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An early reference to perfect numbers? Some notes on Euphorion, SH 4171

Published online by Cambridge University Press:  11 February 2009

J. L. Lightfoot
J. L. Lightfoot
Affiliation:
All Souls College, Oxford
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Extract

Euphorion SH 417 (fr. 36 Van Groningen) deserves to be better known. A curiosity in itself—an apparent poetic reference to number theory—it is also, potentially, one of our earliest references to Euclidean material. On the authority of a late commentator on Aristotle, Euphorion, a mid-third-century b.c. Euboean poet who was also active in Athens and Antioch, is said to have mentioned perfect numbers—i.e. numbers which equal the total of all their factors, including 1 (but obviously excluding the number itself). It is a pity that the context in Euphorion does not survive, and that the line is only preserved, and indeed interpreted, by so late a source. But the wording of the fragment—if Westerink's restoration of its various corruptions (again, a pity) is plausible—would strongly suggest a reference to the notion of perfect number. The fragment has been known since Westerink published it in 1960, and was included both in Van Groningen's edition of Euphorion in 1977 and in Supplementum Hellenisticum (1983). But its implications have still not been discussed, and when David Fowler came to gather the evidence for references to Euclidean material in and after the third century b.c. in The Mathematics of Plato's Academy, his attention, unsurprisingly, was not drawn to it. Euphorion has had a bad press, as a poet of rebarbatively obscure myth and intractable vocabulary; yet he holds some interest, and we may be missing more insights into the intellectual life of the Hellenistic period which the perverse intelligence and baneful wit of the fragments display.

Type
Research Article
Information
The Classical Quarterly , Volume 48 , Issue 1 , May 1998 , pp. 187 - 194
Copyright
Copyright © The Classical Association 1998

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References

2 The fragment was not known to Powell. However I cite the other poetic fragments from Powell's Collectanea Alexandrina and testimonia for Euphorion's life from Van Groningen.

3 Fowler, D. H., The Mathematics of Plato's Academy: A New Reconstruction (Oxford, 1987). The earliest reference to Euclid which he finds is presented on p. 208, and consists of ostraka from Elephantine in Upper Egypt, which deal with the results of El. 13.10 and 16 (Pack2 2323). Like our material, they come from the third quarter of the third century B.C. They do not follow our text of the Elements, but seem to constitute an attempt to work it through to the writer's own personal satisfaction.Google Scholar

4 See Busse, A., Davidis Prolegomena et in Porphyrii Isagogen Commentarium (Berlin, 1904) = CAG XVIII.2. The discussion of perfect numbers is at pp. 22.18–35 Busse.Google Scholar

5 The beginning of the text, including the section on perfect numbers, is represented only by the manuscripts Paris gr. 1939 and Monac. gr. 399; on the manuscripts see Westerink, Pseudo-Elias, pp. vii–viii.

6 This is true of the first five perfect numbers—Nicomachus knew the first four—sbut it ceases to be true afterwards: see Heath, T., History of Greek Mathematics (henceforward HGM) (Oxford, 1921; repr. New York, 1981), vol. I, p. 74Google Scholar

7 [Elias] has been using different verbs and the like.

8 Euclidis Elementa, ed. Heiberg I. L., rev. Stamatis E. S. (Leipzig, 1970), vol. II, p. 105.

9 Heath, HGM, vol. I, pp. 98–99.

10 David 8.10, Nicomachus 1.15.1.

11 Callimachus' Cleombrotus epigram (AP 7.471 = Ep. 23 Pf. = HE 1273–1276) in [Elias] 12.5; David, p. 31.30–31.33 Busse. Theognis 175–176 in [Elias] 13.4; David, p. 32.21–22; Elias, p. 15.17–15.18 Busse in CAG XVIII.l. There are also tags from the tragedians and Homer

12 ([Elias] 19.23) cites Ap. Rhod. 3.1323, also in Et.Mag. 43.47, s.v.

13 E.g. [Iamblichus] Theologoumena Arithmeticae, ed. V. de Falco (Leipzig, 1975), listing (p. 50.2–50.3) and (p. 48.21) among the names for six.

14 When the sum of the sequence of duplications starting from one (1, 2, 4, and so on) is a prime number, then that number, multiplied by the last term of the series, will result in a perfect number. The first two perfect numbers are produced thus: 6 = (1 + 2) x 2, and 28 = (1 + 2 + 4) x 4. On the other hand 1 + 2 + 4 + 8 results in 15, which is not a prime number, so that 1 5 x 8 = 120, which is not perfect.

15 See Heath, HGM, vol. I, pp. 74–75 and his commentary on Euclid El. 7 Def. 23 and 9.36

16 See on his Life and Works in Dictionary of Scientific Biography, ed. Gillespie, C. C. (New York, 1970–1978), vol. II, pp. 414–437 (I. Buhner-Thomas), esp. pp. 414–415, and on the transmission of the Elements, pp. 437–4359 (J. Murdoch), esp. pp. 437–438. Our main sources are Proclus, In Primum Euclidis, p. 68.6–23 Friedlein, and Pappus Collection VII.35, II, p. 678.10–678.12 Hultsch.Google Scholar

17 Treves, Pace P., Euforione e la storia ellenistica (Milan, 1955), pp. 48–54; see the review byGoogle ScholarFraser, P. M., Gnomon 28 (1956), 580–581.Google Scholar

18 See Euphorion's Suda entry (3801) and (a reference to Athenian citizenship) Helladius ap. Photius cod. 279 p. 532 = Testt. 1 and 3 Van Groningen; for Euclid, see Buhner-Thomas, art. cit. (n. 16).

19 An. 1.3 (PCGV 184–185 Euangelus).

20 For instance, see Carm. Aur. 47; Sext. Emp. Adv. Math. 7.94; Hierocles on Carm. Aw. FPGl p. 465 Mullach.

21 Theon of Smyrna, p. 46.12–13, Nicom. ap. Iamblich. Theol. Arithm., p. 44.15.

22 Burkert, W., Lore and Science in Ancient Pythagoreanism (Cambridge, MA, 1972), p. 431, n. 28;AP14.1;Athen. 1.4e.Google Scholar

23 Taisbak, C. M., ‘Perfect Numbers–A Mathematical Pun?’, Centaurus 20 (1976), 269–275CrossRefGoogle Scholar

24 Becker, O., ‘The Theory of Odd and Even in the Ninth Book of Euclid's Elements, An Attempt to Reconstruct its Original Form’, trans, in The Graduate Philosophy Journal 16. 1 (1992), 87–110, originally published as ‘Die Lehre vom Geraden und Ungeraden im neunten Buch der euklidischen Elemente’, Quellen und Studien zw Geschichte der Mathematik, Astronomie undPhysik Abteilung B3 (1934), 533–553.Google Scholar

25 Burkert (n. 22 above), p. 435

26 On Pythagorean ‘pebble arithmetic’, see Huffman, C. A., Philolaus of Croton, Pythagorean and Presocratic (Cambridge, 1993), pp. 185–186.Google Scholar

27 Arist. PA. 14, p. 73a 40–bl and II 13, p. 96a 35–37; Top. 0 2, p. 157a 39; Met. A 6,987b 33

28 Heath, HGM, vol. I, p. 72; O. Becker, Das mathematische Denken der Antike(Gottingen, 1957), pp. 46–47 asserts that Philolaus mentioned prime numbers. The reference to Philolaus is in Speusippus ap. Theol. Arithm. 82.10 (= Vors. I3, p. 304.5, but not included in the sixth edition), but it is far from clear that it is truly well-informed; see Huffman (n. 26 above), pp. 359–363. Thymaridas is also supposed to have made use of the notion, Iambi. In Nicom., p. 27.4.

29 See Waerden, B. L. Van der, Science Awakening (Groningen, 1954), vol. I, p. 111. He argues that the contents of the whole of Book VII of the Elements should be attributed to the Pythagoreans before Archytas.Google Scholar

30 could, of course, refer to the third-person singular: KB 1.602 and Schwyzer II.204–205

31 Philodemus De Piet. 44, p. 16.1 Gomp. = Orph. fr. 36 Kern:

32 See West, M. L., The Orphic Poems (Oxford, 1983), pp. 150–152, 160–163. According to West, it was the Orphic account which included the grave of Dionysus in Delphi and the preservation of the separate heart (cf. also Call. fr. 517/643 Pf.), and the non-Orphic one which had him fitted back together ‘on the same body or at least on the same skeleton’. So both Orphic and non-Orphic accounts are known to Euphorion.Google Scholar

33 Schultze, G., Euphorionea (Diss. StraCburg, 1888), p. 57;Google ScholarScheidweiler, F., Euphorionis Fragmenta (Diss. Bonn, 1908), p. 27.Google Scholar

34 Plat. Leg. Arist. H.A. 486a 9ff.

35 Barigazzi, A., ‘II Dionysos di Euforione’, in Miscellanea di Studi Alessandrini(Torino, 1963), pp. 416–454. He discusses this fragment on pp. 447–448: ‘Una citazione di Euforione , relativa, mi par cosa sicura, allo squartamento di Bacco, al quale vien riportato secondo una tradizione pitagonca I'origine dei numeri perfetti.’Google Scholar

36 Osiris, in many respects the prototype of the dismembered Dionysus, is said to have been cut up into fourteen pieces, a number given a lunar interpretation; but other traditions allude to 16, 26, and even 42 pieces (see Plut. Mor. 358a and 368a, and Gwyn Griffiths' commentary on De hide et Osiride, pp. 338–339).

37 Proclus on Plat. Tim. 35a (II 145.18 Diehl, Orph. fr. 210 Kern).

38 Firm. Mat. De Error. 8.2: alii crudeli morte caesum aut in olla decoquunt aut septem veribus corporis mei membra lacerata subfigunt, cited in Henrichs, A., Die Phoinikika des Lollianos(Bonn, 1972), pp. 67–68, n. 59.Google Scholar

39 See Burkert, W., Homo Necans: The Anthropology of Ancient Greek Sacrificial Ritual and Myth (Berkeley/Los Angeles, 1983), p. 105 with n. 11 on forms of dismemberment which are relics of hunting rituals.Google Scholar