# Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry

Glen Van Brummelen
PRINCETON UNIVERSITY PRESS 2013, 216 PAGES
PRICE £24.95 (HARDBACK) ISBN 978-0-691-14892-2

This book aims to introduce its readers to a mathematical topic that was once widely taught, but that has vanished from textbooks and syllabuses over the past 60 years: spherical trigonometry. The book is intended as a teaching resource; indeed, it appears to have grown out of the author’s summer workshops for teenagers. The author clearly believes that the principles of spherical trigonometry deserve to be better-known, and has set out to impress the reader not only with its great practicality, but also with its elegance.

In Chapter 1, we are given a refresher course on plane trigonometry, which is presented via the problem of determining the distance to the Moon. Here, the author ensures that the reader gains a good impression of the style of the rest of the book. In particular, we learn that almost all of the mathematics that is needed will be derived from first principles – the aim is to view spherical trigonometry through ‘ancient eyes’, i.e., to approach problems in the manner of the mathematicians of centuries past.

Chapter 2 introduces us to the basic features of the celestial sphere (ecliptic, horizon, etc.). There then follow two chapters outlining historical techniques in spherical trigonometry. In the first (Chapter 3), we learn about the ancient Greek approach, with the main theme of the chapter being the derivation of Menelaus’ Theorems for spherical triangles. Chapter 4 then outlines the ways in which mediaeval Arabic and Indian astronomers moved beyond these. As an illustration of the relevant methods, we are taken through the solution to the spherical trigonometric problem of greatest concern to Islamic scholars: the determination of the direction of Mecca.

Chapters 5 and 6 present (early) modern approaches to spherical trigonometry, including (in Chapter 5) the derivation of the ten ‘fundamental identities’ of a right-angled spherical triangle, as well as the various mnemonic schemes that have been devised for remembering these. Chapter 6 turns to oblique spherical triangles, and we are shown how to use the spherical Law of Cosines to determine distances on the Earth’s surface.

In Chapter 7, we find the treatment of certain problems of pure geometric interest, without any view to applications in astronomy or navigation. In particular, spherical trigonometry is used to study regular polyhedra: we see, for example, Legendre’s spherical trigonometric proof of Euler’s polyhedral formula. Returning to more practical matters, Chapter 8 takes us through the details of stereographic projection, whereby we may solve problems in spherical trigonometry by projecting the sphere down onto a plane, applying plane trigonometric identities, and then mapping back up to the sphere. The centre-piece of this chapter is ‘Cesàro’s method’, an elegant ‘royal road to spherical trigonometry’ that was introduced as late as 1905.

The final chapter puts together much of what has been learnt over the course of the book, with an application to navigation by the stars. Using a now little-known trigonometric function, the haversine, certain of whose useful properties have traditionally recommended it to use by sailors, we learn how to construct our own navigational tables, and hence how to determine a ship’s position from the stars.

The book concludes with three appendices: one on Ptolemy’s determination of the Sun’s position, another detailing various textbooks on spherical trigonometry, and a third which provides chapter-by-chapter suggestions for further reading.

This is a lovely book to read. I gained a great deal of satisfaction from working through the proofs and constructions, not least because the author has succeeded in convincing me that spherical trigonometry is not as hard as I once thought it was. The style and level of the book are definitely suited to those secondary school pupils who are prepared to follow the technical details, and perhaps try their hand at the very many exercises that are included. More generally, however, the book would provide a wonderful introduction for anyone who wishes to learn more about this subject. Having read the book, 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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