The Doctrine of Triangles: A History of Modern Trigonometry

Description

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. 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