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THE NORMAL ROAD TO GEOMETRY: δή IN EUCLID'S ELEMENTS AND THE MATHEMATICAL COMPETENCE OF HIS AUDIENCE*

  • Stéphanie van der Pas (a1)
Abstract

Euclid famously stated that there is no royal road to geometry, but his use of δή does give an indication of the minimum level of knowledge and understanding which he required from his audience. The aim of this article is to gain insight into his interaction with his audience through a characterization of the use of δή in the Elements. I will argue that the primary use of δή indicates a lively interaction between Euclid and his audience. Furthermore, the specific contexts in which δή occurs reveal the considerable mathematical competence that Euclid expected from his audience.

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svdpas@math.leidenuniv.nl
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I would like to express my gratitude to Adriaan Rademaker, who commented on earlier versions of this paper, and to the anonymous reader at Classical Quarterly for their helpful suggestions.

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1 Netz R., The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History (Cambridge, 1999, 114–18).

2 Denniston J.D., The Greek Particles (Oxford, 1954 2), 203–4, 229, 236–40.

3 Wakker G.C., ‘The discourse function of particles: some observations on the use of μάν/μήν in Theocritus’, in Harder M.A., Regtuit R.F., and Wakker G.C. (edd.), Theocritus (Groningen, 1996), 247–63. For similar objections, see also Sicking C.M.J. and Ophuijsen J.M. van, Two Studies in Attic Particle Usage: Lysias and Plato (Leiden, 1993), 7, 71–2; Sicking C.M.J., ‘Griekse Partikels: Definitie en Classificatie’, Lampas 19 (1986), 125–41.

4 Wakker G.C. (1997), ‘Modal particles and different points of view in Herodotus and Thucydides’, in Bakker E.J. (ed.), Grammar as Interpretation: Greek Literature in its Linguistic Contexts (Leiden, 1997), 238–42.

5 Sicking and van Ophuijsen (n. 3), 51–3; Sicking (n. 3).

6 Sicking and van Ophuijsen (n. 3), 75.

7 Ibid., 82–3.

8 Heiberg I.L., Euclides Elementa vol. I: libri I–IV cum Appendicibus, ed. Stamatis E.S. (Leipzig, 1969); Heiberg I.L., Euclides Elementa vol. II: libri V–IX cum Appendice, ed. Stamatis E.S. (Leipzig, 1970).

9 The other occurrences of this use are 1.26.37, 1.34.28, 1.46.13, 7.3.11, 7.3.22, 7.3.32, 7.4.10, 7.19.30, 7.28.19, 7.33.15, 7.34.9, 7.34.27, 7.34.32, 7.36.8, 7.36.17, 7.36.21, 7.39.12, 9.8.25, 9.9.12, 9.9.18, 9.9.22, 9.10.28, 9.13.34, 9.15.33, 9.18.9, 9.18.17, 9.19.15, 9.19.33, 9.19.43, 9.19.52, 9.33.7, 9.34.8. Although Heiberg proposes to delete the occurrences at 9.9.12 and 9.9.22, they are structurally similar to (2).

10 It also occurs at 7.24.13.

11 One could wonder why δεῖ δή is not found in every proposition. The reason is the distinction made by Euclid between propositions in which an action (such as a construction or searching for a certain number) needs to be performed and propositions in which a property needs to be proved. In the former case, we always find δεῖ δή; in the latter case, λέγω ὅτι is used.

12 The other uses in this category are at 1.2.4, 1.3.5, 1.9.3, 1.10.2, 1.11.4, 1.12.5, 1.22.9, 1.23.6, 1.31.4, 1.42.4, 1.44.6, 1.45.5, 1.46.2, 7.2.4, 7.3.5, 7.33.4, 7.34.3, 7.36.4, 7.39.2.

13 This use is also found at 1.15.22.

14 The other uses in this category are 1.14.20, 1.15.17, 1.16.23, 1.20.18, 1.27.13, 1.35.9, 1.36.18, 1.39.16, 1.40.16, 1.43.15, 1.47.16, 1.47.36, 7.5.17, 7.6.18, 7.10.19, 7.17.12, 7.18.9, 7.21.19, 7.22.14, 7.26.12, 7.28.15, 7.30.22, 7.33.25, 7.33.29, 9.8.23, 9.8.24, 9.8.37, 9.9.16, 9.9.32, 9.10.25, 9.10.41, 9.12.47, 9.13.27, 9.13.44, 9.13.52, 9.15.32, 9.19.54, 9.24.6, 9.26.6, 9.32.13.

15 See M. Buijs, ‘Clause combining in Ancient Greek narrative discourse’ (Diss., Leiden University, 2003), 199, example 28, for another example in which δή qualifies a genitive absolute.

16 In 7.4, 7.33, 7.34, and 9.19.

17 In 7.4 and 7.32. For example, in 7.32: ἅπας ἀριθμὸς ἤτοι πρῶτός ἐστιν ἢ ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται (‘every number is either prime or is divided by some prime number’).

18 See for example 9.8, 9.9, 9.10, 9.11, 9.12, 9.13, and 9.17.

19 The other uses of δή in exclusive disjunctions are in 7.4.11, 7.36.6, 9.18.6, 9.20.7.

20 The other occurrences of this use are at 1.5.16, 1.5.25, 1.6.13, 1.10.10, 1.11.16, 1.12.21, 1.16.15, 1.24.17, 1.26.23, 1.26.32, 1.26.45, 1.26.52, 1.26.55, 1.27.10, 1.33.12, 1.34.15, 1.34.29, 1.35.12, 1.45.19, 1.47.12, 1.47.20, 1.48.21, 7.3.25, 7.5.15, 7.6.12, 7.8.12, 7.9.15, 7.10.15, 7.15.14, 7.18.10, 7.19.16, 7.19.23, 7.20.13, 7.22.15.

21 This is an unusual proof technique, which Heath T.L., The Thirteen Books of Euclid's Elements: Translated from the Text of Heiberg, with Introduction and Commentary. Volume 1: Introduction and Books I, II (New York, 1956 2), 249–50, notes is not theoretically admissible. It is very visual in nature, as the triangles are imagined to be placed on top of each other. This visual element is present in many of the proofs, especially in examples (11) and (18).

22 A geometric sequence is a sequence of the form: 1, a, a2, a3, a4, ….

23 This visual aspect of the proof is also present in examples (11) and (12).

24 A perfect number is a number whose sum of divisors (not including the number itself) is equal to the number itself. An example is 6 = 1 + 2 + 3.

25 Heiberg (n. 8), 225. F is one of the manuscripts based on Theon's edition. According to Heath (n. 21), 46–7, F is damaged and includes numerous corrections.

* I would like to express my gratitude to Adriaan Rademaker, who commented on earlier versions of this paper, and to the anonymous reader at Classical Quarterly for their helpful suggestions.

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