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ON MENELAUS' SPHERICS III.5 IN ARABIC MATHEMATICS, I: IBN ʿIRĀQ

Published online by Cambridge University Press:  24 January 2014

Roshdi Rashed  and
Athanase Papadopoulos
Roshdi Rashed*
Affiliation:
Université Paris Diderot, Sorbonne Paris Cité, SPHERE, UMR 7219, CNRS, 5 rue Thomas Mann, Bâtiment Condorcet, Case 7093, F-75205 Paris Cedex 13, France
Athanase Papadopoulos*
Affiliation:
Institut de Recherche Mathématique Avancée (Université de Strasbourg et CNRS), 7 rue René Descartes, 67084 Strasbourg Cedex, France
*
Email: [email protected]
Email: [email protected]
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Abstract

This is the first paper in a series in which we provide critical editions of four texts written between the eleventh and the thirteenth century concerning Proposition III.5 of Menelaus' Spherics. The first paper contains introductory material on the work of Ibn ʿIrāq on spherical geometry and a critical edition of his two texts on the rectification of Proposition III.5, together with translations and historical and mathematical commentaries. The second paper contains critical editions of the texts by Naṣīr al-Dīn al-Ṭūsī and Ibn Abī Jarrāda on the same subject, again with translations and historical and mathematical commentaries.

Résumé

C'est le premier d'une série d'articles comportant quatre textes composés entre le XIe et le XIIIe siècle, qui traitent de la proposition 5 du livre III des Sphériques de Ménélaüs. Le premier article comprend des commentaires historiques et mathématiques de l'œuvre d'Ibn ʿIrāq en géométrie sphérique et une édition critique des deux textes qu'il a consacrés à la rectification de la proposition III.5, ainsi que la traduction de ces deux textes. Le second article propose une édition critique des textes de Naṣīr al-Dīn al-Ṭūsī et d'Ibn Abī Jarrāda sur le même sujet, cette fois aussi avec une traduction et des commentaires historiques et mathématiques.

Type
Research Article
Information
Arabic Sciences and Philosophy , Volume 24 , Issue 1 , March 2014 , pp. 1 - 68
Copyright
Copyright © Cambridge University Press 2014 

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References

1 Rashed, Roshdi and Papadopoulos, Athanase, “On Menelaus' Spherics III.5 in Arabic mathematics, II: Naṣīr al-Dīn al-Ṭūsī and Ibn Abī Jarrāda”, forthcoming in ASPGoogle Scholar.

2 Rashed, Roshdi, “Ibn ʿIrāq, Abū Naṣr Manṣūr b. ʿAlī” (in Arabic), in Mawsūʿat aʿlām al- ʿulamāʾ wa-al-udabāʾ (Beirut, 2013), vol. 22Google Scholar; and Geometry and Dioptrics in Classical Islam (London, 2005), chap. VGoogle Scholar.

3 The numbering III.5 (Proposition 5 of Book III) refers to Ibn ʿIrāq's edition (cf. Krause, Max, Die Sphärik von Menelaos aus Alexandrien in der verbesserung von Abū Naṣr Manṣūr b. ʿAlī b. ʿIrāq, Abhandlungen der Gesellschaft der Wissenschaften zu Göttingen, phil.-hist. Klasse, 3, 17 [Berlin, 1936]Google Scholar; MS Leiden 930). In al-Harawī's edition, this is Proposition III.71 in the Leiden MS (Or 399, fol. 101r) and Proposition 10 of Book II in the Istanbul MS (Topkapi Saray, Ahmet III 3464, fol. 97v); see the forthcoming critical edition with English translation of al-Harawī's version of Menelaus' Spherics by Roshdi Rashed and Athanase Papadopoulos.

4 Menelaus used chords and the Arabic commentators used the familiar language of sines.

5 Ibn ʿIrāq states the following consequence of Menelaus' proposition: “The ratio of the sine of the sum of an arc on the ecliptic with its ascension in the right sphere to the sine of their difference is a unique ratio; it is the ratio of the sine of the complement of half of the great inclination to the sine of half of the great inclination multiplied by itself” (see below).

6 Krause, Die Sphärik von Menelaos; Björnbo, Axel A., “Studien über Menelaos' Sphärik. Beiträge zur Geschichte der Sphärik und Trigonometrie der Griechen”, in Abhandlungen zur Geschichte der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, vol. 14 (Leipzig, 1902)Google Scholar; Samsó, J., Estudios sobre Abū Naṣr Manṣūr b. ʿAlī b. ʿIrāq (Barcelona, 1969)Google Scholar and “Manṣūr ibn ʿAlī ibn ʿIrāq, Abū Naṣr”, Dictionary of Scientific Biography, vol. 9 (New York, 1974), pp. 83–5Google Scholar; Rasāʾil Abī Naṣr Manṣūr b. ʿIrāq (Hyderabad, 1948)Google Scholar.

7 Abū ʿAbd Allāh Muḥammad ibn ʿĪsā ibn Aḥmad al-Māhānī (fl. 866).

8 Aḥmad ibn Abī Saʿd al-Harawī (fl. 960).

9 Debarnot, Marie-Thérèse, “Trigonometry”, in Rashed, Roshdi (ed.), Encyclopedia of the History of Arabic Science, 3 vols. (London and New York, 1966), vol. 2, pp. 495–538Google Scholar.

10 Naṣīr al-Dīn al-Ṭūsī, Taḥrīr Kitāb Mānālāūs fī al-ashkāl al-kuriyya (Edition of the Book of Menelaus), MS Sepahsalār 4727, fol. 159.

11 Complete quadrilaterals are considered in Proposition 1 of Book III of Menelaus' Spherics, cf. also Chapter 12 of Book III of Ptolemy's Almagest.

12 Cf. Bellosta, Hélène, “Le traité de Thābit Ibn Qurra sur la figure secteur”, in Rashed (ed.), Thābit ibn Qurra. Science and Philosophy in Ninth-Century Baghdad, Scientia Graeco-Arabica, vol. 4 (Berlin, 2009), pp. 335–90Google Scholar, pp. 339 sqq., and Thābit ibn Qurra. On the Sector-Figure and Related Texts, edited with translation and commentary by Lorch, Richard, Institute for the History of Arabic-Islamic Science (Frankfurt am Main, 2011)Google Scholar.

13 See e.g. Proposition 5 of the treatise of Ibn al-Haytham titled The Configuration of the Motions of Each of the Seven Wandering Stars, pp. 294 sqq., in Rashed, Roshdi, Les mathématiques infinitésimales du IXe au XIe siècle, vol. V: Ibn al-Haytham. Astronomie, géométrie sphérique et trigonométrie (London, 2006)Google Scholar.

14 See Crozet, Pascal, “Thābit ibn Qurra et la composition des rapports”, in Rashed (ed.), Thābit ibn Qurra: Science and Philosophy, pp. 391–535Google Scholar; see also Thābit ibn Qurra. On the Sector-Figure and Related Texts, ed. Lorch.

15 Al-Bīrūnī, Kitāb Maqālīd ʿilm al-hayʾa (The Keys to the Science of Astronomy): La trigonométrie sphérique chez les Arabes de l'Est à la fin du Xe siècle, ed. and transl. by Debarnot, Marie-Thérèse (Damascus, 1985), p. 97Google Scholar.

16 Al-Bīrūnī, Maqālīd ʿilm al-hayʾa, ed. Debarnot, p. 101Google Scholar.

17 Al-Bīrūnī, Maqālīd ʿilm al-hayʾa, ed. Debarnot, p. 133Google Scholar.

18 Risālat Ibn ʿIrāq ilā al-Bīrūnī fī maʿrifat al-qusīy al-falakiyya, MS Khudabakhsh, no. 2519, fol. 101v.

19 Ibid., fol. 100v.

20 Al-Bīrūnī, Maqālīd ʿilm al-hayʾa, p. 111.

21 Cf. Moussa, Ali, The Almagest of Abū al-Wafāʾ al-Būzjānī (in Arabic), Silsilat Tārīkh al-ʿulūm ʿinda al-ʿArab, 10 (Beirut, 2010)Google Scholar.

22 Debarnot, Marie-Thérèse, “Introduction du triangle polaire par Abū Naṣr b. ʿIrāq”, Journal for the History of Arabic Science, vol. 2, no. 1 (1978): 126–36Google Scholar.

23 Cf. Chasles, Michel, Aperçu historique sur l'origine et le développement des méthodes en géométrie (Bruxelles, 1857; repr. Paris, 1989)Google Scholar.

24 Risālat Ibn ʿIrāq ilā al-Bīrūnī fī taṣḥīḥ mā waqaʿ li-al-Khāzin min al-sahw fī Zīj al-ṣafāʾiḥ, MS Khudabakhsh, no. 2519, fol. 75r.

25 See Proposition 11 of the treatise.

26 Menelaus, his translators and his commentators use the term “three-sided figure”. Since in this commentary we use modern notation, we have replaced this expression by the more familiar name “spherical triangle”.

27 One should add here that all the quantities that are involved in these formulae are positive; this follows from the hypothesis.

28 Strictly speaking, for this to make sense, the construction has to be done in a region of the sphere which is contained in an open hemisphere. The reason is that there is no notion of betweenness for three points on a circle, which says in the present setting that given three points on a sphere that lie on the same line (which is a great circle), there is no notion of one point lying between the other two.

29 See Rashed and Papadopoulos, “On Menelaus' Spherics III.5 in Arabic mathematics, II: Naṣīr al-Dīn al-Ṭūsī and Ibn Abī Jarrāda”.

30 This remark should be compared with a remark made by Ibn Abī Jarrāda; see Rashed and Papadopoulos, “On Menelaus' Spherics III.5 in Arabic mathematics, II: Naṣīr al-Dīn al-Ṭūsī and Ibn Abī Jarrāda”.

31 This is Proposition III.8 of Ibn ʿIrāq's Book of Menelaus on Spherical Propositions Rectified. In al-Harawī's edition of the Spherics, this is Proposition III.74 in the Leiden MS (Or 399, fol. 101v) and Proposition 13 of Book II in the Istanbul MS (Ahmet III 3464, fol. 98v); cf. our forthcoming edition.

32 This is Proposition III.22 of Ibn ʿIrāq's Book of Menelaus on Spherical Propositions Rectified; in al-Harawī's version of the Spherics, this is Proposition III.89 in Leiden MS (Or 399, fol. 104v), and Proposition III.8 in Istanbul MS (Ahmet III 3464, fol. 100r).

33 See Rashed, Roshdi, Les Mathématiques infinitésimales du IXeau XIesiècle. Vol. V: Ibn al-Haytham: Astronomie, géométrie sphérique et trigonométrie (London, 2006)Google Scholar, pp. 64 ff.

34 It follows from the fact that CA is less than a quarter of a circle and that the angle C is obtuse that the point K belongs to the segment BC. (The argument is used in Proposition 5 of Book I of Menelaus' Spherics.)

35 Menelaus gives the last statement without proof.

36 This is the heart of the proof: We started with two triangles ABC and DGE with some equalities between two of their angles and with no equalities whatsoever between their sides. Using polarity theory, we end up showing that there are equalities between segments on the polar circle of the vertices G and H, namely, MS = QX and NS = RX and MH = IQ. Then, using some kind of projective geometry (the invariance of a cross ratio), these equalities give equalities between some ratios in the figures, and the proof of the desired relation follows. The reasoning is the same in all three texts that are presented in this paper and in its sequel: those of Ibn ʿIrāq, of Ibn Abī Jarrāda and of Naṣīr al-Dīn al-Ṭūsī.

37 This holds by the invariance of the cross ratio in spherical geometry, see Proposition 3.2 above and the comment following it; see also Proposition 3.18 below.

38 Of this text of Ibn ʿIrāq, there existed only a non-critical edition published at Hyderabad, which nevertheless has done a great service to researchers. J. Samsó translated this text into Spanish and he commented it in his Estudios sobre Abū Naṣr Manṣūr b. ʿAlī b. ʿIrāq. We present here the editio princeps of the text of Ibn ʿIrāq, from the MS Khudabakhsh 2519 (ex 2468), fols 75v–78r.

39 From now on, we translate the expression “a triangle on the surface of the sphere” by “a spherical triangle”.

40 This means a distance equal to the quarter of a circle.

41 Assuming that CD = DA = BD.

42 i.e. proportional means.

43 This text was already established and translated in German by M. Krause, Die Sphärik von Menelaos, pp. 68–72 (Arabic), pp. 202–7 (German). We provide here a new edition from the MS Leiden 930, as well as an English translation.

44 This figure does not exist in the manuscript.