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# Heavenly Mathematics

The Forgotten Art of Spherical Trigonometry

**Glen Van Brummelen **

- Hardcover2012Hardcover42.0034.95ISBN9780691148922208 pp.6 x 9 1/411 color plates. 30 halftones. 93 line illus. 1 table.Live
- Paperback2017Paperback19.9514.95ISBN9780691175997208 pp.6 x 9 1/411 color plates. 30 halftones. 93 line illus.Live

- E-bookISBN9781400844807

*Heavenly Mathematics* traces the rich history of spherical trigonometry, revealing how the cultures of classical Greece, medieval Islam, and the modern West used this forgotten art to chart the heavens and the Earth. Once at the heart of astronomy and ocean-going navigation for two millennia, the discipline was also a mainstay of mathematics education for centuries and taught widely until the 1950s. Glen Van Brummelen explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. He conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation of its elegant proofs and often surprising conclusions. *Heavenly Mathematics* is illustrated throughout with stunning historical images and informative drawings and diagrams. This unique compendium also features easy-to-use appendixes as well as exercises that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

*Glen Van Brummelen*is coordinator of mathematics at Quest University Canada and former president of the Canadian Society for History and Philosophy of Mathematics. He won the 2016 Haimo Award for Distinguished College or University Teaching of Mathematics, awarded by the Mathematical Association of America, and the 2017 3M National Teaching Fellowship, awarded by the Society for Teaching and Learning in Higher Education.

## More about this book

- One of
*Choice*'s Outstanding Academic Titles for 2013 - Shortlisted for the 2013 BSHM Neumann Book Prize, British Society for the History of Mathematics

## Reviews

**--Rosalind Metcalfe,**

*Nature***--Alexander Bogomolny,**

*CTK Insights***--**

*Book News, Inc.***--**

*ThatsMaths.com***--R. Balashankar,**

*Organiser**Heavenly Mathematics*is a truly enjoyable description of the somewhat forgotten science of spherical trigonometry. . . . As readers discover this discipline, they will also appreciate the beauty inherent in the topic."

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*Choice**Heavenly Mathematics*proves the value of bringing a fascinating piece of mathematical history within the grasp of the general reader."

**--Florin Diacu,**

*Literary Review of Canada***--**

*Metascience**Heavenly Mathematics*] is an excellent survey of spherical trigonometry. . . . Simply an appreciation of a beautiful lost subject, with historical overtones. . . . [D]istinguishable for its appealingly fresh style."

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*Mathematical Reviews**Heavenly Mathematics*] is a lovely book to read. . . . [A] wonderful introduction for anyone who wishes to learn more about this subject. . . . I am in full agreement with the author that spherical trigonometry ought to be brought to a wider audience, and I believe that this is the book to do it."

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*Mathematics Today**Heavenly Mathematics*] is excellent."

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*Zentralblatt MATH**Heavenly Mathematics*will be of interest to mathematically inclined historians of science and also to students of mathematics and engineering. Because spherical trigonometry is relevant in applications of modern science, this elegant book may even contribute to a renaissance of the subject."

**--Jan P. Hogendijk,**

*Isis***--Teresa Floyd,**

*Mathematics Teacher***--P.N. Ruane,**

*Mathematical Gazette***--Elizabeth Hamm,**

*Aestimatio Critical Reviews in the History of Science*## Endorsements

*Heavenly Mathematics*is heavenly, is mathematics, and is so much more: history, astronomy, geography, and navigation, replete with historical illustrations, elegant diagrams, and charming anecdotes. I haven't followed mathematical proofs with such delight in decades. If, as the author laments, spherical trigonometry was in danger of extinction, this book will give it a long-lasting reprieve."

**--David J. Helfand, president of the American Astronomical Society**

**--Robin Wilson, president of the British Society for the History of Mathematics, author of**

*Four Colors Suffice: How the Map Problem Was Solved**Heavenly Mathematics*breathes new and interesting life into a topic that has been slumbering for far too long."

**--June Barrow-Green, associate editor of**

*The Princeton Companion to Mathematics**Heavenly Mathematics*is a very good book. It offers an interesting, accessible, and entertaining introduction to spherical trigonometry, which used to be a standard school topic but is now rarely studied. Interesting stories, engaging illustrations, and practical examples come together to enhance the reader's pleasure and understanding."

**--Fernando Q. Gouvêa, Colby College**

**--Victor J. Katz, editor of**

*The Mathematics of Egypt, Mesopotamia, China, India, and Islam*## Table of Contents

*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

## Other Books Written by this Author(s)

- Glen Van Brummelen

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