The Mathematics of the Heavens and the Earth: The Early History of Trigonometry

This book became a classic almost as soon as it appeared. There had never been a comprehensive history of trigonometry available in English before The Mathematics of the Heavens and the Earth was published. Shortly after it appeared, I began to see discussions and disputes in on-line forums being settled by references to it. However, Van Brummelen’s history does far more than simply fill a vacant spot in the historical literature of mathematics. He recounts the history of trigonometry in a way that is both captivating and yet more than satisfying to the crankiest and most demanding of scholars.

Van Brummelen divides the early history of trigonometry into five chapters:

1. The precursors of trigonometry, up to the time of Archimedes,
2. The Alexandrian trigonometry of chords, which largely features Ptolemy’s work, but also includes his predecessors Theodosius and Menelaus,
3. Trigonometry in medieval India, where tables of sines replaced the chords of Alexandria,
4. The many trigonometric innovations and applications of medieval Islam, a chapter that occupies almost a third of the book, and
5. The transmission to the West early in the second millennium CE and the subsequent development of trigonometry in Europe through the early 16^th century.

Van Brummelen covers the period up to 1550, roughly the era of geocentric astronomy. This means that the discoveries and developments he discusses are largely geometric in flavor and were described or proved using the tortuous didactic mathematics that preceded the development of symbolic algebraic methods in the late 16^th and early 17^th centuries. Another innovation of the period following 1550 was the logarithm, which had as great an effect on the application of trigonometry to navigational and other practical matters as the new algebraic methods did on its theoretical development. Fortunately, Van Brummelen plans a second volume on the period after 1550, although he makes no promises as to when it will appear, promising only that “the richness of the story to come will make it worth the wait.”

Van Brummelen has tried to design The Mathematics of the Heavens and the Earth so that the topics are as self-contained as possible, making it useful as a reference work or for the reader with particular interests in just one or two of its chapters. That said, if you have the inclination to read the book from cover to cover, you will find that it draws you along at a surprisingly brisk pace, as long as you are able to resist the temptation to read all of the footnotes on a first reading (it’s not always easy to rise above one’s compulsions). To give an idea of just how densely annotated the text is, the chapter on Alexandrian trigonometry has 128 footnotes over the course of 61 pages. The care that Van Brummelen takes to point the reader to sources in nearly every contentious topic or matter of interpretation makes this book an excellent starting point for deeper research into early trigonometry.

Trigonometry largely developed because of its usefulness in astronomical problems. Therefore, if he wished to make every part of this book completely self-contained, Van Brummelen would have had to repeat the basic set-up for problems involving the heavenly bodies: ecliptic, equator, horizon, celestial latitude and longitude, and so on. So he includes an orientation to the ancient, geocentric conception of the heavens in his first chapter, on the precursors of trigonometry in ancient Egypt, Mesopotamia and Greece. Of course, even to distinguish these early discoveries as precursors to trigonometry, as opposed to trigonometry proper, requires a clear definition of trigonometry. Van Brummelen argues there are two essential elements of trigonometry: “a standard quantitative measurement of the inclination of one line to another; and the capacity for, and interest in, calculating the lengths of line segments.” The mathematical results recounted in his first chapter address only one or the other of these elements, so trigonometry in Van Brummelen’s sense only begins with Ptolemy and his immediate predecessors.

The material of Chapters 2 and 5 — Hellenistic and European trigonometry — is usually reasonably well represented, if in a less comprehensive way, in most standard texts on the history of mathematics. Therefore, the greatest value of Van Brummelen’s book to many readers will be his extensive treatment of trigonometry in India and the Islamic world in the period between the decline of the classical world and the 12^th century European Renaissance.

In Chapter 3, Van Brummelen mentions the question of transmission from the Greek world to India only briefly, referring the reader to the scholarly arguments on both sides. He surveys the many undeniably original contributions of Indian mathematicians from Aryabhata’s sine tables ca. 500 CE through the discovery of the sine and cosine series by Madhava and the Kerala school in the 14^th century. Throughout his book, Van Brummelen generally uses modern notation to explain the mathematics of the past. This seems essential when discussing medieval India, where “formulas” were expressed in verse and calculations were often quite intricate, involving such things as iterative numerical methods and higher order finite differences.

A particularly welcome feature of this book is the use of brief pieces of original source material (in translation) to illustrate the achievements of various early mathematicians in trigonometry. There are 31 of these “Texts” ranging from Problem 58 of the Rhind Mathematical Papyrus (finding the “slope” of a pyramid), through Copernicus’ determination of solar eccentricity. These text passages are illustrated with modern diagrams and followed by careful explanations using modern notation, with Van Brummelen providing additional details as needed. The selection is particularly rich in Chapter 4, from Al-Samaw’ai ibn Yahya al-Maghribi arguing for the division of the circle into 480 parts (because the value of the chord of 1degree can only be approximated, whereas the value of ¾ of a degree can be calculated precisely using the half-angle formula), to al-Sijzi determining the ascension of the signs of the zodiac using a tool called a sine quadrant.

The Mathematics of the Heavens and the Earth should be a part of every university library’s mathematics collection. It’s also a book that most mathematicians with an interest in the history of the subject will want to own. It could be used as a reference text for the standard upper-level history of mathematics course, as well as a source for student projects in such a course. I can also see it being used for a special topics course in the history of trigonometry, especially once the promised companion volume covering the period after 1550 appears. Finally, with 36 pages of references and extensive annotation throughout the book, it seems destined to be the starting point for a lot of new research in the years ahead.

Rob Bradley is professor of mathematics at Adelphi University. His main research interest is the history of mathematics. He is currently the Chairman of the History of Mathematics Special Interest Group of the MAA (HOMSIGMAA) and president of the Euler Society.