5 Islam, neglect and discovery 1 Introduction It should be clear from the present chapter that the traditional view of the Arabs as mere custodians of Greek learning and transmitters of knowledge is a partial and distorted one. (Joseph 1992, p. 344) A number of medieval thinkers and scientists living under Islamic rule, by no means all of them ‘Moslems’ either nominally or substantially,played a useful role of transmitting Greek, Hindu, and other pre-Islamic fruits of knowledge to Westerners. They contributed to making Aristotle known in Christian Europe. But in doing this, they were but transmitting what they themselves had received from non-Moslem sources. (Trifkovic 2002) The history of Islamic mathematics is clearly a contested area, and recent history has if anything sharpened the divisions. The view which Joseph described as ‘partial and distorted’ 13 years ago lives on in some academic circles, as the quote from an admittedly right-wing anti-Islamic columnist illustrates. It is perhaps natural that in the current context even questions about algebra in Baghdad in the ninth century should be charged with political relevance, and voices on the fringe should perpetuate old myths. As far as the mainstream of historians is concerned, the points made by Joseph are almost universally conceded, as Katz’ recent respected textbook makes clear: Islamic mathematicians fully developed the decimal place-value number system to include decimal fractions, system- atized the study of algebra and began to consider the relationship between algebra and geometry, brought the rules of combinatorics from India and reworked them into an abstract system, studied and made advances on the major Greek geometrical treatises of Euclid, Archimedes and Apollonius, and made significant improvements in plane and spherical trigonometry. (Katz 1998, p. 240) The only quibble which could be made against this generous assessment is that Katz does not mention the difficulties which previous scholars have had in getting such reasonable claims accep- ted. The major obstacle has been the viewpoint, referred to by Joseph, which sees the Arabs as transmitters rather than innovators. Why is this? We saw in the last chapter that Chinese math- ematics, obviously outside the Western tradition, could be relegated to the sidelines as a mere collection of isolated problems without coherence and without any idea of proof. With the math- ematics which was developed in the Islamic world from the ninth to thefifteenth century ce, the problem is the opposite. The work could with some justice be seen as a part of ‘Western’ math- ematics, looking back to the Greeks and forward to the European Renaissance, and the existence of influences is not in dispute. However, because it was a specialistfield of study and the original texts were often inaccessible, it was possible to ‘forget’ the ways in which the Islamic writers transformed mathematics and to claim (as Trifkovic does) that they did nothing but pass it on. To undertake a proper discussion of the history as it is now understood, it is useful to look briefly at the West, the Islamic world, and their changing interactions. (Historians have a problem about the choice between ‘Arabic mathematics’ and ‘Islamic mathematics’. Neither is completely accurate for the mathematics practiced in the Islamic world between, say, 800 and 1500 ce Since a choice must be made, we shall opt for the more inclusive ‘Islamic’.) The understanding of Islamic mathematics 102 AHistory ofMathematics in Western Europe has gone through a variety of transformations. In the early Middle Ages, from the eleventh to thirteenth centuries, it was highly regarded, for the good reason that the level of achievement was visibly more sophisticated. Those works which were found most comprehensible or useful were translated from Arabic into Latin as were the contemporaneous translations of the Greek classics into Arabic. By the Renaissance (say by 1550) for complex reasons, there had been a change of view, even though the West had not overall achieved the Islamic world’s level of achievement, much less overtaken it.1 The practice of translation from Arabic was less frequent, while the publication of original Greek texts and their translation, again into Latin, made possible a claim that the Moderns were the direct inheritors of the Ancients. Even though, as far as algebra and the number system were concerned, this was clearly untrue, it was a useful myth in constituting a Renaissance world-view which built on the classics as a source of legitimacy. We shall see later how much the work of Viète, Stevin, Descartes, and their contemporaries owed to Islamic precursors; what is important for the moment is that it was not normal to acknowledge the debt. It is not excessively oversimplifying to say that the broad outlines of the Eurocentric history, which Joseph criticizes were laid down in the sixteenth century, and were the dominant version of history until relatively recently. And yet a number of important, often striking Islamic works have been published and studied in western Europe over the last 200 years. Their understanding, and their incorporation into a general history remained the preserve of specialists with no impact on the mainstream view. A better understanding of what Islamic mathematics was has had to wait for: 1. a political motivation—the demand for recognition from the Islamic world from the 1950s on2; 2. unified research programmes, partly related to that politics, which rapidly deepened and expanded the work of study and translation in the 1950s and 1960s. We shall have more to say about what material is and is not available in Section 2. The important change has been not so much an increasing accessibility of sources as an increasing consciousness of the achievements of the Islamic mathematicians. Twenty years ago,3 Roshdi Rashed, one of the leading historical researchers, made much the same points as Joseph: The same representation is encountered time and again: classical science, both in its modernity and historicity,appears in thefinal count as the work of European humanity alone; furthermore, it is essentially the means by which this branch of humanity is defined. In fact, only the scientific achievements of European humanity are the objects of history. (Rashed 1994, p. 333) New texts, new research, and persuasive arguments by respected scholars have largely allowed Islamic mathematics to take its legitimate place in the histories; and among scholars with any serious academic credentials one will no longerfind it neglected or downgraded. The main problems in building up a proper picture are constitutedfirst by the great gaps in our knowledge—which are, of course, also there for the cultures of Greece and China—and second by the sheer diversity of activity (arithmetic, algebra, classical geometry, astronomy, trigonometry, and much else) over 1. This case is argued by Rashed (1994, appendix 2). The general point is incontestable, although there is disagreement about the detail. 2. Said’s influential book (1978), although quite unrelated to the sciences, played a key part in making academics more self- conscious about how they treated things ‘Oriental’. 3. Rashed’s book dates from 1984, although its English translation is 10 years later. Islam,Neglect andDiscovery 103 what is once again a dauntingly long historical period. It should be easy for the student to approach Islamic mathematics, like Greek, without prejudice and make a fair evaluation. Assuming this possible, one could, if only tofix ideas, pose some questions: 1. Can one give a unified description of ‘Islamic mathematics’, given the length of time and space and the variety offields covered —indeed, should we even try to do so? 2. How would we evaluate the ‘Islamic contribution’ to the development of mathemat- ical thought? 2 On access to the literature One would naturally like to recommend, as a follow-up to the general agreement on the import- ance of Islamic mathematics, that the student could consult texts and histories and examine— for example—the questions raised above. Unfortunately, this is not yet the case; and here an accus- ation of ‘neglect’ can still be made, in that access to the relevant materials remains extremely difficult. If we start with secondary texts, that of Berggren (1986) is full, readable, and well- informed. It is, in our current situation, where any reader should start. Rashed’s work (1994) is more specialist, aimed at the exposition of particular points in arithmetic and algebra; it is also expensive and less often stocked by libraries. And while Youschkevitch’s rather older text (1976) is fuller than either of these and contains much which they exclude, it is (a) in French and (b) long out of print. The situation for the student entering thefield could be worse, but it is not very good. With regard to primary sources, what is available reflects a long and patchy history of transla- tion by individual enthusiasts. The relevant section in Fauvel and Gray, though it contains some essential texts, is relatively brief; and while the works of Euclid, Archimedes, and other major Greek mathematicians can often be found in libraries and are reprinted, this is far from being true of the classics of the Islamic world. One initial problem is that there is no longer a canon of a few great writers, rather a large collection of texts whose differing contributions are still in process of assessment.4 More translation is now in progress, but there are major gaps. To take just a few examples: 1. The earliest, founding book on algebra which underlies all subsequent work is (Muh. ammad ibn M¯usa)al-Khw¯arizm¯i’sH . isab al-jabr wa al-muq¯abala (‘Algebra’, lit. ‘calculating by restoring and comparing’, date about 825). This exists in a translation by F.