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ON THE EVCLIDES LATINVS IN MS VERONA, BIBLIOTECA CAPITOLARE XL (38), AS A WITNESS TO THE GREEK TEXT OF THE ELEMENTS

  • Erik Bohlin (a1)

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In his paper ‘The wrong text of Euclid: on Heiberg's text and its alternatives', published in 1996, W.R. Knorr resuscitated the debate which had taken place in the 1880s between the orientalist M. Klamroth and the editor of Euclid's Elements, J.L. Heiberg. In nuce the debate concerned the fundamental question of which manuscript tradition of the Elements should be assigned textual anteriority: the Greek tradition or the Arabic tradition. Whereas Klamroth argued for the latter position, Heiberg, whose view became the prevailing one, endorsed the anteriority of the Greek tradition.

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1 Knorr, W.R., ‘The wrong text of Euclid: on Heiberg's text and its alternatives’, Centaurus 38 (1996), 208–76. Klamroth, M., ‘Ueber den arabischen Euklid’, Zeitschrift der deutschen morgenländischen Gesellschaft 35 (1881), 270326 . (Note that in his bibliography Knorr erroneously states that Klamroth's article was published in Zeitschrift für Mathematik und Physik.) Heiberg, J.L., ‘Die arabische Tradition der Elemente Euklid's’, Zeitschrift für Mathematik und Physik (historisch-literarische Abteilung) 29 (1884), 122 .

2 For a summary of the debate and its aftermath (with further references), see Knorr (n. 1), 208–12.

3 On the manuscripts of the Greek tradition, see esp. J.L. Heiberg (ed.), Euclidis Elementa (Leipzig, 1883–8), 1.viii–ix, 5.xxiv–xxxv; T.L. Heath, The Thirteen Books of Euclid's Elements. With Introduction and Commentary (Cambridge, 19262 [repr. New York, 1956 and later]), 1.46–50 (with further references). See also Knorr (n. 1), 212–13. On the date of MS P, see e.g. G. Mercati and P. Franchi de'Cavalieri, Codices Vaticani Graeci. Codices 1–329 (Rome, 1923), 1.219.

4 For a general overview—with further references—of the Arabic tradition, see e.g. M. Folkerts, ‘Euclid in Medieval Europe’, in id., The Development of Mathematics in Medieval Europe. The Arabs, Euclid, Regiomontanus (Aldershot, 2006), III.1–64, esp. at 6–12. See also Knorr (n. 1), 213–16.

5 Heiberg (n. 1), 6–22. All references to Heiberg's edition—which will be made according to page and line numbers—henceforth concern the fourth volume, which appeared in 1885; see Heiberg (n. 3).

6 Heiberg, J.L., ‘Paralipomena zu Euklid’, Hermes 38 (1903), 4674 , 161–201, 321–56, at 193–201. It should also be noted that in MS b several propositions are given in an order that is different to that of the other Greek manuscripts.

7 Heiberg (n. 1), 12–14, and Heiberg (n. 6), 198–201.

8 Heiberg (n. 6), 200.

9 Knorr (n. 1), 216–53.

10 Knorr (n. 1), 253–4.

11 Rommevaux, S., Djebbar, A., Vitrac, B., ‘Remarques sur l'histoire du texte des Éléments d'Euclide’, AHES 55 (2001), 221–95, at 227–50. See also B. Vitrac, Euclide d'Alexandrie. Les Éléments. Traduits du texte de Heiberg (Paris, 2001), 4.32–71. Another response and development of Knorr's article is Acerbi, F., ‘Drowning by multiples: remarks on the fifth book of Euclid's Elements, with special emphasis on prop. 8’, AHES 57 (2003), 175242 .

12 M. Geymonat (ed.), Euclidis latine facti fragmenta Veronensia (Milan, 1964).

13 The punctuation in all quotes from L is henceforth my own. It should also be mentioned that, whereas the text is printed in scriptura continua in Geymonat's edition, I have inserted spaces between words so as to facilitate the reading as well as the discussion of the text.

14 Geymonat (n. 12), 39.

15 Heath's translation with my modifications; see Heath (n. 3), 3.373.

16 Heath's translation with my modifications; see Heath (n. 3), 3.374.

17 Bohlin, E., ‘Three notes on the Euclides Latinus preserved in the Verona manuscript, Biblioteca Capitolare XL (38)’, CQ 63 (2013), 455–9, at 457.

18 I have cautiously suggested this earlier; see E. Bohlin, ‘Some notes on the fragmentary Latin translation of Euclid's Elements preserved in the codex palimpsestus Veronensis Bibliothecae Capitularis XL (38)’, in F. Biville, M.-K. Lhommé, D. Vallat (edd.), Latin vulgaire – Latin tardif IX. Actes du IXe colloque international sur le latin vulgaire et tardif, Lyon, 2–6 septembre 2009 (Lyon, 2012), 881–92, at 883.

19 See proposition 4 at p. 397.6 Heiberg and proposition 8 at p. 402.9 Heiberg.

20 Heiberg (n. 3), 4.149 n. 1.

21 Heiberg (n. 3), 5.lxxx–lxxxi.

22 Cf. Heiberg (n. 6), 194, and Knorr (n. 1), 218.

23 The names of the constituent parts of a proposition, as described by Proclus, are henceforth applied so as to facilitate forthcoming references; on the names, see e.g. Heath (n. 3), 1.129–31.

24 A contemporary corrector has deleted –angulas. Geymonat conjectured that –conas was written above the line, producing the corrected reading triconas (= trigonas); cf. fol. 341r col. 2, line 23 – fol. 341v col. 1, line 1 and fol. 338r col. 2, line 11. Cf. also Bohlin (n. 17), 456 with n. 7.

25 As regards in duas sectiones aequales, a contemporary corrector has deleted –as, –tiones, –s and written –o, –menta, –a above the lines, producing the corrected reading in duo secmenta (= segmenta) aequalea. (The corrector apparently failed to correct the letter e to i, presumably because these letters are easily confused in capital script.) Cf. Bohlin (n. 17), 456.

26 A contemporary corrector has deleted –an– in trian– and written –conas above the line; cf. n. 24 above. Geymonat adopted the conjecture <–gulas sedes> and thereby suggested that –gulas had been deleted too. Cf. Bohlin (n. 17), 456 with n. 7. According to Geymonat's commentary, however, a conjecture –gul<as sedes> (with –gul<as> deleted) is ascribed to I. Cazzaniga, which suggests that –gul– can be read (Geymonat [n. 12], 41); apparently Geymonat did not agree with this.

27 Heath (n. 3), 3.378.

28 My translation.

29 Heath's translation with my modification; see Heath (n. 3), 3.378.

30 As noted by Geymonat, tam corresponds to the following adque (= atque, καί) as tam to quam at fol. 331v col. 1, lines 5–6 and 17–18; see Geymonat (n. 12), 40. In addition, Geymonat compares Plaut. Amph. 443 and SHA Sev. 17.6; but in neither of these two examples is tamquam/atque used in the Euclidean sense of τε … καί. On tam … quam/atque, see J.B. Hofmann and A. Szantyr, Lateinische Syntax und Stilistik (Munich, 1965), 590–1.

31 The conjectures were communicated to Geymonat by letter and have been recorded in Geymonat's edition with commentary; see Geymonat (n. 12), 40.

32 Geymonat (n. 12), 40.

33 See Geymonat (n. 12), 63.

34 Geymonat (n. 12), 40.

35 TLL = Thesaurus Linguae Latinae (Leipzig, 1900–).

36 H. Keil (ed.), Grammatici Latini (Leipzig, 1868), 5.95–312.

37 H. Hagen (ed.), Anecdota Helvetica quae ad grammaticam Latinam spectant ex bibliothecis Turicensi Einsidlensi Bernensi collecta […] (Leipzig, 1870), 143–58.

38 Note the ab ‘comparativum’ in the phrase ab eo, quod dimidium esse cernitur (= τὸ ἥμισυ); on this syntactical detail, see Hofmann and Szantyr (n. 30), 111–12.

39 Geymonat (n. 12), 41.

40 O. Keller (ed.), Pseudacronis Scholia in Horatium vetustiora. Vol. I (Leipzig, 1902).

41 W.M. Lindsay (ed.), Isidori Hispalensis episcopi Etymologiarum sive Originum libri XX. Tomus I (Oxford, 1911).

42 Geymonat (n. 12), 41.

43 See Federspiel, M., ‘Sur l'opposition défini/indéfini dans la langue des mathématiques grecques’, LEC 63 (1995), 249–93, at 266–8.

44 Cf. Bohlin (n. 18), 886.

45 Bohlin (n. 18), 886.

46 See Hofmann and Szantyr (n. 30), 478.

47 On the basis of Geymonat's observation that each line in L contains 15–22 letters (Geymonat [n. 12], 61), the illegible letters between poti and aequa should be within the range of 6–13 in number. The suggested pyramidi consists of 8 letters.

48 The illegible letters between aequa and sectiones should be within the range of 10–17 in number. The suggested addition –e in duas aequales consists of 15 letters. Cf. n. 47 above.

49 Bohlin (n. 17), 457–9.

50 Heath's translation with my modifications; see Heath (n. 3), 3.378.

51 I have adopted Κघ, which is Heiberg's correction of the scribal error ΚΛ in MS b (p. 394.3 Heiberg); see Heiberg (n. 6), 195.

52 It could be added that the points bisecting the edges of the pyramid are not labelled in the same way in MS b as in the other Greek manuscripts: for Θ, MS b has घ; for Λ, it has Κ; for Κ, it has Θ; for घ, it has Λ. See K. Saito, ‘Reproduced diagrams of the Elements. Books VI, XI, XII, XIII’, in N. Sidoli, Ch. Li and K. Saito, Reproduced diagrams from Greek and Arabic manuscripts. Research report: ‘Databasing the manuscript diagrams of sources in ancient and medieval mathematics’ (2014), 69–179, at 145. Available at: http://www.greekmath.org/diagrams/Reproduced_Diagrams_from_Greek_and_Arabic_manuscripts_20141214.pdf (accessed 9 February 2015).

53 The lacuna between uide and nientia (line 7), where the illegible letters should be within the range of 11–18 in number, could perhaps be supplied thus: uide<licet latera conue>nientia. The suggested addition consists of 16 letters. Cf. n. 47 above.

54 Heath's translation with my addition; see Heath (n. 3), 3.395.

55 Geymonat's conjecture <triangulas> (line 11), in which –angulas is suggested to have been deleted by a corrector, was conceived from the fact that conas is written above the line; see notes 24 and 26 above.

56 The variant readings reported in Heiberg's edition appear to be of no relevance to this examination.

57 Heath's translation with my modifications; see Heath (n. 3), 3.395–6.

58 I have adopted ἔτι and ΔΒ, ΒΓ, which are Heiberg's corrections of the scribal errors ἐπεί (p. 401.16 Heiberg) and ΑΒ, ΒΓ (p. 401.17 Heiberg) in MS b; see Heiberg (n. 6), 195.

59 In addition, it could be suggested that the letters sis, which were read by Mai immediately before habeant, are the remnants of bases (= βάσεις) (whereby the letter E was misread as an I). The word basis (= βάσις) is attested in L, e.g. in the ἔκϑεσις of proposition 3, at fol. 341r col. 2, line 18; but the Greek term βάσις is also rendered as sedes in L, e.g. in the πρότασις of proposition 3, at fol. 341r col. 2, lines 3 and 6.

60 Heath's translation with my modifications; see Heath (n. 3), 3.396.

61 Cf. the διορισμός of proposition 3 above, at fol. 341r col. 2, line 21, where λέγω ὅτι is rendered as dico quia. The letter I would hardly require much space.

62 I have adopted ΒΑ, which is Heiberg's correction of the scribal error ΒΛ in MS b (p. 401.22 Heiberg); see Heiberg (n. 6), 195.

63 My adaption of Heath's translation; see Heath (n. 3), 3.396.

64 The variant readings reported in Heiberg's edition appear to be of no particular significance.

65 Heath's translation with my modifications; see Heath (n. 3), 3.396.

66 According to Geymonat's edition, the illegible letters should be within the range of 7–14 in number. The suggested addition consists of 7 letters. Cf. note 47 above. Note also the linea nasalis, which is used in L at the end of lines; see Geymonat (n. 12), 62 n. 44.

67 Cf. also fol. 341v col. 2, lines 21–2: ei quod ΑΔΘ de<du>cent.

68 Cf. also Bohlin (n. 18), 885 and 889 n. 34.

69 See Geymonat (n. 12), 42 and 62.

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ON THE EVCLIDES LATINVS IN MS VERONA, BIBLIOTECA CAPITOLARE XL (38), AS A WITNESS TO THE GREEK TEXT OF THE ELEMENTS

  • Erik Bohlin (a1)

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