5 , 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 ‘ 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. Rosen, dated 1831 (The Algebra of Muhammed ben Musa, London, Oriental Translations Fund). It has been reprinted by Olms (1986), and is therefore in a better situation than most (useful extracts are in Fauvel and Gray). 2. Much later, but equally important, is the algebra of Khayyam (‘Umar¯ al-Khayy¯am¯i), dating from about 1070. This has been known about for a long time; while it wasfirst translated in the nineteenth century by Woepcke (into French), there is a more ‘modern’ English translation

4. By an irony in the history of research schools, a large number of very interesting texts were translated into Russian by Youschkevitch and his group in the 1950s and 1960s. Even for the readers, whoever they may be, for whom Russian is an easier option than Arabic, they are not accessible in most libraries. 104 AHistory ofMathematics

(Khayyam 1931). This, however, is long out of print and far from easy tofind. Again, there are good extracts in Fauvel and Gray. 3. A more recentfind is the startlingly innovative algebra text al-B¯ahirfi-l jabr (‘The Shining Treatise on Algebra’) of al-Samaw’al (twelfth century). This has been extensively discussed, and good summaries of what is said in some key passages concerned with sums of series and with polynomials are to be found both in Rashed (1994) and in Berggren (1986). However, while there is a modern Arabic text dating from 1976 with introduction and some foot- notes in French by Rashed, there is no translation, indeed there are no translated extracts. And the edition itself, published in Damascus, is not likely to be stocked outside specialist libraries. 4. Lastly, one of the most famous works, often referred to for its sophisticated calculations— in particular the use of decimal fractions—is al-K¯ash¯i’s Mift¯ah. al-h. is¯ab (‘The Calculator’s Key’), written in Samarkand in thefifteenth century.This has been known and studied for over a century. Besides several editions in Farsi (the work was popular in ), and a translation into Russian by B. A. Rosenfeld in 1956, there is a modern Arabic edition, published in Cairo in 1967, and again long out of print. I know of no English translation, or even of any plans for one; although again one can learn something of the work’s unusual features from descriptions in Berggren (and Youschkevitch).

There is now some serious translation underway; and since thefield is very large, it is bound to be selective. One could single out A. S. Saidan’s version of the (recently discovered) arithmetic of al-Uql¯idis¯i, a fascinating work to which we shall return; and numerous translations into French by Rashed, notably the works of Shar¯afal-D¯in al-T. ¯us¯i (1986), and of ibn al-Haytham (a large project, ongoing). These translators (and others), being active researchers, will necessarily be selecting those authors of most interest to them, so that the act of editing and translating is often part of the construction of a personal ‘canon’ of what the translator considers major works. However, in the impoverished situation already described, any such work is invaluable. It could be argued that a serious research engagement with Islamic science should include the acquisition of the ability to read Arabic (which some readers may have anyway). This seems misconceived, insofar as the works concerned are considered as major historical texts. The time is past when the student was expected to be able to have the leisure to learn languages as part of a general liberal education, and while the specialist might need to read Euclid in Greek or the Principia in Latin, no one would expect it of the student on a history course. In any case, as already stated, modern Arabic editions are not easily available, and the deciphering of the difficult manuscripts which are still our primary sources (Fig. 1) is an advanced research skill comparable to reading Sumerian. If the major works of Islamic mathematicians deserve study on an equal footing with the classics of other times, then they should be equally accessible. Those who research the Greek classics are in a fortunate position, in that critical editions and translations have been made available by scholars who (a century ago) considered it an essential part of their work. A commitment to fair treatment for the Islamic classics is now driving a similar effort as far as they are concerned. In a spirit of optimism, one could hope for a significant part of this vast literature, together with a variety of analytical histories, to be readable by students in 20 years time. (And perhaps a start should be made with al-K¯ash¯i, see item 4.) A good recent bibliography of sources and articles (which omits Russian works, but is other- wise comprehensive) is by Richard Hogendijk at www.math.uu.nl/people/hogend/Islamath.html. Islam,Neglect andDiscovery 105

Fig. 1 MS, page from al-K¯ash¯i.

And many of the out-of-print studies and aricles of the past hundred years are being printed as part of the vast series entitled Islamic Mathematics and Astronomy, by Fuat Sezgin (expensive, and rarely found in even the best libraries). The persistent student canfind a great deal of material, but it may involvefinding a friendly librarian, and possibly some expense. Islam,Neglect andDiscovery 123 the obvious high culture of his milieu, one would like more information on what preceded it and what followed; and one wonders how far the sometimes obsessive accuracy of his calculations is motivated by the demands of practice, by competition, or by a pleasure in the activity of calculating itself.

Exercise 8. (a) Look at the table for al-Samaw’al’s polynomial division, and try to follow through the progress of the division, (b) show that the result of the division is 10x3 x2 4x 10 (8/x 2) (2/x 3). + + + + +

8 The uses of religion

Islam provides a whole set of fundamental values. Among these values onefinds the uniqueness of truth, the lack of contradiction between revelation and reason...These values, among others, have without the least doubt pushed forth research and have fostered the creation of open scientific communities. (Rashed 2003, p. 153) Allah is the ideal merchant. All is counted, everything reckoned... A more simply mathematical ‘body of religion’ than this is difficult to imagine. (C. C. Torrey, cited in Rodinson 1974, p. 81)

Earlier in this chapter it was suggested that the argument for the importance of Islamic mathem- atics, indeed its centrality in a tradition which links Babylonians, Greeks, and ‘Moderns’, is now established beyond argument. The idea that Islam itself played some role in the rapid development of the Abbasid period seems also undeniable; the question is, what was it? The argument (recycled in one of the quotes which opens this chapter) that many or even most ‘Muslim’ scientists were not Muslim at all is easily dismissed. Although a substantial number of important earlyfigures belonged to tolerated non-Muslim religions, this had ceased to be true by about 1000 ce, and many leading mathematicians did more than simply conform, actively working in Islamic law or philosophy. If the Christians, Jews, and star-worshippers of the Fertile Crescent had it in them to create a mathematical revolution, one might ask, why did they have to await the advent of a new religion and social organization to do so? We could simply accept a sociological explanation (a new empire required scientific organization on a large scale—supposing that to be true); but this does not explain the specific value put on learning—which led to the Greek and Indian inputs—or the ways in which it was put to use. We are unfortunately at some distance from ninth-century Islam, which was in many ways still in a state offlux. Either Rashed’s characterization of Islam as promoting reason, or Torrey’s more materialistic view of it as a kind of accountancy have germs of truth, and both were argued in the early conflicts of schools. Was there no conflict between the Qur’an and pagan learning or ‘philo- sophy’ (falsafah)? Had God decided everything and measured it from the beginning? Theologians discussed such points and competed for the favour of the khalifs.

For example, is what can be known in Arabic—the language of the Islamic revelation—different from Greek science and philosophy in part because of its linguistic home? Or does there exist a universal logic of thought that transcends (and is therefore superior to) particular expressions in use in a given culture? The h. ad¯ith, as yet one more category, already contain numerous admonitions about the value of knowledge, its reward and the duty to seek it, to gather and preserve it, to journey abroad in search of it. (McAuliffe (2001–), III, p. 101)

The general question of the relation of Islam to pagan and/or practical knowledge is a large one, and we have neither the space nor the ability to deal with it adequately. However, two points 124 AHistory ofMathematics should be made:

1. Islam did certainly differ from Christianity (for example) in the value placed on knowledge, as the quote above illustrates; and the language of the Qur’an itself is strongly centred on appeals to reason: The Koran is a holy book in which rationality plays a big part. In it, Allah is continually arguing and reasoning. (Rodinson 1974, p. 78 (see also the following pages)) (The reason in question, though, can hardly be equated with mathematical deduction; it is rather the deduction of our obligations to God from the beneficence of his works, and of ethical duties from basic principles.) 2. Høyrup’s point, cited in Section 4: by the ninth century at least, Islam had become codified as a complete system of practice, organizing every sphere of human action; from which the needs not simply for knowledge in itself, but for knowledge to inform practice followed.

Rashed’s very recent interview provides some starting points. By claiming that the values of Islam are specifically favourable to science, he raises the stakes, and makes some statements which even those who are quite committed to promoting better understanding of Islamic science might find difficult to accept. The whole interview is worth reading, since as a scholar he cannot only score good debating points but consider difficult questions such as the ‘decline’ of Islamic mathematics after thefifteenth century (how can it be understood and accounted for?). And he makes a more limited but important point, which has indeed been well appreciated, for example, by Kennedy (1983), that time has a particular value in Islamic observances which calls (one would think) for the application of science.

Science was an important dimension of the Islamic city. One element was the time-keeping () in the mosques. Astronomy was necessary to view the lunar crescent for religious purposes. It must not be forgotten that each of the large mosques had an astronomer associated with it... (Rashed 2003)

In fact, few religions have given practical mathematicians so much to think about as Islam, with its lunar months which start at the moment when the new crescent is visible, its carefully defined five prayer-times a day, and its fast which ends at dusk. Astronomers worked tirelessly on the improvement of their tables, developing the Ptolemaic and Hindu astronomy into a much more efficient instrument; but as early as the time of Th¯abitibn Qurra, who wrote on the difficult question of thefirst visibility of the moon’s crescent, they came to realize that their under- standing of atmospheric phenomena always left some doubt about the key questions of what one could see. The science of time was of course useful beyond a religious context, and similarly mathematics was important to theflourishing societies throughout the Islamic world insofar as it helped with commerce, surveying, architecture, and the various practical arts; and also in geography, the understanding of the known world. In this religion enters again, and the tenth-century univer- salist al-B¯ir¯un¯i can stand as a centralfigure, whose Coordinates of Cities made possible a general understanding of how the various widely scattered centres were related on the globe, using a well-developed understanding of geometry on a sphere. Both al-B¯ir¯un¯i and his modern comment- ators have claimed more; that such knowledge was essential for religious purposes, since to design the layout of a mosque (say in Seville) correctly it was essential to determine the qibla, the direction Islam,Neglect andDiscovery 125 of where the faithful should turn for prayer. As he says:

[L]et us point out the great need for ascertaining the direction of the qibla in order to hold the prayer which is the pillar of Islam and also its pole. God, be He exalted, says: ‘So from wheresoever thou startest forth, turn thy face in the direction of the Sacred Mosque, and wheresoever ye are, turn your face thither.’ (Qur’an, Sura 2:150). (Al-B¯ir¯un¯i 1967, pp. 11–12)

The mathematicians may well have thought their knowledge essential; but mathematicians are not always as important as they think, and George Sarton pointed out in 1933 that many medieval mosques in North Africa and Spain have ‘incorrect’ alignments, despite theflourishing state of mathematics in those countries. This problem has recently been cleared up, it appears, in a detailed study of legal writings and of the mosques themselves by Mónica Rius.12 The answer is interesting for the light it throws on the status of mathematics: in fact, Islamic lawyers pointed out that the complex mathematical methods were (a) sometimes uncertain—particularly in the case of longitude—and (b) not accessible to the mass of the faithful, as they should be. They therefore allowed recourse to simpler definitions, which of course gave more ‘approximate’ directions for prayer. This is not to say that al-B¯ir¯un¯i and others were irrelevant; there must have been cases of mosques where the qibla was determined by mathematics. However, here, as elsewhere, its use could be contested and the idea that it was ‘imposed by religion’ certainly begins to seem simplistic. This example can serve as a cautionary tale on the limits of the usefulness of mathematics, which was certainly important enough in the world of medieval Islam. As we shall see, Marxists tend to claim that mathematics is driven by the demands of society, and mathematicians, when it suits them, claim that they are doing vital and useful work. However, if much of the organization of Islam was favourable to science, there were certainly times and places when science could be dispensed with, even treated with hostility.13 To make a parallel, Descartes, Pascal, and Galileo were no less good Christians than their predecessors. If they found that their religion could be harmonized with a rational and practical scientific outlook, the cause is perhaps to be found in the ideological climate, or what Marxists would call the relations of production. Accordingly, a particular difficulty in the statement with which this section opens is that Rashed seems to be treating Islam, as religion and philosophical outlook, as homogeneous in its positive effect on the sciences (at least during the medieval period). It will be interesting to see how other specialist historians react.

Exercise 9. What would be necessary to know in order to determine the qibla? Given the necessary information, how would you do it?

Appendix A. From al-Khw¯arizm¯i’s algebra

(From Fauvel and Gray 6.B.1) A root is any quantity which is to be multiplied by itself, consisting of units, or numbers ascending, or fractions descending. A square is the whole amount of the root multiplied by itself.

12. La Alquibla en al-Andalus y al-Magrib al-Aqs.à, reviewed in Isis 94 (2003, p. 371). 13. Again, Rashed produces good exmples to show that an anti-science outlook cannot be equated with religious ‘orthodoxy’, but there were trends within orthodoxy which were opposed to science.