Math Through the Ages: A Gentle History for Teachers and Others, Expanded Edition (Mathematical Association of America Textbooks)

287 Pages·2003·7.52 MB·English

by  William P. Berlinghoff

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Math through the Ages
A Gentle History
for Teachers and Others
Expanded Edition
A Joint Publication of Oxton House Publishers
and The Mathematical Association of America
©2004 by
William P. Berlinghoff and Fernando Q. Gouvea
ISBN-13 978-0-88385-736-6 hardback
ISBN-10 0-88385-736-7 hardback
Library of Congress Catalog Card Number: 2003114603
Current Printing (last digit):
10 9 8 7 6 5 4 3
Math through the Ages
A Gentle History
for Teachers and Others
Expanded Edition
William P. Berlinghoff
Fernando Q. Gouvea
A Joint Publication of
Oxton House Publishers
and
The Mathematical Association of America
Committee on Publications
Gerald Alexanderson, Chair
Classroom Resource Materials Editorial Board
Zaven A. Karian, Editor
William Bauldry David E. Kullman
Gerald Bryce Stephen B Maurer
Sheldon P. Gordon Douglas Meade
William J. Higgins Edward P. Merkes
Mic Jackson Judith A. Palagallo
Paul Knopp Wayne Roberts
CLASSROOM RESOURCE MATERIALS
Classroom Resource Materials is intended to provide supplementary class-
room material for students-laboratory exercises, projects, historical infor-
mation, textbooks with unusual approaches for presenting mathematical
ideas, career information, etc.
101 Careers in Mathematics, 2nd edition edited by Andrew Sterrett
Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein
Calculus Mysteries and Thrillers, R. Grant Woods
Combinatorics: A Problem Oriented Approach, Daniel A. Marcus
Conjecture and Proof, Miklds Laczkovich
A Course in Mathematical Modeling, Douglas Mooney and Randall Swift
Cryptological Mathematics, Robert Edward Lewand
Elementary Mathematical Models, Dan Kalman
Environmental Mathematics in the Classroom, edited by B. A. Fusaro and
P. C. Kenschaft
Essentials of Mathematics: Introduction to Theory, Proof, and the Profes-
sional Culture, Margie Hale
Exploratory Examples for Real Analysis, Joanne E. Snow and Kirk E.
Weller
Geometry From Africa: Mathematical and Educational Explorations, Paulus
Gerdes
Identification Numbers and Check Digit Schemes, Joseph Kirtland
Interdisciplinary Lively Application Projects, edited by Chris Arney
Inverse Problems: Activities for Undergraduates, Charles W. Groetsch
Laboratory Experiences in Group Theory, Ellen Maycock Parker
Learn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt
Johansson, and Victor Katz
Math through the Ages: A Gentle History for Teachers and Others (Ex-
panded Edition), William P. Berlinghoff and Fernando Q. Gouvea
Mathematical Evolutions, edited by Abe Shenitzer and John Stillwell
Mathematical Modeling in the Environment, Charles Hadlock
Mathematics for Business Decisions Part 1: Probability and Simulation
(electronic textbook), Richard B. Thompson and Christopher G. Lam-
oureux
Mathematics for Business Decisions Part 2: Calculus and Optimization
(electronic textbook), Richard B. Thompson and Christopher G. Lam-
oureux
Ordinary Differential Equations: A Brief Eclectic Tour, David A. Sanchez
Oval Track and Other Permutation Puzzles, John O. Kiltinen
A Primer of Abstract Mathematics, Robert B. Ash
Proofs Without Words, Roger B. Nelsen
Proofs Without Words II, Roger B. Nelsen
A Radical Approach to Real Analysis, David M. Bressoud
She Does Math!, edited by Marla Parker
Solve This: Math Activities for Students and Clubs, James S. Tanton
Student Manual for Mathematics for Business Decisions Part 1: Probability
and Simulation, David Williamson, Marilou Mendel, Julie Tarr, and
Deborah Yoklic
Student Manual for Mathematics for Business Decisions Part 2: Calculus
and Optimization, David Williamson, Marilou Mendel, Julie Tarr, and
Deborah Yoklic
Teaching Statistics Using Baseball, Jim Albert
Writing Projects for Mathematics Courses: Crushed Cloums, Cars, and
Coffee to Go, Annalisa Crannell, Gavin LaRose, Thomas Ratliff, and
Elyn Rykken
MAA Service Center Oxton House Publishers
P.O. Box 91112 P. O. Box 209
Washington, DC 20090-1112 Farmington, ME 04938
1-800-331-1MAA 1-800-539-7323
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www.maa.org ww.oxtonhouse.com
Preface
This book grew out of a few casual hallway conversations in the
Colby College Mathematics Department about two years ago,
but its roots are much deeper and older than that. For many
years we have been interested in the history of mathematics, both for
its own sake and as an aid in teaching mathematical concepts to a wide
range of audiences. One of us has used it as a major ingredient in several
college mathematics texts for liberal-arts students and as an important
part of his contributions to an NCTM Standards-based high school
mathematics series. The other has done considerable background re-
search in the field, has participated in the Mathematical Association
of America's Institute for the History of Mathematics and its use in
Teaching, and teaches a course in the history of mathematics at Colby.
We are convinced that knowing the history of a mathematical concept
or technique leads to a deeper, richer understanding of the concept or
technique itself.
Unfortunately for teachers and other people with some interest in
mathematical history but relatively little time to pursue it, most books
on the subject are dauntingly large. If you want some historical back-
ground as you prepare to teach quadratic equations or negative num-
bers, or if you are just curious about the history of it or the metric
system or zero, where would you look? The indexes of most history
books will point you to a disjointed scattering of pages, leaving to you
the task of piecing together a coherent picture. A topical search on the
Internet is likely to inundate you with information, some reliable, some
spurious, with little guidance as to which is which.
We decided to write a book with your needs in mind. The main part
of this book is a collection of twenty-five short historical sketches about
some common ideas of basic mathematics. These sketches illustrate the
origins of an idea, process, or topic, sometimes connecting seemingly
distinct things that share common historical roots. They are preceded
by a brief panorama of the history of mathematics, from its earliest days
to the present. This provides a skeletal framework of important people
and events that shaped the mathematics we know today, and it supplies
a unifying context for the separate, self-contained sketches. Of course,
the choice of sketch topics was quite subjective; we were guided partly
by our own interests and partly by our sense of what might interest
vii
viii Preface
teachers and students of mathematics. If you would like to suggest a
sketch topic for the next edition of this book, we invite you to submit
it to Oxton House Publishers, either by mail to the address on the
copyright page of this book or by e-mail to oxtonhse©mainewest.com.
We have made every effort to reflect accurately the historical facts
as they are known today. Nevertheless, history is far from an exact
science, and incomplete or conflicting sources often lead to conflicting
judgments of fact among scholars. Some stories about mathematical
people and events have evolved over many years, creating a body of
"folklore" with very little hard documentary evidence to support it.
Despite their potential to annoy historical scholars, many of these sto-
ries - like folk tales in every culture - are valuable, either as allegories
or as mnemonic "hooks" to help you (or your students) remember a
mathematical idea. Rather than ignore such anecdotes entirely and
lose their value, we have opted to include some of the more interesting
ones, along with appropriate cautions against taking them too literally.
To help you track down more information about any topic that
interests you, the section entitled "What to Read Next" is an annotated
list for further reading. It includes some pointers to reference works,
but its heart is a short "ought-to-read" list of books that we think
anyone interested in the history of mathematics probably would enjoy.
A note about notation: In recent years, some history books have
been using B.C.E. ("before the common era') and c. E. ("the common
era") in place of the more traditional B.c.and A.D., respectively. De-
pending on which historian one consults, this is either (a) the notation
of the future for historical literature, or (b) a passing "politically cor-
rect" fad. Without taking a position on this question, we have opted
for the notation that we believe to be more familiar to most of our
potential readers.
Acknowledgments
We are indebted to many colleagues from near and far for shar-
ing their knowledge and for their forbearance in responding to our
sometimes peculiar questions. In particular, we thank mathematics
education consultant Sharon Fadden in Vermont, Jim Kearns of Lynn-
field High School in Massachussetts, and Bryan Morgan of Oxford Hills
Comprehensive High School in Maine for reading and commenting on
earlier versions of the book. Special thanks also to Georgia Tobin for
creating the TjX symbols for Egyptian and Babylonian numerals, and
to Michael Vulis for converting them to PostScript format; to Robert
Washburn of Southern Connecticut State University for providing some
Preface ix
of the material in Sketch 6; and to Eleanor Robson, who generously gave
us permission to use one of her drawings of Old Babylonian tablets (on
page 63).
We are deeply grateful that one of us was able to participate for
two summers in the MAA's Institute for the History of Mathematics
and its use in Teaching. IHMT helped to transform a lifelong interest
in the history of mathematics into a solid base of knowledge on which
it was easy to build. Special thanks to IHMT organizers Fred Rickey,
Victor Katz, and Steven Schot, to the Mathematical Association of
America, its sponsoring organization, and to all the IHMT colleagues
- an interesting, varied, knowlegeable, and helpful bunch of people.
Many of them answered questions and made useful suggestions while
this book was being written, earning still more of our gratitude in the
process.
Our debt to the many historians of mathematics whose work we read
and used as we were writing this book is enormous. Were it not for the
giants on whose shoulders we attempted to stand, we couldn't possibly
have done the job. We have tried, in bibliographical notes scattered
throughout the book and in the "What to Read Next" section, to point
our readers towards some of their work.
We also would like to thank Don Albers, Martin Davis, David
Fowler, Julio Gonzalez Cabillon, Victor Hill, Heinz Luneburg, Kim
Plofker, Eleanor Robson, Gary Stoudt, Rebekka Struik, and the mem-
bers of the Historia Mathematica group for their answers to our ques-
tions. Of course, any mistakes that remain are our own.
The Expanded Edition
When the original edition of Math through the Ages appeared, some
readers who wanted to use the book as a classroom text asked us what
sorts of questions we might pose to students. In response to those in-
quiries, and with the encouragement of the Mathematical Association
of America, we have added 54 pages of Questions and Projects. This
Expanded Edition is suitable for a fairly broad spectrum of courses,
including many that are designed to prepare future teachers of elemen-
tary or secondary mathematics or are part of in-service professional
development programs. Moreover, even the casual reader can profit
from thinking about some of the questions we pose.
The variability in the mathematical backgrounds of potential read-
ers of this book invites a variety in the kind and level of questions posed
for their consideration. Two pages of Questions and Projects have been
inserted after each topical sketch. The distinction between these is