The Mathematics of the Heavens and the Earth

Name: The Mathematics of the Heavens and the Earth | The Early History of Trigonometry
Brand: Princeton University Press
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The Early History of Trigonometry

Glen Van Brummelen(Autor*in)

Princeton University Press

1. Auflage

Erschienen am 10. August 2021

352 Seiten

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Cover Page
Half-title Page
Title Page
Copyright Page
Dedication Page
Contents
Preface
The Ancient Heavens
1. Precursors
What Is Trigonometry?
The Seqed in Ancient Egypt
Text 1.1 Finding the Slope of a Pyramid
Babylonian Astronomy, Arc Measurement, and the 360° Circle
The Geometric Heavens: Spherics in Ancient Greece
A Trigonometry of Small Angles? Aristarchus and Archimedes on Astronomical Dimensions
Text 1.2 Aristarchus, the Ratio of the Distances of the Sun and Moon
2. Alexandrian Greece
Convergence
Hipparchus
A Model for the Motion of the Sun
Text 2.1 Deriving the Eccentricity of the Sun's Orbit
Hipparchus's Chord Table
The Emergence of Spherical Trigonometry
Theodosius of Bithynia
Menelaus of Alexandria
The Foundations of Spherical Trigonometry: Book III of Menelaus's Spherics
Text 2.2 Menelaus, Demonstrating Menelaus's Theorem
Spherical Trigonometry before Menelaus?
Claudius Ptolemy
Ptolemy's Chord Table
Ptolemy's Theorem and the Chord Subtraction/Addition Formulas
The Chord of 1°
The Interpolation Table
Chords in Geography: Gnomon Shadow Length Tables
Text 2.3 Ptolemy, Finding Gnomon Shadow Lengths
Spherical Astronomy in the Almagest
Ptolemy on the Motion of the Sun
Text 2.4 Ptolemy, Determining the Solar Equation
The Motions of the Planets
Tabulating Astronomical Functions and the Science of Logistics
Trigonometry in Ptolemy's Other Works
Text 2.5 Ptolemy, Constructing Latitude Arcs on a Map
After Ptolemy
3. India
Transmission from Babylon and Greece
The First Sine Tables
Aryabhata's Difference Method of Calculating Sines
Text 3.1 Aryabhata, Computing Sines
Bhaskara I's Rational Approximation to the Sine
Improving Sine Tables
Other Trigonometric Identities
Text 3.2 Varahamihira, a Half-angle Formula
Text 3.3 Brahmagupta, the Law of Sines in Planetary Theory?
Brahmagupta's Second-order Interpolation Scheme for Approximating Sines
Text 3.4 Brahmagupta, Interpolating Sines
Taylor Series for Trigonometric Functions in Madhava's Kerala School
Applying Sines and Cosines to Planetary Equations
Spherical Astronomy
Text 3.5 Varahamihira, Finding the Right Ascension of a Point on the Ecliptic
Using Iterative Schemes to Solve Astronomical Problems
Text 3.6 Paramesvara, Using Fixed-point Iteration to Compute Sines
Conclusion
4. Islam
Foreign Junkets: The Arrival of Astronomy from India
Basic Plane Trigonometry
Building a Better Sine Table
Text 4.1 Al-Samaw'al ibn Yahya al-Maghribi, Why the Circle Should Have 480 Degrees
Introducing the Tangent and Other Trigonometric Functions
Text 4.2 Abu'l-Rayhan al-Biruni, Finding the Cardinal Points of the Compass
Streamlining Astronomical Calculation
Text 4.3 Kushyar ibn Labban, Finding the Solar Equation
Numerical Techniques: Approximation, Iteration, Interpolation
Text 4.4 Ibn Yunus, Interpolating Sine Values
Early Spherical Astronomy: Graphical Methods and Analemmas
Text 4.5 Al-Khwarizmi, Determining the Ortive Amplitude Geometrically
Menelaus in Islam
Text 4.6 Al-Kuhi, Finding Rising Times Using the Transversal Theorem
Menelaus's Replacements
Systematizing Spherical Trigonometry: Ibn Mucadh's Determination of the Magnitudes and Nasir al-Din al-Tusi's Transversal Figure
Applications to Religious Practice: The Qibla and Other Ritual Needs
Text 4.7 Al-Battani, a Simple Approximation to the Qibla
Astronomical Timekeeping: Approximating the Time of Day Using the Height of the Sun
New Functions from Old: Auxiliary Tables
Text 4.8 Al-Khalili, Using Auxiliary Tables to Find the Hour-angle
Trigonometric and Astronomical Instruments
Text 4.9 Al-Sijzi (?), On an Application of the Sine Quadrant
Trigonometry in Geography
Trigonometry in al-Andalus
5. The West to 1550
Transmission from the Arab World
An Example of Transmission: Practical Geometry
Text 5.1 Hugh of St. Victor, Using an Astrolabe to Find the Height of an Object
Text 5.2 Finding the Time of Day from the Altitude of the Sun
Consolidation and the Beginnings of Innovation: The Trigonometry of Levi ben Gerson, Richard of Wallingford, and John of Murs
Text 5.3 Levi ben Gerson, The Best Step Size for a Sine Table
Text 5.4 Richard of Wallingford, Finding Sin(1°) with Arbitrary Accuracy
Interlude: The Marteloio in Navigation
Text 5.5 Michael of Rhodes, a Navigational Problem from His Manual
From Ptolemy to Triangles: John of Gmunden, Peurbach,Regiomontanus
Text 5.6 Regiomontanus, Finding the Side of a Rectangle from Its Area and Another Side
Text 5.7 Regiomontanus, the Angle-angle-angle Case of Solving Right Triangles
Successors to Regiomontanus: Werner and Copernicus
Text 5.8 Copernicus, the Angle-angle-angle Case of Solving Triangles
Text 5.9 Copernicus, Determining the Solar Eccentricity
Breaking the Circle: Rheticus, Otho, Pitiscus and the Opus Palatinum
Concluding Remarks
Bibliography
Index

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