Sherlock Holmes in Babylon and Other Tales of Mathematical History

398 Pages·2004·7.08 MB·English

by  Marlow Anderson

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 [phone]
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