Copyright © 1998 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, Chichester,
West Sussex
All rights reserved
Maor, Eli.
Trigonometric delights / Eli Maor.
p. cm.
Includes bibliographical references and index.
ISBN 0-691-05754-0 (alk. paper)
1. Trigonometry. I. Title.
QA531.M394 1998
516.2402—dc21 97-18001
This book has been composed in Times Roman
Princeton University Press books are printed on acid-free paper and
meet the guidelines for permanence and durability of the Committee
on Production Guidelines for Book Longevity of the Council on
Library Resources
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
Prologue: Ahmes the Scribe, 1650 b.c. 3
RecreationalMathematics in AncientEgypt 11
1. Angles 15
2. Chords 20
Plimpton 322: TheEarliestTrigonometric Table? 30
3. Six Functions Come of Age 35
Johann M¨uller, alias Regiomontanus 41
4. Trigonometry Becomes Analytic 50
François Vi`ete 56
5. Measuring Heaven and Earth 63
Abraham DeMoivre 80
6. Two Theorems from Geometry 87
7. Epicycloids and Hypocycloids 95
Maria Agnesi andHer“Witch” 108
8. Variations on a Theme by Gauss 112
9. Had Zeno Only Known This! 117
10. sin x‘=x 129
11. A Remarkable Formula 139
Jules Lissajous andHis Figures 145
12. tan x 150
13. A Mapmaker’s Paradise 165
14. sin x D 2: Imaginary Trigonometry 181
Edmund Landau: TheMasterRigorist 192
15. Fourier’s Theorem 198
Appendixes 211
1. Let’s Revive an Old Idea 213
2. Barrow’s Integration of sec 218
3. Some Trigonometric Gems 220
4. Some Special Values of sin 222
Bibliography 225
Credits for Illustrations 229
Index 231
This book is neither a textbook of trigonometry—of which there
are many—nor a comprehensive history of the subject, of which
there is almost none. It is an attempt to present selected topics
in trigonometry from a historic point of view and to show their
relevance to other sciences. It grew out of my love affair with
the subject, but also out of my frustration at the way it is being
taught in our colleges.
First, the love affair. In the junior year of my high school we
were fortunate to have an excellent teacher, a young, vigorous
man who taught us both mathematics and physics. He was a
no-nonsense teacher, and a very demanding one. He would not
tolerate your arriving late to class or missing an exam—and you
better made sure you didn’t, lest it was reflected on your report
card. Worse would come if you failed to do your homework or
did poorly on a test. We feared him, trembled when he reprimanded
us, and were scared that he would contact our parents.
Yet we revered him, and he became a role model to many of
us. Above all, he showed us the relevance of mathematics to
the real world—especially to physics. And that meant learning
a good deal of trigonometry.
He and I have kept a lively correspondence for many years,
and we have met several times. He was very opinionated, and
whatever you said about any subject–mathematical or otherwise—
he would argue with you, and usually prevail. Years after
I finished my university studies, he would let me understand
that he was still my teacher. Born in China to a family
that fled Europe before World War II, he emigrated to Israel
and began his education at the Hebrew University of Jerusalem,
only to be drafted into the army during Israel’s war of independence.
Later he joined the faculty of Tel Aviv University and
was granted tenure despite not having a Ph.D.—one of only two
lecture on the history of mathematics, he suddenly collapsed
faculty members so honored. In 1989, while giving his weekly
and died instantly. His name was Nathan Elioseph. I miss him
dearly.
And now the frustration. In the late 1950s, following the early
Soviet successes in space (Sputnik I was launched on October
4, 1957; I remember the date—it was my twentieth birthday)
there was a call for revamping our entire educational system,
especially science education. New ideas and new programs suddenly
proliferated, all designed to close the perceived technological
gap between us and the Soviets (some dared to question
whether the gap really existed, but their voices were swept aside
in the general frenzy). These were the golden years of American
science education. If you had some novel idea about how
to teach a subject—and often you didn’t even need that much—
you were almost guaranteed a grant to work on it. Thus was born
the “New Math”—an attempt to make students understand what
they were doing, rather than subject them to rote learning and
memorization, as had been done for generations. An enormous
amount of time and money was spent on developing new ways
of teaching math, with emphasis on abstract concepts such as set
theory, functions (defined as sets of ordered pairs), and formal
logic. Seminars, workshops, new curricula, and new texts were
organized in haste, with hundreds of educators disseminating
the new ideas to thousands of bewildered teachers and parents.
Others traveled abroad to spread the new gospel in developing
countries whose populations could barely read and write.
Today, from a distance of four decades, most educators agree
that the New Math did more harm than good. Our students
may have been taught the language and symbols of set theory,
but when it comes to the simplest numerical calculations they
stumble—with or without a calculator. Consequently, many high
school graduates are lacking basic algebraic skills, and, not surprisingly,
some 50 percent of them fail their first college-level
calculus course. Colleges and universities are spending vast resources
on remedial programs (usually made more palatable
by giving them some euphemistic title like “developmental program”
or “math lab”), with success rates that are moderate at
best.
Two of the casualties of the New Math were geometry and
trigonometry. A subject of crucial importance in science and
engineering, trigonometry fell victim to the call for change. Formal
definitions and legalistic verbosity—all in the name of mathematical
rigor—replaced a real understanding of the subject.
Instead of an angle, one now talks of the measure of an angle;
instead of defining the sine and cosine in a geometric context—
as ratios of sides in a triangle or as projections of the unit circle
on the x- and y-axes—one talks about the wrapping function
from the reals to the interval ’−1; 1“. Set notation and set language
have pervaded all discussion, with the result that a relatively
simple subject became obscured in meaningless formalism.
Worse, because so many high school graduates are lacking basic
algebraic skills, the level and depth of the typical trigonometry
textbook have steadily declined
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