*The Mathematics of the Heavens and the Earth: the Early History of Trigonometry*, Glenn Van Brummelen, 2009, 352 pp., bibliography, 109 illustrations , 1 table ,cloth, $39.50. I SBN 978-0-691-12973-0, Princeton University Press, 41 William St., Princeton, NJ 08540. www.press.princeton.edu

Recent interest in the history of mathematics has spawned many excellent publications on the subject. The latest contribution to this series of books is Van Brummelen’s history of trigonometry. This is a scholarly, well researched survey of the development of trigonometry from ancient times until the middle of the Renaissance. However, it is a selective survey that focuses on spherical trigonometry and the measurement of the heavens. This choice of course depends on one's definition of trigonometry. The author bases his definition on two conditions: the existence of a quantitative measure for the inclination of a line and the capacity, and interest, to calculate the length of line segments. This approach gives rise to an historical review of trigonometry emanating from: circular arc measurement to the determination of chord length to conception of a sine function and then the tabulation of values for trigonometric functions and computations involving those functions. This progression is undertaken within the investigations of Astronomy. Babylonian, Greek, Indian, Islamic and finally European accomplishments in this field are examined and documented. Little consideration is given to the origins and rise of plane trigonometry as evidenced in land surveying.

While the author gives a brief mention of “shadow reckoning” and the existence of a vertical staff, or *gnomon*, as the oldest trigonometric measuring instrument, this line of investigation is not pursued further. The procedures and results of shadow reckoning concur with the Van Brummelen’s definition of trigonometry. The trigonometric function that implicitly emerges from early shadow reckoning is the tangent, the last to be tabulated under the astronomical approach to trigonometry used in the book. A testimony to the development of trigonometry can be found in the conception and design of trigonometric measuring instruments. A better examination of such instruments could have been made. The material on Indian astronomical computations benefited from the research of Kim Plofker but also reflected the sometimes overly ambitious claims of R. C. Gupta. Despite the accuracy and scope of ancient Chinese star charts no consideration is given to Chinese contributions on heavenly measurements. This, in my opinion, is a glaring omission.

The history of trigonometry is a huge and complex subject. Glen Van Brummelen acknowledges this fact and concedes that a sequel to this book is needed. He is undertaking this task and perhaps, in his new effort, will satisfy some of the issues raised above. I look forward to his continuation of the story of trigonometry. In the meantime, I highly recommend this book for personal reading and library acquisition. Its documentation and bibliography are especially good. This book will provide a basis for both a better understanding of trigonometry as well as further research on the subject.

Frank Swetz, Professor Emeritus, The Pennsylvania State University, Middletown,PA

See also the MAA Review by Rob Bradley.