‘Ibn al-Haytham marshalled arguments to demolish the assertion that place is a surface; he said: if place were a surface, then it would be able to increase while the thing whose place it determines remained as it was, in two cases. A) If we divide a parallelepiped by parallel surfaces that are parallel to the two original surfaces, the surfaces which enclose this body before it is divided up are, without any doubt, smaller than those that enclose it after it has been divided up into many parts, whereas the thing whose place it determines remains as it was. B) If wax is shaped into a sphere, the surface that encloses the wax is smaller than the surface that encloses it when it is shaped into a cube. Since the sphere is the greatest of figures,1 the thing whose place it determines, when it is transformed into a cube, thus remains exactly the same whereas the place increases.
It is possible for the place to remain the same whereas the thing whose place it determines decreases: indeed, the water which is in a goatskin has as its place the inner surface of the goatskin. If we press on the goatskin so that the water overflows through its spout, the surface of the goatskin continues to surround what remains of the water; so the thing whose place is determined has decreased and the place is as it was.
It is possible for the thing whose place is determined to decrease and for the place to increase, as when we make a deep hollow in one of the sides of a cube; its concave surface is then necessarily greater than its plane face, and what remains of the body once it has been hollowed out is much smaller than it was at first. Here the thing whose place is determined has decreased whereas the place has increased. And since the consequences are manifestly inadmissible, the antecedents must be so too.’2
In Volumes II and III we drew attention, under this same title, to the confusion caused by biobibliographers and many historians, from the thirteenth century onwards, regarding the mathematician and the philosopher. We put forward many historical arguments, both regarding technical content and bibliographic, which we still consider to be unanswerable.3 In Volume III, we called upon two important witnesses: ‘Abd al-Laṭīf al-Baghdādī and Fakhr al-Dīn al-Rāzī, both from the end of the twelfth century.
But old habits die hard. Thus, in what is no doubt a desperate attempt to support the identification of al-Ḥasan with his namesake Muḥammad, it has been considered possible to assert that al-Ḥasan’s treatise on place is a revised version of a treatise by Muḥammad, called Treatise on Place and Time according to what he [Muḥammad] Found Following Aristotle’s Opinion on them (Kitāb fī al-makān wa-al-zamān ‘alā mā wajadahu yalzamu ray’ Arisṭūṭālīs fīhimā).
This conjecture is arbitrary in the literal sense, since it is not supported by any argument, whether historical, technical or textual (Muḥammad’s treatise is lost, we have only its title), and it is fraught with implications that are, to say the least, implausible.
1. This title by Muḥammad ibn al-Haytham, reported by the biobibliographer Ibn Abī Uṣaybi‘a from the author’s autobiography, is from a late composition. From the information supplied by Ibn Abī Uṣaybi‘a himself, it appears that this text was composed after January 1027 and before July 1028, that is after Dhū al-Ḥijja 417 of the Hegira and before the end of Jumādā al-ākhira 419 of the Hegira.4 Now, in 417 of the Hegira, Muḥammad, again according to Ibn Abī Uṣaybi‘a, was in his sixty-third (lunar) year. So he would have composed his treatise On Place and Time (now lost, along with the major part of the philosopher’s huge output) at the age of about sixty-five lunar years: so we are not considering a work from his early youth.
2. Between January 1027 and July 1028, Muḥammad had also composed, among other things, A Summary of the Physics of Aristotle (Talkhīṣ al-Samā‘ al-ṭabī‘ī li-Arisṭūṭālīs), A Summary of the Meteorologica of Aristotle (Talkhīṣ Kitāb al-āthār al-‘ulwiyya li-Arisṭūṭālīs), and A Summary of the Book On Animals by Aristotle (Talkhīṣ Kitāb Arisṭūṭālīs fī al-Ḥayawān). To which we must add numerous writings on philosophy, theology, medicine and optics. Moreover, before 1027, Muḥammad ibn al-Haytham had also written a Summary of the Problems of Aristotle’s Physics (Talkhīṣ al-Masā’il al-ṭabī‘iyya li-Arisṭūṭālīs). So it is indeed a philosopher immersed in Aristotelian learning who conceived on this book On Place and Time. It is, moreover, sufficient to run through the titles of many others of his writings on metaphysics, logic and physics for us to register his deep engagement with the work of Aristotle. To confine ourselves to the subject of logic, Muḥammad ibn al-Haytham wrote a summary of Porphyry’s Isagoge, as well as of seven of Aristotle’s logical works; a two-chapter book on the syllogism, a book on proof, and so on. He has also left us a Book to Refute Philoponus’ Criticisms of Aristotle in connection with The Heavens and the World (Fī al-radd ‘alā Yaḥyā al-Naḥwī wa-mā naqaḍahu ‘alā Arisṭūṭālīs … fī al-Samā’ wa-al-‘ālam).
3. We can see clearly the philosophical framework within which Muḥammad ibn al-Haytham was working before and during the period of composition of On Place and Time. The combination of place and time suggests, moreover, that he intended to deal with the ideas of the Physics; and it does not take a great philologist to recognise, simply from the title he gave the work, that Muḥammad had shaped his On Place and Time in accordance with the teachings of Aristotle, or what ‘follows’ from this teaching.
4. If we now return to al-Ḥasan ibn al-Haytham, we have shown that his treatise is resolutely and explicitly anti-Aristotelian. Moreover, in this treatise he formulated the first geometrical theory of place. His anti-Aristotelianism and the originality of his theory did not, indeed, escape his critics, for instance al-Baghdādī at the end of the twelfth century.
Also, in this treatise on place, al-Ḥasan draws extensively on one of his most original and sophisticated mathematical works on isoperimetric and isepiphanic figures and the solid angle (On the Sphere which is the Largest of all Solid Figures having Equal Perimeters and on the Circle which is the Largest of all the Plane Figures having Equal Perimeters — Fī anna alkura awsa‘ al-ashkāl al-mujassama allatī iḥāṭatuhā mutasāwiya wa-anna al-dā’ira awsa‘ al-ashkāl al-musaṭṭaḥa allatī iḥāṭatuhā mutasāwiya).5 This same treatise is mentioned in On Place, as well as in another book by al-Ḥasan: For Resolving Doubts in the Almagest.
Finally, again according to Ibn Abī Uṣaybi‘a and from a list he had found of the writings of al-Ḥasan,6 the text on place (like the majority of al-Ḥasan’s writings) was composed before 1038.
In conclusion, if we accept that Muḥammad and al-Ḥasan are one and the same person, and that the treatise by al-Ḥasan On Place is a revised version of the Treatise on Place and Time according to what he [Muḥammad] Found Following Aristotle’s Opinion on them, we have to accept:
1) that, at the age of sixty-five, al-Ḥasan wrote a treatise on place according to the theory of Aristotle, together with a commentary on the Physics, before changing his opinion completely and turning against Aristotle’s theory. But what event could have led to such a revolution in his thinking, since that is indeed what we are seeing here? Might the composition of his book on isoperimetric and isepiphanic figures and the solid angle (On the Sphere which is the Largest of all Solid Figures having Equal Perimeters) have instigated this conversion? However, a careful reading of this book does not permit us to draw any such conclusion, because the theorem Ibn al-Haytham uses in the treatise On Place can be derived directly from the treatise by al-Khāzin,7 since the mathematician has no need to investigate the solid angle which is the central topic of this book, and since he need not have waited until he was in his sixty-fifth year to turn against Aristotle. On the contrary, as we have shown, the geometrisation of the notion of place can be understood through the advances in geometrical understanding that are accumulated in the other geometrical treatises written by al-Ḥasan ibn al-Haytham. So we should not see the geometer’s anti-Aristotelianism as a sudden change of opinion and still less as simply a philosophical choice; his attitude is, much rather, derived from various works that involve geometrical transformations and movement. In short, al-Ḥasan ibn al-Haytham’s geometrisation of place is an effect of the emergence of geometrical transformations, as operations as well as objects of geometry. In the circumstances, it is hard to see how one could maintain that a mathematician steeped in Aristotelian theory would change his mind because of a result showing that, among solids, it is the sphere that has the minimum surface area, a result known for a long time, that he changed his mind to the point of criticising Aristotle and working out an entirely new theory.
2) that this revolution was not noticed by the man himself, to the point that he did not even refer to it in his second version of the text. This would be the more surprising because it often happens that Ibn al-Haytham returns to an old problem he has already considered, to explore it in a revised text, often a longer one. But in these cases he never failed to refer back to the first version. This is exactly what he did in his treatise On the Figures of Lunes,8 in his treatise On the Construction of the Regular Heptagon9 and in his treatise On the Principles of Measurement,10 among others.
3) that his successors and above all his critics, such as al-Baghdādī, who were conversant with the writings of al-Ḥasan ibn al-Haytham’s times, never even noticed this radical change of position. Surely it is implausible that al-Baghdādī, in particular, would have been unaware of ‘his’ writings on logic to the point of reproaching him for his ignorance of logic; and that because he did not know this first treatise On Place and Time he made no mention of it in his critique of the treatise On Place.
In the absence of historical and textual arguments, any conjectures are possibly correct and there is no limit imposed on implausibility.11 Only an intimate understanding of al-Ḥasan ibn al-Haytham’s mathematical writings can protect one from the temptation to make conjectures like that of there having been a ‘revised version’, a conjecture designed to defend an error made by biobibliographers, and one that has had too long a life.
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1 That is, the sphere is the greatest among solid figures that have equal surface areas.
2 Fakhr al-Dīn al-Rāzī (1149–1210), Al-Mulakhkhaṣ, ms. Teheran, Majlis Shūrā, no. 827, fols 92–93. Cf. above, Ibn al-Haytham’s treatise On Place, pp. 499–500, 511; and the French edition, pp. 670–1 and 955.
3 Les Mathématiques infinitésimales du IXe au XIe siècle, vol. II: Ibn al-Haytham, London, 1993, pp. 8–19; English trans. Ibn al-Haytham and Analytical Mathematics. A History of Arabic Sciences and Mathematics, vol. 2, Culture and Civilization in the Middle East, London, 2012, pp. 11–25. Les Mathématiques infinitésimales du IXe au XIe siècle, vol. III: Ibn al-Haytham. Théorie des coniques, constructions géométriques et géométrie pratique, London, 2000, vol. III, pp. 937–41; English trans. Ibn al-Haytham’s Theory of Conics, Geometrical Constructions and Practical Geometry. A History of Arabic Sciences and Mathematics, vol. 3, Culture and Civilization in the Middle East, London/New York, 2013, pp. 729–34.
4 Ibn Abī Uṣaybi‘a, ‘Uyūn al-anbā’ fī ṭabaqāt al-aṭibbā’, ed. N. Riḍā, Beirut, 1965, p. 558.
5 Les Mathématiques infinitésimales, vol. II, Chap. III.
6 This list is also known through the manuscript in Lahore.
7 Les Mathématiques infinitésimales du IXe au XIe siècle, vol. I: Fondateurs et commentateurs: Banū Mūsā, Thābit ibn Qurra, Ibn Sinān, al-Khāzin, al-Qūhī, Ibn al-Samḥ, Ibn Hūd, London, 1996; English trans. Founding Figures and Commentators in Arabic Mathematics, A History of Arabic Sciences and Mathematics, vol. 1, Culture and Civilization in the Middle East, London, 2012, Chap. IV.
8 Les Mathématiques infinitésimales, vol. II, p. 102.
9 Ibid., vol. III, p. 454.
10 Ibid., vol. III, p. 538.
11 It is with conjectures of this type, and with as little argument for them, that A. Sabra has dedicated himself to defending the identification of the mathematician with the philosopher. The reader will understand why we do not undertake to discuss the conjectures one by one. See A. Sabra, ‘One Ibn al-Haytham or Two? An Exercise in Reading the Bio-Bibliographical Sources’, Zeitschrift für Geschichte der arabischislamischen Wissenschaften, Band 12, 1998, pp. 1–50.