The doctrine of triangles : a history of modern trigonometry / Glen Van Brummelen.

Format:

Book

Author/Creator:

Van Brummelen, Glen, author.

Language:

English

Subjects (All):

Trigonometry--History.

Trigonometry.

History.

Genre:

History.

Physical Description:

xvi, 372 pages : illustrations ; 25 cm

Place of Publication:

Princeton, New Jersey : Princeton University Press, [2021]

Summary:

"An interdisciplinary history of trigonometry from the mid-sixteenth century through to the early twentieth century 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"-- Provided by publisher.

Contents:

Machine generated contents note: 1. European Trigonometry Comes Of Age

What's in a Name?

Text 1.1 Regiomontanus, Denning the Basic Trigonometric Functions

Text 1.2 Reinhold, a Calculation in a Planetary Model Using Sines and Tangents

Trigonometric Tables Evolving

Algebraic Gems by Viete

Text 1.3 Viete, 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 Burgi'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 Pingre, 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.

Notes:

Includes bibliographical references and index.

Other Format:

Online version: Van Brummelen, Glen The doctrine of triangles

ISBN:

9780691179414

0691179417

OCLC:

1201300540

Publisher Number:

99987735941