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View This Volume's Front and Back Matter An Introduction to the History of Algebra Solving Equations from Mesopotamian Times to the Renaissance http://dx.doi.org/10.1090/mawrld/027 MATHEMATICAL WORLD VOLUME 27 An Introduction to the History of Algebra Solving Equations from Mesopotamian Times to the Renaissance Jacques Sesiano Translated by Anna Pierrehumbert Providence, Rhode Island This work was originally published in French under the title: “Une introduction `a l’histoire de l’alg`ebre: R´esolution des ´equations des M´esopotamiens `a la Renaissance”. c 1999 Presses polytechniques et universitaires romandes, Lausanne. All rights reserved. Translated by Anna Pierrehumbert 2000 Mathematics Subject Classification. Primary 01A05, 17–XX, 20–XX, 30–XX, 35–XX, 40–XX. For additional information and updates on this book, visit www.ams.org/bookpages/mawrld-27 Library of Congress Cataloging-in-Publication Data Sesiano, Jacques. [Introduction `a l’histoire de l’alg`ebre. English] An introduction to the history of algebra : solving equations from Mesopotamian times to the Renaissance / Jacques Sesiano ; translated by Anna Pierrehumbert. p. cm. – (Mathematical world ; v. 27) Includes index. ISBN 978-0-8218-4473-1 (alk. paper) 1. Algebra—History. I. Title. QA151.S4513 2009 512.009–dc22 2009008068 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009 Contents Preface vii Chapter 1. Algebra in Mesopotamia 1 1.1. Introduction 1 1.2. Linear Systems 4 1.3. Quadratic Equations and Systems 9 Chapter 2. Algebra in Ancient Greece 17 2.1. Introduction 17 2.2. Common Algebra 19 2.3. Diophantine Algebra 31 Chapter 3. Algebra in the Islamic World 53 3.1. Introduction 53 3.2. Al-Khw¯arizm¯ı56 3.3. Ab¯uK¯amil 63 3.4. The Geometric Construction of Solutions of the Quadratic Equation 79 3.5. The Cubic Equation 80 Chapter 4. Algebra in Medieval Europe 93 4.1. Introduction 93 4.2. The Liber mahameleth 95 4.3. Leonardo Fibonacci 101 4.4. Later Developments 115 Chapter 5. Algebra in the Renaissance 123 5.1. The Development of Algebraic Symbolism 123 5.2. The General Solution of the Cubic and Quartic Equations 125 5.3. The Solution of the Cubic Equation in Italy 129 5.4. The Solution of the Quartic Equation in Italy 134 5.5. Bombelli and Imaginary Numbers 135 5.6. Casus irreducibiles 140 v vi CONTENTS Appendix A. Mesopotamian Texts in Translation 143 Appendix B. Greek and Latin Texts 147 Appendix C. Arabic Texts 157 Appendix D. Hebrew Text 163 Appendix E. French, German, Italian, and Provençal Texts 165 Index 171 Preface The years following 2000 B.C. and the years preceding A.D. 1600 mark two important turning points in the development of algebra. The first witness the solution of equations and systems of the first and second degrees; to the second is associated the solution of the third- and fourth-degree equations, which are also the last in which the unknown may be expressed by means of a general formula. The years between these two turning points also saw the successive extensions of the number system: confined initially to the positive integers and fractions, it next came to include positive irrational numbers; then, towards the end of the Middle Ages, negative numbers made a timid appearance, and this was followed a little later by the first use of complex numbers. This book does not aim to give an exhaustive survey of the history of algebra up to early modern times, but merely to present some significant steps in solving equations and, wherever applicable, to link these develop- ments to the extension of the number system. Various examples of problems, with their typical solution methods, are analyzed, and sometimes translated completely. Indeed, it is another aim of this book to ease the reader’s access to modern editions of old mathematical texts, or even to the original texts; to this end, some of the problems discussed in the text have been reproduced in the Appendix in their original language (Greek, Latin, Arabic, Hebrew, French, German, Provençal, and Italian) with explicative notes. Bibliographical references have been restricted to the particular subjects considered. General information may be found in classical textbooks on the history of mathematics, such as K. Vogel’s Vorgriechische Mathematik II , Hannover/Paderborn 1959 (Mesopotamian mathematics), T. Heath’s two- volume History of Greek mathematics, Oxford 1921 (reprinted), M. Cantor’s four-volume Vorlesungen über Geschichte der Mathematik, Leipzig 1900–13 (reprinted), J. Tropfke’s seven-volume Geschichte der Elementar-mathematik in systematischer Darstellung, Leipzig 1921–24 and (vol. 1-4 only) 1930–40. While preparing this English edition of the original French book pub- lished by the Presses polytechniques et universitaires romandes, I had the pleasure to work with Dr. Ina Mette, in Book Acquisitions at the American vii viii PREFACE Mathematical Society, and with Anna Pierrehumbert, in charge of the trans- lation and typesetting of the book. I wish to express to them my sincerest thanks for their patience and kindness. I also wish to express my indebt- edness to my student Christophe Hebeisen, who has been of constant help whenever technical difficulties appeared in relation to the production of the text. Lausanne, Switzerland September, 2008 Index ‘Abd al-H. am¯ıd ibn Turk 59–61 Bubnov, N. 21n, 148n Abel, N. 140–41 Busard, H. 94n Ab¯uK¯amil 46–49, 55, 61, 63–79, 94, Byzantine contributions 18, 32, 94–95, 96, 112 103, 106n, 110, 112 Ab¯u’l-J¯ud 85–86 Akkadian 1 Alcuin 78, 93, 104 Cajori, F. 125n algebra (origin of the name) 34 Cantor, M. v. algebraic identities see identities Cardano, G. 123, 126–27, 131–35, 136 algebraic language and symbols see casus irreducibilis 139 symbolism cattle problem of Archimedes 27–31 algo 163 (n. 212) censo 123 algorithmus 56 (origin of the name), census 96, 123, 157 (n. 170) 138n Chinese contributions 78, 104, 112 Almoravides 100 Christianidis, J. 5n,44n Amthor, A. 31 Chuquet, N. 116–17, 123, 125 Anbouba, A. 46n Colla, G. 132, 134 Apollonius 54 Compendion del art de algorisme application of areas 79 116–17 Archimedes 17, 27–28, 31, 54, 82–83, complex numbers 63, 89, 135–40 85. See also cattle problem conic sections 81, 82, 85, 88–89, 160n Aristotle 71n continued fractions 31n Arrighi, G. 120n cosa 118, 123–24, 130 cubic equation 63, 80–83, 84–86, Bachet de M´eziriac, C. G. 38n,49 88–91, 95, 118–21, 125–28, Bah¯a’ al-D¯ın al-‘Amil¯¯ ı63 129–34, 135–40, 140n, 168–70 bayt al-h. ikma 53 cubo 123 bird problems (mas¯a’il al-t.uy¯ur) 75–79 cubus 96, 123 B¯ır¯un¯ı 53, 86 Blume, F. 21n Boethius 93 Dardi of Pisa 119–20 Bolognetti, P. 129–30 decagon 84 Bombelli, R. 49, 123–24, 135–39, 169n decimal number system 1–4 Boncompagni, B. 95n Delian problem see duplication of the Bortolotti, E. 124n, 130n, 132n, 138n, cube 169n Descartes, R. 125, 139 Bruins, E. 148n difference of two squares 36, 39–42 171 172 INDEX Diophantus and his Arithmetica 17, H¯ajj¯ıKhal¯ıfa 61n 24–26, 31–46, 49–50, 54–55, 110, Halma, N. 4n 112, 115–16, 158 (n. 181), 165n Heath, T. v, 37n Djebbar, A. 96n Heiberg, J. 4n,22n,85n duplication of the cube 81–82, 85 Henry, C. 38n heptagon 83, 85 van Egmont, W. 118n, 119n Heron of Alexandria 22, 54 enneagon 83, 86 hexagon 83 equations (simplification of) 9, 34, 62; Hippocrates of Chios 81–82 (six standard forms) 57, 60–61, 64, Hogendijk, J. 85n 69–70, 88, 88n; (systems of) 1, 4–9, Høyrup, J. 5n 11–15, 25–31, 40–49, 61–63, 67–70, Hypatia 32 75–79, 98, 100–1, 103–19, 126, 130, hyperbola 88–91 134–36. See also quadratic, cubic, quartic, quintic; geometric solving; Iamblichus 26–27 indeterminate; inconsistent, Ibn al-Haytham 83 unsolvable Ibn Khald¯un 63 Eratosthenes 27 identities, use of (in algebra) 1, 11–12, Euclid and his Elements 54–55, 64–66, 13, 15–16, 19, 20–25, 65–66, 98–99, 71, 79–81, 83, 93, 96, 98–99, 152 126, 130 Euler, L. 24, 49–51, 124, 165n al-ikm¯al 62, 69 imaginary numbers 135–140. See also de Fermat, P. 31, 38, 44, 49, 83n complex numbers Ferrari, L. 132n, 133–34 impossible, equation or problem see dal Ferro, S. 129–31, 134 unsolvable Fibonacci, L. 63, 95, 101–15, 117 inconsistent equations 108–9 Fibonacci sequence 102–3 indeterminate equations and systems Fior, A.
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