The Doctrine of Triangles: A History of Modern Trigonometry
392
The Doctrine of Triangles: A History of Modern Trigonometry
392Hardcover
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Overview
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.
Product Details
| ISBN-13: | 9780691179414 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 06/08/2021 |
| Pages: | 392 |
| Product dimensions: | 9.30(w) x 6.20(h) x 1.30(d) |
About the Author
Table of Contents
Preface xi
1 European Trigonometry Comes of Age 1
What's in a Name? 3
Text 1.1 Regiomontanus, Defining the Basic Trigonometric Functions 4
Text 1.2 Reinhold, a Calculation in a Planetary Model Using Sines and Tangents 6
Trigonometric Tables Evolving 16
Algebraic Gems by Viète 25
Text 1.3 Viète, Finding a Recurrence Relation for sin nθ 25
New Theorems, Plane and Spherical 30
Text 1.4 Snell on Reciprocal Triangles 37
Consolidating the Solutions of Triangles 39
Widening Applications 45
Text 1.5 Clavius on a Problem in Surveying 49
Text 1.6 Gunter on Solving a Right-Angled Spherical Triangle with His Sector 56
2 Logarithms 62
Napier, Briggs, and the Birth of Logarithms 62
Text 2.1 Napier, Solving a Problem in Spherical Trigonometry with His Logarithms 65
Interlude: Joost Bürgi's Surprising Method of Calculating a Sine Table 69
The Explosion of Tables of Logarithms 71
Computing Tables Effectively: Logarithms 76
Computing Tables Effectively: Interpolation 78
Text 2.2 Briggs, Completing a Table Using Finite Difference Interpolation 81
Napier on Spherical Trigonometry 84
Further Theoretical Developments 91
Developments in Notation 97
Practical and Scientific Applications 99
Text 2.3 John Newton, Determining the Declination of an Arc of the Ecliptic with Logarithms 100
3 Calculus 110
Quadratures in Trigonometry Before Newton and Leibniz 110
Text 3.1 Pascal, Finding the Integral of the Sine 118
Tangents in Trigonometry Before Newton and Leibniz 120
Text 3.2 Barrow, Finding the Derivative of the Tangent 122
Infinite Sequences and Series in Trigonometry 126
Text 3.3 Newton, Finding a Series for the Arc Sine 129
Transforming the Construction of Trigonometric Tables with Series 135
Geometric Derivatives and Integrals of Trigonometric Functions 143
A Transition to Analytical Conceptions 145
Text 3.4 Cotes, Estimating Errors in Triangles 149
Text 3.5 Jakob Kresa, Relations Between the Sine and the Other Trigonometric Quantities 155
Euler on the Analysis of Trigonometric Functions 161
Text 3.6 Leonhard Euler, On Transcendental Quantities Which Arise from the Circle 165
Text 3.7 Leonhard Euler, On the Derivative of the Sine 175
Euler on Spherical Trigonometry 177
4 China 185
Indian and Islamic Trigonometry in China 185
Text 4.1 Yixing, Description of a Table of Gnomon Shadow Lengths 188
Indigenous Chinese Geometry 191
Text 4.2 Liu Hui, Finding the Dimensions of an Inaccessible Walled City 192
Indigenous Chinese Trigonometry 198
The Jesuits Arrive 202
Trigonometry in the Chongzhen lishu 204
Logarithms in China 208
The Kangxi Period and Mei Wending 213
Dai Zhen: Philology Encounters Mathematics 222
Infinite Series 227
Text 4.3 Mei Juecheng, On Calculating the Circumference of a Circle from Its Diameter 228
Text 4.4 Minggatu, On Calculating the Chord of a Given Arc 231
5 Europe After Euler 243
Normal Science: Gap Filling in Spherical Trigonometry 244
Text 5.1 Pingré, Extending Napier's Rules to Oblique Spherical Triangles 245
Symmetry and Unity 253
The Return of Stereographic Projection 255
Surveying and Legendre's Theorem 260
Trigonometry in Navigation 264
Text 5.2 James Andrew, Solving the PZX Triangle Using Haversines 268
Tables 273
Fourier Series 281
Text 5.3 Jean Baptiste Joseph Fourier, A Trigonometric Series as a Function 287
Concerns About Negativity 290
Hyperbolic Trigonometry 294
Text 5.4 Vincenzo Riccati, The Invention of the Hyperbolic Functions 294
Education 303
Concluding Remarks 314
Bibliography 317
Index 363
What People are Saying About This
"The Doctrine of Triangles adheres to Van Brummelen’s signature scholarly excellence and lucid style. Although emphasizing European developments, this history of trigonometry devotes considerable space to China and also considers the history of logarithms and of trigonometry education. It will be a standard reference on the subject for years to come."—Victor Katz, author of A History of Mathematics"Exploring the creation of modern trigonometry from 1550 through to 1900, The Doctrine of Triangles succeeds brilliantly in making this history accessible. No other book covers this subject with such astoundingly thorough and deep scholarship. It is a monumental accomplishment and pleasure to read."—Dennis Duke, Florida State University"The Doctrine of Triangles displays Van Brummelen’s mastery in engaging communication. There is no doubt in my mind that this rich and far-reaching book will be well-received by specialists and generalists alike. A veritable feast, it will be heavily referenced and inspire generations to come."—Clemency Montelle, author of Chasing Shadows