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

Glen van Brummelen
Princeton University Press
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Rob Bradley
, on

This book became a classic almost as soon as it appeared. There had never been a comprehensive history of trigonometry available in English before The Mathematics of the Heavens and the Earth was published. Shortly after it appeared, I began to see discussions and disputes in on-line forums being settled by references to it. However, 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.

Van Brummelen divides the early history of trigonometry into five chapters:

  1. The precursors of trigonometry, up to the time of Archimedes,
  2. The Alexandrian trigonometry of chords, which largely features Ptolemy’s work, but also includes his predecessors Theodosius and Menelaus,
  3. Trigonometry in medieval India, where tables of sines replaced the chords of Alexandria,
  4. The many trigonometric innovations and applications of medieval Islam, a chapter that occupies almost a third of the book, and
  5. The transmission to the West early in the second millennium CE and the subsequent development of trigonometry in Europe through the early 16th century.

Van Brummelen covers the period up to 1550, roughly the era of geocentric astronomy. This means that the discoveries and developments he discusses are largely geometric in flavor and were described or proved using the tortuous didactic mathematics that preceded the development of symbolic algebraic methods in the late 16th and early 17th centuries. Another innovation of the period following 1550 was the logarithm, which had as great an effect on the application of trigonometry to navigational and other practical matters as the new algebraic methods did on its theoretical development. Fortunately, Van Brummelen plans a second volume on the period after 1550, although he makes no promises as to when it will appear, promising only that “the richness of the story to come will make it worth the wait.”

Van Brummelen has tried to design The Mathematics of the Heavens and the Earth so that the topics are as self-contained as possible, making it useful as a reference work or for the reader with particular interests in just one or two of its chapters. That said, if you have the inclination to read the book from cover to cover, you will find that it draws you along at a surprisingly brisk pace, as long as you are able to resist the temptation to read all of the footnotes on a first reading (it’s not always easy to rise above one’s compulsions). To give an idea of just how densely annotated the text is, the chapter on Alexandrian trigonometry has 128 footnotes over the course of 61 pages. The care that Van Brummelen takes to point the reader to sources in nearly every contentious topic or matter of interpretation makes this book an excellent starting point for deeper research into early trigonometry.

Trigonometry largely developed because of its usefulness in astronomical problems. Therefore, if he wished to make every part of this book completely self-contained, Van Brummelen would have had to repeat the basic set-up for problems involving the heavenly bodies: ecliptic, equator, horizon, celestial latitude and longitude, and so on. So he includes an orientation to the ancient, geocentric conception of the heavens in his first chapter, on the precursors of trigonometry in ancient Egypt, Mesopotamia and Greece. Of course, even to distinguish these early discoveries as precursors to trigonometry, as opposed to trigonometry proper, requires a clear definition of trigonometry. Van Brummelen argues there are two essential elements of trigonometry: “a standard quantitative measurement of the inclination of one line to another; and the capacity for, and interest in, calculating the lengths of line segments.” The mathematical results recounted in his first chapter address only one or the other of these elements, so trigonometry in Van Brummelen’s sense only begins with Ptolemy and his immediate predecessors.

The material of Chapters 2 and 5 — Hellenistic and European trigonometry — is usually reasonably well represented, if in a less comprehensive way, in most standard texts on the history of mathematics. Therefore, the greatest value of Van Brummelen’s book to many readers will be his extensive treatment of trigonometry in India and the Islamic world in the period between the decline of the classical world and the 12th century European Renaissance.

In Chapter 3, Van Brummelen mentions the question of transmission from the Greek world to India only briefly, referring the reader to the scholarly arguments on both sides. He surveys the many undeniably original contributions of Indian mathematicians from Aryabhata’s sine tables ca. 500 CE through the discovery of the sine and cosine series by Madhava and the Kerala school in the 14th century. Throughout his book, Van Brummelen generally uses modern notation to explain the mathematics of the past. This seems essential when discussing medieval India, where “formulas” were expressed in verse and calculations were often quite intricate, involving such things as iterative numerical methods and higher order finite differences.

A particularly welcome feature of this book is the use of brief pieces of original source material (in translation) to illustrate the achievements of various early mathematicians in trigonometry. There are 31 of these “Texts” ranging from Problem 58 of the Rhind Mathematical Papyrus (finding the “slope” of a pyramid), through Copernicus’ determination of solar eccentricity. These text passages are illustrated with modern diagrams and followed by careful explanations using modern notation, with Van Brummelen providing additional details as needed. The selection is particularly rich in Chapter 4, from Al-Samaw’ai ibn Yahya al-Maghribi arguing for the division of the circle into 480 parts (because the value of the chord of 1degree can only be approximated, whereas the value of ¾ of a degree can be calculated precisely using the half-angle formula), to al-Sijzi determining the ascension of the signs of the zodiac using a tool called a sine quadrant.

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. It could be used as a reference text for the standard upper-level history of mathematics course, as well as a source for student projects in such a course. I can also see it being used for a special topics course in the history of trigonometry, especially once the promised companion volume covering the period after 1550 appears. Finally, with 36 pages of references and extensive annotation throughout the book, it seems destined to be the starting point for a lot of new research in the years ahead.

Rob Bradley is professor of mathematics at Adelphi University. His main research interest is the history of mathematics. He is currently the Chairman of the History of Mathematics Special Interest Group of the MAA (HOMSIGMAA) and president of the Euler Society.


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