The Doctrine of Triangles: A History of Modern Trigonometry

With the publication of this book, Glen van Brummelen, the noted Canadian historian and specialist in mathematical astronomy, has completed a comprehensive, scholarly, and engaging two-volume “scientific history” of the origins and development of trigonometry. The first of these volumes,

The Mathematics of the Heavens and the Earth, was published by Princeton University Press in 2009, and it recounted an essentially chronological development of the subject from ancient times into the mid-sixteenth century.  See our review of the earlier book. One of the chief reasons for the success of that first volume was its exposition of vast amounts of historical scholarship that have come to light in recent decades concerning mathematical achievements from the non-Western world; these were brought together in one place for the first time by this author. The approach, there and here in this second volume, is less sociological than has become typical in contemporary historical scholarship, but as the author recognizes, there is “a lack of documentary evidence to support a detailed social narrative over [many of] the cultures and times being described” (p. xiii). Nonetheless, he does assert that he “has moved a little in the direction of social history.” The narrative may focus on ideas, methods, theorems and applications, but it is a narrative primarily about the work of individuals in particular cultural milieus dealing with specific mathematical problems.

 

The current volume picks up where the first left off, with a demarcation at the introduction of logarithms in Europe in the 1500s. While today we generally don’t link logarithms with trigonometry in typical presentations to students of mathematics, other than noting that the logarithm is another (quite central) transcendental function useful in analysis, one must recognize that in the sixteenth century they were introduced with a different purpose in mind. Back then logarithms were heralded as a new technology for speeding calculations for practical astronomy, specifically for practitioners working with trigonometric formulas in spherical trigonometry based on ancient geometric models for arcs connecting points in a spherical sky. Logarithms were used to process observational data, to assist in the design of newer and ever more accurate instruments for observation and to make accurate predictions about what they observed in the heavens above, whence they rightly belong in a history of trigonometry.

 

In the first chapter of The Doctrine of Triangles, van Brummelen documents the critical period in the latter half of the sixteenth century in Europe, when trigonometric methods were expanded: to deal with mensuration of planar (as opposed to spherical) triangles in advance of an explosion of activity in new applications to geodesy, surveying and cartography; to standardize the collection of the now-familiar six trigonometric quantities of sine, tangent, secant and their respective co-measures; in the production of accurate handy tables; and not least, the coining of the word “trigonometry” (in Latin, that is) by Bartholomew Pitiscus in the title of his 1595 work Trigonometriae.

 

The author notes appropriately that these developments “were part of a larger movement [in the years 1650–1850] of mathematics away from geometric conceptions and towards analysis, caused by the emergence of calculus” (p. 110). Work of Kepler, Roberval, James Gregory, Pascal, Isaac Barrow, Newton and Leibniz mark how the trigonometric quantities were central from the beginning to the development of infinitesimal analytical methods, and were folded in with algebraic (i.e., non-transcendental) quantities in the steady rise of the powerful use of symbolic methods during these centuries.

 

The book provides numerous helpful illustrative excerpts drawn from key texts in the historical literature (all in English translation, but often accompanied by facsimile images of the original sources). The excerpts also come with explanatory commentary that allow readers to become immersed first-hand in thel development of these mathematical concepts and methods. 

 

A significant interruption of the historical timeline takes place in the middle of the book when the reader is transported to ancient and medieval China. As van Brummelen notes (p. xv), “the placement of this chapter was a problem with no solution; it would disrupt the narrative wherever I included it. Since it deals occasionally with the introduction of concepts from European trigonometry,” part of the legacy of European Jesuit missionaries into China in the sixteenth century who attempted to gain influence with the Emperor’s court by sharing Western scientific methods, the author decided it best to place this chapter here. We find in it “a great deal of originality in ancient Chinese mathematics that might be interpreted as something related to trigonometric activity” (p. 243). Coupled with a summary of how Chinese scholars made use of Western methods, these pages provide one of the few convenient places to learn about any of this.

 

The latter part of this story brings the reader to the early years of the twentieth century, illustrating how trigonometry continued to be expanded by the addition of new theorems and computational techniques – notably, by infinite series representations – as well as how, from the time of Euler, trigonometric functions found an important place in mathematical models of periodic phenomena besides those in astronomical settings. The central role of trigonometric functions in Fourier analysis is another key element of this development.

 

There are very few sources available that tell the rich and engaging story that is the development of trigonometry. The last comprehensive history to be published was another two-volume history by Anton von Braunmühl, and that was more than a century ago (and in German, not English).  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.

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Daniel Otero is an Associate Professor of Mathematics at Xavier University.