The Doctrine of Triangles

A History of Modern Trigonometry

Glen Van Brummelen

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English

Princeton University Press
24 September 2025

Trigonometry; History of mathematics; Popular science

The Doctrine of Triangles offers an interdisciplinary history of trigonometry that spans four centuries, starting in 1550 and concluding in the 1900s. Glen Van Brummelen tells the story of trigonometry as it evolved from an instrument for understanding the heavens to a practical tool, used in fields such as surveying and navigation. In Europe, China, and America, trigonometry aided and was itself transformed by concurrent mathematical revolutions, as well as the rise of science and technology.

Following its uses in mid-sixteenth-century Europe as the 'foot of the ladder to the stars' and the mathematical helpmate of astronomy, trigonometry became a ubiquitous tool for modeling various phenomena, including animal populations and sound waves. In the late sixteenth century, trigonometry increasingly entered the physical world through the practical disciplines, and its societal reach expanded with the invention of logarithms. Calculus shifted mathematical reasoning from geometric to algebraic patterns of thought, and trigonometry's participation in this new mathematical analysis grew, encouraging such innovations as complex numbers and non-Euclidean geometry. Meanwhile in China, trigonometry was evolving rapidly too, sometimes merging with indigenous forms of knowledge, and with Western discoveries. In the nineteenth century, trigonometry became even more integral to science and industry as a fundamental part of the science and engineering toolbox, and a staple subject in high school classrooms.

A masterful combination of scholarly rigour and compelling narrative, The Doctrine of Triangles brings trigonometry's rich historical past full circle into the modern era.

By: Glen Van Brummelen
Imprint: Princeton University Press
Country of Publication: United States
Dimensions: Height: 235mm, Width: 156mm,
ISBN: 9780691270432
ISBN 10: 0691270430
Pages: 392
Publication Date: 24 September 2025
Audience: College/higher education , Professional and scholarly , Primary , Undergraduate
Format: Paperback
Publisher's Status: Forthcoming

Glen Van Brummelen is dean of the faculty of natural and applied sciences at Trinity Western University, and a historian of mathematics and astronomy. His many books include The Mathematics of the Heavens and the Earth: The Early History of Trigonometry and Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry (both Princeton).

Reviews for The Doctrine of Triangles: A History of Modern Trigonometry

""Glen van Brummelen has prepared a highly recommended, accessible and definitive history of the subject that will serve as a resource for scholars for decades to come.""---Daniel Otero, MAA Reviews ""The Doctrine of Triangles is an informative and valuable reference work. ""---Wallace A Ferguson, Institute of Mathematics and its Applications ""A guided tour through the museum of mathematics. . . . [The Doctrine of Triangles] takes the history of trigonometry, which is a formidable subject in its scope and size, and transforms it into something readable.""---Daniel Mansfield, The Mathematical Intelligencer ""Very easy to read, and there are lots of helpful diagrams, especially for the spherical trigonometry . . . [The Doctrine of Triangles] is deeply enriched by extracts from contemporary texts, given first in fairly literal English translations, often accompanied by the original diagrams, and then explained in modern terms. So mathematical readers (and, I hope, their students) can experience a little of what trigonometry was actually like at each stage in its history.""---John Hannah, Aestimatio