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Encyclopedia of the History of Arabic Science. Volume 2, Mathematics

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Encyclopedia of the History of Arabic Science. Volume 2, Mathematics Encyclopedia of the History of Arabic Science Encyclopedia of the History of Arabic Science Volume 2 Edited by ROSHDI RASHED in collaboration with RÉGIS MORELON LONDON AND NEW YORK First published in 1996 by Routledge 11 New Fetter Lane, London EC4P 4EE 29 West 35th Street, New York, NY 10001 This edition published in the Taylor & Francis e-Library, 2009. To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk. Structure and editorial matter © 1996 Routledge The chapters © 1996 Routledge All rights reserved. No part of this book may be reprinted or reproduced or utilized 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. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Library of Congress Cataloguing-in-Publication Data A catalogue record for this book is available on request. ISBN 0-203-40360-6 Master e-book ISBN ISBN 0-203-71184-X (Adobe ebook Reader Format) ISBN 0-415-12411-5 (Print Edition) 3 volume set ISBN 0-415-02063-8 Contents VOLUME 1 Contents v Preface ix 1 General survey of Arabic astronomy Régis Morelon 1 2 Eastern Arabic astronomy between the eighth and the eleventh centuries 20 Régis Morelon 3 Arabic planetary theories after the eleventh century AD 58 George Saliba 4 Astronomy and Islamic society: Qibla, gnomonics and timekeeping 128 David A.King 5 Mathematical geography 185 Edward S.Kennedy 6 Arabic nautical science 202 Henri Grosset-Grange (in collaboration with Henri Rouquette) 7 The development of Arabic science in Andalusia 243 Juan Vernet and Julio Samsó 8 The heritage of Arabic science in Hebrew 276 Bernard R.Goldstein 9 The influence of Arabic astronomy in the medieval West 284 Henri Hugonnard-Roche Bibliography 306 VOLUME 2 10 Numeration and arithmetic 1 Ahmad S.Saidan 11 Algebra 17 Roshdi Rashed 12 Combinatorial analysis, numerical analysis, Diophantine analysis and number theory 44 Roshdi Rashed 13 Infinitesimal determinations, quadrature of lunules and isoperimetric problems 85 Roshdi Rashed vi Contents 14 Geometry 115 Boris A.Rosenfeld and Adolf P.Youschkevitch 15 Trigonometry 160 Marie-Thérèse Debarnot 16 The influence of Arabic mathematics in the medieval West 204 André Allard 17 Musical science 242 Jean-Claude Chabrier 18 Statics 274 Mariam Rozhanskaya (in collaboration with I.S.Levinova) 19 Geometrical optics 299 Roshdi Rashed 20 The emergence of physiological optics 324 Gül A.Russell 21 The Western reception of Arabic optics 363 David C.Lindberg Bibliography 374 VOLUME 3 22 Engineering 751 Donald R.Hill 23 Geography 796 André Miquel 24 Botany and agriculture 813 Toufic Fahd 25 Arabic alchemy 853 Georges C.Anawati 26 The reception of Arabic alchemy in the West 886 Robert Halleux 27 Medicine 903 Emilie Savage-Smith 28 The influence of Arabic medicine in the medieval West 963 Danielle Jacquart 29 The scientific institutions in the medieval Near East 985 Françoise Micheau 30 Classifications of the sciences 1008 Jean Jolivet Postface: approaches to the history of Arabic science 1026 Muhsin Mahdi Bibliography 1045 1067 Index 10 Numeration and arithmetic AHMAD S.SAIDAN The earliest Arabic works of arithmetic ever written are those by al-Khw!rizm", ibn M#s! (ninth century). He wrote two tracts: one, on Hindu arithmetic, has come down to us in a Latin tradition,1 and the other, named wa al-Tafr!q (‘Augmentation and diminution’), is mentioned in Arabic bibliographies,2 and we have a quotation of it in an Arabic work (al-Baghd!d", al-Takmila). The earliest Arabic arithmetic that has come to us in full is the arithmetic of al-Uql"dis", ibn Ibr!h"m (tenth century) (al-Uql"dis", p. 349). It discusses a Hindu system of calculation and refers to two others, namely, finger-reckoning and the sexagesimal system. These three systems, together with the Greek arithmetica, which is in fact rudiments of number theory, formed the elements of arithmetic and were amalgamated and developed. THE SCALE OF SIXTY In Arabic books, the scale of sixty is called the method of astronomers. The sexagesimal system comprises the major arithmetical operations in the scale of sixty. It came down to the Islamic world from the ancient Babylonians through Syriac and Persian channels. We have no early work devoted to it, but we find it in all arithmetic works amalgamated with one or both of the Hindu system and finger-reckoning. Later works that stress it contain hardly anything purely arithmetical which is not involved in other works nor bearing traces of Islamic development. Scholars today find it easier than the decimal system when treating medieval astronomical calculations. It has now disappeared from use, but leaves its trace in our submultiples of the degree and hour. FINGER-RECKONING In Arabic works, finger-reckoning is called the arithmetic of the R#m (i.e. the Byzantines) and the Arabs. When and how it came down to the Islamic world cannot be known for sure. But we can guess that before Islam the Arab merchants learnt to count by fingers from their neighbours. The system seems to have been spread all over the then civilized world. Some prophetic sayings involve allusion to denoting numbers by finger signs, which is characteristic of the system. Basically the system is mental. Addition and subtraction involve no serious difficulty. Multiplication, division and ratio can be more complicated. Hence these form most of the works in this system. Too many schemes of multiplication are given; but the main ones are shortcuts that are still used. False position and double-false position, which implies the principle of linear interpolation, are used in the case of ratio and proportion. 2 Encylopedia of the history of arabic science: volume 2 Extraction of square roots seems to be done mainly by rough approximation. The working is done mentally. But intermediate results need to be temporarily ‘held’. This is done by bending the fingers of the two hands in distinct poses that can denote numbers from 1 up to 9999. In The Arithmetic of al-Uql!dis! the different poses are given. These are what Arabic works call (knots) and therefore the whole system is called i.e. the arithmetic of knots (finger joints). Numerals in this system are the letters of the Arabic alphabet in a fixed order called jummal, which gives another name to the system, namely jummal arithmetic. The letters of the alphabet in the Eastern order, with the numbers they denote, are given below. 1 A 8 H 60 S 400 T 2 B 9 I 70 O 500 U 3 C 10 J 80 P 600 V 4 D 20 K 90 Y 700 Z 5 E 30 L 100 Q 800 W 6 F 40 M 200 R 900 I! 7 G 50 N 300 X 1000 O! Thus to denote 1111 they wrote O!QJA; for 2000 they wrote BO!; for 1000000 O!O!. Thus theoretically they could express any number in this system of numeration. But big numbers do not appear in the extant works on finger-reckoning, because they made free use of the scale of sixty, for which letters from A to N are used. In the Western wing of the Muslim world, the jummal order of letters is different, but the difference comes beyond the letter N and thus does not affect the sixty scale. The earliest known work on finger-reckoning is that of al-B#zj!n" (tenth century).3 Not much later appeared al-K"f! f! by al-Karaj".4 These are the only two worthwhile works on this system. It started to disappear as the Hindu system spread, leaving only shortcuts of multiplication and division together with an Arabic concept of fractions. Although finger-reckoning came down to the Arabic-speaking world from others who spoke mostly Syriac, as it appears in the works of and al-Karaj" it looks as if made to fit exactly the Arabic language, especially when we consider the concept of fractions. In Arabic there are nine single words to denote nine distinct fractions of the unit numerator; these are These are the only kus#r (i.e. fractions) in the system; each is kasr (fraction). Even …are each kus#r, plural of kasr. An expression like is pronounced as part of and in calculations should be expressed as Fractions of denomination other than 2, 3, 5 and 7, e.g. and are considered surds ( literally deaf) and should better be changed by approximation into spoken fractions. In arithmetic many pages are devoted to showing how best to turn such ‘parts’ into fractions. The key method Numeration and arithmetic 3 is to exploit the scale of sixty. Thus of a degree was the only acceptable ‘fraction’ with a numerator other than unity. The concept makes practical calculation easy. But it is naive, anti-generalizational and not mathematical. There were several fractional systems in finger-reckoning, the foremost being the scale of sixty, i.e. the submultiples of the degree. But any scale of measurement, of length, area, volume, or size, or transaction could be used as a fractional scale. Thus where 1 dirham=24 kirats, 1 kirat stands for These systems faded away, and the concept of the general fraction a/b established itself in the Islamic era, apparently with the spread of the Hindu system. But the trend to express a fraction like continued and can still be noticed among illiterate people.
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