The Mathematics of the Heavens and the Earth

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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

There does not seem to have been a book-length history of trigonometry in English before this fine book. Van Brummelen takes us from the unnamed Egyptians and Babylonians who created trigonometry to the subject's first few centuries in Europe. In between, he deftly traces how it was studied by the astronomers Hipparchus and Ptolemy in classical Greece, and later by a host of scholars in India and the Islamic world. -- John H. Conway, coauthor of "The Book of Numbers" This book is the first detailed history of trigonometry in more than half a century, and it far surpasses any earlier attempts. The Mathematics of the Heavens and the Earth is an extremely important contribution to scholarship. It will be the definitive history of trigonometry for years to come. There is nothing like this out there. -- Victor J. Katz, professor emeritus, University of the District of Columbia A pleasure to read. The Mathematics of the Heavens and the Earth is destined to become the standard reference on the history of trigonometry for the foreseeable future. Although other authors have attempted to tell the story, I know of no other book that has either the breadth or the depth of this one. Van Brummelen is one of the leading experts in the world on this subject. -- Dennis Duke, Florida State University Van Brummelen presents a history of trigonometry from the earliest times to the end of the sixteenth century. He has produced a work that rises to the highest standards of scholarship but never strays into pedantry. His extensive bibliography cites every work of consequence for the history of trigonometry, copious footnotes and diagrams illuminate the text, and reproductions from old printed works add interest and texture to the narrative. -- J. Lennart Berggren, professor emeritus, Simon Fraser University This book presents, for the first time in more than a century, a concise history of plane and spherical trigonometry, an important field within applied mathematics. It will appeal to a wide audience thanks to the pleasant style in which it is written, but at the same time it adheres to a very high scholarly standard. -- Benno van Dalen, Ludwig Maximilians University, Munich

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".

"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 "[T]his new and comprehensive history of trigonometry is more than welcome--even more so because it is the first in English... [T]his book will be appreciated by many with an interest--general or more specific--in the history of mathematics."--Steven Wepster, Centaurus "[T]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 "I highly recommend the book to all those interested in the way in which the ancient people solve their practical problems and hope that the next volume of this interesting history of spherical and plane trigonometry will appear soon."--Cristina Blaga, Studia Mathematica

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