶν εἰκόνων. ΚΛ. Ὀρθότατα λέγεις. ΑΘ. Οὐκοῦν αὖ ἥ γε µηδέποτε ὡσαύτως µηδὲ κατὰ τὰ αὐτὰ µηδὲ ἐν ταὐτῷ µηδὲ περὶ ταὐτὰ µηδὲ πρὸς ταὐτὰ µηδ’ ἐν ἑνὶ φεροµένη µηδ’ ἐν κόσµῳ µηδ’ ἐν τάξει µηδὲ ἔν τινι λόγῳ κίνησις ἀνοίας ἂν ἁπάσης εἴη συγγενής; 58

The division is set out in 893b6–894c9.

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ΚΛ. Εἴη γὰρ ἂν ἀληθέστατα. ATH. Let us consider as image the one of the ten motions distinguished above to which reason is similar. I will remind you of it, and we will give the answer together. CL. Very nice. ATH. Do we then recall at least this distinction of those made above, that, of all things, some move and some are at rest? CL. Yes. ATH. And that, of those things that move, some move in one place and some in many? CL. Yes. ATH. Of those motions, therefore, the one in the same place must always be about a center, of course, an imitation of the wheels that have been turned on the lathe, and must have the closest possible kinship and similarity to the motion of reason. CL. What do you mean? ATH. If we said that both of them, reason and motion in one place, being likened to the rotation of a sphere that has been turned on the lathe, moved in the same manner and in the same place and about the same things and in relation to the same things and with a single pattern and order, then we would never turn out to be bad makers of beautiful images in speech. CL. You speak most truly. ATH. And would not a motion be akin to complete irrationality, therefore, if it moved never in the same manner nor in the same place nor about the same things nor in relation to the same things nor in one place nor with order nor with some pattern? CL. Undoubtedly it would.

Plato has the Athenian dwell on circular and uniform motion as an image, a sign that we should take this seriously into account. The motion of reason must be an image, too, in view of the stranger’s disavowal of knowledge about the nature of reason and the connection of staticity with reason in Sph. 249b12–c5: it can only be an image of the way by which a divine rational soul brings about motion.59 The circularity and uniformity of this motion is an appropriate image of the hallmark of a soul’s rationality, static and symmetric order, thanks to the exceptional symmetry of the circle and the orderliness of uniform motion. Plato clearly thinks that the symmetry and orderliness of divine reason that will turn out to be in charge of celestial motions outstrips by far the symmetry and orderliness of the circle and uniform motion in a circle, whose symmetry and orderliness are the highest that are accessible to our limited human minds. Cleinias agrees with the Athenian stranger that the motion of the heavens is caused by a rational and good soul or souls (898c1–8): ΑΘ. Νῦν δὴ χαλεπὸν οὐδὲν ἔτι διαρρήδην εἰπεῖν ὡς, ἐπειδὴ ψυχὴ µέν ἐστιν ἡ περιάγουσα ἡµῖν πάντα, τὴν δὲ οὐρανοῦ περιφορὰν ἐξ ἀνάγκης περιάγειν φατέον ἐπιµελουµένην καὶ κοσµοῦσαν ἤτοι τὴν ἀρίστην ψυχὴν ἢ τὴν ἐναντίαν. ΚΛ. Ὦ ξένε, ἀλλὰ ἔκ γε τῶν νῦν εἰρηµένων οὐδ’ ὅσιον ἄλλως λέγειν ἢ πᾶσαν ἀρετὴν ἔχουσαν ψυχὴν µίαν ἢ πλείους περιάγειν αὐτά. ATH. There is no difficulty any more to declare that, since we consider soul to be the mover of all things, we must say that the rotation of the heavens is necessarily brought about by an ordering and caring soul either fully good or the opposite. 59

By being itself in motion, an image from our everyday experience; cf. 898e5–899a10.

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3. The myth about Plato’s role in the development of mathematics CL. But, stranger, what we have just said implies that denying that these things are carried round by one or more perfectly good souls is actually impious.

If the rationality and goodness of the soul(s) responsible for the motion of the heavens is to follow from the similarity of this motion to the motion of reason, we need as tacit premise the circularity and uniformity of the motion of the heavens, i.e. of the diurnal rotation:60 it is similar to the motion of reason, hence caused by a rational soul. Plato has then the Athenian stranger lead Cleinias to agree that, if the motion of the heavens is due to a rational and good soul or souls, so is the motion of the Sun, the Moon and each of the other celestial objects, for they are all carried round by the heavens that are moved by soul (898c9–d5): ΑΘ. Κάλλιστα, ὦ Κλεινία, ὑπήκουσας τοῖς λόγοις· τόδε δὲ προσυπάκουσον ἔτι. ΚΛ. Τὸ ποῖον; ΑΘ. Ἥλιον καὶ σελήνην καὶ τὰ ἄλλα ἄστρα, εἴπερ ψυχὴ περιάγει πάντα, ἆρ’ οὐ καὶ ἓν ἕκαστον; ΚΛ. Τί µήν; ATH. You have followed the argument very carefully, Cleinias. Listen now to this, too. CL. What is this? ATH. Are not the Sun and the Moon and the other celestial bodies carried round each by soul if it is soul that carries round the sum total of things? CL. Certainly.

What is implicitly denied in this passage is that the motion of the Moon, the Sun and each of the planets said in Lg. 7 to appear as a constant roaming in many paths is truly such: for, if it were, only an irrational and bad soul could possibly cause these wanderings.61 The conclusion Plato has the Athenian stranger and Cleinias draw is indeed that the Sun and the other celestial bodies are gods (898d6–899b2), whence follows the necessity of the belief in the existence of gods and their involvement in human affairs (899b3–d6): ΑΘ. Ἄστρων δὴ πέρι πάντων καὶ σελήνης, ἐνιαυτῶν τε καὶ µηνῶν καὶ πασῶν ὡρῶν πέρι, τίνα ἄλλον λόγον ἐροῦµεν ἢ τὸν αὐτὸν τοῦτον, ὡς ἐπειδὴ ψυχὴ µὲν ἢ ψυχαὶ πάντων τούτων αἴτιαι ἐφάνησαν, ἀγαθαὶ δὲ πᾶσαν ἀρετήν, θεοὺς αὐτὰς εἶναι φήσοµεν, εἴτε ἐν σώµασιν ἐνοῦσαι, ζῷα ὄντα, κοσµοῦσιν πάντα οὐρανόν, εἴτε ὅπῃ τε καὶ ὅπως; ἔσθ’ ὅστις ταῦτα ὁµολογῶν ὑποµενεῖ µὴ θεῶν εἶναι πλήρη πάντα;

60

61

This tacit premise must be observational; see ch. 1.4 and cf. Arist. Cael. Α 5, 272a5–7, and Β 10, 291a34–b1. It cannot be provided by Lg. 7, 822a4–c6, as Jirsa (2008) 252 thinks, for at issue in this passage are only the wanderings of the Moon, the Sun and the five planets, not, as in the argument discussed here, ἡ σύµπασα οὐρανοῦ ὁδὸς ἅµα καὶ φορὰ καὶ τῶν ἐν αὐτῷ ὄντων ἁπάντων (897c4–5), i.e. the diurnal rotation. Moreover, that the wanderings of the Moon, the Sun and the five planets are apparent, each of these celestial objects moving always in the same circular path, has not been shown in Lg. 7, hence it cannot be used tacitly as premise in the argument that the heavens with their contents are moved by a rational and good soul. On the contrary, it must be part of the demonstrandum: for, if the Moon, the Sun and the planets wander, their celestial wanderings must be caused by an irrational and bad soul or souls. Cf. 898b5–c5, quoted and translated above.

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ΚΛ. Οὐκ ἔστιν οὕτως, ὦ ξένε, παραφρονῶν οὐδείς. ΑΘ. Τῷ µὲν τοίνυν µὴ νοµίζοντι θεοὺς ἐν τῷ πρόσθεν χρόνῳ, ὦ Μέγιλλέ τε καὶ Κλεινία, εἰπόντες ὅρους ἀπαλλαττώµεθα. ΚΛ. Τίνας; ΑΘ. Ἢ διδάσκειν ἡµᾶς ὡς οὐκ ὀρθῶς λέγοµεν τιθέµενοι ψυχὴν γένεσιν ἁπάντων εἶναι πρώτην, καὶ τἆλλα ὁπόσα τούτων συνεπόµενα εἴποµεν, ἢ µὴ δυνάµενον βέλτιον λέγειν ἡµῶν, ἡµῖν πείθεσθαι καὶ ζῆν θεοὺς ἡγούµενον εἰς τὸν ἐπίλοιπον βίον. ὁρῶµεν οὖν εἴτε ἱκανῶς ἤδη τοῖς οὐχ ἡγουµένοις θεοὺς εἰρήκαµεν ὡς εἰσὶν θεοί, εἴτε ἐπιδεῶς. ΚΛ. Ἥκιστά γε, ὦ ξένε, πάντων ἐπιδεῶς. ΑΘ. Τούτοις µὲν τοίνυν ἡµῖν τὸ λόγων τέλος ἐχέτω· τὸν δὲ ἡγούµενον µὲν θεοὺς εἶναι, µὴ φροντίζειν δὲ αὐτοὺς τῶν ἀνθρωπίνων πραγµάτων, παραµυθητέον. ATH. As regards all celestial bodies and the Moon, and as regards the years and the months and all seasons, are we to say something else or the same–that, since the causes of all these have turned out to be one or more perfectly good souls, we will declare them gods who order the whole heavens either by dwelling in bodies as living beings or in whatever manner and by whatever means? Will anyone agree with this and be able to deny that all things are full of gods? CL. No one, stranger, will be so out of his right mind. ATH. Let us then set terms, Megillus and Cleinias, to the person who till now does not believe in gods and get done with it. CL. What terms? ATH. Either he will explain to us that we are wrong to assume that soul is the first cause of all things and to accept all that follow from this or, if he is unable to say something better, he will have to accept our argument and live believing in gods for the rest of his life. So, let us see now if we have addressed satisfactorily those who deny the existence of gods or not. CL. Most satisfactorily, stranger. ATH. Let us then put an end to our argument against the unbelievers. We must now address gently the man who thinks that there are gods but denies that they are concerned with human affairs.

As far as the fixed stars are concerned, that they are moved by a soul or souls and must be considered gods follows thus: what holds of the diurnal rotation holds of them, too, since the diurnal rotation is their motion in parallel circles extended to the sphere whose circles of latitude are these paths.62 That the Sun, the Moon and the planets, which are observed to wander, are also moved by a rational and good soul or souls and must also be considered gods, hence their wanderings are only apparent, for a rational and good soul or souls cannot possibly bring about irrational motion, in all probability follows thus: sharing in the diurnal rotation, which must be brought about by a rational and good soul or souls, is a prominent element of the complex motions at issue here, so the latter must be brought about by the same cause in their entirety, for it is either this cause or its opposite that brings about a motion (this assumption is stated in 898c1–5 for the diurnal rotation). Although the stranger has demonstrated that the Sun, the Moon and each planet cannot wander, he has not shown that each of these seven celestial objects moves uniformly in one and the same circular path but only that its motion is caused by one or more souls that are rational and good. Such motion, however, is not necessarily uniform motion in one and the same circular path: this type of 62

Cf. the passages from Euclid’s Phaenomena and Ptolemy’s Almagest quoted in ch. 1.4.

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motion is just an image of rational orderliness. Therefore, when in Lg. 7 the Athenian says that the Sun, the Moon and the planets do not wander but move each in same circular orbit, he does not speak literally but uses an image: no matter what the shape of their paths, the motions at issue are not irrational wanderings but orderly and rational. It is easy to see how the passages from Lg. 7 discussed above could be taken out of context and give rise to the pseudo-biographical story recorded by Simplicius, according to which Eudoxus’ theory of homocentric spheres arose as solution to a problem posed by Plato to contemporary astronomers: to save the phenomena of the Moon, the Sun and the planets by hypothesizing circular, uniform and orderly motions of these celestial objects, for all celestial motion must be circular, uniform and orderly. But what is set out in Lg. 7 as demonstrandum, that each of these celestial objects actually moves always in one and the same circular path and only seems to wander, is not presented as a conjecture posed to astronomers, which could lead to the intricate theory of homocentric spheres: it is instead set out as an important but elementary fact of astronomy, one of whose truth, though, the two scientifically illiterate Dorian characters of the dialogue ought to be convinced. Moreover, when the time comes in Lg. 10 for its implicit proof, it seems to be tacitly weakened in the following way: each of the so-called wandering celestial objects actually moves always with a motion which is like uniform circular motion, and only appears to wander. Uniform circular motion is an image, hence Plato is not firmly committed to it but to what it depicts: rational orderliness. In the light of R. 7, ultimately this is best understood not as a property of some actual motions other than, and bringing about, the wanderings at issue, perhaps even the zodiacal motions of the Sun and the Moon, but of static mathematical objects, whatever and however complex they might be, that are mirrored in the paths of these sensible motions: the goal of astronomy ought to be the ascent from the realm of the senses to those static objects by discovering and studying other such objects, more and more akin to them (see ch. 1.6.1–2). Since they belong to elementary astronomy, the circular paths mentioned in Lg. 7, one for each of the so-called wandering celestial objects, can only be the seven circles centered at the Earth in which the Moon, the Sun and the planets of our solar system known in antiquity are assumed to move zodiacally in Ti. 38b6–39b2, the direction of motion being to the left (apparently for an observer in the Earth’s northern hemisphere facing to the south). All these seven circles are inclined to the celestial equator, and are assumed to be coplanar, as if not only the Sun but also the Moon and the five planets were always on the ecliptic, whose angle with the celestial equator is not specified. This omission and the assumption of coplanarity, however, is easily accounted for by the fact that the circles are simply images, not real paths. Moreover, the many paths in which the Sun, the Moon and the planets appear each to wander must be the coils of the spirals which the celestial objects at issue are each observed to describe in a period of time as they also participate in the opposite diurnal rotation. As remarked above in 3.3.1, this phenomenon is explained in the passage of the Timaeus just cited; the explanation is based on the simple image of circular zodiacal motion, and in the same passage Plato also mentions the second fact in

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which, according to Lg. 7, the teaching of elementary astronomy in the future Cretan colony must culminate (if he draws implicitly a parallel in Lg. 7 between the explanation of the phenomena of spiral motion in astronomy and incommensurability in geometry as both being long-established at the time but still not widely known in Greece, it is unlikely that the former was first detected by him). The many paths, along which the wandering celestial objects are said in Lg. 7 to roam, cannot also include the retrograde paths of the planets since retrogradations are not accounted for by the image of circular zodiacal paths of the planets painted in Ti. 38b6–39b2, as Plato himself implicitly acknowledges in 40c3–d5.63

63

If the seven coplanar circles, along which the Moon, the Sun and the five planets of our solar system known in antiquity are assumed to move zodiacally in Ti. 38b6–39b2, are nothing but mere images, partial representations of a complex situation, there is no point in attempting to imagine how Plato would get planetary retrogradations out of them; see e.g. Knorr (1990) 313–317, Cornford (1997) 137 and the detailed discussion of the issue in Gregory (2000) 128– 153. These images can account only for the spiral motion of the Moon, the Sun and the five planets. In Ti. 40c3–d5 Plato admits his lack of “visible imitations”, δι᾽ ὄψεως µιµήµατα, for a host of celestial phenomena, among which planetary retrogradations (ἐπανακυκλήσεις) are most probably included, though Bowen (2013) 230–232 denies it (see ch. 1.6.2): if what is called in Lg. 10 “image” is called in Ti. 40c3–d5 “visible imitation”, then the seven concentric and coplanar circles of the Moon, the Sun and the planets in Ti. 38b6–39b2, if they are also tacitly thought to be images, as in Lg. 7 and 10, cannot account for retrogradations (recall that these circles are tellingly introduced as “paths” of a rational soul’s “motions”). Gregory (2000) ch. 4 has argued that in the Laws and the Timaeus Plato considers all celestial motions as regular, stable and amenable to precise mathematical prediction, unlike in R. 7. Prediction does not seem to have been a desideratum for Plato or contemporary astronomers, however, although he does seem to have followed contemporary astronomers in firmly believing that even the mysteriously complicated motions of the planets are amenable to precise mathematical description in the sense elaborated above in ch. 1. Moreover, nothing in the Laws suggests that Plato abandoned his view in R. 7, 530a4–b5, that celestial bodies cannot possibly move in an undeviatingly regular manner since they are material. As for the Timaeus, since in 47e5–48a5 reason is said to fail in completely persuading material necessity to submit to it (cf. 29d7–30a6), it is not easy to see how celestial motions, which are undergone by material objects, can be exempted from this general principle even if the incomplete persuasiveness of reason over material necessity is assumed to mean the incomplete instantiation of teleology due to initial conditions and logical possibility, as Gregory (2000) 113–115 suggests addressing this passage.

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INDEX OF PASSAGES Aeschylus

Cael.

frr. 181a–182 Radt: 37 n. 40 Alexander of Aphrodisias

Α 5 272a5–7: 128 n. 60 B 6 288a13–17: 51 n. 61 10 291a34–b1: 128 n. 60

in Metaph. (Hayduck)

GA

79.8–11: 14

A 3 717a23–29: 54

Anonymous Prolegomena to Platonic Philosophy (Westerink)

Metaph.

5.13–24: 59 n. 82, 112, 115 Archimedes Eratosth. (Heiberg) 2.430.1–9: 29 n. 31 Spir.

Α 6 987b14–18: 13 B 2 996b20–21: 76 n. 9 997b12–30: 55 997b12–998a6: 15, 35, 48 n. 56, 49, 49 n. 57, 53–54 3 999a6–10: 101 n. 50 E 1 1026a23–27: 82–83

Def. 1: 50 n. 60 Archytas

K 7 1064b6–9: 82–83

DK 47 A 19: 75 A 23a: 108 n. 12 B 1: 45, 73 B 4: 45, 75

M 2 1077a9–10: 83 1077b17–22: 83 6 1080a12–35: 13 7 1081a5–12: 13–14 1082b24–28: 14

Aristotle

PA

APr.

Γ 14 675b22–27: 54

A 23 41a21–32: 76, 80, 97

Ph.

APo.

Θ 10 267a21–b6: 54 n. 67

A5 7 9 13

Fragments

74a17–25: 83 75a38–b20: 96–97 76a4–15: 97 n. 43 78b34–79a16: 97 n. 43

211 Rose: 54 n. 67 489 Rose: 110

de An.

[Arist.]

B 2 413a17–20: 76 n. 9

Mech.

EE

851b34–35: 48 n. 56

A 8 1218a24–30: 100 n. 49

Index of passages Pr.

137

XVI.9, 915a25–32: 108 n. 12

1.216.3–18: 104 1.207.7–10: 21 n. 15 and 17 2.532.19–533.3: 104 n. 2

Aristoxenus

Dicaearchus

Harm. (da Rios)

fr. 85 Wehrli = 74 Mirhady: 110

39.8–40.4: 100

Diodorus Siculus

Autolycus

12.36.2–3: 21 n. 17

de Ort.

Diogenes Laertius

1 Def. 1–2: 30 Prop. 1: 26–27, 29 n. 30

8.13: 110 8.86: 70 n. 97

2 1: 29 n. 30 2: 29 n. 30 4: 28 n. 28 5: 28 n. 28 9: 28 n. 28

Euclid

de Sphaera

Elementa 1 Def. 2: 14 Def. 4: 14 Def. 19: 15 Def. 22: 15 Prop. 19: 17

Def. 1–3: 30 Prop. 1: 31 Prop. 2: 30 n. 33

2 14: 76 n. 9, 115 n. 43

Boethius

3 Prop. 16: 16

Mus. (Friedlein)

4 Prop. 16: 102 n. 51

285.7–286.19: 75

6 13: 76 n. 9, 115 n. 43

Callimachus

7 Def. 1: 18, 43 Def. 2: 18, 43

Ap. 60–63: 110 Censorinus De die natali 18.5: 21 n. 17 Claudius Ptolemy Alm. (Heiberg) 1.10.4–11.13: 33–34 1.13.10–12: 34 1.205.15–21: 21 n. 15

8 12: 120 n. 55 9 4: 97 n. 45 11 Def. 14: 17 Def. 18: 17 Def. 21: 18 12 7: 29 n. 31 10: 29 n. 31 Phaen. (Menge) 2.1–6.14: 32

138

Index of passages

Eudemus

Plato

fr. 133 Wehrli: 107 n. 8

Hp.Ma.

fr. 149 Wehrli: 21 n. 16 Eutocius in Sph. Cyl. (Heiberg) 3.56.2–10: 58 n. 81 3.56.14–58.14: 59 n. 82, 111 3.78.14–84.7: 58 n. 81 3.84.13–88.2: 58 n. 81 3.88.13–90.11: 58 n. 78, 110 3.88.18–23: 111 3.90.8–11: 59 n. 82, 112 Geminus Elem. Astr. 1.19–21: 107 8.50: 21 n. 17 Heraclitus DK 22 B 84a: 50 Hesiod Op. 383–387: 23–24 564–570: 24 597–599: 24 609–623: 24–25 Hippocrates of Chios

285b5–c5: 124 Lg. 5 747a1–5: 76 6 759c6–d1: 118 7 817e5–818a1: 120 817e5–818a3: 90 n. 28 817e5–822d1: 90 n. 27 818b7–e2: 123 818e3–4: 123 818e5–7: 124 818e8–819a1: 124 819d5–820c9: 116 819d5–820e7: 82, 123 820e7–821b4: 124 821b5–c5: 121, 123, 124 821c6–822a3: 123 822a4–c6: 121–122, 123, 128 n. 60 822c7–d1: 122 10 893b6–894c9: 126 n. 58 896d5–897c3: 124–125 897c4–d2: 125–126 897d3–e3: 126 897e4–898b9: 126–127 898c1–8: 127–128 898c9–d5: 128 898d6–899b2: 128 898e5–899a10: 127 n. 59 899b3–9: 118 899b3–d6: 128–129

DK 42 A 4: 111

12 963a1–964d9: 90 n. 28 967d4–968a4: 89–90

Philodemus

Men.

Acad.Hist. = PHerc 1021

81c5–d5: 93 82b9–e3: 115 82b9–85b7: 93 86b6–c2: 115–116

col. Y.3–17: 107 Philolaus DK 44 A 7a: 74

R. 4 430d3–432b2: 87 n. 25

Index of passages 430d3–433b4: 86 n. 23 433a8–b4: 87 n. 24 442b10–d7: 86–87 443c9–444a2: 86 n. 23, 87–88 5 462a2–b3: 88 473c11–e4: 12, 119 473e5–480a13: 12 475e6–476b11: 37 478e7–479c5: 37, 39 479a5–c5: 38 6 500b8–d10: 89 504e3–509d5: 43 504e6–505b3: 100 506d5–e1: 100 507b1–9: 13 n. 1 509b6–c2: 100 n. 47 509d6–510b1: 11 510b2–511a9: 72 510b2–511c2: 11, 14, 18 510b2–511d5: 98 510c2–5: 72 510c2–d3: 30, 31, 32–33 510c2–511a3: 48 n. 55 510d5–511a3: 35 511a10–b1: 72 511b2–c2: 72 511c3–e5: 40, 98 511d6–e5: 43 7 514a1–521b11: 36 517a8–c4: 36 519e1–520a5: 88 520a6–d6: 88 520a6–e3: 86 521b1–531c8: 71 521b1–531d8: 18 521c1–d7: 20, 34, 36 521d8–12: 19, 37 521e1–2: 57 522b1–c9: 19, 37 522b5–523a3: 78 522c1–11: 81, 85 522c1–e2: 19 522c5–7: 37 522c7–d9: 37 522e1–4: 37 522e5–524c9: 37 522e5–525c7: 54–55 523d3–4: 40 524b3–5: 39 524c10–d5: 38, 39

139

524d6–525a3: 38 525a4–c7: 39 525b9–c6: 40, 44 525c8–d3: 41 525c8–526a7: 41–42, 73 525d5–526c6: 41, 44 526b1–3: 44 526c7–d7: 19 526c11–e5: 37 527a1–b11: 35, 42, 46, 49 n. 57, 73 527b2–11: 49, 50 527b6–c11: 78 527d1–4: 19, 71 n. 1 527d1–528b1: 77 527d5–528a3: 20, 34 528a4–5: 20 528a6–10: 28–29, 31, 35, 46, 68 528a6–c7: 56, 68, 107, 114, 119 528a9–10: 71 528a10–c7: 18 528a6–d11: 20, 34 528b1–2: 47, 77 528b3–4: 47, 77 528b5–c7: 91–92, 101 n. 50 528d1–10: 69 528d1–530c4: 78 528e1–529a2: 34 528e1–530c4: 20 528e1–531c8: 18, 73 529a3–c2: 35–36, 44–45, 54, 73 529c3–5: 46 529c6–7: 79 529c6–d6: 47, 49, 50, 51, 52, 54, 59, 66, 76, 78, 104 529c6–530a2: 101 529d7–530a3: 48, 55 530a4–b5: 67 n. 90 530b6–c2: 49 n. 57, 67 530b6–c4: 19, 48, 79, 120 530c3–4: 47 530d6–10: 45, 46, 71, 73, 74 530d7: 71 530e1–531c8: 46 531b2–c5: 19, 56 n. 71, 92 n. 34, 120 531b2–c8: 99, 101 n. 50 531b8–c4: 68 531c9–d5: 71, 98, 99 533a10–c6: 12, 85 533a10–d9: 42 533c8–534a8: 40, 42, 44 533d6–9: 40

140 534b3–6: 40 534b3–d2: 37 n. 41 534e2–535a2: 84 536d4–537a3: 84 n. 21 537b7–c3: 11, 73 537b7–c8: 83–84, 98 537e1–539d7: 40 10 596a6–b3: 102 Phd. 74d4–75a4: 48 n. 55 78c10–79a5: 13 n. 1 84c8–85b9: 70 98a6–b3: 100 101c9–e3: 44

Index of passages 38b6–39b2: 53, 120, 130, 131 38c6: 53 38d2–4: 52 n. 63 39a5–b2: 53 n. 64 39c5–d2: 52 39d1–2: 79 39e3–40d5: 118 n. 51 40c3–d5: 131 47e5–48a5: 131 n. 63 48e2–52d1: 13 n. 4, 102 53c4–56c7: 49 n. 56 55c4–6: 49 n. 56 62b6–63e8: 102 67a7–c3: 71 n. 1 79e10–80b8: 71 n. 1 [Plato]

Phdr.

Epin.

265e1–3: 102

246a7–c3: 13 n. 2 249b12–c5: 127

986a8–987a6: 118 987a6–7: 118 987d3–988a5:118 989e1–991b4: 80 990b8–c5: 81 990d1–4: 80, 117 990d4–6: 117 990d6–e1: 80, 117 990e1–991b4: 81, 117–118 990e1–2: 117 991b5–c1: 81 991c2–6: 81 991d4–5: 114 991d5–992a3: 81 992a3–6: 114

Smp.

Ep.

210e2–211b5: 13 n. 3, 41

7 342a7–344d2: 14 n. 8 13 360c2–4: 110 n. 16

Phlb. 56d4–e6: 44 Plt. 262a8–263a1: 102 299e1–2: 76 Sph.

Tht. 147d3–148b2: 75, 115 n. 42, 116 n. 44 156c8: 47 n. 51 156d2–3: 47 n. 51 Ti. 29d7–30a6: 131 n. 63 31c2–32a7: 87 n. 25 31c2–4: 120 31c4–32c4: 120

Plutarch Moralia 386E3–10: 110 578E9–579B1: 109 n. 14 579B1–D3: 109–110 718E7–F4: 59 n. 82, 111–112 Dio 19.4: 110 n. 16

Index of passages Marc. 14.9–11: 59 n. 82, 111–112 Thes. 2.21: 110 Proclus in Euc. (Friedlein) 7.13–8.20: 82 43.22–44.9: 82 54.1–8: 18 64.16–68.6: 107 n. 8 66.8–68.4: 107 77.7–78.13: 68 n. 91 185.8–19: 50–51 185.19–25: 51 212.24–213.11: 58 n. 79, 111 n. 21 269.8–21: 102 n. 52 Protagoras DK 80 B 7: 15 n. 10 Simplicius in Cael. (Heiberg) 488.10–24: 103–104 492.31–493.11: 105–106 497.17–22: 21 n. 16 Theodosius of Bithynia De diebus 2.18: 21 n. 17 Theon of Smyrna Expositio (Hiller) 2.1–2: 113 2.3–12: 110, 114 2.13–14: 110 2.13–16.2: 114

141

In his Republic Plato considers grasping the unity of mathematics as the ultimate goal of the mathematical studies in which the future philosopher-rulers must engage before they turn to philosophy. How the unity of mathematics is supposed to be understood is not explained, however. This book argues that Plato conceives of the unity of mathematics in terms of the mutually benefiting links between its branches, just as he conceives of the unity of the state outlined in the Republic in terms of the common benefit for all citizens. Evidence for this view is provided by a discussion of his conception of astronomy as a propedeutic to philosophy,

which can be best understood as hinting at a historically possible link between fourth-century-BC astronomy and solid geometry. The monograph also includes a detailed discussion of two well-known stories about Plato: not only he motivated Greek mathematicians to solve a difficult problem in solid geometry with his interpretation of a Delphic oracle given to the inhabitants of the island of Delos but he also posed the question which led to the development of the astronomical theory of homocentric spheres. It is argued that these stories are best understood as fictional episodes in Plato’s life, constructed from passages in his works.

www.steiner-verlag.de Franz Steiner Verlag

ISBN 978-3-515-11076-1