HistoryMath-Burton.85.pdf

 Revised Con rming Pages The History of Mathematics AN INTRODUCTION Seventh Edition David M. Burton...

 Revised Con rming Pages The History of Mathematics AN INTRODUCTION Seventh Edition David M. Burton University of New Hampshire bur83155 fm i-xii.tex i 01/13/2010 16:12  Revised Con rming Pages THE HISTORY OF MATHEMATICS: AN INTRODUCTION, SEVENTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright  c 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Previous editions c 2007, 2003, and 1999. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 0 DOC/DOC 1 0 9 8 7 6 5 4 3 2 1 0 ISBN 978–0–07–338315–6 MHID 0–07–338315–5 Editorial Director: Stewart K. Mattson Sponsoring Editor: John R. Osgood Director of Development: Kristine Tibbetts Developmental Editor: Eve L. Lipton Marketing Coordinator: Sabina Navsariwala-Horrocks Project Manager: Melissa M. Leick Senior Production Supervisor: Kara Kudronowicz Design Coordinator: Brenda A. Rolwes Cover Designer: Studio Montage, St. Louis, Missouri (USE) Cover Image: Royalty-Free/CORBIS Senior Photo Research Coordinator: John C. Leland Compositor: Laserwords Private Limited Typeface: 10/12 Times Roman Printer: R. R. Donnelley All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. Library of Congress Cataloging-in-Publication Data Burton, David M. The history of mathematics : an introduction / David M. Burton.—7th ed. p. cm. Includes bibliographical references and index. ISBN 978-0-07-338315-6 (alk. paper) 1. Mathematics–History. I. Title. QA21.B96 2011 510.9–dc22 2009049164 www.mhhe.com bur83155 fm i-xii.tex ii 01/13/2010 16:15  Revised Con rming Pages A ll these were honored in their generations, and were the glory of their times. T here be of them, that have left a name behind them, that their praises might be reported. A nd some there be, which have no memorial; who are perished, as though they had never been; and are become as though they had never been born; and their children after them. E C C L E S I A S T I C U S 4 4: 7–9 bur83155 fm i-xii.tex iii 01/13/2010 16:14 This page intentionally left blank  Revised Con rming Pages Contents Early Egyptian Multiplication 37 The Unit Fraction Table 40 Representing Rational Numbers 43 2.3 Four Problems from the Rhind Papyrus 46 The Method of False Position 46 A Curious Problem 49 Preface x–xii Egyptian Mathematics as Applied Arithmetic 50 2.4 Egyptian Geometry 53 Approximating the Area of a Circle 53 Chapter 1 The Volume of a Truncated Pyramid 56 Early Number Systems and Speculations About the Great Pyramid 57 Symbols 1 2.5 Babylonian Mathematics 62 A Tablet of Reciprocals 62 1.1 Primitive Counting 1 The Babylonian Treatment of Quadratic Equations 64 A Sense of Number 1 Two Characteristic Babylonian Problems 69 Notches as Tally Marks 2 2.6 Plimpton 322 72 The Peruvian Quipus: Knots as Numbers 6 A Tablet Concerning Number Triples 72 1.2 Number Recording of the Egyptians and Greeks 9 Babylonian Use of the Pythagorean Theorem 76 The History of Herodotus 9 The Cairo Mathematical Papyrus 77 Hieroglyphic Representation of Numbers 11 Egyptian Hieratic Numeration 15 Chapter 3 The Greek Alphabetic Numeral System 16 The Beginnings of Greek 1.3 Number Recording of the Babylonians 20 Mathematics 83 Babylonian Cuneiform Script 20 Deciphering Cuneiform: Grotefend and Rawlinson 21 3.1 The Geometrical Discoveries of Thales 83 The Babylonian Positional Number System 23 Greece and the Aegean Area 83 Writing in Ancient China 26 The Dawn of Demonstrative Geometry: Thales of Miletos 86 Chapter 2 Measurements Using Geometry 87 3.2 Pythagorean Mathematics 90 Mathematics in Early Pythagoras and His Followers 90 Civilizations 33 Nicomachus’s Introductio Arithmeticae 94 2.1 The Rhind Papyrus 33 The Theory of Figurative Numbers 97 Egyptian Mathematical Papyri 33 Zeno’s Paradox 101 A Key to Deciphering: The Rosetta Stone 35 3.3 The Pythagorean Problem 105 2.2 Egyptian Arithmetic 37 Geometric Proofs of the Pythagorean Theorem 105 v bur83155 fm i-xii.tex v 01/13/2010 16:43  Revised Con rming Pages vi Contents Early Solutions of the Pythagorean Equation 107 The Almagest of Claudius Ptolemy 188 The Crisis of Incommensurable Quantities 109 Ptolemy’s Geographical Dictionary 190 Theon’s Side and Diagonal Numbers 111 4.5 Archimedes 193 Eudoxus of Cnidos 116 The Ancient World’s Genius 193 3.4 Three Construction Problems of Antiquity 120 Estimating the Value of ³ 197 Hippocrates and the Quadrature of the Circle 120 The Sand-Reckoner 202 The Duplication of the Cube 124 Quadrature of a Parabolic Segment 205 The Trisection of an Angle 126 Apollonius of Perga: The Conics 206 3.5 The Quadratrix of Hippias 130 Rise of the Sophists 130 Chapter 5 Hippias of Elis 131 The Twilight of Greek The Grove of Academia: Plato’s Academy 134 Mathematics: Diophantus 213 Chapter 4 5.1 The Decline of Alexandrian Mathematics 213 The Waning of the Golden Age 213 The Alexandrian School: The Spread of Christianity 215 Euclid 141 Constantinople, A Refuge for Greek Learning 217 4.1 Euclid and the Elements 141 5.2 The Arithmetica 217 A Center of Learning: The Museum 141 Diophantus’s Number Theory 217 Euclid’s Life and Writings 143 Problems from the Arithmetica 220 4.2 Euclidean Geometry 144 5.3 Diophantine Equations in Greece, India, Euclid’s Foundation for Geometry 144 and China 223 Postulates 146 The Cattle Problem of Archimedes 223 Common Notions 146 Early Mathematics in India 225 Book I of the Elements 148 The Chinese Hundred Fowls Problem 228 Euclid’s Proof of the Pythagorean Theorem 156 5.4 The Later Commentators 232 Book II on Geometric Algebra 159 The Mathematical Collection of Pappus 232 Construction of the Regular Pentagon 165 Hypatia, the First Woman Mathematician 233 4.3 Euclid’s Number Theory 170 Roman Mathematics: Boethius and Cassiodorus 235 Euclidean Divisibility Properties 170 5.5 Mathematics in the Near and Far East 238 The Algorithm of Euclid 173 The Algebra of al-Khowârizmı̂ 238 The Fundamental Theorem of Arithmetic 177 Abû Kâmil and Thâbit ibn Qurra 242 An Infinity of Primes 180 Omar Khayyam 247 4.4 Eratosthenes, the Wise Man of Alexandria 183 The Astronomers al-Tûsı̂ and al-Kashı̂ 249 The Sieve of Eratosthenes 183 The Ancient Chinese Nine Chapters 251 Measurement of the Earth 186 Later Chinese Mathematical Works 259 bur83155 fm i-xii.tex vi 01/13/2010 16:43  Revised Con rming Pages Contents vii Chapter 6 Cardan’s Solution of the Cubic Equation 320 Bombelli and Imaginary Roots of the Cubic 324 The First Awakening: 7.4 Ferrari’s Solution of the Quartic Equation 328 Fibonacci 269 The Resolvant Cubic 328 6.1 The Decline and Revival of Learning 269 The Story of the Quintic Equation: The Carolingian Pre-Renaissance 269 Ruffini, Abel, and Galois 331 Transmission of Arabic Learning to the West 272 Chapter 8 The Pioneer Translators: Gerard and Adelard 274 6.2 The Liber Abaci and Liber Quadratorum 277 The Mechanical World: The Hindu-Arabic Numerals 277 Descartes and Newton 337 Fibonacci’s Liber Quadratorum 280 8.1 The Dawn of Modern Mathematics 337 The Works of Jordanus de Nemore 283 The Seventeenth Century Spread of Knowledge 337 6.3 The Fibonacci Sequence 287 Galileo’s Telescopic Observations 339 The Liber Abaci’s Rabbit Problem 287 The Beginning of Modern Notation: Some Properties of Fibonacci Numbers 289 François Vièta 345 6.4 Fibonacci and the Pythagorean Problem 293 The Decimal Fractions of Simon Stevin 348 Pythagorean Number Triples 293 Napier’s Invention of Logarithms 350 Fibonacci’s Tournament Problem 297 The Astronomical Discoveries of Brahe and Kepler 355 Chapter 7 8.2 Descartes: The Discours de la Méthode 362 The Writings of Descartes 362 The Renaissance of Mathematics: Inventing Cartesian Geometry 367 Cardan and Tartaglia 301 The Algebraic Aspect of La Géométrie 372 7.1 Europe in the Fourteenth and Fifteenth Descartes’s Principia Philosophiae 375 Centuries 301 Perspective Geometry: Desargues and Poncelet 377 The Italian Renaissance 301 8.3 Newton: The Principia Mathematica 381 Artificial Writing: The Invention of Printing 303 The Textbooks of Oughtred and Harriot 381 Founding of the Great Universities 306 Wallis’s Arithmetica Infinitorum 383 A Thirst for Classical Learning 310 The Lucasian Professorship: Barrow and Newton 386 7.2 The Battle of the Scholars 312 Newton’s Golden Years 392 Restoring the Algebraic Tradition: Robert Recorde 312 The Laws of Motion 398 The Italian Algebraists: Pacioli, del Ferro, and Later Years: Appointment to the Mint 404 Tartaglia 315 8.4 Gottfried Leibniz: The Calculus Controversy 409 Cardan, A Scoundrel Mathematician 319 The Early Work of Leibniz 409 7.3 Cardan’s Ars Magna 320 Leibniz’s Creation of the Calculus 413 bur83155 fm i-xii.tex vii 01/13/2010 16:43  Revised Con rming Pages viii Contents Newton’s Fluxional Calculus 416 Scientific Societies 497 The Dispute over Priority 424 Marin Mersenne’s Mathematical Gathering 499 Maria Agnesi and Emilie du Châtelet 430 Numbers, Perfect and Not So Perfect 502 10.2 From Fermat to Euler 511 Chapter 9 Fermat’s Arithmetica 511 The Development of Probability The Famous Last Theorem of Fermat 516 Theory: Pascal, Bernoulli, The Eighteenth-Century Enlightenment 520 and Laplace 439 Maclaurin’s Treatise on Fluxions 524 Euler’s Life and Contributions 527 9.1 The Origins of Probability Theory 439 10.3 The Prince of Mathematicians: Carl Graunt’s Bills of Mortality 439 Friedrich Gauss 539 Games of Chance: Dice and Cards 443 The Period of the French Revolution: The Precocity of the Young Pascal 446 Lagrange, Monge, and Carnot 539 Pascal and the Cycloid 452 Gauss’s Disquisitiones Arithmeticae 546 De Méré’s Problem of Points 454 The Legacy of Gauss: Congruence Theory 551 9.2 Pascal’s Arithmetic Triangle 456 Dirichlet and Jacobi 558 The Traité du Triangle Arithmétique 456 Mathematical Induction 461 Chapter 11 Francesco Maurolico’s Use of Induction 463 9.3 The Bernoullis and Laplace 468 Nineteenth-Century Christiaan Huygens’s Pamphlet on Probability 468 Contributions: Lobachevsky to The Bernoulli Brothers: John and James 471 Hilbert 563 De Moivre’s Doctrine of Chances 477 11.1 Attempts to Prove the Parallel Postulate 563 The Mathematics of Celestial Phenomena: The Efforts of Proclus, Playfair, and Wallis 563 Laplace 478 Saccheri Quadrilaterals 566 Mary Fairfax Somerville 482 The Accomplishments of Legendre 571 Laplace’s Research in Probability Theory 483 Legendre’s Eléments de géométrie 574 Daniel Bernoulli, Poisson, and Chebyshev 489 11.2 The Founders of Non-Euclidean Geometry 584 Gauss’s Attempt at a New Geometry 584 Chapter 10 The Struggle of John Bolyai 588 Creation of Non-Euclidean Geometry: Lobachevsky 592 The Revival of Number Theory: Models of the New Geometry: Riemann, Fermat, Euler, and Gauss 497 Beltrami, and Klein 598 10.1 Marin Mersenne and the Search Grace Chisholm Young 603 for Perfect Numbers 497 11.3 The Age of Rigor 604 bur83155 fm i-xii.tex viii 01/13/2010 16:43  Revised Con rming Pages Contents ix D’Alembert and Cauchy on Limits 604 Zermelo and the Axiom of Choice 701 Fourier’s Series 610 The Logistic School: Frege, Peano, and Russell 704 The Father of Modern Analysis, Weierstrass 614 Hilbert’s Formalistic Approach 708 Sonya Kovalevsky 616 Brouwer’s Institutionism 711 The Axiomatic Movement: Pasch and Hilbert 619 11.4 Arithmetic Generalized 626 Chapter 13 Babbage and the Analytical Engine 626 Extensions and Generalizations: Peacock’s Treatise on Algebra 629 Hardy, Hausdorff, The Representation of Complex Numbers 630 and Noether 721 Hamilton’s Discovery of Quaternions 633 Matrix Algebra: Cayley and Sylvester 639 13.1 Hardy and Ramanujan 721 Boole’s Algebra of Logic 646 The Tripos Examination 721 The Rejuvenation of English Mathematics 722 Chapter 12 A Unique Collaboration: Hardy and Littlewood 725 India’s Prodigy, Ramanujan 726 Transition to the Twentieth 13.2 The Beginnings of Point-Set Topology 729 Century: Cantor and Frechet’s Metric Spaces 729 Kronecker 657 The Neighborhood Spaces of Hausdorff 731 12.1 The Emergence of American Mathematics 657 Banach and Normed Linear Spaces 733 Ascendency of the German Universities 657 13.3 Some Twentieth-Century Developments 735 American Mathematics Takes Root: 1800–1900 659 Emmy Noether’s Theory of Rings 735 The Twentieth-Century Consolidation 669 Von Neumann and the Computer 741 12.2 Counting the Infinite 673 Women in Modern Mathematics 744 The Last Universalist: Poincaré 673 A Few Recent Advances 747 Cantor’s Theory of Infinite Sets 676 Kronecker’s View of Set Theory 681 Countable and Uncountable Sets 684 General Bibliography 755 Transcendental Numbers 689 Additional Reading 759 The Continuum Hypothesis 694 The Greek Alphabet 761 12.3 The Paradoxes of Set Theory 698 Solutions to Selected Problems 762 The Early Paradoxes 698 Index 777 bur83155 fm i-xii.tex ix 01/13/2010 16:43  Revised Con rming Pages Since many excellent treatises on the history of mathe- Preface matics are available, there may seem to be little reason for writing another. But most current works are severely technical, written by mathematicians for other mathe- maticians or for historians of science. Despite the ad- mirable scholarship and often clear presentation of these works, they are not especially well adapted to the undergraduate classroom. (Perhaps the most notable exception is Howard Eves’s popular account, An Introduction to the History of Mathematics.) There is a need today for an undergraduate textbook, which is also accessible to the general reader interested in the history of mathematics. In the following pages, I have tried to give a reasonably full account of how mathematics has developed over the past 5000 years. Because mathematics is one of the oldest intellectual instruments, it has a long story, interwoven with striking personalities and outstanding achievements. This narrative is chronological, beginning with the origin of mathematics in the great civilizations of antiquity and progressing through the later decades of the twentieth century. The presen- tation necessarily becomes less complete for modern times, when the pace of discovery has been rapid and the subject matter more technical. Considerable prominence has been assigned to the lives of the people responsible for progress in the mathematical enterprise. In emphasizing the biographical element, I can say only that there is no sphere in which individuals count for more than the intellectual life, and that most of the mathematicians cited here really did tower over their contemporaries. So that they will stand out as living gures and representatives of their day, it is necessary to pause from time to time to consider the social and cultural framework in which they lived. I have especially tried to de ne why mathematical activity waxed and waned in different periods and in different countries. Writers on the history of mathematics tend to be trapped between the desire to interject some genuine mathematics into a work and the desire to make the reading as painless and pleasant as possible. Believing that any mathematics textbook should concern itself primarily with teaching mathematical content, I have favored stressing the mathematics. Thus, assorted problems of varying degrees of dif culty have been interspersed throughout. Usually these problems typify a particular historical period, requiring the procedures of that time. They are an integral part of the text and, in working them, you will learn some interesting mathematics as well as history. The level of maturity needed for this work is approximately the mathematical background of a college junior or senior. Readers with more extensive training in the subject must forgive certain explanations that seem unnecessary. The title indicates that this book is in no way an encyclopedic enterprise: it does not pretend to present all the important mathematical ideas that arose during the vast sweep of time it covers. The inevitable limitations of space necessitate illuminating some out- standing landmarks instead of casting light of equal brilliance over the whole landscape. A certain amount of judgment and self-denial has been exercised, both in choosing mathematicians and in treating their contributions. The material that appears here does re ect some personal tastes and prejudices. It stands to reason that not everyone will be satis ed with the choices. Some readers will raise an eyebrow at the omission of some household names of mathematics that have been either passed over in complete silence or shown no great hospitality; others will regard the scant treatment of their favorite topic as an unpardonable omission. Nevertheless, the path that I have pieced together x bur83155 fm i-xii.tex x 01/13/2010 16:14  Revised Con rming Pages Preface xi should provide an adequate explanation of how mathematics came to occupy its position as a primary cultural force in Western civilization. The book is published in the modest hope that it may stimulate the reader to pursue more elaborate works on the subject. Anyone who ranges over such a well-cultivated eld as the history of mathematics becomes much indebted to the scholarship of others. The chapter bibliographies represent a partial listing of works that in one way or another have helped my command of the facts. To the writers and many others of whom no record was kept, I am enormously grateful. Readers familiar with previous editions of The History of Mathematics New to This Edition will nd that this seventh edition maintains the same overall organi- zation and content. Nevertheless, the preparation of a seventh edition has provided the occasion for a variety of small improvements as well as several more signi cant ones. The most notable difference is an enhanced treatment of American mathematics. Section 12.1, for instance, includes the efforts of such early nineteenth-century gures as Robert Adrain and Benjamin Banneker. Because the mathematically gifted of the period often became observational astronomers, the contributions of Simon Newcomb, George William Hill, Albert Michelson, and Maria Mitchell are also recounted. Later sections consider the work of more recent mathematicians, such as Oswald Veblen, R. L. Moore, Richard Courant, and Walter Feit. Another noteworthy difference is the attention now paid to several mathematicians passed over in previous editions. Among them are Lazar Carnot, Herman Günther Grass- mann, Andrei Kolmogorov, William Burnside, and Paul Erdös. Beyond these modi cations, there are some minor changes: biographies are brought up to date and certain numerical information kept current. In addition, an attempt has been made to correct errors, both typographical and historical, which crept into the earlier editions. If you or your students are ready for an alternative version of the Electronic Books traditional textbook, McGraw-Hill has partnered with CourseSmart and VitalSource to bring you innovative and inexpensive electronic textbooks. Students can save up to 50 percent off the cost of a print book, reduce their impact on the environment, and gain access to powerful Web tools for learning including full text search, notes and highlighting, and email tools for sharing notes between classmates. eBooks from McGraw-Hill are smart, interactive, searchable, and portable. To review complimentary copies or to purchase an eBook, go to either www.CourseSmart.com or www.VitalSource.com. Many friends, colleagues, and readers—too numerous to mention Acknowledgments individually—have been kind enough to forward corrections or to of- fer suggestions for the book’s enrichment. My thanks to all for their collective contributions. Although not every recommendation was in- corporated, all were gratefully received and seriously considered when deciding upon alterations. bur83155 fm i-xii.tex xi 01/13/2010 16:14  Revised Con rming Pages xii Preface In particular, the advice of the following reviewers was especially helpful in the creation of the seventh edition: Victor Akatsa, Chicago State University Carl FitzGerald, The University of California, San Diego Gary Shannon, California State University, Sacramento Tomas Smotzer, Youngstown State University John Stroyls, Georgia Southwestern State University A special debt of thanks is owed my wife, Martha Beck Burton, for providing assis- tance throughout the preparation of this edition. Her thoughtful comments signi cantly improved the exposition. Finally, I would like to express my appreciation to the staff members of McGraw-Hill for their unfailing cooperation during the course of production. Any errors that have survived all this generous assistance must be laid at my door. D. M. B. bur83155 fm i-xii.tex xii 01/13/2010 16:14  Con rming Pages CHAPTER 1 Early Number Systems and Symbols To think the thinkable—that is the mathematician’s aim. C. J. K E Y S E R The root of the term mathematics is in the Greek word math- 1.1 Primitive Counting emata, which was used quite generally in early writings to indicate any subject of instruction or study. As learning ad- A Sense of Number vanced, it was found convenient to restrict the scope of this term to particular elds of knowledge. The Pythagoreans are said to have used it to describe arithmetic and geometry; previously, each of these subjects had been called by its separate name, with no designation common to both. The Pythagoreans’ use of the name would perhaps be a basis for the notion that mathematics began in Classical Greece during the years from 600 to 300 B.C. But its history can be followed much further back. Three or four thousand years ago, in ancient Egypt and Babylonia, there already existed a signi cant body of knowledge that we should describe as mathematics. If we take the broad view that mathematics involves the study of issues of a quantitative or spatial nature—number, size, order, and form—it is an activity that has been present from the earliest days of human experience. In every time and culture, there have been people with a compelling desire to comprehend and master the form of the natural world around them. To use Alexander Pope’s words, “This mighty maze is not without a plan.” It is commonly accepted that mathematics originated with the practical problems of counting and recording numbers. The birth of the idea of number is so hidden behind the veil of countless ages that it is tantalizing to speculate on the remaining evidences of early humans’ sense of number. Our remote ancestors of some 20,000 years ago—who were quite as clever as we are—must have felt the need to enumerate their livestock, tally objects for barter, or mark the passage of days. But the evolution of counting, with its spoken number words and written number symbols, was gradual and does not allow any determination of precise dates for its stages. Anthropologists tell us that there has hardly been a culture, however primitive, that has not had some awareness of number, though it might have been as rudimentary as the distinction between one and two. Certain Australian aboriginal tribes, for instance, counted to two only, with any number larger than two called simply “much” or “many.” South American Indians along the tributaries of the Amazon were equally destitute of number words. Although they ventured further than the aborigines in being able to count to six, they had no independent number names for groups of three, four, ve, or six. In 1 bur83155 ch01 01-32.tex 1 11/03/2009 18:11  Con rming Pages 2 Chapter 1 Early Number Systems and Symbols their counting vocabulary, three was called “two-one,” four was “two-two,” and so on. A similar system has been reported for the Bushmen of South Africa, who counted to ten (10 D 2 C 2 C 2 C 2 C 2) with just two words; beyond ten, the descriptive phrases became too long. It is notable that such tribal groups would not willingly trade, say, two cows for four pigs, yet had no hesitation in exchanging one cow for two pigs and a second cow for another two pigs. The earliest and most immediate technique for visibly expressing the idea of number is tallying. The idea in tallying is to match the collection to be counted with some easily employed set of objects—in the case of our early forebears, these were ngers, shells, or stones. Sheep, for instance, could be counted by driving them one by one through a narrow passage while dropping a pebble for each. As the ock was gathered in for the night, the pebbles were moved from one pile to another until all the sheep had been accounted for. On the occasion of a victory, a treaty, or the founding of a village, frequently a cairn, or pillar of stones, was erected with one stone for each person present. The term tally comes from the French verb tailler, “to cut,” like the English word tailor; the root is seen in the Latin taliare, meaning “to cut.” It is also interesting to note that the English word write can be traced to the Anglo-Saxon writan, “to scratch,” or “to notch.” Neither the spoken numbers nor nger tallying have any permanence, although nger counting shares the visual quality of written numerals. To preserve the record of any count, it was necessary to have other representations. We should recognize as human intellectual progress the idea of making a correspondence between the events or objects recorded and a series of marks on some suitably permanent material, with one mark representing each individual item. The change from counting by assembling collections of physical objects to counting by making collections of marks on one object is a long step, not only toward abstract number concept, but also toward written communication. Counts were maintained by making scratches on stones, by cutting notches in wooden sticks or pieces of bone, or by tying knots in strings of different colors or lengths. When the numbers of tally marks became too unwieldy to visualize, primitive people arranged them in easily recognizable groups such as groups of 5, for the ngers of a hand. It is likely that grouping by pairs came rst, soon abandoned in favor of groups of 5, 10, or 20. The organization of counting by groups was a noteworthy improvement on counting by ones. The practice of counting by ves, say, shows a tentative sort of progress toward reaching an abstract concept of “ ve” as contrasted with the descriptive ideas “ ve ngers” or “ ve days.” To be sure, it was a timid step in the long journey toward detaching the number sequence from the objects being counted. Notches as Tally Marks Bone artifacts bearing incised markings seem to indicate that the people of the Old Stone Age had devised a system of tallying by groups as early as 30,000 B.C. The most impressive example is a shinbone from a young wolf, found in Czechoslovakia in 1937; about 7 inches long, the bone is engraved with 55 deeply cut notches, more or less equal in length, arranged in groups of ve. (Similar recording notations are still used, with the strokes bundled in ves, like. Voting results in small towns are still counted in the manner devised by our remote ancestors.) For many years such notched bones were bur83155 ch01 01-32.tex 2 11/03/2009 18:11  Con rming Pages Primitive Counting 3 interpreted as hunting tallies and the incisions were thought to represent kills. A more recent theory, however, is that the rst recordings of ancient people were concerned with reckoning time. The markings on bones discovered in French cave sites in the late 1880s are grouped in sequences of recurring numbers that agree with the numbers of days included in successive phases of the moon. One might argue that these incised bones represent lunar calendars. Another arresting example of an incised bone was unearthed at Ishango along the shores of Lake Edward, one of the headwater sources of the Nile. The best archeological and geological evidence dates the site to 17,500 B.C., or some 12,000 years before the rst settled agrarian communities appeared in the Nile valley. This fossil fragment was probably the handle of a tool used for engraving, or tattooing, or even writing in some way. It contains groups of notches arranged in three de nite columns; the odd, unbalanced composition does not seem to be decorative. In one of the columns, the groups are composed of 11, 21, 19, and 9 notches. The underlying pattern may be 10 C 1, 20 C 1, 20  1, and 10  1. The notches in another column occur in eight groups, in the following order: 3, 6, 4, 8, 10, 5, 5, 7. This arrangement seems to suggest an appreciation of the concept of duplication, or multiplying by 2. The last column has four groups consisting of 11, 13, 17, and 19 individual notches. The pattern here may be fortuitous and does not necessarily indicate—as some authorities are wont to infer—a familiarity with prime numbers. Because 11 C 13 C 17 C 19 D 60 and 11 C 21 C 19 C 9 D 60, it might be argued that markings on the prehistoric Ishango bone are related to a lunar count, with the rst and third columns indicating two lunar months. The use of tally marks to record counts was prominent among the prehistoric peoples of the Near East. Archaeological excavations have unearthed a large number of small clay objects that had been hardened by re to make them more durable. These handmade artifacts occur in a variety of geometric shapes, the most common being circular disks, triangles, and cones. The oldest, dating to about 8000 b.c., are incised with sets of parallel lines on a plain surface; occasionally, there will be a cluster of circular impressions as if punched into the clay by the blunt end of a bone or stylus. Because they go back to the time when people rst adopted a settled agricultural life, it is believed that the objects are primitive reckoning devices; hence, they have become known as “counters” or “tokens.” It is quite likely also that the shapes represent different commodities. For instance, a token of a particular type might be used to indicate the number of animals in a herd, while one of another kind could count measures of grain. Over several millennia, tokens became increasingly complex, with diverse markings and new shapes. Eventually, there came to be 16 main forms of tokens. Many were perforated with small holes, allowing them to be strung together for safekeeping. The token system of recording information went out of favor around 3000 b.c., with the rapid adoption of writing on clay tablets. A method of tallying that has been used in many different times and places involves the notched stick. Although this device provided one of the earliest forms of keeping records, its use was by no means limited to “primitive peoples,” or for that matter, to the remote past. The acceptance of tally sticks as promissory notes or bills of exchange reached its highest level of development in the British Exchequer tallies, which formed an essential part of the government records from the twelfth century onward. In this instance, the tallies were at pieces of hazelwood about 6–9 inches long and up to an inch thick. Notches of varying sizes and types were cut in the tallies, each notch representing a xed amount of money. The width of the cut decided its value. For example, the notch of £1000 bur83155 ch01 01-32.tex 3 11/03/2009 18:11  Con rming Pages 4 Chapter 1 Early Number Systems and Symbols was as large as the width of a hand; for £100, as large as the thickness of a thumb; and for £20, the width of the little nger. When a loan was made the appropriate notches were cut and the stick split into two pieces so that the notches appeared in each section. The debtor kept one piece and the Exchequer kept the other, so the transaction could easily be veri ed by tting the two halves together and noticing whether the notches coincided (whence the expression “our accounts tallied”). Presumably, when the two halves had been matched, the Exchequer destroyed its section—either by burning it or by making it smooth again by cutting off the notches—but retained the debtor’s section for future record. Obstinate adherence to custom kept this wooden accounting system in of cial use long after the rise of banking institutions and modern numeration had made its practice quaintly obsolete. It took an act of Parliament, which went into effect in 1826, to abolish the practice. In 1834, when the long-accumulated tallies were burned in the furnaces that heated the House of Lords, the re got out of hand, starting a more general con agration that destroyed the old Houses of Parliament. The English language has taken note of the peculiar quality of the double tally stick. Formerly, if someone lent money to the Bank of England, the amount was cut on a tally stick, which was then split. The piece retained by the bank was known as the foil, whereas the other half, known as the stock, was given the lender as a receipt for the sum of money paid in. Thus, he became a “stockholder” and owned “bank stock” having the same worth as paper money issued by the government. When the holder would return, the stock was carefully checked and compared against the foil in the bank’s possession; if they agreed, the owner’s piece would be redeemed in currency. Hence, a written certi cate that was presented for remittance and checked against its security later came to be called a “check.” Using wooden tallies for records of obligations was common in most European countries and continued there until fairly recently. Early in this century, for instance, in some remote valleys of Switzerland, “milk sticks” provided evidence of transactions among farmers who owned cows in a common herd. Each day the chief herdsman would carve a six- or seven-sided rod of ashwood, coloring it with red chalk so that incised lines would stand out vividly. Below the personal symbol of each farmer, the herdsman marked off the amounts of milk, butter, and cheese yielded by a farmer’s cows. Every Sunday after church, all parties would meet and settle the accounts. Tally sticks—in particular, double tallies—were recognized as legally valid documents until well into the 1800s. France’s rst modern code of law, the Code Civil, promulgated by Napoleon in 1804, contained the provision: The tally sticks which match their stocks have the force of contracts between persons who are accustomed to declare in this manner the deliveries they have made or received. The variety in practical methods of tallying is so great that giving any detailed account would be impossible here. But the procedure of counting both days and objects by means of knots tied in cords has such a long tradition that it is worth mentioning. The device was frequently used in ancient Greece, and we nd reference to it in the work of Herodotus ( fth century B.C.). Commenting in his History, he informs us that the Persian king Darius handed the Ionians a knotted cord to serve as a calendar: The King took a leather thong and tying sixty knots in it called together the Ionian tyrants and spoke thus to them: “Untie every day one of the knots; if I do not return before the bur83155 ch01 01-32.tex 4 11/03/2009 18:11  Con rming Pages Primitive Counting 5 Three views of a Paleolithic wolfbone used for tallying. (The Illustrated London News Picture Library.) bur83155 ch01 01-32.tex 5 11/03/2009 18:11  Con rming Pages 6 Chapter 1 Early Number Systems and Symbols last day to which the knots will hold out, then leave your station and return to your several homes.” The Peruvian Quipus: Knots as Numbers In the New World, the number string is best illustrated by the knotted cords, called quipus, of the Incas of Peru. They were originally a South American Indian tribe, or a collection of kindred tribes, living in the central Andean mountainous highlands. Through gradual expansion and warfare, they came to rule a vast empire consisting of the coastal and mountain regions of present-day Ecuador, Peru, Bolivia, and the northern parts of Chile and Argentina. The Incas became renowned for their engineering skills, constructing stone temples and public buildings of a great size. A striking accomplishment was their creation of a vast network (as much as 14,000 miles) of roads and bridges linking the far- ung parts of the empire. The isolation of the Incas from the horrors of the Spanish Conquest ended early in 1532 when 180 conquistadors landed in northern Peru. By the end of the year, the invaders had seized the capital city of Cuzco and imprisoned the emperor. The Spaniards imposed a way of life on the people that within about 40 years would destroy the Inca culture. When the Spanish conquerors arrived in the sixteenth century, they observed that each city in Peru had an “of cial of the knots,” who maintained complex accounts by means of knots and loops in strands of various colors. Performing duties not unlike those of the city treasurer of today, the quipu keepers recorded all of cial transactions concerning the land and subjects of the city and submitted the strings to the central government in Cuzco. The quipus were important in the Inca Empire, because apart from these knots no system of writing was ever developed there. The quipu was made of a thick main cord or crossbar to which were attached ner cords of different lengths and colors; ordinarily the cords hung down like the strands of a mop. Each of the pendent strings represented a certain item to be tallied; one might be used to show the number of sheep, for instance, another for goats, and a third for lambs. The knots themselves indicated numbers, the values of which varied according to the type of knot used and its speci c position on the strand. A decimal system was used, with the knot representing units placed nearest the bottom, the tens appearing immediately above, then the hundreds, and so on; absence of a knot denoted zero. Bunches of cords were tied off by a single main thread, a summation cord, whose knots gave the total count for each bunch. The range of possibilities for numerical representation in the quipus allowed the Incas to keep incredibly detailed administrative records, despite their ignorance of the written word. More recent (1872) evidence of knots as a counting device occurs in India; some of the Santal headsmen, being illiterate, made knots in strings of four different colors to maintain an up-to-date census. To appreciate the quipu fully, we should notice the numerical values represented by the tied knots. Just three types of knots were used: a gure-eight knot standing for 1, a long knot denoting one of the values 2 through 9, depending on the number of twists in the knot, and a single knot also indicating 1. The gure-eight knot and long knot appear only in the lowest (units) position on a cord, while clusters of single knots can appear in the other spaced positions. Because pendant cords have the same length, an empty position (a value of zero) would be apparent on comparison with adjacent cords. bur83155 ch01 01-32.tex 6 11/03/2009 18:11  Con rming Pages Primitive Counting 7 Also, the reappearance of either a gure-eight or long knot would point out that another number is being recorded on the same cord. Recalling that ascending positions carry place value for successive powers of ten, let us suppose that a particular cord contains the following, in order: a long knot with four twists, two single knots, an empty space, seven clustered single knots, and one single knot. For the Inca, this array would represent the number 17024 D 4 C (2 Ð 10) C (0 Ð 102 ) C (7 Ð 103 ) C (1 Ð 104 ): Another New World culture that used a place value numeration system was that of the ancient Maya. The people occupied a broad expanse of territory embracing southern Mexico and parts of what is today Guatemala, El Salvador, and Honduras. The Mayan civilization existed for over 2000 years, with the time of its greatest owering being the period 300–900 a.d. A distinctive accomplishment was its development of an elaborate form of hieroglyphic writing using about 1000 glyphs. The glyphs are sometimes sound based and sometimes meaning based: the vast majority of those that have survived have yet to be deciphered. After 900 a.d., the Mayan civilization underwent a sudden decline— The Great Collapse—as its populous cities were abandoned. The cause of this catastrophic exodus is a continuing mystery, despite speculative explanations of natural disasters, epidemic diseases, and conquering warfare. What remained of the traditional culture did not succumb easily or quickly to the Spanish Conquest, which began shortly after 1500. It was a struggle of relentless brutality, stretching over nearly a century, before the last unconquered Mayan kingdom fell in 1597. The Mayan calendar year was composed of 365 days divided into 18 months of 20 days each, with a residual period of 5 days. This led to the adoption of a counting system based on 20 (a vigesimal system). Numbers were expressed symbolically in two forms. The priestly class employed elaborate glyphs of grotesque faces of deities to indicate the numbers 1 through 19. These were used for dates carved in stone, commemorating notable events. The common people recorded the same numbers with combinations of bars and dots, where a short horizontal bar represented 5 and a dot 1. A particular feature was a stylized shell that served as a symbol for zero; this is the earliest known use of a mark for that number. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 The symbols representing numbers larger than 19 were arranged in a vertical column with those in each position, moving upward, multiplied by successive powers of 20; that is, by 1, 20, 400, 8000, 160,000, and so on. A shell placed in a position would indicate the absence of bars and dots there. In particular, the number 20 was expressed by a shell at the bottom of the column and a single dot in the second position. For an example bur83155 ch01 01-32.tex 7 11/03/2009 18:11  Con rming Pages 8 Chapter 1 Early Number Systems and Symbols Thirteenth-century British Exchequer tallies. (By courtesy of the Society of Antiquaries of London.) of a number recorded in this system, let us write the symbols horizontally rather than vertically, with the smallest value on the left: For us, this expression denotes the number 62808, for 62808 D 8 Ð 1 C 0 Ð 20 C 17 Ð 400 C 7 Ð 8000: Because the Mayan numeration system was developed primarily for calendar reckoning, there was a minor variation when carrying out such calculations. The symbol in the third position of the column was multiplied by 18 Ð 20 rather than by 20 Ð 20, the idea being that 360 was a better approximation to the length of the year than was 400. The place value of each position therefore increased by 20 times the one before; that is, the multiples are 1, 20, 360, 7200, 144,000, and so on. Under this adjustment, the value of the collection of symbols mentioned earlier would be 56528 D 8 Ð 1 C 0 Ð 20 C 17 Ð 360 C 7 Ð 7200: Over the long sweep of history, it seems clear that progress in devising ef cient ways of retaining and conveying numerical information did not take place until primitive people abandoned the nomadic life. Incised markings on bone or stone may have been adequate for keeping records when human beings were hunters and gatherers, but the food producer required entirely new forms of numerical representation. Besides, as a means for storing information, groups of markings on a bone would have been intelligible only to the person making them, or perhaps to close friends or relatives; thus, the record was probably not intended to be used by people separated by great distances. Deliberate cultivation of crops, particularly cereal grains, and the domestication of animals began, so far as can be judged from present evidence, in the Near East some bur83155 ch01 01-32.tex 8 11/03/2009 18:11  Con rming Pages Number Recording of the Egyptians and Greeks 9 10,000 years ago. Later experiments in agriculture occurred in China and in the New World. A widely held theory is that a climatic change at the end of the last ice age provided the essential stimulus for the introduction of food production and a settled village existence. As the polar ice cap began to retreat, the rain belt moved northward, causing the desiccation of much of the Near East. The increasing scarcity of wild food plants and the game on which people had lived forced them, as a condition of survival, to change to an agricultural life. It became necessary to count one’s harvest and herd, to measure land, and to devise a calendar that would indicate the proper time to plant crops. Even at this stage, the need for a means of counting was modest; and tallying techniques, although slow and cumbersome, were still adequate for ordinary dealings. But with a more secure food supply came the possibility of a considerable increase in population, which meant that larger collections of objects had to be enumerated. Repetition of some fundamental mark to record a tally led to inconvenient numeral representations, tedious to compose and dif cult to interpret. The desire of village, temple, and palace of cials to maintain meticulous records (if only for the purposes of systematic taxation) gave further impetus to nding new and more re ned means of “ xing” a count in a permanent or semipermanent form. Thus, it was in the more elaborate life of those societies that rose to power some 6000 years ago in the broad river valleys of the Nile, the Tigris-Euphrates, the Indus, and the Yangtze that special symbols for numbers rst appeared. From these, some of our most elementary branches of mathematics arose, because a symbolism that would allow expressing large numbers in written numerals was an essential prerequisite for computation and measurement. Through a welter of practical experience with number symbols, people gradually recognized certain abstract principles; for instance, it was discovered that in the fundamental operation of addition, the sum did not depend on the order of the summands. Such discoveries were hardly the work of a single individual, or even a single culture, but more a slow process of awareness moving toward an increasingly abstract way of thinking. We shall begin by considering the numeration systems of the important Near Eastern civilizations—the Egyptian and the Babylonian—from which sprang the main line of our own mathematical development. Number words are found among the word forms of the earliest extant writings of these people. Indeed, their use of symbols for numbers, detached from an association with the objects to be counted, was a big turning point in the history of civilization. It is more than likely to have been a rst step in the evolution of humans’ supreme intellectual achievement, the art of writing. Because the recording of quantities came more easily than the visual symbolization of speech, there is unmistakable evidence that the written languages of these ancient cultures grew out of their previously written number systems. The writing of history, as we understand it, 1.2 Number Recording of the Egyptians is a Greek invention; and foremost among the and Greeks early Greek historians was Herodotus. Herodotus (circa 485–430 B.C.) was born at Halicarnassus, a The History of Herodotus largely Greek settlement on the southwest coast of Asia Minor. In early life, he was involved in political troubles in his home city and forced to ee in exile to the island of Samos, and thence to Athens. From there Herodotus set out on travels whose leisurely character and bur83155 ch01 01-32.tex 9 11/03/2009 18:11  Con rming Pages 10 Chapter 1 Early Number Systems and Symbols broad extent indicate that they occupied many years. It is assumed that he made three principal journeys, perhaps as a merchant, collecting material and recording his impres- sions. In the Black Sea, he sailed all the way up the west coast to the Greek communities at the mouth of the Dnieper River, in what is now Ukraine, and then along the south coast to the foot of the Caucasus. In Asia Minor, he traversed modern Syria and Iraq and traveled down the Euphrates, possibly as far as Babylon. In Egypt, he ascended the Nile River from its delta to somewhere near Aswan, exploring the pyramids along the way. Around 443 B.C., Herodotus became a citizen of Thurium in southern Italy, a new colony planted under Athenian auspices. In Thurium, he seems to have passed the last years of his life involved almost entirely in nishing the History of Herodotus, a book larger than any Greek prose work before it. The reputation of Herodotus as a historian stood high even in his own day. In the absence of numerous copies of books, it is natural that a history, like other literary compositions, should have been read aloud at public and private gatherings. In Athens, some 20 years before his death, Herodotus recited completed portions of his History to admiring audiences and, we are told, was voted an unprecedentedly large sum of public money in recognition of the merit of his work. Although the story of the Persian Wars provides the connecting link in the History of Herodotus, the work is no mere chronicle of carefully recorded events. Almost anything that concerned people interested Herodotus, and his History is a vast store of information on all manner of details of daily life. He contrived to set before his compatriots a general picture of the known world, of its various peoples, of their lands and cities, and of what they did and above all why they did it. (A modern historian would probably describe the History as a guidebook containing useful sociological and anthropological data, instead of a work of history.) The object of his History, as Herodotus conceived it, required him to tell all he had heard but not necessarily to accept it all as fact. He atly stated, “My job is to report what people say, not to believe it all, and this principle is meant to apply to my whole work.” We nd him, accordingly, giving the traditional account of an occurrence and then offering his own interpretation or a contradictory one from a different source, leaving the reader to choose between versions. One point must be clear: Herodotus interpreted the state of the world at his time as a result of change in the past and felt that the change could be described. It is this attempt that earned for him, and not any of the earlier writers of prose, the honorable title “Father of History.” Herodotus took the trouble to describe Egypt at great length, for he seems to have been more enthusiastic about the Egyptians than about almost any other people that he met. Like most visitors to Egypt, he was distinctly aware of the exceptional nature of the climate and the topography along the Nile: “For anyone who sees Egypt, without having heard a word about it before, must perceive that Egypt is an acquired country, the gift of the river.” This famous passage—often paraphrased to read “Egypt is the gift of the Nile”—aptly sums up the great geographical fact about the country. In that sun-soaked, rainless climate, the river in over owing its banks each year regularly deposited the rich silt washed down from the East African highlands. To the extreme limits of the river’s waters there were fertile elds for crops and the pasturage of animals; and beyond that the barren desert frontiers stretched in all directions. This was the setting in which that literate, complex society known as Egyptian civilization developed. The emergence of one of the world’s earliest cultures was essentially a political act. Between 3500 and 3100 B.C., the self-suf cient agricultural communities that clung to the strip of land bordering the Nile had gradually coalesced into larger units until there bur83155 ch01 01-32.tex 10 11/03/2009 18:11  Con rming Pages Number Recording of the Egyptians and Greeks 11 The habitable world according to Herodotus. (From Stories from Herodotus by B. Wilson and D. Miller. Reproduced by permission of Oxford University Press.) were only the two kingdoms of Upper Egypt and Lower Egypt. Then, about 3100 B.C., these regions were united by military conquest from the south by a ruler named Menes, an elusive gure who stepped forth into history to head the long line of pharaohs. Protected from external invasion by the same deserts that isolated her, Egypt was able to develop the most stable and longest-lasting of the ancient civilizations. Whereas Greece and Rome counted their supremacies by the century, Egypt counted hers by the millennium; a well-ordered succession of 32 dynasties stretched from the uni cation of the Upper and Lower Kingdoms by Menes to Cleopatra’s encounter with the asp in 31 B.C. Long after the apogee of Ancient Egypt, Napoleon was able to exhort his weary veterans with the glory of its past. Standing in the shadow of the Great Pyramid of Gizeh, he cried, “Soldiers, forty centuries are looking down upon you!” Hieroglyphic Representation of Numbers As soon as the uni cation of Egypt under a single leader became an accomplished fact, a powerful and extensive administrative system began to evolve. The census had to be taken, taxes imposed, an army maintained, and so forth, all of which required reckoning with relatively large numbers. (One of the years of the Second Dynasty was named Year of the Occurrence of the Numbering of all Large and Small Cattle of the North and South.) As early as 3500 B.C., the Egyptians had a fully developed number system that would allow counting to continue inde nitely with only the introduction from time to time of a new symbol. This is borne out by the macehead of King Narmer, one of the most remarkable relics of the ancient world, now in a museum at Oxford University. bur83155 ch01 01-32.tex 11 11/03/2009 18:11  Con rming Pages 12 Chapter 1 Early Number Systems and Symbols This scene is taken from the great stone macehead of Narmer, which J. E. Quibell discovered at Hierakonpolis in 1898. There is a summary of the spoil taken by Narmer during his wars, namely “cows, 400,000, goats, 1,422,000, , and captives, 120,000,.” Scene reproduced from the stone macehead of Narmer, giving a summary of the spoil taken by him during his wars. (From The Dwellers on the Nile by E. W. Budge, 1977, Dover Publications, N.Y.) Near the beginning of the dynastic age, Narmer (who, some authorities suppose, may have been the legendary Menes, the rst ruler of the united Egyptian nation) was obliged to punish the rebellious Libyans in the western Delta. He left in the temple at Hierakonpolis a magni cent slate palette—the famous Narmer Palette—and a ceremonial macehead, both of which bear scenes testifying to his victory. The macehead preserves forever the of cial record of the king’s accomplishment, for the inscription boasts of the taking of 120,000 prisoners and a register of captive animals, 400,000 oxen and 1,422,000 goats. Another example of the recording of very large numbers at an early stage occurs in the Book of the Dead, a collection of religious and magical texts whose principle aim was to secure for the deceased a satisfactory afterlife. In one section, which is believed to date from the First Dynasty, we read (the Egyptian god Nu is speaking): “I work for you, o ye spirits, we are in number four millions, six hundred and one thousand, and two hundred.” The spectacular emergence of the Egyptian government and administration under the pharaohs of the rst two dynasties could not have taken place without a method of writing, and we nd such a method both in the elaborate “sacred signs,” or hieroglyphics, and in the rapid cursive hand of the accounting scribe. The hieroglyphic system of writing is a picture script, in which each character represents a concrete object, the signi cance of which may still be recognizable in many cases. In one of the tombs near the Pyramid bur83155 ch01 01-32.tex 12 11/03/2009 18:11  Con rming Pages Number Recording of the Egyptians and Greeks 13 of Gizeh there have been found hieroglyphic number symbols in which the number one is represented by a single vertical stroke, or a picture of a staff, and a kind of horseshoe, or heelbone sign \ is used as a collective symbol to replace ten separate strokes. In other words, the Egyptian system was a decimal one (from the Latin decem, “ten”), which used counting by powers of 10. That 10 is so often found among ancient peoples as a base for their number systems is undoubtedly attributable to humans’ ten ngers and to our habit of counting on them. For the same reason, a symbol much like our numeral 1 was almost everywhere used to express the number one. Special pictographs were used for each new power of 10 up to 10,000,000: 100 by a curved rope, 1000 by a lotus ower, 10,000 by an upright bent nger, 100,000 by a tadpole, 1,000,000 by a person holding up two hands as if in great astonishment, and 10,000,000 by a symbol sometimes conjectured to be a rising sun. 1 10 100 1000 10,000 100,000 1,000,000 10,000,000 or Other numbers could be expressed by using these symbols additively (that is, the number represented by a set of symbols is the sum of the numbers represented by the individual symbols), with each character repeated up to nine times. Usually, the direction of writing was from right to left, with the larger units listed rst, then the others in order of importance. Thus, the scribe would write to indicate our number 1 Ð 100;000 C 4 Ð 10;000 C 2 Ð 1000 C 1 Ð 100 C 3 Ð 10 C 6 Ð 1 D 142;136: Occasionally, the larger units were written on the left, in which case the symbols were turned around to face the direction from which the writing began. Lateral space was saved by placing the symbols in two or three rows, one above the other. Because there was a different symbol for each power of 10, the value of the number represented was not affected by the order of the hieroglyphs within a grouping. For example, all stood for the number 1232. Thus the Egyptian method of writing numbers was not a “positional system”—a system in which one and the same symbol has a different signi cance depending on its position in the numerical representation. Addition and subtraction caused little dif culty in the Egyptian number system. For addition, it was necessary only to collect symbols and exchange ten like symbols for the next higher symbol. This is how the Egyptians would have added, say, 345 and 678: bur83155 ch01 01-32.tex 13 11/03/2009 18:11  Con rming Pages 14 Chapter 1 Early Number Systems and Symbols 345 678 1023 This converted would be and converted again, Subtraction was performed by the same process in reverse. Sometimes “borrowing” was used, wherein a symbol for the large number was exchanged for ten lower-order symbols to provide enough for the smaller number to be subtracted, as in the case 123 45 78 which, converted, would be bur83155 ch01 01-32.tex 14 11/03/2009 18:11  Con rming Pages Number Recording of the Egyptians and Greeks 15 Although the Egyptians had symbols for numbers, they had no generally uniform nota- tion for arithmetical operations. In the case of the famous Rhind Papyrus (dating about 1650 B.C.), the scribe did represent addition and subtraction by the hieroglyphs and , which resemble the legs of a person coming and going. Egyptian Hieratic Numeration As long as writing was restricted to inscriptions carved on stone or metal, its scope was limited to short records deemed to be outstandingly important. What was needed was an easily available, inexpensive material to write on. The Egyptians solved this problem with the invention of papyrus. Papyrus was made by cutting thin lengthwise strips of the stem of the reedlike papyrus plant, which was abundant in the Nile Delta marshes. The sections were placed side by side on a board so as to form a sheet, and another layer was added at right angles to the rst. When these were all soaked in water, pounded with a mallet, and allowed to dry in the sun, the natural gum of the plant glued the sections together. The writing surface was then scraped smooth with a shell until a nished sheet (usually 10 to 18 inches wide) resembled coarse brown paper; by pasting these sheets together along overlapping edges, the Egyptians could produce strips up to 100 feet long, which were rolled up when not in use. They wrote with a brushlike pen, and ink made of colored earth or charcoal that was mixed with gum or water. Thanks not so much to the durability of papyrus as to the exceedingly dry climate of Egypt, which prevented mold and mildew, a sizable body of scrolls has been preserved for us in a condition otherwise impossible. With the introduction of papyrus, further steps in simplifying writing were almost inevitable. The rst steps were made largely by the Egyptian priests who developed a more rapid, less pictorial style that was better adapted to pen and ink. In this so-called “hieratic” (sacred) script, the symbols were written in a cursive, or free-running, hand so that at rst sight their forms bore little resemblance to the old hieroglyphs. It can be said to correspond to our handwriting as hieroglyphics corresponds to our print. As time passed and writing came into general use, even the hieratic proved to be too slow and a kind of shorthand known as “demotic” (popular) script arose. Hieratic writing is child’s play compared with demotic, which at its worst consists of row upon row of agitated commas, each representing a totally different sign. In both of these writing forms, numerical representation was still additive, based on powers of 10; but the repetitive principle of hieroglyphics was replaced by the device of using a single mark to represent a collection of like symbols. This type of notation may be called “cipherization.” Five, for instance, was assigned the distinctive mark instead of being indicated by a group of ve vertical strokes. 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 1000 bur83155 ch01 01-32.tex 15 11/03/2009 18:11  Con rming Pages 16 Chapter 1 Early Number Systems and Symbols The hieratic system used to represent numbers is as shown in the preceeding table. Note that the signs for 1, 10, 100, and 1000 are essentially abbreviations for the pic- tographs used earlier. In hieroglyphics, the number 37 had appeared as but in hieratic script it is replaced by the less cumbersome The larger number of symbols called for in this notation imposed an annoying tax on the memory, but the Egyptian scribes no doubt regarded this as justi ed by its speed and conciseness. The idea of ciphering is one of the decisive steps in the development of numeration, comparable in signi cance to the Babylonian adoption of the positional principle. The Greek Alphabetic Numeral System Around the fth century B.C., the Greeks of Ionia also developed a ciphered numeral system, but with a more extensive set of symbols to be memorized. They ciphered their numbers by means of the 24 letters of the ordinary Greek alphabet, augmented by three obsolete Phoenician letters (the digamma for 6, the koppa for 90, and the sampi for 900). The resulting 27 letters were used as follows. The initial nine letters were associated with the numbers from 1 to 9; the next nine letters represented the rst nine integral multiples of 10; the nal nine letters were used for the rst nine integral multiples of 100. The following table shows how the letters of the alphabet (including the special forms) were arranged for use as numerals. 1 Þ 10  100 ² 2 þ 20  200 ¦ 3  30 ½ 300 − 4 Ž 40 ¼ 400 × 5 " 50 ¹ 500  6 60 ¾ 600  7  70 o 700 8  80 ³ 800 ! 9 90 900 Because the Ionic system was still a system of additive type, all numbers between 1 and 999 could be represented by at most three symbols. The principle is shown by ³ Ž D 700 C 80 C 4 D 784: For larger numbers, the following scheme was used. An accent mark placed to the left and below the appropriate unit letter multiplied the corresponding number by 1000; thus 0 þ represents not 2 but 2000. Tens of thousands were indicated by using a new letter bur83155 ch01 01-32.tex 16 11/03/2009 18:11  Con rming Pages Number Recording of the Egyptians and Greeks 17 M, from the word myriad (meaning “ten thousand”). The letter M placed either next to or below the symbols for a number from 1 to 9999 caused the number to be multiplied by 10,000, as with Ž ŽM; or M D 40;000; ²¹ ²¹M; or M D 1;500;000: With these conventions, the Greeks wrote − ¼"M 0 þ²¼Ž D 3;452;144: To express still larger numbers, powers of 10,000 were used, the double myriad MM denoting (10,000)2 , and so on. The symbols were always arranged in the same order, from the highest multiple of 10 on the left to the lowest on the right, so accent marks sometimes could be omitted when the context was clear. The use of the same letter for thousands and units, as in Ž¦ ½Ž D 4234; gave the left-hand letter a local place value. To distinguish the numerical meaning of letters from their ordinary use in language, the Greeks added an accent at the end or a bar extended over them; thus, the number 1085 might appear as 0 Þ³ " 0 Þ³ ": 0 or The system as a whole afforded much economy of writing (whereas the Greek alphabetic numerical for 900 is a single letter, the Egyptians had to use the symbol nine times), but it required the mastery of numerous signs. Multiplication in Greek alphabetic numerals was performed by beginning with the highest order in each factor and forming a sum of partial products. Let us calculate, for example, 24 ?