¹ “Combinatoire et métaphysique: Ibn Sīnā, al-Tūsī et al-Halabī” in Rashed, R. and Biard, J. (ed.), Les doctrines de la science (Leuven, 1999), pp. 61–86.Google Scholar

² See Rashed, R., “Optique géométrique et doctrine optique chez Ibn al-Haytham”,Archive for the History of Exact Sciences, 6 (1970): 271–98 and “Lumière et vision: l'application des mathématiques dans l'optique d'Alhazen”,CrossRefGoogle Scholar in Taton, R. (ed.), Roemer et la vitesse de la lumière (Paris, 1978): 19–44; repr. in Optique et Mathématiques: Recherches sur l'histoire de la pensée scientifique en arabe, Variorum Reprints (Aldershot, 1992).Google Scholar

³ Ibid. Cf. also Nazif, M., Ibn al-Haytham, Buhūthuhu wa-kushūfuhu al-basariyya (Cairo, 1942), vol. I, pp. 121–35Google Scholar and “Arā' al-falāsifa al-islāmiyyīn fi al-haraka”, Majallat al-Jam ‘iyya al-misriyya li-tārīkh al-‘ulūm, n° 2 (1943): 45–64.Google Scholar

⁴ On the life and mathematical work of al-Qūhī, see Rashed, R., Les mathématiques infinitésimales du IX^e au XI^e siècle (London, 1995), vol. I, chap. 5.Google Scholar

⁵ Rashed, R., Géométrie et dioptrique au X^e siècle. Ibn Sahl, al-Qūhī et Ibn al-Haytham (Paris, 1993).Google Scholar

⁶ Hayyān, Abū al-Tawhīdī, al-Imtā‘ wa-al-mu’ānasa, ed. Amīn, A. and al-Zayn, A. (reprod. Būlāq, n.d.), vol. I, p. 38.Google Scholar

⁷ Cf. note 9.Google Scholar

⁸ Sayili, Aydin, “Al-Qūhī on the possibility of infinite motion in finite time”, Belleten, 21 (1957): 489–95. The regretted Professor Sayili edited the text as well as an English and a Turkish translation, thereby attracting attention to this important document. On this occasion we honor the memory of our departed friend.Google Scholar

⁹ Physique, Book VI, 7, 238a, 20–37.Google Scholar Cf. the Arabic translation, ed. ‘Badawī, A., Aristūtālīs, al-Tabi'a (Cairo, 1964–1965), vol. II, p. 699. Cf. also the commentary of this passage by Abū al-Faraj ibn al-Tayyib as well as that of Yahyā (probably Johannes Philoponus), pp. 697–8.Google Scholar

¹⁰ Cf. Phys., ed. ‘Badawī, A. (Cairo, 1964–1965), p. 634. It is to be noted that we deal here with one of the longest glosses of the Arabic version, among those explicitly attributed to Alexander.Google Scholar

¹¹ His paraphrase (Them. in Ar. Phys. Paraphrasis, ed. Schenkl, H. [Berlin, 1900], C.A.G. V, 2, 188Google Scholar, 10ff) reviews practically word for word Aristotle's phrase: ή αύτή δ έάπόδειξιζ χαì εí τò μχοζ πειον ύποθοιμεθα, τòν χρόνον δπεπερασμνον.

¹² Simplicius, , In Ar. Phys. Comm., ed. Diels, H. [Berlin, 1895], C.A.G. X, 951Google Scholar, 22ff: Δειξαχ δέ οűτωξ, őτι άδύνατον èν άπείρẉχρόντò πεπεραάμένον μέγεθοζ χινθήνατ, τοιζ αùτοīζ χρωένουζ νεστιν, φησιν, δεīξαι őτι μηδé τò πειρον έν πεπερασμένω χρόνω διελθειν δυνατόν. For memory, we add that the conserved (partial) version of Philoponus' commentary does not comment upon this passage (cf. Ioan. Philop., In Ar. Phys. Comm., ed. Vitelli, H. [Berlin, 1888], C. A. G. XVII, p. 864).Google Scholar

¹³ Cf. Kraus, P., Jābir ibn Hayyān, Contribution à l'histoire des idées scientifiques dans l'Islam, Jābir et la science grecque (repr. Paris, 1986), pp. 320–1: “in the course of the 3rd/9th century the text and the ancient commentaries (Alexander, Porphyry, Themistius, Johannes Philoponus) of the Physics were translated many times, and paraphrases and summaries were put together” (and the notes).Google Scholar

¹⁴ This concerns notably the twelfth proposition of the Elements of Physics (Procli Diadochi Lycii Elementatio Physica, ed. Ritzenfeld, A. [Leipzig, 1912] p. 37)Google Scholar: ένπεπερασμνχρόντό πειρον χινεīσθαι ούχ σττ.

¹⁵ We showed that the mathematician al-Sijzī knew Proclus' Elements of Physics which he cited. Cf. Rashed, R., “Al-Sijzī et Maïmonide: Commentaire mathématique et philosophique de la proposition II-14 des Coniques d'Apollonius,” Archives Internationales d'Histoire des Sciences, 37, n^° 119 (1987): 263–96;Google Scholar English translation in Fundamenta Scientiae, 8, n^° 3/4 (1987): 241–56. See Pines, S., “Hitherto unknown Arabic extracts from Proclus' Stoicheiōsis Theologikē and Stoicheiōsis Physikē” in The Collected Works of Shlomo Pines, vol.II: Studies in Arabic Versions of Greek Texts and in Mediaeval Science (Jerusalem, 1986), pp. 287–93.Google Scholar

¹⁶ Cf. Sayili, “Al-Qūhī on the possibility of infinite motion in finite time.”Google Scholar

¹⁷ Physique 262a, 7–8; Arabic transl., vol. II, pp. 892–3.Google Scholar

¹⁸ According to Simplicius, Alexander elaborates on this thesis: “But Alexander interprets ‘the middle plays the role of both extremes in respect to each of them (Phys. VIII 8, 262a 20)’ … as meaning that since the middle is opposed to each <of the extremes> as being the other one, it becomes both of them. But if one also takes each of the extremes in both senses (i. e. both as beginning and end) — e. g. in the straight motion from high to low, the high is a beginning in respect to that whose motion proceeds from it but an end for that which arrives to it (and the same applies to the low point) —, the middle also plays both roles in respect to each in thought (for in reality, it is numerically one). But for that which concerns the points on the straight line intermediary between the two extremes, as long as one takes the straight line as one and continuous, they exist only potentially, and since they exist only potentially, they cannot be ‘middles’ in the proper sense. In the proper sense, nothing exists which does not exist in actuality. The middle becomes in actuality, when the mobile object on the straight line divides it at one of its points by becoming stationary. […] For it is impossible that simultaneously and in the same instant something comes-to-be in and out of the same thing — as <Aristotle> makes clear a little later — or that <the mobile object> arrives to and departs from <the same point>. For it would then be and not be simultaneously in the same thing. It must consequently <arrives> at one instant and <depart> at another. If this is the case, then, and if between two instants there exists an intermediary time, it is during this time that there exists a rest in the thing in which and out of which something comes-to-be. But everything that moves in a continuous manner cannot come-to-be at one of the points of the straight line. In respect to these points it would not be said to be in time, but at the instants of the time, which are the limits of the time and not the time <itself>. For if it was in some time, il would be necessary that it stops in that time. At every instant, then, <the mobile object> is at a different point of the straight line. The mobile object A cannot come-to-be in or from the point B: it is at this position only in the instant and not in some time. For there is an analogy between the intermediary points in the straight line and the intermediary instants in the time. Then that which moves in a continuous manner is only potentially in the intermediary points and in the intermediary instants” (In Phys. 1281,37–1282-31).

¹⁹ Ibid., 262a, 5–6.

²⁰ Many sources underline the importance of this Karnīb family, as much in mathematics as in philosophy. At present, aside from the citations reported by their successors such as Ibrāhīm ibn Sinān, none of their writings has come down to us. This one is Abū Ahmad ibn al-Husayn, the philosopher and contemporary of Thābit ibn Qurra, who was, as al-Nadim describes him “of great eminence, possessing a knowledge of and was familiar with the ancient physical sciences; among his books one finds Reply to Abū al-Hasan ibn Thābit ibn Qurra concerning the denial of the existence of rest between two contrary movements”Google Scholar (al-Nadim, Kitāb al-fihrist, ed. Tajaddud, R. [Tehran, 1971], p. 321). Al-Qiftī cites this title in a way which is very probably faulty.Google Scholar Cf. also Morelon, R., “Les deux versions du traité de Thābit b. Qurra Sur le mouvement des deux luminaires”, MIDEO, 18 (1988): 9–44, p. 42.Google Scholar

²¹ It is precisely this experiment that the philosopher of the twelfth century, Abū al-Barakāt al-Baghdādī mentions and attributes to “certain eminent scholars” (ba ‘d al-fudalā’), Kitāb al-Mu'tabar (Hyderabad, 1358), vol. II, p. 97. See the citation below, pp. 23–24.Google Scholar

²² Al-Qūhī gives, in effect, a counter-example to refute Aristotle. He knew without doubt that there exist many examples of contrary movements between which there is a “rest”. With the same construction that he uses one can give the example of the lead-weight defined in respect to hole O by y = a + a sin t, its speed is then a cos t. This speed becomes zero and changes its sign for t = π/2. In this case, one can easily show that the movement of the hand over the ruler would be defined in respect to hole O by:

and

. The speed of the hand is therefore always positive and equal to a │ cos t │ it becomes zero for t = π/2.

²³ Implied: perpendicular to the plane ABC.Google Scholar

²⁴ This word was added by Muhammad ibn Sartāq al-Marāghī, a mathematician of the first half of the 14th century, author of the book al-Ikmāl, commentator of Ibn Hūd's Istikmāl, and numerous glosses on the writings of al-Qūhī contained in this collection.Google Scholar

²⁵ In other words, the shadow of point E.Google Scholar

²⁶ The text has been edited and summarised in English by Schacht, J. and Meyerhof, M., The Medico-Philosophical Controversy Between Ibn Butlan of Baghdad and Ibn Ridwan of Cairo. A Contribution to the History of Greek Learning Among the Arabs (Cairo, 1937), pp. 65–6. The English summary includes some errors which hinder the understanding of al-Qūhī's line of thought.Google Scholar

²⁷ Persian festival in the first days of October.Google Scholar

²⁸ I.e., without stopping and of the same movement throughout.Google Scholar

²⁹ (Hyderabad, 1358), vol. II, p. 97.Google Scholar