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Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry

Glen Van Brummelen
Princeton University Press
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Ezra Brown
, on

Where was this book when I needed it?

Picture this: As a kid, maps and the night-time sky get their hooks into you and never let go. You wonder why the maps of the world are wrong: Greenland looks lots bigger than Africa, but the reality is the opposite. These maps may also show air routes from, say, San Francisco to Tokyo, that don’t look anything like straight-line routes — but they are. You wonder how anybody could possibly navigate an ocean voyage, using only the sky and a chronometer, without getting totally lost — when being off by a thousand yards is a life-and-death matter. You learn the constellations, the planets and the brightest stars and you wonder how it’s possible to determine distances to faraway objects in the sky, to predict eclipses, and to track the motions of celestial objects to great accuracy.

Eventually, you take plane trigonometry in high school and ask your trig teacher about these matters, and she tells you that it’s trig that can help you answer your questions. “But for most of your questions,” she says, “you need to learn spherical trigonometry. “Great!” you say, and a trip to the library rewards you with a book on that subject that was quite old even in those ancient times (the 1950s). You open this book, begin to read, and encounter a bewildering list of terms and theorems with few explanations and no proofs. Even the worked-out examples are unenlightening. You long for something better.

This book is “something better” — and how! Right away, on page 2, you read the following: “Our mission is as follows: Accepting nothing but the evidence of our senses and simple measurements we can take ourselves, determine the distance to the Moon.” With that, you’re hooked.

The author’s approach is a mix of the historical, the mathematical, and the practical. You learn about the works of Hipparchus, Ptolemy, Menelaus, al-Kuhi, al-Biruni, Rheticus, Napier, and a great many more. You begin with plane trigonometry and plane geometry, including Euclid’s proof of a disguised form of the planar Law of Cosines, which statement, you are told, Euclid could not have given because trig did not appear on the planet for another century. [Please note that on page 96, the first two appearances of “AD” should read “CD”; this misprint is subsequently corrected.] You learn the great theorems of spherical geometry and spherical trigonometry, such as the Spherical Law of Cosines and the Angle Sum Theorem; you learn about the first correct proof of Euler’s polyhedral formula by Legendre — who used spherical geometry in his proof. You learn about the azimuth and the celestial equator and the ecliptic and right ascensions and declinations. You learn that spherical distance is measured both in nautical miles and in angles. And in the final chapter, you learn about navigating by the stars.

The author’s audience is not only mathematicians, but also nonspecialists with an interest in the subject. He takes you through the technical topics, indicating proofs (which he says you may skip) with bold arrows. He provides plenty of exercises for you to work. Now, going through the book is very much like an ocean voyage: clear sailing at times, rough waters at others. Spherical trig is not to be picked up and learned in an evening’s reading, but it is worth it. Place yourself in Van Brummelen’s caring hands, work through the proofs and examples, be patient, and enjoy the ride. Oh, and you can find the distance to the Moon on page 17.

As I said at the beginning, “Where was this book when I needed it?”

Ezra Brown teaches at Virginia Tech, where he is currently Alumni Distinguished Professor of Mathematics. He is a number theorist and a frequent contributor to the MAA journals, and he plays the piano, sings, kayaks, and takes long walks for fun. His email is and his home page is

Preface vii
1 Heavenly Mathematics 1
2 Exploring the Sphere 23
3 The Ancient Approach 42
4 The Medieval Approach 59
5 The Modern Approach: Right- Angled Triangles 73
6 The Modern Approach: Oblique Triangles 94
7 Areas, Angles, and Polyhedra 110
8 Stereographic Projection 129
9 Navigating by the Stars 151
Appendix A. Ptolemy's Determination of the Sun's Position 173
Appendix B. Textbooks 179
Appendix C. Further Reading 182
Index 189