CHAPTER III
IBN AL-HAYTHAM AND THE GEOMETRISATION OF PLACE
As we have seen, the emergence of geometrical transformations – the operations as well as the transformed objects of geometry – spurred Ibn al- Haytham into conceiving a new mathematical discipline: the discipline of the knowns. This discipline was designed to justify the operations, and to provide a basis for the existence of the objects by its introduction of motion. And the same is true of its method: the technique of analysis. In truth Euclid’s Common Notions, Postulates and Definitions seem no longer to be enough, being unsuitable for the new representation of the objects of geometrical knowledge that are envisaged. In the Elements, this object was simply the figure, without any consideration either of its place or, in general, of the space that contained it. From now on, figures are no longer the sole objects of geometry; moreover, a figure can move, undergo translation, dilatation, contraction, inversion and projection. Thus, the figure moves, and motion even comes into the way the object is conceived. For example, it is relevant to the concept of parallelism, and for the process of deriving figures by the transformation of other figures. So it is clear that it was no longer possible to think about the relationships between elements of a single figure, any more than about the relationships between figures and still less about the relationships used in finding these figures, without raising questions about the notion of spatial relationships itself. This is precisely what Ibn al-Haytham is doing in the treatise On Place.
But, in the tenth century, reflections on the nature of spatial relationships were common currency among philosophers1 and philosopher-theologians.2 It was, clearly, not an idea of space, such as we encounter after Newton, but rather a reflection on place and the void. Indeed, it is in connection with these two concepts that ideas about spatial relationships were formulated. The framework was already to be found in Aristotle’s Physics and it remained current because of that work’s longevity: a theory of place based on the everyday experience that every perceptible body is in some location. In any case, it is in this same intellectual context and in the same language that Ibn al-Haytham returns to the question of space, but he bases himself upon the new concerns generated by a considerable revival in geometry.
The notions of place and of the void had been discussed by Aristotle in several treatises and in some detail in Book IV of the Physics.3 From then on, treatises on physics include a chapter about place and the void in which Aristotle’s theory is adopted, improved upon or refuted. Among the ancients, we have Alexander of Aphrodisias, Themistius, Philoponus and Simplicius; in Ibn al-Haytham’s time the important names are those of al- Fārābī, Avicenna and the members of the Baghdad School, as well as the philosopher-theologians.4 The ancients’ principal writings on this subject were known in Arabic5 and were thus, like the works composed by the philosophers of their own time, available to Ibn al-Haytham and his contemporaries. It is no doubt the wide diffusion of these theories, and the interest taken in ideas about place and the void, that allowed Ibn al-Haytham to dispense with setting out their nature in detail. He confines himself with merely referring to them.
Of these numerous theories and their ramifications, Ibn al-Haytham chose only the two principal ones, whose theses he summarises very briefly, without the argumentation that supports them. He begins by returning to Aristotle’s theory, according to which the place of a body is the enclosing surface that envelopes the body.6 The second thesis considered is that of one of Aristotle’s critics, Philoponus, who maintains that the place of a body is the void filled by the body. Having described these theories, Ibn al-Haytham at once goes on to say that the question of place has not yet received the rigorous attention it deserves and to prove, by his critique, that both of these theses seem equally ill adapted to his purpose and his requirements. And indeed, while these theories were an integral part of books and commentaries on physics, Ibn al-Haytham does not intend that his own treatise shall be a work concerned only with physics, but writes above all as a mathematician; and it is a mathematical notion of place that he proposes to develop in his treatise. It is with exactly this purpose that he composes one of the first treatises entirely and exclusively devoted to the notion of place, at the least the first of its kind. Although his use of philosophers’ terms – that is of the language of the time – has been a cause of misunderstanding,7 Ibn al-Haytham nevertheless does not go so far that he disguises the novelty of what he proposes to do, the more so since his text does not contain anything beyond mathematics. Thus, we find no allusion to a substantial discussion of the perception of place that he had already completed, in his famous Book on Optics.
In constructing his mathematical theory of place, Ibn al-Haytham begins by criticising the Aristotelian thesis. His intention is, however, less that of pointing out its weaknesses than that of laying the foundations for his own theory. When we come to examine it we see that, even if it sometimes does not genuinely refute Aristotle, this critique allows Ibn al-Haytham, as a mathematician, to free the notion of place from any discussion of material existence, that is to free it of its physical and cosmological connections. In contrast, the refutation of Philoponus’ thesis seems to have two purposes. Ibn al-Haytham seems to want to, as it were, put us on our guard against rapidly assimilating his own thesis into that theory. Indeed it is as if he wished to warn us in advance against a mistake that would in reality be made by some commentators, such as ‘Abd al-Laṭīf al-Baghdādī8 and other more recent ones.9 But thanks to this critique he can also determine the conditions for constructing the geometrical concept of place, and it is at this stage that Ibn al-Haytham’s intention becomes clear: we have nothing less than the first known geometrisation of the notion of place. The project is so innovative and so strange in its time that even an Aristotelian philosopher saw it that way, though without being able to understand its exact significance.10 Let us examine Ibn al-Haytham’s procedure.
In Aristotle, place is the place of a body, and its existence is given as something immediately obvious that does not stand in need of proof. To convince oneself of this, it is enough to show what it is not and then go on to investigate what it is, its particular attributes: all of them refer to something that exists. Upwards and downwards are not merely relative to one another, but represent the places towards which certain bodies naturally move. So the true difficulty in the problem of understanding place is connected not with its existence, but with its essence and its definition. Accordingly, we must start by looking for attributes of the essence: all of them arise from a primary relationship ‘between what is contained and what contains it, that is between two things united by a relationship of exteriority; so it is this relationship that makes it possible to determine the essence of place’.11 Thus, Aristotle finds the essence in this primary relationship between the container and that which is contained, what encloses and what is enclosed, and defines place as the first envelope of each body, which does not belong to the body itself but has another body of its own that encloses the former one. Or, as he expresses it: ‘the limiting surface of the body continent – the content being a material substance susceptible of movement by transference’.12 So we are concerned with the inner surface of the containing entity lying next to that which is contained, in which the body is positioned, in accordance with its nature and in accordance with the order of the cosmos, even if the body can be removed from its place. In short, as Aristotle says, ‘Further, place is coincident with the thing, for boundaries are coincident with the bounded’.13 The image of the inner surface of a vase provides a good illustration of such a representation of place. Thus, place is the whole of the surface adjacent to what encloses the whole of the body whose place it is.
Ibn al-Haytham marshals several arguments against Aristotle’s theory – counter-examples, most of them mathematical. We observe that, in all these counter-examples, the only property of the body that the mathematician retains is extension, itself conceived as made up of distances, which is a preparation for a formal idea of place. In short, place becomes ontologically neutral.14
Let us begin by examining the least mathematical of all these counterexamples, one that might be found in the writings of commentators on Aristotle and those of his critics. We take a goatskin filled with water; if we press it, the water overflows through the spout and the surface of the goatskin encloses the remainder of the water. If we repeat this several times, the surface of the goatskin will enclose less and less water and thus will be the place of several volumes of water. Thus, we have the same place for different volumes. Even though an Aristotelian philosopher can always reply that, in this case, the form of the goatskin changes, the argument is not entirely lacking in force and, at the least, refers us back to a difficulty in the theory of combined matter and form.
All the other counter-examples are geometrical in nature and reduce to the fact that a body can have a change in the surface that bounds it without changing in volume, or can even increase in its outer surface while diminishing in volume.
The first example is that of a parallelepiped that we cut into slices with faces parallel to two of its original faces; we rearrange these slices so that the parallel faces make up the faces of a new parallelepiped. The volume remains unchanged, whereas the area of the outer surface that encloses it, and thus the place, has greatly increased.
Furthermore, if we consider a body with plane faces, which we hollow out so as to give its interior, for example, the form of a concave sphere its volume diminishes whereas its enclosing surface increases. If on the other hand we consider a wax cube that we model into a sphere, its surface area diminishes without its changing its volume, in accordance with the properties of bodies with the same surface area (isepiphanic bodies) established by Ibn al-Haytham in another treatise.15
If, again, we model the cube into a regular polyhedron with twelve faces, then this polyhedron has a surface area – and thus a place – greater than that of the initial cube. Ibn al-Haytham had in fact proved that if there are two regular polyhedra with similar faces that have the same total area, then the polyhedron that has the greater number of faces has the greater volume.16 So if the cube and the polyhedron with similar faces have the same surface area, the volume of the polyhedron would be greater than that of the cube, which is contrary to what was assumed.
An Aristotelian would certainly not find himself at a loss as to how to reply to Ibn al-Haytham’s criticisms. He could indeed object that the ‘individual’ body is no longer the same since in one case it is the form that has been altered and in the other case it is the matter that has been changed. This is indeed the way that the philosopher and physician ‘Abd al-Laṭīf al-Baghdādī replies to the mathematician.17 But this reply would not have affected Ibn al-Haytham’s opinion, which was based on other grounds, lying outside Aristotelianism. We have seen that he gives a different meaning to the word ‘body’; he also gives another meaning to the expression ‘adjacent surface’. Like the body, this entity has in fact no other quality except extension in three dimensions. The body and the adjacent surface have now been stripped of any physical or cosmological quality. So there is every indication that in his critique of the Aristotelian theory, Ibn al-Haytham is setting out not so much to mount an effective attack but rather to prepare the ground for a deliberately more abstract conception of the idea of place. It is in the course of his critique of the theory of place of the type put forward by Philoponus that Ibn al-Haytham set about constructing his own concept of place.
We may first observe that it is in the light of this theory, but also predominantly against it, that Ibn al-Haytham constructs his concept of place. Nevertheless, Ibn al-Haytham does not approach the theories as a historian and it can happen that in criticising certain notions he introduces into them meanings slightly different from their original ones. All the same, since he does not cite any names or any book titles, we need to be prudent.
In his commentary on Aristotle’s Physics and especially in his Corollaries on Place and the Void,18 Philoponus develops the theory that place is an extension in three dimensions, empty by definition, and thus distinct from the bodies that may occupy it. He expresses his idea as follows:
That place is not the limit enclosing a body is adequately clear from what has just been said; that it is a certain three-dimensional interval, distinct from the bodies that are to be found in it (because place and the void are in reality the same in regard to their substance), we might show this by elimination of the other possibilities: indeed if it is not the matter nor the form nor the limit of the enclosing body, there remains only that place is the interval.19
As to the meaning of this key concept of extension, Philoponus writes:
And I certainly do not say that this interval has ever been or could be empty of all body. Absolutely not, but I state that it is something other than the bodies that are to be found in it, that it is empty as regards its own definition, but that it is never separated from a body, rather as we say that matter is something other than forms, but that it can nevertheless never be separated from a form. So in this way we agree that an interval is something other than any body, empty as regards its own definition but there are, continually, new bodies which come to be found in it, itself remaining immobile, as a whole and in regard to its parts, as a whole because the cosmic interval which admitted the body of the whole Universe can never be moved, in regard to its parts because it is impossible that the interval, incorporeal and empty by its own definition, can move.20
For Philoponus, extension exists (‘an interval is something other than the bodies that are to be found in it, but it is never without bodies’);21 it is empty by definition. To sum up, what Philoponus means by ‘place’ is extension in three dimensions, empty but possessing existence, even if one might say that existence is not ‘in actu’.
There remains the question of knowing how, starting from dimensions that are empty and of necessity abstract, we can observe a variety of different bodies. It is this question and the difficulty that it raises which, it seems, persuaded Ibn al-Haytham to move away from Philoponus’ theory. This theory is, indeed, incapable of explaining how an extension, defined in this way, is the place even of a body – if not of a family of different bodies – unless we suppose that we are concerned simply with extension conceived in relation to the body. A static theory, if one may call it so, that Ibn al-Haytham makes an effort to turn into a dynamic one, but at the cost of subjecting it to considerable modifications.
From the theory of place put forward by Philoponus, Ibn al-Haytham retains the idea of empty extension and that of the existence of place independently of any body to be found in it. But, as a mathematician, Ibn al-Haytham gives these two ideas a sense different from that given by the philosopher of nature. He begins by assigning empty extension a level of existence, that of mathematical concepts: it is ‘imagination’ which, as we have already seen, for Ibn al-Haytham is an act of thinking by which, stating from the traces left by objects, we separate out intellectual forms that are unchanging.22 So we are concerned with an ‘imagined void’, apprehended by this act that starts from the traces of bodies that move from one position to another. After that we can take this position to be empty, even if it is never empty because it will at once be filled by another body. The act of imagination separates out unchanging intellectual form from this void: the distances between all the imagined points, distances that are themselves imagined because they are not material entities; they are in fact the imagined distances between all the points of the surface of a region of space. This manner of conceiving extension presents two advantages: Ibn al-Haytham does not need to give a purely conventional definition of the void; on the other hand he is in a position to present a mathematical notion of the void without having to believe in the existence of a physical void. So, by employing the adjective ‘imagined’, Ibn al-Haytham ensures that the mathematical notion of place has a level of existence.
But we need to know how this imagined void becomes the place of a body, or of a variety of bodies. In this, Ibn al-Haytham clearly departs from all his predecessors. He does not propose a single set of imagined distances, but two. First, the distances that are ‘fixed, intelligible, imagined’ (althābita al-ma‘qūla al-mutakhayyala)23 in this void-extension, in this region of space. On the other hand, the set of imagined distances between all the points of an arbitrary body. For Ibn al-Haytham these distances, those in both sets, are segments of straight lines. So we shall say that an imagined void is the place of a certain body if and only if the imagined distances from this body ‘can be superimposed on and can be identified with’ the distances from the imagined void.
These two sets of distances and this ‘perfect superposition’ are the essential elements of this new conception of place. The end result of the superposition is another set of distances, since we are dealing with segments of straight lines, and thus with lengths without breadth; or, as we shall read later:
But if on every imagined distance we superimpose an imagined distance, together they will be a single distance, because the imagined distance is simply the straight line which is a length without width. Now if on the straight line which is a length without width we superimpose a straight line which is a length without width, together they become a single straight line, because from their superposition there results no width, nor a length which exceeds the length of one of them. If one of the two imagined straight lines is superimposed upon the other, they become a single straight line which is a length without width. The imagined void filled by the body is thus imagined distances, on which there are superimposed the distances of the body, and which have become the single and same distances.24
Ibn al-Haytham’s conception is unequivocal, and is sharply different from that of Philoponus. We now know why, from the beginning of the treatise, he has made a point of warning us against hastily assuming the two are the same. Let us assess Ibn al-Haytham’s ideas on this matter using words other than his own, to try to make clear both Ibn al-Haytham’s intentions and the nature of his contribution.
Ibn al-Haytham simply jettisons the idea found in all his predecessors, that of regarding a body as a whole, and substitutes for it a view of a body as a set of points joined up by segments of straight lines. Thus, of all the qualities a body can have, he retains only its extension, itself understood as a set of line segments. Moreover, the imagined void is also a set of unchanging line segments joining the points of a region of three-dimensional space, independent of any body. Thus, the imagined void, that is place, is conceived from the first as a region of Euclidean space with its intrinsic metric. In other words, let C be the body concerned; with it we associate an abstract construct – place – which is V, the set of the distances (V is the imagined void), with a one-to-one correspondence C → V. The distances which define V do not depend on the body C that fills them: they are unchanging in magnitude and in position. This place is called the place of the body C if and only if we establish the isometric one-to-one correspondence mentioned above. Place has a reality that is independent of any body: it is the family of imagined distances. This latter is obviously conceived more geometrico, within the framework of Euclidean geometry. Finally, the place of a body is defined, as we shall see later, as being the metric of the part of Euclidean space occupied by the body, which is itself conceived in the same way, and the two are connected by an isometric one-to-one correspondence. In such a scheme, it is clear that Euclidean space, a universal void, serves as a substrate for the unchanging distances between all the points, even though that is not stated in so many words. This substrate is indispensable for the coherence of the fixed distances that are considered in one or another region of the space, that is at the point, and thus to the conception of places as regions, or parts, of this space. It was not until Descartes, it seems, that it was to be stated, this time explicitly, that space is logically anterior to points.25 Although it is succinct, Ibn al-Haytham’s treatise geometrises the notion of place and mathematises the notions connected with it. It is, as far as I know, the first treatise that includes an attempt of this kind, and this is the direction that will later be followed by seventeenth-century mathematicians, notably by Descartes and Leibniz.26
Now this conception allows Ibn al-Haytham to do what was forbidden to his predecessors: he can compare different geometrical solids, and also diverse figures, which occupy the same place, as well as the places that they occupy. From now on it is legitimate for him to think about their observable relationships, their positions, their shapes and their magnitudes, as he planned to do in The Knowns. It is now possible to make a rigorous comparison between a solid – for example a sphere – or an arbitrary figure, such as a circle, and so on, and its transformed version, as well as to compare their respective places, as it is possible to compare each to the other or to a third entity in a different place. He needed something like this new conception of place in order to investigate geometrical transformations.
So Ibn al-Haytham’s treatise has close links with the new discipline of the Knowns. It is a book on geometry, or, we might say, on the philosophy of geometry. It consciously positions itself outside the tradition of investigations of place as they appear in Aristotle’s Physics, or in the works of his critics and commentators, Greek and Arabic. So we should be risking serious misunderstanding if we were to discuss Ibn al-Haytham’s theses without being aware that we need to see him as deliberately setting out to conceive place in terms that are mathematical and abstract. This kind of mistake was, however, made by, for example, ‘Abd al-Laṭīf al-Baghdādī.
HISTORY OF THE TEXT
The treatise On Place (Fī al-makān) by Ibn al-Haytham appears on the lists of his writings drawn up by al-Qifṭī and Ibn Abī Uṣaybi‘a.27 In this same treatise, Ibn al-Haytham refers to his book on isoperimetric figures. Further, in a treatise we have edited in Les Mathématiques infinitésimales (vol. IV, Appendix III), al-Baghdādī quotes at length from this text by Ibn al-Haytham. The philosopher-theologian Fakhr al-Dīn al-Rāzī also refers to it more than once. That is to say that we have a superabundance of evidence for the authenticity of the attribution of this text.
The treatise itself has come down to us in five manuscripts.
The first, which we shall call C, belongs to the collection 2823 in Dār al-Kutub in Cairo, fols 1v–5v. This same collection includes another treatise by Ibn al-Haytham, on the direction of the Qibla (Fī samt al-qibla). This latter is copied in the same hand; we read in the colophon ‘copied from a copy in the hand of Qāḍī Zādeh’ (fol. 5v), that is the hand of the famous astronomer and mathematician in the employ of Ulugh Beg, during the first half of the fifteenth century. The writing is in nasta‘līq. We can find four omissions of a word and two omissions of a sentence of more than three words.
The second manuscript, which we shall call T, belongs to the collection 2998, fols 166–74, in the library of Majlis Shūrā in Tehran. This collection also includes several other treatises on optics by Ibn al-Haytham: On Light, On Shadows, On the Light of the Heavenly Bodies. This collection is in the same hand, in nasta‘līq script.
We note five omissions of a word and one omission of a sentence of three words, with a relatively large number of errors. C and T have two omissions of a word in common.
The third manuscript forms part of the collection 2196, fols 19v–22r, in the Salar Jung Museum at Hyderabad (India), we shall call it H.
The fourth manuscript – called L – belongs to the collection no. 1270, fols 25v–27v in the India Office Library in London. We do not know the date of the copy, which could be the tenth century of the Hegira. Examination shows that it has one omission of a word and six errors. Apart from these omissions peculiar to each of the manuscripts H and L, they share three omissions of a word and twenty errors.
The fifth manuscript belongs to the collection Fātiḥ 3439, fols 136v–138r, in the Süleymaniye library in Istanbul – we have called it F. This collection includes several treatises by Ibn al-Haytham. The manuscript was copied in 806/1403–4. It is difficult to read because the ink has faded and it contains a significant number of omissions.
Comparing these five manuscripts two by two, we can divide them into two groups: H and L on the one hand and C and T on the other, while F, on account of its omissions and its errors, remains separate. The probable stemma is particularly simple merely on account of our drastic lack of information.
This text by Ibn al-Haytham was published at Hyderabad, by the Oriental Publications Bureau, in a non-critical edition by Osmania based only on manuscript L.
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1 For example al-Fārābī’s text on the void: Risāla fī al-khalā’, edited and translated by Necati Lugal and Aydin Sayili in Türk tarikh yayinlarindan, XV.1, Ankara, 1951, pp. 21–36. In his lost Physics, al-Fārābī must certainly have dealt with this subject. See also the Physics of al-Shifā’ by Avicenna, ed. Ja‘far Āl-Yāsīn, Beirut, 1996, Chapters 5 to 9; and al-Najāt by Avicenna, ed. M. S. al-Kurdī, Cairo, 1938, pp. 118–24.
2 Later philosopher-theologians were able to pass on to their own time the theses discussed by their predecessors, notably the discussion in the School of Baṣra from Abū al-Hudhayl al-‘Allāf and his nephew al-Naẓẓām, as well as later Abū ‘Alī al-Jubbā’ī and his son Abū Hāshim. See for example Ibn Mattawayh, al-Tadhkira, ed. Samīr Naṣr Luṭf and Faysal Badīr ‘Aūn, Cairo, 1975, especially p. 116. See also Abū Rashīd al-Nīsābūrī, Kitāb al-Tawḥīd, ed. Muḥammad ‘Abd al-Hādī Abū Rīda, Cairo, 1965, pp. 416 sqq. See Alnoor Dhanani, The Physical Theory of Kalām, Leiden, 1994, pp. 62–89.
3 It is above all in the first six chapters if Book IV of the Physics that Aristotle develops his arguments and his theory of place. See Aristotle, Physics, Books III and IV, trans. by E. Hussey, Clarendon Aristotle Series, Oxford, 1983 and trans. P. H. Wicksteed and F. M. Cornford, London and Cambridge (Mass.), 1970, 211a–213a. On the important problem of the place of the Whole, see Marwan Rashed, ‘Alexandre et la “magna quaestio”’, Les Études classiques, 63, 1995, pp. 295–351, especially pp. 303–5. On the problem of place in Aristotle, see the now classic study by V. Goldschmidt, ‘La théorie aristotélicienne du lieu’, in Écrits, Paris, 1984, vol. I, pp. 21–61.
5 See the translation of Aristotle’s Physics with commentaries by Ibn al-Samḥ, by Mattā ibn Yūnus, by Ibn ‘Adī and by Abū al-Faraj ibn al-Ṭayyib, edited by ‘A. Badawi in Arisṭūṭālīs, al-Ṭabī‘a, vol. I, Cairo, 1964; vol. II, Cairo, 1965, especially Book IV, vol. I, pp. 271 sqq. See also E. Giannakis, ‘Yaḥyā ibn ‘Adī against John Philoponus on Place and Void’, Zeitschrift für Geschichte der arabisch-islamischen Wissenschaften, Band 12, 1998, pp. 245–302.
6 See Aristotle, Physics IV, 212a; cf. trans. Hussey, p. 28.
7 In fact, here, as in the Analysis and Synthesis as well as in The Knowns, that is in the treatises preceded by a theoretical introduction containing a mixture of philosophical and mathematical considerations, Ibn al-Haytham makes use of the language of the philosophy of his time, which has an Aristotelian look. We encounter terms such as ‘essence’, ‘in actu’, ‘in potentia’, ‘form’, ‘place’, ‘demonstrative syllogism’, and so on. Although for a historian versed in Ibn al-Haytham’s mathematics, optics or astronomy this terminology does not impede understanding Ibn al-Haytham’s actual intentions and ideas, it may happen that it misleads a historian of philosophical theories. Such a reader can indeed see here a trace of Aristotelianism, whereas Ibn al-Haytham thinks very differently. This is exactly the mistake made by the philosopher al-Baghdādī, see p. 500.
8 See Appendix III in Les Mathématiques infinitésimales, vol. IV.
9 Thus, A. Dhanani writes: ‘In the end he [Ibn al-Haytham] endorsed this view of space (which derives ultimately from John Philoponus)’ (The Physical Theory of Kalām, p. 69). This error does not affect the value of A. Dhanani’s work, because Ibn al-Haytham is not an example of the theologian philosopher – the true subject of the book.
10 Al-Baghdādī, like al-Rāzī, has not grasped the essential point of Ibn al-Haytham’s theory, namely the one-to-one correspondence between two sets of different distances.
11 ‘entre le contenu et le contenant, c’est-à-dire entre deux choses unies par une relation d’extériorité; c’est donc celle-ci qui permettra de déterminer l’essence du lieu’ (V. Goldschmidt, ‘La théorie aristotélicienne du lieu’, p. 28).
12 Aristotle, Physics, Book IV, 212a, trans. P. H. Wicksteed and F. M. Cornford, p. 313; cf. trans. E. Hussey, p. 28.
13 Aristotle, Physics IV, 212a29–30, ed. and trans. Ross.
14 We may, moreover, ask ourselves whether one or another of Ibn al-Haytham’s mathematical predecessors had already made a move towards stripping the idea of place of ontological elements. In other words, we may wonder whether there was a movement to ‘de-ontologise’ place, a movement of which Ibn al-Haytham was part. This conjecture stems from a thesis attributed to Thabit ibn Qurra, to be found in a text that is no longer extant.
According to the testimony of the philosopher-theologian Fakhr al-Dīn al-Rāzī, Thābit ibn Qurra, unlike the philosophers, was notable for having a thesis of his own: one that contradicted the Aristotelian theory of natural place. Al-Rāzī wrote as follows: ‘the philosophers are in agreement about it (every body has a natural place); nevertheless I have seen in some chapters attributed to Thābit ibn Qurra a surprising theory which he has chosen for himself’ (al-Mabaḥīth al-mashriqiyya, Tehran, 1966, vol. II, p. 63). Al-Rāzī quotes Thābit ibn Qurra before going on to criticise this theory: ‘Thābit ibn Qurra has said: he who believes that the Earth is seeking for the place in which it is to be found, has a false opinion; because there is no need to imagine in any place whatsoever a state that is proper to it making it unlike others. But, on the contrary, if one had imagined all places to be empty and then that the whole Earth arrives at any one of them, it necessarily stops there and does not move away towards another [place], because this one and all the places are equivalent’ (ibid., p. 63).
On the theory of natural place and the attraction of the Earth, see Marwan Rashed, ‘Kalām e filosofia naturale’, in R. Rashed (ed.), Storia della scienza, vol. III: La civiltà islamica, Rome, Istituto della Enciclopedia Italiana, 2002.
15 Les Mathématiques infinitésimales du IXe au XIe siècle, vol. II: Ibn al-Haytham, London, 1993, Chap. III; 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. 289–95.
16 Ibid., vol. II, p. 339 and pp. 444–51; English trans. pp. 249 and 336–9.
17 See Appendix III in Les Mathématiques infinitésimales, vol. IV.
18 See Ioannis Philoponi in Aristotelis Physicorum libros quinque posteriores commentaria, ed. H. Vitelli, CAG XVII, Berlin, 1888.
19 Philoponus, In Phys. 567, 29–568, 1.
20 Philoponus, In Phys. 569, 7–17.
21 Philoponus, In Phys. 569, 19–20.
22 See Introduction, pp. 11–12.
25 Descartes writes in his Discours de la méthode: ‘[…] the geometers’ object, which I conceived as being a continuous body, or a space indefinitely extended in length, width and height or depth, divisible into various parts, which could have various shapes and magnitudes, and could be moved or transposed in any way’ (Œuvres de Descartes, publiées par C. Adam and P. Tannery, Paris, 1965, vol. VI, p. 36).
26 One simply cannot avoid noticing that it is this direction that the seventeenth-century mathematicians will take, each in his own way, and with differences that need to be pointed out on each occasion. For example, let us look at what Leibniz wrote in Geometrical Characteristic, where he represents place as a fragment of geometrical space. Place is a situs in Leibniz’s sense, that is a relation between different points of a configuration (of an object) and Leibniz indicates it with a ‘.’. As for example A.B: ‘A.B represents the mutual situation of the points A and B, that is an extensum (rectilinear or curvilinear, it does not matter which) that connects them and remains the same as long as that situation does not vary’ (La Caractéristique géométrique, text edited and with introduction and notes by Javier Echeverría; French translation with notes and postface by Marc Parmentier, coll. Mathesis, Paris, 1995, p. 235).
27 See Les Mathématiques infinitésimales, vol. II, p. 524; English trans. p. 408.