Teaching and Learning the Trigonometric Functions through Their Origins: Episode 6 – Regiomontanus and the Beginnings of Modern Trigonometry

Daniel E. Otero (Xavier University)

In the preface to the first book of his two-volume history of trigonometry, The Mathematics of the Heavens and the Earth [2009] and The Doctrine of Triangles [2021], Glen Van Brummelen discussed how he decided where to divide the two volumes:

Trigonometry proper began with the origins of the Ptolemaic geocentric system, and a nice breaking point is found with the beginning of Copernicus’s heliocentric model [Van Brummelen 2009, p. xv].

But where Van Brummelen sees the transition from the early history of trigonometry to its modern phase in the shift of perspective between “the Earth-centered and Sun-centered universe of Copernicus’ (literally) world-changing De revolutionibus (1543), the classroom unit Regiomontanus and the Beginnings of Modern Trigonometry featured in this article highlights a pedagogically more useful transition in the history of trigonometry through excerpts from a much less “revolutionary” work by Johannes Müller, better known by the name Regiomontanus (1436–1476).

Regiomontanus Woodcut from Nuremberg Chronicles
Johannes Müller / Regiomontanus (1436–1476).
Woodcut from the Nuremberg Chronicle. Public domain.

RegiomontanusDe triangulis omnimodis (On Triangles of Every Kind) was compiled decades earlier than De revolutionibus, in 1464, but it was not published until 1533. While it contained little brand-new mathematics, On Triangles was the first fully comprehensive treatment of trigonometry to appear in Europe. What makes it truly noteworthy, however, is what one will not find in its pages: astronomical problems. As Van Brummelen makes clear in The Mathematics of the Heavens and the Earth, the history of trigonometry up to the sixteenth century is to a large extent the history of astronomy. What we know today as trigonometry began as a collection of mathematical techniques, both geometrical and arithmetical, for addressing astronomical problems, such as: how to measure the elevation of the Sun in the sky; how to predict when a given star will rise, based on the regularity of its revolutions about an Earth-based observer; or when an eclipse (solar or lunar) should occur. Ancient mathematician-astronomers recognized that these questions could be handled by coordinating the measures of arcs along circles with the measures of chords across those arcs. Later, they found that it was more efficient to tabulate not the measures of chords, but measures of half-chords (sines).

From the title page of De triangulis (Basel, 1533). Mathematical Treasures, Convergence.

What Regiomontanus accomplished in On Triangles was essentially a reorientation of trigonometry away from its primary connection with astronomy and toward a resettlement within mathematics, specifically the Euclidean geometry of triangles and circles. He saw that many trigonometric problems could be rephrased in terms of finding one or more missing angles or side lengths in a right triangle, so he recast the subject as the branch of mathematics whose function was to solve triangles.[1] As a result, one can draw a direct connection between On Triangles and modern trigonometry textbooks: both introduce the sine by means of right triangles. This is precisely the connection made in the classroom unit Regiomontanus and the Beginnings of Modern Trigonometry. The unit also presents examples of solving triangles from On Triangles that ask students to familiarize themselves with calculator functions for sine and inverse sine.

This classroom unit Regiomontanus and the Beginnings of Modern Trigonometry is one of a series of six units in the Convergence series Teaching and Learning the Trigonometric Functions through Their Origins that attempts to provide students with a rich motivation for the study of trigonometry by presenting a series of episodes from the long history of the subject. These episodes work to contextualize the main ideas of trigonometry in the questions that earlier mathematicians addressed in developing the subject over thousands of years.

The unit Regiomontanus and the Beginnings of Modern Trigonometry (pdf) is ready for student use. It is meant to be completed in two 50-minute classroom periods, plus time in advance for students to do some initial reading and time afterwards for them to write up their solutions to the tasks. A brief set of instructor notes offering additional background and practical advice for the use of these materials in the classroom is appended at the end of the student version of the project.

Regiomontanus and the Beginnings of Modern Trigonometry is the sixth (and final) episode in the Convergence series Teaching and Learning the Trigonometric Functions through Their Origins. Although these classroom projects are posted here as parts of a series, each episode has been redesigned to stand alone. Readers who want to see the entire Primary Source Project (PSP) from which these units are drawn can obtain that PSP, A Genetic Context for Understanding the Trigonometric Functions, from the website of the NSF-funded project TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS). The LaTeX source code of all TRIUMPHS projects, including the units appearing in this series, are available from the project authors by request.

[1] To be sure, On Triangles was written for astronomers, but only as a mathematical preparation for doing astronomy. Indeed, in the note “To Readers” at the beginning of On Triangles, Regiomontanus  remarked that:

 [N]o one can bypass the science of triangles and reach a satisfying knowledge of the stars. . . . You, who wish to study great and wonderful things, who wonder about the movement of the stars, must read these theorems about triangles. Knowing these ideas will open the door to all astronomy and to certain geometric problems

It was only in the sixteenth century that the term trigonometry (literally “the measure of triangles”) began to be used to describe the subject, for by then it was differentiated from its astronomical “womb.”