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Mathematics of the Heavens and the Earth: The Early History of Trigonometry

by Glen Van Brummelen Princeton University Press
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Hbk 352 pages
AU$119.00 NZ$123.48
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The Mathematics of the Heavens and the Earth is the first major history in English on the origins and early development of trigonometry. Glen Van Brummelen identifies the earliest known trigonometric precursors in ancient Egypt, Babylon, and Greece, and he examines the revolutionary discoveries of Hipparchus, the Greek astronomer believed to have been the first to make systematic use of trigonometry in the second century BC while studying the motions of the stars. The book traces trigonometry’s development into a full-fledged mathematical discipline in India and Islam; explores its applications in the European Middle Ages and Renaissance to such areas as geography and seafaring navigation; and shows how trigonometry retained its ancient roots, yet transformed into an important part of the foundation for modern mathematics.

The Mathematics of the Heavens and the Earth looks at the controversies as well, including disputes over whether Hipparchus indeed was the father of trigonometry, whether Indian trigonometry is original or derived from the Greeks, and the extent to which Western science is indebted to Islamic trigonometry and astronomy. The book also features extended excerpts of translations of original texts, and accessible yet detailed explanations of the mathematics in them to aid the reader.

No other book on trigonometry offers the historical breadth, analytical depth, and coverage of non-Western mathematics that readers will find in The Mathematics of the Heavens and the Earth.

Preface xi
The Ancient Heavens 1

Chapter 1: Precursors 9
What Is Trigonometry? 9
The Seqed in Ancient Egypt 10
* Text 1.1 Finding the Slope of a Pyramid 11
Babylonian Astronomy, Arc Measurement, and the 360° Circle 12
The Geometric Heavens: Spherics in Ancient Greece 18
A Trigonometry of Small Angles? Aristarchus and Archimedes on Astronomical Dimensions 20
* Text 1.2 Aristarchus, the Ratio of the Distances of the Sun and Moon 24

Chapter 2: Alexandrian Greece 33
Convergence 33
Hipparchus 34
A Model for the Motion of the Sun 37
* Text 2.1 Deriving the Eccentricity of the Sun's Orbit 39
Hipparchus's Chord Table 41
The Emergence of Spherical Trigonometry 46
Theodosius of Bithynia 49
Menelaus of Alexandria 53
The Foundations of Spherical Trigonometry: Book III of Menelaus's Spherics 56
* Text 2.2 Menelaus, Demonstrating Menelaus's Theorem 57
Spherical Trigonometry before Menelaus? 63
Claudius Ptolemy 68
Ptolemy's Chord Table 70
Ptolemy's Theorem and the Chord Subtraction/Addition Formulas 74
The Chord of 1° 76
The Interpolation Table 77
Chords in Geography: Gnomon Shadow Length Tables 77
* Text 2.3 Ptolemy, Finding Gnomon Shadow Lengths 78
Spherical Astronomy in the Almagest 80
Ptolemy on the Motion of the Sun 82
* Text 2.4 Ptolemy, Determining the Solar Equation 84
The Motions of the Planets 86
Tabulating Astronomical Functions and the Science of Logistics 88
Trigonometry in Ptolemy's Other Works 90
* Text 2.5 Ptolemy, Constructing Latitude Arcs on a Map 91
After Ptolemy 93

Chapter 3: India 94
Transmission from Babylon and Greece 94
The First Sine Tables 95
Aryabhata's Difference Method of Calculating Sines 99
* Text 3.1 Aryabhata, Computing Sines 100
Bhaskara I's Rational Approximation to the Sine 102
Improving Sine Tables 105
Other Trigonometric Identities 107
* Text 3.2 Varahamihira, a Half-angle Formula 108
* Text 3.3 Brahmagupta, the Law of Sines in Planetary Theory? 109
Brahmagupta's Second-order Interpolation Scheme for Approximating Sines 111
* Text 3.4 Brahmagupta, Interpolating Sines 111
Taylor Series for Trigonometric Functions in Madhava's Kerala School 113
Applying Sines and Cosines to Planetary Equations 121
Spherical Astronomy 124
* Text 3.5 Varahamihira, Finding the Right Ascension of a Point on the Ecliptic 125
Using Iterative Schemes to Solve Astronomical Problems 129
* Text 3.6 Paramesvara, Using Fixed-point Iteration to Compute Sines 131
Conclusion 133

Chapter 4: Islam 135
Foreign Junkets: The Arrival of Astronomy from India 135
Basic Plane Trigonometry 137
Building a Better Sine Table 140
* Text 4.1 Al-Samaw'al ibn Yahya al-Maghribi, Why the Circle Should Have 480 Degrees 146
Introducing the Tangent and Other Trigonometric Functions 149
* Text 4.2 Abu'l-Rayhan al-Biruni, Finding the Cardinal Points of the Compass 152
Streamlining Astronomical Calculation 156
* Text 4.3 Kushyar ibn Labban, Finding the Solar Equation 156
Numerical Techniques: Approximation, Iteration, Interpolation 158
* Text .4 Ibn Yunus, Interpolating Sine Values 164
Early Spherical Astronomy: Graphical Methods and Analemmas 166
* Text 4.5 Al-Khwarizmi, Determining the Ortive Amplitude Geometrically 168
Menelaus in Islam 173
* Text 4.6 Al-Kuhi, Finding Rising Times Using the Transversal Theorem 175
Menelaus's Replacements 179
Systematizing Spherical Trigonometry: Ibn Mucadh's Determination of the Magnitudes and Nasir al-Din al-Tusi's Transversal Figure 186
Applications to Religious Practice: The Qibla and Other Ritual Needs 192
* Text 4.7 Al-Battani, a Simple Approximation to the Qibla 195
Astronomical Timekeeping: Approximating the Time of Day Using the Height of the Sun 201
New Functions from Old: Auxiliary Tables 205
* Text 4.8 Al-Khalili, Using Auxiliary Tables to Find the Hour-angle 207
Trigonometric and Astronomical Instruments 209
* Text 4.9 Al-Sijzi (?), On an Application of the Sine Quadrant 213
Trigonometry in Geography 215
Trigonometry in al-Andalus 217

Chapter 5: The West to 1550 223
Transmission from the Arab World 223
An Example of Transmission: Practical Geometry 224
* Text 5.1 Hugh of St. Victor, Using an Astrolabe to Find the Height of an Object 225
* Text 5.2 Finding the Time of Day from the Altitude of the Sun 227
Consolidation and the Beginnings of Innovation: The Trigonometry of Levi ben Gerson, Richard of Wallingford, and John of Murs 230
* Text 5.3 Levi ben Gerson, The Best Step Size for a Sine Table 233
* Text 5.4 Richard of Wallingford, Finding Sin(1°) with Arbitrary Accuracy 237
Interlude: The Marteloio in Navigation 242
* Text 5.5 Michael of Rhodes, a Navigational Problem from His Manual 244
From Ptolemy to Triangles: John of Gmunden, Peurbach, Regiomontanus 247
* Text 5.6 Regiomontanus, Finding the Side of a Rectangle from Its Area and Another Side 254
* Text 5.7 Regiomontanus, the Angle-angle-angle Case of Solving Right Triangles 255
Successors to Regiomontanus: Werner and Copernicus 264
* Text 5.8 Copernicus, the Angle-angle-angle Case of Solving Triangles 267
* Text 5.9 Copernicus, Determining the Solar Eccentricity 270
Breaking the Circle: Rheticus, Otho, Pitiscus and the Opus Palatinum 273

Concluding Remarks 284
Bibliography 287
Index 323

Fans of the history of mathematics will be richly rewarded by this exhaustively researched book, which focuses on the early development of trigonometry. . . . Finally, the generous and lucid explanations provided throughout the text make Van Brummelen's history a rewarding one for the mathematical tourist. -- Mathematics Teacher his new and comprehensive history of trigonometry is more than welcome--even more so because it is the first in English. . . . his book will be appreciated by many with an interest--general or more specific--in the history of mathematics. -- Steven Wepster, Centaurus his book will have wide appeal, for students, researchers, and teachers of history and/or trigonometry. The excerpts selected are balanced and their significances well articulated. . . . It is a book written by an expert after many years of exposure to individual sources and in this way Van Brummelen uniquely advances the field. The book will no doubt become a necessary addition to the libraries of mathematicians and historians alike. -- Clemency Montelle and Kathleen M. Clark, Aestimatio Van Brummelen's history does far more than simply fill a vacant spot in the historical literature of mathematics. He recounts the history of trigonometry in a way that is both captivating and yet more than satisfying to the crankiest and most demanding of scholars. . . . The Mathematics of the Heavens and the Earth should be a part of every university library's mathematics collection. It's also a book that most mathematicians with an interest in the history of the subject will want to own. -- Rob Bradley, MAA Reviews
Glen Van Brummelen is professor of mathematics at Quest University Canada. He is the coeditor of Mathematics
and the Historian’s Craft and the coauthor of Calculus Explorations Powered by Technology: Tales of History and Imagination.