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Preface xi The Ancient Heavens 1 Chapter 1: Precursors 9 What IsTrigonometry? 9 The Seqed in Ancient Egypt 10 * Text 1.1 Finding the Slopeof a Pyramid 11 Babylonian Astronomy, Arc Measurement, and the 360 Circle12 The Geometric Heavens: Spherics in Ancient Greece 18 A Trigonometry ofSmall Angles? Aristarchus and Archimedes on Astronomical Dimensions 20 *Text 1.2 Aristarchus, the Ratio of the Distances of the Sun and Moon 24Chapter 2: Alexandrian Greece 33 Convergence 33 Hipparchus 34 A Model forthe Motion of the Sun 37 * Text 2.1 Deriving the Eccentricity of the Sun'sOrbit 39 Hipparchus's Chord Table 41 The Emergence of SphericalTrigonometry 46 Theodosius of Bithynia 49 Menelaus of Alexandria 53 TheFoundations of Spherical Trigonometry: Book III of Menelaus's Spherics 56 *Text 2.2 Menelaus, Demonstrating Menelaus's Theorem 57 SphericalTrigonometry before Menelaus? 63 Claudius Ptolemy 68 Ptolemy's Chord Table70 Ptolemy's Theorem and the Chord Subtraction/Addition Formulas 74 TheChord of 1 76 The Interpolation Table 77 Chords in Geography: GnomonShadow Length Tables 77 * Text 2.3 Ptolemy, Finding Gnomon Shadow Lengths 78Spherical 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 thePlanets 86 Tabulating Astronomical Functions and the Science of Logistics 88Trigonometry in Ptolemy's Other Works 90 * Text 2.5 Ptolemy, ConstructingLatitude Arcs on a Map 91 After Ptolemy 93 Chapter 3: India 94Transmission from Babylon and Greece 94 The First Sine Tables 95Aryabhata's Difference Method of Calculating Sines 99 * Text 3.1 Aryabhata,Computing Sines 100 Bhaskara I's Rational Approximation to the Sine 102Improving Sine Tables 105 Other Trigonometric Identities 107 * Text 3.2Varahamihira, a Half-angle Formula 108 * Text 3.3 Brahmagupta, the Law ofSines in Planetary Theory? 109 Brahmagupta's Second-order InterpolationScheme for Approximating Sines 111 * Text 3.4 Brahmagupta, InterpolatingSines 111 Taylor Series for Trigonometric Functions in Madhava's KeralaSchool 113 Applying Sines and Cosines to Planetary Equations 121 SphericalAstronomy 124 * Text 3.5 Varahamihira, Finding the Right Ascension of aPoint on the Ecliptic 125 Using Iterative Schemes to Solve AstronomicalProblems 129 * Text 3.6 Paramesvara, Using Fixed-point Iteration to ComputeSines 131 Conclusion 133 Chapter 4: Islam 135 Foreign Junkets: TheArrival of Astronomy from India 135 Basic Plane Trigonometry 137 Building aBetter Sine Table 140 * Text 4.1 Al-Samaw'al ibn Yahya al-Maghribi, Why theCircle Should Have 480 Degrees 146 Introducing the Tangent and OtherTrigonometric Functions 149 * Text 4.2 Abu'l-Rayhan al-Biruni, Finding theCardinal Points of the Compass 152 Streamlining Astronomical Calculation 156* Text 4.3 Kushyar ibn Labban, Finding the Solar Equation 156 NumericalTechniques: Approximation, Iteration, Interpolation 158 * Text .4 Ibn Yunus,Interpolating Sine Values 164 Early Spherical Astronomy: Graphical Methodsand Analemmas 166 * Text 4.5 Al-Khwarizmi, Determining the Ortive AmplitudeGeometrically 168 Menelaus in Islam 173 * Text 4.6 Al-Kuhi, Finding RisingTimes Using the Transversal Theorem 175 Menelaus's Replacements 179Systematizing Spherical Trigonometry: Ibn Mucadh's Determination of theMagnitudes and Nasir al-Din al-Tusi's Transversal Figure 186 Applications toReligious Practice: The Qibla and Other Ritual Needs 192 * Text 4.7Al-Battani, a Simple Approximation to the Qibla 195 AstronomicalTimekeeping: Approximating the Time of Day Using the Height of the Sun 201New Functions from Old: Auxiliary Tables 205 * Text 4.8 Al-Khalili, UsingAuxiliary Tables to Find the Hour-angle 207 Trigonometric and AstronomicalInstruments 209 * Text 4.9 Al-Sijzi (?), On an Application of the SineQuadrant 213 Trigonometry in Geography 215 Trigonometry in al-Andalus 217Chapter 5: The West to 1550 223 Transmission from the Arab World 223 AnExample of Transmission: Practical Geometry 224 * Text 5.1 Hugh of St.Victor, Using an Astrolabe to Find the Height of an Object 225 * Text 5.2Finding the Time of Day from the Altitude of the Sun 227 Consolidation andthe Beginnings of Innovation: The Trigonometry of Levi ben Gerson, Richard ofWallingford, and John of Murs 230 * Text 5.3 Levi ben Gerson, The Best StepSize 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 FromPtolemy to Triangles: John of Gmunden, Peurbach, Regiomontanus 247 * Text5.6 Regiomontanus, Finding the Side of a Rectangle from Its Area and AnotherSide 254 * Text 5.7 Regiomontanus, the Angle-angle-angle Case of SolvingRight 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 theCircle: Rheticus, Otho, Pitiscus and the Opus Palatinum 273 ConcludingRemarks 284 Bibliography 287 Index 323