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

**by Glen Van Brummelen**

*Princeton University Press*

- Pub Date:
- 01/2009
- ISBN:
- 9780691129730
- Format:
- Hbk
*352 pages* - Price:
**AU$119.00***NZ$123.48*

**Product Status:**

*Available in Approx 9 days***Instructors**

& Academics:

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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

and the Historian’s Craft and the coauthor of Calculus Explorations Powered by Technology: Tales of History and Imagination.