i i \master" | 2011/4/5 | 12:53 | page i | #1 i i Sherlock Holmes in Babylon and Other Tales of Mathematical History i i i i i i \master" | 2011/4/5 | 12:53 | page ii | #2 i i c 2004by (cid:13) TheMathematicalAssociationof America(Incorporated) LibraryofCongressCatalogCardNumber2003113541 Print ISBN:978-0-88385-546-1 Electronic ISBN:978-1-61444-503-6 Printedin theUnitedStatesofAmerica Current Printing (last digit): 109 8765432 i i i i i i \master" | 2011/4/5 | 12:53 | page iii | #3 i i Sherlock Holmes in Babylon and Other Tales of Mathematical History Edited by Marlow Anderson Colorado College Victor Katz University of the District of Columbia Robin Wilson Open University Published and Distributedby The Mathematical Association of America i i i i i i \master" | 2011/4/5 | 12:53 | page iv | #4 i i Committee on Publications Gerald L. Alexanderson, Chair Spectrum EditorialBoard Gerald L. Alexanderson, Chair Robert Beezer Russell L. Merris WilliamDunham Jean J. Pedersen Michael Filaseta J. D. Phillips Erica Flapan Marvin Schaefer Eleanor Lang Kendrick Harvey Schmidt Jeffrey L. Nunemacher Sanford Segal Ellen Maycock Franklin Sheehan John E. Wetzel i i i i i i \master" | 2011/4/5 | 12:53 | page v | #5 i i SPECTRUM SERIES The Spectrum Series of the Mathematical Association of America was so named to reflect its purpose: to publish a broad range of books including biographies, accessible expositions of old or new mathematical ideas, reprintsand revisions of excellentout-of-printbooks, popular works, and other monographs of high interest that will appeal to a broad range of readers, including students and teachers of mathematics, mathematical amateurs, and researchers. 777 MathematicalConversationStarters, by John de Pillis All the MathThat’s Fit to Print, by Keith Devlin CarlFriedrich Gauss:TitanofScience, byG. WaldoDunnington,withadditionalmaterialbyJeremyGray and Fritz-EgbertDohse The ChangingSpace of Geometry, editedby ChrisPritchard Circles: A MathematicalView, by Dan Pedoe Complex Numbers and Geometry, by Liang-shin Hahn Cryptology, by AlbrechtBeutelspacher Five HundredMathematicalChallenges,Edward J.Barbeau, MurrayS. Klamkin,andWilliamO.J. Moser From Zero to Infinity,by Constance Reid The Golden Section, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance of Jean Pedersen. I Wantto Be a Mathematician,by Paul R. Halmos Journey intoGeometries, by Marta Sved JULIA: a life in mathematics, by Constance Reid The LighterSideofMathematics:Proceedings oftheEuge(cid:18)ne StrensMemorialConference onRecreational Mathematics& Its History, editedby Richard K. Guy and Robert E. Woodrow Lure of the Integers, by Joe Roberts Magic Tricks, Card Shuffling,andDynamic ComputerMemories: The Mathematicsof the Perfect Shuffle, by S. Brent Morris The Math ChatBook, by Frank Morgan MathematicalApocrypha, by Steven G. Krantz MathematicalCarnival, by Martin Gardner MathematicalCircles Vol I: In MathematicalCircles QuadrantsI, II, III, IV, by Howard W. Eves MathematicalCirclesVolII:MathematicalCirclesRevisitedandMathematicalCirclesSquared,byHoward W. Eves MathematicalCirclesVolIII:MathematicalCirclesAdieuandReturntoMathematicalCircles,byHoward W. Eves MathematicalCircus, by Martin Gardner MathematicalCranks, by Underwood Dudley MathematicalEvolutions, editedby Abe Shenitzer and John Stillwell MathematicalFallacies, Flaws, and Flimflam,by Edward J. Barbeau MathematicalMagic Show, by Martin Gardner MathematicalReminiscences, by Howard Eves MathematicalTreks: From Surreal Numbers to Magic Circles, by Ivars Peterson Mathematics:Queen and Servant of Science, by E.T. Bell MemorabiliaMathematica,by Robert Edouard Moritz New MathematicalDiversions, by Martin Gardner Non-EuclideanGeometry, by H. S. M. Coxeter i i i i i i \master" | 2011/4/5 | 12:53 | page vi | #6 i i Numerical Methods That Work, by Forman Acton Numerology or What Pythagoras Wrought, by Underwood Dudley Out of the Mouthsof Mathematicians,by Rosemary Schmalz Penrose Tiles to Trapdoor Ciphers ...and the Return of Dr. Matrix, by Martin Gardner Polyominoes, by George Martin Power Play, by Edward J. Barbeau The Random Walks of George Po(cid:19)lya, by Gerald L. Alexanderson Remarkable Mathematicians,from Euler to von Neumann, Ioan James The Search for E.T. Bell, also known as John Taine, by Constance Reid ShapingSpace, editedby Marjorie Senechal and George Fleck SherlockHolmes inBabylonandOtherTalesofMathematicalHistory,editedbyMarlowAnderson,Victor Katz, and Robin Wilson Student Research Projects in Calculus, by Marcus Cohen, Arthur Knoebel, Edward D. Gaughan, Douglas S. Kurtz, and David Pengelley Symmetry, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance of Jean Pedersen. The Trisectors, by Underwood Dudley Twenty Years Before the Blackboard, by Michael Stuebenwith Diane Sandford The Words of Mathematics, by Steven Schwartzman MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 800-331-1622 FAX 301-206-9789 i i i i i i \master" | 2011/4/5 | 12:53 | page vii | #7 i i Introduction For the past one hundred years, the Mathematical Association of America has been publishing high-qualityarticlesonthehistoryofmathematics,somewrittenbydistinguishedhistorianssuchas Florian Cajori, Julian Lowell Coolidge, Max Dehn, David Eugene Smith, Carl Boyer, and others. Many well-known historians of the present day also contribute to the MAA’s journals. Some years ago, Robin Wilson and Marlow Anderson, along with the late John Fauvel, a distinguished and sorely missed historian of mathematics, decided that it would be useful to reprint a selection of these papers and to set them in the context of modern historical research, so that current mathematicians can continue to enjoy them and so that newer articles can be easily compared with older ones. After John’s untimely death, Victor Katz was asked to fill in and help bring this project to completion. Acareful readingof someof theolderpapersinparticular showsthatalthoughmodernresearch has introduced some new information or has fostered some new interpretations, in large measure they are neither dated nor obsolete. Nevertheless, we have sometimes decided to include two or more papers on a single topic, written years apart, to show the progress in the history of mathematics. The editors hope that you will enjoy this collection covering nearly four thousand years of history, from ancient Babylonia up to the time of Euler in the eighteenth century. We wish to thank Don Albers, Director of Publication at the MAA, and Gerald Alexanderson, chair of the publicationscommittee of the MAA, for their support for the history of mathematicsat the MAA ingeneral,andfor thisprojectinparticular. Wealso wanttothankBeverly Ruedifor hertechnical expertise in preparing this volume for publication. vii i i i i i i \master" | 2011/4/5 | 12:53 | page viii | #8 i i i i i i “master” — 2003/10/27 — 15:43 — page ix — #9 i i Contents Introduction vii Ancient Mathematics Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Sherlock Holmes in Babylon, R. Creighton Buck . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Words and Pictures: New Light on Plimpton 322, Eleanor Robson . . . . . . . . . . . . . . . . 14 Mathematics, 600 B.C.–600 A.D., Max Dehn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Diophantus of Alexandria, J. D. Swift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Hypatia of Alexandria, A. W. Richeson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Hypatia and Her Mathematics, Michael A. B. Deakin . . . . . . . . . . . . . . . . . . . . . . . . 52 The Evolution of Mathematics in Ancient China, Frank Swetz . . . . . . . . . . . . . . . . . . 60 Liu Hui and the First Golden Age of Chinese Mathematics, Philip D. Straffin, Jr. . . . . . . . . 69 Number Systems of the North American Indians, W. C. Eells . . . . . . . . . . . . . . . . . . . 83 The Number System of the Mayas, A. W. Richeson . . . . . . . . . . . . . . . . . . . . . . . . . 94 Before The Conquest, Marcia Ascher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Medieval and Renaissance Mathematics Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha, Ranjan Roy . . 111 Ideas of Calculus in Islam and India, Victor J. Katz . . . . . . . . . . . . . . . . . . . . . . . . 122 Was Calculus Invented in India?, David Bressoud . . . . . . . . . . . . . . . . . . . . . . . . . 131 An Early Iterative Method for the Determination of sin 1 , Farhad Riahi . . . . . . . . . . . . 138 ◦ Leonardo of Pisa and his Liber Quadratorum, R. B. McClenon. . . . . . . . . . . . . . . . . . . 143 The Algorists vs. the Abacists: An Ancient Controversy on the Use of Calculators, Barbara E. Reynolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Sidelights on the Cardan-Tartaglia Controversy, Martin A. Nordgaard . . . . . . . . . . . . . . . 153 Reading Bombelli’s x-purgated Algebra, Abraham Arcavi and Maxim Bruckheimer . . . . . . . 164 The First Work on Mathematics Printed in the New World, David Eugene Smith . . . . . . . . . 169 Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 The Seventeenth Century Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 An Application of Geography to Mathematics: History of the Integral of the Secant, V. Frederick Rickey and Philip M. Tuchinsky . . . . . . . . . . . . . . . . . . . . . . . . . 179 Some Historical Notes on the Cycloid, E. A. Whitman . . . . . . . . . . . . . . . . . . . . . . . 183 Descartes and Problem-Solving, Judith Grabiner . . . . . . . . . . . . . . . . . . . . . . . . . . 188 ix i i