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Published online by Cambridge University Press: 11 February 2009
Ever since Proclus wrote his commentary on Plato's Republic, repeated attempts have been made to find a hidden number of cosmic significance in Rep. 8.546. For the Neo-Platonist it was natural to look for esoteric secrets in ancient works; among the men of the New Learning at the end of the Middle Ages there were enough astrologers and necromancers to ensure respect for the proposition; we are now again enamoured of irrationality. But the scholars who attempted such calculations around 1900 must have considered Plato himself a mystery-monger.
In this article I propose: (i) to show why such attempts are mistaken, (ii) to discuss what early writers who mention the passage say about its meaning, (iii) to provide a mathematician's translation that fits the context, and to comment on it; for the currently accepted explanation is unsatisfactory.
‘There is fairly widespread agreement that the geometrical number is 12,960,000 = 3,6002 = 4,800 × 2,700, but on the method by which this number is reached the widest divergence exists’ or, from an earlier, different guess: ‘…one can, so to speak, state a priori that Plato's number is a multiple of 19 ten thousands’, i.e. the text is approached with a ready-made answer in mind. There are three further objections to the conventional view, (a) It takes the text out of its context as if it had strayed from the Timaeus.
1 Thomas, I., Greek Mathematical Works i (Loeb), 400n.Google Scholar; cf. Bury, R. G. in CR 33 (1919), 45–6Google Scholar on A. G. Laird, Plato's Geometrical Number and the Comment of Proclus. ‘While accepting Adam's solution…he maintains that “his method of reaching the 36002 is wrong…” …On Mr Laird's view, the number 216, on which Adam set such store, seems to disappear, and with it, apparently, much of the pertinence of the whole passage to the subject of “better and worse births”.’
2 Dupuis, J., Théon de Smyrne (Paris, 1892), 372nGoogle Scholar.
3 Denkinger, M., REG 68 (1955), 38, 71CrossRefGoogle Scholar.
4 Ad Att. 7.13.5.
5 Inst. Orat. 1.3.13.
6 In Plat. Rep., ed. Kroll, W., 2.8.4fGoogle Scholar.
7 Cod. Theod. 9.13; cf. 16.25; 62. Coll. Leg. Mos. et Rom. Tit. 15 (in Riccobono, , FIRA. 2.578)Google Scholar.
8 Pauli Sent. 5.21.3 (FIRA 2.407)Google Scholar.
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10 Tac. Hist. 1.22.
11 6.562; cf. Suet. Vitell. 14.3.
12 In Plat. Rep. 2.1.3Google Scholar.
13 ibid. p. 36.
14 Adam, James, The Nuptial Number of Plato (London, 1891), 11 n. 1Google Scholar.
15 In Plat. Rep. 2.26, 16fGoogle Scholar.
16 Ed. Hiller, 43f.
17 Théon de Smyrne 380Google Scholar.
18 Proclus, , In Plat. Rep. ed. Kroll, , 2.394fGoogle Scholar.
19 Cf. Heath, T. L. in The Legacy of Greece, ed. Livingston, R. W. (1922), 110Google Scholar.
20 Proclus 2.405.
21 Oeuvres de Platon, traduites par V. Cousin, quoted from Dupuis, op. cit. 370.
22 Pol. 5.1316a.
23 De Is. et Os. 373fGoogle Scholar. Oddly enough in view of this and of Plato's own words (Rep. 546c fin.) J. Dupuis objects to the appellation ‘marriage number’ (op. cit. 388).
24 E.g. Aristides Quintilianus 3.23; Iamblichus, , Vita Pyth. 27.130Google Scholar; Proclus, , In Eucl. (ed. Friedlein, ) 427fGoogle Scholar.
25 Loc. cit.
26 The Nuptial Number of Plato, 23Google Scholar.
27 Loc. cit.
28 Loc. cit.; also Alex. Aphrod. In Metaph. 56.19fGoogle Scholar. (Hayduck 75.27f.).
29 Loc. cit.
30 Diels, , Vorsokrat. 1.421.5Google Scholar; Arist. Quint., loc. cit.
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32 Probl. 15.3, 910b36.
33 Index Aristotelicus.
34 De mundio 2.391b 19, prob. spurious.
35 In the same sense in De caelo 2.4, 286b13; 3.5, 304a15; 306b7.
36 Physica 1.5, 188a22f.
37 De Anim. 1.2, 404b23f.
38 E.g. Metaph. 1.21, 8, 15f.Google Scholar; 24, 8, 30f.; Plato, , Rep. 534cGoogle Scholar; Philo, De opif. mundi 92 (Cohn-Wendland 1.3i, 26)Google Scholar; ibid. 98; Iamblichus, , Prot. (Pistelli) 119–20Google Scholar; 124; Comm. Math. 36f., where it is ascribed to Archytas; Euseb. Praep. Ev. 2.9.3, 524a (Mras 2.24.10f.).
39 De opif. mundi 49Google Scholar.
40 Plato, I.e.'s ‘geometrical number’, Rep. 546c finGoogle Scholar.
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42 Loc. cit. (n. 22).
43 Rep. 459d fGoogle Scholar.
44 The Exact Sciences of Antiquity 2 (Brown Univ. Press, 1957), 27Google Scholar.
45 Heath, T. L., The Legacy of Greece, 103Google Scholar.
46 E.g. Plutarch, , Quaest. Rom. 264aGoogle Scholar; 288d;, De Is. et Os. 373fGoogle Scholar.
47 So according to Baltes, M., Kommentar zu Timaios Lokros (Leiden, 1972), 43Google Scholar; but Plato comes near to it in Politicus 262e.
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49 Op. cit. 41.
50 Macrob. Somn. Scip. 2.11; Dupuis, op. cit. 371, 383.
51 REG 63Google Scholar.
52 CRAI 1933, 228–35Google Scholar, quoted after Denkinger, loc. cit. Cf. Platon, Oeuvres 6 (Paris, 1967)Google Scholar, Rep. 8, p. 9n. 1Google Scholar.
53 In Rep. 2.36, 13fGoogle Scholar.
54 Ed. Hiller, 9.41.
55 Arithm. Introd. 17.6.
56 In Nic. Arithm. 93–4.
57 Vita Pyth. 27.130Google Scholar.
58 Plut. De E apud Delph. 388c; Quaest. Rom. 264aGoogle Scholar; Arist. Quint. De Mus. 3.23; Theon 102.5, with a different explanation.
59 E.g. Theon 102. 1f.
60 Thomas, I., Greek Mathematical Works (Loeb), 399 n.bGoogle Scholar.
61 Brandwood, L., Word Index to Plato (Leeds, 1976) s.vGoogle Scholar.
62 In Metaph. 56.19f.Google Scholar, (Hayduck 75.31).
63 Iamblich., loc. cit.
64 REG 68 (1955), 38fCrossRefGoogle Scholar.
65 De opif. mundi 102, 47 (Cohn-Wendland 1.35.15.10f.)Google Scholar. Cf. Plutarch, , Epit. 1.3Google Scholar, in Diels, , Doxogr. Graeci, p. 282Google Scholar.
66 De decalogo27, 4.274.
67 Arist. Quint. De Mus. 3.23; Alex. Aphrod., In Metaph. 56.19fGoogle Scholar.
68 Aristoxenux, , Harmonica, ed. Macran, H. S. 1.21, 20Google Scholar; 1154; 46, 1; Plato, , Tim. 36Google Scholar; Diels, , Vorsokrat. 1.409f.Google Scholar; Philolaos frg. 6; Philo, De opif. mundi 96 fin.Google Scholar; Arist. Quint, loc. cit.; Macrob. Somn. Scip. 2.1.
69 De opif. mundi 97, 1.33Google Scholar; Vita Mosis 2.79 (C.-W. 4.219).
70 De Is. et Os. 373fGoogle Scholar.
71 Tim. 54a fGoogle Scholar.
72 Reidemeister, K., Das exakte Denken der Griechen, 50 (commenting on a different passage)Google Scholar.
73 Hero Alex. Geometr. 8.1; also found in Proclus, , Eucl. (Friedlein) 424Google Scholar.
74 Neugebauer, , The Exact Sciences in Antiquity 36fGoogle Scholar. The interest in and knowledge of ‘Pythagorean’ numbers was not confined to the Near East. Unfortunately I cannot find the source, published some years ago, for the following information about an Orkney Henge: ‘The [inaccurate] circle was in fact an ellipse, described from a triangle with sides in the ratio 5:12:13, and with diameters in integral numbers of megalithic yards.’
75 Cf. Plato, , Tim. 54Google Scholar, where the similar right-angled isosceles triangles are all ‘of the same nature’.
76 Op. cit. 103.
77 Geometr. 9.1; repeated by Proclus, , Eucl. (Friedlein) 467Google Scholar.
78 Théon de Smyrne, 370Google Scholar. See Heath, T. L., The Legacy of Greece 117Google Scholar.
79 ‘Artio-perissoi’ (see any ancient Arithmetic), Plato, , Parmen. 143eGoogle Scholar.
80 Nic. Geras. 2.17: Even numbers partake of the nature of the 2.
81 Cf. Plato, , Legg. 759dGoogle Scholar.
82 In a lecture.
83 Op. cit. 43f.
84 10.28a.
85 2.24, 10 fin., transl. by M. L. D. Ooge.
