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Published online by Cambridge University Press: 12 March 2013
Ptolemy presents only one argument for the eccentricity in his models of the superior planets, while each one of them has two eccentricities: one for center of the uniform motion, the other for the center of the constant distance. To take into account the first eccentricity, he introduces the equant point, but he provides no argument for the eccentricity of the center of the deferent. Why is the second eccentricity different from the first one? The 13th century astronomer Quṭb al-Dīn al-Shīrāzī, a member of the famous school of Marāgha, who was interested in this problem, suggests the “retrograde arcs” as the empirical origin of the second eccentricity and develops an argument to justify this conjecture. Although his argument is not without difficulty, his suggestion is in line with the suggestions made by some historians of astronomy in recent decades.
Ptolémée ne donne qu'un seul argument pour expliquer dans son système l'excentricité des planètes supérieures, alors que chacune d'elles a deux excentricités: l'une par rapport au centre du mouvement uniforme, l'autre par rapport au centre de la distance constante. Pour rendre compte de la première excentricité, il introduit le point équant, mais il ne donne en revanche aucun argument pour l'excentricité par rapport au centre du cercle déférent. Or, pourquoi la seconde excentricité est-elle différente de la première? Quṭb al-Dīn al-Shīrāzī, astronome du xiiie siècle membre de l'école de Marāgha, qui s'est intéressé à cette question, a fait l'hypothèse que les “arcs de rétrogradation” constituent l'origine empirique de cette seconde excentricité. Bien que l'argument sur lequel il appuie cette hypothèse ne soit pas exempt de difficultés, sa suggestion rejoint celles faites par des historiens de l'astronomie durant les dernières décennies.
1 Toomer, G.J., Ptolemy's Almagest (New York and Berlin, 1984), pp. 472–4Google Scholar.
2 Ibid, p. 480.
3 Ibid, pp. 484–98, 507–19, 525–38.
4 Pedersen, Olaf, A Survey of the Almagest (Odense, 1974)Google Scholar; Swerdlow, Noel M., “The empirical foundations of Ptolemy's planetary theory”, Journal for the History of Astronomy, XXXV (2004): 249–71CrossRefGoogle Scholar; Evans, James, “On the function and probable origin of Ptolemy's equant”, American Journal of Physics, LII (1984): 1080–9CrossRefGoogle Scholar; Jones, Alexander, “A route to the ancient discovery of non-uniform planetary motion”, Journal for the History of Astronomy, XXXV (2004): 375–86Google Scholar.
5 Pedersen, A Survey, p. 277–8.
6 Swerdlow, “The empirical foundations”, p. 260.
7 See Saliba, George, “Arabic planetary theories after the eleventh century AD”, in Rashed, Roshdi (ed.), Encyclopedia of the History of Arabic Science (London and New York, 1996), vol. I, pp. 58–127Google Scholar.
8 Saliba, George, “The original source of Quṭb al-Dīn al-Shīrāzī's planetary model”, Journal for the History of Arabic Science, III (1979): 3–18Google Scholar.
9 Al-Ṭūsī, Taḥrīr al-Majisṭī, MS Astan Quds Library, Shelfmark 5452, f. 73v.
10 The Greek term used by Ptolemy is έπικύκλου προηγήσεν (Claudii Ptolemaei opera quae exstant omnia. Volumen I. Syntaxis mathematica. Edidit J. L. Heiberg, Pars I. Libros I.–VI. continens [Leipzig, 1895], p. 317). It has been literally translated into “epicycle's regression” by Taliaferro, Robert Catesby in The Almagest: 10.6, in Great Books of Western World, vol. 16, Encyclopedia Britannica (Chicago, 1952), p. 322Google Scholar. The two well-known Arabic versions of Almagest are those of al-Ḥajjāj ibn Yūsuf and Isḥāq ibn Ḥunayn. The last sentence of the above-mentioned text of Ptolemy is translated by Isḥāq (MS Mutahhari Library, Tehran, 3145, f. 127v) as: 
and by al-Ḥajjāj (MS Leiden, Or. 680, f. 156v) as: 
So, in both translations the term has been rendered by a paraphrase.
11 Al-ʿUrḍī, The Astronomical Work of Muʾayyad al-Dīn al-ʿUrḍī: A Thirteenth Century Reform of Ptolemaic Astronomy: Kitāb al-Hayʾa, edited by Saliba, George (Beirut, 1990), p. 211Google Scholar. (English translation is ours.)
12 Toomer, Ptolemy's Almagest, p. 581.
13 Toomer, Ptolemy's Almagest, pp. 563–81.
14 In Almagest 12.2, Ptolemy remarks that “if the epicycle center had no motion towards the rear”, these angles would be the retrograde arcs. But since the epicycle moves, these angles are greater than the retrograde arcs, according to the ratio of the speed of the epicycle to the speed of the planet (Toomer, p. 565). Al-Shīrāzī makes this mistake despite the fact that he seems to be inspired by Almagest 12.2.
15 Although Ptolemy mentions the name of Apollonius in Almagest 12.1 (Toomer, p. 555), al-Shīrāzī doesn't mention his name.
16 Al-Shīrāzī's uses the language of the “data”: in the right angle OC1I, the angle C1OI and the hypotenuse OC1 are “given”, so the sides IO and IC are also given. Here we use trigonometric functions to reconstruct his argument.
17 This theorem is proved, in the general case, in Almagest 10.6 (Toomer, p. 481).
18 Ptolemy presents his calculation of retrograde arcs of Mars in Almagest 12.4 (Toomer, pp. 572–5). He remarks at the end of his treatment of retrogradation (p. 581) that his values agree with the phenomena. According to the interpretation of Swerdlow, Ptolemy's remark implies that he had some sort of values against which he checked his results (Swerdlow, p. 260).
19 This figure is drawn on the basis of Swerdlow, “The empirical foundations”, p. 261.
20 Evans, “On the function”.
21 Swerdlow, “The empirical foundations”.
22 Qurra, Thābit ibn, Œuvres d'astronomie, texte établi et traduit par Régis Morelon (Paris, 1987)Google Scholar; see also Morelon, Régis, “The astronomy of Thābit Ibn Qurra”, in Rashed, Roshdi (ed.), Thābit Ibn Qurra: Science and Philosophy in Ninth-Century Baghdad (Berlin and New York, 2009)Google Scholar.
23 Here is the starting point of manuscript [ق]: + “A Chapter about the Way by which Ptolemy Understood that the Deferent Center of Each One of the Outer Planets is Located at Mid-way between the Center of the Ecliptic and the Equant Point, and I Think that it is by Thābit Ibn Qurra al-Ḥarrānī”. This title, which is only in manuscript [ق], does not exist in the original text of Nihāya.
24 Here is the end point of manuscript ق.
25 1 وإن اشتهر: ناقصة [ت] / حکم: يحکم [ق، م] / مركز: مرکزي [م] – 4 کذلك: لذلك ]ت] – 6 يحيل: يخيل [م]، قد غُيّرت في [ت] بــ ”يخيل“ ولکن ”يحيل“ أصح – 8 إن: ههنا بدائة نسخة [ق]: + ” فصــــل في الطريق الذي به علم بطلميوس أن مرکز الحامل في کل واحد من الکواکب العلوية علی منتصف ما بين مرکزي البروج ومعدل المسير وأظن أنه لثابت بن قرة الحراني. الطريق فيه“ – 9 يرجع فيها: فيها يرجع [ق] / من الفلك البروج: ناقصة [ق] – 11 وبالأعظم: مالأعظم [ت، م] - 12 کذلك: لذلك [ت] / لو زال: اذازلّ [ق]
26 1 محسوس: ناقصة [ق] – 2 بالطريق الذي أسلکُه: بمثل ما وصفه في هذا المثال [ق] – 3 
[م] – 4 و
: + و [م] – 5 واحدة: واحد [م] / أن: + النا [م] – 7 
[ق] – 8 وکذا: کذلك [ت] – 9 
[ق] – 10 قطري: قطر [ت] نصف قطر [ق] / 
: ث ن ه [م]
27 1 كنسبة: + ص [م] – 2 بطلميوس: ناقصة [م، ت] – 3 خطاءٍ: خطا [م] – 4 فکل: كل [ت] – 6 فنخرج: يخرج [ق] – 9 الحضيض: + وهي [ق] – 11 الأول: الثاني [ق] / وهو نصف قطر التدوير بما به ه ر ستون: ناقصة [ق] – 12 الثاني: الاول [ق] / وهو نصف قطر التدوير بمابه 
ستون: ناقصة [ق] – 13 
: ناقصة [م] – 14 الرجوع: + ”حيث علم التعديل الاول منفردا عن الثاني ان قدمنا ؟؟ ينفرد اذا کان المرکز في الاوج والحضيض“ (مخدوش) [ق] – 16 لمسيرها: لمسيره [ت] بمسيرها [ق] / عند: عن [ق]
28 1 
: + الذي هو [ق] / ت: ث [ق] – 2 الاول: الثاني [ق] – 4-3 أعني .... التعديل: ناقصة [ق] - 4 أعظم ما يکون: الاعظم [ق] / وصار: فصار [ت، ق] – 6 بما به 
ستون: ناقصة [ق] – 8 المعلوم: المقدم [م] – 9 النسبة: النسب [ق] – 10 ذلك: هذه هذا [ق] / بالتقريب: + ”لا بالتحقيق وهو أعلم بالسرائر“؛ ههنا نهاية نسخة [ق] – 13–14 إنما دلّ علی أن: ناقصة [ت]
