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COUNTING ON EPIC: MATHEMATICAL POETRY AND HOMERIC EPIC IN ARCHIMEDES' CATTLE PROBLEM

Published online by Cambridge University Press:  27 November 2015

Max Leventhal
Max Leventhal*
Affiliation:
Corpus Christi College, Cambridgeml649@cam.ac.uk
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Extract

In 1773, the celebrated enlightenment thinker G.E. Lessing discovered in Wolfenbüttel's Herzog August Library a manuscript which contained a previously unknown Ancient Greek poem. The manuscript identified the author as Archimedes (c.287-212 BCE), and the work became known as the Cattle Problem (henceforth CP). On the surface, its twenty-two couplets capitalise on Homer's depiction of the ‘Cattle of the Sun’ in Book 12 of the Odyssey and its numerical aspect. A description of the related proportions of black, white, brown and dappled herds of cattle, which are then configured geometrically on Sicily, creates a strikingly colourful image. The author's decision to encode a number into the figure of the Cattle of the Sun styles the poem as a response to, and expansion of, Homer's scene. Reading through the work, though, it becomes clear that the mathematics is more complex than that of Homer's Odyssey.

Type
Research Article
Information
Ramus , Volume 44 , Issue 1-2: New Essays on Homer: Language, Violence, and Agency , December 2015 , pp. 200 - 221
Copyright
Copyright © Aureal Publications 2015 

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References

1. This article started life as an M.Phil. thesis, submitted to Cambridge's Faculty of Classics in June 2013. I am indebted to the guidance of my supervisor Liba Taub, and the advice of my examiners Oliver Thomas and Richard Hunter. Others who deserve mention for sharing with me their time and thoughts are James Halladay, Alex Matthews, Francesca Middleton and Lucia Prauscello. Naturally, any errors are my own.

2. The specifics of the ratios of the cattle need not detain us here. I offer at the end an edition of the full text (Appendix I), a translation (Appendix II) and the traditional mathematical delineation of the ratios (Appendices III and IV).

3. According to Hermann (1831), 230, C.F. Gauss was reported to have worked on the problem, although Krumbeigel (1880), 123, doubts Gauss's involvement. The key advance towards a solution is found in Wurm (1830), 194f., later developed in Nesselmann (1842), 484, and finalised in the form we have in Amthor (1880), 154f. It was he who found a method for calculating the solution's large size, expressing only the first four significant digits of a number containing hundreds of thousands of digits.

4. That is to say, the number was fully expressed. See Williams, German and Zarnke (1965) and in a more manageable form, Nelson (1981).

5. Cf. e.g. Heath (1921), 14.

6. I will not discuss the question of authenticity at length, but I take it as quite possible that the epigram is genuine. For further discussions cf. Struve and Struve (1821); Nesselmann (1842), 481f.; Krumbeigel (1880), 125. The most recent and balanced approach can be found in Fraser (1972), 407. I hope however that the discussion I offer here can at least be used to shed light on that debate.

7. It is still unclear how ancient mathematicians would begin to think about solving the problem, nor is it known if the creator of the mathematical problem knew the quantities beforehand, although another Archimedean work, the Sand-Reckoner, does develop a system for coping with large numbers. Cf. Vardi (1998), 318.

8. Although it is far from epistolary in content; cf. Rosenmeyer (2002). A subsequent and no doubt purposefully ambiguous question then arises as to whether Eratosthenes is a foreigner (ξεῖνος) or a guest-friend (ξεῖνος).

9. This appears to be the default position, although, as Sourvinou-Inwood (1995), 279f., admits, it is often unstated. Cf. Tueller (2010), 59f.

10. The ideas of playfulness, generic awareness and supplementation have been a fruitful area of research in recent years. Cf. Bing (1995), 115-31; Bing (1998); Selden (1998), 307-19; Gutzwiller (2002); Fantuzzi and Hunter (2004), 291-306.

11. Tueller (2008), 66-94.

12. Stewart (1976), 75f., and Murnaghan (1987), 91f., still offer the best discussions of disguise, recognition and guest-friendship in the Odyssey.

13. For inscriptions and re-performance cf. Day (1989), 16-20; Day (2007), 39-47; Day (2010), 14; Vestrheim (2010), 75-78; Tsagalis (2008b), 44-48. For the literary epigram and the symposium cf. Giangrande (1967).

14. Many such epigrams do in fact survive in the Palatine Anthology 9.457-80.

15. See also, for example, Thebaid Fr.1.

16. The clearest discussion of this is still Lenz (1980), 21-41.

17. Benson (2014). The issue of elegy was also highlighted by the anonymous reviewer. Benson's article came to my attention too late to remodel and incorporate fully his ideas herein. Needless to say, I encourage the reader to consult his work also.

18. See Benson (2014), 178-82.

19. Generally, his analysis of the structural similarities is strong. Antimachus, Hermesianax and Callimachus all employ elegy in catalogue form, and this may well have influenced Archimedes. His argument that something like the tradition of the Seven Sages exists behind the CP does not persuade.

20. E.g. Odyssey 6.191, 7.193, 8.301.

21. On the literal and figurative movements of reading epigrams see Höschele (2007).

22. Fr. 9 R = Strabo 1.2.37 Radt. For Strabo's positive view of Homer see most recently Kim (2010), 47-84.

23. Fr. 8 R = Strabo 7.3.6-7 Radt. The particular naming and concretisation of this theory as ἐξωϰεανισμός, however, comes only later with Crates of Mallos; cf. Crates frr. 44 and 77 Broggiato; Roller (2010), 120-23; Walbank (1979), 586f.

24. Gomme, Andrew & Dover (1970), 211.

25. Although he was not against Homer's poetry per se.

26. Cf. Pindar Isthm. 1.22; Nem. 3.67; Ol. 2.5, 13.14.

27. On both similes see most recently Graziosi and Haubold (2010), 226f.

28. Schol. in Ap. Rhod. 4.989i Wendel = fr. 16. Dettori. Lightfoot (2009), 79.

29. Cf. Dettori (2000), 21; Lightfoot (2009), 2.

30. Callim. frr. 1.9, 287 and 357 Pf.; Lycophr. 619 and 1263; Ap. Rh. 4.988. Eratosth. frr. XVIII-XIX Hiller = fr. 16 Powell. Eratosthenes’ Hermes was a learned and recondite composition; the most recent and illuminating contributions are by Scanzo (2002) and Di Gregorio (2010). It is intriguing that the adjective appears at significant points in these two acquaintances’ works.

31. Dettori (2000), 122f.

32. Ibid , 121.

33. LSJ s.v. ὄμπνιος.

34. Cf. LSJ ibid. Odysseus alone refrains from meat (Odyssey 12.391-400). This may suggest a playful meaning of ἐνὶ πραπίδεσσιν, where the Homeric πραπίδες means both ‘intellect’ and ‘midriff’ or broadly ‘stomach’; a notable change from φροντίς (line 2). See LSJ s.v. πραπίδες. However, the allusion is by no means certain.

35. Translation by Kwapisz (2013), 156.

36. Bing (1986), 224.

37. Kwapisz (2013), 156. Cf. Cerri (2005).

38. It is mentioned at Od. 5.64 and 239.

39. Bing (2009), 147-74, does in fact consider insightfully the difference between general and specific allusions.

40. Translations from the Palatine Anthology are adapted from Paton (1918).

41. Morgan (1998), 74f.; cf. Thompson (1994), 67f., and Cribiore (2001), 225f.

42. Although this compilation is undoubtedly Byzantine, it is now agreed that many of the epigrams, if not from the Hellenistic period, at least echo Hellenistic models. Cf. Cameron (1993), 268.

43. MacLachlan (1993), 51 n.23.

44. Mojsik (2011), 74f.

45. Cf. schol. ad Od. 24.1 Dindorf.

46. Netz (2009), 168, where Fantuzzi's thought per litteras is noted. See also Gow (1952), 128.

47. Sistakou (2008), 50-54.

48. Ibid ., 54 n.95.

49. Yatsuhashi (2010), 125-77.

50. Cribiore (1994); Cribiore (1996); Cribiore (2001), 194f.

51. The translation is my own.

52. Cf. most recently Murray (1981), 90, and Graziosi and Haubold (2010), 2. This is no doubt an over-simplification of the passage. Further discussion can be found in Ford (1991), 57-89.

53. Purves (2010a), 8, makes a similar suggestion.

54. Or they are not concerned in a scientific way at least. For numbers in epic see the meticulous study of Rubincam (2003).

55. It appears only twelve times in the Iliad, four as verse initial.

56. The most recent, illuminating and extensive discussion of this idea can be found in Squire (2011), 102-10 and 247-83.

57. We may compare it to the equally ambitious inscription found at Salamacis on the history of Halicarnassus, on which most recently see Gagné (2006).

58. West (1971), 93. Intriguing, though not to be covered here, is the echo of language from both the Tabulae Iliacae, especially the ἀδαής of CP 30 and the repeated use of δαείς in the inscriptions, as well as in the Eudoxus papyrus. For now, see Petrain (2014), 54-59, and Squire (2011), 102-10.

59. This text of Archimedes is from Lloyd-Jones and Parsons (1983), 77f.

60. The translation is adapted from Thomas (1941), 202-05.