An interdisciplinary history of trigonometry from the mid-sixteenth century to the early twentieth
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 rigor and compelling narrative, The Doctrine of Triangles brings trigonometry’s rich historical past full circle into the modern era.
Author(s): Glen Van Brummelen
Edition: 1
Publisher: Princeton University Press
Year: 2021
Language: English
Pages: 392
City: Princeton, NJ
Cover
Contents
Preface
1. European Trigonometry Comes of Age
What’s in a Name?
Text 1.1 Regiomontanus, Defining the Basic Trigonometric Functions
Text 1.2 Reinhold, a Calculation in a Planetary Model Using Sines and Tangents
Trigonometric Tables Evolving
Algebraic Gems by Viète
Text 1.3 Viète, Finding a Recurrence Relation for sin nθ
New Theorems, Plane and Spherical
Text 1.4 Snell on Reciprocal Triangles
Consolidating the Solutions of Triangles
Widening Applications
Text 1.5 Clavius on a Problem in Surveying
Text 1.6 Gunter on Solving a Right-Angled Spherical Triangle with His Sector
2. Logarithms
Napier, Briggs, and the Birth of Logarithms
Text 2.1 Napier, Solving a Problem in Spherical Trigonometry with His Logarithms
Interlude: Joost Bürgi’s Surprising Method of Calculating a Sine Table
The Explosion of Tables of Logarithms
Computing Tables Effectively: Logarithms
Computing Tables Effectively: Interpolation
Text 2.2 Briggs, Completing a Table Using Finite Difference Interpolation
Napier on Spherical Trigonometry
Further Theoretical Developments
Developments in Notation
Practical and Scientific Applications
Text 2.3 John Newton, Determining the Declination of an Arc of the Ecliptic with Logarithms
3. Calculus
Quadratures in Trigonometry Before Newton and Leibniz
Text 3.1 Pascal, Finding the Integral of the Sine
Tangents in Trigonometry Before Newton and Leibniz
Text 3.2 Barrow, Finding the Derivative of the Tangent
Infinite Sequences and Series in Trigonometry
Text 3.3 Newton, Finding a Series for the Arc Sine
Transforming the Construction of Trigonometric Tables with Series
Geometric Derivatives and Integrals of Trigonometric Functions
A Transition to Analytical Conceptions
Text 3.4 Cotes, Estimating Errors in Triangles
Text 3.5 Jakob Kresa, Relations Between the Sine and the Other Trigonometric Quantities
Euler on the Analysis of Trigonometric Functions
Text 3.6 Leonhard Euler, On Transcendental Quantities Which Arise from the Circle
Text 3.7 Leonhard Euler, On the Derivative of the Sine
Euler on Spherical Trigonometry
4. China
Indian and Islamic Trigonometry in China
Text 4.1 Yixing, Description of a Table of Gnomon Shadow Lengths
Indigenous Chinese Geometry
Text 4.2 Liu Hui, Finding the Dimensions of an Inaccessible Walled City
Indigenous Chinese Trigonometry
The Jesuits Arrive
Trigonometry in the Chongzhen lishu
Logarithms in China
The Kangxi Period and Mei Wending
Dai Zhen: Philology Encounters Mathematics
Infinite Series
Text 4.3 Mei Juecheng, On Calculating the Circumference of a Circle from Its Diameter
Text 4.4 Minggatu, On Calculating the Chord of a Given Arc
5. Europe After Euler
Normal Science: Gap Filling in Spherical Trigonometry
Text 5.1 Pingré, Extending Napier’s Rules to Oblique Spherical Triangles
Symmetry and Unity
The Return of Stereographic Projection
Surveying and Legendre’s Theorem
Trigonometry in Navigation
Text 5.2 James Andrew, Solving the PZX Triangle Using Haversines
Tables
Fourier Series
Text 5.3 Jean Baptiste Joseph Fourier, A Trigonometric Series as a Function
Concerns About Negativity
Hyperbolic Trigonometry
Text 5.4 Vincenzo Riccati, The Invention of the Hyperbolic Functions
Education
Concluding Remarks
Bibliography
Index
