Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry

208 Pages·2012·6.49 MB·English

by  Glen Van Brummelen

Heavenly Mathematics
Heavenly Mathematics
The Forgotten Art of Spherical Trigonometry
Glen Van Brummelen
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright © 2013 by Princeton University Press Published by Princeton University Press, 41 William
Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW
Jacket art: Painting of Euclid by Justus von Ghent. © Bettmann/Corbis press.princeton.edu
All Rights Reserved
Library of Congress Cataloging-in-Publication Data Van Brummelen, Glen
Heavenly mathematics : the forgotten art of spherical trigonometry / Glen Van Brummelen.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-691-14892-2 (hardcover : alk. paper) 1. Spherical trigonometry. 2. Trigonometry. I. Title.
QA535.V36 2013
516.24—dc23
2012023216
British Library Cataloging-in-Publication Data is available This book has been composed in Minion Pro
Printed on acid-free paper. ∞
Printed in the United States of America
1 3 5 7 9 10 8 6 4 2
Contents
Preface
1 Heavenly Mathematics
2 Exploring the Sphere
3 The Ancient Approach
4 The Medieval Approach
5 The Modern Approach: Right-Angled Triangles
6 The Modern Approach: Oblique Triangles
7 Areas, Angles, and Polyhedra
8 Stereographic Projection
9 Navigating by the Stars
Appendix A. Ptolemy’s Determination of the Sun’s Position
Appendix B. Textbooks
Appendix C. Further Reading
Index
Preface
Mathematical subjects come and go. If you glance at a textbook from a century
ago you may recognize some of the contents, but some will be unfamiliar or
even baffling. A high school text in analytic geometry, for instance, once
contained topics like involutes of circles, hypocycloids, and auxiliary circles of
ellipses: topics that most college students today will never see. But spherical
trigonometry may be the most spectacular example of changing fashions in the
20th-century mathematics classroom. Born of the need to locate stars and planets
in the heavens, for more than 1500 years it was the big brother to the plane
trigonometry that high school students slog through today. Navigators on the
open seas relied on spherical trigonometry to find their way; lives were lost
when their skills failed them. Its dominance continued through the early 20th
century: editions of Euclid’s Elements that were designed for classrooms often
included appendices devoted to this now-forgotten subject.
During World War II the popularity of spherical trigonometry remained high.
Applications in naval and military settings were touted as motivations, and were
given a prominent place in the exercises. Through the early 1950s textbooks
continued to be published, although gradually spherical trigonometry found itself
relegated to the last major section in a textbook mostly devoted to plane
trigonometry. Suddenly, mid-decade, it disappeared, dropped in a pedagogical
tide that was heading in other directions. Today almost no trigonometry texts
even mention the existence of a spherical counterpart. The only book on the
subject continuously in print (Clough-Smith 1966) is difficult to obtain and
available only from nautical booksellers. This paucity comes strangely at a time
when new applications of spherical trigonometry are being found. GPS devices
have some of its formulas built in. It’s amusing to see bibliographies of research
papers in computer graphics and animation (for use in movies like those made
by Pixar) referring to nothing older than last week, except for that stodgy old
spherical trig text.
So if mathematics teachers have long since given up on spherical
trigonometry, why bring it back? I’m not advocating that everyone should dust
off the covers of their grandparents’ textbooks (although, come to think of it …),
but a treasure would be lost if no one did. The applications are interesting and
substantial: finding the distance between two points on the Earth’s surface, such
as how far the Titanic sailed before it sank; navigating a sailboat or airplane by
the stars; predicting the Sun’s altitude so that a faithful Muslim will know when
to perform his daily prayers. Much better, and much more genuine, than the
obviously contrived examples found in today’s plane trigonometry classroom:
finding heights of trees, or distances of motorboats as they speed across
unnamed lakes.
Figure 0.1. Image from the title page of Bonnycastle’s Treatise on Plane and Spherical Trigonometry. In
the drawing, one gentleman is measuring the height of a church spire presumably in preparation for a plane
trigonometry problem; the other is measuring the altitude of the sun for a spherical problem. Western
Archives, Western University.
But I admit that for me, the navigation, astronomy, and geography are only the
icing on the cake. I appreciate spherical trigonometry mostly because it’s
beautiful. The theorems are elegant and often surprising. Even the ordinary
results deepen our understanding of the trigonometry that we already know;
many of the identities in plane trigonometry are only flattened images of their
spherical counterparts. The proofs, especially the geometric ones, can be
unexpected and are sometimes breathtaking.
Figure 0.2. Title page of John Keill’s Elements of Plain and Spherical Trigonometry, 3rd edition, 1726. The
Thomas Fisher Rare Book Library, University of Toronto.
Shocking as it may sound, spherical trigonometry was not always a charming
mathematical diversion. Early 20th-century Canadian humorist Stephen Leacock
seems to have had a special relationship with it, mentioning it four times in his
writings. His experience seems not to have been pleasant. In his story “The Man
of Asbestos” (Nonsense Novels) Leacock imagines a future where education
happens not in schools, but by brain surgery. Even this method seems to have
been distasteful for the ingestion of several subjects, our favorite among them:
“It was a mere nothing; an operation of a few minutes would suffice to let in poetry or foreign languages
or history or anything else that one cared to have. Here, for instance,” he added, pushing back the hair at
the side of his head and showing a scar beneath it, “is the mark where I had my spherical trigonometry let
in. That was, I admit, rather painful, but other things, such as English poetry or history, can be inserted
absolutely without the least suffering. When I think of your painful, barbarous methods of education
through the ear, I shudder at it. Oddly enough, we have found lately that for a great many things there is
no need to use the head. We lodge them—things like philosophy and metaphysics, and so on—in what
used to be the digestive apparatus. They fill it admirably.”
Figure 0.3. Frontispiece of Kells/Kern/Bland’s 1942 Spherical Trigonometry with Naval and Military
Applications. Courtesy Lockheed Martin Corporation.
Now, I have no wish to cut a hole in anyone’s scalp. Sometimes, though, there
is benefit to be gained by struggling with a text for a little while. If education
were always a downhill journey, the pearls grown by the irritation of a properly
placed grain of sand would never exist. Leacock imagined such schooling:
Within recent years it is becoming clear that a university is now a superfluous institution. College
teaching is being replaced by such excellent little manuals as the Fireside University Series, the World’s
Tiniest Books, the Boys Own Conic Sections, and the Little Folks Spherical Trigonometry. Thanks to
books such as these no young man in any station of life need suffer from an unsatisfied desire for
learning. He can get rid of it in a day. In the same way any business man who wishes to follow the main
currents of history, philosophy and radio-activity may do so while changing his shirt for dinner.
Within these pages you will find what might have been expected in Little
Folks Spherical Trigonometry, but also plenty of room for mathematical brain
surgery. The experience of wrestling with mathematics (provided that it meets
with at least occasional success) can be one of the world’s greatest pleasures. At
the difficult moments the reader may at least be consoled that I have not
speculated on the nature of the “cubular trigonometry” that tortures the school
boy residents in David Foster Wallace’s Infinite Jest.