How does one estimate the size of the Earth? The distance to the Moon? More to the point, how could scientists accomplish these feats at a time when the only available instruments were as simple as those of ancient surgeons: no X-rays, CAT scans or MRIs; just scalpels, spatulas and forceps?

Scientists in the ancient world, and the successors who extended the techniques in the following centuries, did so using nothing but geometrical reasoning and simple computations. Taking a scenic journey through the history of mathematics, Glen Van Brummelen’s *Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry *shows exactly how they did so.

Euclidean geometry is a product of the ancient Greek world that has been studied and held in high esteem ever since. One branch, planar geometry, focuses on flat surfaces and explores the relationships between the sides and the angles of regular triangles. But to understand the cosmos as conceived by the ancients—a sphere on which the heavenly objects were assumed to move—a separate allied subject was needed. By the second century AD, this evolving field of study, known as spherical trigonometry, was enabling the growth of astronomy and navigational theory. Important refinements continued in succeeding centuries, first in India (where theoretical developments were at least partly independent from advances in the West) and the Islamic world, and then in medieval and modern Europe.

During the 19th century, the rise of modern geometries challenged mathematicians to go beyond the intuition they had built for Euclidean space. The study of spherical phenomena from this new more all-encompassing perspective left little room for traditional spherical trigonometry. Nevertheless, the subject remained in school textbooks. Then in the mid 20th century, the curricula of the western world dropped it within a decade.

Planar trigonometry was kept: it still forms basic knowledge in mathematics. But spherical trigonometry proved dispensable. Those few who need it now can learn it in a matter of weeks if their geometric background is solid enough. Even without this background, readers with basic mathematical skills, and more importantly the will to actively apply them, can learn to scale some of the subject’s greatest heights. *Heavenly Mathematics* shows that the intellectual reward is considerable. Not least, the logic that underpinned past practices in astronomy and navigation comes gratifyingly to light.

This is not a scholarly work in the history of mathematics. It does not contain footnotes, does not profess to tell the whole story … This is simply an appreciation of a beautiful lost subject, with historical overtones and a few subtly placed messages that I’m sure you will recognize. Take it for what it is, and enjoy.

Readers should be warned that mathematics can rarely be absorbed linearly. More commonly it resembles a cliff. Sometimes those who attempt to scale it reach a dead end and are forced to return to an intermediate point to start afresh. They may also need to stop and reflect occasionally, using pencil and paper to clarify their understanding of the path being taken. To make the ascent more bearable, and to provide occasions to breathe between steep climbs through mathematical proofs, Van Brummelen intermingles each new theorem with historical narrative. As the president of the Canadian Society for History and Philosophy of Mathematics, he specializes in the astronomy and trigonometry of ancient Greece and medieval Islam, and these subjects are at the core of his book. He even gives thrill-seekers the chance to attack new cliffs alone, with carefully chosen end-of-chapter exercises.

*Heavenly Mathematics* is a work of love, with a passion for its subject permeating every line. This passion is one reason Van Brummelen is such an excellent guide, but this is still mathematics, so there is no royal road. His main goal is to introduce the mixture of geometric foundations and astronomic questions that led to the systematic exploration of spherical trigonometry over many centuries. Those familiar with only Euclidean geometry will be surprised by how different everything is when one moves from the flat surface of a geometric plane to a sphere. In the plane, we measure distances along straight lines. On the sphere we do so along arcs of so-called great circles such as the Equator or meridians of longitude. To fly from Paris to Toronto, for example, an airplane must stay as close as possible to the great circle that passes through these cities and whose geometric plane, if extended through the Earth’s core, would pass directly through its centre.

Such simple observations open a new world. A spherical triangle is formed by three arcs of great circles that connect three points. For example, two points could be on the Equator and one could be the North Pole. So two of this triangle’s arcs follow longitudinal meridians, which form 90 degree angles with the equator. Given these two right angles, the three angles of this illustrative spherical triangle (and indeed of all spherical triangles) must add to more than 180 degrees—a feature that is impossible in regular Euclidian geometry, where a triangle can have only one right angle, and where the sum of all three of a triangle’s angles always sums to 180 degrees.

This is only one of the numerous ways that spherical triangles differ from their planar counterparts, and it is these differences that give spherical trigonometry its mysterious quality in the eyes of most laypeople. It was also a keen understanding of these distinctive features that allowed generations of early mathematicians to answer a host of practical questions, including the relatively exact determination of a certain location’s latitude and—after the invention of accurate chronometers in the 18th century—longitude as well. Other intriguing questions were dealt with along the way—for example, helping believers in the Islamic world ensure they were correctly aligned to Mecca during their daily prayers.

As adept as Van Brummelen is in explaining such solutions, unfortunately he never puts spherical trigonometry fully in perspective by summarizing how its evolution fits into the broader development of geometry. A few pages explaining where spherical trigonometry belongs in mathematics, how it relates to other branches and why it is no longer studied would have helped make this book complete. For example, to say, as Van Brummelen does, that the decline of spherical trigonometry was merely a matter of fashion is an oversimplification of the truth. A couple of pages that mention the contribution of Gauss, Bolyai, Lobachevsky, Riemann and Klein—the extraordinary journey from classical to non-Euclidean and, finally, to the modern geometries of the Erlangen Program—would have clarified why spherical trigonometry is not studied anymore.

But such criticism does not detract from Van Brummelen’s achievement. *Heavenly Mathematics* proves the value of bringing a fascinating piece of mathematical history within the grasp of the general reader. At a time when the impact of mathematical advances in the broad sweep of intellectual history is all too often ignored or misunderstood, one can only hope that this author’s talents will be put to further use.

Florin Diacu is a professor of mathematics at the University of Victoria and author of *The Lost Millennium: History’s Timetables under Siege*, whose second edition was published in 2011 by Johns Hopkins University Press.

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