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Towards a Resolution of the Problem of τά ένί διαστηματι γ ραφόμενα In Pappus' Collection Book VIII

Published online by Cambridge University Press:  11 February 2009

D. E. P. Jackson
D. E. P. Jackson
Affiliation:
University of St. Andrews
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Extract

The phrase τά ένί διαστηματι γ ραφόμενα occurs in that part of Pappus' Collection Book VIII which deals with instrumental solutions to problems more practical than purely geometrical. In the preceding section an instrumental solution for the problem of doubling the cube has been propounded, which is dependent on the use of a ruler passing through a point about which it is turned in the generation of the locus of points known as the cissoid, and in the subsequent section a solution is propounded for the problem of finding the diameter of a cylinder of which only an incomplete portion of the side remains, leaving neither base intact.

Type
Research Article
Information
The Classical Quarterly , Volume 30 , Issue 2 , December 1980 , pp. 523 - 533
Copyright
Copyright © The Classical Association 1980

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References

1 Hultsch, F., Pappi Alexandrini Collectionis quae supersunt (Berlin, 1878), vol. iii (1), p. 1074.Google Scholar

2 Ibid., pp. 1070–2.

3 Ibid., pp. 1074 ff.

4 Arabic: bucd. cf. Heath, T. L., The Thirteen Books of Euclid's Elements (New York, 1956), Postulate 3, i. 199.Google Scholar

5 Hultsch, 10C. cit.

6 Ibid., p. 1075.

7 Zeitschrift fiir Math. u. Phys. XXIV, Hist.-Lit. Abt., p. 128. cf. also ver Eecke, P., Pappus d'Alexandrie. La Collection mathimatique avec une introduction et notes (Paris 1933), ii. 845 n. 3 ‘, expression obscure … visant, sans doute, soit une seule et même ouverture de cornpas, soit une seule et même distance prise sur la règle mobile …’Google Scholar

8 W. M. Kutta, ‘Zur Geschichte der Geometric mit constanter Zirkelöffnung’, in Nova Acta, Abh. der Kaiser'. Carol, Leop.. deutschen Akademie der Naturforscher (Halle, 1897), lxxi. 3. 72–4.Google Scholar

9 Hultsch, op. cit. i. 224–6.

10 See D. E. P. Jackson, ‘The Arabic Translation of a Greek Manual of Mechanics’ Islamic Quarterly, xvi, 1–2 (January-June 1972), 96–103. Since this article a further manuscript of this work has come to light: MS. Aya Sofya 3624. See Sezgin, Fuat, Geschichte des cirabischen Schrifttums (Leiden, 1974), v. 175. The Arabic text, with translation etc., is due to be published by Springer-Verlag.Google Scholar

11 Treweek, A. P., ‘Pappus of Alexandria. The Manuscript Tradition of the “Collectio Mathematics”’, Scriptorium xi. 2 (1957), 197.Google Scholar

12 Hultsch, op. cit., iii. 1102–3 (between props. 19 and 20).

13 A translation from the Arabic of these constructions follows pp. 325–8.

14 Hultsch, op. cit., iii. 1074.

15 e.g. Hultsch, op. cit., i, Liber III, pp. 66–8, and vol. iii, Liber VIII, pp. 1070–72.

16 See n. 9 above.

17 Cf. L. Nix and W. Schmidt, Herons von Alexandria Mechanik und Katoptrik, herausg. und übersetzt von L. Nix und W. Schmidt (Leipzig, 1900); also Drachmann, A. G., The Mechanical Technology of Greek and Roman Antiquity (Copenhagen, 1963).Google Scholar

18 See the article ‘Pappos’ in RE xviii (2) (1949), 1091–1106.

19 See §F, p. 527.

20 See Heath, T. L., The Thirteen Books of Euclid's Elements i. 241–3.Google Scholar

21 In Fig. 5, following the manuscript, AG BD are perpendicular to AB, where strictly, by the preceding construction AZG BHD are equilateral triangles.

22 By the preceding construction.

23 I am indebted for consultations and assistance to Professor G. J. Toomer, of Brown University, Providence, Rhode Island, to Dr. A. G. Molland, of the University of Aberdeen, and to Emeritus Professor E. T. Copson and Mr. A. J. T. Davie, both of the University of St. Andrews.