| Preface |
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xi | |
| The Ancient Heavens |
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1 | (8) |
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9 | (24) |
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9 | (1) |
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The Seqed in Ancient Egypt |
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10 | (2) |
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Text 1.1 Finding the Slope of a Pyramid |
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11 | (1) |
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Babylonian Astronomy, Arc Measurement, and the 360° Circle |
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12 | (6) |
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The Geometric Heavens: Spherics in Ancient Greece |
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18 | (2) |
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A Trigonometry of Small Angles? Aristarchus and Archimedes on Astronomical Dimensions |
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20 | (13) |
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Text 1.2 Aristarchus, the Ratio of the Distances of the Sun and Moon |
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24 | (9) |
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33 | (61) |
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33 | (1) |
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34 | (3) |
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A Model for the Motion of the Sun |
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37 | (4) |
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Text 2.1 Deriving the Eccentricity of the Sun's Orbit |
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39 | (2) |
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41 | (5) |
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The Emergence of Spherical Trigonometry |
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46 | (3) |
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49 | (4) |
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53 | (3) |
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The Foundations of Spherical Trigonometry: Book III of Menelaus's Spherics |
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56 | (7) |
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Text 2.2 Menelaus, Demonstrating Menelaus's Theorem |
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57 | (6) |
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Spherical Trigonometry before Menelaus? |
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63 | (5) |
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68 | (2) |
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70 | (4) |
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Ptolemy's Theorem and the Chord Subtraction/Addition Formulas |
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74 | (2) |
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76 | (1) |
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77 | (1) |
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Chords in Geography: Gnomon Shadow Length Tables |
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77 | (3) |
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Text 2.3 Ptolemy, Finding Gnomon Shadow Lengths |
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78 | (2) |
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Spherical Astronomy in the Almagest |
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80 | (2) |
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Ptolemy on the Motion of the Sun |
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82 | (4) |
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Text 2.4 Ptolemy, Determining the Solar Equation |
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84 | (2) |
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The Motions of the Planets |
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86 | (2) |
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Tabulating Astronomical Functions and the Science of Logistics |
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88 | (2) |
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Trigonometry in Ptolemy's Other Works |
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90 | (3) |
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Text 2.5 Ptolemy, Constructing Latitude Arcs on a Map |
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91 | (2) |
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93 | (1) |
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94 | (41) |
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Transmission from Babylon and Greece |
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94 | (1) |
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95 | (4) |
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Aryabhata's Difference Method of Calculating Sines |
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99 | (3) |
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Text 3.1 Aryabhata, Computing Sines |
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100 | (2) |
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Bhaskara I's Rational Approximation to the Sine |
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102 | (3) |
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105 | (2) |
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Other Trigonometric Identities |
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107 | (4) |
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Text 3.2 Varahamihira, a Half-angle Formula |
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108 | (1) |
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Text 3.3 Brahmagupta, the Law of Sines in Planetary Theory? |
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109 | (2) |
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Brahmagupta's Second-order Interpolation Scheme for Approximating Sines |
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111 | (2) |
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Text 3.4 Brahmagupta, Interpolating Sines |
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111 | (2) |
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Taylor Series for Trigonometric Functions in Madhava's Kerala School |
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113 | (8) |
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Applying Sines and Cosines to Planetary Equations |
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121 | (3) |
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124 | (5) |
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Text 3.5 Varahamihira, Finding the Right Ascension of a Point on the Ecliptic |
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125 | (4) |
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Using Iterative Schemes to Solve Astronomical Problems |
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129 | (4) |
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Text 3.6 Paramesvara, Using Fixed-point Iteration to Compute Sines |
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131 | (2) |
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133 | (2) |
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135 | (88) |
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Foreign Junkets: The Arrival of Astronomy from India |
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135 | (2) |
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137 | (3) |
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Building a Better Sine Table |
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140 | (9) |
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Text 4.1 Al-Samaw'al ibn Yahya al-Maghribi, Why the Circle Should Have 480 Degrees |
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146 | (3) |
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Introducing the Tangent and Other Trigonometric Functions |
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149 | (7) |
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Text 4.2 Abu'l-Rayhan al-Biruni, Finding the Cardinal Points of the Compass |
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152 | (4) |
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Streamlining Astronomical Calculation |
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156 | (2) |
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Text 4.3 Kushyar ibn Labban, Finding the Solar Equation |
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156 | (2) |
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Numerical Techniques: Approximation, Iteration, Interpolation |
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158 | (8) |
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Text 4.4 Ibn Yunus, Interpolating Sine Values |
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164 | (2) |
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Early Spherical Astronomy: Graphical Methods and Analemmas |
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166 | (7) |
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Text 4.5 Al-Khwarizmi, Determining the Ortive Amplitude Geometrically |
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168 | (5) |
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173 | (6) |
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Text 4.6 Al-Kuhi, Finding Rising Times Using the Transversal Theorem |
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175 | (4) |
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179 | (7) |
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Systematizing Spherical Trigonometry: Ibn Mucadh's Determination of the Magnitudes and Nasir al-Din al-Tusi's Transversal Figure |
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186 | (6) |
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Applications to Religious Practice: The Qibla and Other Ritual Needs |
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192 | (9) |
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Text 4.7 Al-Battani, a Simple Approximation to the Qibla |
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195 | (6) |
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Astronomical Timekeeping: Approximating the Time of Day Using the Height of the Sun |
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201 | (4) |
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New Functions from Old: Auxiliary Tables |
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205 | (4) |
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Text 4.8 Al-Khalili, Using Auxiliary Tables to Find the Hour-angle |
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207 | (2) |
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Trigonometric and Astronomical Instruments |
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209 | (6) |
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Text 4.9 Al-Sijzi (?), On an Application of the Sine Quadrant |
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213 | (2) |
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Trigonometry in Geography |
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215 | (2) |
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Trigonometry in al-Andalus |
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217 | (6) |
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223 | (61) |
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Transmission from the Arab World |
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223 | (1) |
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An Example of Transmission: Practical Geometry |
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224 | (6) |
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Text 5.1 Hugh of St. Victor, Using an Astrolabe to Find the Height of an Object |
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225 | (2) |
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Text 5.2 Finding the Time of Day from the Altitude of the Sun |
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227 | (3) |
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Consolidation and the Beginnings of Innovation: The Trigonometry of Levi ben Gerson, Richard of Wallingford, and John of Murs |
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230 | (12) |
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Text 5.3 Levi ben Gerson, The Best Step Size for a Sine Table |
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233 | (4) |
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Text 5.4 Richard of Wallingford, Finding Sin(1°) with Arbitrary Accuracy |
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237 | (5) |
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Interlude: The Marteloio in Navigation |
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242 | (5) |
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Text 5.5 Michael of Rhodes, a Navigational Problem from His Manual |
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244 | (3) |
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From Ptolemy to Triangles: John of Gmunden, Peurbach, Regiomontanus |
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247 | (17) |
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Text 5.6 Regiomontanus, Finding the Side of a Rectangle from Its Area and Another Side |
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254 | (1) |
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Text 5.7 Regiomontanus, the Angle-angle-angle Case of Solving Right Triangles |
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255 | (9) |
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Successors to Regiomontanus: Werner and Copernicus |
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264 | (9) |
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Text 5.8 Copernicus, the Angle-angle-angle Case of Solving Triangles |
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267 | (3) |
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Text 5.9 Copernicus, Determining the Solar Eccentricity |
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270 | (3) |
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Breaking the Circle: Rheticus, Otho, Pitiscus and the Opus Palatinum |
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273 | (11) |
| Concluding Remarks |
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284 | (3) |
| Bibliography |
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287 | (36) |
| Index |
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323 | |