2 3 4 TAMING THE UNKNOWN 5 TAMING THE UNKNOWN A History of Algebra from Antiquity to the Early Twentieth Century VICTOR J. KATZ AND KAREN HUNGER PARSHALL PRINCETON NIERSITY PRESS PRINCETON AND OXFORD 6 Copyright © 2014 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire, OX20 1TW Jacket images: Background images: Title page of Discours de la method by René Descartes, from the Plimpton Collection of the Rare Book and Manuscript Library, Columbia University. Problem on parabolas from Elementa curvarum linearum by Jan de Witt, in the 1659 edition of Descartes’s Geometria, from the David Eugen Smith Collection of the Rare Book and Manuscript Library, Columbia University. Mathematicians, left to right: Luca Pacioli, Pierre de Fermat, Évariste Galois, James Joseph Sylvester, Georg Frobenius, Emmy Noether. Detail portrait of the mathematician Luca Pacioli and an unknown young man by Jacopo de’Barbari. Museo Nazionale di Capodimonte, Naples, Italy. © Scala/Ministero per I Beno e le Attivitá culturali/Art Resources, NY. Image of Noether courtesy of Drs. Emiliana and Monica Noether. All Rights Reserved Library of Congress Cataloging-in-Publication Data Katz, Victor J. Taming the unknown: history of algebra from antiquity to the early twentieth century / Victor J. Katz and Karen Hunger Parshall. pages cm Includes bibliographical references and index. ISBN 978-0-691-14905-9 (hardcover: alk. paper) 1. Algebra–History. I. Parshall, Karen Hunger, 1955- II. Title. QA151.K38 2014 512–dc23 2013024074 7 British Library Cataloging-in-Publication Data is available This book has been composed in ITC New Baskerville Printed on acid-free paper. ∞ press.princeton.edu Typeset by S R Nova Pvt Ltd, Bangalore, India Printed in the United States of America 2 4 6 8 10 9 7 5 3 1 8 We dedicate this book to our spouses, PHYLLIS KATZ and BRIAN PARSHALL, and to our parents, BESS and MARTIN KATZ (in loving memory) and JEAN KAY “MIKE” and MAURICE HUNGER 9 Contents Acknowledgments xi 1 Prelude: What Is Algebra? 1 Why This Book? 3 Setting and Examining the Historical Parameters 4 The Task at Hand 10 2 Egypt and Mesopotamia 12 Proportions in Egypt 12 Geometrical Algebra in Mesopotamia 17 3 The Ancient Greek World 33 Geometrical Algebra in Euclid’s Elements and Data 34 10 Geometrical Algebra in Apollonius’s Conics 48 Archimedes and the Solution of a Cubic Equation 53 4 Later Alexandrian Developments 58 Diophantine Preliminaries 60 A Sampling from the Arithmetica: The First Three Greek Books 63 A Sampling from the Arithmetica: The Arabic Books 68 A Sampling from the Arithmetica: The Remaining Greek Books 73 The Reception and Transmission of the Arithmetica 77 5 Algebraic Thought in Ancient and Medieval China 81 Proportions and Linear Equations 82 Polynomial Equations 90 Indeterminate Analysis 98 The Chinese Remainder Problem 100 6 Algebraic Thought in Medieval India 11 6 Algebraic Thought in Medieval India 105 Proportions and Linear Equations 107 Quadratic Equations 109 Indeterminate Equations 118 Linear Congruences and the Pulverizer 119 The Pell Equation 122 Sums of Series 126 7 Algebraic Thought in Medieval Islam 132 Quadratic Equations 137 Indeterminate Equations 153 The Algebra of Polynomials 158 The Solution of Cubic Equations 165 8 Transmission, Transplantation, and Diffusion in the Latin West 174 The Transplantation of Algebraic Thought in the Thirteenth 12 Century 178 The Diffusion of Algebraic Thought on the Italian Peninsula and Its Environs from the Thirteenth Through the Fifteenth Centuries 190 The Diffusion of Algebraic Thought and the Development of Algebraic Notation outside of Italy 204 9 The Growth of Algebraic Thought in Sixteenth-Century Europe 214 Solutions of General Cubics and Quartics 215 Toward Algebra as a General Problem-Solving Technique 227 10 From Analytic Geometry to the Fundamental Theorem of Algebra 247 Thomas Harriot and the Structure of Equations 248 Pierre de Fermat and the Introduction to Plane and Solid Loci 253 Albert Girard and the Fundamental Theorem of Algebra 258 René Descartes and The Geometry 261 Johann Hudde and Jan de Witt, Two Commentators on The 13 Geometry 271 Isaac Newton and the Arithmetica universalis 275 Colin Maclaurin’s Treatise of Algebra 280 Leonhard Euler and the Fundamental Theorem of Algebra 283 11 Finding the Roots of Algebraic Equations 289 The Eighteenth-Century Quest to Solve Higher-Order Equations Algebraically 290 The Theory of Permutations 300 Determining Solvable Equations 303 The Work of Galois and Its Reception 310 The Many Roots of Group Theory 317 The Abstract Notion of a Group 328 12 nderstanding Polynomial Equations in n nknowns 335 Solving Systems of Linear Equations in n Unknowns 336 14 Linearly Transforming Homogeneous Polynomials in n Unknowns: Three Contexts 345 The Evolution of a Theory of Matrices and Linear Transformations 356 The Evolution of a Theory of Invariants 366 13 nderstanding the Properties of “Numbers” 381 New Kinds of “Complex” Numbers 382 New Arithmetics for New “Complex” Numbers 388 What Is Algebra?: The British Debate 399 An “Algebra” of Vectors 408 A Theory of Algebras, Plural 415 14 The Emergence of Modern Algebra 427 Realizing New Algebraic Structures Axiomatically 430 The Structural Approach to Algebra 438 References 15 449 Index 477 16 Acknowledgments The present book came about almost serendipitously. In 2006, the first author reviewed nknown Quantity: A Real and Imaginary History of Algebra, a work on the history of algebra by John Derbyshire.1 Although very well written, that book contained so many errors of fact and inadequacies of interpretation that it would have been a disservice to let it go unchallenged. In January of 2008, the first author heard the second author’s invited hour lecture on the history of algebra at the Joint Mathematics Meetings in San Diego. Realizing that her ideas were similar to those he had expressed in his own invited lecture on the history of algebra at the International Congress of Mathematics Education (ICME) in Copenhagen in 2004, he asked if she would be interested in jointly writing a book on the history of algebra. Curiously, although we live only about two hours apart, we only finally met to outline our ideas and to agree to submit the project to the Princeton University Press when we both attended a meeting of the International Study Group on the Relations Between History and Pedagogy of Mathematics in Mexico City in July of 2008. The present book is the fruit of our ideas developed over long careers teaching and working on the history of mathematics and, in particular, on the history of algebra. The first author would like to thank Clemency Montelle and John Hannah for inviting him to the University of Canterbury in Christchurch, New Zealand in the spring of 2011 to lecture on the history of algebra. He was able to develop some of his ideas during those lectures. In addition, both Clemency and John graciously read drafts of several of the chapters 17 of the book and offered many suggestions for improvement. The first author is also grateful to Mogens Niss of Roskilde University, Denmark for inviting him to present a lecture at ICME 10 that gave him the chance further to organize his ideas on the history of algebra. His greatest debt of gratitude, however, is to his wife, Phyllis, for her encouragement over the years, for her many conversations at every hour of the day and night, and for everything else. We embarked on this book project at the same moment that the second author began a three-year term as Associate Dean for the Social Sciences at the University of Virginia. She would like to thank both Meredith Woo, the Dean of the College and Graduate School of Arts and Sciences, for asking her to serve and all of her colleagues in the Dean’s Office and in the Division of Social Sciences for their support as she wore the very distinct hats of administrator and researcher. Her special thanks go to Rick Myers and Nancy Bertram for their endless patience and sage advice as she learned the administrative ropes; to Cristina Della Coletta and Jim Galloway, the Associate Deans of Arts and Humanities and of Sciences, respectively, for their friendship and sense of common cause; and to Seth Matula for always reminding her when it was Thursday. Three years of Thursdays and a year-long sabbatical leave later, the book is finished. The second author is also indebted to her friends and colleagues, Joe Kett in the Corcoran Department of History and Fred O’Bryant in the Brown Science and Engineering Library, who, although not mathematicians, seemingly never tired of listening to tales of the latest twists and turns in the research and writing that went into the book over what for her, at least, were restorative lunches; to Ivor Grattan-Guinness, Professor Emeritus at Middlesex University and doyen of the history of mathematics, who always challenged her with questions she had not thought to ask; and to her husband and colleague in the Department of 18 Mathematics, Brian Parshall, who read and critiqued the book in manuscript with his algebraist’s eye. Both authors extend their heartiest thanks to Greg Hays, John Miller, and Tony Woodman in the Department of Classics and to Elizabeth Meyer in the Corcoran Department of History, all at the University of Virginia, for their help with questions involving both Latin and Greek; to Lew Purifoy and his coworkers in Interlibrary Loan at the University of Virginia for never failing to secure even the hardest-to-find sources; to Jennifer Lee of the Rare Book and Manuscript Division of the Columbia University Library for her assistance in finding most of the manuscript and book pages that serve as illustrations in the text; to Julie Riddleberger, the TEX guru of Virginia’s Department of Mathematics, for cheerfully and masterfully transforming our often crude sketches into the myriad figures that now illuminate our text; and to the Press’s readers, who gave us numerous valuable suggestions for improvement.