Studies in the Making of Islamic Science: Knowledge in Motion Volume 4 9781315242170, 9780754629160

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Studies in the Making of Islamic Science: Knowledge in Motion

Islam and Science: Historic and Contemporary Perspectives Titles in the Series: Studies in the Islam and Science Nexus Volume 1 Muzaffar Iqbal Contemporary Issues in Islam and Science Volume 2 Muzaffar Iqbal New Perspectives on the History of Islamic Science Volume 3 Muzaffar Iqbal Studies in the Making of Islamic Science: Knowledge in Motion Volume 4 Muzaffar Iqbal

Studies in the Making of Islamic Science: Knowledge in Motion Volume 4 Edited by

Muzaffar Iqbal Center for Islam and Science, Canada

O Routledge

S^^ Taylor & Francis Group LONDON AND NEW YORK

First published 2012 by Ashgate Publishing Published 2016 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN 711 Third Avenue, New York, NY 10017, USA Routledge is an imprint of the Taylor & Francis Group, an informa business Copyright © Muzaffar Iqbal 2012. For copyright of individual articles please refer to the Acknowledgements. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Wherever possible, these reprints are made from a copy of the original printing, but these can themselves be of very variable quality. Whilst the publisher has made every effort to ensure the quality of the reprint, some variability may inevitably remain. British Library Cataloguing in Publication Data Studies in the making of Islamic science : knowledge in motion. - (Islam and science ; v. 4) 1. Islam and science. I. Series II. Iqbal, Muzaffar, 1954297.2'65-dc23 Library of Congress Control Number: 2011935822 ISBN 9780754629160 (hbk)

Publisher’s Note The publisher has gone to great lengths to ensure the quality of this book but points out that some imperfections from the original may be apparent.

Contents Acknowledgements Editor's A cknowledgem ents Introduction

vii ix xi

PART I GREEK INTO ARABIC 1 Paul Kunitzsch (2003), 'The Transmission of Hindu-Arabic Numerals Reconsidered', in J.P. Hogendijk and A.I. Sabra (eds), The Enterprise of Science in Islam: New Perspectives, Cambridge, MA: The MIT Press, pp. 3-21. 2 Juan Vernet and Julio Samso (1996), 'The Development of Arabic Science in Andalusia', Encyclopedia of the History of Arabic Science (Vol. 1), London: Routledge, pp. 243-75. 3 Bernard R. Goldstein (1996), 'The Heritage of Arabic Science in Hebrew', Encyclopedia of the History of Arabic Science (Vol. 1), London: Routledge, pp. 276-83. 4 J.L. Berggren (1991), 'Greek and Islamic Elements in Arabic Mathematics', published in a special edition ofApeiron: A Journal for Ancient Philosophy and Science, 24, pp. 195-217.

3 23 65 75

PART II NATURALIZATION, TRANSFORMATION, AND ORIGINALITY 5 A.I. Sabra (1987), 'The Appropriation and Subsequent Naturalization of Greek Science in Medieval Islam: A Preliminary Statement', History of Science, 25, pp. 223^2. 6 A.I. Sabra (1996), 'Situating Arabic Science: Locality versus Essence', Isis, 87, pp. 654-70. 7 J. Len Berggren (2001), 'Historical Reflections of Scientific Knowledge: The Case of Medieval Islam', in Ruth Hayhoe and Julia Pan (eds), Knowledge Across Cultures: A Contribution to Dialogue Among Civilizations, Comparative Education Research Centre: The University of Hong Kong, pp. 127-38. 8 George Saliba (1994), 'Early Arabic Critique of Ptolemaic Cosmology: A Ninth-Century Text on the Motion of the Celestial Spheres', Journal for the History of Astronomy, 25, pp. 115^1. 9 George Saliba (1993), 'Al-QushjT's Reform of the Ptolemaic Model for Mercury', Arabic Sciences and Philosophy, 3, pp. 161-203. 10 George Saliba (1989), 'A Medieval Arabic Reform of the Ptolemaic Lunar Model', Journal for the History of Astronomy, 20, pp. 157-64.

101 119

137 149 177 221

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11 George Saliba (1996), 'Arabic Planetary Theories after the Eleventh Century AD', Encyclopedia of the History of Arabic Science (Vol. 1), London: Routledge, pp. 58-127. 12 Roshdi Rashed (1993), 'Al-Kindfs Commentary on Archimedes' "The Measurement of the Circle'", Arabic Sciences and Philosophy, 3, pp. 7-53. 13 A.I. Sabra (1994), 'Science and Philosophy in Medieval Islamic Theology: The Evidence of the Fourteenth Century', Zeitschrift fur Geschichte der Arabisch-Islamischen Wissenschaften, 9, pp. 1^2. (First presented at the Symposium on Science and Theology in Medieval Islam, Judaism, and Christendom, held at Madison, Wisconsin, April 1993). 14 David A. King (1996), 'Astronomy and Islamic Society: Qibla, Gnomonics and Timekeeping', Encyclopedia of the History of Arabic Science (Vol. 1), London: Routledge, pp. 128-84. 15 David A. King (2000), 'Too Many Cooks ... A New Account of the Earliest Muslim Geodetic Measurements', Suhayl, 1, pp. 201-41. PART III

301

349

391 451

ISLAMIC SCIENCE TO THE WEST

16 Henri Hugonnard-Roche (1996), 'The Influence of Arabic Astronomy in the Medieval West', Encyclopedia of the History of Arabic Science (Vol. 1), London: Routledge, pp. 284-305. 17 Fuat Sezgin (2002/3), 'Arab Origin of European Maps', Zeitschrift fur Geschichte der Arabisch-Islamischen Wissenschaften, 15, pp. 1-23. 18 Roshdi Rashed (2008), 'Science as a Western Phenomenon', in Helaine Selin (ed.), Encyclopedia of the History of Science, Technology, and Medicine in Non-Western Cultures, New York: Springer, pp. 1927-33. Name Index

229

489 513

537 545

Acknowledgements The editor and publishers wish to thank the following for permission to use copyright material. Barcelona University Arabic Department for the essay: David A. King (2000), 'Too Many Cooks ... A New Account of the Earliest Muslim Geodetic Measurements', Suhayl, 1, pp. 207^2. Brill for the essay: A.I. Sabra (1987), 'The Appropriation and Subsequent Naturalization of Greek Science in Medieval Islam: A Preliminary Statement', History of Science, 25, pp. 22342. Cambridge University Press for the essays: George Saliba (1993), 'Al-QushjT's Reform of the Ptolemaic Model for Mercury', Arabic Sciences and Philosophy, 3, pp. 161-203. Copyright © 1993 Cambridge University Press; Roshdi Rashed (1993), 'Al-KindT's Commentary on Archimedes' "The Measurement of the Circle'", Arabic Sciences and Philosophy, 3, pp. 7-53. Copyright © 1993 Cambridge University Press. Institut fur Geschichte der Arabisch-Islamischen Wissenschaften for the essays: A.I. Sabra (1994), 'Science and Philosophy in Medieval Islamic Theology: The Evidence of the Fourteenth Century', Zeitschriftfur Geschichte der Arabisch-Islamischen Wissenschaften, 9, pp. 1^2. (First presented at the Symposium on Science and Theology in Medieval Islam, Judaism, and Christendom, held at Madison, Wisconsin, April 1993.); Fuat Sezgin (2002/3), 'Arab Origin of European Maps', Zeitschrift fur Geschichte der Arabisch-Islamischen Wissenschaften, 15, pp. 1-23. MIT Press for the essay: Paul Kunitzsch (2003), 'The Transmission of Hindu-Arabic Numerals Reconsidered', in J.P. Hogendijk and A.I. Sabra (eds), The Enterprise of Science in Islam: New Perspectives, Cambridge, MA: The MIT Press, pp. 3-21. Copyright © 2003 Massachusetts Institute of Technology, by permission of The MIT Press. Science History Publications for the essays: George Saliba (1994), 'Early Arabic Critique of Ptolemaic Cosmology: A Ninth-Century Text on the Motion of the Celestial Spheres', Journal for the History of Astronomy, 25, pp. 115-41. Copyright © 1994 Science History Publications Ltd; George Saliba (1989), 'A Medieval Arabic Reform of the Ptolemaic Lunar Model', Journal for the History of Astronomy, 20, pp. 157-64. Copyright © 1989 Science History Publications Ltd. Springer Science and Business Media for the essay: Roshdi Rashed (2008), 'Science as a Western Phenomenon', in Helaine Selin (ed.), Encyclopedia of the History of Science, Technology, and Medicine in Non-Western Cultures, New York: Springer, pp. 1927-33. Copyright © 2008 Springer-Verlag: Berlin, Heidelberg, New York.

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Taylor and Francis for the essays: Juan Vernet and Julio Samso (1996), 'The Development of Arabic Science in Andalusia', Encyclopedia of the History of Arabic Science (Vol. 1), London: Routledge, pp. 243-75. Copyright © 1996 Routledge; Bernard R. Goldstein (1996), 'The Heritage of Arabic Science in Hebrew', Encyclopedia of the History of Arabic Science (Vol. 1), London: Routledge, pp. 276-83. Copyright © 1996 Routledge; George Saliba (1996), 'Arabic Planetary Theories after the Eleventh Century AD', Encyclopedia of the History of Arabic Science (Vol. 1), London: Routledge, pp. 58-127. Copyright © 1996 Routledge; David A. King (1996), 'Astronomy and Islamic Society: Qibla, Gnomonics and Timekeeping', Encyclopedia of the History of Arabic Science (Vol. 1), London: Routledge, pp. 128-84. Copyright © 1996 Routledge; Henri Hugonnard-Roche (1996), 'The Influence of Arabic Astronomy in the Medieval West', Encyclopedia of the History of Arabic Science (Vol. 1), London: Routledge, pp. 284-305. Copyright © 1996 Routledge. University of Chicago Press for the essay: A.I. Sabra (1996), 'Situating Arabic Science: Locality versus Essence', Is is, 87, pp. 654-70. Copyright © 1996 by the History of Science Society. Every effort has been made to trace all the copyright holders, but if any have been inadvertently overlooked the publishers will be pleased to make the necessary arrangement at the first opportunity.

Editor's Acknowledgements The four volumes in this series owe a great deal to the painstaking work of a small group of historians of science who have studied numerous manuscripts, treatises, and instruments over the last four decades and whose work has been instrumental in revising our understanding of the Islamic scientific tradition. This series was made possible by their vigor and insights. It provides new perspectives on Islamic scientific tradition by presenting their work in a certain thematic order. I am grateful to all the authors and researchers whose work is included in this series. I wish to express my love and thanks to my son, Basit Kareem Iqbal, whose thoughtful critique of the four introductions has been helpful in reformulating certain arguments. His interest in various academic debates on themes related to Islamic scientific tradition and attention to detail and academic rigor has been inspiring. Needless to say that only I am responsible for the shortcomings in selection or presentation. A work of this nature cannot be free of editorial biases, even though one tries to present a balanced view of the fields. One hopes, nevertheless, that this series provides a broad spectrum of views on various aspects of Islamic scientific tradition and contributes to a richer understanding of the field in some small way. Wuddistan 9 Dhul-hijja, 1432/5 November 2011

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Introduction Two movements of knowledge, in different times and across several geographical regions, shaped the contours of the scientific and philosophical traditions of the pre-modern world. The first movement began in the middle of the eighth century and lasted, with intermittent breaks, for approximately two hundred years; the second started around the middle of the twelfth century and remained an ongoing process until the seventeenth century. The first was a focused, well-organized, and well-funded translation project of enormous scope and ambition, primarily located in Baghdad. During this sustained effort, almost all scientific and philosophical Greek works were translated into Arabic, as well as most of the available Syriac, Sanskrit, and Persian texts. The second movement was less focused and less organized, and was spread over a large geographical area that included Spain, Italy, France, and to a lesser extent, England. Both movements were intimately connected with the social, political, economic, and intellectual milieus of the empires of the times. Both present fascinating stories of human relations and cross-civilizational journeys of ideas, information, and techniques. This volume presents some of the recent seminal studies on these two movements which transmitted knowledge across civilizations. The translation movement, which eventually brought a large number of Greek, Syriac, Sanskrit, and PahlavT works into Arabic, had been in the making when the 'Abbasid revolution was consolidated by al-Mansur (r. 136-37/753-54), the second Caliph of the emerging c Abbasid dynasty and the famed builder of Baghdad, but it had yet to be organized into a focused and sustained activity. We now know that, even before the arrival of the c Abbasids on the political scene, various academies of ancient learning had been brought into the Muslim realm. Harran, situated on the small Jullab river at the intersection of important caravan routes to Asia Minor, Syria, and Mesopotamia, was conquered in 19/640. According to al-Blrunl, it resembled the shape of the moon, and its inhabitants were star-worshippers; Ibn Abi Usaybi'a tells us that the Umayyad Caliph £Umar II (r. 99-101/717-20) had transferred a school of medicine from Alexandria to Harran.1 It is also well established that Arabic translations from PahlavT, Sanskrit, and Syrian texts preceded translations from Greek. This early translation movement, however, remained unorganized until the time of alMansur, who was deeply interested in astrology, which he believed provided foreknowledge of events to come and which he elevated to a court-sponsored activity. He appointed Nawbakht (d. ca. 160/776-77), a Persian astrologer and engineer, as his court astrologer and asked him to cast the horoscope for laying the foundation of Baghdad. Nawbakht decided the date best suited was 30 July 762, and also foretold that a revolt against Mansur would be led by Ibrahim b. 'Abdallah, an £AlId contender. Nawbakht was succeeded at court by his son Abu Sahl. The construction of Baghdad was supervised by a Persian, Khalid ibn Barmak, who brought with him a vast store of Persian knowledge about city-building. Al-Mansur also established a 1

See Fehervari (1960-2005). The reference to Ibn Abi Usaybi'a is from his well-known 'Uyun al-anba * fltabaqat al-atibba' quoted by Fehervari.

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Baghdad-Jundishapur axis by bringing Jurjis b. Bakhtishuc, a Syrian Christian physician, to his court in 147^8/764-65. Jurjis had been the head of the hospital at Jundishapur and the Bakhtishu' family had long been associated with the medical tradition. Jurjis was made court physician and he was succeeded in this position by his family members. With the construction of Baghdad and consolidation of 'Abbasid rule, the translation movement gained a home and a proper place. The fifty-year period between 134/750 and 184/800 is enormously important in understanding the social and intellectual milieu which gave birth to the translation movement. During this half-century, five 'Abbasid caliphs ruled the new empire2 and the MalikI legal school (madhhab) emerged in Islam. Based on the works of a number of scholars, this school was crystallized through the writings of Malik b. Anas (93-179/712-95), the compiler of Kitab al-Muwatta\ one of the earliest collections of the Prophetic traditions. Kitab al-Kharaj, one of the first legal treatises on taxation, was written by an able student of Abu Hanlfa (d. 151/768), QadI Abu Yusuf (113-183/731-99), who was appointed as the first supreme judge (Qadi al-Quda5). Arabic grammar was systematized, as al-Kitab (The Book), one of the first books on Arabic grammar, was written by Abu Bishr (or Abu-1 Hasan) £ Amr ibn 'Uthman ibn Qanbar (d. c. 179/795), who himself was a Persian student of al-Khalll ibn Ahmad (d. c. 175/791), the founder of the science of Arabic grammar and metrics. The translation movement was thus born at time of intellectual fervour and great achievements. Translation activity gained momentum with the ascendancy of Caliph al-Ma'mun (r. 198-218/813-33), who is credited with establishing the famous House of Wisdom (Bayt alHikma) - although Dimitri Gutas has seriously challenged the notion that any such 'House' was actually 'founded' (Gutas, 1993, pp. 53-60). Whether or not there was an actual 'House of Wisdom', historical sources provide concrete evidence that translation activity greatly intensified in Baghdad during the reign of al-Mansur. Major early translators included Ibn alMuqaffac (d. 139/756) and his son, Ibn Na'ima (ft. 2nd/8th century), Theodore Abu Qurra (d. c. 211/826), the disciple of John of Damascus (d. 749) who held a secretarial post under the Umayyad Caliphs, the Sabian mathematician Thabit ibn Qurra (d. 289/901) (al-Nadim, 1994) and Eustathius (ft. 3rd/9th century) (ibid., p. 304), both of whom translated for al-Kindi, and Ibn al-Bitnq (263-328/877-944), who was a member of the circle of the Caliph al-Ma'mun, whose accession marks the beginning of the second phase of the translation movement. This second phase of the translation movement brought a host of new translators who worked under the able guidance of Hunayn ibn Ishaq. It was during this period that more polished translations of previously translated material were produced. In addition, this activity now covered a whole range of ancient texts, from Aristotle to Galen. The third and the final phase of the translation movement, approximately 288^11/900-1020, marks the appearance of revised versions of older translations and a vast corpus of commentary. This period also witnessed the emergence of textual criticism and greater command of the translated texts. By the middle of the eleventh century the translation movement had culminated, leaving behind Arabic versions of all existing and available works of Greek science and philosophy. This included the entire corpus of Aristotle; the subjects of the quadrivium (that is, arithmetic, geometry, astronomy, and alchemy); books on all branches of the health sciences, including 2

Abu'l 'Abbas al-Saffah (750-754), Abu Ja'far al-Mansur (754-75), al-Mahdl (775-85), al-Hadl (785-86), and Harun al-Rashld (786-809).

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medicine, pharmacology, and veterinary science; handbooks and tables of astronomy; books on the construction of scientific instruments and mechanical devices; popular works on wisdom, falconry, and animal husbandry; and literary works of all sorts. The impact of this translated material on Islamic scientific tradition is variously understood by historians, but, as A.I. Sabra emphasizes in his important study on 'The Appropriation and Subsequent Naturalization of Greek Science in Medieval Islam' (presented here as Chapter 5), this was not a passive or reactive process. The 'reception' of Greek science is better understood as an 'appropriation' of Greek science, given that the latter term denotes purposive activity.3

II

Part I of this volume (Chapter 1^) opens a small window onto various dimensions of our current understanding of the movement of scientific knowledge from Greek and Hindu sources into Arabic, as well as the movement of Islamic science from the eastern part of the Muslim world to the Maghrib (the Muslim West) and into Hebrew. In addition to describing various aspects of this knowledge transmission, these papers also provide insights into the social and intellectual contexts of this cross-cultural process; they enrich our understanding of the historical forces at work, and their detailed description of the primary sources from which they draw their content has broadened the field of research. For instance, 'The Transmission of Hindu-Arabic Numerals Reconsidered', a fascinating essay by Paul Kunitzsch (Chapter 1), is not only replete with textual evidence for the transmission of Hindu numerals, but also provides material helpful in understanding how these numerals were received. In one example cited by Kunitzsch, we see the working of a poet's imagination: he saw in the number 1 the shape of the letter alif\ in the number 2, the shape of the final ya'; 3 looked like a combination of ha '-jim; he likened 4 to the combination of 'ayn-waw; 5 to 'ayn; 6 to ha'; 7 to a khuttqf(an iron hook); 8 was like two zeros, one above the other; and 9 was like a wow (p. 15 below). In Chapter 2, while describing the development of Arabic science in Andalusia, Juan Vernet and Julio Samso simultaneously explore differences between the scientific tradition of the Maghrib and the East and note that there remained a 'modestly important Latin-VisigothicMozarabic science (and culture) which dominated until about the middle of the ninth century and survived until at least the eleventh century' (p. 24 below). Attending to the local influences on the development of science in Andalus enables comparisons to the cultural, linguistic, and other influences in other parts of the Muslim world - and then furthers our understanding of how the translated scientific knowledge from Greek, Hindu, and PahlavT sources changed due to these influences, a theme more fully explored in the above-mentioned landmark paper by A.I. Sabra included here as Chapter 5. In addition to the broad contours of the 'easternization' of Andalusian science, Vernet and Samso also present a summary of indigenous developments in Andalusian science and classify various trends over a long period of scientific activity. The Islamic scientific tradition in al-Andalus not only received knowledge from the eastern lands, it also transmitted it to the medieval Christian West as well as to various Jewish communities via translations into Hebrew. Bernard Goldstein presents a summary of this 3

See below for further comments on Sabra's essay, which is also reprinted with other papers in his 1994 collection Optics, Astronomy and Logic: Studies in Arabic Science and Philosophy.

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fascinating aspect of knowledge in motion in his 'The Heritage of Arabic Science in Hebrew' (Chapter 3). This is still an understudied area of history of science and most of the relevant Hebrew texts remain scattered in libraries in manuscript form. At the turn of the twentieth century, there were more than 200,000 such manuscripts in the Cairo Geniza alone; these documents, dating from the tenth to the nineteenth century, were transferred to European and American libraries at that time. Despite the lack of textual evidence from a fully studied tradition, Goldstein believes that sufficient material has been examined to allow certain initial generalizations. He notes that 'many Arabic texts were copied in Hebrew characters, a common practice among Arabic-speaking Jews, and, in some cases, this is their only surviving form' (p. 65 below). The subjects most widely studied by the Jewish community included astronomy, mathematics, and medicine. In 1982, Richler published a study in which he listed over 100 copies of various Hebrew versions of Ibn Slna's Qanun (Canon of Medicine) (p. 66 below). One fascinating aspect of these studies in the history of transmission of scientific knowledge is that one often comes across remarks of pre-modern Jews, Christians, and Muslims in primary sources that provide significant indications of their more general attitudes towards nature. Maimonides, for instance, mentions criticism of Ptolemaic astronomy by Ibn Bajja and Jabir ibn Aflah and then adds his own criticisms, partly based on the work of al-QablsI, and concludes by saying that 'man's faculties are too deficient to comprehend even the general proof the heavens contain for the existence of Him who sets them in motion. It is in fact ignorance or a kind of madness to weary our minds with finding out things which are beyond our reach, without the means of approaching them.'4 J.L. Berggren's research in Islamic mathematics has considerably expanded our understanding of this tradition. The essay selected here as Chapter 4 is a survey of recent literature that 'tellfs] of the different ways in which Greek mathematics influenced the Arabic as well as how Arabic mathematics differentiated itself from that of the Greeks in ways that make it possible to speak of Islamic mathematics' (p. 76 below). Berggren also comments on certain themes that belong to the discourse on the nature of the relationship between Islam and science, which is the focus of the first two volumes of this series. For instance, using the case of mathematics, he finds the 'marginality thesis' (see below and the introductions to Volumes I and II of this series) a problematic construction, whereas a more representative model is to understand 'Islam as a scientifically creative culture, incorporating Greek material and transforming it in the process. In many cases this second view correctly captures the relationship of Islamic science to its Greek predecessors' (p. 81 below). His creative description of the five ways 'in which Arabic mathematics related to its Greek predecessor: preservation, extension, criticism, systematization, and philosophical reflection', is an important advance in our understanding of this relationship. He also examines the process and ways through which the transmitted knowledge was transformed, in other words, 'what made it become not just an Arabic version of Greek mathematics but, in fact, Islamic mathematics' (p. 95 below).

4

Maimonides, The Guide for the Perpelexed, ch. 24; quoted from Goldstein's essay in this volume (Chapter 3, p. 68 below).

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III

It is now generally accepted that the large amount of Persian, Indian, and Greek knowledge brought into Arabic created a dynamic process of assimilation, sifting, and response. New concepts, philosophical ideas, scientific knowledge, and their underpinning assumptions and beliefs also created tensions when they conflicted with Islamic beliefs and practices, and this started a process of appropriation and transformation of the received knowledge. This aspect of the flow of Greek, Persian, and Indian thought into the Islamic tradition has received ample attention by historians of science, sociologists, and general historians. The existing opinions range from 'reductionism' to 'precursorism', two explanatory terms used by A.I. Sabra in 1987. Part II of the volume opens with his seminal essay (here Chapter 5) which redefined, in many ways, the discourse on the relationship between Islamic scientific tradition and what it received from other sources. Reductionism, in this context, refers to the 'view that the achievements of Islamic scientists were merely a reflection, sometimes faded, sometimes bright, or more or less altered, of earlier (mostly Greek) examples; precursorism, on the other hand, reads the future into the past, with a sense of elation' (pp. 101-2 below). Historians of science are still not able to reconstruct with confidence a complete picture of the enterprise of science in Islamic civilization or even describe all aspects of the complex phenomena of cross-cultural transmission, but they are certainly able to assert that it was not a passive reception of material into one civilization from another. Rather, it was an enormously complex and creative process that transformed the material in the very act of appropriation. Let us also note that it was not only the Islamic scientific tradition that was affected by the influx of new material; the entire intellectual tradition had to contend with the received material, which was regarded by some to be dangerous to faith and extraneous to Islam. This reaction is sometimes used to draw the reductionist conclusion that the scientific tradition in Islam was nothing but a 'foreign' entity that somehow survived despite the opposition it faced and then died as soon as orthodoxy was able to gain an upper hand. This view has been succinctly called 'the marginality thesis' and its validity has been seriously challenged on the basis of sound historical evidence (pp. 107-8 below); the case for appropriation, rather than passive reception, is now well established.5 The process which has produced new perspectives on the history of science in Islamic civilization has a history of its own and, in retrospect, one can see certain landmarks which have helped to refine the field; A.I. Sabra's lecture delivered to the annual meeting of the History of Science Society, held in Minneapolis, Minnesota, on 28 October 1995, is one such episode. In this lecture, entitled 'Arabic Science: Locality versus Essence' (here Chapter 6), Sabra spoke about some of the most fundamental aspects related to the study of science in Islamic civilization and disarmed numerous flawed constructions which had remained the 5

The case for the originality of Islamic science has been convincingly made by a number of historians of science; see, for instance, Pines (1986); also see various studies on Islamic astronomy by David King, E.S. Kennedy, and David Pingree. An interesting inside account is to be found in the letter written by the fifteenth-century Persian scientist Jamshld ibn Mas'ud ibn Mahmud Ghiyath al-Dln al-Kashl (or KashanI) to his father after his arrival in Samarqand; Kasha's work found in a manuscript, Nuzhat al-Hada 'iq, was translated by E.S. Kennedy as The Planetary Equatorium (1960). Amore recent account of the discovery of two world maps is also important; see King (1999); also of interest is King (1987).

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staple of general histories for almost a century. He began by defining the nature of history itself, clarified a term which has remained problematic for decades ('Arabic science'), and then situated science in a broad context that encompassed social, political, economic, and intellectual aspects of Islamic civilization: The powerful drive that eventually led to the transfer of the bulk of Greek science and philosophy (as well as elements of the scientific thought of India and Persia) to Islam was launched as a massive translation effort that took place in the context of empire and under the patronage of the confident Abbasid court in Baghdad ... One might then say, and with much justification, that the stage was set, at a certain place and time, for the translation movement that quickly acquired unprecedented proportions - unprecedented not only in the Middle East but in the world at large. But in order to explain the momentum, scope, and multiple dimensions of that movement, it is necessary to go beyond the availability of favorable conditions, and even beyond the important consideration of practical expectations that must have loomed large at least in the minds of the Muslim patrons, (p. 123 below)

During the course of his lecture, Sabra spoke of 'a huge intellectual ferment' in a multicultural setting that provided an unprecedented impetus to the nascent scientific tradition. He 'borrowed' an obsolete term, aspecting, 'in order to refer to the way in which individuals in a given culture aspect another culture as they direct their gaze to the other from their own location'. As Sabra explained: Aspecting in this sense is conditioned both by the interests, aspirations, and aptitudes of the aspecting individuals and by the accessible aspects of the viewed culture, that is to say, the aspects that happen to be disclosed to them by the accidents of history or by their further, determined effort. Thus, for example, through the Sabians of Harran, Muslim thinkers were able to view facets of Hellenistic thought that might not have been available to them by way of the Christian theologians, who had already made their own choices from their own standpoints. And, as has been plausibly suggested, the absence of Greek literature and Greek historiography from the translated corpus may be attributable to a lack of acquaintance or serious interest on the part of the Christian translators.' (pp. 123-4 below)

IV

J.L. Berggren's essay 'Historical Reflections of Scientific Knowledge' (here Chapter 7) not only recounts the movement of mathematical knowledge across civilizations, it also points out certain creative processes involved in this transference. 'The transmission of ancient sciences to medieval Islam late in the eighth century and throughout the ninth ... took place in a learned environment,' he notes (p. 137 below). Berggren makes several insightful remarks with respect to translations, their reception, assimilation, and impact. The case of Euclid's Elements, which he cites, is not an isolated example of the deep interest and care involved in translation of scientific texts from other civilizations: The greatest work of Euclid, the Elements, was translated twice by Hajjaj b. Matar (for Harun alRashld and for al-Ma'mun). It was again translated by Ishaq b. Hunayn. And one sign of the care with which such things were done is that when Thabit b. Qurra came to revise this translation he again went back to the Greek manuscripts ... in addition to the translations there was an astounding amount

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of work aimed at a wide variety of audiences: the beginning students, the professional researchers and the philosophers, (p. 138 below)

Just as Euclid's Elements provided a foundation for geometry, Ptolemy's Almagest became the basic text for astronomy, the queen of the sciences during the pre-modern era. Recent studies in the history of Islamic scientific tradition have paid considerably more attention to astronomy than other branches, and this is also reflected in the selection of articles for the present volume. Several articles discuss various aspects of Islamic astronomical tradition, especially in reference to various efforts to reform and refine Greek astronomical models. George Saliba, who has made important contributions to this subfield, describes in a paper not included here the reaction of scholars in the field to the discovery of a manuscript by Ibn alShatir - a manuscript that allowed historians to seriously challenge the entrenched notion that the torch of science was ignited in ancient Greece and was passed on to Europe by some 'gobetweens', who 'were seen as mere scribes, faithfully translating and preserving the ancient texts of Greek astronomy, philosophy and medicine until such time as Europe would reawaken from its darkness, pick up the books, and once again carry the light' (Saliba, 2002, p. 360). This manuscript, discovered in 1957, suggested that there was, in fact, a 'hidden connection between the works of Copernicus and those of medieval Islamic astronomers' - but instead of welcoming this refreshing lead, the new hypothesis was 'greeted with responses that ranged from total disbelief to total denial - some historians even walked out of public lectures when the manuscript was mentioned' (ibid.). This episode underscores the fact that the history of Islamic scientific tradition is a work in progress, fraught by ideological pressures of many sorts, and it may be a long time before we have an adequate history rather than caricatures drawn on sand. The contours of what may eventually be a definitive history of science in Islamic civilization are being outlined through studies on specific aspects of the tradition. Chapter 8, Saliba's 'Early Arabic Critique of Ptolemaic Cosmology', is an example of one such specific study. In this essay, Saliba shows that the theoretical aspects of Ptolemaic astronomy were being investigated in earnest as early as the ninth century by studying an Arabic text by Qutb al-Dln al-ShlrazI (d. 1311), written before 1306, and bearing an 'awkward title,/a laltu fa-la talum ('I did what I had to do, so do not blame [me]')'. Saliba comments: This text gains greater importance when we realize that it was composed just about the same time as the Greek Ptolemaic texts of the Planetary hypotheses and the Almagest were being translated into Arabic. More significantly, it also demonstrates the direction taken by the critical Arabic tradition from the very beginning, in that it illustrates how this tradition was originally conceived along lines that emphasized the similarities between celestial and terrestrial physics. In contrast to the repeated Ptolemaic appeals to consider celestial physics as something separate from terrestrial physics, which we can observe and examine, this ninth-century text requires that the motion of the celestial spheres be analysed from the perspective of terrestrial physics, and that this motion be subjected to the rules of such physics. This result indeed contradicts the commonly held opinion that the abolition of such dichotomies between celestial and terrestrial physics was drawn 'perhaps for the first time' only in medieval and Renaissance Europe, (p. 150 below)

Saliba returns to the theme of reform of Greek astronomy in three other essays selected here. In the first (Chapter 9), the focus is on the reform of the Ptolemaic model for Mercury suggested by the fifteenth-century astronomer 'Ala al-DTn CA1T b. Muhammad al-QushjT

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(d. 879/1474) in his Risala fi hall ishkal al-mu'addil li-l-masir ('A Treatise Regarding the Solution of the Equant Problem'). QushjT describes the motions of the planet Mercury with a model that does not suffer from the shortcomings of Ptolemy's model. Unlike Ptolemy, QushjT adheres to the paramount medieval astronomical principle of uniform motion insofar as all the spheres employed in QushjT's model revolve at uniform speeds around their own centres, and not around fictitious equant points, as in Ptolemy's model. In the second, 'A Medieval Arabic Reform of the Ptolemaic Lunar Model' (Chapter 10), Saliba studies the reform proposed by the Damascan astronomer Mu'ayyad al-Dln al-£UrdI (d. 1266) and situates it in the larger reform tradition. His third essay (Chapter 11), a chapter-length survey of the field, focuses on the Arabic planetary theories after the eleventh century. This survey allows him to pinpoint two major achievements of the reform efforts - 'Urdi's lemma and the TusT Couple: With the help of these two theorems, and with the technique of dividing the eccentricities of the Ptolemaic models, it was possible to transfer segments of these models from the central parts to the peripheries and back. This freedom of movement not only allowed the retention of the effect of the equant in the Ptolemaic models, but also allowed the development of sets of uniform motions that would not violate any physical principles. The TusT Couple allowed, in addition, the production of linear motion as a combination of circular motions, and thus allowed someone like Ibn al-Shatir, and after him Copernicus, to create the effect of enlarging the size of the epicyclic radius and of shrinking it by using uniform circular motion only or combinations thereof, (p. 296 below)

Furthermore, Saliba is able to draw upon the large body of research to state with confidence that the tradition of criticism of Ptolemaic astronomy became a well-established tradition after the thirteenth century, and that very few astronomers could do any serious work without attempting some reform of Greek astronomy on their own. Ironically, this period of original production in Arabic is usually thought of as a period of decadence in Islamic science, and little effort is spent to study it in any depth (ibid.).

V

Next, in Chapter 12, Roshdi Rashed explores numerous interrelated aspects in al-Kindi's work. These include the relationship between mathematics and theoretical philosophy, al-Kindi's treatment of infinity, plurality, the sphericity of the elements, the shape of the universe, his use of the axiomatic method and of proofs in theoretical philosophy, and, notably, his argument in favour of proof by reductio ad absurdum. A notable aspect of al-Kindi's work in general is that for him 'a proposition in theoretical philosophy required a proof as tightly argued as a mathematical one' (pp. 301-2 below). Rashed also notes that: the connection between al-Kindi's philosophy and his mathematics is essential for a full understanding of his general system of thought. The inter-relationship is explicitly stated by al-Kindi himself when he writes a treatise entitled Philosophy can only be Acquired by Mathematical Discipline and when he champions mathematics as a prerequisite subject for philosophical education in his letter On the Number of the Books of Aristotle. In the latter, he goes so far as to warn the student of philosophy that he faces a stark choice: either learn mathematics before approaching the works of Aristotle (listed in al-Kindi's own preferred order), and only then hope to become a true philosopher; or else avoid

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mathematics and become at best a philosophical regurgitator, strictly limited by what he could retain in his memory, (p. 303 below)

Rashed's important study on al-Kindl serves several purposes in this series: it acts as a bridge between the history of science proper and studies which explore the Islam and science nexus; it restores to al-Kindl the double role of a scientist and a philosopher; it provides insights into how the relationship between revelation and rational sciences was understood by al-Kindl and his contemporaries; and it provides a fascinating study of a newly discovered text by al-Kindl - his commentary on the third proposition of Archimedes' Measurement of the Circle, which relates to the approximation of the ratio of the circumference to the diameter. As Rashed notes, this discovery has sharpened our image of al-Kindl and enlarged our view of his influence. 'We can see the philosopher actually engaged in mathematical research, and ascertain that his knowledge of mathematics, though not equal to that of a Muhammad ibn Musa ibn Shakir, was nevertheless sufficiently advanced for him to tackle directly the mathematical subjects being studied in his own day' (p. 325 below). Next, the connection between science and philosophy in what is called 'Islamic theology' in Western scholarship is explored by A.I. Sabra from a vantage point different from that used by Rashed. In Chapter 13, 'Science and Philosophy in Medieval Islamic Theology: The Evidence of the Fourteenth Century', Sabra examines various connections between theology, science, and falsafa, the latter of which of course emerged in urban centres such as Basra and Baghdad well before the translation movement. The common domain between 'theology' and falsafa included doctrines concerning a comprehensive array of subjects including God, His relation to man and the world, questions of epistemology, and those related to morality and political leadership. In the last two chapters in Part II, David King provides fascinating studies on what has remained an area of sustained interest for generations of Muslim scientists and scholars through to our own times: the direction of qibla and the times of prayer. King's specialized studies in instruments and mathematical astronomy have massively advanced our understanding of the history of Islamic science in these areas.6 The two essays complement each other. First, Chapter 14, 'Astronomy and Islamic Society: Qibla, Gnomonics and Timekeeping', surveys efforts made by Muslims to find the direction of qibla and determine prayer times. The qibla, the direction to which Muslims turn for prayer, is that of the Ka c ba, the cube-shaped House of Allah (B ay t Allah), in the heart of Makka. All mosques are oriented towards the Ka c ba, with the mihrab (prayer-niche) inside indicating the qibla, the local direction of Makka. The direction of the qibla is also important for other religious rites, such as burial and the ritual slaughter of animals. Muslims also face this direction while reciting the Qur'an and for calling to prayer (adhan); and they avoid the direction of qibla for certain acts, such as when they answer the call of nature. Thus, the daily lives of Muslims are spiritually and physically oriented with respect to the Ka c ba - and have been since the beginning of Islam. Initially, the qibla direction was determined by lay persons, but with the geographical expansion of the Muslim state, the need to find the qibla from remote areas could not be fulfilled by guesswork 6

Most of King's work has been collected in five Variorum volumes: Islamic Mathematical Astronomy (1986/1993); Islamic Astronomical Instruments (1987/1995); Astronomy in the Service of Islam (1993); Astrolabes from Medieval Europe (2011); and Islamic Astronomy and Geography (forthcoming 2012).

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and thus astronomers and mathematicians were pressed into service. From the late eighth century onward, they devised methods to compute the qibla from available geographical data, treating the determination of the qibla as a problem of mathematical geography. This practice was not always followed or correct, hence many medieval mosques are not properly aligned, and King provides a helpful survey of the field. The second part of this chapter is a study on timekeeping and regulating the times of prayer to which Muslim astronomers and mathematicians made substantial contributions. Instrument-makers were also involved in these efforts, and by the late medieval period there were sundials of one form or another in most major mosques in the Muslim world. King provides a brief history of how Muslims came into contact with the sundial with the expansion of their empire into the Greco-Roman world, an account of the earliest surviving Arabic treatise on sundials, and some details regarding the theoretical mathematics used for the construction of sundials. Then, in Chapter 15, 'Too Many Cooks ... A New Account of the Earliest Muslim Geodetic Measurements', King studies four different accounts of the measurement of the length of one degree on the meridian by astronomers commissioned by the c Abbasld Caliph al-Ma'mun in Baghdad around 830. Different versions of the observations are recorded and the details of the actual observations remain far from clear, especially due to the presence of those versions in which two groups of astronomers laden with instruments are reported heading off in opposite directions along a meridian in the middle of the desert. King provides a new account by Yahya ibn Aktham, the judge appointed by al-Ma'mun to oversee the observations, and compares two reports in the light of the later accounts of Ibn Yunus and al-Blrunl.

VI

The first two chapters of Part III provide a small window into certain aspects of the transmission of scientific knowledge from Islamic civilization to Europe, while the third and final chapter is a reflective piece which attempts to correct the erroneous view that science is a Western phenomenon. Henri Hugonnard-Roche's 'The Influence of Arabic Astronomy in the Medieval West' (Chapter 16) is a general survey of Muslim contributions to the development of astronomy in Europe. Using the five tasks which Kepler associated with the profession of an astronomer, Hugonnard-Roche examines key elements of the transmission of knowledge to Europe under each of these aspects: (i) historical, to do with the recording and classification of observations; (ii) optical, to do with the shaping of hypotheses; (iii) physical, dealing with the causes underlying hypotheses; (iv) arithmetical, concerned with tables and computation; and (v) mechanical, relating to instruments. Kepler considered the first three areas to mainly involve theory. Hugonnard-Roche makes the bold claim that 'the contribution of Arabic science was essential to the birth and subsequent development of astronomy in the Latin West. Prior to this contribution, there was indeed no astronomy of any advanced level in those countries' (p. 489 below). Using historical examples, Hugonnard-Roche confirms the lack of direct observation in European astronomy of the early Middle Ages, and, though his essay does not cover all the changes produced in the Latin West by successive translations of Arabic works, nor the influence of Islamic scientific tradition on the development of trigonometry, instruments, Latin catalogues of stars, and Latin astrology, he does provide a broad outline of essential aspects

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of Islamic influence on astronomical theory. He also gives certain details of this influence on the growth and development of the stereographic astrolabe, which reached its definitive form in the middle of the twelfth century - the century during which a considerable collection of Arabic texts were translated into Latin. This opened to Latin astronomers a much wider field of study in the form of 'astronomical tables', a term designating a huge variety of material including elements relating more or less directly to the astronomy of the prime mover (tables of right and oblique ascensions, of declinations, of the equation of time); planetary tables; and sundry other tables relating to conjunctions of the sun and moon, eclipses, parallaxes, and the visibility of the moon and other planets. Hugonnard-Roche identifies three principal sources which served to introduce Latin astronomers to all these subjects, and describes how the very large number of manuscripts of the Toledan tables, dating from the fourteenth and even the fifteenth century, also influenced almanacs which were designed not to provide means for calculating planetary positions but to give the positions themselves. He also explores links between the Copernican Revolution and Arabic astronomy in some detail, noting that with Copernicus the long period of influence of Arabic astronomy in the Latin West came to an end as he was the last to make constant use of observational results taken from Arab authors. Fuat Sezgin's pioneering work on the history of Islamic science has been globally recognized. The essay selected for this volume as Chapter 17 is part of his specialized studies on the history of cartography, which is generally understood as a Greco-Roman invention and which from the sixteenth century onward is often presented as an accompaniment to the 'miracle' of expanding European technology. Sezgin shows how this practice is linked to the Arabic tradition. He considers maps to be graphic representations, facilitating a spatial understanding of things, concepts, conditions, processes, or events in the human world, and draws lines of continuity which connect European maps to those produced in the Muslim world. The final chapter of Part III sums up various themes of this volume through a reflective essay which questions the general perception held by philosophers, historians, and sociologists that science is essentially a Western phenomenon. In Chapter 18, Roshdi Rashed underscores the entrenched nature of this perception by stating that almost all philosophers have uncritically accepted this notion; these include 'Kant, Comte, the neo-Kantians as well as the neopositivists, Hegel as well as Husserl, the Hegelians and the phenomenologists as well as the Marxists, all acknowledge this postulate as the basis of their interpretations of Classical Modernity' (p. 537 below). The same postulate pervades general, non-specialized histories of science. Even in debates that sharply divide historians of science into two groups - those advocating continuity with and those seeing a break from the Middle Ages - this concept remains an accepted notion, despite the work of numerous scholars on the history of Arabic and Chinese science. In spite 'of the wide representation of non-Western scientists in Dictionary of Scientific Biography, the works of the historians rest on an identical fundamental concept: in its modernity as well as in its historical context, classical science is a work of European man alone' (ibid.). Rashed candidly places this notion in a wider framework and shows how occasional acknowledgement of the existence of some science in non-Western cultures is placed 'outside history', as it were, 'or is integrated into it only to the extent of its contributions to the essentially European sciences'. He comments:

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These are only technical supplements which do not modify the intellectual configuration or the spirit of the latter in any way. The image given of Arabic science constitutes an excellent illustration of this approach. Essentially it consists of a conservatory of the Greek patrimony, transmitted intact or enriched by technical innovation to the legitimate heirs of ancient science. In all cases, scientific activity outside Europe is badly integrated into the history of the sciences; rather, it is the object of an ethnography of science whose translation into university study is nothing more than Orientalism, (pp. 537-8 below)

After exploring this issue from various perspectives, including debates about modernism and tradition, and sketching the history of this view of European science, Rashed then critically examines it. He shows how most later historians of science simply borrowed earlier representations of the Western essence as well as methods for describing and commenting on the evolution of science, and thereby perpetuated these erroneous notions. He pinpoints five essential elements of the entrenched narrative: (i) just as science in the East did not leave any consequential traces in Greek science, Arabic science has not left any traces of consequence in classical science; (ii) science subsequent to that of the Greeks is strictly dependent upon it; (iii) whereas Western science addresses itself to theoretical fundaments, Oriental science, even in its Arabic period, is defined essentially by its practical aims; (iv) the distinctive mark of Western science is its conformity to rigorous standards; in contrast, Oriental science in general, and Arabic science in particular, lets itself be carried away by empirical rules and methods of calculation, neglecting to verify the soundness of each step on its path; and (v) the introduction of experimental norms which, according to historians, totally distinguishes Hellenistic science from classical science is solely the achievement of Western science. Rashed then provides counterfactual evidence from the history of science, especially that of algebra, to debunk these notions. He also shows how experimentation was not the deciding factor between two periods of Western science (Greek and Renaissance); rather, it was the Arabs who introduced experimentation, as is acknowledged by some nineteenth-century historians and philosophers such as Alexander von Humboldt in Germany and Cournot in France. His postscript, written 26 years after the publication of the original French article, is equally important in the context of this series on new perspectives on the history of Islamic science. He notes that the quarter-century that has passed since its first publication has witnessed remarkable growth - in fact, a rebirth - of the discipline of history of Islamic science, offering 'historians the possibility of developing and comparing their research findings with facts.' Even though the 'task remains huge, and we are only at the beginning ... at least this new growth of information puts to rest the argument of ignorance' (p. 543 below). The hope Rashed articulated in 2008 was expressed despite certain recent setbacks which brought various factors into the academic discourse on the history of Islamic science, and which arose from 'reasons which are extraneous to science and its history'. These setbacks arose through the ahistorical projection of Islamic society - even Islam itself- as irrational and intolerant (and thus somehow hostile to science). This has resulted in a resurrection of the tired model of 'Islam versus foreign sciences' (see the introduction to Volume I of this series). As Rashed notes, the 'aging doctrine' was buttressed by 'the doctrine of double marginality: with regard to the society which saw the development of science, and with regard to the history of the sciences. Thus, one could still write in 1992, "We must remember that at an advanced level the foreign sciences had never found a stable institutional home in Islam," or "Greek learning never found a secure institutional home in Islam, as it was eventually to do

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in the universities of medieval Christendom" ... Thus, we are back where we started and the doctrine of the Westernness of science is saved' (ibid.). It is hoped that the four volumes of this series will help to dispel these erroneous ideological views, and, ultimately, help facilitate a new understanding of the history of science in Islamic civilization.

Wuddistan 5 Dhul Qa da 14327 3 October 2011 c

References Fehervari, G. (1960-2005), 'Harran', in Encyclopedia of Islam, 2nd edn, Leiden: Brill, pp. 227-30. Gutas, D. (1998), Greek Thought, Arabic Culture, London and New York: Routledge. Ibn al-Nadlm (1994), al-Fihrist, ed. I. Ramadan, Beirut: Dar al-Ma'rifa. al-Kashl, J. (1960), Nuzhat al-Hada^iq, translated by E.S. Kennedy as The Planetary Equatorium, Princeton: Princeton University Press. King, D.A. (1986/1993), Islamic Mathematical Astronomy, London: Variorum; 2nd rev. edn, Aldershot: Variorum. King, D.A. (1987/1995), Islamic Astronomical Instruments, London: Variorum; repr. Aldershot: Variorum. King, D.A. (1993), Astronomy in the Service of Islam, Aldershot: Variorum. King, D.A. (1999), World-Maps for Finding the Direction and Distance to Mecca, Leiden: EJ. Brill. King, D.A. (2011), Astrolabes from Medieval Europe, Farnham: Variorum. King, D.A. (forthcoming 2012), Islamic Astronomy and Geography, Farnham: Variorum. Pines, S. (1986), 'What was Original in Arabic Science?' in S. Pines, Studies in Arabic Versions of Greek Texts and in Mediaeval Science, Leiden: EJ. Brill, pp. 181-205. Sabra, A.I. (1994), Optics, Astronomy and Logic: Studies in Arabic Science and Philosophy, Aldershot: Variorum. Saliba, G. (2002), 'Greek Astronomy and the Medieval Arab Tradition', American Scientist, 90, pp. 360-67.

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THE TRANSMISSION OF HINDU-ARABIC NUMERALS RECONSIDERED Paul Kunitzsch

For the last two hundred years the history of the so-called Hindu-Arabic numerals has been the object of endless discussions and theories, from Michel Chasles and Alexander von Humboldt to Richard Lemay in our times. But I shall not here review and discuss all those theories. Moreover I shall discuss several items connected with the problem and present documentary evidence that sheds light—or raises more questions—on the matter. At the outset I confess that I believe the general tradition, which has it that the nine numerals used in decimal position and using zero for an empty position were received by the Arabs from India. All the oriental testimonies speak in favor of this line of transmission, beginning from Severus Sebokht in 662] through the Arabic-Islamic arithmeticians themselves and to Muslim historians and other writers. I do not touch here the problem whether the Indian system itself was influenced, or instigated, by earlier Greek material; at least, this seems improbable in view of what we know about Greek number notation. The time of the first Arabic contact with the Hindu numerical system cannot safely be fixed. For Sebokht (who is known to have translated portions of Aristotle's Organon from Persian) Fuat Sezgin2 assumes possible Persian mediation. The same may hold for the Arabs, in the eighth century. Another possibility is the Indian embassy to the caliph's court in the early 770s, which supposedly brought along an Indian astronomical work, which was soon translated into Arabic. Such Indian astronomical handbooks usually contain chapters on calculation3 (for the practical use of the parameters contained in the accompanying astronomical tables), which may have conveyed to the Arabs the Indian system. In the following there developed a genre of Arabic writings on Hindu reckoning (fi l-hisab al-hindl, in Latin de numero Indorum), which propagated the new system and the operations to be made with it. The oldest known text of this kind is the book of al-Khwarizmi (about 820, i.e., around fifty years or more after the first contact), whose Arabic text seems to be lost, but which can very well be reconstructed from the surviving Latin adaptations of a Latin translation made in Spain in the twelfth century. Similar texts by al-Uqlldisi (written in 952/3), Kushyar ibn Labban (2nd half of the 10th

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century) and cAbd al-Qahir al-Baghdadi (died 1037) have survived and have been edited.4 All these writings follow the same pattern: they start with a description of the nine Hindu numerals (called ahruf, plural of harf; Latin litterae), of their forms (of which it is often said that some of them may be written differently), and of zero. Then follow the chapters on the various operations. Beside these many more writings of the same kind were produced,5 and in later centuries this tradition was amply continued, both in the Arabic East and West. All these writings trace the system back to the Indians. The knowledge of the new system of notation and calculation spread beyond the circles of the professional mathematicians. The historian al-YaequbI describes it in his Tdrlkh (written 889)—he also mentions zero, sifr, as a small circle (dd'ira saghira).6 This was repeated, in short form, by al-Mascudi in his Muruj.1 In the following century the encyclopaedist Muhammad ibn Ahmad alKhwarizmi gave a description of it in his Mafdtlh al-culum (around 980); also he knows the signs for zero (asfar, plural) in the form of small circles (dawd'ir sighdr).8 That the meaning of sifr is really "empty, void" has been nicely proved by August Fischer,9 who presents a number of verses from old Arabic poetry, where the word occurs in this sense. It may thus be regarded as beyond doubt that sifr, in arithmetic, indeed renders the Indian sunya, indicating a decimal place void of any of the nine numerals. Exceptional is the case of the Fihrist of Ibn al-Nadim (around 987, that is, contemporaneous with the encyclopaedist al-Khwarizml). This otherwise well-informed author apparently did not recognize the true character of the nine signs as numerals; he treats them as if they were letters of the Indian alphabet.10 He juxtaposes the nine signs to the nine first letters of the Arabic abjad series and says that, if one dot is placed under each of the nine signs, this corresponds to the following (abjad) letters yd' to sad, and with two dots underneath to the remaining (abjad) letters qdf to zd3 (with some defect in the manuscript transmission). This sounds as if he understood the nine signs and their amplification with the dots as letters of the Indian alphabet. Even a Koranic scholar, Abu cAmr 'Uthman al-Danl (in Muslim Spain, died 1053), knows the zero, sifr, and compares it to the common Arabic orthographic element sukun}1 (For all these authors it must be kept in mind that the manuscripts in which we have received their texts date from more recent times and therefore may not reproduce the forms of the figures in the original shape once known and written down by the authors.) Of some interest in this connection are, further, two quotations recorded by Charles Pellat: the polymath al-Jahiz (died 868) in his Kitdb al-mucallimin advised schoolmasters to teach finger reckoning (hisdb al-'aqd) instead of hisdb al-hind, a method needing "neither spoken word nor writing"; and the historian and literate Muhammad ibn Yahya al-SulI (died 946) wrote in his

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A dab al-kuttab: "The scribes in the administration refrain, however, from using these [Indian] numerals because they require the use of materials [writingtablets or paper?] and they think that a system which calls for no materials and which a man can use without any instrument apart from one of his limbs is more appropriate in ensuring secrecy and more in keeping with their dignity; this system is computation with the joints (caqd or cuqad) and tips of the ringers (banari), to which they restrict themselves."12 The oldest specimens of written numerals in the Arabic East known to me are the year number 260 Hijra (873/4) in an Egyptian papyrus and the numerals in MS Paris, BNF ar. 2457, written by the mathematician and astronomer al-Sijzl hi Shiraz between 969 and 972. The number in the papyrus (figure l.l)13 may indicate the year, but this is not absolutely certain.14 For an example of the numerals in the Sijzi manuscript, see figure 1.2. It is to be noted that here "2" appears in three different forms, one form as common and used in the Arabic East until today, another form resembling the "2" in some Latin manuscripts of the 12th century, and a form apparently simplified from the latter; also "3" appears in two different forms, one form as common in the East and used in that shape until today, and another form again resembling the "3" in some Latin manuscripts of the 12th century.

Figure 1.1 Papyrus PERF 789. Reproduced from Grohmann, PL LXV, 12

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Figure 1.2 MS Paris, B. N. ar 2457, fol 85v. Copied by al-SijzI, Shiraz, 969-972

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7

This leads to the question of the shape of the nine numerals. Still after the year 1000 al-Biruni reports that the numerals used in India had a variety of shapes and that the Arabs chose among them what appeared to them most useful.15 And al-Nasawl (early eleventh century) in his al-Muqnicft l-hisdb alhindl writes at the beginning, when describing the forms of the nine signs, "Les personnes qui se sont occupees de la science du calcul n'ont pas ete d'accord sur une partie des formes de ces neuf signes; mais la plupart d'entre elles sont convenues de les former comme il suit"16 (then follow the common Eastern Arabic forms of the numerals). Among the early arithmetical writings that are edited al-Baghdadi mentions that for 2,3, and 8 the Iraqis would use different forms.17 This seems to be corroborated by the situation in the Sijzi manuscript. Further, the Latin adaptation of al-Khwarizmi's book says that 5, 6, 7, and 8 may be written differently. If this sentence belongs to al-KhwarizmTs original text, that would be astonishing. Rather one would be inclined to assume that this is a later addition made either by Spanish-Muslim redactors of the Arabic text or by the Latin translator or one of the adapters of the Latin translation, because it is in these four signs (or rather, in three of them) that the Western Arabic numerals differ from the Eastern Arabic ones.18 Another point of interest connected with Hindu reckoning and the use of the nine symbols is: how these were used and in what form the operations were made. Here the problem of the calculation board is addressed. It was especially Solomon Gandz who studied this problem in great detail and who arrived at the result that the Arabs knew the abacus and that the term ghubdr commonly used in Western Arabic writings on arithmetic renders the Latin abacus.19 As evidence for his theory he also cites from Ibn al-Nadim's Fihrist several Eastern Arabic book titles such as Kitdb al-hisab al-hindl bi-l-takht (to which is sometimes added wa-bi-l-mil ), "Book on Hindu Reckoning with the Board (and the Stylus)." I cannot follow Gandz in his argumentation. It is clear, on the one side, that all the aforementioned eastern texts on arithmetic, from alKhwarizmi through al-Baghdadi, mention the takht (in Latin: tabula) and that on it numbers were written and—in the course of the operations—were erased (mahw, Latin: delere). It seems that this board was covered with dust (ghubdr, turdb) and that marks were made on it with a stylus (mil). But can this sort of board, the takht (later also lawh, Latin tabula), be compared with the abacus known and used in Christian Spain in the late tenth to the twelfth centuries? In my opinion, definitely not. The abacus was a board on which a system of vertical lines defined the decimal places and on which calculations were made by placing counters in the columns required, counters that were inscribed with caracteres, that is, the nine numerals (in the Western Arabic style) indicating

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the number value. The action of mahw, delere, erasing, cannot be connected with the technique of handling the counters. On the other side, the use of the takht is unequivocally connected with writing down (and in case of need, erasing) the numerals; the takht had no decimal divisions like the abacus, it was a board (covered with fine dust) on which numbers could be freely put down (Ibn al-Yasamin speaks of naqashd) and eventually erased (mahw, delere). Thus it appears that the Arabic takht and the operations on it are quite different from the Latin abacus. Apart from the theoretical descriptions in the arithmetical texts we have an example where an astronomer describes the use of the takht in practice: al-Sijz! mentions, in his treatise Ft kayfiyat san(atjamlc alasturlabdt, how values are to be collected from a table and to be added, or subtracted, on the takht.20 Furthermore it is worth mentioning that al-Uqlldisi adds to his arithmetical work a Book IV on calculating bi-ghayr takht wa-la mahw bal bi-dawat wa-qirtas, "without board and erasing, but with ink and paper," a technique, he adds, that nobody else in Baghdad in his time was versant with. All this shows that the takht, the dust board of the Arabs, was really used in practice—though for myself I have some difficulty to imagine what it looked like—and that it was basically different from the Latin abacus. Let me add here that the Eastern Arabic forms of the numerals also penetrated the European East, in Byzantium. Woepcke has printed facsimiles of the Arabic numerals appearing in four manuscripts of Maximus Planudes' treatise on Hindu reckoning, Psephophoria kat'Indous.21 So far, at least for the Arabic East, matters appear to be reasonably clear. But now we have to turn to the Arabic West, that is, North Africa and Muslim Spain. Here we are confronted with two major questions, for only one of which I think an answer is possible, whereas the second cannot safely be answered for lack of documentary evidence. Question number one concerns the notion ofghubar. This term, meaning "dust" (in reminiscence of the dust board), is understood by most of the modern authorities as the current designation for the Western Arabic forms of the numerals; they usually call them "ghubar numerals." It is indeed true that the term ghubar—as far as I can see—does not appear in book titles on Hindu reckoning or applied to the Hindu-Arabic numerals in the arithmetical texts of the early period in the Arabic East. On the contrary, in the Arabic West we find book titles like hisdb al-ghubar (on Hindu reckoning) and terms like hurilf al-ghubar or qalam al-ghubar for the numerals used in the Hindu reckoning system. The oldest occurrence so far noticed of the term is in a commentary on the Sefer Yesira by the Jewish scholar Abu Sahl Dunas ibn Tamlm. He was active in Kairouan and wrote his works in Arabic. This commentary was written in 955/6. In it Dunas says the following: "Les Indiens ont

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imagine neuf signes pour marquer les unites. J'ai parle suffisamment de cela dans un livre que j'ai compose sur le calcul indien connu sous le nom de hisab al-ghubdr, c'est-a dire calcul dugobar ou calcul de poussiere."22 The next work to be cited in this connection is the Talqlh al-afkdr fi 'arnal rasrn al-ghubdr by the North African mathematician Ibn al-Yasamln (died about 1204). Two pages from this text were published in facsimile in 1973;23 on page 8 of the manuscript (= page 232 in the publication) the author presents the nine signs (ashkal) of the numerals which are called ashkdl al-ghubar, "dust figures"; at first they are written in their Western Arabic form, then the author goes on: wa-qad takunu ay dan hakadha [here follow the Eastern Arabic forms] wa-ldkinna l-nas 'indana in conformity with the motion of the deferent, in direction as well as magnitude, in the upper half, then the motion of the encircling < sphere > would be uniform with respect to the center of the Director; its distances from a point whose distance from the center of the deferent, in the direction of the deferent's apogee, is always equal to the distance between the deferent's center and that of the Director, as it was demonstrated by the Principle of the Encompassing < sphere >. The center of the small encircling < sphere > is then moved by the two motions of the large encircling < sphere > and the Inclined, along a circle equal to the cincture (mintaqd) of the deferent and whose center is moved by the motion of the Director along a circular path whose radius is three parts. [17] Then if we assume the motion of the small encircling < sphere > to be equal to the difference between the motion of its center, which is uniform with respect to the center of the Director and in the direction of the order < of the signs >, and the motion of the Director, which is opposite to the order , I mean that it is equal to the motion of the center of the sun, then the motion of the epicyclic center will be uniform with respect to the center of the equant < sphere >, as it was demonstrated by the Principle of the Encompassing < sphere >. [18] No one should criticize the details of the proof of the Principle of the Encompassing < sphere > in regard to the fact that the center of the encircling < sphere > approaches the center of the Director and draws away from it. The center of the epicycle will necessarily move along the circumference of a circle equal to the cincture of the deferent, whose center moves, in the direction opposite to that of the order < of the signs >, by an amount equal to the motion of the Director along a circular path, whose center is a point at a distance from the center of the Director, in the direction of its apogee, of one part and a

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half, I mean the center of the Director according to the common opinion (*ind al-jumhuf), and whose radius is three parts. That circle (i.e. the first circle) is the cincture of the deferent according to common opinion ('aid ra'y al-jumhur). [19] To demonstrate that, we assume a point on the cincture of the small encircling < sphere > which moves along that < cincture > by an amount equal to the motion of the epicyclic center, in a direction opposite to it, and coincides with it when the latter reaches the apex or the perigee of the small encircling < sphere >, then that point will always be along the circumference of that circle. You should know that, for it has a geometric proof which becomes evident upon reflection. [20] When the epicyclic center is at the apogee or the perigee of the Director, it would then be along that circle. At positions other than those two, it deviates from it. But it does not deviate except by a very small amount, the maximum of which is reached when it reaches the mean quadrature from the apogee. That is so because the line connecting the center of that circle and the center of the epicycle is always greater than the radius of that circle by an amount equal to the square of one of the chords of the small encircling < sphere>, but at that point (i.e. at quadrature) by the square of its diameter. The maximum deviation is in the amount of four minutes. [21] According to common opinion (al-jumhur), while assuming the epicyclic center to adhere always to the cincture of the deferent whose center is in motion, they consider for the computation of the distances of Mercury the distance between the epicyclic center and the equant center, when it (i.e. the epicyclic center) reaches the mean quadrature from the apogee, to have the value of fifty-seven parts, and its distance from the center of the world to have the value of the square root of the sum of the two squares of fifty-seven and three, as they are in this Principle . This necessitates that its distance from the center of the deferent would be sixty parts and four minutes, as it was necessitated by this Principle. Therefore if this Principle were to be adhered to it would not differ in any way from the commonly held opinion. [22] Here, there is a subtle remark that should be observed, namely that the small encircling moves accidentally through the motion of the large encircling < sphere > by a daily amount equal to twice the motion of the center of the sun. We

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should then assume that its particular motion, in the upper half, is in the direction opposite to that of the order and equal to the daily motion of the center of the sun. What we have just stated is necessary in order to have the difference between its accidental and particular motions equal to the motion of the center of the sun. And it is indeed so. [23] Similarly, the epicycle moves accidentally through the motion of the small encircling < sphere >. We should then assume its particular motion to be equal to the excess of the motion in anomaly (harakat al-ikhtilaf) over that of the center in order that it would move as a result of its particular and accidental motions by a daily amount equal to the motion in anomaly. [24] The treatise is complete, with the help of God. APPENDIX

[From the text of al-Takmila fi Shark al-Tadhkira, by Muhammad b. Ahmad al-Khafri (c. 1522), Zahiriyya Manuscript No. 6782, fols. 199r-199v. Numbers enclosed within the square brackets are references to the corresponding paragraphs in Qushji's text.] [15] One of the moderns has another method for the solution of this problem. That is, we assume that the center of Mercury's epicycle is at the apex (dhurwa) of a small encircling < epicycle > whose radius is one part and a half. And let the center of the encircling < sphere > be at the apex (dhurwa) of another large encircling < sphere > whose radius is also one part and a half. Let the center of the large encircling < sphere > be at the apogee (awj) of the deferent/The center of the deferent is at a distance of one part and a half from the center of the Director, in the direction of its apogee. The center of the Director is at a distance of four parts and a half from the center of the world. All of these < measures > are in the same units that make the radius of the deferent sixty. [16] Now, if the motion of the Director is assumed to be equal to that of the motion of the sun, in the direction opposite to that of the order < of the signs >, and the motion of the deferent equal to twice that, in the same direction as that of the order < of the signs >, while the motion of the large encircling

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< sphere > is equal to the motion of the deferent in value and in direction in its upper half, then the motion of the center of the small encircling < sphere > would be uniform with respect to the center of the Director; its distances from a point whose distance from the center of the deferent, in the direction of the deferent's apogee, is always equal to the distance between the deferent's center and that of the Director, as it was demonstrated by the Principle of the Encompassing < sphere >. The center of the small encircling < sphere > is then moved by the two motions of the large encircling < sphere > and the deferent, along a circle equal to the cincture (mintaqd) of the deferent, and whose center is moved, by the motion of the Director around its own center, along a circular path whose radius is three parts. [17] Then if the motion of the small encircling is assumed, along the upper part, to be in the direction opposite to that of the order < of the signs >, and equal to the difference between the motion of its center, which is uniform with respect to the center of the Director and in the direction of the order < of the signs >, and the motion of the Director, which is opposite to the order < of the signs >, I mean that it is equal to the motion of the center of the sun, then the motion of the epicyclic center will be uniform with respect to the center of the equant < sphere > as it was demonstrated by the Principle of the Encompassing < sphere >. [18] No one should criticize the details of the proof of the Principle of the Encompassing < sphere > in regard to the fact that the center of the encircling < sphere > approaches the center of the Director and draws away from it. The center of the epicycle will necessarily move along the circumference of a circle equal to the cincture of the deferent approximately, whose center moves, in the direction opposite to that of the order , by an amount equal to the motion of the Director along a circular path, whose center is a point at a distance from the center of the Director, in the direction of its apogee, of one part and a half, I mean the center of the Director according to the common opinion Cind al-jumhur), and whose radius is three parts. That circle (i.e. the first circle) is the cincture of the deferent according to common opinion ('aid ra'y aljumhur). [19] To demonstrate that, we assume a point on the cincture of the small encircling < sphere > which moves along that

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< cincture > by an amount equal to the motion of the epicyclic center, in a direction opposite to it, and coincides with it when the latter reaches the apex or the perigee of the small encircling < sphere >, then that point will always be along the circumference of that circle, as was demonstrated by geometric proof. [20] When the epicyclic center is at the apogee or the perigee of the Director, it would then be along that circle. At positions other than those two, it deviates from it. But it does not deviate except by a very small amount, the maximum of which is reached when it reaches the mean quadrature from the apogee. The maximum deviation is in the amount of four minutes. That is so because the line connecting the center of that circle and the center of the epicycle is always greater than the radius of that circle and three parts of parts. [21] According to common opinion (al-jumhur), while assuming the epicyclic center to adhere always to the cincture of the deferent whose center is in motion, they consider for the computation of the distances of Mercury the distance between the epicyclic center and the equant center, when it (i.e. the epicyclic center) reaches the mean quadrature from the apogee, to have the value of fifty-seven parts and three parts, as it is according to this Principle. Its distance from the center of the deferent would be sixty parts and four minutes, as it was necessitated by this Principle. Therefore does not differ in any way from the commonly held opinion. [23] According to this solution, you should assume as well that the motion of the anomalistic epicycle (al-tadwir dhi alkhdssa) is equal to the excess of the anomaly (khassa) over the motion of the epicyclic center, in the direction of the order . The author of this solution could take pride in it. But you are thoroughly knowledgeable about its inclusion of the two excessive complements which speak against the solutions mentioned before which were of God's inspiration to me. They (i.e the proposed solutions previously mentioned) are more appropriate to select. COMMENTARY

[1-4] As mentioned before, these paragraphs are taken up by the flowery introduction of Qushji, which, as it was said, was

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typical of the literary style of the period. I only wish to note here that in the very first paragraph, a section usually devoted to the praise of the lord, the author goes to great pain to describe the lord in adjectives drawn from the technical vocabulary of astronomical literature. He systematically refers to the lord as the "director" (mudir), a term used to describe the sphere in Mercury's model that carries the deferent; the "adjuster of their motion" (al-mu'addil li-masiriha), which is the technical term for the equant sphere; the "one who completes his blessings upon any bearer" (al-mutammim nVamihi 'aid hull hdmil), where mutammim is a term used to describe the portions of a sphere remaining outside or inside an eccentric shell embedded in it, and hdmil is the technical name of the deferent; "favors with excessive gifts those who are outside (khdrij) the circle of evil and inclined (ma'il) towards virtuous acts," where both khdrij and md'il are the technical terms for the eccentric and the inclined spheres respectively. Three other terms, found in the technical vocabulary of astrology, are also included in the formulaic blessings usually afforded the prophet of Islam, namely "ascending" (tali*), "conjunction" (muqdrana), and "opposition" (muqdbald). Paragraph [2] is typically devoted to the patron, in this case extolling the virtues of Ulugh Beg, without forgetting to assert that his throne should be set "above the northern stars" (fawq farq al-farqadayn), another reminder of the author's skill in merging astronomical terminology with literary hyperbole. Paragraph [3] confirms the information available from other sources about the relationship between Qushji and Ulugh Beg. Here Qushji clearly admits that he was especially favored by being allowed to study with Ulugh Beg. It is this relationship that gives us some indication of the difference in their ages, which was noted above. It also confirms the sincere interest of Ulugh Beg in the science of astronomy, and corroborates the testimony to the flourishing status of astronomy during his reign. The remainder of the paragraph describes the personal fortunes of Qushji in human terms, which shed some light upon the intrigues and jealousies that were endemic to the court of Ulugh Beg. It also confirms the information given by Ta§kopriilu-Zade about Qushji's travels, his distance from the court, as well as his return and his reception. Paragraph [4] testifies to the fact that Qushji thought of the

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present treatise as an appropriate gift to Ulugh Beg to signal his return and his acceptance at the court. It is this paragraph that casts doubt on the report given by Ta§kopriilu-Zade about Qushjf s other presumed treatise "Concerning the Solution of the Problem of the Moon" which was mentioned above. If Ta§koprulu-Zade's report is correct, and Ulugh Beg had asked Qushji to read to him the treatise about the Moon's problem that he had composed during his absence from the court, and that Ulugh Beg had indeed liked the treatise, then it is very strange that Qushji should say nothing about this encounter in this paragraph, where one might expect to find it. The fact that the title of the present treatise does not include the word Mercury in it may have been responsible for its attribution to the Moon, and might thus have given rise to the report of Ta§kopriilu-Zade. As was stated earlier, the ultimate resolution of this problem could only come from the discovery of a similar treatise devoted to the solution of the lunar model, for whose existence there is no firm evidence. [5-12] This section deals with the development of the Ptolemaic model for Mercury, as described in the Almagest IX, 7-9. Figure 1 is a schematic representation of such a model, where the deferent with center F is carried in a direction opposite to that of the order of the signs by a director sphere with center M, and moves at a speed equal to the mean motion of the sun. The deferent itself moves in the opposite direction such that at any time the position of the epicycle carried by this deferent will look as if it is moving uniformly with respect to a point E, half-way between the center of the director sphere and the center of the world, at a speed equal to that of the mean motion of the sun as well. Finally, the planet Mercury moves on its own epicycle, at its own speed, in the direction of the order of the signs. The net effect of this model is that it allows the planet Mercury to reach a minimum distance from the earth twice in one revolution: once when the center of the epicycle is at about 120° from the slow-moving apogee determined by the direction OEM, and once when it is at the symmetrical position at about 240° from that same apogee. With the appropriate dimensions, the model also accounts for the observational data gathered by Ptolemy for the planet Mercury when it is at the apogee or at the point diametrically opposite to it. [13-14] Qushji notes correctly that such a model violates the

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Figure 1 Mercury's Model according to Ptolemy

principle of uniform motion on two counts: First [13], the deferent does not move uniformly around its own center F, but rather around another point JE, which sometimes coincides with point F, but mostly does not. In paragraph [14] Qushji raises a more formal objection to the way in which the motion of the "center" of the planet is calculated in such a model. The comparison he obviously has in mind is the model for the upper planet where such a motion is measured from the apogee of the planet and then added to the longitude of the apogee itself to obtain the mean longitude. The model of the upper planets

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does not contain a director sphere, and thus the measurements could be taken along the same circle. While here, in the case of Mercury, the introduction of the director sphere introduces another circle, and thus the measurements of the motion of the deferent sphere and that of the director sphere are no longer along the same circle. [15-24] In this section Qushji develops his own model, which actually responds to the first objection that he had raised quite successfully against the Ptolemaic model. In Figure 2,1 superimpose Qushjfs model over that of Ptolemy, in order to show that the projected path of the center of the epicycle in Qushji's model comes very close to that projected by the Ptolemaic model.

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Ptolemy Qushji

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Qushji starts by dividing the eccentricity of the Ptolemaic model at point N, and by taking that point to be the center of a new director sphere. The director sphere moves in the same direction stipulated by Ptolemy, and by the same amount, thus bringing the center of the deferent to point H rather than F, and the apogee to point A9 rather than'A. Now he stipulates that the deferent should move uniformly around its own center H, in the opposite direction, at twice the speed of the director sphere. This motion brings the line HA9 to the direction HB. At point JB, where HB = R, Qushji posits the existence of a small epicyclet, of a radius equal to half the eccentricity of the Ptolemaic model, and whose motion is equal to that of the deferent and in the same direction. By applying 'Urdfs Lemma17 once, Qushji realizes that this combination of motions makes point D look as if it is moving uniformly around point N at the same speed as the mean motion of the sun. At point D, Qushji proposes to place another epicyclet of the same size as the first but whose motion is opposite to that of the first and is equal to the mean motion of the sun only. With such motion, line BD will then be in the direction of DG, and point G will be so close to C that, for all practical purposes, it will be indistinguishable from it (the distance between the two is intentionally exaggerated in the diagram). Now, if the epicyclic center is carried by this second epicyclet at point G, then the behavior of the model proposed by Qushji would duplicate very closely the behavior of the Ptolemaic model. This is a necessary result of a second application of 'Urdi's Lemma where it can be1 shown that since lines DG and NE are equal and angles END and NDG are also equal, then line GE will always be parallel to DN. This, in effect, means that point G, or the center of the epicycle in Qushji's model, will look as if it is moving uniformly around the Ptolemaic equant point E, as was required by the observations. At the critical points, i.e. the points at which the Ptolemaic model was tested, namely at the apogee, the point opposite to 17 'Ural's Lemma, here called the Principle of the Encompassing (sphere) as it was called by Qutb al-Din al-Shirazi at various places in his Nihayat al-idrak fi dirayat al-aflak, and al-Tuhfa al-shahiyya, stipulates that if two equal lines, such as NH, BD, are attached to the extremity of one line such as HB, and are made to move at the same speed, such that angles NHB, and DBH are always equal, then line DN will always be parallel to line HB. For the text of this Lemma and a more detailed description, see Saliba, "The Original Source."

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it, and the two perigees around 120° and 240° from the apogee, Qushji's model yields identical results to those produced by the Ptolemaic model. At other less important points there is obviously some divergence. In fact, Qushji calculates the divergence in paragraph [20] to be around four minutes, when the epicyclic center reaches a distance of 90° from the apogee. In order to test the behavior of the model at all points, Table 1 was calculated for every five degrees, starting from the apogee, for the following variables: The first column gives the value of the independent variable a which is equal to the mean motion of the sun. The next two columns give the distances of the epicyclic center from the center of the world O, and the equant center E, respectively, according to the Ptolemaic model. The next two columns give the same distances according to Qushjf s model. The column under "Eq. dif" gives the difference, in minutes, between the two distances from the equant center, while the one under "Ea. dif" gives the difference, also in minutes, between the two distances from the center of the world. Finally, the last column gives the differences, in minutes as well, at the various distances in the maximum elongation of the planet Mercury as observed from the earth. From this table, we can confirm Qushji's computation that the maximum variation around quadrature is indeed in the order of four minutes. That variation produces an angle of elongation at the earth of less than two minutes. But from the same table, it becomes obvious that there is a much larger variation of about fourteen (14) minutes that takes place when the epicyclic center is about 55° from the apogee. Even at that point, where the variation in the distances is at its maximum, the angular variation for an observer on the earth is less than five mmutes, which is well within the tolerance of medieval observational machines. The most remarkable result of the new model is that it produces no variations at all at the critical points, namely when a = 0°, 120° or its symmetric point 240° and 180°, as can be easily observed from the table.

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TABLE 1 Mercury Model of Qushji compared to that of Ptolemy showing distances of epicyclic centers and their differences in minutes from the Earth and the Equant as well as the difference in the maximum angle seen on Earth.

alpha

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180

Ptolemy 69.00 68.93 68.71 68.35 67.87 67.27 66.57 65.79 64.95 64.06 63.15 62.23 61.33 60.46 59.63 58.86 58.17 57.54 57.00 56.55 56.19 55.91 55.71 55.60 55.56 55.59 55.67 55.79 55.95 56.13 56.31 56.50 56.66 56.80 56.91 56.98 57.00

66.00 65.94 65.75 65.45 65.04 64.54 63.96 63.31 62.62 61.90 61.18 60.46 59.77 59.13 58.54 58.02 57.57 57.20 56.92 56.73 56.63 56.61 56.67 56.80 57.00 57.25 57.55 57.87 58.21 58.56 58.89 59.20 59.47 59.70 59.86 59.97 60.00

Qushji 69.00 68.93 68.73 68.40 67.94 67.38 66.71 65.96 65.15 64.28 63.38 62.46 61.55 60.67 59.82 59.03 58.30 57.65 57.08 56.60 56.22 55.92 55.72 55.60 55.56 55.59 55.67 55.80 55.96 56.14 56.32 56.50 56.67 56.81 56.91 56.98 57.00

66.00 65.94 65.77 65.50 65.12 64.65 64.10 63.48 62.82 62.12 61.41 60.69 60.00 59.34 58.73 58.18 57.70 57.31 57.00 56.78 56.66 56.63 56.68 56.80 57.00 57.25 57.55 57.88 58.22 58.57 58.90 59.21 59.48 59.70 59.86 59.97 60.00

Eq.Dif

Ea.Dif

angle

0 0 1 2 4 6 8 10 11 13 13 13 13 12 11 9 7 6 4 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0;0 0 18 1 11 2 35 4 21 6 21 8 24 10 18 11 56 13 7 13 48 13 56 13 30 12 35 11 16 9 40 7 55 6 9 4 29 3 2 1 51 0 58 0 23 0 5 0 0 0 3 0 12 0 23 0 31 0 36 0 36 0 31 0 24 0 15 0 7 0 2 0 0

0 0 0 0 1 1 2 3 3 4 4 4 4 4 4 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 18 11 35 22 21 24 19 56 8 49 56 31 36 17 40 55 9 30 2 51 58 23 5 0 3 12 23 31 36 36 31 24 15 7 2 0

0 5 20 45 17 55 35 15 52 23 45 57 57 46 23 52 15 35 56 20 49 26 10 2 0 1 5 10 14 16 16 14 10 6 3 0 0

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[10] A MEDIEVAL ARABIC REFORM OF THE PTOLEMAIC LUNAR MODEL GEORGE SALIBA, Columbia University Introduction In a previous article, published in 1979, I have described in some detail the planetary model of Mu'ayyad al-Dm al-'Ur

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computations in Book X of the Almagest were specifically carried out to determine the relative dimensions of the model of each planet in such a way as to satisfy the observational data that Ptolemy was trying hard to save. Moreover, if Abu 'Ubayd's model were to work, Ptolemy would have been the first to adopt it, for it only seems to replace an eccentric sphere, the deferent, by a concentric one and a secondary epicycle. This equation was very well known to Ptolemy who further attributed it to Apollonius in Book XII, 1, of the Almagest, and used it efficiently in Books III, 3, and IV, 6 (See Neugebauer 1959). It would be naive, therefore, to assume with Abu 'Ubayd that the observational problem of the equant could be solved simply by replacing the eccentric hypotheses by the epicyclic hypotheses as Ptolemy would have called this transformation. The problem, therefore, was still to find a model that preserved both the Ptolemaic deferent distance and the effect of the equant, and would still be the result of the motion of spheres that move uniformly around their own centres. Mu'ayyad al-Dm al-cUrdi30 Taking advantage of the fact that one could transfer motion on an eccentric to a motion along a concentric with an epicycle - the Apollonius equation 104

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referred to above - 'Urdl's problem was to devise such a motion so that point B (Figure 3.17) in JuzjanT's model could be brought closer to Ptolemy's deferent, if possible to coincide with Z. This does not necessarily mean that eUrdI was trying to emend the model of JuzjanI directly, for he does not mention JuzjanI at all, and he could have been working directly with the Apollonius equation. But it was a stroke of genius to realize that one does not have to transfer the whole eccentricity TD = BH to the secondary epicycle, but instead accept a compromise and transfer only half of that eccentricity KD = NB. To do so, and approximate Ptolemy's deferent as closely as possible, 'UrdI found that the epicyclet BOH must revolve in the same direction and by the same amount as the new deferent with centre K that he had just introduced. Only then will the combined motion of the deferent with centre K and the epicyclet with centre N produce a resultant path marked by point O which hugs very closely the Ptolemaic deferent EZH. Once this technique had been discovered by 'UrdI, it was used by every astronomer who came after him to adjust the Ptolemaic model in one way or another. A

iL

B NT

H

E O

Z

x & &

K

vXi #»n2 K^

T

e

0

Figure 3.17

But to preserve the effect of the equant as well, eUrdI had to show that the resultant motion of point O had to look as if it is itself uniform with respect to point D, the equant. This is tantamount to proving that under

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the stated conditions - namely, the epicyclet moving by the same angle x as the proposed deferent and in the same direction - the lines OD and NK will always be parallel. To do so, cUrdI stated the problem in the form of a general lemma, namely: Every straight line upon which we erect two equal lines on the same side so that they make two equal angles with the (first) line, be they corresponding or interior, if their edges are connected, the resulting line will be parallel to the line upon which they were erected.

(Kitab al-Hafa, p. 220)

Figure 3.18 is taken from 'Urdl's text in which he shows that line GD is always parallel to AB in all the cases where AG and BD describe equal angles with line AB. It is also assumed that AG = BD. The proof is then straightforward both when the corresponding angles DBE and GAB are equal or the interior angles DBA and GAB are equal, for with the construction of line DZ parallel to AG, both cases become identical and require only Elements I, 27-33, to be proved. D

E

A

Z

B

r>

E

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E

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A

B D

G

B

G

D

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E Figure 3.18

B

G

A

Now that the line OD (Figure 3.19) was shown to be always parallel to NK, point O could then be taken as the centre of the planet's epicycle and the Ptolemaic conditions would be very closely approximated. eUrdI was quite aware of the fact that the path resulting from the motion of O coincides exactly with the Ptolemaic deferent only at the apogee E and the

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.B, Q

Ptolemy 'Urd! Ibn al-Shatir Copernicus Figure 3.23

Apollonius equivalence to produce his own model. We noted that Ibn alShatir knew of the works of cUrdI and took issue with him for retaining the eccentrics in his model. Then it is understandable that he did not feel obliged to prove the parallelism of OD and NK (Figure 3.23), for it was already proved by eUrdI with the general lemma (Figure 3.18). Similarly, Copernicus did not prove that parallelism either, and it was Maestlin who explicitly proved it again in his letter to Kepler (see Graf ton 1973: 528f). The question of the explicit relationship between Copernicus and his Muslim predecessors, especially Ibn al-Shatir, remains open, and further research will have to be done before it can be decisively established one way or the other. What is clear, however, is that the equivalent model of Ibn alShatir seems to have had a well-established history within the results reached by earlier Muslim astronomers, and could therefore be historically explained as a natural and gradual development that had started some three centuries earlier. The same could not be said of the Copernican model. But some more research has to be done on the Arabic sources themselves before 113

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their inner relationships can be fully understood and exploited in this regard, and on the Byzantine sources for a possible connection between Copernicus and his Muslim predecessors. The planetary model of TusI In terms of its relationship to the Copernican model for the upper planets, Tusl's model represents a tradition different from that of Ibn al-Shatir. Rather than bisecting the Ptolemaic eccentricity, in the tradition of eUrdI, TusI generalizes his own lunar model (Figure 3.24) and allows a 'Couple' to move in such a way that the centre of the epicycle would also be moved closer to the equant when the epicycle is at the Ptolemaic apogee, and farther away when the epicycle is at perigee. The 'Couple' itself is carried by a deferent that is now concentric with the equant. All motions therefore would be uniform around the centres of the spheres concerned, and would

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.R e

e

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Ptolemy TusI Figure 3.24

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produce none of the contradictions that were assumed in the Ptolemaic model. The model for the motion of Mercury The Ptolemaic model for the motion of Mercury as described above (Figure 3.4) is very similar to that of the moon. In effect, Ptolemy uses the same crank-like mechanism that allows the planet to come close to the earth at two points instead of one, hence accounting for what Ptolemy observed to be the greatest elongations from the sun, and thus assumed the existence of two perigees for Mercury. The equant for Mercury, on the other hand, is now placed on the line of centres in between the centre of the universe and that of the eccentric, when the diameter of the eccentric is still in the direction of the apogee, instead of its being at twice that distance away from the centre of the universe as in the case of the upper planets. Unlike the lunar model, -the model for Mercury requires the planet to move uniformly around the equant point instead of the centre of the universe as in the case of the moon. The first astronomer known to have proposed an alternative model that would answer the objections to the Ptolemaic model is the same Mu'ayyad al-DIn al-'Urdl whose work we have seen above in connection with the models for the moon and the upper planets. The Mercury model of cUrdI e

UrdI devotes two different chapters to the discussion of Mercury's model, in addition to the various remarks that he makes about it in connection with the other planets. Chapter 4432 contains a straightforward description of the spheres of Mercury with brief remarks about the motions of these spheres. Whenever it was appropriate, eUrdI would correct the Ptolemaic description to fit the new observations. He, at one point, reminds the reader that 'it is no longer necessary to add the conditions that were assumed by Ptolemy for these motions, for it was found that the solar apogee [assumed fixed by Ptolemy] indeed moves at the same rate as the apogee of the director (al-mudir) which is in Libra'. Chapter 48,33 as its title Islah Ha fat *Utarid ('A Reform of Mercury's Model') implies, is devoted to a reconstruction of Mercury's model in such a way that the two main problems in the Ptolemaic model are solved. These problems were, as in the case of the lunar model, (1) a deferent that moves uniformly around an axis that does not pass through its centre, and (2) an equant point which is neither the deferent's centre nor the centre around which the deferent describes equal motion. 115

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In Ptolemy's model (Figure 3.25), the deferent is moved uniformly by the director around centre B in a direction contrary to the direction of the order of the signs, in order that it brings the apogee to point T. The deferent itself is made to move in the opposite direction around its own centre G, to carry the epicyclic centre to point C, but seems to be moving uniformly at equal and opposite speed around point E, the equant. It is necessary, therefore, that the deferent describe an irregular motion around its own centre G, a clear violation of the uniform motion principle. A

T

Ptolemy's apogee

hw

Epicycle

c

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'Urdl's deferenl Ptolemy e UrdI

H

Director sphere Figure 3.25

In response to that, 'Urdi states the following: This total configuration resulted from several considerations. Among them are the observations, the proof that is based on the observations, the periodic

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Studies in the Making of Islamic Science ARABIC PLANETARY THEORIES movements, the model [hay3a] that he [i.e. Ptolemy] conjectured [hadasa], and the directions of these movements. No one should be critical of the observations, the proof, or the periodic movements, for there has not come to light anything to contradict them. As for the method of conjecture, he [i.e. Ptolemy] should have no priority over anyone else, especially that his error had been made evident. And if anyone were to find another proposition [amr] that agrees with the principles, and matches the particular movements of the planet that are found by observation, then that person should be deemed as closer to the truth [awla bi-isabat al-haqq\. And now that we have seen his erroneous opinion and sought to emend it [islah] as we did in the case of the other planets, we found that we could do so if we reversed the directions of the two movements mentioned above, that is, the movement of the director and that of the deferent. Let us assume then, that the director moves in the direction of the order of the signs by as much as three times the mean motion of the sun, and that the deferent moves in the opposite direction [i.e. contrary to the direction of the order of the signs] by twice that motion. Then the resulting motion of the epicyclic center is in the direction of the order of the signs by as much as the mean motion of the sun, which is the same as in his [i.e. Ptolemy's] model.

Translated into the diagram (Figure 3.25) and superimposed, not to scale, over Ptolemy's model, 'Urdi's model describes the motion of Mercury by letting the director move uniformly, like Ptolemy's lunar deferent, in the direction of the order of the signs around centre B to carry the apogee to point S. Then the deferent should also move uniformly, but in the opposite direction, around its own centre K to bring the epicyclic centre back to I. The resultant motion of the epicyclic centre would then be parallel to that of Ptolemy and is very close to it, as in the figure. Moreover, 'Urdl's model will agree with the principles of uniform motion and will match the results of observations very closely, thereby, in 'Urdl's words, 'varying only slightly (from that of Ptolemy) by an amount that could escape the observer'. eUrdI then continues to say: 'Our method, on the other hand, is free from doubt and contradiction, and is therefore clearly superior to any other'.34 The next astronomer to propose an alternative model for Mercury was Qutb al-DIn al-ShlrazI, a student of TusI, for TusT himself clearly admitted in his Tadhkira that he had not yet devised a model for Mercury, and that he would describe it once he did.35 The status of our present research does not indicate that he ever did.

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The Mercury model of Qutb al-DIn al-ShlrazI A brief description of ShlrazTs model has already been given by E. S. Kennedy (see Kennedy 1966: esp. pp. 373-5), and the following is mainly derived from his work and from the work of ShlrazT in the Tuhfa. ShlrazT proposed to replace the Ptolemaic model with one of his own (Figure 3.26), which was composed of six spheres: (1) a deferent with a radius r\ equal to 60, eccentric to the centre of the universe by the same eccentricity as that of Ptolemy, and whose centre B does not move as in the Ptolemaic model, thus eliminating the need for a 'director'; (2-5) two sets of Tusi Couples whose smaller spheres have radii r2 = r^ - r* = rs equal to half the Ptolemaic eccentricity; and (6) a final sphere of radius r& equal to the eccentricity. The motions of these spheres as described by Kennedy, and by Qutb alDIn in his Tuhfa, are such that the deferent moves uniformly at the same

2v

C

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G

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G

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G

F C

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Figure 3.26

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speed x as the mean motion of the sun, in the same direction as that of the order of the signs. This deferent carries with it all other spheres, the two TusI Couples and the sixth sphere of radius equal to the eccentricity. The first of the two Couples moves in such a way that the larger of the two spheres moves at the same speed as the mean motion of the sun but in the direction opposite to that of the order of the signs. This means that the smaller sphere will move at twice this speed but in the opposite direction, thus keeping the point of original tangency along the diameter of the larger sphere, which is in turn the radius of the deferent. This point F which has to oscillate along the radius of the deferent is taken to be the centre of the larger sphere of the second Couple. The second Couple will then take over and move in the opposite direction to the first Couple at twice the speed, which means that it will generate its own point G which will oscillate along the diameter of the larger sphere, which, in turn, is along the radius of the deferent. The effect of both Couples is to keep the centre of the sixth sphere G along the radius of the deferent, but to allow it to oscillate nearer to the earth and farther away from it. With this motion the radius of the sixth sphere GC = r6 will, together with line BE, satisfy the conditions for 'Urdl's lemma, thus allowing the centre of the epicycle to describe a curve that looks like an egg-shape but pressed in at the waist, so to speak, when the centre of the epicycle is at the two perigees. To describe these motions by using modern vector terminology, and if we assume that the deferent had already moved by an angle equal to x, we could then take the radius of the deferent (Figure 3.26) to be a vector r\ that has been moved by an angle x, and vector ri, the radius of the smaller sphere in the first Couple, to have been moved by the larger sphere in the opposite direction by an angle equal to x. By the motion of the smaller sphere vector r3 would have been moved in the direction opposite to r2 by an angle equal to 2 x. Now, in the second Couple, vector r* would be moved by the larger sphere through an angle equal to 2 x, measured from the direction of ri, and 7*5 would be moved in the opposite direction to r* by the second smaller sphere through an angle equal to 4 x measured from the direction of r4. Finally r$ would be moved by its own sphere through an angle equal to x, measured from the direction of r\. Perceived as such, the sum of vectors ri, ra, r* and rs will allow the centre of the sixth sphere G, i.e. the origin of vector r^, to always oscillate along the radius of the deferent. In this model the centre of the deferent is fixed at a distance from the centre of the universe equal to twice the Ptolemaic eccentricity. Now, since vector r$ will always move at an angle equal to that through which the deferent moves, and in the same direction, it means that the tip of that vector will seem as if it is always moving uniformly around the equant centre, as it would have been predicted by the lemma proposed 119

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by 'Urdi in the model for the upper planets, and as it would have been required by the Ptolemaic observational data. What Qutb al-DIn seems to have done is to use the results reached by TusI and 'UrdI and to develop his own model by using both techniques that were developed earlier, namely TusFs Couple and 'Urdl's lemma. Ibn al-Shatir's model for Mercury To account for a uniform motion of Mercury with respect to the equant, and for the larger elongations from the sun at distances symmetrically placed at around 120° on both sides of the apogee, both facts supported by Ptolemaic observations, Ibn al-Shatir devised a model in which all these facts can be accommodated as resulting from uniform motions of spheres around their respective centres. Like Qutb al-DIn, he too, as we shall see, used the results reached by TusI and eUrdI, namely the TusI Couple and 'Urdl's lemma. Ibn al-Shatir used the same techniques he had used in the lunar and planetary models described above. And here too, he started constructing his model with the assumption that it should be strictly geocentric, so that he would not have to use eccentric deferents which he thought that others had erroneously used.36 To make the model strictly geocentric he assumed (Figure 3.27) the existence of an inclined sphere, of radius r\ = 60 parts, which is concentric with the centre of the universe O, and which moves in the direction of the order of the signs at the same speed as the mean motion of the sun. That inclined sphere is supposed to carry at its cincture (mintaqa) another sphere, called the deferent (al-hamil), whose radius TI is 4; 5 parts and which moves at the same speed as the inclined sphere but in the opposite direction. The deferent carries, in the same manner, a third sphere, called the director (al-mudir), whose radius r3 is 0; 50 parts and whose motion is like that of the inclined sphere in the direction of the order of the signs, but at twice the mean daily motion of the sun. The director then carries the epicycle whose radius r* is 22; 46 parts and whose motion is equal to that of the anomaly of Mercury. At the cincture of the epicycle there is a fifth sphere, called the encompassing sphere (al-muhit or alshamil), whose radius rs is 0; 33 parts and whose motion is equal to twice that of the daily mean motion of the sun and in the same direction as the order of the signs. In turn the encompassing sphere carries an identical sixth sphere, called the preserver (al-hqfi%), whose radius r6 is the same as that of the fifth sphere and whose motion is four times as much as the daily mean motion of the sun, but in a direction opposite to the direction of the order of the signs. The planet Mercury is immersed at the cincture of this sixth sphere. 120

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/5

J6

2x

7 /4

2* /*3

A: /*2

vn

X

O

Figure 3.27

To use modern vector terminology, we let the radius of the inclined sphere be a vector ri, 60 parts long. Its motion would then be equal to that of the daily mean motion of the sun in the same direction as the order of the signs. At the tip of this vector, another one, r2, will represent the deferent and whose length will therefore be 4; 5 parts. Its motion will be equal and opposite to the motion of r\. This means that r2 will continuously be displaced in such a way that it will remain parallel to the line of apsides, and will in effect carry an amount of the eccentricity equal to 4; 5 from the centre to the periphery. The vector that represents the director, r3, will move at twice the speed of r\ and in the same direction. It can be easily shown by 'Urdl's lemma that the tip of 7-3 will seem to move uniformly around a point on the apsidal line at a distance from the centre of the universe equal to 4; 5 - 0; 50 = 3; 15 parts. Since the tip of r$ is actually the centre of the epicycle in the Ptolemaic model, this displacement will in effect produce the same effect as that of the motion of the centre of the epicyclic centre around the equant which is 3 parts away from the centre of the universe in the Ptolemaic model. Thus far the problem of the equant is resolved.

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The last two vectors rs and re are supposed to answer the second requirement of the Ptolemaic model, namely to create the effect of enlarging the epicycle of Mercury when the planet is about 90° away from the apogee. This will be achieved if we assume those two vectors to represent the two radii of the small circle of a TusI Couple,37 whereby the diameter of the larger circle will be along the direction of the epicyclic diameter, and thus allowing the latter to be reduced in length by 0; 66 parts, or enlarged by the same amount. With that solved, the two main requirements of the Ptolemaic model were met, and the contradictions were solved. As stated above, the model takes advantage of the two important results which were achieved by 'UrdI and TusI. Ibn al-Shatir was, therefore, unlike Copernicus who used the same model for the motion of Mercury - once without understanding it fully as in the Commentariolus (Swerdlow 1973: 504), and once in the De Revolutionibus (Swerdlow and Neugebauer 1984: 403f), where it was better described - a true heir to a long tradition of Arabic astronomy, which supplied him with such techniques that he only had to put them together, as he did in the model for the upper planets, and then add the requirement of making the whole model strictly geocentric. The Mercury model of Sadr al-Shari'a In his Kitab al-tcfdil, Sadr al-Shanea describes the Ptolemaic model for Mercury (fols 32r-33v) and concludes the section by a statement of the inadequacies of this model. He then repeats the statement of TusI in his Tadhkira where he admitted that he had not yet developed a model for the motion of Mercury. Sadr al-Shari'a then claims that, with God's help, he had been successful where TusI had failed. He goes on to describe a model which was essentially a modification of the lunar model proposed by Qutb al-DIn and is described above. Sadr al-Shanca proposes a new eccentric deferent whose centre F (Figure 3.28) is to be placed at a distance ej2 from the centre of the director, i.e. above the Ptolemaic equant in the direction of the apogee by one and a half times the Ptolemaic eccentricity, and whose motion is taken to be twice that of the director and in the opposite direction to that of the director, i.e. in the same direction as the order of the signs. He then uses 'Urdl's lemma, by affixing an epicyclet to the cincture of this deferent of a radius r\ equal to e/29 and allows this epicyclet to move at the same speed and direction as the deferent. The actual epicycle of the planet is supposed to be carried at the cincture of this epicyclet. By 'Urdl's lemma, the centre of the actual epicycle H will seem as if it is describing equal arcs in equal times, i.e. moving uniformly, around the centre of the director B. Moreover, the 122

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K

r\ H

c

2*

rT x. e\t

R

F

e E

^ O

Figure 3.28

centre of the actual epicycle H would then be in the same direction from the centre of the director as the centre of the epicycle in the Ptolemaic model is from the equant. Since all these motions were described as mean motions, then to have the centre of the epicycle move in a direction parallel to the one that would have been anticipated in the Ptolemaic model must have satisfied Sadr al-Shariea, for he claimed that he had found an equivalent model that did not suffer from the Ptolemaic inconsistencies. e

Ala9 al-DIn al-QushjI (d. 1474)

In an anonymous treatise kept at the Asiatic Society Library in Calcutta (No. A1482), which, according to the present author, was written by QushjT, we find yet another attempt to resolve the problem of Mercury. After presenting the Ptolemaic model for Mercury, and criticizing it, Qushjl goes on to present his own solution of the problems entailed by the Ptolemaic model. He first assumes (Figure 3.29) the centre C (or G) of the Ptolemaic epicycle to be carried by an epicyclet with centre D, whose radius is half the Ptolemaic eccentricity, and which is in turn carried by another epicyclet, with centre B, of identical radius. Next he assumes that the epicyclet B is carried by a new deferent whose centre is point H, at a 123

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B>

2a

as

->

A

2a

A'

r*

R

OL

M

a

F H

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Ptolemy QushjT Figure 3.29

distance equal to half the Ptolemaic eccentricity from the centre of the director N after defining this new centre of the director to be at a distance from the centre of the universe equal to one and a half times the Ptolemaic eccentricity. The motions of these spheres are then described as in the Figure to be as follows. The director carries the deferent with it in the direction opposite to that of the order of the signs at a speed equal to the mean daily motion of the sun to bring the apogee to point A'. The deferent moves in the opposite direction at twice that speed, bringing point B, the centre of one of the epicyclets, to the direction HB. The epicyclet with centre B moves in the same direction as the deferent and with the same speed, thus bringing point D, the centre of the second epicyclet, to look as if it is moving uniformly 124

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around centre N, the centre of the new director. The second epicyclet then carries the centre of the epicycle G by moving back in the same direction as the director and at the same speed as the director. This combination of motions will ensure that G will always be in line with C, and on the extension of the line connecting C to the equant E, thus making point G look as if it is always moving uniformly around the equant as it should. Any close consideration of this model will immediately reveal its indebtedness to 'Urdl's lemma, used once to align D with N, and another time to align G with E, to Qutb al-DIn's lunar model, by retaining the crank-like mechanism of Ptolemy but also bisecting the eccentricity, and to Sadr alShari'a's more rudimentary form of the same model. CONCLUSION After this general review of the planetary theories which were developed by Arabic writing astronomers after the twelfth century, it has become clear that the two major achievements of this long tradition were, after disregarding motion in latitude and planetary distances for they are less important for that tradition, essentially two mathematical theorems: one that we referred to above as eUrdI's lemma and the other being the so-called TusI Couple. With the help of these two theorems, and with the technique of dividing the eccentricities of the Ptolemaic models, it was possible to transfer segments of these models from the central parts to the peripheries and back. This freedom of movement not only allowed the retention of the effect of the equant in the Ptolemaic models, but also allowed the development of sets of uniform motions that would not violate any physical principles. The TusI Couple allowed, in addition, the production of linear motion as a combination of circular motions, and thus allowed someone like Ibn al-Shatir, and after him Copernicus, to create the effect of enlarging the size of the epicyclic radius and of shrinking it by using uniform circular motion only or combinations thereof. The other result that has become clear from this overview is that the tradition of criticism of Ptolemaic astronomy became a well-established tradition after the thirteenth century, and very few astronomers could do any serious work without attempting some reform of Greek astronomy on their own. Ironically, this period of original production in Arabic is usually thought of as a period of decadence in Islamic science and little effort is spent to study it in any depth. But recent scholarship on Copernican astronomy, especially that of Swerdlow and Neugebauer, has left no doubt that this Arabic tradition in astronomy must have had an impact on Copernicus himself, and only

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future research will reveal the exact nature of the channels of transmission from the East to the West that were responsible for this impact. NOTES 1 For a complete statement of these problems and their solutions see the fuller discussion that follows. 2 For a description of the direction of motion and the problems associated with it, see Toomer (1984: 20 and 221) where he says that a point A moves 'clockwise' as being in 'advance [i.e. in the reverse order]of the signs'. 3 See, for example, Petersen (1969), Pedersen (1974: 167-95) and Neugebauer (1975: 68f). 4 For a geometric description of Mercury's model in the Almagest, see Toomer (1984: 444-5). 5 We know about this anonymous author from a treatise titled simply Kitab alhay'a, which seems to have survived in a unique copy at the Osmania Library in Hyderabad (India), and is summarized later in this chapter. 6 For the contents of this work I use the Cairo edition. There is a preliminary English translation of this text that was completed as a dissertation by Dan Voss at the University of Chicago under Noel Swerdlow's supervision (unpublished). 7 Shukuk, p. 23. For the clumsy derivation of the limits of eclipses by Ptolemy in the Almagest, VI, 5, see Pedersen (1974:'227f). 8 The actual statement of Ptolemy is: 'Now it is our purpose to demonstrate for the five planets, just as we did for the sun and the moon, that all their apparent anomalies can be represented by uniform circular motions, since these are proper to the nature of divine beings, while disorder and non-uniformity are alien (to such things)' (Toomer 1984: 420). 9 Shukuk, pp. 48-58. See also page 60 for the comparison between the conditions of the spheres and the shells of spheres. 10 The work of Jabir ibn Aflah has not yet been fully analysed. That of al-Bitruji was published by Goldstein (1971), and the work of Averroes in conjunction with that of al-Bitruji was first analysed by Gauthier (1909) and more recently by Sabra (1984). 11 For a full description of the problem of the planetary distances in the Ptolemaic works, see Swerdlow (1968). 12 Escurial Ms. Arab. 910, fols 78v-79r. 13 Cf., for example, Kennedy et al. (1983), passim. 14 Ibn al-Shatir, Nihayat al-Sul, Bodleian Library Ms., Marsh 139, fol. 4 V . 15 ibid., fol. 10r. 16 The present author has completed a critical edition of this text of Ibn al-Shatir, which is now being prepared for the press. The references given here, however, are to the Bodleian Arabic manuscript Marsh 139. 17 For the date of 'Urdl's work, see Saliba (1979a), and for the edition of the text, see al-'Urdl, Mu'ayyad al-DIn: Kitab al-Hay'a (Tarikh film al-falak al-arabi). 18 A translation of that chapter, including the intended lemma, has been given, in French, by Baron Carra de Vaux (1893), and more recently in English by Faiz Jamil Ragep (1993), pp. 194-223.

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Studies in the Making of Islamic Science ARABIC PLANETARY THEORIES 19 We use for this study the Koprulii Ms 657, which is dated 20, Jumada I, 681 AH (27 August 1282) within the lifetime of ShirazI (d. 1311). 20 Ms. fol. 61V. 21 Ms. fol. 66r, the last sentence in the quotation is the same one quoted by 'Urdi, Kitab al-Hay'a, p. 136. See also p. 118 of the same text where 'Urdi states that he had gone against the opinion of all astronomers in matters relating to the directions of the motions of the lunar spheres and the magnitudes of these motions (khalafna fihi jam? a ashab eilm al-hay'a). In a forthcoming article the present author will show the exact indebtedness of Shlrazi to 'Urdl in regard to the lunar model. 22 For this astronomer, see Suter (1900: 165, n. 404). The work used for this study is Sadr al-ShanVs Kitab al-Taedll fl al-hay'a, British Museum Add. 7484, fol. 27rsqq. 23 Ibid. 24 For a brief description of this model see Roberts (1957: esp. pp. 430-2). 25 In an earlier version of Nihayat al-Siil (The Final Quest'), the book in which Ibn al-Shatir proposed his new astronomy, the radius of this sphere is taken to be sixty-seven. 26 This measurement was not given in the early version of the Nihaya. 27 Again, these measurements were not given in the earlier version of the Nihaya. 28 In the earlier version of the Nihaya he adds a note to the effect that this sphere should not be confused with the commonly known epicyclic sphere for they are not the same. 29 For a general survey of these solutions see Saliba (1984). The rest of this section depends heavily on this article. 30 For the edition of this author's work, see al-'Urdi, Kitab al-Hay'a, and Saliba (1979a). 31 For the derivation of the greatest distance between the two, see Swerdlow (1973: esp. p. 469). 32 See 'Urdi, Kitab al-Hay'a, pp. 235-8. The following citation is found on p. 237. 33 Ibid., pp. 246-57. The following citation is found on pp. 250-1. 34 Ibid., p. 257. 35 In the Tadhkira, Leiden Ms. Or. 905, fol. 47r, he says: 'As for [the model of] Mercury, I have not yet been able to imagine that in the proper manner. For it is difficult to imagine the cause for the uniform motion of a [body] around a point by having [that body] move with a complex composite motion closer to that point or away from it. If God were to grant me success in that, I would append it at the appropriate place, if God wills it'. 36 See Ibn al-Shatir's attack against earlier astronomers who used eccentric deferents at the beginning of his Nihayat al-Siil, ch. 2. 37 Ibn al-Shatir speaks of two spheres having identical radii, one carried at the cincture of the other. This could only mean that he was thinking of a TusI Couple, and not of two intersecting circles, for in medieval terminology such spheres would have had to intersect with each other which was never allowed.

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Studies in the Making of Islamic Science BIBLIOGRAPHY Gauthier, L. (1909) *Une reforme du systeme astronomique de Ptolemee', Journal Asiatique 10 ser. 14: 483-510. Goldstein, B. R. (1971) Al-Bitruji: On the Principles of Astronomy, 2 vols, New Haven. Grafton, A. (1973) 'Michael Maestlin's account of Copernican planetary theory', Proceedings of the American Philosophical Society 117: 523-50. Ibn al-Haytham (1971) Al-Shukuk 'ala Batlamiyus, edited by A. I. Sabra and N. ShehabI, Cairo. Kennedy, E. S. et al. (1983) Studies in the Islamic Exact Sciences, Beirut. Neugebauer, O. (1959) The equivalence of eccentric and epicyclic motion according to Apollonius', Scripta Mathematica 24: 5-2preprinted in Neugebauer, O. (ed.) Astronomy and History: Selected Essays, New York, pp. 335-51. Pedersen, O. (1974) A Survey of the Almagest, Odense. Petersen, V. M. (1969) The three lunar models of Ptolemy', Centaurus 14: 142-71. Ragep, F. J. (1993) Nasir al-Dm al-Tusi's Memoir on Astronomy (al-Tadhkira fi e llm al-Hay*a), with translation and commentary, 2 vols, New York/Berlin/Heidelberg. Roberts, Victor (1957) The solar and lunar theory of Ibn ash-Shatir', Isis 48: 428-32. Sabra, A. I. (1984) The Andalusian revolt against Ptolemaic astronomy: Averroes and al-Bitrujr, in E. Mendelsohn (ed.) Transformation and Tradition in the Sciences: Essays in Honor of L Bernard Cohen, Cambridge, pp. 133-53. Saliba, George (1979a) The first non-Ptolemaic astronomy at the Maraghah School', Isis 10: 571-6. (1980) 'Ibn Sina and Abu eUbayd al-Juzjanl. The problem of the Ptolemaic equant', Journal for the History of Arabic Science 4: 376-403. (1984) * Arabic astronomy and Copernicus', Zeitschrift fur Geschichte der Arabisch-Islamischen Wissenschaften 1: 73-87. (1987a) Theory and observation in Islamic astronomy: the work of Ibn alShatir of Damascus (d. 1375)', Journal for the History of Astronomy 18: 35-43. Suter, Heinrich (1900) Die Mathematiker und Astronomen der Araber und Ihre Werke, Leipzig, 'Abhand. zu Gesch. der Math. Wiss.', X. Swerdlow, Noel M. (1968) Ptolemy's Theory of the Distances and Sizes of the Planets: A Study of the Scientific Foundations of Medieval Cosmology, Ph.D. thesis, Yale University, 1968, University Microfilms International 69-8442. (1973) The derivation and first draft of Copernicus's planetary theory. A translation of the Commentariolus with commentary', Proceedings of the American Philosophical Society 117: 423-512. Swerdlow, N. M. and Neugebauer, O. (1984) Mathematical Astronomy in Copernicus's De Revolutionibus, 2 vols, New York. Toomer, G. J. (1984) Ptolemy's Almagest, New York. al-'Urdl, Mu'ayyad al-DTn, Kitab al-Hay'a (Tarikh *ilm al-falak al-arabi), edited by G. Saliba, Beirut, 1990; coll. 'Silsila Tarikh al-'Ulum 'Inda-l-'Arab, 2'. English translation and commentary in press.

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[12] AL-KINDI'S COMMENTARY ON ARCHIMEDES' 'THE MEASUREMENT OF THE CIRCLE'* ROSHDI HASHED I INTRODUCTION

Al-Kindfs work as a whole raises a question about the relationship between science and philosophy or, more specifically, the relationship between mathematics and theoretical philosophy. One has only to consider the titles of his writings to realise that this 'first philosopher' in Arabic was also one of the first scientists. The list which al-Nadlm compiled in 987 of al-Kindi's writings,1 and which has been the basis of all bibliographical research in both ancient and modern times,2 shows very clearly that al-Kindi devoted a great deal more of his writing to the various mathematical, astronomical, optical, medical and other branches of knowledge of his time than to philosophy in the strict sense. No less than sixty titles are attributed to him on mathematics alone. The bibliographies aside, there are at least two reasons for discussing the relationship between the sciences and philosophy in al-Kindi's work. Firstly, the historian of al-Kindf s thought cannot avoid recognising a connection between some of his questions and themes, as well as his methods of exposition and argumentation, and the sciences, particularly mathematics. Consideration of al-Kindi's mathematical knowledge is essential for understanding properly his treatment, for example, of infinity, plurality, the sphericity of the elements, the shape of the universe, his use of the axiomatic method and of proofs in theoretical philosophy, and, notably, his argument in I am grateful to A.W. Lloyd for his translation of this article and to C. Burnett for his help in improving the translation. 1 Al-Nadim, Kitab al-fihrist, ed. R Tagaddud (Tehran, 1971), pp. 315-20. 2 For the ancient bibliographers, see esp. al-Qifti, Ta'rih al-hukamd', ed. J. Lippert (Leipzig, 1903), pp. 366-78; Ibn Abi Usaybi'a, Vyun al-anba* fi tabaqat alatibba', ed. N. Ri^a (Beirut, 1965), pp. 285-93. For the modern bibliographers, cf. esp. R.J. McCarthy, al-Ta$anif al-mansuba ila faylasuf al-'arab (Baghdad, 1963); N. Rescher, AL-Kindl, An Annotated Bibliography (Pittsburgh, 1964).

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favour of proof by reductio ad absurdum. We need look no further than his writings on theoretical philosophy to see the force of this fact. In these writings, he regularly employs certain proofs where his method is quite clearly derived from the Elements of Euclid. Even when he writes to a poet, f All ibn alGahm, he writes as a mathematician rather than as a rhetorician.3 For al-Kindi a proposition in theoretical philosophy required a proof as tightly argued as a mathematical one. Secondly, al-Kindi himself states a number of times in his scientific writings that his aim was to achieve two goals: to extend and complete the scientific knowledge inherited from the ancients,4 and to offer a propaedeutic for the teaching of philosophy The connection between al-Kindi's philosophy and his mathematics is essential for a full understanding of his general system of thought. The inter-relationship is explicitly stated by al-Kindi himself when he writes a treatise entitled Philosophy can only be Acquired by Mathematical Discipline* and when he champions mathematics as a prerequisite subject for philosophical education in his letter On the Number of the Books of Aristotle? In the latter, he goes so far as to warn the student of philosophy that he faces a stark choice: either learn mathematics before approaching the works of Aristotle (listed 3

Al-Kindi's letter to the poet 'AH ibn al-Gahm, On the Uniqueness of God and on the Finitude of the Universe^ cf. Rasa'il al-Kindl al-falsafiyya, ed M.A. Abu Rida, 2 vols. (Cairo, 1950-3), vol. I, pp. 201-7 (hence forward Rasa'il). 4 This project had been taken up by al-Kindi several times for philosophy as well as for the sciences, something which has already been pointed out [see J. Jolivet and R. Hashed, (Al-Kindi', Dictionary of Scientific Biography (New York, 1978), vol. XV, Suppl. I, pp. 261-7]. By way of example, what he writes in First Philosophy may be cited: 'It is good... that we start off this book, according to our practice whatever the subject, by recalling what the Ancients said everything about, by means of the shortest and easiest methods for those of us who are researching into it, and by completing anything that the Ancients did not say everything about, according to the usage of the (Arabic) language and the custom of the time, to the extent that we are able to do so*; cf. Rasa'il, I, 103. Let us also quote an example from his scientific writings, such as Optics: 'In our desire to embrace the mathematical sciences, to explain the first results that the Ancients have left on this subject, to develop what they have begun and the points where they have allowed us to harvest the full fruits of the spirit... (et augere quod inceperunt et in quibus fuerunt nobis occasiones adipiscendi universas bonitates animates)'; cf. V.A. Bjornbo and S. Vogl, *Al-Kindi, Tideus und Pseudo-Euclid: Drei optische Werke', Abhandlungen zur Geschichte der mathematischen Wissenschaften, XXVI, 3 (1912): 3-41, p. 3, 5 This treatise was mentioned in the ancient bibliographies; for example in that of al-Nadim, al-Fihrist, p. 316. 6 Rasa'il, I, 363-84.

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in al-Kindfs own preferred order), and only then hope to become a true philosopher; or else avoid mathematics and become at best a philosophical regurgitator, strictly limited by what he could retain in his memory.7 For al-Kindi mathematics are fundamental to any course of study in philosophy. Historians have usually characterised al-Kindi's work in two very different ways. One view sees al-Kindi as a Muslim representative of the Aristotelian tradition of Neo-Platonism, a philosopher not of late, but of very late, Antiquity. The second view sees him as basically a speculative theologian (mutakallim) who changed his language and learned to speak the language of Greek philosophy. But the real character of alKindi's position will become clear only when we restore to mathematics their proper role in his philosophy. On the one hand, he sees revelation, following the tradition of Islamic speculative theology, as the means for finding truth, the source of reason. On the other hand, he finds in the Elements of Euclid a model and a method from which to derive a rational approach, where the starting point is the basic 'truths of reason'. These are quite independent of revelation and must answer to the criteria imposed by geometrical proofs. The basic concepts and postulates of these 'truths of reason' are supplied in al-Kindfs time by the Aristotelian tradition of NeoPlatonism. To the extent that they satisfy the demands of geometric thought and make possible the use of the axiomatic method, it is these 'truths' which are chosen to replace those which revelation supplied from the tradition of speculative theology.8 It is thus that 'mathematical investigation' (al-fahs alriyadi) becomes the instrument of metaphysics. 7 Al-Kindi writes, having given his various groupings of the books of Aristotle: 'These, then, are the books [of Aristotle] which we have previously mentioned; and these are the books of which the perfect philosopher must have knowledge after the study of mathematics, that is to say, those which I have specified by their names; for anyone without the knowledge of mathematics - and by this I include arithmetic, geometry, astronomy and music - trying to use these books throughout his life, will be unable to perfect his understanding of them, and all his efforts will only lead to his being able to repeat those of their contents he can recount from memory. As far as acquiring a deep knowledge and understanding of them, this is simply unattainable if he does not have a sound basis in mathematics', ibid., I, 369-70. 8 This is in fact the case for the letters about theoretical philosophy, such as First Philosophy, On the Explanation of the Finitude of the Universe, etc. To take the case of this latter text, al-Kindi proceeds by the 'ordered' method to demonstrate the inconsistency in the concept of a body of infinite size. He begins by defining the basic

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In any case, the investigation of the role of mathematics in al-KindTs philosophy should establish more definitely the actual character of his philosophic method. But this task (to which we intend to return), as well as the related question of the relationship of al-Kindl's mathematics to the Elements of Theology of Proclus,9 cannot be undertaken before performing a preparatory task, namely that of restoring and analysing alKindi's own mathematical knowledge. Only then will we know the kind of mathematical subject matter that he could handle, and equally, we will be in a better position to understand certain features of the process by which Greek mathematics were transmitted into the Arabic world, if we could see al-Kindf s time for the turning point that it was. We already know that al-Kindi was a witness to the first translations of Greek mathematical texts into Arabic in the first half of the ninth century. We also know that, among other subjects, he concerned himself with the isoperimetric problem,10 the approximation of the ratio of the circumference of the circle to its diameter, several solid problems involving threedimensions (trisection of the angle, two mean proportionals between two straight lines), the projection of a sphere, some problems of spherical geometry, as well as the subjects which he took over from the Greek mathematicians and which were at terms 'magnitude' and 'homogeneous magnitudes'. He then introduces what he calls the 'truth proposition* (qadiyya haqq) [Rasa'il, I, 188], or, as he explains it, 'the true premises conceived directly* (al-muqaddamdt al-uwwal) ['First Philosophy', Rasa'il, I, 114], or 'the self-evident premises conceived directly* ['Regarding the Essence of That Which Cannot Be Infinite', Rasa'il, I, 202]; that is to say, tautological propositions. These are formulated in terms of basic concepts, of relations of order on themselves, of operations of reunion and separation on themselves, and also in terms of the predications 'finite' and 'infinite*. It is a question of propositions such as this one: homogeneous magnitudes of which the ones which are not greater than the others are equal; or, such as this one: if one adds to one of the equal homogeneous magnitudes another magnitude which is homogeneous to it, then the magnitudes will he unequal [Rasa'il, I, 188]. Finally, al-Kindi continues with a demonstration, with the help of reductio ad absurdum, in using the following hypothesis: the part of an infinite magnitude is necessarily finite. 9 Some extracts of which had been translated into Arabic. See the edition of 'A. Badawi in al-Aflatuniyya al-muhdata *inda al-rarab (Cairo, 1955), pp. 4-33. 10 Al-Nadim, and others after him, mention a work by al-Kindi: 'The sphere is the greatest of the solid figures, and the circle is the greatest of the plane figures', alFihrist, p. 316. Further, in his Great Art, al-Kindi reminds himself that he has written a book 'on the sphere, and the solids, knowledge of which is connected with that of the sphere...'. Cf. al-Kindi, Fl al-sinara al-*uzmd, ed. 'Azml Taha al-Sayyid Afcmad (Cyprus, 1987), p. 120.

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the time considered an area of geometrical research. These intimations of al-Kindf s mathematical activity, which we owe either to the author himself or the bibliographers,11 are all confirmed by several mathematicians who had access to certain of al-Kindi's works, such as al-Farganl and al-Biruni.12 According to them, al-Kindi was responsible for inventing a device for making zenithal and equidistant projection as part of a design for an astrolabe.13 This information, as well as the remnants of his work which have come down to us including his writings on optics, catoptrics, music and astronomy, altogether undermine the picture of this philosopher as simply a spectator while others did mathematical research. Rather, the titles of his mathematical writings which we have been discussing suggest that al-Kindi was one of the pioneers of this type of research in Arabic mathematics, together with his 'enemies' the Banu Musa. Unfortunately, the provisional or permanent loss of most of al-Kindi?s mathematical writings (apart from his writings on optics and catoptrics which, far from being works of popularisation,14 are a strict application of his programme, namely, to 11

Al-Nadim, al-Fihrist, pp. 316-7. In his treatise on The Plane Projection of the Figures and of the Spheres (Tastlh al-$uwar wa tdbtlh al-kuwar)y al-Biruni writes: *It is possible to transfer what belongs to a sphere onto a plane by another method attributed by Abu al-*Abbas al-Farganl in several copies of his book entitled al-Kamil to Yaf qub ibn Ishaq al-Kindi and hi several other copies to IJalid ibn 'Abd al-Malik alMarwarrudi, which is called the astrolabe in the shape of a melon'; cf. R. Hashed, Geometric et dioptrique au 2C siecle: Ibn Sahl, al-Quhi, Ibn al-Haytham (Paris, 1993), p. CIV n. 16. This projection, and the type of astrolabe which it allows one to construct, are due to al-Kindi or at the very least, were improved by him. Moreover, al-Biruni comes back to this in his book Istl'ab al-wuguh al-mumkina fi $an'at alasturldb (Concerning all the Possible Ways of Making an Astrolabe), [MS Leiden 1066, fol. 89v-90r], and recalls some criticisms made by Muhammad ibn Musa ibn Sakir. Al-Biruni does not support this criticism which he finds at the very least without import or, as he puts it: 'Muhammad ibn Musa ibn Sakir has in this matter done no more than attack the person who made it and slander the person who invented it'. According to al-Biruni, Muhammad's criticism was a consequence of the hostility between the Banu Musa and al-Kindi. The evidence of al-Fargani and of al-Biruni, as well as the attack by Muhammad ibn Musa, are further indications that contemporary mathematicians and those who came after al-Kindi did not regard him simply as a philosopher, but also as one of their own. 13 This refers to the astrolabe in the shape of a melon. See note 12 above. 14 This opinion has been supported by F. Rosenthal, 'Al-Kindi and Ptolemy', Studi Orientalistici in onore di Giorgio Levi della Vida, 2 vols. (Rome, 1956), vol. II, pp. 436-56. 12

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recapitulate concisely the teaching of the ancients while filling in the gaps as much as possible) may prevent us from arriving at a true appreciation of his contribution. We have undertaken research into the works of al-Kindi in order to group his mathematical materials before examining the connections between his mathematics and philosophy. This research has led to an important first result, namely the discovery of al-Kindi's commentary on the third proposition of Archimedes' Measurement of the Circle, a proposition relating to the approximation of the ratio of the circumference to the diameter. This commentary, of which we are giving here the editio princeps, shows us al-Kindi actually engaged in mathematical research, and allows us to examine the state of his mathematical knowledge. It also illuminates the extent to which contemporary mathematicians knew Archimedes's work. Before this discovery of al-Kindi's commentary, we really had from this period only the commentary of the Banu Musa, which is part of their treatise On the Measurement of Plane and Spherical Figures. The newly-discovered text of al-Kindi is at least contemporary with that of Banu Musa, if not earlier than it. It is an important link in the long history of commentaries on The Measurement of the Circle of Archimedes.15 O THE TEXT

The text of al-Kindi is a commentary on the third proposition of Archimedes' Measurement of the Circle™ The first question to ask is whether al-Kindi had direct access to an Arabic version of this treatise; or whether he knew of it indirectly through the writings of Theon of Alexandria, whose Commentary on the First Book of the Almagest he very probably knew? Al-Kindi himself gives an unequivocal answer to this question in his Great Art, which he devotes to a commentary on 15 The history of these commentaries in Arabic has not yet been written. For mathematical texts in Latin, see M. Clagett, Archimedes in the Middle Ages, 5 vols. (Madison, 1964-1984), vol. I. 16 This text of al-Kindi, hitherto unknown, is part of MS n° 7073 in the Tehran University Library. The colophon shows us that al-Kindi's letter was copied in 1036 A.H. (1626 A.D.). The folios are not numbered; the copy is in nastallq, and there is nothing in the margin which indicates that the copyist compared his copy with the original master manuscript.

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certain chapters of the first book of the Almagest. These are his own words: What the ancients have written apropos the approximation of the ratio of the diameter of a circle to its circumference is that it is the same as the ratio between the numbers seven and twenty-two. The last of the ancient Greeks who devoted themselves to this question was Archimedes.17

Al-Kindi continues a little further on: We have shown in our book On Spherics that if one multiplies the circumference of a circle by its diameter, the rectangle obtained is four times the area of the circle, and that the product of the diameter of the circle by a quarter of its circumference is equal to the area of the circle; therefore, the ratio between the square of the diameter of the circle and the area of the circle is the same as the ratio between the numbers fourteen and eleven.18

Finally, al-Kindi gives a demonstration of this last property.19 These quotations show well that al-Kindi knew all three propositions of Archimedes' treatise, of which the treatise of Theon mentioned above contains only one. We come now to the title of text. In al-Nadim's list of alKindf s works we find two titles bearing on exactly the same subject matter, but whose formulation leaves much to be desired. This is what we read: 1) Fl taqrib qawl Arsimldis fi qadr qutr al-da'ira min muhltiha, of which the literal translation is: 'On the approximation of the proposition (sic) of Archimedes on the ratio between the diameter of a circle and its circumference'. 2) Fi taqrib watar al-da'ira, 'On the approximation of the diameter of the circle'. These titles cannot be the ones given by al-Kindi himself, because the first is flawed and the second does not make any sense. However, once corrected, the first becomes: 'The proposition of Archimedes on the approximation...(qawl Arsimldis ft 17

The Great Art, p. 174. Ibid., p. 175. 19 Ibid., pp. 175-6. In his demonstration, al-Kindi begins with this statement: *It has been said that the circumference of the circle is approximately the triple of its diameter plus a seventh again'. 16

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taqrib qadr...) . And if this title was in fact a phrase taken word-for-word from the preamble of the text of al-Kindi we are currently establishing, we would have the following: The proposition of Archimedes on the approximation of the ratio between the circumference of a circle and its diameter'. The second title is clearly truncated. For it to have any meaning, two words, which have disappeared from al-Nadlm's version, would need to be re-instated. We would then have from this bibliographer two disfigured titles relating to one and the same text - the one we now have before us. We may note here that al-Qifti and Ibn Abi Usaybi'a, the thirteenth-century successors of al-Nadim, only mention the second title in its incomplete form. The recovered text, then, is not simply endowed with the title which the bibliographers ought to have given it, but it is a title that can be explicitly attributed to al-Kindi himself. A lexical and stylistic analysis leaves no doubt about the authenticity of the attribution. As it happens, the majority of al-KindTs writings are letters addressed to princes, to the Caliph himself, and to men of letters and scientists, which unfold in a manner which might be called 'canonical'. Al-Kindi begins with invocations and dedications, introduces the proposition for discussion, discusses it and then concludes, albeit more briefly, with further invocations and tributes. These letters can, moreover, be either a few pages in length, or go on for several tens of pages, making up substantial treatises. Our text is no exception to this genre: it is a letter addressed to none other than Yuhanna ibn Masawayh,20 the famous doctor and translator of Baghdad, to whom al-Kindi addresses not only these pages, but also a text on the soul and one on scapulomancy21 Moreover, the letter in question was written (as was often the case) in response to a request from his correspondent: Ibn Masawayh had asked alKindi to explain Archimedes' proposition on the approximation of TT. Further, the preamble of this letter distinctly recalls the first 20 On the life of Ibn Masawayh and his activities, see particularly al-Nadim, alFihrist, pp. 295-6; Ibn Abi Usaybfa, *Uyun al-anba't pp. 246-55. See also the article by J.C. Sournia and G. Troupeau, 'Medecine arabe: biographies critiques de Jean Mesue (VHP siecle) et du pr&endu "Mesue le Jeune" (Xe siecle)', Clio Medica, 3 (1968): 109-17. 21

These are: Risdla fi al-nafs wa afaliha, and Risala fi Urn al-katif.

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lines of other letters of al-Kindi: e.g. On the Intellect, On the Art of Removing Melancholy, On the Uniqueness of God and on the Finitude of the Universe. Also, we find in the preamble of our text the very same words and expressions found in other letters.22 All these elements serve to confirm the authenticity of the text. We know then that al-Kindi had collaborative links with Ibn Masawayh, such that he sent him at least three letters, and on markedly different subjects; we can also say that an Arabic version of Archimedes' Measurement of the Circle was circulating in mathematical and medical circles towards the middle of the ninth century at the latest. Ibn Masawayh was born in 786 in Baghdad, where he died in 857. He held there a maglis, at one and the same time an intellectual salon and a kind of teaching forum, which some of his pupils - including the celebrated Hunayn ibn Ishaq - attended. It is noteworthy that a doctor of Ibn Masawayh's calibre regarded al-Kindi as a 'savant' sufficiently well-versed in mathematics to be worth consulting on Archimedes. In addition, al-Kindi kept company with other translators such as Yahya ibn al-Bitriq, Ibn Na'ima al-Himsi and Ustat (or Astat), who were working on translations of philosophical texts. It has even been suggested recently that we are dealing here with a circle of intellectuals which had formed around the figure of al-Kindi himself.23 We can see, then, that al-Kindf s collaborations were many, and varied according to the field of research; and it would be unwise to limit the number of translators with whom he had relations to those whom we have just listed. Al-Kindi also had connections with another great translator, Qusta ibn Luqa, several of whose scientific translations he had revised.24 The name of Qusta ibn Luqa deserves special attention, for he might have been the one who translated Archimedes' The Sphere and the Cylinder, as is suggested by 22

Cf. the Supplementary notes to the text. This thesis has been presented by G. Endress, Proclus Arabus; Zwanzig Abschnitte aus der Institutio Theologica in ardbischer Ubersetzung, Beiruter Texte und Studien, 10 (Wiesbaden, 1973), pp. 66-193; note particularly pp. 101-5, 192 and 242-5. 24 By way of example, al-Kindi revised the translation by Ibn Luqa of Anaphorikos (Kitab al-matdli*) of Hypsicles. Al-Nadim, moreover, attributes other revisions to alKindi. 23

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ROSHDI HASHED 25

Ibn Abi Usaybi'a; a suggestion which is strongly reinforced by other sources.26 Could Qusta ibn Luqa have also been responsible for translating The Measurement of the Circle*! Was it this translation that al-Kindi had at his disposal? And, finally, was it the one which was known to his contemporaries, the Band Musa? These are unresolved problems, which may well remain so for some time yet. Let us now consider this Arabic tradition of The Measurement of the Circle. We have one surviving Arabic translation of it, which was made from a Greek manuscript, which itself stems from an archetype shared by a surviving Greek text. The author of this Arabic translation remains anonymous. Any comparison between this Arabic translation of Archimedes' text and al-KindTs commentary to identify the text which he consulted can only have limited results, because we can do it only for the third proposition of Archimedes, the single one which al-Kindi examined. In addition, al-Kindi himself never cites the text on which he is making his commentary. However, we can pick out some lexical differences which are not lacking in interest. The algebraic style of certain expressions in al-KindTs commentary constitutes the most striking feature of the comparison. But it is still not possible to know with certainty whether the difference between al-Kindf s text and the actually available Arabic version is due to al-Kindi himself, or whether he consulted a different Arabic translation.

25

Ibn Abi Usaybi'a in fact attributes On the Shape of the Sphere and the Cylinder to Ibn Luqa himself (cf. Vyun al-anba*, p. 330). It is most probable, however, that he is referring to Archimedes' treatise and a translation of it, and not to an independent work. To this we must add the testimony of the Hebrew translator of the Arabic version of The Sphere and the Cylinder of Archimedes, Kalonymos b. Kalonymos, who attributes the Arabic translation to Qusta ibn Luqa. See M. Steinschneider, Die arabischen Ubersetzungen aus dem Griechischen> repr. (Graz, 1960), p. 174. 26 R. Hashed, 'Archimede dans les mathematiques arabes' in Idem, Optique et mathematiques. Recherches sur I'histoire de la pensee scientifique arabe, Variorum, Collected Studies Series, 378 (London, 1992), DC.

Studies in the Making of Islamic Science 'THE MEASUREMENT OF THE CIRCLE' ARCHIMEDES' TEXT27

ARABIC TRANSLATION28

AL-KINDIM

TOU tdkXou f) TTeptiierpoc

a/-£o# al-mufyit bi-al-da'ira the line which encompasses the circle

initially dawr which turns around

311

17

then muhif al-dd'ira the circumference of the circle TeTp,f)a0u) ... 8ixa

qusima bi~ni$fayn divided in two halves

bi-ni$fayn separated in two halves and sometimes qusima bi~ni$fayn divided in two halves

8vva[i€i

ft al-quwwa potentially

duriba fi-nafsihi multiplied by itself murabba* squared

fi al-tul in length

g\4r root

rov TToXxrycjv^ov -rrXevpd irXeupac ixoi/TOC **

olil' al-sakl al-kaftr al-zawdya di al-sitt wa al-tis'lna zawiya the side of the polygon having 96 angles

$ilr dl al-sitt wa al-tis'ina qa *ida the side of that one having 96 bases

8idp.eTpoc (the circle)

qujr diameter

watar chord (once only)

|JLf|K€L

qufr diameter

27 Archimedis Opera Omnia cum Commentariis Eutocii, ed. J.L. Heiberg, 3 vols. (Leipzig, 1880), vol. I, pp. 257-71 and vol. Ill (1881) (Eutocius' commentary), pp. 263-303. 28 We have established the text of this version from the two existing manuscripts; it is this edition which we translate here. 28 Cf. below.

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ROSHDI HASHED

The arguments pointing to the existence of another Arabic version,30 which al-Kindi might have used, are not without substance. We will not repeat here the arguments concerning the first proposition of Archimedes' treatise, the algebraic aspect of the vocabulary, the modification of the ratio (1,172 + \ ) to 153, given by Archimedes which was replaced in al-Kindi's text and that of his contemporaries - the Banu Musa - with the ratio (1,172 +j ) to 153.31 None of these elements is decisive on its own, but taken together they make up a plausible case for the existence of another Arabic translation of Archimedes' treatise.

m AL-KINDI'S COMMENTARY

In the third proposition of The Measurement of the Circle, Archimedes shows that "for any circle, the circumference exceeds three times its diameter by an amount less than one seventh of the diameter and more than ten parts of seventy-one parts of the diameter'. Thus, if p is the circumference and d is the diameter, Archimedes demonstrates the following:

{3 + ^d511; thus BC > 571 BG 153' from which, if we assume that BG = 153, then we have BC > 571. The segment BG is half of the side of the dodecagon circumscribing the circle. If one draws the line CH bisecting the angle BCG, the segment BH will be half of one side of the 24-sided polygon circumscribing the circle. We can write as before CB =CG =CB + CG BH GH BG '

But

CG2 = BC2 + GB\

so then

CG2 > 5712 + 1532, 5712 = 326,041 and 1532 = 23,409; then and consequently

CG2 > 349,450,

CG>591 + - andBC + CG> 1,162 + -. 8

8

We have then BC_ BH

> >

*'162 + I 153

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'THE MEASUEEMENT OF THE CIRCLE'

21

from which, if one assumes that BH = 153, we have BC> 1,162 + A2 2 2 2 2 ? andCtf = BC +BH > 153 + 1,162 + 8± .

v

But

y

1532 =23,409 and 1.162 + 8-M > 1,350,534

v

from which

y

CH2 > 1,373,943 and CH > 1,172 + ±. 8

If we draw the line C7 bisecting the angle BCH, the segment 57 will be half of one side of the 48-sided polygon circumscribing the circle. We have BC = CH ^BC + CH but

BI

~HI~

BH

'

SC + C77>l,162 + ± + l,172 + i; BC + CH> 2,334 + ^, then nr

^c 57

>

2,334 + 1 4 153

from which, if one assumes that BI = 153, we have BC>2,334 + j andC72 =BI2+BC2 > 1532 +f 2,334 + ^1 ; 4 . . I 4j let C72 > 23,409 + 5,448,723 = 5,472,132 from which C7> 2,339 + - and CB + CI> 4,673 +i. 4 2 If we draw the bisecting line CJ of the angle Bti, the segment BJ will be half of one side of the 96-sided polygon circumscribing the circle. We have

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ROSHDI HASHED

BC = CI = BC + CI

BJ

IJ

BI

4,673 + ^

153

2

'

The perimeter Pj of the 96-sided polygon circumscribing the circle is 2BJ. 96, so we then have PI = 2BJ.96 < 153.96 AB 2BC 4 R 7 3 + 2

FL < 14,688 < 3 + l 7 4,673 + -

AB

b)

Polygons inscribed in a circle

Let BK be the side of the regular hexagon inscribed in the circle of diameter AB and let AL be the line bisecting the angle BAK, where L is the middle of the arc BK, AL cutting BK at point O (cf. the figure of the text p. 37). Let us draw successively AM as the line bisecting BAL, AN as the line bisecting BAM, and AS as the line bisecting BAN. The straight lines AM, AN and AS respectively cut the lines BL, BM and BN at the points P, U and Q. The chord BL is the side of the dodecagon, BM is that of the 24-sided polygon, BN that of the 48-sided polygon and BS that of the 96-sided polygon (regular polygons inscribed in the circle). Let us assume AB = 1,560; then CB = BK = 780 and AK2 =AB2-BK* = 2,433,600 - 608,400 = 1,825,200 from which

AK < 1,351 and AK + AB < 2,911.

In the triangle AKB, we have AK= Afi = AK±AB KO BO KB ' and moreover, the two right-angled triangles AKO and ALB

Studies in the Making of Islamic Science 'THE MEASUREMENT OF THE CIRCLE'

317 23

have an equal acute angle, they are similar and we have (the £=• = ===• equality does not arise) AB AO AL=AK

LB

KO

from which

^ ^ AK+AB < 2,911 LB KB 780 If we assume that LB = 780, we have AL < 2,911. But

from which

AB2 = LB2 +AL2< 7802 + 2,9112 AB2 < 608,400 + 8,473,921 = 9,082,321 ABP > P2, from which 3+ I > P L > ^ _ > ^ > 3 + 1 0 ,

7

but

thus

AB

AB

AB

71'

3 + 1 = 3+10+^^,33 7 71 497' ,

ri, 3+7ii_^-M) sin(SR) " sin(90° - h) sin(90° + TR) ' Later Muslim astronomers also used the sine rule and the tangent rule to solve the problem in essentially the same way. The most popular procedure involving spherical trigonometry was known as the 'method of the z//es'. It is recorded in several works from the ninth to the fifteenth century and simply involves finding the azimuth of the zenith of Mecca on the meridian and then on the local horizon. In Figure 4.8 we draw EZMF perpendicular to the meridian and then determine ZMF = AZ/ and QF = 0', called the modified longitude difference and the modified latitude, respectively. These two quantities are found by two successive applications of the sine rule, as follows. From right triangles Z]\iFP and TQP we have sin(ZMF) = sin(ZMP) sin(TQ) sin(TP) '

i.e. sin A I/ = sin(90° - 0M) sin AL sin 90° and from right triangles FQE and ZMTE we have sin(FQ) = sin(FE) sin(ZMT) ~ sin(ZME) '

i.e. sin' sin 90° = sin(90° - 0 M ) ~ sin(90° - AL') ' Then we determine FZ = A' =

^>A

S

Yi Y O A' N Figure 4.17 An analemma construction for finding the hour-angle from the observed altitude of the sun or any fixed star. The three-dimensional celestial sphere of unit radius is first projected orthogonally into the plane of the meridian SQBN. Then OQ, A'B and SN represent the celestial equator, the day-circle and the horizon, and X' is the projection of X. The altitude circle (arc ZXK in Figure 4.16) is folded about its radius ZO into the plane of the meridian: X moves to Xi such that XiS = h, so that XiYi = sin h. Note also that BY = sin H. The day-circle is then folded about its diameter through B into the plane of the meridian (yielding an arc of a circle radius cos d). X (in Figure 4.16) moves perpendicular to BA' to X2 (in this figure) and 6X2 measures the hour-angle t and BA the half arc of daylight D. Note that triangles BZX' and BYA' are similar. Therefore,

BX' vers t BZ sin H- sin h BA' ~vers~D~BY~ sin// ' whence the standard medieval formula for t(h,H)

Indian sources, is (in modern notation) sin h vers D vers t = vers(D - T) = vers D sin// This could be derived with facility by reducing the three-dimensional problem on the celestial sphere to two dimensions (Figures 4.16 and 4.17). The equivalent modern formula for the hour-angle t can also be derived by such procedures. It is . sin h - sin d sin cos t = COS 0 COS

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Studies in the Making of Islamic Science DAVID A. KING

and is used in a form equivalent to this by later Muslim astronomers (see al-KhaljH below). Several Islamic tables were universal in the sense that they served all terrestrial latitudes. From the ninth century onwards we find descriptions of how to find the time of day or night using an analogue computer such as an astrolabe or using a calculating device such as a sine quadrant. In the first case there is no need to know the formula; in the second case one uses the formula to compute specific examples. Likewise numerous Islamic instruments were devised to be universal, serving all terrestrial latitudes. e AH ibn Amajur also compiled a table of T(h,H) for Baghdad based on an accurate trigonometric formula. Some anonymous prayer-tables for Baghdad are preserved in a thirteenth-century Iraqi zij\ these display, for example, the duration of twilight in addition to the times of the daylight prayers for each day of the year, and are probably another Abbasid production, dating perhaps from the tenth century. Certainly quantitative estimates of the angle of depression of the sun at nightfall and daybreak occur in the zij of the ninth-century astronomer Habash al-Hasib. Isolated tables displaying the altitudes of the sun at the zuhr and casr prayers and the duration of morning and evening twilight occur in several other early medieval Islamic astronomical works, usually of the genre known as zij* Several examples of extensive tables for reckoning time by day for the solar altitude, or for reckoning time of night from altitudes of certain prominent fixed stars, have come to light. All of these tables were computed for a specific locality, and display either T(h,H) or T(h9 X), where X is the solar longitude. To use any of them, one needed an instrument, such as an astrolabe, to measure celestial altitudes or the passage of time. There is no evidence that these early tables were widely used. Of particular interest was the development in the ninth and tenth centuries of auxiliary trigonometric tables for facilitating the solution of problems of spherical astronomy, though not especially those of timekeeping. The auxiliary tables of Habash (see above) and Abu Nasr (Jl. Central Asia, c. 1000) are the most impressive of these from a mathematical point of view, and al-KhaliU's universal tables for timekeeping (see below) should be considered in the light of these earlier developments. THE INSTITUTION OF THE MUWAQQIT In practice, at least before the thirteenth century, the regulation of the prayer-times was the duty of the muezzin (Arabic, mu'adhdhin). These individuals were appointed for the excellence of their voices and their character, and they needed to be proficient in the rudiments of folk astronomy. They needed to know the shadows at the zuhr and the easr for 176

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each month, and which lunar mansion was rising at daybreak and setting at nightfall, information which was conveniently expressed in the form of mnemonics; they did not need astronomical tables or instruments. The necessary techniques are outlined in the chapters on prayer in the books of sacred law and the qualifications of the muezzin are sometimes detailed in works on public order (hisba or ihtisab). In the thirteenth century there occurred a new development, the origins of which are obscure. In Egypt at that time we find the first mention of the muwaqqit, a professional astronomer associated with a religious institution, whose primary responsibility was the regulation of the times of prayer. Simultaneously, there appeared astronomers with the epithet mlqati who specialized in spherical astronomy and astronomical timekeeping, but who were not necessarily associated with any religious institution. TIMEKEEPING IN MAMLUK EGYPT In Cairo in the late thirteenth century, a mlqati named Abu 'All al-Marrakushl compiled a compendium of spherical astronomy and instruments from earlier sources which was to set the tone of "Urn al-mlqat for several centuries. His treatise, appropriately entitled Jam? al-mabad? wa-l-ghayat fi eilm al-mlqat, (An A to Z of Astronomical Timekeeping), was first studied by the Sedillot pere et fils in the nineteenth century. Al-Marrakushl's contemporary, Shihab al-Dln al-MaqsI, compiled a set of tables displaying the time since sunrise as a function of solar altitude and longitude for the latitude of Cairo (apparently based on an earlier, perhaps less extensive, set by the tenth-century astronomer Ibn Yunus). In the fourteenth century these tables were expanded and developed into a corpus covering some 200 manuscript folios and containing over 30,000 entries. The Cairo corpus of tables for timekeeping was used for several centuries and survives in numerous copies, no two of which contain the same tables. Besides tables displaying the time since sunrise, the hour-angle (time remaining until midday) and the solar azimuth for each degree of solar longitude, which with about 30,000 entries make up the bulk of the corpus (Plate 4.16), there are others displaying the solar altitude and hour-angle at the easr, the solar altitude and hour-angle when the sun is in the direction of the qibla (see section (a)), and the duration of morning and evening twilight. In some late copies of the Cairo corpus there are tables for regulating the time when the lamps on minarets during Ramadan should be extinguished and when the muezzin should pronounce a blessing on the Prophet Muhammad. In some copies, early and late, there is a table for orienting 177

Studies in the Making of Islamic Science DAVID A. KING

Plate 4.16 An extract from the Cairo corpus of tables for timekeeping. This particular sub-table displays values of three functions - the hourangle, time since sunrise and azimuth — for each degree of solar longitude when the sun has altitude above the horizon 15°. Taken from MS Cairo Dar al-Kutub mlqat 690, fols 15v-16r, with kind permission of the Director of the Egyptian National Library

the large ventilators which throughout the medieval period were a prominent feature of the Cairo skyline. These were aligned with the roughly orthogonal street plan of the medieval city, itself astronomically aligned towards winter sunrise. Al-MaqsI also compiled an extensive treatise on sundial theory, including tables of co-ordinates for making the curves on horizontal sundials for different latitudes and vertical sundials at any inclination to the local meridian for the latitude of Cairo. The latter were particularly useful for constructing sundials on the walls of mosques in Cairo, and the special curves for the zuhr and *asr enabled the faithful to see how much time remained until the muezzin would announce the call to prayer. A contemporary of al-Marrakushi and al-MaqsT, the Cairo astronomer Najrn al-DIn, compiled a table of timekeeping which would work for any latitude and which could be used for the sun by day and the stars by night. 178

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The function tabulated is T(h, H, D), where D was the half arc of visibility of the celestial body in question over the horizon, and the entries number over a quarter of a million. This table was not widely used (if at all) and is known from a unique copy, perhaps in the hand of the compiler. Another region of the Islamic world in which the writings of alMarrakushi and the output of the early Cairo muwaqqits were influential was the Yemen. Under the Rasulids mathematical astronomy was patronized and practised. In particular the Sultan al-Ashraf (reg. 1295-6) compiled a treatise on instrumentation inspired by that of al-Marrakushl. The Yemeni astronomer Abu al-'Uqul, who worked for the Sultan alMu'ayyad in Taiz, compiled a corpus of tables for timekeeping by day and night which is the largest such corpus compiled by any Muslim astronomer, containing over 100,000 entries. In Cairo in the fourteenth century there were several muwaqqits producing works of scientific merit, but the major scene of e/7m al-miqat during this century was Syria. TIMEKEEPING IN FOURTEENTH-CENTURY SYRIA The Aleppo astronomer Ibn al-Sarraj, who is known to have visited Egypt, devised a series of universal astrolabes and special quadrants and trigonometric grids, all for the purpose of timekeeping: his works represent the culmination of the Islamic achievement in astronomical instrumentation. Two other major Syrian astronomers, al-MizzT and Ibn al-Shatir, studied astronomy in Egypt. Al-MizzI returned to Syria and compiled a set of hourangle tables and prayer-tables for Damascus modelled after the Cairo corpus. Ibn al-Shatir compiled some prayer-tables for an unspecified locality, probably the new Mamluk city of Tripoli. Al-MizzT also compiled various treatises on instruments, but Ibn al-Shatir turned his attention to theoretical astronomy and planetary models. This notwithstanding, he also devised the most splendid sundial known from the Islamic Middle Ages. It was a colleague of al-MizzI and Ibn al-Shatir named Shams al-DIn alKhalill who made the most significant advances in *ilm al-miqat. Al-Khalfll recomputed the tables of al-MizzI for the new parameters (local latitude and obliquity of the ecliptic) derived by Ibn al-Shatir (Plate 4.17). His corpus of tables for timekeeping by the sun and regulating the times of prayer for Damascus was used there until the nineteenth century. He tabulated the following functions for each degree of solar longitude A: the solar meridian altitude; half the diurnal arc; the number of hours of daylight; the solar altitude at the beginning of the easr; the hour-angle at the beginning of the *asr\ the time between the beginning of the 'asr and sunset; the time between midday and the end of the *asr\ the duration of night; the duration of 179

Studies in the Making of Islamic Science DAVID A. KING

Plate 4.17 An extract from the prayer-tables for Damascus prepared by al-KhalilT. This particular sub-table serves solar longitudes in Aquarius and Scorpio, and the twelve functions are tabulated for each degree of longitude across the double-page. Taken from MS Paris B.N. ar. 2558, fols 10v-llr, with kind permission of the Director of the Bibliotheque Nationale

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evening twilight; the duration of darkness (from nightfall to daybreak); the duration of morning twilight; and the time remaining until midday from the moment when the sun is in the same direction as Mecca. Entries for all but the third function are in equatorial degrees and minutes (where 1° corresponds to 4 minutes of time). These tables contain 2,160 entries. AlKhalHl also tabulated the hour-angle t as a function of solar altitude h and solar longitude X for the latitude of Damascus. His tables of t(h, X) contain about 10,000 entries. In addition, al-KhalUT compiled some tables of auxiliary trigonometric functions for any latitude considerably more useful than the earlier tables of this kind by Habash (see above). The functions tabulated are *Sin0

ft* A\ /(,h) -/(0,6)], h). These tables serve to solve numerically any problem which can, in modern terms, be solved by means of the spherical cosine formula. Al-Khafill also compiled a table displaying the qibla or local direction of Mecca as a function of terrestrial longitude and latitude. He appears to have used the universal auxiliary tables to compile this qibla table. Some of the activities of the Damascus school became known in Tunis in the fourteenth and fifteenth centuries. Extensive auxiliary tables and prayer tables for the latitude of Tunis were compiled there by astronomers whose names are not known to us. Prayer-tables were also prepared for various latitudes in the Maghrib.

181

Studies in the Making of Islamic Science DAVID A. KING TIMEKEEPING IN OTTOMAN TURKEY More significant was the influence of the Cairo and Damascus schools on the development of e/7m al-mlqat in Ottoman Turkey. The Damascus astronomers of the fourteenth century had already prepared a set of prayertables for the latitude of Istanbul, but several new sets of tables were prepared by Ottoman astronomers for Istanbul and elsewhere in Turkey after

Plate 4.18 An extract from the prayer-tables for Istanbul prepared by Darendell. This sub-table serves the two zodiacal signs Aries and Virgo. Note that the entries are written in Indian numerals, rather than the alphanumerical (abjad) notation which was more usual for astronomical tables even under the Ottomans. Taken from MS Cairo Tal'at mlqat turki 29, fol. 44r, with kind permission of the Director of the Egyptian National Library

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the model of the corpuses for Cairo and Damascus. Prayer-tables for Istanbul are contained in the very popular almanac of the fifteenth-century Sufi Shaykh Vefa and in the less widely distributed almanac of the sixteenth-century scholar Darendell (Plate 4.18). The latter displays the lengths of daylight and night, as well as the times (expressed in the Turkish convention, see below) of midday, the first and second *asr9 nightfall and daybreak, the moment when the sun is in the qibla and a morning institution called the zahve (related to the duha, see above). These two sets of tables remained in use until the nineteenth century. Large sets of tables for timekeeping by the sun and/or stars were prepared for Istanbul and for Edirne. One set for the sun was compiled by TaqI al-DTn ibn Ma'ruf, director of the short-lived Istanbul Observatory in the late sixteenth century. In the eighteenth century the architect Salih Efendl produced an enormous corpus of tables for timekeeping which was also very popular amongst the muwaqqits of Istanbul. A feature distinguishing some of these Ottoman tables from the earlier Egyptian and Syrian tables is that values of the time of day are based on the convention that sunset is 12 o'clock. This convention, inspired by the fact that the Islamic day begins at sunset (because the calendar is lunar and the months begin with the sighting of the crescent shortly after sunset), has the disadvantage that clocks registering 'Turkish' time need to be adjusted by a few minutes every few days. Prayer-tables based on this convention were compiled all over the Ottoman Empire and beyond: examples have been found in the manuscript sources for localities as far apart as Algiers and Yarkand and Crete and Sanaa. In the late Mamluk and Ottoman periods the muwaqqits compiled numerous treatises on the formulae for timekeeping and the procedures for computing the time of day or night, or the prayer-times, using either an almucantar quadrant (modified from the astrolabe) or a sine quadrant. MODERN TABLES FOR THE PRAYER TIMES In the nineteenth and twentieth centuries, the times of prayer have been or still are tabulated in annual almanacs, wall-calendars and pocket-diaries, and the times for each day are listed in newspapers. In Ramadan, special sets for the whole month are distributed. These are called imsakiyas, and indicate in addition to the times of prayer, the time of the early morning meal called the suhur and the time shortly before daybreak when the feast should begin, called the imsak. Modern tables are prepared either by the local surveying department or observatory or by some other agency enjoying the approval of the religious authorities; usually they display the times of the five prayers and sunrise. Recently, electronic clocks and watches have 183

Studies in the Making of Islamic Science DAVID A. KING

appeared on the market which are programmed to beep at the prayer-times for different localities, and to pronounce a recorded prayer-call. FURTHER READING On the prayers in Islam see the article 'Salat9 in the Encyclopaedia of Islam (2nd edn, Leiden, 1960 onwards). For an overview of Islamic timekeeping see the article 'Mikaf in the Encyclopaedia of Islam, reprinted in King (1993). On the definitions of the times of prayer as they appear in the astronomical sources, see Wiedemann and Frank (1926). For al-BIrunl's discussion, see Kennedy et al. (1983: 299-310). On the origin of these definitions see King, 'On the Times of Prayer in Islam', to appear. On the procedures advocated by the legal scholars and in treatises on folk astronomy see King (1987a). On the formulae for timekeeping used by the Muslim astronomers see the papers by Davidian, Nadir and Goldstein reprinted in Kennedy et al. (1983: 274-96) and the studies listed below. On solutions (i.e. tables and instruments) serving all latitudes see King (1987c, 1988, 1993). On the earliest known tables for regulating the prayer times and reckoning time of day from solar altitude, see King (1983d: esp. 7-11). On alMarrakushl and his treatise see the section 'Gnomonics' in this chapter and also King (1983c: esp. 539-40 and 534-5). On the institution of the professional mosque timekeepers see King, 'On the role of the Muezzin and the Muwaqqit in medieval Islamic society', to appear in S. Livesey and J. F. Ragep, eds., Proceedings of the Conference 'Science and Cultural Exchange in the Premodern World' in Honor of A. I. Sabra, University of Oklahoma, Norman, Ok., Feb. 25-27, 1993, Leiden: E. J, Brill, 1995. On the corpuses of tables for Cairo, Taiz, Damascus and Jerusalem, Tunis and Istanbul, see respectively, King (1973a; 1979: esp. 63; 1976), King and Kennedy (1982: esp. 8-9) and King (1977a). Each of these papers is reprinted in King (1987b). On the auxiliary tables of Habash, Abu Nasr and al-KhalHl see, respectively, Irani (1956), Jensen (1971) and King (1973b). For an analysis of all available tables see King, Studies in Astronomical Timekeeping in Islam, I: A Survey of Tables for Reckoning Time by the Sun and Stars, and //: A Survey of Tables for Regulating the Times of Prayer (forthcoming). On the Ottoman convention of reckoning sunset as 12 o'clock, see Wurschmidt (1917). On the muvakkithanes, the buildings adjacent to the major Ottoman mosques which were used by the muwaqqits, see Unver (1975). 184

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Studies in the Making of Islamic Science BIBLIOGRAPHY Barmore, F. E. (1985) Turkish mosque orientation and the secular variation of the magnetic declination', Journal of Near Eastern Studies 44: 81-98. Bel, A. (1905) Trouvailles archeologiques a Tlemcen: Un cadran solaire arabe', Revue Africaine 49: 228-31. Berggren, J. L. (1980) 'A comparison of four analemmas for determining the azimuth of the qibla', Journal for the History of Arabic Science 4: 60-80. (1981) 'On al-Bfrum's method of the zljes for the qibla', Proceedings of the XVIth International Congress for the History of Science, Bucharest, pp. 237-45. (1985) The origins of al-BIrunl's method of the zljes in the theory of sundials', Centaurus 28: 1-16. Bonine, M. E. (1990) The sacred direction and city structure: A preliminary analysis of the Islamic cities of Morocco', Muqarnas 7: 50-72. Brice, W., Imber, C. and Lorch, R. (1976) The Da'ire-yi Mu'addal of Seydl 'All Re Is', Seminar on Early Islamic Science\ 1, Manchester. Dallal, A. (1995b) 'Ibn al-Haytham's universal solution for finding the direction of the qibla by calculation', Arabic Sciences and Philosophy 5: 145-93. Dizer, M. (1977) The Da'irat al-Mueaddal in the Kandilli Observatory ...', Journal for the History of Arabic Science 1: 257-62. Garbers, Karl (1936) 'Ein Werk Thabit b. Qurra's iiber ebene Sonnenuhren', Quellen und Studien zur Geschichte der Mathematik, Astronomic und Physik A (4): 1-80. See also Thabit, Treatise 9. Hawkins, G. S. and King, D. A. (1982) 'On the orientation of the Ka'ba', Journal of the History of Astronomy 13: 102-9. Irani, R. A. K. (1956) The Jadwal al-Taqwlm of Habash al-Hasib', unpublished Master's dissertation, American University of Beirut. Janin, L. (1971) 'Le cadran solaire de la Mosquee Umayyade a Damas', Centaurus 16: 285-98; reprinted in E. S. Kennedy and I. Ghanem (eds) (1976) The Life and Work of Ibn al-Shatir: An Arab Astronomer of the Fourteenth Century, Aleppo, pp. 107-21. (1977) 'Quelques aspects recents de la gnomonique tunisienne', Revue de I'Occident Musulman et de la Mediterranee 24: 202-21. Janin, L. and King, D. A. (1977) 'Ibn al-Shatir's Sanduq al-Yawaqit: an astronomical Compendium', Journal for the History of Arabic Science 1: 187-256; reprinted in King, D. A. (1987b) Islamic Astronomical Instruments, London, Variorum Reprints, XII. (1978) 'Le cadran solaire de la Mosquee d'Ibn Tulun au Caire', Journal for the History of Arabic Science, 2: 331-57; King, D. A. (1987b) Islamic Astronomical Instruments, London, Variorum Reprints, XVI. Jensen, C. (1971) 'Abu Nasr's approach to spherical trigonometry as developed in his treatise The Tables of Minutes', Centaurus 16: 1-19. Kennedy, Edward S. (1973) A Commentary upon Blruni's Kitab Tahdid al-Amakin: An llth Century Treatise on Mathematical Geography, Beirut. Kennedy, E. S. and Id, Y. (1974) 'A letter of al-BIrunl: Habash al-Hasib's analemma for the qibla9, Historia Mathematica 1: 3-11; reprinted in E. S. Kennedy et #/.(1983) Studies in the Islamic Exact Sciences, Beirut, pp. 621-9.

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Studies in the Making of Islamic Science BIBLIOGRAPHY Kennedy, E. S. et al. (1983) Studies in the Islamic Exact Sciences, Beirut. King, D. A. (1973a) 'Ibn Yunus' Very useful tables for reckoning time by the sun', Archive for History of Exact Sciences 10: 342-94. (1973b) 'Al-KhamTs auxiliary tables for solving problems of spherical astronomy', Journal for the History of Astronomy 4: 99-100; reprinted in King, D. A. (1986b), XI. (1975) 'Al-KhaliK's qibla table', Journal of Near Eastern Studies 34, 81-122; reprinted in King, D. A. (1986b), XIII. (1976) * Astronomical timekeeping in fourteenth-century Syria', Proceedings of the First International Symposium for the History of Arabic Science, Aleppo, II, pp. 75-84 and plates. (1977a) * Astronomical timekeeping in Ottoman Turkey', in M. Dizer (ed.) (1980) Proceedings of the International Symposium on the Observatories in Islam, Istanbul 1977, Istanbul, pp. 245-69. (1977b) 'A fourteenth-century Tunisian sundial for regulating the times of Muslim prayer', in W. Saltzer and Y. Maeyama (eds) Prismata: Festschrift fur Willy Hartner, Wiesbaden, pp. 187-202; reprinted in King, D. A. (1987b), XVIII. (1978a) 'Three sundials from Islamic Andalusia', Journal for the History of Arabic Science 2: 358-92; reprinted in King, D. A. (1987b) XV. (1978b) 'Some medieval values of the qibla at Cordova', Journal for the History of Arabic Science 2: 370-87; reprinted in King, D. A. (1987b) XV. (1979) 'Mathematical astronomy in medieval Yemen', Arabian Studies 5: 61-5. (1983a) 'Astronomical alignments in medieval Islamic religious architecture', Annals of the New York Academy of Sciences 385: 303-12. (1983b) 'Al-Bazdawi on the qibla in early Islamic Transoxiana', Journal for the History of Arabic Science 7: 3-38. (1983c) 'The astronomy of the Mamluks', Isis 74: 531-55; reprinted in King, D. A. (1986b), III. (1983d) 'Al-Khwarizml and new trends in mathematical astronomy in the ninth century', Occasional Papers on the Near East, New York University, Hagop Kevorkian Center for Near Eastern Studies, 2. (1984) 'Architecture and astronomy: the ventilators of Cairo and their secrets', Journal of the American Oriental Society 104: 97-133. (1985b) 'The sacred direction in medieval Islam: a study of the interaction of science and religion in the Middle Ages', Interdisciplinary Science Reviews 10: 315-28. (1986a) 'The earliest mathematical methods and tables for finding the direction of Mecca', Zeitschrift fur Geschichte der Arabisch-Islamischen Wissenschaften 3: 82-146; with corrections, 4 (1987). (1987a) 'A survey of medieval Islamic shadow schemes for simple timereckoning', Zeitschrift fur Geschichte der Arabisch-Islamischen Wissenschaften 4. (1987b) Islamic Mathematical Instruments, London, Variorum Reprints.

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Studies in the Making of Islamic Science BIBLIOGRAPHY (1987c) 'Universal solutions in Islamic astronomy', in J. L. Berggren and B. R. Goldstein (eds) From Ancient Omens to Statistical Mechanics. Essays on the Exact Sciences Presented to Asger Aaboe, Copenhagen, pp. 121-32. (1988) 'Universal solutions to problems of spherical astronomy from Mamluk Egypt and Syria', in F. Kazemi and R. B. McChesney (eds) Islam and Society: Arabic and Islamic Studies in Honor of Bayly Winder, New York/London, pp. 153-84. (1993) Astronomy in the Service of Islam, Aldershot: Variorum. King, D. A. and Kennedy, E. S. (1982) 'Indian astronomy in fourteenth-century Fez', Journal for the History of Arabic Science 6: 3-45. Lorch, R. P. (1980) 'The qibla-table attributed to al-Khazinl', Journal for the History of Arabic Science 4: 259-64. (1982) 'Nasr b. 'Abdallah's instrument for finding the qibla\ Journal for the History of Arabic Science 6: 125-31. Luckey, P. (1937-8) Thabit b. Qurra's Buch iiber die ebenen Sonnenuhren', Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik (4): 95-148. Michel, H. and Ben-Eli, A. (1965) 'Uncadran solaireremarquable', Cielet Terrell. Rosenfeld, B. A. (1983) Muhammad ibn Musa al-Khorezmi, Moscow. Schoy, Karl (1921) 'Abhandlung des al-Hasan ibn al-Hasan ibn al-Haitham (Alhazen) iiber die Bestimmung der Richtung der Qibla9, Zeitschrift der Deutschen Morgenldndischen Gesellschaft 75: 242-53. (1922) 'Abhandlung von al-Fadl b. Hatim al-NayrizI iiber die Richtung der qibla\ Sitzungsberichte der math.-phys. Klasse der Bayerischen Akademie der Wissenschaften zu Munchen: 55-68. (1923) 'Gnomonik der Araber', in E. von Bassermann-Jordan, (ed.) Die Geschichte der Zeitmessung und der Uhren, vol. IF, Berlin/Leipzig. (1924) 'Sonnenuhren der spatarabischen Astronomic', Isis 6: 332-60. Sedillot, J.-J. (1834-5) Traite des Instruments Astronomiques des Arabes compose au treizieme siecle par Aboul Hhassan All du Maroc ..., 2 vols, Paris; reprinted, Frankfurt, 1984. Unver, A. S. (1975) 'Osmanli Tiirkler'inde ilim Tarihinde Muvakkithaneler', Atatiirk Konferenslan 5: 217-57. Wiedemann, E. and Frank, J. (1926) 'Die Gebetszeiten im Islam', Sitzungsberichte der phys.-med. Sozietdt zu Erlangen 58: 1-32; reprinted in E. Wiedemann (1970), Aufsatze zur arabischen Wissenschaftsgeschichte, 2 vols, Hildesheim, II, pp. 757-88. Wurschmidt, J. (1917) 'Die Zeitrechnung im Osmanischen Reich', Deutsche optische Wochenschrift: 88-100.

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[15] Too Many Cooks ***

& New Account of the Earliest Muslim Geodetic Measurements a A, King

Table of contents: Introduction, 1: Ibn Yunus* account 2: Al~BMm's account, 3: Hie account of Habash. 4: The account of Yahyl ibn Aktham. 5: Concluding femari(^. Ap|&ndix: Aratoie texts. Bibliography and bibliographical abbreviations. Introduction

The measurement of the length of one degree on the meridian by astronomers cormnissioned by the Abbasid Caliph al-Ma'mun in Baghdad a*. 830 is in om sense wett fcnovra, md in amsiier, (dooied in ote Different versions of the observations ai* iwmled by Ibn Ytttius $> Cairo m. 990) and iJ^Binirir^ Centra! Asia mf 1025), as well as by varioos later writers; Piawiciilarly those versions in which two grcwps of astronomers laden with instruments are reported heading off in opposite directions along a meridian in the middle of the desert simply boggle the mind, There is a substantial secondary literature on these reports, including important contributions by C A, HMliw, S, H, Baiam, A* Sipli, 1/S,

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Kennedy, R. P. Mercier, and F. JL Ragep.1 In this paper these well-known accounts will be discussed, but only briefly, for my main purpose is to present a new contemporaneous account of the measurements. The earliest Muslim latitude and longitude measurements in Mecca and Baghdad, also commissioned by al-Ma'mim, are less well known, and they too are clouded in obscurity* Until recently they were known only from passing remarks by al~BIranT,2 and it was Aydm Sayili who first drew attention to that scholar's brief mention of the simultaneous lunar eclipse observations in the two cities.3 It was clear that the main purpose of the measurements was to derive the qibla at Baghdad, but details were lacking.4 In 1985 Dr, Y, Trvi Langermann of the Hebrew University in Jerusalem published a newly-discovered treatise on both sets of observations by the contemporary astronomer Habash,5 and I have come across a different

r

See Naliino, Serini, V, pp. 408-457; Barani, "Muslim Geodesy'*; Sayih, The Observatory in Islam, pp. 85-86; Kennedy, al-BfrOnr$ Tahdtd, pp, 131-136; and Che recent critical survey of Ac evidence in Mercier, "Muslim Geodesy", pp. 178-18L On Ragep's astounding findings see n. 67 below. See also the forthcoming volumes of Sezgin, GAS, relating 10 geography, The various places mentioned in this article are shown on a map in Mercier, * Muslim Geodesy *t p. ISO.

2

On al-BMnl see the article by E. S* Kennedy in DSB, His treatise on mathematical geography has been published by P. Bulgakov, translated by J. All, and analyzed by E, S. Kennedy - see the bibliography for details,

* Sayili, The Observatory in hUm, p. 85. 4

For a general survey of the qibla problem see King, "Sacred Direction", and the articles "Kibla. IL Astronomical aspects* and "Makka. iv. As centre of the world** in E/2, reprinted in King, Studies, C-IX and €-X, as well as King & Lorch, *Qibla Charts". See also n. 60 below on non-mathematical methods that were widely used in medieval Islam.

5

On Habash see the articles by W. Hartner in E? and S, Tekeii in DSB, and the recent study of his iff in Debarnot, **Haba$h*s Zy (Istanbul MS)*» (The later recension extant in a Berlin manuscript is currently being investigated by Dr. Benno van DaSen (Utrecht and Frankfurt).) On the title of his treatise, see Sezgin, GAS, V, p. 276, no. 5, where no manuscripts are listed: the treatise is now published in Langermann, "Habash on Distances".

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account by Yahya ibn Aktham,* the judge appointed by al~Ma'mun to oversee the observations. In this paper I shall compare both of these reports in the light of the later accounts of Ibn Yunus and al-BIrum, Mercier's recent overview of the evidence available to Mm, which included the treatise of Habash, draws attention to the obvious inconsistencies in the sources. From the outset it must be borne in mind that this newlydiscovered report of Yahya ibn Aktham raises yet more questions and casts but little light on the confusion. In the sequel I use the following notation freely: q $Wa(IoeaI direction of Mecca, measured from the local meridian) L terrestrial longitude AL longitude difference from Mecca ALB longitude difference between Baghdad and Mecca A B-VE, pp, 250-252, It is now being studied by Margarita Casiells of Barcelona,

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Later in the same treatise al-Biruni again cites the same work of Habash:29 The observations (commissioned) by al~Ma'mun (took place) only when he (had) read in the books of the Greeks that the equivalent (hissd) of one degree was five hundred stades, which is the unit of theirs that they used for measuring distances, and when he did not find amongst the translators sufficient knowledge about its length compared with other known (units). At that time - according to what Habash related on the authority of Khalid al-Marwarrudh! and a group of scholars of (instrument) construction and expert constructors from amongst the carpenters and brass-workers ~ (al-Ma'mun) ordered the construction of the instruments and the selection of a place for this survey. There was chosen a location in the desert of Sitijar in the area (min hud&d) of Mosul, nineteen^mi&te from the city (min qasabatihi) and forty-three farsdkhs from Samarra, where they were satisfied that (the ground was) level. They transported the instruments there and they selected a place where they observed the solar altitude at midday* Then two groups set forth (in two different directions)* Khalid and a group of surveyors and instrument-makers (al-mass&h wa-'l-sunntf) headed in the direction of the northern (celestial) pole, and CA1T ibn clsa al-Asturlab! and Ahmad ibn al-Buhturi the surveyor with a(nother) group towards the south (celestial) pole. Each of the two groups observed the altitude of the sun at midday until they found that it had changed by one degree, apart from the change that resulted in the solar declination (siwa *l-taghayyur al~ h&dith mina 'l-mayt). They measured the track on their way out {/T dhihabihim) and set up markers (sih&m, lit,, arrows) as they went, and as they returned they investigated (Ftabaru) the distance (al-mis&ha) for the second time* The two groups met again at the place from which they had set out, and they found that one degree of the terrestrial meridian is equivalent to fifty-six miles* (Habash) claimed that he had heard Khalid dictating that number to the Qadf Yahya ibn Aktham, so (Habash himself) heard it from the horse's mouth tfa~ 'Itaqatahu minhu mmcfan). 29

al~BlWnfs Tahdfd: text, pp< 213-214; translation, pp, 178-179; and comincntaryt p. 133* Sec also Barani* "Muslim Geodesy*, pp* 11-14,

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Abu Hamid al-Sagharii, who heard it from Thabit ibn Qurra, told me the same thing. (On the other hand) it is related of al-Fargharil (that he reported) twothirds of a mile in addition to the (56) miles mentioned above. Similarly I found all of the records confirming these (additional) two-thirds, and I may not attribute that to their having dropped out of the manuscript of the Kitab al-Abc8d wa- 'l~ajram because Habash derived from that (value) the circumference and diameter of the earth and all of the (planetary) distances. When I investigated (these) I found that they result from fiftysix miles only {that is, without the two-thirds}. It is preferable to imagine that (these different results for the length of one degree) derive from two accounts by two teams. (Certainly) this is a subject surrounded with confusion, which should inspire renewed investigations and observations. Who is prepared to help me with this (project)? ... ,„. Elsewhere al-BTrum deals in general terms with the problem of determining longitude differences from simultaneous observations of a lunar eclipse,30 but does not mention the Abbasid observations again,

Commentary Mansur ibn Talha was a philosopher, mathematician, astronomer and music theorist, who died ca* 855.3I Khalid al-MarwarrudhT was one of the leading astronomers of al-Ma*mun: he apparently authored a zfj that has been lost without a trace?2 Of particular interest in this account are the coordinates of the locations in the plain of Sinjar where the two groups of observers set forth in two different directions. This information does not stem from the treatise of

30

al~BrrM's Talutid: see text, pp, 166-169, translation, pp, 129-132, and commentary, pp. KXMOU ^)d my review of the last in Ctntaurus 19 (1974), pp, 320-323, especiaJly p. 322, See also Mercier, ^Muslim Geodesy*, p. 176.

31

Sezgin, GAS, V, p. 245, and VI, p, 145.

* Ibid., V, p. 244, and VI, p, 139.

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Habash as it has come down to us - see the next section. What is disappointing is that there is no record of two different results - such as one would expect from any scientific mission - which could then have been compared and used to produce a "final" result. Rather, in this case, the "final" result is presented as a fait accompli, Having briefly discussed these two later accounts, both of which have been known to previous researchers, we are now in a position to investigate the more original accounts „ 3. The account of Habash The first two parts of the treatise of Habash, preserved in a precious I7th(?)-century astronomical compendium given to Rabbi Josef Kafah by a Muslim Yemeni colleague before he left Sanaa for Jerusalem,3* provide considerably more information - see Arabic text no. L The text translates:34 In the Name of God the Merciful and Compassionate, from whom we seek help. The Book of Sizes and Distances by Habash ibn cAbdallah the astronomer (al-hasib). He said that the Commander of the Faithful, al-Ma'mun, wanted to know the size of the earth, so he made some investigations about this and found that Ptolemy had stated in one of his books that the circumference of the earth was so many thousand stades. Thereupon he asked the interpreters about the meaning of stades and they gave different interpretations, [He said of their interpretatioiB:] "They do not dispense with (?) what we wanted to know/1 {The text is corrupt here,}

J>

This is yet additional proof of the importance of Yemeni manuscripts for the history of Islamic astronomy. See King, Astronomy in the Yemen, for a first look ac over !00 Yemeni astronomical manuscripts, now supplemented by Langennann, "Bibliographical Notes*, based on Jewish sources.

M

Langermann, *Habash on Distances'*, pp. 115-116 and 122-123* Some problems in the published text have been fixed, and the translation is my own, I am grateful to Dr. Langermann for photocopies of the relevant sections of the original manuscript.

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(Al-Ma'mun) therefore sent out KMlid ibn cAbd al~MaIik aI~Marwamidht, CA1T ibn *Isa al-Asturlabl ami Ahmad ibn al-Butotun the surveyor together with a group of surveyors and craftsmen including carpenters and brass-workers, to make correctly the instruments which they would need, transporting all of them to a place which they selected in the desert of Sinjar. Next Khalid and one group headed towards the North Pole of the Banat Nacsh (— Ursa Minor), and CA1T and Ahmad and a(nother) group headed towards the South (celestial) Pole. They {i.e., the former group ?} continued until they finally found that the (solar) meridian altitude had decreased and changed from what they had found in the place where they had separated, by the amount of one degree, after they had subtracted from that (the change in) solar declination during the time it took (for them) to cover that distance. (As they went) they set up markers (siham), and then they retraced their steps using the markers and checked the distance a second time. They found the amount of one degree of the earth's surface to be fifty-six miles, where one mile is four thousand black cubits, this being the cubit established by al-Ma'mun for measuring clothes, surveying buildings and reckoning distances between stations on the pilgrimage road (qismat al~manazit). I heard this (information) which I have mentioned in my book {sc. in this book of mine?} from Khalid ibn cAbd aWVialik al-Marwarrudhl as he was conveying it to the Qftdi Yahya ibn Aktham, Yahya [had beenj ordered [by al-Ma'mfin] {the text appears to be corrupt here} to write down for him {sc, for al-Ma'mun} all that Khalid told him, so he wrote (it) for him. I have written what I heard from KMlid himself. The Commander of the Faithful al-Ma'mun - may God be pleased with him - (also) wanted to measure the azimuth of the qibla* So he sent out (someone) at the time of a lunar eclipse to measure the longitude between Mecca and Baghdad, (That person) found that the meridian of Mecca was west of that of Baghdad by approximately three degrees. If we modify one degree of the celestial equator (the expression al-falak al~mustaqfm, meaning sphaera recta or right sphere, is usually used for

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astronomical ascensiom^ - conipre the teimiiKriogy of Yahya ibn Aktham) we find that the andutfint on the celestial diy-circle corresponding to the parallel of Mecca ^ al^ffiak #1ma$im*al& *&rtf Jtf^ fifty-six minutes. So if we rtmltiply tile distance between the meridiam M Mecca and Baghdad, (i^^ly^:,titeee degrees, by 56 minutes wh^rc nd$^ 12 seconds* This we divide by 11 decrees and 45 minutes, which Is the diflemice between the two latitudes, [the quotient is Q;^(,l,l), We add this to the 11-45 and obtain] for the distance between Baghdad and Mecca 12 degrees and 5 mifiutes. This we multiply by 56 miles and obtain appraxiinately six hundred and seventy-six and two-thitds miles as the distance between Baghdad and Mecca as the crow flies (cala *t~$ahm^ When KhllKi Ibn ^Abw flies {al-sahm offifc^^ of the road was thirty-five miles and oi^ihii^ of a mile. Al-Ma'mun said that this was not to be regarded as excessive because th^e must be (m addition to) the flat 31

See the article ^Maiali^ m &,

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parts, inclines up or down on (the road) amounting to this (difference). Commentary In his Planetary Hypotheses, Ptolemy stated that the circumference of the earth was 180,000 stades (ast&dMyya)?* The determination of terrestrial latitude is best achieved by measuring the altitude of the Pole Star or by measuring the meridian altitudes of the sun. Our text asserts that the second method was used, As John Britton has shown, there are problems inherent in the use of the meridian quadrant which affect the accuracy of the altitude measurements.37 Furthermore, given the problems associated with obtaining a reasonable result from such a crude procedure, it is hardly necessary to make a correction for the change in solar declination during the time the team moved the fifty-odd miles. The report attributed to Yahya ibn Aktham - see below - states that the observations were made at the summer solstice - when the daily change in declination is a minimum - but adds that the declination at that time is zero, which is clearly absurd„ Does the text mean that one can find the meridian by standing perpendicular to the direction of the setting sun at the equinoxes? The information on the so-called "black* cubit is interesting: W. Hinz has stated in one place that this unit was standardized by al~Ma'mun and elsewhere that it was not?8 Now we know from a contemporary source that it was. It is the account of the simultaneous lunar eclipse observations and latitude measurements in Baghdad and Mecca which adds to the information recorded by al-BuCUL The simultaneous observation of a lunar eclipse in two localities is a very sensitive kind of observation even with

M

See Goldstein, "Ptolemy's Planetary Hypotheses", p, 1.

J7

See Britton, "Ptolemy on the Obliquity", for details.

38

See Hinz, Mafte void Gewichte, pp, 60-61, and the same author's article *Eftur§c* in Ef, as well as other sources cited in Barani, "Muslim Geodesy*, pp. 44-45* See also Mercier, "Muslim Geodesy", p> 177, n, 13, pointing to errors made by Hinz,

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sophisticated instruments, and it is particularly difficult to obtain good results when the longitude difference is so small Problems include the determination of the exact moment of a particular eclipse phase and the determination of the time at that moment. An error of 1° in the measurement of the altitude of a star at one end could of itself lead to an error of the same order in the time estimate, and hence in the longitude difference. The details of the necessary observations are recorded by Ibn Yunus and al-BTruriL39 Habash unfortunately gives no information of consequence on the simultaneous eclipse observations; he simply reports that the astronomers measured the distance between the meridian of the two localities as 3°, and that they then "adjusted* this to 2;48°. Indeed, the length of the arc corresponding to a longitude difference of &L on a parallel of latitude through Mecca is AL cos 1/2 * 1/4, as stated in the text Note, however, that the results of al-Ma'iMa's finll survey of the road betwecm Me^ dad as reported here are also at variance wth what was recorded by IJabash. The reader will observe ttet m value is presented for the qihla at Baghdad, Clearly, though, the meitod usM was equivalent to: siiiq - &*^fa-l-'^\£^*

55

4- ^Lco$^)^J .

On Ibp ai-A^fam $ee Se^m, GAS* V, :g. 30^, and VI, pp. 21S41& and on his lost astfdiK)mtcal tables see Ko«ifi Arch^^^ d'Mttomdes Sdmcts 39 (1989), pp. 3-2L f lament Almamic: Daniel M^ Vari$oovM^^ omsf Mwmc Science •«• The Almmc •$•& Yemeni Mum, (llniversity of Wasl^l vMderWaettl^ R* L> van der Waerden, (^^rymdAl^ in Ancient (^itisations, Berlin, arc,: Springer, 1981 Vernet, "T^ml^ Probataew: luan Vernet, ^1^$ Tabulae Probatae", in

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Homenaje a MtlUs-VaUicrasa, II, Barcelona, 1956, pp. 501-522, reprinted in idem, Estudios sobre historia de la ciencia medieval, Barcelona: Universidad de Barcelona & Bellaterra: Univemdad Autonoma de Barcelona, 1979, pp. 191-212, Youschkevitch, Mathematiques arabes: Adolf P. Yooschkevitch, Les mathematigues arabes (VIIF - XV* si£cles)> translated by M. Cazenave and K. Jaouiche, Paris: J. Vrin, 1976« [See also Juschkevitch.]

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Part III Islamic Science to the West

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[16] The influence of Arabic astronomy in the medieval West HENRI HUGONNARD-ROCHE

At the beginning of his Epitome astronomiae Copernicanae, Kepler lists the following components of astronomy, all of which he considers necessary to the science of celestial phenomena (Kepler 1953: 23). The astronomer's task, he says, consists of five main parts: historical, to do with the recording and classification of observations; optical, to do with the shaping of the hypotheses; physical, dealing with the causes underlying hypotheses; arithmetical, concerned with tables and computation; and mechanical, relating to instruments. The first three areas, adds Kepler, involve mainly theory; the last two are more concerned with practical aspects. In each of the areas identified by Kepler, the contribution of Arabic science was essential to the birth and subsequent development of astronomy in the Latin West. Prior to this contribution, there was indeed no astronomy of any advanced level in those countries.1 What was understood by astronomy was scarcely more than a collection of imprecise cosmological ideas concerning the shape and size of the world, and some basic notions about the movements of celestial bodies, principally concerning synodical phenomena, such as heliacal risings and settings. The needs of the Church with regard to the regulation of the calendar had nourished a tradition of chronological calculation following the De temporum ratione of Bede (d. 735). But this literature of computation, with which the names of Raban Maur, Dicuil or Garlande are associated, was not based on any mathematical treatment of the phenomena. A single example will suffice: in Bede, the planetary movements are represented by simple eccentrics, and the second planetary anomaly thus remains unexplained. In short, the science of the heavens in the early Middle Ages lacked observations, geometrical analysis

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of celestial phenomena and reflection on the foundations of hypotheses, in other words, the three areas that Kepler related to astronomical theory. Practical astronomy was no better represented: tables were inexistent and instruments (gnomons, sundials) were very basic. This chapter obviously cannot detail, or even list, all the changes produced in the Latin West by successive translations of Arabic works, nor can it cite all the translations or all the medieval authors who may have been influenced by them.2 We shall omit, among other things, Arabic influence on the development of trigonometry in the West, on instruments and on the Latin catalogues of stars,3 as well as the considerable influence exerted on Latin astrology by treatises such as the Introductorium mains or the De magnis coniunctionibus of Abu Ma'shar (end of the ninth century).4 This chapter will focus instead on the problems of astronomical theory proper, in order to reveal some essential aspects of Arabic influence on the growth and development of this theory in the medieval West. THE ASTROLABE AND THE A S T R O N O M Y OF THE PRIME M O V E R The first evidence of the penetration of Arabic astronomy in the Latin West relates to the stereographic astrolabe. The properties and advantages of stereographic projection, on which this instrument was based, had already been described by Ptolemy in his Planisphere, but this text was not known in the Latin world until the twelfth century, through the translation by Hermann of Dalmatia (1143) of a critical Arabic revision of the text by Maslama al-Majrlti (c. 1000). However, scholars in the north of the Iberian peninsula, who were in contact with Islam, became familiar with this instrument and the treatises relating to it from the end of the tenth century. At this period, the first technical literature appeared in Latin under the names of Gerbert (the future Pope Sylvester II), Llobet of Barcelona and Hermann the Lame. This literature consists of texts describing applications or construction, or construction followed by applications, which are extracts or revisions of earlier Arabic treatises that have still not been clearly identified.5 A new series of translations in the twelfth century, such as the translation of the treatise of Ibn al-Saffar (d. AH 426 (AD 1035)) by Plato of Tivoli (fl. 1134-45), and various original Latin works, such as those of Adelard of Bath (c. 1142-6), Robert of Chester (1147) and Raymond of Marseilles (before 1141), gave the Latin West definitive mastery of the instrument. In addition, the inclusion of the astrolabe in university teaching programmes reinforced the educational role of this instrument until the end of the Middle Ages and ensured the success of the Latin translation by John of

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Seville (fl. 1135-53) of a work attributed to Masha'allah (end of the eighth century). The astrolabe was not only the educational instrument par excellence of the Middle Ages, but also an instrument of calculation, permitting the rapid geometrical solution of the principal problems of spherical astronomy. The astrolabe provided an easy demonstration of the daily and annual motions of the sun and of the combination of their effects, covering right and oblique ascensions, the duration of irregular hours, the heliacal rising of stars and the position of the celestial houses in astrology. Bearing in mind the traditional medieval division of astronomy into two distinct areas - the astronomy of the daily motion of the heavenly vault, on the one hand, i.e. the astronomy of the prime mover, and planetary astronomy, on the other - treatises on the astrolabe obviously dealt only with the first of these. Consequently, they contain few technical data: apart from the positions of stars, these are confined to the obliquity of the ecliptic, the location on the zodiac of the apogee of the sun and the position of the first point of Aries (spring equinox) in the calendar, which is associated with the movement of precession. Raymond of Marseilles's treatise on the astrolabe6 - the oldest Latin text on the subject that is not a pure adaptation from the Arabic contains two tables of stars, one drawn from the treatises of Llobet of Barcelona and Hermann the Lame, and the other derived from al-Zarqallu (d. 1100). Raymond demonstrates a marked enthusiasm for this last author, and also borrows from him the position of the apogee of the sun at 17; 50° of Gemini and the value of the obliquity of the ecliptic estimated as 23; 33,30°, which he prefers to that of Ptolemy (23; 50°). This example already enables us to identify two notable aspects of Arabic influence on Latin astronomy: the major role played by the work of al-Zarqallu and the questioning of Ptolemaic parameters in relation to the sun. THE TOLEDAN TABLES AND P L A N E T A R Y ASTRONOMY By the time the treatise of the astrolabe had reached its definitive form, in the middle of the twelfth century, it was far from being the Latin world's only means of access to technical astronomy. A considerable collection of Arabic texts were translated in the course of that century, which opened up to Latin astronomers a much wider field of study in the form of astronomical tables. This designation covers a huge variety of material, which can be divided schematically into three groups: the first comprises elements relating more or less directly to the astronomy of the prime mover (tables of right and oblique ascensions, of declinations, of the equation of time);

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the second comprises the planetary tables and is made up of four parts (chronological tables, tables of mean co-ordinates, tables of equations and tables of latitudes); the third group consists of disparate tables relating to conjunctions of the sun and moon, eclipses, parallaxes, the visibility of the moon and other planets, etc. Three principal sources served to introduce the Latin astronomers to all these subjects: first, the canons and tables of al-Khwarizml (c. 820), as revised by the Andalusian astronomer Maslama al-Majritl and translated by Adelard of Bath (c. 1126); next, the tables of al-Battanl (d. AH 317 (AD 929)), first translated by Robert of Chester in a text that remains unfound, and then in a version by Plato of Tivoli, of which only the canons have been preserved;7 lastly, the tables of al-Zarqallu, which form the basis of the collection known as the tables of Toledo from their meridian of reference. Translated by Gerard of Cremona (d. 1187), the Toledan tables achieved widespread diffusion throughout the Latin West.8 One of the first Latin authors to use tables of Arabic origin was Raymond of Marseilles. In 1141 he composed a work on the motions of the planets, consisting of tables preceded by canons and an introduction in which he claims to draw on al-Zarqallu. In fact, his tables are an adaptation of those of al-Zarqallu to the Christian calendar and the longitude of Marseille. As in his treatise on the astrolabe, Raymond utilizes the value of 23; 33,30° for the obliquity of the ecliptic, which he took from al-Zarqallu. Furthermore, he is av/are of the proper motion of the apogee of the sun as demonstrated by al-Zarqallu and he reproduces the Arab astronomer's table for the positions of the apogees of the sun and other planets. Appearing some thirty years before the translations of Ptolemy's Almagest and the Toledan tables, by Gerard of Cremona,9 Raymond's work was the first to introduce to the Latin world, through the perspective of a borrowing from al-Zarqallu, the Ptolemaic method of calculating planetary positions (Figure 9.1), which consists in finding the algebraic sum of the mean motion, the equation of the centre and the equation of the argument, correcting the equation of the argument by means of proportional parts. From his study of the tables of al-Zarqallu, Raymond of Marseilles understood, however, the clearly stated notion that astronomical tables demand continual correction. Astronomers throughout the Middle Ages found themselves faced with these corrections and the theoretical problems that they involved, and it was one of the aims of Copernicus finally to establish tables that would be permanently valid. The adaptation of Arabic tables, and particularly the Toledan tables, continued in various parts of the Christian world throughout the twelfth and thirteenth centuries (Millas Vallicrosa 1943-50: 365-94). Thus one can cite tables for the meridian of Pisa compiled around 1145 by Abraham ibn Ezra, tables for the meridian of London in 1149-50 by Robert of Chester 287

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A_

X >

Xj

7

S>

\0i

x

^J

p

x"

E }

yt

0)

'V M

xl

D

T

Figure 9.1 Ptolemaic theory of the motion of the planets in longitude (general case: upper planets and Venus). Medieval nomenclature: T, centre of the earth or the world; D, centre of the deferent; E, centre of the equant (TD = DE); O, centre of the epicycle; P, planet; X, origin of the co-ordinates on the ecliptic (first point of Aries); A, apogee on the ecliptic; o>, longitude of the apogee; ji, mean motion; 7, mean centre; a, mean argument; x, equation of the centre; y, equation of the argument; X, true locus

and in 1178 by Roger of Hereford, and further anonymous tables, for London (1232), Malines, Novara, Cremona, etc. The tables for Toulouse seem to have been particularly well used, notably by Parisian astronomers, because of the proximity of the meridians of Toulouse and Paris. The large number of manuscripts of the Toledan tables dating from the fourteenth and even the fifteenth century testify to their continued use even after the Alphonsine tables had become the preferred source of astronomical reformers in Paris at the beginning of the fourteenth century. The Toledan tables also influenced the almanacs, which were designed not to provide the means of calculating planetary positions but to give the positions themselves. This is the case, for example, in the Almanack compiled for Montpellier for the years 1300 and following by Profatius (d. c. 1307), who himself records his debt to the Toledan tables.10 288

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The Toledan tables are a composite collection, including the parts taken from the tables of al-Zarqallu alongside extracts from al-Khwarizml (notably the planetary latitudes), elements from al-Battani (in particular, the tables of planetary equations), and yet other parts derived from the Almagest or the Handy Tables of Ptolemy, and from the De motu octavae spherae, which was attributed in the Middle Ages to Thabit ibn Qurra.n This diversity of composition means that the Toledan tables do not have a coherent underlying astronomical schema and that certain computations are based on different, incompatible parameters. For example, the tables of differences of ascension are calculated for an obliquity of the ecliptic equal to 23; 51°, the value which appears in the Handy Tables, whilst the table of right ascension is calculated with the value of 23; 35° used by al-Battani. As another example, some of the columns which make up the table of equation of Venus are calculated using two different eccentricities for the planet. The absence of any geometrical analysis of planetary motions in the canons, which are limited to stating the methods of calculation to be applied, must have made it more difficult for the early Latin users of the Toledan tables to be critical of them, and they tacitly accepted the new parameters. The characteristic features of the twelfth- and thirteenth-century Latin tables are therefore identical with those of the Toledan tables, and essentially reflect the modifications applied to Ptolemaic theory by the Arab astronomers of the ninth century. These modifications mainly affected solar parameters, whose definition by Ptolemy had proved very unsatisfactory. Observations made in the East in the ninth century - some 700 years after Ptolemy - had led to different estimations from Ptolemy's12 for the length of the tropical year, for the speed of precessional movement, for the obliquity of the ecliptic (23; 33° according to the astronomers of al-Ma'mun and 23; 35° according to al-Battani, instead of 23; 51,20° in the Almagest), for the eccentricity of the sun (2; 4,45 p for al-Battani, 2; 29,30P for Ptolemy) and for the position of the solar apogee (at 65; 30° from the first point of Aries according to Ptolemy, at 82; 17° according to al-Battani, and 82; 45° according to the De anno solis attributed to Thabit ibn Qurra13). The discovery of these divergences between the results obtained by Ptolemy and their own findings confronted the Arab astronomers with a delicate problem which echoed on until the time of Copernicus: were these divergences due to errors of observation or to long-term variations in the parameters which would therefore indicate the existence of movements not so far observed? Both interpretations were put forward in the ninth century. The first was supported by al-Battani, who did not question the kinematic models of Ptolemy, merely adopting a more rapid precessional movement than Ptolemy's (1° in 66 years rather than 1° in 100 years). The second 289

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interpretation was represented by the author of De motu octavae spherae, who postulated further that the presumed variations of the solar parameters were periodic: to account for this, he imagined a model14 which produced simultaneously a periodic variation in precession, and thus in the length of the tropical year, and a periodic variation of the obliquity of the ecliptic. Briefly, this model consisted of two ecliptics: a fixed ecliptic inclined at 23; 33° to the equator, which it bisects at two points, the first point of Aries and the first point of Libra; these two points are taken as the centres of two small circles described by the first point of Aries and the first point of Libra of a moving ecliptic (but fixed in relation to the stars), which, in turn, bisects the equator at the equinoctial points. When the moving first point of Aries, which is the origin of the sidereal co-ordinates, completes a revolution on its small circle, the vernal point is drawn into an oscillatory motion on the equator. The parameters of the model were chosen to produce the maximum effect (distance between the first point of moving Aries and the vernal point) equal to ± 10; 45°, and the periodicity of the oscillatory movement was 4163.3 Arabic years (4039.2 Christian years). The tables of De motu that correspond to this geometrical model were included without amendment in the Toledan tables, thus ensuring until the end of the thirteenth century the unchallenged success of this theory of the oscillatory motion of the equinoxes, known in medieval times as the motion of accession and recession (accessio and recessio translating the Arabic terms iqbal and tdbar).15 The calculation of planetary motions contained in the Toledan tables is based on three quantities: the mean motion and the two corrections, known as the equation of the centre and the equation of the argument. These two corrections are the translation into computational terms of the irregularities produced by the presence in the Ptolemaic geometrical models of eccentricities and epicycles. They are therefore a function, for each planet, of the eccentricity and of the relation between the radius of the epicycle and the radius of the deferent. It is remarkable that, although the mean co-ordinates given in the Toledan tables (mean motion of the upper planets and mean argument of the lower planets) appear to have been established independently of earlier known tables, the tables of equations are essentially the same as those of al-Battanl and derive from the Handy Tables of Ptolemy. The principal exception to the Ptolemaic origin of the tables of planetary equations is the table of equation of the centre of Venus, which is similar to al-Battanl's but completely different from that of the Handy Tables. The reason is that, in the table of al-Battanl, the centre of the epicycle of Venus was assumed to coincide with the mean sun and thus the eccentricity of Venus had to be the same as that of the sun. This was the concept - generally accepted by Arab astronomers, according to 290

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al-BIrunl (d. 1048) (Toomer 1968: 65) - that was duly adopted by the author of the Toledan tables. With the exception of Venus, therefore, the preservation of equation tables of Ptolemaic origin indicates that the structure of the geometrical planetary models underlying the Toledan tables, and the Latin tables that derived from them, had remained the same since Ptolemy. By contrast, the setting of those models within the reference system of a solar theory associated with the theory of the motion of the fixed stars had involved a complete modification of the Ptolemaic concept. In fact, the Arab astronomers of the ninth century had shown that the position of the solar apogee is variable (in tropical co-ordinates) and had found a value for its movement similar to that for the precessional movement (1° in 66 years). They had therefore assumed that these two movements were identical, i.e. that the solar apogee was fixed, not in relation to the equinox (as Ptolemy had thought) but relative to the sphere of the stars. As a result, the sphere of the stars served as the reference for planetary motions from that time on. Thus, whereas the Ptolemaic tables had been expressed in tropical coordinates, the Toledan tables were expressed in sidereal co-ordinates. It was therefore only after having found the true positions of the planets on the sphere of the fixed stars (the eighth sphere in medieval terms), by algebraic summation of the mean motion and the equations, that the positions on the ninth sphere (or sphere of the stationary ecliptic) could be calculated by adding the equation of the motion of accession and recession, to take account of the motion of 'trepidation' of the stars, and consequently of the planetary apogees, in relation to the vernal point. This procedure, inherited from the Toledan tables, was constantly used in Latin astronomy until the end of the thirteenth century. PLANETARY THEORIES AND THE GEOMETRICAL ANALYSIS OF APPEARANCES Although the astronomical tables could satisfy the practising astronomer by enabling him to find the position of a celestial body in longitude and latitude at any particular moment, they did not provide any direct information in two of the areas defined by Kepler as constituting the theory of astronomy, i.e. the study of hypotheses and of their causes. These two areas of study developed in the Latin West in the thirteenth century, and once again Arabic influence had a considerable part to play in them. The development of this new field of research was made possible by the appearance of a new type of astronomical text, the theoricae planetarum, whose aim was to set forth kinematic models that would represent the celestial motions as faithfully as possible. Instead of the highly technical demonstrations in the 291

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Almagest, Latin astronomers preferred more basic descriptions of the world system according to Ptolemy, as epitomized in two Arabic treatises: the introduction to Ptolemaic astronomy by al-Farghanl, entitled Difference scientie astrorum in the translation of 1137 by John of Seville and Liber de aggregationibus scientiae stellarum in the translation by Gerard of Cremona; and second, an analogous treatise composed by Thabit ibn Qurra (d. AH 288 (AD 901)), also translated by Gerard of Cremona and known as De hits que indigent antequam legatur Almagesti.16 In the same way as these two treatises, the theoricae planetarum of the Latin Middle Ages usually restricted themselves to explaining basic astronomical concepts and the general organization of the circles used to represent planetary motions. A notable example of this approach is the most widely known of all the medieval theoricae, called the Theoricaplanetarum Gerardi,ll whose author is unknown but which probably dates from the beginning of the thirteenth century. The geometrical models described in this Theorica conform to Ptolemaic constructs, with the exception of those concerning the erroneous determination of planetary stations by the tangents and the theory of planetary latitudes. On the second point, two traditions were known in the Middle Ages: the first was represented by the Almagest and followed by al-Battanl and an anonymous translation of the Toledan tables; the other tradition, derived from Indian methods, came into the West via the tables of al-Khwarizml and the translation of the Toledan tables by Gerard of Cremona. Based on a model of the inclinations of the planes of the various circles representing planetary motions which differed from that of Ptolemy, this second method led naturally to different computational procedures from those in the Almagest. These are the procedures discussed in Theorica Gerardi, and that work was largely responsible for their dissemination until the beginning of the fourteenth century, at which time the Alfonsine tables restored the primacy of Ptolemaic methods. A concise example of the medieval theoricae, the Theorica planetarum Gerardi, gave no indication of the parameters of the geometrical constructions, nor of the periods of revolution of their moving elements. A more elaborate theorica, the Theorica planetarum of Campanus of Novara (composed between 1261 and 1264), by contrast, combined a detailed theoretical expose of the Plotemaic kinematics of planetary motions with a description of the appropriate equipment to represent those motions - the first Latin treatise on the equatorium. Included in university programmes during the fourteenth century, the Theorica of Campanus aided the widespread diffusion of ideas drawn from the work of al-Farghanl which was, after Ptolemy, its major source. Like al-Farghanl, Campanus augmented his summary of the Almagest with information concerning the system of celestial spheres: he completed the description of each planetary model with an 292

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evaluation of the dimensions of each part of the model. He was himself the author of astronomical tables for the town of Novara, which were based on the Toledan tables, and he took quite a lot of his parameters from the latter. Thus all the parameters of the planetary apogees were drawn from the Toledan tables, including the solar apogee, which is subject to precessional movement, as for the Arab astronomers. Equally, Campanus adopted the Toledan values for the mean motions of the upper planets and for the mean argument of Mercury, but he used the value from his own Novara tables for the mean argument of Venus. For the distances between station and apogee, he again followed the Toledan tables. Like them also, he adopted Ptolemaic parameters for the eccentricities and the magnitudes of the radii of the epicycles (except in the case of Mars where the difference is probably due to error). With regard to the dimensions of the world, Campanus derived the basic elements of the comparative dimensions of the spheres of the earth, moon and sun from Ptolemy, and adopted the Ptolemaic principle of the contiguity of the celestial spheres which permits the calculation, step by step, of the relative dimensions of the planetary spheres up to Saturn and, from there, to the fixed stars. However, Campanus based all his estimations in absolute values on the evaluation of the length of a terrestrial degree of latitude (56| miles) that he took from al-Farghanl and introduced into the Ptolemaic calculations of basic parameters (the diameters of the earth and sun, the distance from the earth to the sun, etc.). By also using the magnitudes of the planetary bodies provided by al-Farghanl, Campanus was able to calculate the dimensions of all the parts of the world system. To summarize very broadly, we can say that medieval astronomy in the thirteenth century, as exemplified by the Theoricae planetamm of Campanus, was dominated by three major influences: the influence of Ptolemy on the geometrical models and their parameters; the influence of the Toledan tables on the mean co-ordinates of the moving elements in those models; and the influence of al-Farghanl (and through him the influence of Ptolemy's Planetary Hypotheses) on the cosmological constitution of the universe. Within this framework, two principal questions remained: the problem of the motion of the sphere of the stars, merely alluded to by Campanus in a reference to both the Ptolemaic movement of 1 ° in 100 years and the movement of accession and recession (not quantified) attributed to Thabit; and the question of the actual reality of Ptolemy's kinematic models.

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THE PROBLEM OF THE FOUNDATION OF THE HYPOTHESES At the same time as the Latin West discovered, through the theoricae, the Ptolemaic hypotheses implicit in the tables and their canons, they learned, through the translations of Michael Scot (d. c. 1236), of the commentaries of Averroes (d. 1198) in which those hypotheses were strongly criticized.18 Aristotelian physics required that the celestial substance undergo no other movement than the uniform rotation of homocentric spheres. It was therefore easy for Averroes to show the contradictions between this physics and the astronomy of eccentrics and epicycles. Simultaneously with the radical criticism by Averroes, the Latin West acquired Michael Scot's 1217 translation of the De motibus celorum of al-BitrujI (c. 1200), in which the author attempted to reformulate astronomy in accordance with the physics of Aristotle. In principle, the models of al-BitrujI can be seen as a kind of reworking of the homocentric models of Eudoxus - accepted by Aristotle - with the innovation that the inclinations of the axes of the planetary spheres were made variable, the movement of each sphere being governed by that of its pole, which described a small epicycle in the neighbourhood of the pole of the equator. The discovery of these texts initiated a lengthy medieval debate on the foundation of these hypotheses (Duhem 1913-59: 3, pp. 241-498 passim). As early as 1230 echoes of the work of al-Bitruji - albeit still confused could be found in the writings of William of Auvergne (1180-1249), and a little later in the work of Robert Grosseteste (1175-1253). Albertus Magnus (d. 1280), for his part, was fascinated by a very simplified model of the theory of al-Bitruji, i.e. the attempt to explain all celestial appearances by means of a single driving force that would carry all the celestial bodies in a more or less rapid motion towards the west, which would account for their apparent proper motions towards the east. At the conclusion of his discussion, Albert rejects the criticism of Averroes concerning the eccentrics and epicycles, for the reason that celestial bodies differ from terrestrial bodies in matter and in form. He also rejects the astronomy of homocentric spheres, for 'this astronomy', he says, 'has not been completed by observation of the magnitude of the motions'. He thus gives prominence to the inability of this astronomy to account for appearances quantitatively, a failing that was constantly cited against the hypothesis of al-Bitruji in the Middle Ages and which explains the indifference of astronomers toward it. The doubts and criticisms concerning Ptolemy raised by the works of Averroes and al-Bitruji, by contrast, prompted a deepening reflection on the status of astronomical theory and led to the appearance of theses which would be studied anew in the sixteenth century as part of the polemic

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between Ptolemaic and Copernican hypotheses. These theses were clearly articulated by Thomas Aquinas (1225-74), when he stated that the suppositions imagined by the astronomers were not necessarily true even if they seemed to explain appearances, for it may be possible to explain those appearances by some other process not yet conceived. Thomas thus contrasted two ways of explaining a phenomenon: sufficient proof of a principle from which the phenomenon follows, or the demonstration of agreement between the phenomenon and a principle advanced beforehand. Astronomy, according to Thomas, uses the second method, which suffices to explain the most obvious appearances. In this debate between physics and astronomy - championed at the time of Simplicius by Aristotle and Ptolemy and revived in the guise of the opposition between Ptolemy and al-Bitruji — certain Latin scholastics found the germ of a solution in the work of another Arab author: the treatise on the Configuration of the World attributed to Ibn al-Haytham (d. c. 1041), of which three anonymous Latin translations survive (one dated 1267).19 The work is a cosmography without any mathematical treatment in which Ibn al-Haytham returns to the arrangement of solid orbs imagined by Ptolemy in his Planetary Hypotheses. Schematically, the sphere of each planet was seen as composed of an orb concentric with the earth into which there is fitted an eccentric orb containing the deferent and the epicycle: the two parts of the concentric orb, which are respectively interior and exterior to the eccentric orb, are of unequal thickness and function, as it were, to 'compensate' the eccentricity and to make the whole of the planetary sphere concentric with the world. Presented by Roger Bacon (d. 1294) in his Opus tertium as an ymaginatio modernorum created to avoid the difficulties of eccentrics and epicycles, this physical interpretation of Ptolemaic astronomy invalidates the objections of Averroes, according to the author. Conversely, the variations of planetary distances and the non-uniformity of their motions appeared to him to confirm the hypotheses of Ptolemy. This was also the opinion of numerous great medieval scholars, such as Bernard of Verdun, Richard of Middleton and Duns Scotus. The inability of the system of al-BitrujI to account for simple observations concerning, for example, the eccentricity of the planets - an inability again denounced at the end of the Middle Ages, by Regiomontanus - and, conversely, the ability of the ymaginatio inherited from Ibn al-Haytham to respond to the criticisms of Averroes, ensured the triumph of the Ptolemaic hypotheses and their physical interpretation by means of the orbs of Ibn alHaytham. The most thorough exposition of this interpretation appeared at the end of the Middle Ages in the Theoricae novae planetarum, written in 1454 by Georg Peurbach: the description of the celestial orbs contained in this treatise served as an authoritative account of the structure of the 295

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heavens until Tycho Brahe (1546-1601) rejected the very existence of the celestial spheres. THE PROBLEM OF PRECESSION AND THE A B A N D O N M E N T OF THE TOLEDAN TABLES The second major problem encountered by the medieval astronomers, that concerning the movement of precession, was more difficult to overcome. In his commentary (probably dated 1291) on Gerard of Cremona's translation of the canons of al-Zarqallu regarding the Toledan tables, the Parisian astronomer John of Sicily20 enumerated the various hypotheses that he knew relating to precession: the uniform motion estimated by Ptolemy as 1° in 100 years and by al-Battanl as 1° in 66 years; the to-and-fro motion of 1° in 80 years and of 8° amplitude rejected by al-Battanl; and the movement of accession and recession of the De motu octavae spherae attributed to Thabit ibn Qurra. He rejected, for his part, the movement of accession and recession and adhered to the Ptolemaic concept of uniform motion, while regarding its exact magnitude as uncertain. In this respect, John of Sicily is representative of the mistrust of Parisian astronomers of the time regarding the theory of De motu and, more generally, the Toledan tables. At the end of the thirteenth century, indeed, the divergence between the positions calculated from these tables or the Latin tables derived from them - notably the tables for Toulouse - and the observed positions of the celestial bodies had become inadmissible. Thus, on the basis of personal observations made to establish his Almanack, William of Saint-Cloud21 estimated the difference between the positions of the moving apogees and those of the fixed apogees on the eighth sphere as 10; 13° for 1290 and 10; 15° for 1292. Noting that this difference was nearly 1° greater than the value which would have resulted from the calculation made according to the law of motion proposed in the De motu octavae spherae, he concluded that this law should be rejected, and he accepted that the movement of precession must be considered, at least provisionally, to be uniform at one minute per year (i.e. a value close to that obtained by al-Battanl). Concerning the mean motions of the planets, on the other hand, William supplied empirical corrections to the radices of the Toledan tables, adding or subtracting fixed quantities as follows: +1;15° for Saturn, -1° for Jupiter, -3° for Mars and +0; 22° for the moon. These same corrections were also proposed by two other Parisian authors, Peter of Saint-Omer and G. Marchionis (Poulle 1980a: 205-9, 260-5) in their treatises concerning equatoria, written in 1294 and 1310 respectively. In addition, Peter of SaintOmer evaluated the difference between the fixed apogees and the moving apogees at 10; 10°, by reference to the estimations of precessional motion 296

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by William of Saint-Cloud, which also seem likely to have inspired Profatius in his treatise on the equatorium written between 1300 and 1306. A collection of texts from the very end of the thirteenth century thus attests to the ending of the comprehensive influence of the Toledan tables: the astronomers of this era no longer considered them sufficient, and they rejected in particular the movement of accession and recession, preferring instead a uniform motion of precession. The influence of these criticisms was, however, short-lived. At the beginning of the fourteenth century, Latin astronomy replaced the Toledan tables with the Alfonsine tables. The latter were drawn up in Spanish between 1252 and 1272 for Alfonso X of Castile, and only the original canons survive. However, the Latin version, which appeared in Paris around 1320, dominated tabular astronomy from then until the publication of the De revolutionibus of Copernicus in 1543. In the first known essay concerning the new astronomy, the Expositio tabularum Alfonsi regis Castelle,22 written in 1321, John of Murs did not refer to planetary parameters, eccentricities and magnitudes of epicycles, but concentrated his study on the values given in the Alfonsine tables for the mean motion of the sun and the movement of the auges of the planets. It was the treatment of the movement of precession, in fact, that most clearly differentiated the Alfonsine tables from the earlier tables. As John of Murs said, they represented an attempt to reconcile the Ptolemaic theory of uniform precessional motion with the Arabic theory of the movement of accession and recession. According to the Alfonsine theory, the motion of the apogees and the stars was made up of two components: a uniform motion in the order of the signs, for which the period was 49,000 years (1° in just over 136 years) and a movement of accession and recession relative to the intersection of the zodiac and the equator, for which the period was 7000-years, with a maximum effect of 9°. The movement of accession and recession of De motu was thus conserved as the component that causes the velocity of precessional motion of the apogees and the stars to vary. In addition, this movement of precession was taken into account from the start of operations to compute the planetary positions and not, as in the Toledan tables, at the end when it was necessary to transpose the positions obtained on the sphere of the fixed stars into tropical co-ordinates. More generally, the Alfonsine tables were designed to give the true positions of the planets on the ninth sphere directly, i.e., in tropical co-ordinates. As far as the planetary equations were concerned,23 the Alfonsine astronomers made only slight modifications to the Toledan tables, except in the cases of the sun, Venus and Jupiter. The change in the maximal equation of the sun (and consequently in its equation table) arose from the tacit modification (nowhere explained in the canons) of the eccentricity of the 297

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sun, which varied from 2; 6 P in the Toledan tables (2; 30P according to Ptolemy) to 2; 15p in the Alfonsine tables. The eccentricity (of the deferent) of Venus being traditionally taken as half that of the sun - i.e. 1; 8 P for the Alfonsine astronomers (instead of 1;15P for Ptolemy and 1;3P in the Toledan tables) - the maximal equation of Venus and the corresponding table of equation were similarly modified. Finally, in the case of Jupiter, the increase in the maximal equation, which changed from 5; 15P in the Ptolemaic and Toledan tables to 5;57 P in the Alfonsine tables, corresponded to an increase in the eccentricity from 2; 45p to 3; 7 p . With regard to the radii of the epicycles, by contrast, parameters derived (by modern computation) from the tabulated values of the equation of the argument show that the Alfonsine tables were based on similar values to those used for the Toledan and Ptolemaic tables. In short, the new hypotheses did not change the structure of the Ptolemaic planetary models, except as far as the eccentricity of the sun and of Venus and Jupiter were concerned. Once again it was the theory of motion of the sun and the directly linked theory of the movement of the fixed stars that were the essential subject of modification. On this point, the concepts of the De motu octavae spherae again played a key role: although they no longer served to describe the actual motion of the equinoxes, they served to describe variations in the velocity of that motion. THE COPERNICAN REVOLUTION AND ARABIC ASTRONOMY Once the astronomical tables had been updated by the Alfonsine reforms, the attention of the leading astronomers of the late Middle Ages turned to the analysis of Ptolemy's kinematic models. This was the task, in particular, of the Theoricae novae planetarum of Peurbach and the Epitome in Almagestum Ptolemaei, started by Peurbach and completed by Regiomontanus (d. 1476). The latter work, which contained a highly detailed analysis of Ptolemy's treatise, was the principal source for Copernicus concerning the results obtained by the Arab astronomers, notably al-Battanl and alZarqallu. In the former work Copernicus could become familiar with the constitution of the solid spheres, as inherited from Ptolemy's Hypotheses and Ibn al-Haytham's Configuration of the World. There too he could read the description of the movement of accession and recession according to the De motu octavae spherae in a chapter on this subject added by Peurbach after his original draft. He could discover there also the representation of the deferent of Mercury as an oval figure, the first mention of which occurs in the treatise on the equatorium of al-Zarqallu, which had become known 298

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in the West through the Spanish translation in the Libros del Saber compiled for Alfonso X and which was probably Peurbach's ultimate source.24 The question of Arabic influence on Copernican texts25 focuses on two groups of problems which relate, on the one hand, to the theory of precession and solar theory, and, on the other hand, to planetary theory. As we have seen, the problem of the motion of the sun and stars was the major stumbling block for Latin astronomers throughout the Middle Ages, and it is therefore not surprising that the prime merit ascribed to Copernicus by his disciple Rheticus was to have solved this problem. The long medieval debate on the solar parameters (eccentricity, position of the apogee and obliquity of the ecliptic) and on the precession or the trepidation of the equinoxes appeared in a new light in the Copernican system, once the earth was seen as responsible not only for the diurnal revolution but also the annual revolution and even, through the motion of its axis, for the westward slide of the equinoxes with respect to the fixed stars and thus the difference in length of the sidereal and the tropical years. Taking into consideration, in his Commentariolus, the lengths of the tropical year given by Ptolemy, al-Battanl and the Alfonsine tables, and the corresponding values for precession obtained by the same sources, Copernicus concluded that in all cases the calculation gave a constant sidereal year of 365 days 6| hours. The model conceived in the Commentariolus to account for this result, i.e. the westward movement of the earth's axis accomplishing its revolution in a tropical year, while the great orb carrying the earth turned to the east in a sidereal year, still produced only a uniform precession, because Copernicus, by his own admission, had not at that date discovered the law of precessional motion. It none the less indicated that the sphere of stars is fixed, that the lines of planetary apsides are fixed with respect to it and that it is the motion of the earth's axis which displaces the equinox with respect to the ecliptic. It also demonstrated the return by Copernicus to the concepts of the Arab astronomers, for whom, since the time of Thabit ibn Qurra and al-Battanl, the sidereal year had been constant and the periods of planetary motion had been fixed with respect to the stars. The analogy does not stop there. When he turned his attention, in the De revolutionibus, to a more accurate description of the inequalities in the motions of the earth, Copernicus carried out a historical assessment of the data obtained by his predecessors for the precession, the obliquity of the ecliptic, and the eccentricity and position of the solar apogee, and he took the results of al-Battanl and of al-Zarqallu for the medieval period.26 In view of the diversity of values that emerged, Copernicus found himself facing exactly the same problem as the Arab astronomers of the ninth century with their new data for the parameters in question: were the discrepancies in the findings due to error or to variations in the parameters over a 299

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long period? In other words, should certain values be rejected, or should they all be integrated in the laws of motion to be determined? On this point, Copernicus was inspired by the example of De motu octavae spherae. Like the author of that treatise, Copernicus assumed that the combined observations reflected periodic variations in the relevant motions, and he constructed a model which, like that of the De motu, combined a uniform sidereal year and a trepidation of the equinoxes. For Copernicus, however, the trepidation was not a simple one but was composed, as in the Alfonsine tables, of a secular term and a periodic term (having periods of 25,816 and 1717 years of 365 days respectively). According to Copernicus, however, the variation of the degree of precession was insufficient to explain the variation in the length of the year. It was also necessary to incorporate two long-term inequalities which, according to his assessment, affected the motion of the sun, i.e. a decrease in the eccentricity and a non-uniform motion of the line of aspides. It was in the work of al-Zarqallu that the Latin astronomers had first discovered the affirmation of the solar apogee's own (but uniform) motion and a clear distinction of the anomalous year confused until then with the tropical year (Ptolemy) or the sidereal year (Thabit, al-Battanl). It was from al-Zarqallu too - through the intermediary of the Epitome of Regiomontanus - that Copernicus adopted27 the mechanism designed to account for both the variation of the eccentricity (the period of which he assumed to be equal to that of the variation in obliquity of the ecliptic) and the inequality of motion of the line of the apsides: all that was required was to let the centre of the terrestrial orbit, i.e. the mean sun, move on a small circle around a point removed from the real sun by a distance equal to the mean eccentricity in the relevant period (3434 years of 365 days). It may also be from al-Zarqallu that Copernicus drew the principle for his model representing the concomitant variations of precession and obliquity of the ecliptic. In fact, al-Zarqallu had succeeded in making these two variations independent of each other by using, in one case, an epicycle placed around the equinox to make the precession vary (following the method of the De motu), and in the other case, a polar epicycle (with its centre on a deferent concentric with the pole of the ecliptic) to make the obliquity of the ecliptic vary.28 The method of polar epicycles was later generalized by al-BitrujI, who employed it for all planetary motion but with the disastrous consequence that the latitude depended on the longitude (or more accurately, on the argument of the planet). Copernicus, in turn, took up this method of polar epicycles, as part of a complex solution permitted by the fact that these two variations of precession and obliquity could be treated as two perpendicular oscillations of the axis of the terrestrial equator: each of the two variations was then given a small polar circle of 300

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appropriate diameter, the earth's axis was made to oscillate back and forth along the diameters of these circles and the two oscillations were combined so as to occur in perpendicular planes and in the relevant periods. The technical procedure used by Copernicus to obtain each of the oscillations is described by Naslr al-DIn al-TusI (1201-74) in his major treatise alTadhkira fl eilm al-hay*a> and has consequently become known to modern scholars as the TusI Couple'. This procedure, used by TusI in planetary theory, thus leads us to the second group of problems relating to Arab influence on Copernican astronomy. This set of problems is not concerned with the second planetary anomaly, which relates to proving the heliocentric theory, but with the first anomaly, which is explained in Ptolemaic theory by the uniform motion of the eccentric deferent around a point that is not its own centre but the centre of the equant. Such a movement had been strongly criticized as contrary to the principles of physics by Ibn al-Haytham and then by the astronomers associated with the observatory of Maragha (founded by Hulagu in 1259), such as Naslr al-DIn al-TusT, Mu'ayyad al-DIn al-eUrdi (d. 1266) and Qutb al-DIn al-ShlrazI (1236-1311), as well as by the Damascene astronomer Ibn al-Shatir (1304-75).29 The method employed by these scientists to avoid the difficulty consisted of breaking down the motion around the centre of the equant into two or more components which were circular motions and which controlled the direction and the distance of the centre of the epicycle in such a way that the centre was as close as possible to the position that it would have occupied in the Ptolemaic model. The Eastern astronomers used two technical procedures to achieve this end: the addition of epicycles to give the Ptolemaic effect of bisection of the eccentricity, and the TUSI Couple'. This model permits a rectilinear motion to be produced from circular motions in the following manner (Figure 9.2): if two equal circles rotate around their respective centres D and F so that the circle of centre F revolves in the opposite direction and twice as fast as the circle of centre D, the point H (such that GFH= -2CDF) on the circumference of the circle of centre F describes with an oscillatory motion (or motion of libration in the terminology of Copernicus) the diameter AB of a large circle (with centre D and radius double that of the small circles). If this model is in plane, it produces a rectilinear oscillation of H. If it is drawn on a sphere, the diameter AB described by H will be an arc of large circle (provided that the oscillation is weak). These two technical procedures, the TusI Couple' and the addition of epicycles, were put to work by Copernicus. He used the first, as we have seen, to account at one and the same time for the inequality of the precession and the variation in the obliquity of the ecliptic. For this he used not one, but two, TusI models, in such a way that the diameters described by 301

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GX

H \F>

"c \/ 'D

E

R

Figure 9.2 Copernicus, De revolutionibus, Nuremberg, 1543, fol. 67v

the two resulting oscillations are in perpendicular planes and intersect at the mean North pole to the equator (the radii of the circles and the speeds of rotation being chosen, of course, so that the two oscillatory motions have the necessary amplitude and periodicity). Copernicus also used the TusI model, as did the author of the Tadhkira, to account for the oscillations of the orbital planes in the theory of latitudes. More striking still is the similar use made by Copernicus and Ibn al-Shatir (in his treatise Nihayat al-Sulfl Tashih al-Usitl) of the other procedure (the addition of epicycles) to represent planetary motion in longitude while avoiding the problems associated with the Ptolemaic equant. Thus, in the CommentarioluSy all the planetary models are similar, with regard to the first anomaly, to those of Ibn al-Shatir in which ,the combination of a deferent and two epicycles is substituted for the movement of the deferent with respect to the centre of the equant. The only differences between the two authors lie in the values attached to the parameters and, of course, in the fact that the earth was at the centre of the planetary models for Ibn alShatir while it was the mean sun for Copernicus. A further similarity brings together the models of Copernicus and Ibn al-Shatir: both place a TusI Couple' at the tip of the deferent radius of Mercury in such a way as to vary the length of the orbital radius of this planet, by imposing at the centre of the first epicycle an oscillatory motion along a line directed always towards 302

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the centre of the deferent. A final similarity is the following: the model of the moon in the Commentariolus and the De revolutionibus is the same, except for parameters, as the model of Ibn al-Shatir. These numerous analogies suggest that Copernicus was influenced by the Eastern astronomers of the thirteenth and fourteenth centuries. It is true that we do not know of any Latin translation of their works, nor even of any reference to them in the Latin literature of the late Middle Ages. However, it seems that the transmission of certain of these Arabic texts to the Latin West may have been achieved through the intermediary of Byzantine sources which reached Italy in the fifteenth century. Thus a manuscript (Vat. Gr. 211 which was in Italy by 1475) contains a treatise dealing with planetary theory (in a Greek translation, made around 1300 by Chioniades from the original Arabic), that contains TusT's lunar model and an illustration showing the TusI Couple'. Further evidence of the use of the TusI Couple' is found in the treatise of Giovanni Battista Amico entitled De motibus corporum coelestium iuxta principia peripatetica sine excentricis et epicyclis, published in Venice in 1536, in which the author endeavours to revive homocentric astronomy with the aid of models which are all based on the use of Tusl's mechanism.30 THE END OF THE INFLUENCE OF ARABIC ASTRONOMY IN THE LATIN WEST Copernicus marks the end of the long period of influence of Arabic astronomy in the Latin West. He was the last to make constant use of observational results taken from Arab authors, results which helped him to elaborate his estimations of the long-term variations in solar parameters. He was the last, also, to choose the thesis based on the De motu octavae spherae, which involved serious use of the collected observations of the past to formulate the laws of motion being sought, rather than using new observations to refute pre-existing theories. Remembering Kepler's three-way division of theoretical astronomy, we note that shortly after Copernicus, the abundant and accurate observations of Tycho Brahe made all reference to the history of ancient observations irrelevant. As for the Ptolemaic geometrical models and their Arabic or Latin variations, Kepler put an end to them. All that remained was the requirement to account physically for the phenomena, which Ibn al-Haytham and the Eastern astronomers of the thirteenth and fourteenth centuries had striven to do: nevertheless, after the refutation of the existence of solid spheres by Tycho Brahe, this requirement was no longer linked by Kepler with an Aristotelian vision of the world but rather with a vision inspired by a Platonic mathematical tradition. 303

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NOTES 1 On the astronomy of the Middle Ages before the arrival of Arabic science in the West, see the synthesis and study by Pedersen (1975). 2 The most recent study of the transmission of Arabic science to the Latin world, with an extensive bibliography, is by Vernet (1985). Despite its age, Raskins (1927) remains useful. See also Carmody (1956). 3 On this last point, see Kunitzsch (1959, 1966). 4 See Lemay (1962). The doctrine of De magnis coniunctionibus (translated by John of Seville from Kitab al-qiranat) which exposes the effects of planetary combinations on the rise and fall of dynasties and earthly kingdoms exerted a persistent influence in the Middle Ages, whose traces can still be found in Rheticus, pp. 47-8, 98-9. 5 The classic study on this subject is in Millas Vallicrosa (1931). See also the work of synthesis entitled 'Las primeras traducciones cientificas de origen oriental hasta mediados del siglo XII' in Millas Vallicrosa (1960: 79-115). 6 See the edition of this treatise by Poulle (1964) (with a list of existing editions of Latin treatises on the astrolabe, pp. 870-2). See also Poulle, 'Raymond of Marseilles', in Dictionary of Scientific Biography, XI, 1975, pp. 321-3. 7 There is no modern edition of Plato of Tivoli's translation, which was published in Nuremberg in 1537 under the title De scientiis astrorum. 8 There is no modern edition of the Toledan tables, but see the detailed analysis by Toomer (1968). 9 An annotated list of the Latin translations attributed to Gerard of Cremona can be found in Lemay, 'Gerard of Cremona', Dictionary of Scientific Biography, XV, 1978, pp. 173-92. For the Arabic-Latin tradition of the Almagest, see Kunitzsch (1974). 10 The planetary positions calculated from the Toledan tables do in fact coincide well with the values given by Profatius, as demonstrated by Toomer (1973). 11 The Arabic text of this treatise has not been found. The Latin version by Gerard of Cremona appears in Millas Vallicrosa (1943-50: 487-509) (reprinted in Millas Vallicrosa 1960: 191-209) and in Carmody (1960). The attribution of this work, which is definitely not by Thabit, is currently disputed: Millas Vallicrosa has rejected the attribution to al-Zarqallu, supported by Duhem (1913-59: II, 246f); the attribution to Ibrahim b. Sinan, the grandson of Thabit b. Qurra, is supported by Ragep (1993: 400-08). An annotated translation can be found in Neugebauer (1962b). 12 Most of the values that follow are taken from Hartner, 'Al-Battanf, in Dictionary of Scientific Biography, I, 1970, pp. 507-16. 13 The Latin version of this treatise has been edited by Carmody (1960), who attributes it to Gerard of Cremona. This attribution is considered doubtful by Morelon, who also thinks that the original Arabic text came from the circle of the Banu Musa and not from Thabit: see Thabit ibn Qurra, pp. XLVI-LII. 14 On this model, and on theories of precession generally in the Middle Ages, see Mercier (1976-7), Goldstein (1964a). 15 Analysis of some texts relating to this tradition can be found, for example, in North (1976), vol. 3, pp. 238-70. 304

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Studies in the Making of Islamic Science INFLUENCE OF ARABIC ASTRONOMY IN THE MEDIEVAL WEST 16 This translation is published in Carmody (1960). The original Arabic text, with French translation and commentary by Morelon, is in Thabit ibn Qurra. 17 See Gerardus. An English translation by Pedersen is published in Grant (1974: 451-65). 18 The passages of commentary on the treatises of Aristotle in which Averroes criticizes Ptolemaic astronomy are collected in Carmody (1952). On the criticism of Ptolemy by the Arab scholars of Spain, see Sabra (1984). 19 One of these translations, which seems to have been made from a Spanish version (now lost) compiled for Alfonso X, has been published by Millas Vallicrosa (1942: 285-312). On the astronomical concepts of Ibn al-Haytham, see Sabra (1978). 20 See Poulle, 'John of Sicily', in Dictionary of Scientific Biography, VII, 1973, pp. 141-2. 21 On this astronomer and the values quoted, see Poulle, 'William of Saint-Cloud', in Dictionary of Scientific Biography, XIV, 1976, pp. 389-91, and Poulle (1980a: 68, 209). 22 This important treatise has been published by Poulle (1980b). See also Poulle, 'John of Murs', in Dictionary of Scientific Biography, VII, 1973, pp. 128-33. 23 The information that follows has been taken from Poulle (1980a: 26-7, 767-9). 24 Concerning solid spheres and the representation of the deferent of Mercury according to Peurbach (and his Arabic sources), see Hartner (1955). 25 An overall survey of the influence of Arabic astronomy on Copernicus can be found in Swerdlow and Neugebauer, pp. 41—8. For the Commentariolus, see also Swerdlow (1973: passim.) 26 A good summary of this historical assessment and of the conclusions drawn by Copernicus can be found in Rheticus, pp. 94-8. 27 On the solar theory of al-Zarqallu and its transmission to the Latin West, see Toomer (1969). 28 See Goldstein (1964a) and the same author's edition of al-BitrujI, On the Principles of Astronomy. 29 From the extensive literature on this aspect of Arabic astronomy, we only mention here the studies directly concerned with the comparison of the Arabic and Copernican models: Kennedy (1966), Kennedy and Roberts, Hartner (1971). 30 These two references are taken from Swerdlow and Neugebauer, pp. 47-8. On Amico, see Swerdlow (1972).

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Studies in the Making of Islamic Science BIBLIOGRAPHY Carmody, Francis J. (1952) The planetary theory of Ibn Rushd', Osiris 10: 556-86. (1956) Arabic Astronomical and Astrological Sciences in Latin Translation. A Critical Bibliography, Berkeley/Los Angeles. (1960) The Astronomical Works of Thabit b. Qurra, Berkeley/Los Angeles. Duhem, P. (1913-59) Le Systeme du Monde. Histoire des Doctrines Cosmologiques de Platon a Copernic, 10 vols, Paris. Gerardus (1942) Theorica Planetarum Gerardi, ed. by F. J. Carmody, Berkeley. Goldstein, B. R. (1964a) 'On the theory of trepidation according to Thabit b. Qurra and al-Zarqallu and its implications for homocentric planetary theory', Centaurus 10: 232-47. Grant, Edward (1974) A Source Book in Medieval Science, Cambridge, Mass. Hartner, W. (1955) The Mercury horoscope of Marcantonio Michiel of Venice: a study in the history of Renaissance astrology and astronomy', Vistas in Astronomy I: 84-138; reprinted with additions in W. Hartner (1968) OriensOccidens, Hildesheim, pp. 440-95. Haskins, Charles Homer (1927) Studies in the History of Mediaeval Science', reprinted, New York, 1960. Kennedy, Edward S. (1966) 'Late medieval planetary theory', Isis 51: 365-78. Kepler, Gesammelte Werke, Bd. VII, edited by M. Caspar, Munich, 1953. Kunitzsch, P. (1959) Arabische Sternnamen in Europa, Wiesbaden. (1966) Typen von Sternverzeichnissen in astronomischen Handschriften des zehnten bis vierzehnten Jahrhunderts, Wiesbaden. (1974) Der Almagest. Die Syntaxis Mathematica des Claudius Ptolemdus in arabisch-lateinischer Uberlieferung, Wiesbaden. Lemay, R. (1962) Abu Ma'shar and Latin Aristotelianism in the Twelfth Century, Beirut. Mercier, R. (1976—7) 'Studies in the medieval conception of precession', Archives Internationales d'Histoire des Sciences 26 (1976): 197-220; 27 (1977): 33-71. Millas Vallicrosa, Jose M a (1931) Assaig d'Historia de les Idees Fisiques i Matematiques a la Catalunya Medieval, Barcelona. (1942) Las Traducciones Orientates en los Manuscritos de la Biblioteca Catedral de Toledo, Madrid. (1943-50) Estudios sobre Azarquiel, Madrid/Granada. (1960) Nuevos Estudios Sobre Historia de la Ciencia Espanola, Barcelona, Consejo Superior de Investigaciones Cientificas. Neugebauer, O. (1962b) Thabit ibn Qurra On the solar year and On the motion of the eighth sphere', Proceedings of the American Philosophical Society 106: 264-99. North, J. (1976) Richard of Wallingford, 3 vols., Oxford. Pedersen, O. (1975) The corpus astronomicum and the traditions of mediaeval Latin astronomy', Colloquia Copernicana, Wroclaw, coll. 'Studia Copernicana, XIII', pp. 57-96.

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Studies in the Making of Islamic Science BIBLIOGRAPHY Poulle, E. (1964) 'Le traite d'astrolabe de Raymond de Marseille', Studi Medievali 5: 866-904. (1980a) Les Instruments de la Theorie des Planetes selon Ptolemee: Equatoires et Horlogerie Planetaire du XIII* au XVIe siecle, 2 vols, Paris, coll. 'Hautes etudes medievales et modernes, 42'. (1980b) 'Jean de Murs et les tables alphonsines', Archives d'Histoire Doctrinale et Litteraire du Moyen Age 47: 241—71. Ragep, F. J. (1993) Nasir al-Dln al-Tusi's Memoir on Astronomy (al-Tadhkira fi f llm al-Hay'a), with translation and commentary, 2 vols, New York/Berlin/Heidelberg. Rheticus, G. J. (1982) Narratio prima, critical edition, French translation and commentary by H. Hugonnard-Roche and J. P. Verdet (in collaboration with M. P. Lerner and A. Segonds), Wroclaw, coll. 'Studia Copernicana XX'. Sabra, A. I. (1984) 'The Andalusian revolt against Ptolemaic astronomy: Averroes and al-BitrujI', in E. Mendelsohn (ed.) Transformation and Tradition in the Sciences: Essays in Honor of I. Bernard Cohen, Cambridge, pp. 133-53. Swerdlow, Noel M. (1972) 'Aristotelian planetary theory in the Renaissance: Giovanni Battista Amico's homocentric spheres', Journal for the History of Astronomy 3: 36-48. (1973) 'The derivation and first draft of Copernicus's planetary theory. A translation of the Commentariolus with commentary', Proceedings of the American Philosophical Society 117: 423-512. Swerdlow, N. M. and Neugebauer, O. (1984) Mathematical Astronomy in Copernicus's De Revolutionibus, 2 vols, New York. Thabit ibn Qurra, (Euvres d'Astronomie, edited and translated by R. Morelon, Paris, 1987. Toomer, G. J. (1968) 'A survey of the Toledan tables', Osiris 15: 5-174. (1969) 'The solar theory of az-Zarqal. A history of errors', Centaurus 14: 306-36. (1973) 'Prophatius Judaeus and the Toledan tables', Isis 64: 351-5. Vernet, J. (1985) Ce que la culture doit aux Arabes d'Espagne, translated from Spanish by G. M. Gros, Paris; German translation, Die spanisch-arabische Kultur in Orient und Okzident, Zurich/Munich, 1984.

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FUAT SEZGIN The cartographic image of the Earth's surface which we inherited in the 20th century has doubtless attained a high degree of precision. However, its accuracy has yet to be verified. Only now is it becoming possible, thanks to advances made in sciences associated with today's image of the world, viz. the observations and measurements resulting from space technology, to complete this unfinished business. Even if corrections to this image need to be undertaken, they will not impair the general validity of the image, which is the joint inheritance of all humanity. Our predecessors did not yet have the advantage of this experience in the second half of the 19th century. The work of the fledgeling discipline of the historiography of cartography, which involves appropriately describing, as far as possible, the individual stages of development and the contributions made by different cultures, is immensely difficult. No doubt it will never be revealed to us when and where the first attempt was made to represent pictorially part of the Earth's surface by human hands. Fortunately, attempts by the Babylonians and Ancient Egyptians to sketch out their perception of the inhabited world are known to us. We also know that as long ago as 530 B.C. the Carthaginian Hanno was able to sail from his native city and penetrate the interior of the Gulf of Guinea, approximately down to the Equator. Herodotos tells us of a Phoenician circumnavigation of Africa commissioned by Pharaoh Nekho (ca. 596-584 B.C.). This Pharaoh is alleged to have ordered his sailors to sail southwards from the Red Sea along the coasts until they passed through the pillars of Hercules and returned via the Mediterranean to Egypt. They are said to have completed the task within three years. The beginnings of mathematical geography in Greece The Greeks created the basis for mathematical documentation of the known surface of the Earth by postulating that the Earth was a * Translated from the German original (published in Forschung Frankfurt 2002, no. 4, pp. 22-31) by Geoff Sammon.

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sphere in the 5th and 4th centuries B.C., by making the first attempt to measure the Earth in the 3rd century B.C. and by transferring the Babylonian division of the firmament into 360° in the great circle onto the Earth. They also contributed both the idea of longitudes in the sense of the time difference between places by simultaneously observing lunar eclipses and the proposition of the identity of the geographical latitude of a place and the altitude of the pole, which is fimdamental to determining locations. Hipparchos, one of the greatest of Greek astronomers, found that it was not yet feasible in the third quarter of the 2nd century B.C. to draw a map which was substantiated by mathematical and astronomical data. He regarded the cartographic achievements of geographic science to date as premature and unsuccessful, counselling patience and the collection of adequately precise co-ordinates. The compilation of a map was, he averred, a task for the future, which could not be implemented until a large number of scholars in various countries had completed the spadework required. There is no doubt that the Greeks had at their disposal a difference in longitude: it had been calculated in 331 B.C. by the method of observing lunar eclipses between Carthage and Arbela, and was approximately 11° too large. Latitudes calculated over the course of time, measurements of distances obtained on ships' voyages or by the Roman army, and data gained by other means in route books resulted in the first half of the 2nd century A.D. in a map of the known world using an orthogonal projection. Its creator was Marines of Tyre. His younger contemporary Ptolemy leads us to traces of Marinos's long lost map. To all appearances, this map and its accompanying text was the sole basis for Ptolemy's geography. As we have learnt, Marinos had based his map of the known world on a graticule whose longitude was 225°, i.e. about 80° to 90° too large. His successor Ptolemy felt moved to compile a book using the data and information about degrees which he obtained from this map of the known world (possibly also from the regional maps enclosed) and the accompanying text. This book was to serve later generations as a basis for compiling new editions of the map. While working on his predecessor's data he came to the conclusion that the data on routes, especially the east-west ones in terms of longitudes, were excessively large. He therefore systematically reduced in proportion the size of the parts involving Asia. While retaining the longitude of the grand axis of the Mediterranean at 63° (ca. 21° too large), he reduced the longitude of the known world to 180° (still ca.

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40° too much). Ptolemy appears not to have enclosed a map in his book. It is astonishing that his text conveys the picture of a continuous landmass, in which the North Atlantic and the Indian Ocean are shown as inland seas. The oldest known world map with a globular projection Marinos's cartographic achievement and Ptolemy's Geography reached the Arab-Islamic world in the early 9th century, at a time when this extended from the Atlantic across to India and its inhabitants were in the process of acquiring the learning of other peoples, being on the threshold of their creative period. Caliph al-Ma'mun, who sponsored all the areas of learning of his time, commissioned a large group of scholars to create a new 'Geography' and a world map. It goes without saying that in carrying out this task the scholars mainly had to follow in the footsteps of the achievements of their Greek mentors. Fortunately, some parts of the atlas and its accompanying geographical work, which were the result of this assignment, still survive. From the point of view of the history of mathematical geography and cartography it is of outstanding importance that the world map of the Ma'mun geographers re-emerged during the 1980s in a copy from the year 1340. It is without doubt a copy of what was once a magnificent original, albeit rather distorted as a result of repeated copying (Fig. 1). However, thanks to a surviving table of co-ordinates, which had been simultaneously excerpted from the original map, it proves to be a unique cartographic monument: it uses a globular projection. It shows a west-east extension of the inhabited world which is reduced by 15°-20°, with at the same time a longitudinal axis of the Mediterranean reduced by 10°. Furthermore, it is very significant that the MarinosPtolemaic notion of a continuous landmass has been replaced by a new version. This involves the inhabited world being surrounded by an 'Encompassing Ocean1, which in turn is surrounded by an 'Ocean of Darkness1. The Atlantic and the Indian Ocean are no longer inland seas, forming part of the Encompassing Ocean (Fig. 2). The attempts by the Greeks to achieve a precise cartographic representation of the Earth's surface and the mathematical-astronomical methods used to this end, which had culminated in the work of Marinos and Ptolemy (Fig. 3), and had at the same time reached the limits of their potential for development in their own culture, entered into a

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new evolutionary phase due to the work of Caliph al-Ma'mun's geographers. We are experiencing the most recent period of this phase in our own times. In my recently published book Mathematische Geographic und Kartographie im Islam und ihr Fortleben im Abendland (vols. X-XII of my Geschichte des arabischen Schrifttums)1 I have attempted to convey to a specialist readership the phenomena of an uninterrupted continuous development which were revealed to me. In what follows I would like to point to several aspects of this development process which seem important to me. Expansion of mathematical geography into an independent discipline The geographic positioning of places which was carried on intensively and with scholarly meticulousness in the Islamic world led in the first quarter of the 11th century to the emergence of mathematical geography as an independent discipline. The credit is due to al-Birum, one of the most important scholars of the Arab-Islamic world. He made the attempt, unique in the history of geography, to determine the longitudes and latitudes of major places located between Ghazna (in modern Afghanistan) and Baghdad within a radius of approx. twice 2,000 kms) on the basis of astronomical observation, the measurement of routes and the use of the rules of spherical trigonometry (Fig. 4). The errors he made in the longitudes of about 60 places, when compared with modern values, only amount to between 6 and 40 minutes. His data were used as the basis for the determination of co-ordinates which was continued unbroken for centuries in the eastern part of the Islamic world. The additional corrections to longitudes made in the part of the Islamic world west of Baghdad resulted as early as the first half of the llth century in reducing the east-west axis of the Mediterranean to 44°-45° (now 42°), and as a consequence of this to a relocation of the zero meridian into the Atlantic at 17°30' west of the Canary Islands, i.e. 28°30f west of Toledo. The first Arab maps in Europe There are several Arab and European maps surviving which reveal the after-effects of the Ma'mun geography. They include the world 1

Frankfurt 2000. Vol. I: Historische Darstellung (Teil 1), Vol. II: Historische DarstellungWeil 2), Vol. HI: Kartenband.

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map and regional maps of the geographer al-Idrisi (Fig. 5) of 1154. The maps and the geographical work by this aristocrat from Ceuta, which he compiled in Sicily at the behest of the Norman king Roger II, show a high degree of similarity to the maps of the Ma'mun geographers, although they also involve a not inconsiderable expansion and improvement to the Mediterranean and particularly to North-East, Eastern and Central Asia. One fact which has not been given its due attention in the history of cartography is that around 1265 in South-West Europe a world map emerged which shows striking divergences from contemporary European maps, revealing an astonishing similarity to the world maps of the Ma'mun geographers and al-Idrisi (Fig. 6). About a third of a century later, as the 13th century turned into the 14th, a series of maps emerged which give an almost correct representation of the Mediterranean and the Black Sea. Historians of cartography called them, not quite accurately, portolan maps. The question of their origin has been discussed for about 150 years. According to some scholars they are said to have emerged suddenly; their originators, they claim, were European navigators. Some other historians of cartography associate them with various older cultures. Basing his assertions on the rudimentary knowledge of Arab geography of the time, Joachim Lelewel (ca. 1850), the first or one of the first scholars to discuss the origin of these maps, was convinced that the maps were derivatives of the map and the geographical work of al-Idrisi (Fig. 7). Emergence of a new type of map in Europe A thorough study of this question in the light of the history of the mathematical geography and cartography of the Arab-Islamic world shows that not only these so-called portolan maps but also the European world and regional maps which begin to appear shortly afterwards have direct or indirect links with maps from the Arab-Islamic world down to the 18th century. Research in the history of cartography dealing with the origin of both the so-called portolan maps and the representations of Asia and Africa on world and regional maps in the course of the subsequent period has always treated these questions in isolation, as separate issues, and in almost total ignorance of the mathematical geography and cartography of the Arab-Islamic world. Whereas the issue of the origin of the portolan maps is regarded as an unsolved mystery, the important new parts of the inhabited world and

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their topographic elements, which crop up for the first time on the world and regional maps, arc explained as the achievements of European mapmakers, which were made possible by the explorations of travellers and their travel reports. According to this view, a mapmaker located in, say, Venice, in Genoa or on Majorca is supposed to have been capable of drawing a near-perfect outline of the Caspian Sea, the Indian subcontinent or even of a relatively small lake such as Lake Urmia, simply on the basis of travellers' reports or explorations. Does this not mean that we are ascribing superhuman ability to a mapmaker, are we not expecting from him a feat which he could not possibly achieve? Would it not be more acceptable and logical to entertain the possibility that this or that mapmaker might perhaps have got hold of a map which had been produced locally and which could only have been made in the course of centuries, as the result of the work of several generations? Influence of Ptolemaic geography on cartography in Europe In the last quarter of the 15th century a new tendency arose in European cartography, as a result of the printing of the Latin translation of Ptolemy's Geography. Many maps bearing Ptolemy's Latinised name which were not completely identical with the content of his Geography managed to come into circulation (Fig. 8). These and others based on them, which emerged over a period of about 50 years, were provided with graticules on which the longitude of the Mediterranean, for example, was 63°, and the southern tip of the Indian subcontinent was 125°. Whereas this 'Ptolemaic1 graticule was able to survive on some world maps down to the mid-16th century and for a few years after, from about 1510 it was superseded on most world maps in the dimensions mentioned by the graticule of the Ma'mun world map, where the longitude of the Mediterranean was 52° or 53° and the longitude of the southern tip of India 115°. Break with Ptolemaic geography The three-part map of Asia and the new world map published in 1560 and 1561 by Giacomo Gastaldi had a revolutionary effect. This Italian engineer and cartographer, who had dedicated himself for about 30 years to drawing 'Ptolemaic' maps, now published maps of a completely different type, with a different graticule, different outlines,

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a new topography and toponymy. How and why did this happen? He gave no explanation for it himself. Several years later his two colleagues Abraham Ortelius (Fig. 9) and Gerard Mercator, the most famous cartographers of their times, published their own version of Gastaldi's map of Asia, with some alterations or additions. What criteria did they have to assume that this map was correct or more correct than others? Where did Gastaldi's co-ordinates come from? Ortelius thought that he had got to the bottom of the secret. In the bottom right-hand corner of his map he noted down: 'We hereby offer our revered readers a new representation of Asia, which Jacobus Gastaldus, a man who has rendered sterling service to geography, compiled according to the tradition of the Arab cosmographer Abu 1-FidaV Ortelius is referring here to the book of comparative tables of co-ordinates by the Arab geographer Abu 1-Fida' (d. 1331), a manuscript of which had been brought back from Istanbul to France by the French Orientalist Guillaume Postel in 1524. Although the book contained co-ordinates which in the Islamic world were long since outdated and replaced by more correct ones, in Europe the author was celebrated in the second half of the 16th century as a new Ptolemy, familiarity with his book was hailed as 'venit divinamente in luce ...' or 'coming divinely to light in our time'. In reality neither would the co-ordinates of Abu 1-Fida's book have sufficed to draw the configuration of the Gastaldi map, nor did the map agree with the book's data. My view is that Gastaldi must have used a general map or several regional maps from the Arab-Islamic world as his sources. How expertly he used these is a separate question. Not only the incorrect explanation which Ortelius gave for the origin of the Gastaldi map permits us to conclude that those geographers in Europe who were the leading exponents of their subject in their day were not aware of how their source maps had originated and where they came from, apart from the fact that they did not know, or rather could not have known which of the source maps known to them corresponded best to reality. A cartographer produced a map because he was interested in it himself, or for commercial purposes, or because he had been commissioned to do so, following a source map which happened to be available or one which was aesthetically particularly pleasing or one which had just arrived from the Arab-Islamic world. The choice was arbitrary. One aspect of the methods of work used by a European cartographer between the 14th and 18th centuries is that he had no compunction

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about inserting a regional map which had come to his attention into a general map or world map, without being able to assess whether the result was correct. The cartographic history of the Caspian provides us with an interesting example of this. It is an amazing fact that the Caspian Sea circulates on regional maps in Europe from the 14th century onwards in the near-perfect shape which had been attained in the 13th century in the Arab-Islamic world. In the 14th and 15th centuries it is represented largely accurately on European world maps, but then disappears (with a few exceptions) from the mapmakers' field of vision in the 16th and 17th centuries, only to resurface in the first quarter of the 18th century. Relationship of maps to co-ordinates in Europe This observation is closely connected with the fact that the maps of the Old World compiled in Europe up to the 18th century had not yet been drawn up using co-ordinates, being inserted into existing graticules by graphically transferring the data of the relevant source maps. Although there were many tables of co-ordinates in existence in the West which had been taken from the Arab-Islamic world or even been compiled in Europe, they remained, with the exception of some parts of Europe, without any influence on the maps being compiled there. The only attempt known to us, that by Johannes Kepler, to create a link between the co-ordinates of the tables known to him and the representation of the Old World, was a failure. Wilhelm Schickard in the 1630s seems to have been the first scholar to come to the conclusion that the maps of the Old World in circulation in Europe, especially as regards the representation of Asia and Africa, were seriously flawed and that he could compile a more accurate map on the basis of Arabic tables of co-ordinates and on the data contained in Arabic geographical works. My view is that in connection with this it is very significant what the Dutch geographer Willem Janszoon Blaeu wrote to Schickard in 1634: 'What you noticed about the longitude between Alexandria and Rome is what I have always thought to be true, in accordance with the observations of our compatriots, that in fact the whole of Europe was represented as too long.1 The attempts made by Schickard over many years to find out the co-ordinates of the book of tables by Abu l-Fida>, so as then to be able to draw up, by using additional Arab geographical works, a more accurate map of the Old World than those current in Europe, show

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that he did not consider the possibility that it might be more expedient to obtain maps from the Arab-Islamic world and publish them at his own discretion. There can be no doubt that he knew as little as his predecessors and successors about how and under what conditions the maps circulating in Europe had originated. Indeed, he could not have known that these originally derived from maps from the Arab-Islamic world which represented different stages of development and reached Europe more by accident, via many different types of contact - wars, travellers and seafarers, the Crusades or ambassadors. Although there are older Portuguese, Spanish, Italian or Dutch sources which provide us with clues to this reality, these have not yet had sufficient impact on the awareness of historians of cartography or have in some cases been subjected to arbitrary interpretation and relegated to the realm of legend. Deliberate transmission of Arab maps to Europe The period of deliberate transmission of maps from the Arab-Islamic world began a few years after the attempt by Schickard mentioned above. To our present knowledge it was the German scholar Adam Olearius who was the first to unambiguously state that he had converted maps from Arabic script into Latin. Those in question were a map of Persia and one of Anatolia to which his attention had been drawn in 1637 during his stay in Shamakhia (in the Caucasus), together with additional regional maps (Fig. 10). This kind of transmission of maps from the Arab-Islamic world intensified in Paris between ca. 1650 and 1750 and is thereby linked to the beginning of the creative period of European cartography. In this I am ignoring the repeated clear references by Portuguese seafarers since Vasco da Gama that they saw, captured, copied or brought back Arab maps or nautical charts, likewise the remark by the Dutch cartographer Jan Huygen van Linschoten (Fig. 11) that he had translated the map of South-West Asia and India going by his name from a local language into his own. The maps by Olearius, those of the Paris School and many of the preceding world maps up to 1560 lead us directly or indirectly to a graticule on which they are based, whose prime meridian lies 28°30f west of Toledo, just as it was fixed half a millennium previously in the Islamic world. If historians of cartography had paid due attention to the indications pointing to this in the graticules of the maps of Adam Olearius, Nicolas Sanson, Adrian Reland (Fig. 12), Guillaume

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Delisle, Joseph-Nicolas Delisle (Fig. 13), Jean-Baptiste Bourguigoon d'Anville, Emmanuel Bowen, James Rennell and others and if some of the tables of co-ordinates accessible in European languages had been compared with the corresponding maps still surviving from the Arab-Islamic world, our discipline would have been spared much futile effort and fruitless discussions.

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Fig. 1: The world map commissioned by Caliph aUMa'mun in Ihc first third of the 9lh century, in a copy daling back to 1340. Its outstanding feature-apart from its globular projection - is a continuous ocean surrounding the landmasscs, showing Africa as a continent which can be circumnavigated and the Indian Ocean - in contrast to Its Ptolemaic representation as an inland sea - as an open sea.

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Fig, 2: Reconstruction of the world map of Caliph al-Ma'mun according 10 the data in the surviving book of co-ordinates of one of the Ma'mfln geographers. A comparison with the surviving map [Fig. 1] shows that they are basically identical and that in addition in several details the reconstruction convevs a more precise idea of the lost original than the surviving version, which has been modified by repeated copying.

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Fig. 3: World map from Ptolemy's 'Geography* in a manuscript from the first half of the 14th century, reconstructed by the Byzantine scholar Maximos Planudes, In contrast to the Ma'mun geography [Figs. 1 and 2), the Indian Ocean and the North Atlantic are still shown here as inland seas.

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Fig. 4: Schematic diagram of the routes measured by al-Biruni in the first quarter of the 11th century and of latitudes ascertained astronomically for calculating the longitudes of ca> 60 places between Baghdad and Ghazna.

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Fig. 5: World map of al-Idris! (compiled 1154), copy from 1500. The map is largely based on the Ma'mun map [Figs. 1 and 2J, Whai is striking is the substantially improved representation of North and North-East Asia, which for centuries had a significant influence on later European maps of Asia.

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Fig. 6: The oldest known European imitation of the world maps of the Ma'mun geographers [Figs. 1 and 2] and al-ldris! [Fig. 5], preserved in the encyclopedic work Tresor' by Brunette Latini (ca. 1265), where there is no connection between the text of the book and the map, the latter being an exotic insertion in the book.

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Fig. 7: World inap by Marino Sanuto - Petrus Vesconte (ca, 1320), an imitation of the world map of al-Idrisi [Fig. 5], as is clearly evident from the basic outlines and the details.

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Fig. 8: Pseudo-Ptolemaic world map from Ptolemy's 'Geography', Strasbourg 1513. Africa appears in near^perfect shape, while South-East Asia is represented in a very old-fashioned way reminiscent of the Ma'mun geography |Figs. 1 and 2\. Neither can be reconciled with the Ptolemaic image of the world.

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Fig. 9: Map of Asia by Abraham Ortelius (Antwcq), 1561), published as a new edition of the Gastaldi map. In the bottom right-hand corner Ortelius remarks that Gastaldi compiled this map in the Arab tradition.

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5 Pig. \0: Persia and adjacent areas*, compiled by Adam Olearius in 1637 on the basis of two Arabic regional maps and transliterated into Latin script, as he clearly indicates in his 'Vennehrte Moscowitische und Persianische Reisebeschreibung1 IScWcswig, 1656, p. 434].

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*£' I Fig, 11: Map of India and adjacent areas by the Dutchman Jan Huygen van Lmschoien (1596), which he himKself states was transliterated into Latin script from an Oriental source map, Topography and toponymy of the map leave no room for doubi that the source was an Arabic map.

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Fig, 12: Illustration of the Persian Empire from the writings of the greatest Arab and Persian geographers' by Adrian Reland (Amsterdam, 1705), one of the European cartographers who expressly mention their Oriental sources. The reason why the northern part of the Caspian Sea. which was not part of the Persian Empire, is missing on the map is probably the fact that Reland used a Persian map as his source.

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Pig. 13: Exact Ottoman map of the Black Sea» whose zero meridian lies 28°30f west of Toledo in the Atlantic, in accordance with Arab-Persian tradition. The longitudes and latitudes given in the margins prove that the representation of the sea by the Ottoman geographers has almost attained perfect dimensions. The French cartographer G. Delisle made use of a copy or of the original of this map, which had reached Paris before 1700.

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[18] Science as a Western Phenomenon ROSHDI RASHED Philosophers, historians, and sociologists of science all accept as a basic postulate that science is essentially Western. This postulate is still conditioning contemporary scientific ideologies. This article analyzes the characteristics, history, and validity of this doctrine by means of a confrontation with one of the non-Western scientific contributions: science written in Arabic. Classical science is essentially European, and its origins are directly traceable to Greek philosophy and science; this tenet has survived intact through numerous conflicts of interpretation over the last two centuries. Almost without exception, the philosophers accepted it. Kant, as well as Comte, the neo-Kantians as well as the neopositivists, Hegel as well as Husserl, the Hegelians and the phenomenologists as well as the Marxists, all acknowledge this postulate as the basis of their interpretations of Classical Modernity. Even until our time, the names of Bacon, Descartes, and Galileo (sometimes omitting the first, and sometimes adding a number of others) are cited as so many markers on the road to a revolutionary return to Greek science and philosophy. This return was understood by all to be both the search for a model and the rediscovery of an ideal. One might impute this unanimity to the philosophers' zeal to pass beyond the immediate data of history, to their wish for radical insight, or to their effort to seize what Husserl describes as "the original phenomenon (Urphdnomen) which characterizes Europe from the spiritual point of view." One would expect that the position taken by those who have stuck

more closely with the facts of the history of science would be quite different, but such is not the case. This same postulate is adopted by the historians of science as a point of departure for their work, and especially for their interpretations. Whether they interpret the advent of classical science as the product of a break with the Middle Ages, whether they defend the thesis of continuity without breaking or cutting, or whether they adopt an eclectic position, the majority of historians agree in accepting this postulate more or less implicitly. Today, in spite of the works of many scholars on the history of Arabic and Chinese science, in spite of the wide representation of non-Western scientists in Dictionary of Scientific Biography, the works of the historians rest on an identical fundamental concept: in its modernity as well as in its historical context, classical science is a work of European man alone. Furthermore, it is essentially the means by which this branch of humanity is defined. Occasionally the existence of a certain practical science in other cultures might be acknowledged; nevertheless, it rests outside history, or is integrated into it only to the extent of its contributions to the essentially European sciences. These are only technical supplements which do not modify the intellectual configuration or the spirit of the latter in any way. The image given of Arabic science constitutes an excellent illustration of this approach. Essentially it consists of a conservatory of the Greek patrimony, transmitted intact or enriched by technical

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innovation to the legitimate heirs of ancient science. In all cases, scientific activity outside Europe is badly integrated into the history of the sciences; rather, it is the object of an ethnography of science whose translation into university study is nothing more than Orientalism. The effects of this doctrine are not limited to the domain of science, its history, and its philosophy. It is at the center of the debate between modernism and tradition. As was the case in eighteenth-century Europe, we find, in certain Mediterranean and Asian countries of today, that science (which is qualified as European) is identified with modernism. Our purpose here is not to redress wrongs, nor to oppose to that science qualified as European an alleged Eastern science. It is simply a matter of understanding the significance of the European determination of the concept of classical science, grasping the reasons for it and measuring its importance. We shall begin by sketching the history of this view of European science and then estimate its effects. We shall limit ourselves to posing the problem and advancing several hypotheses, and we also add these two restrictions: the only non-European science considered is one which was produced by various cultures, by scholars of different beliefs and religions, all of whom wrote their science principally, if not exclusively, in Arabic. As for the tenets of the history of the sciences, we shall most often cite those of the French historians. The concept of a European science is already present in the works of the historians and philosophers of the eighteenth century. In the debate of the Ancients and the Moderns, scholars and philosophers referred to science to define modernity where one combines reason and experience. Historical induction intended to give its concrete determinations to this dogmatic debate, so as to render the superiority of the Moderns indisputable. But the West was already being identified as Europe, and "Oriental wisdom" was already counterpoised against the natural philosophy of the post-Newtonian West, such as we find in Montesquieu's Persian Letters (1721). Classical science is European and Western only to the degree that it represents a stage in the continuous and regular development of humanity. The Discours Preliminaire of Abbe Bossut in Diderot's Encyclopedie Methodique offers an illustration of this concept. Dividing the history of the progress of the exact sciences into three periods, this tableau allows conjecture, alleged facts, and facts to intermingle. Its initial postulate is that ".. .all of the eminent peoples of the ancient world liked and cultivated mathematics. The most distinguished among them are the Chaldeans, the Egyptians, the Chinese, the Indians, the Greeks, the Romans, the Arabs, etc... in modern times, the western

nations of Europe." Classical science is European and Western because, writes Abbe Bossut, ".. .the progress made by the western nations of Europe in the sciences from the sixteenth century to our times utterly effaces those of other peoples." The concept of Western science changed in nature and extent at the turn of the nineteenth century. With what Edgar Quinet called the "Oriental Renaissance", the conceptualization was completed in its anthropological dimension in the last century. This Oriental Renaissance ended by discrediting science in the East. If it is true that the eighteenth-century concept still survived here and there, from the first years of the nineteenth century the materials and ideas of Oriental studies contributed the most to the makeup of the historical themes of the different philosophies. In Germany as well as in France, the philosophers adopted Oriental studies for diverse reasons in accordance with an identical representation: the East and the West do not oppose each other as geographical, but as historical positivities; this opposition is not limited to a period of history, but goes back to the essence of each term. In this regard, Lessons on the History of Philosophy and other works of Hegel can be invoked. Also at this time, as is shown by the French Restoration philosophers, the themes of the "call of the East" and the "return to the Orient" appear, which translate as a reaction against science, and more generally, against Rationalism. But it is with the advent and growth of the German philological school that the notion of science as a Western phenomenon was regarded as having been endowed with the scientific, and no longer purely philosophical, support which had been lacking until then. This influence also extended into mythological and religious studies. For example, Friedrich von Schlegel distinguishes two classes of language: the flexional IndoEuropean languages, and others. The former are "noble," the latter less perfect. Sanskrit, and consequently, German, considered the closest to it, is ".. .a systematic language and perfect from its conception"; it is "...the language of a people not composed of brutes, but of clear intelligence." There is nothing surprising in this; with the advent of the German school we are akeady in the realm of classifying mentalities. From now on everything is in place for effecting the passage from the history of languages to history through languages. The comparative study of religions and myths is developed around the middle of the century by A. Kuhn and Max Miiller in particular. The classification of mentalities is perfected. It is from the basis of these tenets and dating from this period that one of the most important efforts to establish the notion of science as Western and European in an allegedly scientific manner is elaborated. This project achieves its full extent in France in the work of Ernest Renan.

Studies in the Making of Islamic Science Science as a Western phenomenon

For Renan, civilization is divided between Aryans and Semites; the historian only has to evaluate their contributions in a differential and comparative manner. The notion of race would constitute the foundation of historiography. By race, one meant the whole of the "...aptitudes and instincts which are recognizable solely through linguistics and the history of religions." In the last analysis, it is for reasons attributable to the Semitic languages rather than the Semites themselves that they did not and could not have either philosophy or science. "The Semitic race," writes Renan, "is distinguished almost exclusively by its negative traits: it has neither mythology, epic poetry, science, philosophy, fiction, plastic arts nor civil life." The Aryans, whatever their origin, define the West and Europe at one and the same time. Arabic science is, ".. .a reflection of Greece, combined with Persian and Indian influences." The historians of science borrowed not only their representation of the Western essence of science from this tradition, but also some of their methods for describing and commenting on the evolution of science. Thus, they applied themselves to discovering the concepts and methods of science and to following their genesis and propagation through philological analyses of the terms and on the basis of the texts at their disposal. Like the historian of myths or of religion, the historian of the sciences must be a philologist as well. In France, the situation is such that philosophers borrow Renan's interpretation and, often, even his terminology. Even though this brand of anthropology has already been abandoned by historians, they nevertheless preserve and propagate a series of inferences engendered by it. These can be enumerated as follows: 1. Just as science in the East did not leave any consequential traces in Greek science, Arabic science'has not left any traces of consequence in classical science. In both cases, the discontinuity was such that the present could no longer recognize itself in its abandoned past. 2. Science subsequent to that of the Greeks is strictly dependent upon it. According to Duhem, "...Arabic science only reproduced the teachings that it received from Greek science." In a general fashion, Tannery reminds us that the more one examines the Hindu and Arabic scholars, ".. .the more they appear dependent upon the Greeks.. .(and).. .quite inferior to their predecessors in all respects." 3. Whereas Western science addresses itself to theoretical fundaments, Oriental science, even in its Arabic period, is defined essentially by its practical aims. 4. The distinctive mark of Western science is its conformity to rigorous standards; in contrast, Oriental science in general, and Arabic science in particular,

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lets itself be carried away by empirical rules and methods of calculation, neglecting to verify the soundness of each step on its path. The case of Diophantus illustrates this idea perfectly: as a mathematician, said Tannery, "...Diophantus is hardly Greek." But when he compares the Arithmetics of Diophantus to Arabic algebra, Tannery writes that the latter ".. .in no way rises above the level achieved by Diophantus." 5. The introduction of experimental norms which, according to historians, totally distinguishes Hellenistic science from classical science, is solely the achievement of Western science. Thus it is to Western science alone that we owe both the concept and experimentation. We are not going to oppose this ideology to another. We propose simply to confront some of these elements with the facts of the history of science, beginning with algebra and concluding with the crucial problem of the relationships between mathematics and experimentation.

Algebra As with the other Arabic sciences, algebra had practical aims, a flair for calculation, and an absence of rigorous standards. It is precisely this that allowed Tannery to make his statement. Bourbaki took this as his authorization to exclude the Arabic period when he retraced the evolution of algebra. The historical writings of the modern mathematician Dieudonne are significant; between the Greek prehistory of algebraic geometry and Descartes, he finds only a void, which, far from being frightening, is ideologically reassuring. Some historians cite al-Khwarizml, his definition of algebra and his solution of the quadratic equation, but it is generally to reduce Arabic algebra to its initiator. This restriction misconstrues the history of algebra, which in actuality does not show a simple extension of the work of al-Khwarizml in the West, but an attempt at theoretical and technical overtaking of his achievements. Moreover, this overtaking is not the result of a number of individual works, but the outcome of genuine traditions. The first of these traditions had conceived the particular project of arithmetizing the algebra inherited from al-Khwarizml and his immediate successors. The second one, in order to surmount the obstacle of the solution by radicals of third and fourth degree equations, formulated in its initial stage a geometric theory of equations, subsequently to change viewpoint and study known curves by means of their equations. In other words, this tradition engaged itself explicitly in the first research in algebraic geometry. As we have said, the first tradition had proposed arithmetizing the inherited algebra. This theoretical program was inaugurated at the end of the tenth century

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by al-KarajT, and is thus summarized by one of his successors, al-Samaw'al (d. 1176): "to operate on unknowns as the arithmeticians work on known quantities." The execution is organized into two complementary stages. The first is to apply the operations of elementary arithmetic to algebraic expressions systematically; the second is to consider algebraic expressions independently from that which they can represent so as to be able to apply them to operations which, up to that point, had been restricted to numbers. Nevertheless, a program is defined not only by its theoretical aims, but also by the technical difficulties which it must confront and resolve. One of the most important of these was the extension of abstract algebraic calculation. At this stage, the mathematicians of the eleventh and twelfth centuries obtained some results which unjustly are attributed to the mathematicians of the fifteenth and sixteenth centuries. Among these are the extension of the idea of an algebraic power to its inverse after defining the power of zero in a clear fashion, the rule of signs in all its general aspects, the formula of binomials and the tables of coefficients, the algebra of polynomials, and above all, the algorithm of division, and the approximation of whole fractions by elements of the algebra of polynomials. In a second period, the algebraists intended to apply this same extension of algebraic calculation to irrational algebraic expressions. Al-KarajT questioned how to operate by means of multiplication, division, addition, subtraction, and extraction of roots on irrational quantities. To answer this question the mathematicians gave, for the first time, an algebraic interpretation of the theory contained in Book X of the Elements. This book was regarded by Pappus, as well as by Ibn al-Haytham much later, as a geometry book, because of the traditional fundamental separation between continuous and discontinuous magnitudes. With the school of al-KarajT, a better understanding of the structure of real algebraic numbers is achieved. In addition, the works of this algebraic tradition opened the route to new research on the theory of numbers and numerical analysis. An examination of numerical analysis, for example, reveals that after renewing algebra through arithmetic, the mathematicians of the eleventh and twelfth centuries also effected a return movement to arithmetic to search for an applied extension of the new algebra. It is true that the arithmeticians who preceded the algebraists of the eleventh and twelfth centuries extracted square and cube roots, and had formulas of approximation for the same powers. But, lacking an abstract algebraic calculation, they could generalize neither their results, their methods, nor their algorithms. With the new algebra, the generalization of algebraic calculation became a constituent of numerical analysis which, until

then, had only been a sum of procedures, if not prescriptions. It is in the course of this double movement which is established between algebra and arithmetic that the mathematicians of the eleventh and twelfth centuries achieved results which are still wrongly attributed to the mathematicians of the fifteenth and sixteenth centuries. This is the case with the method attributed to Viete for the resolution of numerical equations, the method ascribed to RuffiniHorner, the general methods of approximation, in particular that which D. T. Whiteside designates by the name of al-Kashl-Newton, and finally, the theory of decimal fractions. In addition to methods, which were to be reiterative and capable of leading in a recursive manner to approximations, the mathematicians of the eleventh and twelfth centuries also formulated new procedures of demonstration such as complete induction. We have just seen that the concept of polynomials is among those elaborated by the algebraist arithmeticians from the end of the tenth century. This tradition of algebra as the "arithmetic of unknowns", to use the expression of the time, opened the road toward another algebraic tradition which was initiated by cUmar al-Khayyam (eleventh century), and renewed at the end of the twelfth century by Sharaf al-Dm al-Tusi. While the former formulated a geometric theory of equations for the first time, the latter left his mark on the beginnings of algebraic geometry. The immediate predecessors to al-Khayyam, such as al-Blrum, al-Maham, and Abu al-Jud, had already been able, in contrast to the Alexandrian mathematicians and precisely because of the concept of the polynomial, to treat the problems of solids in terms of third degree equations. But al-Khayyam was the first to address these unpondered questions: can one reduce the problems of straight lines, planes, and solids to equations of a corresponding degree, on the one hand, and on the other, reorder the group of third degree equations to seek, in the absence of a solution by factoring, solutions which can be reached through means of the intersection of auxiliary curves? To answer these questions, al-Khayyam is led to formulate the geometric theory of equations of a third or lesser degree. His successor, al-Tusi, did not delay in changing perspective; far from adhering to geometric figures, he thought in terms of functional relations and studied curves by means of equations. Even if al-Tusi still solved equations by means of auxiliary curves, in each case the intersection of the curves is demonstrated algebraicly by means of their equations. This is important, since the systematic use of these proofs introduces into the practice instruments which were already available to the mathematical analysts of the tenth century: affine transformations, the study of the maxima of algebraic expressions, and with the aid of what will later be regarded as

Studies in the Making of Islamic Science Science as a Western phenomenon

derivatives, the study of the upper bounds and lower bounds of roots. It is in the course of these studies and in applying these methods that al-Tusi grasps the importance of the discriminant of the cubic equation and gives the so-called Cardan formula just as it is found in the Ars Magna. Finally, without enlarging any further on the results which were obtained, we can say that both on the level of results as well as that of style, we find al-Khayyam and al-Tusi fully in the field allegedly pioneered by Descartes. If we exclude these traditions and justify this exclusion by invoking the practical and computational aims of the Arab mathematicians and an absence of rigorous standards of proof in their work, we can say that the history of classical algebra is the work of the Renaissance. Among the mathematical disciplines, algebra is not a unique case. To varying degrees, trigonometry, geometry, and infinitesimal determinations are likewise illustrative of the preceding analysis. In a more general sense, optics, statistics, mathematical geography, and astronomy are also no exception. Recent works in the history of astronomy render Tannery's understanding of the Arab astronomers and the interpretations which he gives of them manifestly outmoded, if not erroneous. But since we assigned ourselves the task of examining the doctrine of the Western nature of classical science, we shall restrict our discussion to an essential component of this doctrine, experimentation.

Experimentation In fact, is not the cleavage between the two periods of Western science, the Greek period and the Renaissance, often marked by the introduction of experimental norms? Undoubtedly the general agreement of the philosophers, historians, and sociologists of science stops here; the divergences become apparent as soon as they attempt to define the meaning, the implications, and the origins of these experimental norms. The origins are linked in one case to the current of AugustinianPlatonism, in another to the Christian tradition, and particularly to the dogma of Incarnation, in a third case to the engineers of the Renaissance, in a fourth to the Novum Organum of Francis Bacon, and finally, in a fifth, to Gilbert, Harvey, Kepler, and Galileo. Some of these attitudes superimpose upon one another, become entangled or contradictory, but they all converge on one point: the occidental nature of the new norms. Nevertheless, as early as the nineteenth century, historians and philosophers such as Alexander von Humboldt in Germany and Cournot in France diverge from this predominating position to attribute to the Arab period the origins of experimentation. It is difficult to analyze the origins or the beginnings of

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experimentation correctly, since no study has been made of the interrelations of the different traditions and the different themes to which the concept of experimentation has been applied. Perhaps it would be in writing such a history, especially a history of the term itself, that one could give an accounting of the multiplicity of uses and ambiguities of the concept. For this analysis two histories are needed: the history of the relationship between art and science and that of the links between mathematics and physics. With the history of the relationship between science and art, we are in a position to understand when, why, and how it became accepted that knowledge can emanate from demonstrations and from the rules of practice at the same time, and that a body of knowledge possesses the stature of a science while, at the same time, it is conceived in its possibilities of practical realization with an external purpose. The traditional opposition between science and art seems likely to be the work of the intellectual currents of the Arabic period. Certainly one fact is striking: whether we are dealing with Muslim traditionalists, rationalist theologians, scholars of different fields, or even philosophers of the Hellenistic tradition such as al-KindT or al-Farabl, all contribute to the weakening of the traditional differentiation between science and art. In other respects, this general trait is at the origin of the opinion of some historians regarding the practical spirit and realistic imagination of the Arab scholars. Knowledge is accepted as scientific without its conforming either to the Aristotelian or to the Euclidean scheme. This new concept of the stature of science promoted the dignity of scientific understanding of disciplines which traditionally were confined to the domain of art, such as alchemy, medicine, pharmacology, music, or lexicography. Whatever might be the importance of this concept, it could only lead to an extension of empirical research and to a diffuse notion of experimentation. One does witness the multiplication and systematic use of empirical procedures: the classifications of the botanists and the linguists, the control experiments of the doctors and alchemists, and the clinical observations and comparative diagnostics of the physicians. But it was not until new links were established between mathematics and physics that such a diffuse notion of experimentation acquired the dimension that determines it, a regular and systematic component of the proof. Primarily it is in Ibn alHaytham's work in the field of optics where the emergence of this new dimension can be perceived. With Ibn al-Haytham the break is established with optics as the geometry of vision or light. Experimentation had indeed become a category of the proof. The successors of Ibn al-Haytham, such as al-FarisT, adopted experimental norms in their optical research,

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such as that performed on the rainbow. What did Ibn al-Haytham understand by experimentation? We will find in his work as many meanings of this word and as many functions served by experimentation as there are links between mathematics and physics. A thorough look at his texts indicates that the term and its derivatives belong to several superimposed systems, and are not likely to be discerned through simple philological analysis. But if attention is fixed on the content rather than the lexical form, one can distinguish several types of relationships between mathematics and physics which allow one to spot the corresponding functions of the idea of experimentation. In fact, the links between mathematics and physics are established in several ways; even if they are not specifically treated by Ibn al-Haytham, they underlie his work and are amenable to analysis. As for the field of geometric optics, which was reformed by Ibn al-Haytham himself, the only link between mathematics and physics is a similarity of structures. Owing to his definition of a light ray, Ibn al-Haytham was able to formulate his theory on the phenomena of propagation, including the important phenomenon of diffusion, so that they relate perfectly to geometry. Then several experiments were devised to assure technical verification of the propositions. Experiments were designed to prove the laws and rules of geometrical optics. The work of Ibn al-Haytham attests to two important facts which are often insufficiently stressed: first of all, some of his experiments were not simply designed to verify qualitative assertions, but also to obtain quantitative results; in the second place, the apparatus devised by Ibn al-Haytham, which was quite varied and complex, is not limited to that of the astronomers. In physical optics one encounters another type of relationship between mathematics and physics and therefore a second meaning of experimentation. Without opting for an atomistic theory, Ibn al-Haytham states that light, or as he writes, "the smallest of the lights", is a material thing, external to vision, which moves in time, changes its velocity according to its medium, follows the easiest path, and diminishes in its intensity depending on its distance from its source. Mathematics is introduced into physical optics at this stage by means of analogies established between the systems of movement of a heavy body and those of the reflection and refraction of light. This previous mathematical treatment of the concepts of physics permitted them to be transferred to an experimental plane. Although this situation on the experimental level might be somewhat approximate in nature, it nevertheless furnishes a level of existence to ideas which are syntactically structured, but semantically indeterminate, such as Ibn al-Haytham's scheme of the movements of a projectile.

A third type of experimentation, which was not practiced by Ibn al-Haytham himself but was made possible by his own reform and his discoveries in optics, appears at the beginning of the fourteenth century in the work of his successor al-FarisI. The links established between mathematics and physics aim to construct a model and to reduce by geometric means the propagation of light in a natural object to its propagation in an artificial object. The problem is to define for propagation, between the natural and the artificial object, some analogical correspondences which were genuinely certain of mathematical status. For example, they built a model of a massive glass sphere filled with water to explain the rainbow. In this case, experimentation serves the function of simulating the physical conditions of a phenomenon that can be studied neither directly nor completely. The three types of experimentation studied all reveal themselves both as a means of verification and as furnishing a plane of material existence to ideas which are syntactically structured. In the three cases, the scientist must realize an object physically in order to handle it conceptually. Thus, in the most elementary example of rectilinear propagation, Ibn al-Haytham does not consider any arbitrarily chosen opening of a black box, but rather specific ones, in accordance with specific geometric relationships, in order to realize as precisely as possible his concept of a ray. To recapitulate several points: 1. The tenet of the Western nature of classical science which was launched in the eighteenth-century owes to the Orientalism of the nineteenth century the image that we now recognize. 2. On the one hand, the opposition between East and the West underlies the critique of science and rationalism in general; on the other, it excludes the scientific production of the East from the history of science both de facto and de jure. An absence of rigor is invoked, as well as the computational attributes and the practical aims of science written in Arabic, to justify this effective debarment from the history of science. 3. This tenet reveals a disdain for the data of history as well as a creative capacity for ideological interpretation, which are admitted as evidence for ideas that raise many more problems than they solve. Thus we have the notion of a Scientific Renaissance, when in several disciplines everything indicates that there was merely a reactivation. These pieces of pseudoevidence quickly become conceptual bases for a philosophy or sociology of science, as well as the departure point for theoretical elaborations in the history of science. We must ask ourselves if the moment has not arrived to abandon this characterization of classical science and

Studies in the Making of Islamic Science Science as a Western phenomenon its still lively traces in the writing of history, to restore to the profession of the historian of science the objectivity required of it, to ban the clandestine importation and diffusion of uncontrolled ideologies, to refrain from all reductionist tendencies which favor similarities at the expense of differences, and to be wary of miraculous events in history. The neutrality of the historian is not an a priori ethical value; it can only be the product of patient work which will not be duped by the myths which the East and West have engendered. Above all, it is necessary to cast out the periodization everywhere admitted in the history of science. The term used for classical algebra or classical optics, for example, will integrate the works which extended from the tenth to the seventeenth centuries. Consequently, it will realign not only the idea of the classical sciences, but also that of medieval science. The classical sciences will then reveal themselves as the product of the Mediterranean which was the hub of exchanges among all civilizations of the ancient world. Only then will the historian of science be able to enlighten the debate over modernism and tradition.

Science as a Western Phenomenon: Postscript The 26 years since the publication of the original French edition of the text above has been a very fertile period for the study of the history of Islamic science. Indeed, we have witnessed an unprecedented rebirth of this discipline. Texts have been written and translated, new collections have appeared, and specialized reviews and journals have been published. These have offered historians the possibility of developing and comparing their research findings with facts. The task remains huge, and we are only at the beginning, but at least this new growth of information puts to rest the argument of ignorance. With all this new information, one would have expected historians and philosophers to rectify the impressions and ideas they had inherited from the nineteenth century. One would have thought that the doctrine of Western science which we have described and analyzed here would have disappeared along with the props on which that doctrine was based. Indeed we were beginning to see a growing tendency to break with this doctrine and its implications. Then, for reasons which are extraneous to science and its history, images of Islamic society - if not of Islam itself - arose, according to which it was seen as irrational and intolerant and thus a society foreign to science. The aging doctrine was naturally given new life because of these images. How is it possible, under these conditions, to reconcile such an image of Islamic society with scientific results obtained from the heart of that

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same society? It was enough to back up the preceding doctrine with another, the doctrine of double marginality: with regard to the society which saw the development of science, and with regard to the history of the sciences. Thus, one could still write in 1992, "We must remember that at an advanced level the foreign sciences had never found a stable institutional home in Islam," or "Greek learning never found a secure institutional home in Islam, as it was eventually to do in the universities of medieval Christendom" (Lindberg 1992). As for the second marginality, we have already described how it works. Thus, we are back where we started and the doctrine of the Westernness of science is saved. Undoubtedly, these ideological views are beginning to give way, weakened by new research findings. And even if they are still capable of slowing down the acceptance of facts, it is not widespread, and it certainly will not last much longer. See also: ^Western Dominance, ^Ibn al-Haytham, ^cUmar al-Khayyam

References Al-Khayyam. L'algebre d'Omar Alkhayydmi. Ed. F. Woepcke. Paris: B. Duprat, 1851. Al-Samaw'al. al-Bdhiren algebre. Ed. R. Rashed. Damascus: University of Damascus, 1972. Cournot, A. A. Considerations sur la marche des idees et des evenements dans les temps modernes. Paris: Vrin, 1973. Demeunier, Jean Nicolas. Encyclopedic methodique. Paris: Pancoucke, 1784. Dieudonne, J. Cours de geometric algebrique. Paris: Presses universitaires de France, 1974. Duhem, P. Le systeme du monde. 10 vols. Paris: A. Hermann, 1913-1959. Hegel, G. W. F. Lectures on the History of Philosophy. New York: Humanities Press, 1968. Kojeve, A. The Christian Origin of Modern Science. Melanges Alexandre Koyre. Vol. 2. Paris: Hermann, 1964. 295-306. Lindberg, David. The Beginning of Western Science. Chicago: University of Chicago Press, 1992. 182. Montesquieu. Oeuvres completes. Paris: Gallimard, 1964. Rashed, R. Resolution des equations numeriques et algebre: Sharaf al-DI al-Tasi, Viete. Archive for History of Exact Sciences 12.3 (1974): 244. —. L'extraction de la racine nieme et 1'invention des fractions decimales. Archive for History of Exact Sciences 18.3 (1978): 191. Renan, E. Nouvelles considerations sur le caractere generate des peuples semitiques. Paris: Imprimerie imperiale, 1859. —. Histoire general et systeme compare des langues semitiques. Paris: Imprimerie imperiale, 1863. Sarton, G. The Incubation of Western Culture in the Middle East. Washington: Library of Congress, 1951. Schlegel, F. Essai sur la langue et la philosophic des Indiens. London: Bell & Daldy, 1871. Tannery, P. La Geometric grecque. Paris: Gauthier-Villars, 1887.

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Name Index 'Abd al-Rahman I 28 'Abd al-Rahman II, Emir 28, 29, 32 'Abd al-Rahman III, Caliph 31 'Abd Allah 26 'Abdallah, Ibrahim xi al-Abhari, Athir al-Din 111 Abu Hanlfa, xii Abu Ishaq, Shah 362 Abu Kamil 93 AbuMa'shar 29, 490 Abu Sa'd, 146 Abu al-Salt of Denia 45 Abu'l-'Uqul, 456 'Ali, 'Abd 413 Adelard of Bath 10, 32, 138, 139, 143, 490, 492 al-Afdal, al-Malik 37 Africanus, Leo 51 al-Akhawayn 230, 231, 234, 247 Alexander of Aphrodisias 48 Alexander the Great 122 Alfonso X of Castile 24, 26, 35, 36, 40, 46, 49, 502, 504 Ali, J. 420 Alonso, M. 44 Alquie, F. 378, 379 Amico, Giovanni Battista 508 Amir-Moez, A.R. 91 Anatolios of Berito, Vindanios 27 Anbouba, A. 91 al-Andalusi, Ibn Juljul 26 al-Andalusi, Muhyi al-Din al-Maghribi 131 al-Andalusi, Sa'id 9, 10, 11, 12 Anthemius of Tralles 75, 76, 78, 87 Apollonius of Perga 80, 81, 93, 95, 139, 140, 232, 275, 276, 281, 283, 284 Aquinas, Thomas 500 Arberry, Arthur John 76 Archimedes 77, 83, 84, 91, 139, 140, 141, 306, 307, 308, 309, 310, 312, 313, 320, 323, 324, 326, 335, 345 Aristotle xii, xviii, 3, 42, 46, 48, 49, 68, 89, 94, 108, 125, 128, 140, 145, 147, 302, 346, 351, 360, 365, 499, 500

al-Ash'ari 360, 367, 373, 374, 381 al-Ashraf, Sultan 442 al-Asturlabi, 'Ali ibn 'Isa 454, 459, 462 Artie, B. 41 Ausejo, E. 46 Awmatiyus 40 Bacon, Francis 537, 541 Bacon, Roger 147, 500 al-Baghdadl, 'Abd al-Qahir 4, 7 al-Baghdadi, Abu al-Barakat 47, 49 Bakhtlshu, Jurjls b. xii al-Baqillani, Abu Bakr 360, 375 Baqir, Muhammad 414 bar Hiyya of Barcelona, Abraham 66, 69 Barani, S.H. 451, 457 Barcelo, M.C. 12, 36, 50 Barmore, F.E. 419 al-Battani 10, 34, 35, 39, 66, 69, 70, 143, 144, 405, 457, 470, 492, 494, 495, 497, 501, 503, 504, 505 al-Baydawi, Nasir al-Din 361 Baznad of Valencia 51 Bede, Venerable 489 Beg Gurgan, Ulugh 70, 110, 130, 180, 181, 182, 183, 185, 196, 197, 230 Bel, A. 433 ben Eliyahu of Saloniki, Shelomo 70 ben Gerson, Levi 68, 69, 70 ben Judah of Marseilles, Samuel 68, 69 ben Kalonymos, Kalonymos 69 Ben Sham'un, Joseph ben Yehudah 46 ben Tibbon, Moshe 68 Ben-Eli, A. 433 Benedetti, Giovanni Battista 49 Berggren, J. Len xiv, xvi, 75-97, 137-48, 419 Berman, L.V 69 Bernard of Verdun 500 Bishr, Abu xii al-Birjandi, Abu 'Ali 230 al-Blrunlxi, xx, 7, 87, 90, 104, 130, 142, 229, 230, 305, 410, 416, 420, 447, 451, 452,

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453, 457, 458, 459, 460, 464, 465, 466, 473, 474, 496, 516, 540 al-Bitruji 45, 46, 47, 48, 49, 68, 89, 102, 255, 256, 499, 500, 505 Bjornbo,A. 35 Bolens, Lucie 43 Bonfils of Tarascon, Emmanuel 70 Bonine, M.E. 419 Borro, Girolamo 49 Bossut, Abbe 538 Bott, G. 420 Bourbaki, Nicolas 539 Bourguignon d'Anville, Jean-Baptiste 522 Bowen, Emmanuel 522 Brahe, Tycho 70, 114, 501, 508 Brahmagupta 31 Brice, W. 433 Britton, John 464 al-Buhturi, Ahmad 459 Bulgakov, P. 420 al-Buzjani, Abu al-Wafa 229 Calvo, E. 52 Campanus of Novara 138, 497, 498 Capella, Martianus 48 Carabaza, J.M. 41 Carandell, J. 36, 53 Carmody, Francis J. 46 Caro Baroja, J. 43 Casanova, Giacomo 433 Casulleras, J. 36 Catala, M.A. 35 Chabas, J. 45 Chasles, Michel 3 ChehabEddin, 10 Chioniades 508 Clagett, M. 320, 321, 323 Cluzan, S. 420, 433 Comes, M. 35, 36, 38, 45 Comte, Auguste xxi, 537 Copernicus, Nicolaus xvii, xviii, 39, 70, 104, 143, 145, 149, 178, 271, 274, 278, 279, 282, 283, 284, 285, 293, 296, 385, 386, 492, 494, 503, 504, 505, 506, 507, 508 Cournot, A.A. xxii, 541 Cruz Hernandez, M. 44 DaGama, Vasco 421 al-Dabbi, Ishaq 25 al-Dakhil, 'Abd al-Rahman I 24

Dallal,A. 270, 419 al-Danl, Abu 'Amr 'Uthman 4, 12 al-Dani, Abu al-Salt Umayya 43 DarendelT 446 Al-Dashtaghi, Mansur ibn Muhammad 230 Davidian, Marie-Louise 447 de Perceval, Caussin 456 de Premare, Alfred-Louis 43 de Vaux, Carra 104 Delisle, Guillaume 522 Delisle, Joseph-Nicolas 522 Delpont, E. 420, 433 Democritus 48 Descartes, Rene 377, 378, 379, 537, 539, 541 Dicuil 489 Diderot, Denis 538 Dietrich, A. 44 Dieudonne, J. 539 Dijksterhuis, EJ. 345 Al-Dimyati 398 Diophantus 78, 86, 92, 93, 539 Dioscorides 31, 32, 44, 51 Dizer, M. 433 Djebbar, Ahmed 38, 82, 85, 94 Dubler, C.E. 31 Duhem,R 47, 104, 499, 539 Ecchellensis, Abraham 139 Effendi, Salih 446 Eilmer of Malmesbury 29 el Arabigo, Bernardo 50 El-Faiz,M. 41 Empedocles 4 Endress, G. 95 Euclid xvi, xvii, 38, 66, 80, 84, 85, 86, 90, 91, 92, 95, 138, 139, 140, 143, 145, 146, 147, 157, 302, 303, 312, 323, 324, 346, 347 Eudoxus 47 Eulogius of Cordoba 26 Eustathius xii Eutocius 87, 313, 324 Euxodus 499 al-Farab l49, 95, 113, 127, 541 al-Farghani 81, 112, 305, 457, 497, 498 al-Farisi, Kamal al-Din 82, 85, 147, 541, 542 al-Fatih, Sultan Muhammad 179 Favaro, A. 345 al-Fazari 421,426, 457 al-Fida, Abu 474, 519, 520

Studies in the Making of Islamic Science Fischer, August 4 Folkerts, M. 12 Frank, J. 447 Freudenthal, Hans 95 Galenxii, 29, 30, 42, 111, 140, 245 Galileo, Galilei 49, 537, 541 Gandz, Solomon 7, 47, 67 Garbers, Karl 432 Garcia Ballester, Luis 51 Garcia Sanchez, E. 41, 42 Garijo, I. 32 Garlande, Johannes de 489 Gastaldi, Giacomo 518, 519 Gastaldus, Jacobus 519 Gauthier, L. 44 Gerard of Cremona 46, 145, 320, 323, 346, 492, 497, 501 al-Ghafiqi 44, 51 al-Gharbal 37 al-Gharnati, Abu Hamid 43 al-Ghazal, Yahya 29 al-Ghazali, Abu Hamid 110, 115, 116, 360, 365, 367, 368, 370, 376, 377, 378, 379, 380, 382 Gilbert, William 541 Glick, T.F. 42 Goblot, H. 42 God 115, 125, 128, 152, 184, 185, 187, 189, 193, 195, 254, 258, 293, 326, 335, 336, 346, 351, 353, 357, 360, 364, 367, 369, 373, 374, 376, 378, 379, 381, 388, 391, 395, 461, 462 Goldstein, Bernard R. xiv, 33, 40, 44, 45, 47, 65-72, 90, 252, 447 Goldziher, Ignaz 107, 108, 110 Grafton, A. 278, 284 Grant, Edward 49 Grosseteste, Robert 499 Gutas, Dimitri xii 29 al-Hadib 71, 72 Hadrian 126 al-Haggag 347 al-Hakam II 28, 30 al-Hakim, Caliph 418, 454, 455, 474 HammadI 151 Hanno 513 al-Harawi, Abu '1-Fadl 473 Harvey, William 541

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al-Hashimi, 'Ali ibn Sulayman 33 al-Hasib, Habash 30, 36, 87, 229, 407, 439, 444, 458, 459, 460, 461, 465, 466, 468, 469, 471, 472 Hawkins, G.S. 419 Hegel, G.WF. xxi, 537, 538 Heiberg, J.L. 81, 345 Hercules 513 Hermann of Carinthia 141 Hermann of Dalmatia 35, 490 Hermann the Lame 490, 491 Hermelink, H. 38 Herodotus 513 Heron of Alexandria 92 Herzfeld, E. 473 Hinz, W 464 Hipparchus 39, 155, 223, 514 Hippocrates of Chios 26, 30, 32, 42, 92 Hisham I, Emir 25 Hogendijk, J.R 34, 38, 80, 82, 91 Holmyard, E.J. 40 H0yrup, Jens 92, 95, 96 Hugonnard-Roche, Henri xx, 45, 489-508 Hulagu 132, 229, 506 Hunayn, Ishaq b. xvi Husayn, Muhammad 414 Husayn, Shah 416 Husserl, Edmund xxi, 537 Hypsicles 80 Ibn 'Abd al-'Aziz, Caliph 'Umar 420 Ibn 'Abdallah, Ahmad 455 Ibn 'Abdallah, Habash 461 Ibn Abi 'Amir, al-Mansur 26 Ibn Abi Mansur, Yahya 30, 456, 467, 468 IbnAblUsaybi'axi, 308, 310 Ibn Aflah, Jabir xiv, 44, 45, 67, 69, 71, 255 Ibn Ahmad, al-KhalTl xii Ibn Aktham, Yahya xx, 453, 459, 462, 463, 464, 465, 466, 467, 472, 474 Ibn al-A'lam 470 Ibn 'Ali, Sanad 112, 454, 469, 473, 474 Ibn Anas, Malik xii Ibn 'Asim, 'Abd Allah b. Husayn 37 Ibn al-'Awwam 27, 41, 42, 43, 44, 52 Ibn al-Adami 423 Ibn al-Banna of Marrakesh 15, 52 Ibn al-Baytar 44, 50, 51, 52 Ibn al-Bitriq, Yahya xii, 309 Ibn al-Buhturi, 'Ali 454

548

Studies in the Making of Islamic Science

Ibn al-Buhturi, Ahmad 462, 467 Ibn al-Gahm, 'All 302 Ibn al-Ha'im of Seville 45, 53 Ibn al-Hajj, Muhammad 51 Ibn al-Haytham, al-Hasan 38, 69, 77, 82, 84, 89, 91, 130, 132, 133, 139, 144, 145, 146, 147, 150, 162, 223, 229, 245, 246, 247, 248, 249, 250, 251, 252, 253, 255, 256, 385, 388, 410, 419, 500, 503, 506, 508, 540, 541, 542 Ibn al-Kammad 45 Ibn al-Khatib 49, 51, 52, 53 Ibn al-Majdi, Shihab al-Din 10 Ibn al-Muqaffa' xii Ibn al-Muthanna, Muhammad 33, 67 Ibn al-Nadlm xii, 4, 7, 80 Ibnal-Nafisll3 Ibn al-Qifti 9 Ibn al-Qutiyya 24 Ibn al-Raqqam 50, 53, 72 Ibn al-Saffar 32, 34, 36, 426 Ibn al-Samh 34, 36, 45 Ibn al-Sarraj 442 Ibn al-Shaddad, Abu al-Qasim 427 Ibn al-Shamir 29 Ibn al-Shatir of Damascus, 'Ala al-Din xvii, xviii, 37, 70, 71, 96, 104, 110, 113, 116, 117, 133, 134, 135, 145, 150, 178, 229, 230, 257, 258, 259, 260, 261, 270, 271, 273, 274, 279, 281, 282, 283, 284, 285, 291, 293, 296, 385, 387, 388, 428, 429, 430, 431, 433, 442, 506, 507, 508 Ibn al-Wardi 402 Ibn al-Yasamin, Abu Paris 8, 9, 10, 12 Ibnal-Zayyat 125 Ibn Amajur, 'Ali 436, 437, 439 Ibn Ayyub, Ya'qub 52 Ibn Badr 52 Ibn Bajja xiv, 24, 38, 43, 44, 46, 47, 48, 49, 67 Ibn Barmak, Khalid xi Ibn Bas, Husayn b. 52 Ibn Basil, Istifan 31 Ibn Baso, Hasan b. Muhammad 52, 53 IbnBassa l37, 41, 42 Ibn Buklarish 44 IbnBulbul, Saqr, Isma'il 161, 162 Ibn Butlan 44 Ibn Ezra, Abraham 33, 47, 67, 71, 492 Ibn Firnas, 'Abbas 28, 29, 30 IbnHajjaj 27, 37

Ibn Hasan, Ahmad 52 IbnHazm l09 IbnHibinta 33 Ibn Hudhayl, Abu Zakariyya 50 IbnHunayn, Ishaq 77, 88, 137, 138, 139, 161 Ibn 'Iraq, Abu Nasr 86 Ibn Iraq, Mansur ibn Nasr 229 Ibn'Isa, Ali 141, 467 Ibn Ishaq, Hunayn xii, 137, 309 Ibn Ishaq, Yahya 29 Ibn Jazla 44 Ibn Juljul 29, 32, 44 Ibn Khalaf, 'Ali 37 IbnKhaldun 114, 355, 359, 360, 361, 365, 372 Ibn Khurradadhbih 400 Ibn Labban, Kushyar 3, 71 IbnLuqa, Qusta 77, 87, 153, 154, 156, 158, 309, 310 IbnLuyun 51, 52 Ibn Ma'ruf, Taqi al-Din 446 Ibn Masawayh, Yuhanna 308, 309, 320 IbnMatar, Hajjaj 138 IbnMun'im 85, 94 Ibn Musa, Ahmad 152, 153, 154, 157, 158, 161, 162 Ibn Musa, Hasan 157 Ibn Musa, Muhammad 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 165 Ibn Na'ima al-Himsi xii, 309 IbnNasih 29 Ibn Qanbar, 'Amr ibn 'Uthman xii Ibn Qunfudh 52 Ibn Quraish, Hasan 162 Ibn Qurrah, Thabit xii, xvi, 77, 79, 85, 86, 88, 90, 93, 137, 138, 139, 143, 162, 229, 422, 423, 460, 494, 497, 501, 504, 505 Ibn Qutayba 37 Ibn Rushd 43, 44, 45, 46, 48, 49, 69, 104, 108, 113, 130, 255, 256, 370, 499, 500 Ibn Sa'd, 'Ali b. Muhammad 41 Ibn Sa'Id,'Anb 30, 31, 32 Ibn Sakir, Muhammad ibn Musa 325 Ibn Sayyid 38, 82 Ibn Shakir, Muhammad ibn Musa xix IbnShakir, Musa 112, 157 Ibn Sid, R. Ishaq 41 Ibn Slnaxiv, 50, 66, 95, 113, 127, 130, 138, 274, 383, 388 Ibn Suraqa 400, 401, 402 Ibn Talha 460, 465

Studies in the Making of Islamic Science Ibn Tamim, Abu Sahl Dunas 8, 12 IbnTariq,Ya'qub 421 IbnTaymiyya 109 Ibn Thabit, Ibrahim ibn Sina 139 Ibn Tufay 143, 46 Ibn Turk 92 Ibn Wand 41, 42, 44 Ibn Wahshiyya 41 Ibn Yahya, Abu Ishaq Ibrahim 254 Ibn Yunus of Mawsil, Kamal al-Din xx, 71, 111, 113, 130, 410, 418, 440, 451, 453, 454, 455, 456, 457, 465, 474 IbnZayd, Rabi 30, 31 Id, Y 419 al-Idrisi 142, 517 al-Iji, 'Adud al-Din 358, 361, 362, 363, 364, 366, 367, 368, 369, 370, 371, 372, 374, 375, 380, 381, 382, 383, 384, 385, 386, 387 Irani, R.A.K. 447 al-Ishbili, Abu al-Khayr 42 Isidor 11 Isidore of Miletus 87 Isidore of Seville 25, 26 al-Islami 52 al-Istiji, 'Ubayd Allah 26 Israeli of Toledo, Isaac 68 Jaghmml, Mahmud ibn Muhammad ibn Umar 230 al-Jahiz 4, 125 al-Jahiziyya 125 Janin, L. 420, 433 Janszoon Blaeu, Willem 520 Jaouiche, Khalil 79, 90, 91 al-Jayyanl, Ibn Mu'adh 24, 38, 45, 68, 69 Jensen, C. 447 John of Damascus xii JohnofMurs 502 John of Seville 491, 497 John of Sicily 501 al-Jud, Abu 540 al-Jurjani, al-Sharif 358, 362, 368, 369, 374, 387 al-Juwayni, Abu al-Ma'ali 360, 382 al-Juzjani, Abu 'Ubayd 229, 274, 275, 276, 283 Kafah, Rabbi Josef 461 Kant, Immanuel xxi, 537 al-Karaji 93, 94, 540 al-Kashi, Jamshid b. Giyath 97, 148, 181, 230 Kennedy, Edward S. 33, 34, 37, 45, 69, 87, 89, 90, 114, 289, 419, 447, 452

549

Kepler, Johannes xx, 114, 147, 278, 284, 489, 490, 496, 508, 520, 541 al-Khafri, Muhammad b. Ahmad 181, 193 al-Khalili, Shams al-Din 96, 410, 439, 442, 444, 447 Khattabi, M.A. 42 al-Khayr, Abu 41 al-Khayyam, 'Umar 82, 90, 91, 93, 97, 130, 140, 141, 540, 541 al-Khazin, Abu Ja'far 36 al-Khwarizmi, Muhammad ibn Ahmad 3, 4, 7, 9, 10, 11, 13, 28, 30, 32, 33, 34, 38, 67, 92, 130, 229, 230, 421, 432, 436, 457, 470, 492, 494, 497, 539 al-Khwarizmi, Muhammad ibn Musa 143 al-Kindi, Abu Yusuf Ya'qub ibn Ishaq xii, xviii, xix 32, 44, 102, 103, 112, 125, 126, 127, 128, 145, 146, 147, 301-47 passim, 388, 541 King, David A. xix, xx, 34, 36, 53, 96, 117, 133, 134, 391-447, 451-75 Knorr, Wilbur 77, 79, 86 Kohen of Toledo, Yahuda ben Solomon 68 al-Kufa 125 al-Kuhi, Abu Sahl 77, 83, 84, 87, 95, 139, 140, 229 Kuhn, A. 539 Kuhne, R. 27 Kunitzsch, Paul xiii, 3-16, 34, 66, 87, 88, 161 Labarta,A. 12, 36 Langermann, Y Tzvi 80, 452, 466 Lelewel, Joachim 517 Lemay, Richard 3, 12 Lettinck, P. 49 Livesey, S. 447 Llobert of Barcelona 490, 491 Lorch, R.P. 35, 45, 77, 80, 86, 87, 420 Luckey, P. 432 Lucretius 48 al-Ma'mun, Caliph xii, xvi, xx, 33, 40, 41, 90, 105, 106, 108, 111, 112, 124, 125, 128, 130, 132, 138, 162, 254, 451, 452, 453, 454, 455, 457, 458, 460, 461, 462, 463, 464, 467, 468, 469, 471, 473, 475, 494, 515, 516, 517, 518 Maestlin, Michael 278, 279, 284 Magnus, Albertus 499 al-Maghribi 50

550

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al-Mahalli 15 al-Mahani 77, 540 al-Majriti, Abu Maslama 29, 32, 33, 34, 35, 36, 37, 38, 39, 40, 141, 143, 490, 492 Maimonides, Ibn Maymun, Musa xiv, 43, 44, 46, 67, 71, 386 Makdisi, George 109 al-Mansur xi, xii al-Maqsi 424, 426, 431, 440, 441 Marchionis, G. 501 al-Maridlnl, Muhammad Sibt 10 Marin, M. 25 Marines of Tyre 514, 515 Marti, R. 35 al-Marrakushi, Abu 'Ali 424, 425, 426, 430, 432, 440, 441, 442, 447 al-Marwarrudhi, Khalid ibn 'Abd al-Malik 454, 456, 459, 460, 462, 463, 467 Masha'allah 35, 66, 71, 137, 491 al-Maridlnl, Muhammad Sibt 10 Matar, Hajjaj b. xvi Matvievskaya 91 Menelaus of Alexandria 35, 86, 89, 92, 143, 408 Mercator, Gerard 519 Mercier, R.P. 39, 452, 453, 457, 473 Mersenne, Father 377, 379, 380 Meyerhof, M. 29, 31, 44 Michel, H. 433 Millas Vallicrosa, Jose M. 66, 492 Millas Vendrell, E. 33 al-Mizzi 442 Molina 28 Montesquieu 538 Moody, E.A. 49 Moulierac, J. 420, 433 al-Mu'ayyad, Sultan 442 al-Mu'taman of Saragossa, King Yusuf 3 al-Mu'tasim, Caliph 125 al-Mundhir 26 Mu'tazila 108, 112 Muhammad 76, 107, 184, 336, 393, 440 Muhammad I, Emir 30, 51 Muhammad II 50, 51 Muhammad, 'Abd al-Rahman b. 'Isa 38 Muller, Max 538 Munoz, R. 25 Minister 346 al-Muqaddasi 400 al-Muradi, Muhammad ibn Khalaf 41 al-Mutawakkil 111

al-Nabati, Abu al-'Abbas 44, 52 al-Nadim 152, 153, 154, 156, 157, 301, 308, 347 Nadir, Nadi 447 Nagy, A. 346 Naini 85 Najm al-Din, 441 Nallino, C.A. 451, 457 al-Nasawi 7 Nasih, 'Abbas b. 28 al-Nasir, 'Abd al-Rahman III 26, 27 Nasr, Abu 50, 439, 447 Nawbakht xi al-Nayrizi 408, 470, 472 Nekho, Pharoah 513 Neugebauer, O. 32, 67, 232, 293, 296 Nicolosi 87, 142 Nicomachus 85 Nizam al-Mulk 110 al-Numayri, Ibn Arqam 52 Olearius, Adam 521 Oliver Asin, J. 42 Orosius, Paulus 26 Ortelius, Abraham 519 Osiander, Andreas 386 Palacios, Asin 42, 44 Pappus 87, 345, 540 Pecham, John 147 Pedal, Menendez 26 Pellat, Charles 4 Peter of Saint-Omer 501 Peurbach, Georg 500, 503 Philoponus, John 47, 49, 383 Pines, S. 69 Pingree,D. 33, 71 Plato of Trivoli 143, 490, 492 Plooij 38, 91 Postel, Guillaume 519 Poulle, E. 50, 501 Proclus 304 Profatius 493, 502 Ptolemy xvii, 10, 33, 34, 35, 37, 39, 40, 45, 47, 48, 66, 67, 68, 69, 70, 84, 86, 87, 89, 90, 141, 142, 143, 144, 145, 146, 152, 155, 159, 160, 161, 162, 165, 179, 180, 183, 187, 188, 190, 199, 200, 202, 222, 223, 224, 225, 226, 227, 231, 232, 233, 235, 236, 237, 240 241, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254,

Studies in the Making of Islamic Science

255, 256, 258, 259, 260, 261, 263, 264, 265, 266, 269, 271, 273, 274, 275, 276, 278, 279, 281, 282, 283, 286, 287, 288, 289, 296, 383, 385, 404, 423, 461, 464, 469, 474, 475, 490, 491, 492, 494, 495, 496, 497, 498, 499, 500, 501, 503, 504, 505, 514, 515, 518, 519 Puig,R. 51, 52 Pythagorus 466 al-QablsI xiv, 68 Qadizade Rumi 181 al-Qalasadl 9, 15, 50 al-Qasim, Abu 32 Qayini, Qasim 'All 416 al-Qazwini 402 al-Qifti 154, 156, 157, 308 Quinet, Edgar 538 Qurra, Thabit b. Sinan b. Thabit 29, 39 Qurra, Theodore Abu xii al-QushjT, 'Ala al-Dln 'All b. Muhammad xvii, 177-219 passim, 230, 294 Raban Maur 489 Rabbihi, Sa'idb. 'Abd27 Ragep, FJ. 452 Ragep, J.F. 447 Rashed, Roshdi xviii, xix, xxi, xxii, 78, 82, 85, 86, 91, 94, 95, 301-47, 537-43 al-Rashld, Harun xvi, 105, 138 Raymond of Marseilles 490, 491, 492 al-Razi,AbuBakr ll3, 245 al-Razi, Fakhr al-Din 25, 159, 360, 363, 368, 373, 381, 382 Recemund 30 Regiomontanus 46, 500 Reinert 86 Reland, Adrian 521 Renan, Ernest 66, 538, 539 Renaud, H.PJ. 51, 52 Rennell, James 522 Rheticus, FJ. 504 Ribera, J. 28, 50 Richard of Middleton 500 Richler, B. xiv, 66 Richter-Bernburg, Lutz 39 al-Riquti, Muhammad 50 Robert of Chester 490, 492 Rodgers, R.H. 27 Roger II, King 142, 517

551

Roger of Hereford 493 Romano, D. 68 Rosenfeld, B.A. 90, 432 Rosinska 70 Ruman, Khalid b. Yazid b. 27 al-Rummani 109 al-Sabi, Abu Ishaq 95 Sa'id of Toledo 26, 34, 37, 38 Sabra, A.I. xiii, xv, xvi, xix, 46, 76, 77, 84, 96, 101-117, 119-35, 349-90, 447 al-Safa, Ikhwan 29 al-Safaqusi 402 al-Saghani, Abu Hamid 86, 460 al-Sakhawi 15 Saliba, George xvii, xviii, 89, 149-71, 177-219, 221-7, 229-97 al-Samarqandi, Sa'id ibn Khafif 423 al-Samaw'al 93 Samso, Julio xiii, 23-53 al-San'an! 24 Sanchez Perez 52 Sanson, Nicolas 521 Sarfatti, G.B. 68 Sarre, F. 473 Sarton, George 44, 107 Sayih, A. 451, 452 Sayili, A. 44 Schickard, Wilhelm 520, 521 Schoy, Carl 414, 419, 432 Scot, Michael 499 Scotus, Duns 500 Sebokht, Severus 3 Sedillot, J.-J. 425, 432 Sedillot, L.A.R 425, 432 Sesiano 86, 93, 94 Sezgin, Fuatxxi, 3, 66, 77, 79, 90, 153, 513-35 al-Shafi'i, Husayn ibn Muhammad al-Mahalli 15 al-Shafra, Muhammad 50, 51 Sharaf al-Din 97 al-Shari'a, 'Ubaydallah b. Mas'ud b. 'Umar Sadr 270, 279, 283, 293, 294, 296 al-ShlrazI, Qutb al-Dlnxvii, 46, 130, 151, 152, 153, 154, 155, 157, 158, 159, 162, 163, 221, 229, 267, 268, 269, 270, 279, 283, 288, 289, 291, 293, 296, 506 al-SijzT 5, 8, 95 Simplicius 500 Sisebut, King 25 Snell, Willebrord 146

552

Studies in the Making of Islamic Science

Sobhy, G.R 44 Southern, Richard 122 Spengler, Oswald 126 Steinschneider, Moritz 66 Stern, S.M. 29 al-Suli, Muhammad ibn Yahya 4 al-Suyuti, Jalal al-Din 383 Suter, Heinrich 32, 35, 67 Swerdow, Noel M. 45, 282, 293, 296 Sylvestre II, Pope 490 al-Tabari, 'Umar b. Farrukhan 29 al-Tadili, Ibn al-Zayyat 48 al-Tahiri, Mansur ibn Talha 458 Tannery, P. 539, 541 al-Tantawi 429, 430 Taskoprulu-Zade 180, 182, 183, 196, 197 Teres,E. 28, 31 Theodoric of Freiburg 147 Theodosius 86, 92, 143 Theon of Alexandria 48, 306, 307 Theophrastus 42 Thorndike, L. 50 al-Tignari 41, 42 Toomer, GJ. 39, 45, 80, 248, 496 Torres B albas, L. 43 Torricelli, Evangelista 69 al-Tusi, Nasir al-Din xviii, 69, 77, 88, 89, 90, 97, 104, 130, 142, 144, 151, 152, 229, 230, 241, 264, 265, 266, 268, 269, 285, 288, 291, 293, 296, 385, 387, 475, 506, 507, 508 al-Tusi, Sharaf al-Din 52, 82, 93, 540, 541 Twer sky, L 71 Ubba, HamdTn b. 26 Unver,A.S. 447 al-UqlIdisl3, 8 al-'Urdi of Damascus, Mu'ayyad al-Din xviii, 131, 162, 221, 222, 223, 224, 225, 226, 227, 229, 256, 261, 262, 263, 264, 268,

269, 270, 275, 276, 277, 278, 279, 281, 282, 283, 284, 285, 286, 287, 288, 290, 291, 292, 296, 506 al-Urmawi, al-Qadi 369 'Umar II, Caliph xi Vallicrosa, Millas 34, 36, 39, 40 van Berchem, Max 110 Van Linschoten, Jan Huygen 521 Vernet, Juan xiii, 23-53 Viladrich, M. 35, 36 Villuendas, M.V. 38 von Humbolt, Alexander xxii, 3, 541 von Schlegel, Friedrich 538 al-Wafa,Abu 410, 469 al-Wafa'i 431 al-Wahid, 'Abd 25 al-Wathiq 125 Whiteside, D.T. 540 Wiedemann, E. 447 William of Auvergne 499 William of Saint-Cloud 501, 502 Woepcke, F. 8, 9, 10, 12, 82 Wolfson, Harry 128, 372 Wurschmidt, J. 447 al-Ya'qubl 4 Yazdegerd III 33 al-Yazdi 77 Yazdi, Muhammad Baqir 86, 414 Yazdi, Muhammad Husayn ibn Muhammad Baqir 416 Yusuf, QadlAbuxii, 51 al-Zahrawi, Abu al-Qasim 32, 36, 37, 42 al-Zanji, Sultan Nur al-Din 430 al-Zarqael 254 al-Zarqallo 25, 34, 37, 38, 39, 40, 44, 45, 47, 48, 50, 53, 72, 491, 492, 494, 503, 504, 505