In his magisterial Aperçu historique, Michel Chasles, having given the titles of books and referred to the achievements of the great geometers of the Hellenistic period, writes:
[…] then, for two or three centuries more, there came the writers of commentaries, who have passed on to us the works and the names of the geometers of Antiquity; then finally the centuries of ignorance, during which Geometry slumbered among the Arabs and the Persians until the Renaissance of learning in Europe.1
In this peremptory judgement, Chasles, whose good faith cannot be doubted, was setting out what was known by historians in the mid nineteenth century rather than describing historical facts. At the time, investigations into the history of geometry in classical Islam were few and scattered, and it is not surprising that Chasles’ judgement became the received opinion. And indeed we find it repeated tirelessly in the historical introductions to manuals of geometry, such as that by Robert Deltheil and Daniel Caire,2 as well as in writings penned by historians of geometry, even sometimes up to the present day. Nevertheless, a little later a better, though still far from satisfactory, grasp of the historical facts struck the first blows against this general prejudice. Today, for the majority of historians of mathematics, the opinion for which Chasles was the spokesman has given way to a different one, less absolute without however being more accurate, which could be summarised as follows: Arab geometers, while they never reached the high level of the geometry of the classical Greeks, did at least have the merit of recognising the importance of their work and of having preserved both its spirit and its matter, even going so far as to add several notable details. For this, the names that are mentioned are those of Thābit ibn Qurra and of Naṣīr al-Dīn al-Ṭūsī. Although it is more nuanced, but also more eclectic, this way of looking at things in fact derives from the same logic: to stop at the threshold of questions, without indicating the criteria and setting out the reasoning that could have led to this modest contribution to geometry. It is not clear why, according to the proponents of this theory, the geometers of classical Islam should have thus confined themselves to the role of conscientious preservers of the Hellenistic geometrical heritage, while they were making major advances in all the other disciplines: algebra, number theory, trigonometry, and so on. It is inexplicable that the notable developments in these latter disciplines and in mathematised disciplines such as astronomy and optics should have had so little effect on work in geometry. It is not clear why the single exception made by the historians of mathematics should be for the development of the theory of parallel lines.
To understand how such an opinion came into being, we may point to historians’ ideology, the failings of historical research in this area and the huge size of the field of investigation, which is often examined only in part and in a narrowly focussed way; geometers are considered one at a time and their individual contributions are often divided up, which makes it difficult to perceive the underlying mathematical rationality, the more so because the development of geometry in classical Islam may appear in a somewhat paradoxical light.
The geometers of classical Islam are the heirs of the Greek geometers and, one might say, of them alone. Geometry, from the ninth century onwards, is incomprehensible without a knowledge of the works of Euclid, Archimedes, Apollonius, Menelaus and others, which were translated into Arabic. But to understand the linkage between the two phases of the history we first need to make a critical examination of how, from the ninth century on, the geometers took possession of this immense heritage.
This task is enormous; it amounts to finding the relation between Greek and Arabic geometrical work. We hope that the volumes of this book will contribute to carrying out this task, because establishing this relationship is not only necessary for grasping the history of geometry from the Greeks to at least the eighteenth century, but also we cannot manage without it if we wish to make a rigorous assessment of what was contributed by Arabic geometry. This is also the method that must be adopted if we wish to avoid writing in the very worst style for history, namely that of eclecticism: in this style, work written in Arabic is reduced to versions of Greek geometry, or again we discern in it the seeds of future geometry, but always in small pieces and in particular cases.
To look at the wider picture, it seems to be best to look back from the twelfth century, when geometrical research had already been carried out in Arabic for three centuries. Now this picture, while very different from that of the third century BC, is also much larger. In the twelfth century the domain of geometry includes all the area of Greek geometry, but we also find territory that is almost virgin: algebraic geometry represented by the works of al-Khayyām and Sharaf al-Dīn al-Ṭūsī; Archimedean geometry given renewed vigour by more substantial use of arithmetical sums and the employment of geometrical transformations; and it extended into domains hardly glanced at earlier: solid angles, lunes, and so on; the geometry of projections, that is to say the study of projections as a complete subject area within geometry, as it is presented in the works of al-Qūhī and Ibn Sahl; trigonometry (for example, in al-Bīrūnī); the theory of parallel lines, and so on. Some of these subject areas were known to Greek geometers, others were hardly suspected and there were others whose very existence was inconceivable for the Greeks. But it is difficult to draw a map of such a continent. One risks going astray if one proceeds author by author and relies on the books that are available. We need to begin with the research traditions: first to identify them, to make rough reconstructions of them, accepting that the description will need to be filled out later, and to recognise variations and individual styles. Without employing this method, the historian cannot find the patterns of reasoning which govern research work in geometry. If we do not know how to formulate things, the history is obscured and it is impossible to recognise the lines of division that run through it. So we do not see epistemological analysis as an optional luxury: it provides our only means of identifying traditions and styles. This is the task we set ourselves in the volumes of this series. In the first two volumes, we tried to reconstruct the subject area of ‘infinitesimal geometry’, with its dominant type of reasoning. We shall not repeat here, in summary form, what we set out there in detail; we shall merely note that these mathematicians combined infinitesimal arguments with projection, and infinitesimal arguments with point-to-point geometrical transformations. Moreover, they brought together geometry of position and geometry of measurement, to an incomparably greater extent than had been done in the past. In other works, we have considered Ibrāhīm ibn Sinān, al-Qūhī and Ibn Sahl, who all lived in the tenth century, and we have noted the same things in relation to projections, transformations, geometry of position and geometry of measure. In the third volume of the present series, we have proceeded in the same way: reconstructing the tradition that led to the opening up of a new area in geometry, ‘geometrical constructions by means of the conic sections’, new criteria for constructability and new means for carrying out constructions (notably the use of transformations).
The introduction of the concepts of transformation and projection as concepts proper to geometry, and (a fortiori) the concept of motion, the use of motion in definitions and proofs encouraged geometers to make more extensive use of transformations – which is what Ibn al-Haytham later does in his treatise On the Properties of Circles, translated in this volume – and to examine methods for discovery and proof and also to give justifications for making use of these concepts, particularly that of motion. This again is exactly what Ibn al-Haytham turns to in his treatise Analysis and Synthesis and in his book The Knowns, and it is what explains why he needed to geometrise the concept of place; which he did.
But, to assess these works, and all the others that the reader will find here translated into English for the first time and the subject of commentaries, it was better not to isolate the works from their context and the other writings in the tradition to which they belong. So the reader will find here two texts – one by Thābit ibn Qurra, the other by al-Sijzī – to which Ibn al-Haytham’s treatise Analysis and Synthesis is related. These two texts were already published, in an unsatisfactory edition, so we made critical editions of them, as rigorously as possible, as well as a French translation that was as precise as possible. Here the two works appear in English for the first time. In the same spirit, we have also included another text by al-Sijzī.
Ibn al-Haytham’s On Place was the target of thunderbolts hurled by the Aristotelian philosopher ‘Abd al-Laṭīf al-Baghdādī, who devoted a complete book to his criticisms. We gave the editio princeps and the first translation of this text in Les Mathématiques infinitésimales (vol. IV, 2002).
As has been the rule in this series of volumes, Christian Houzel, Directeur de recherche at the Centre National de la Recherche Scientifique, reread the analyses and historical and mathematical commentaries that I wrote to accompany all these texts.
Pascal Crozet, Chargé de recherche at the Centre National de la Recherche Scientifique, did the same for the analysis and commentaries on the texts by al-Sijzī. Badawi El-Mabsout, Professor at the University of Paris VI, has read our commentary on the geometry of triangles. I thank them all very much for their comments and criticisms, which have been of considerable benefit to this work.
I am also grateful to Aline Auger, Ingénieur d’Études at the Centre National de la Recherche Scientifique, who has prepared this book for printing and compiled the indexes.
I also thank Professors S. Demidov and M. Rozhanskaya who helped to arrange for me to visit St Petersburg where I was able to work on the manuscript of Ibn al-Haytham’s book On the Properties of Circles; Professor B. Rosenfeld who, more than forty years ago, courteously sent me photographs of a very significant part of this St Petersburg manuscript; the Nabī Khān family and Obaidur-Rahman Khān for having allowed me to work on the manuscript of al-Sijzī’s texts; and finally Professor Y. T. Langermann for having sent me a microfilm of the text by al-Baghdādī.
Roshdi RASHED
Bourg-la-Reine, December 2001