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Published online by Cambridge University Press:  13 February 2015

Roshdi Rashed*
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*
Institut de Recherche Mathématique Avancée (Université de Strasbourg et CNRS), 7 rue René Descartes, 67084 Strasbourg Cedex, France


In his Sphaerica, Menelaus did not prove Proposition III.5 which is particularly important. He only gave an outline of a proof. Once the Sphaerica were translated into Arabic, mathematicians, starting from the end of the 9th century on, took up this proof. That was made possible to Ibn ʿIrāq thanks to the development of spherical geometry. A first paper contained the history of his contribution. Two other mathematicians, from the 13th century – Naṣīr al-Dīn al-Ṭūsī and Ibn Abī Jarrāda – worked out again the proof of the proposition with the help of Menelaus' book and of the new acquisitions of Ibn ʿIrāq. This is the subject of this second paper.


Dans les Sphériques, Ménélaüs ne démontre pas l'importante proposition III.5, mais propose seulement une esquisse de démonstration. Une fois le livre des Sphériques traduit en arabe, les mathématiciens, à partir de la fin du IXe siècle, ont voulu en donner une démonstration complète. Le développement de la géométrie sphérique a permis à Ibn ʿIrāq de parvenir au but. Un premier article a été consacré à sa contribution. Deux mathématiciens du XIIIe siècle – Naṣīr al-Dīn al-Ṭūsī et Ibn Abī Jarrāda – ont repris la démonstration de cette même proposition, à partir de Ménélaüs et des acquis d'Ibn ʿIrāq. C'est le sujet du présent article.

Research Article
Copyright © Cambridge University Press 2015 

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1 On Menelaus' Spherics III.5 in Arabic mathematics, I: Ibn ʿIrāq”, Arabic Sciences and Philosophy, 24.1 (2014): 168CrossRefGoogle Scholar.

2 For the statement of Menelaus' Proposition III.5, see the beginning of the next section.

3 MSS Istanbul, ‘Āṭif 1712, fol. 190v; Tehran, Sepahsalar 4727, fol. 136.

4 A proof of this fact was given by Ibn ʿIrāq, see Proposition 3.11 (“On Menelaus' Spherics III.5, I: Ibn ʿIrāq”, p. 23).

5 See the proof of al-Ṭūsī below.

6 This step follows from Proposition 2.3 below.

7 This is the construction of the polar triangle, which al-Ṭūsī borrows from Ibn ʿIrāq, see our first article “On Menelaus' Spherics III.5, I: Ibn ʿIrāq”, p. 12.

8 We described this theorem in “On Menelaus' Spherics III.5, I: Ibn ʿIrāq”, Section 2.

9 Ibn ʿIrāq, in the first text we presented in the first paper, indicated a proof of this proposition without the hypothesis that AC and DG are smaller than a quarter, see “On Menelaus' Spherics III.5, I: Ibn ʿIrāq”, p. 25.

10 See “On Menelaus' Spherics III.5, I: Ibn ʿIrāq”.

11 This text is edited from the manuscripts: Istanbul, Atif 1712, fols. 213r–215v, [A, ا]; British Museum, fols. 59v–61v [B, ب]; Tehran, Sepahsalar 4727 [C, ج], fols. 142–145, Columbia, Plimpton Or. 306, fols. 47v–49v [K, ك]; Istanbul, Aya Sofia 2758, fols. 60r–61r [S, ص]; Teheran, Danishka 2432, fols. 283r–284r [T, ط].

12 He means that we take CK = CA = CL.

13 He means equal to a quarter of a circle.

14 Al-Ṭūsī is referring to two manuscripts of the Arabic edition of Menelaus' book by Ibn ʿIrāq. He writes in the preface of his edition of the Spherics: “This treatise consists of three books in some of the copies, and of two in some others. Concerning the three books, in the majority of cases, the first one contains thirty-nine propositions, the last one contains twenty-five propositions, and the middle one, in many copies, contains twenty-four propositions, and in the copy of Ibn ʿIrāq, twenty-one propositions. And in some of them, the first one contains sixty-one propositions, the second one nineteen propositions, and the third one twelve propositions. Regarding the copies with two books, the first one contains sixty-one propositions and the last one thirty propositions. And there are differences in some propositions, because some of them made two propositions a single one and conversely. But in general, the total number of the propositions in the treatise is between eighty-five and ninety-one. I referred in the margin to some of the books and the number of propositions they contain, in red and in black, and sometimes I did it in the text.” (MSS Istanbul, ‘Āṭif 1712, fol. 190v; Tehran, Sepahsalar 4727, fol. 136.)

15 Follows a lemma involving a solid geometry construction which cannot be used in the proof, as al-Ṭūsī says. We refer for this lemma to our edition of al-Harawī’s work: R. Rashed and A. Papadopoulos, Menelaus' Spherics, al-Harawī's Edition, English translation with a mathematical and historical commentary, to appear.

16 This text is edited from the manuscript Manisa 1706, fols. 89v–90v.

17 This means that al-Harawī made Euclidean solid geometry reasonings. In his proof, he drew lines, planes, angles, etc. outside the sphere and not only on the sphere.

18 This means that he worked with intersections of the sphere with planes.

19 The “necessity” means here the conclusion of the theorem (Proposition III.5). Thus, Ibn Abī Jarrāda says that he can prove the theorem in this other situation in which he is modifying the hypothesis of Menelaus.

20 Each of these equalities is equivalent to the fact that two supplementary angles have the same sines.

6-7 أحد ... بينهما: في الهامش مع صح [ص] - 6 جيب مجموع: مجموع جيب، ثم ضرب على ”مجموع“ بالقلم وكتبها في الهامش وأشار إلى موضعها [ب] – 7 الضلعين: ضلعين [ا] – 8 د؟ه؟ز: د؟ه [ا] – 13 فلنخرج: في الهامش مع صح [ص] – 20 لدائرة: الدائرة [ا].

2 ولأن: فلأن [ا] / ل؟ﺟ؟س: ناقصة [ا] – 3 قائمة: في الهامش [ط] / ولأن: فلأن [ص] – 3-4 ح؟ن ... ق قطب: في الهامش [ا] – 5 فكل: وكل [ا] / من: فوق السطر [ب] / وح؟م: ح؟م [ا] – 7 د؟ز؟ه: في الهامش [ص] / أنصافهما: ايضا فيها [ا] – 11 وأما: اما [ا] – 13 ما بين القوسين هو ما نجده في نسخة ابن عراق. فالنسخة التي بالحمرة هي نسخة ابن عراق / ولأن: نجد ”كان“ فوقها [ص] – 19 ل؟ﺟ: كتب ﺟ فوق السطر [ط].

2 أيضًا: ناقصة [ﺟ] – 11 بالضلع ... الموترة: في الهامش مع صح [ص] – 13 مثلث: المثلث [ﺟ] – 17 د؟ه: ه؟د [ا] ناقصة [ﺟ] ز؟ح، وكتب الصواب فوقها [ص] – 18 يلتقيا: صحح عليها وكتب يتلاقيا [ص] – 21 ز؟ﺟ: نجد تحتها ا؟ﺟ [ص].

1-4 صار ... متساويان: في الهامش [ﺟ] – 5 ه؟د في جيب: ناقصة [ا] – 6-7 ولكن ... جيب ﻛ؟ا: في الهامش [ك] – 7 جيب (الثانية): ناقصة [ط] تحت السطر [ك] – 10 يتبين: تبين [ا] تبين، ثم صحح عليها [ص] / ساوت: تساوت [ا، ص، ط] – 12 الشكل: في الهامش [ﺟ] – 13 المتقدمين: المقدمين، ثم صحح عليها [ص].

1 زاوية: ناقصة [ا] – 1-3 زاوية ا؟ب؟ل ... ب؟ا؟ل إلى جيب زاوية: في الهامش [ص] - 2 وبتبادل: وتبادل [ا] – 5 زاوية ﻛ؟ﺟ؟ا ... في: في الهامش [ص] – 6 فالنسبة: والنسبة [ط] – 9 زاوية ﻛ؟ب؟ا إلى جيب: كررها ثم ضرب عليها بالقلم [ب] – 9-11 ﻛ؟ا؟ب ... إلى جيب زاوية: ناقصة [ص] – 17 جيب: ناقصة [ا] – 18 تكافأتا: تكافات [ا، ب، ص، ط، ك].

4 مساويًا: مساو [ا، ب، ﺟ، ص، ط، ك] – 5 جيب (الأولى): ناقصة [ص] – 6 السياقة: أشار هنا إلى تعليق في الهامش [ب] – 11 تحيّروا: تحيّرو [ا] – 15 وأنا: واما [ا] / بشرح: لشراب [ا] – 16 خير الجزاء: خيرا كثيرا [ب].

13 د؟ز: د؟ه – 14 ف: قد تقرأ ”ب“ وكذلك فيما بعد – 20 م؟ﺟ ﺟ؟ن: م؟ﺟ؟ن.

1 ق؟ز: ف؟ز.

3 ب؟ﻛ: ه؟ﻛ – 8 ش؟ق: س؟ن – 12 وع؟ه: وع؟ح / ا؟ﺟ: ا؟ﺟ؟ب.

5 د؟ز ز؟ه : د؟ز؟ه.