The History of Mathematics: A Brief Course, 3rd EditionISBN: 9781118217566
648 pages
November 2012

Praise for the Second Edition
"An amazing assemblage of worldwide contributions in mathematics and, in addition to use as a course book, a valuable resource . . . essential."
—CHOICE
This Third Edition of The History of Mathematics examines the elementary arithmetic, geometry, and algebra of numerous cultures, tracing their usage from Mesopotamia, Egypt, Greece, India, China, and Japan all the way to Europe during the Medieval and Renaissance periods where calculus was developed.
Aimed primarily at undergraduate students studying the history of mathematics for science, engineering, and secondary education, the book focuses on three main ideas: the facts of who, what, when, and where major advances in mathematics took place; the type of mathematics involved at the time; and the integration of this information into a coherent picture of the development of mathematics. In addition, the book features carefully designed problems that guide readers to a fuller understanding of the relevant mathematics and its social and historical context. Chapterend exercises, numerous photographs, and a listing of related websites are also included for readers who wish to pursue a specialized topic in more depth. Additional features of The History of Mathematics, Third Edition include:
 Material arranged in a chronological and cultural context
 Specific parts of the history of mathematics presented as individual lessons
 New and revised exercises ranging between technical, factual, and integrative
 Individual PowerPoint presentations for each chapter and a bank of homework and test questions (in addition to the exercises in the book)
 An emphasis on geography, culture, and mathematics
In addition to being an ideal coursebook for undergraduate students, the book also serves as a fascinating reference for mathematically inclined individuals who are interested in learning about the history of mathematics.
PREFACE xxiii
Changes from the Second Edition xxiii
Elementary Texts on the History of Mathematics xxiv
PART I. WHAT IS MATHEMATICS?
Contents of Part I 1
1. Mathematics and its History 3
1.1. Two Ways to Look at the History of Mathematics 3
1.2. The Origin of Mathematics 5
1.3. The Philosophy of Mathematics 8
1.4. Our Approach to the History of Mathematics 11
2. Protomathematics 14
2.1. Number 14
2.2. Shape 16
2.3. Symbols 18
2.4. Mathematical Reasoning 20
PART II. THE MIDDLE EAST, 2000–1500 BCE
Contents of Part II 25
3. Overview of Mesopotamian Mathematics 27
3.1. A Sketch of Two Millennia of Mesopotamian History 27
3.2. Mathematical Cuneiform Tablets 29
3.3. Systems of Measuring and Counting 30
3.4. The Mesopotamian Numbering System 31
4. Computations in Ancient Mesopotamia 38
4.1. Arithmetic 38
4.2. Algebra 40
5. Geometry in Mesopotamia 46
5.1. The Pythagorean Theorem 46
5.2. Plane Figures 48
5.3. Volumes 49
5.4. Plimpton 322 49
6. Egyptian Numerals and Arithmetic 56
6.1. Sources 56
6.2. The Rhind Papyrus 58
6.3. Egyptian Arithmetic 58
6.4. Computation 59
7. Algebra and Geometry in Ancient Egypt 66
7.1. Algebra Problems in the Rhind Papyrus 66
7.2. Geometry 68
7.3. Areas 69
PART III. GREEK MATHEMATICS FROM 500 BCE TO 500 CE
Contents of Part III 77
8. An Overview of Ancient Greek Mathematics 79
8.1. Sources 80
8.2. General Features of Greek Mathematics 82
8.3. Works and Authors 87
9. Greek Number Theory 91
9.1. The Euclidean Algorithm 92
9.2. The Arithmetica of Nicomachus 93
9.3. Euclid’s Number Theory 97
9.4. The Arithmetica of Diophantus 97
10. FifthCentury Greek Geometry 103
10.1. “Pythagorean” Geometry 103
10.2. Challenge No. 1: Unsolved Problems 106
10.3. Challenge No. 2: The Paradoxes of Zeno of Elea 107
10.4. Challenge No. 3: Irrational Numbers and Incommensurable Lines 108
11. Athenian Mathematics I: The Classical Problems 115
11.1. Squaring the Circle 116
11.2. Doubling the Cube 117
11.3. Trisecting the Angle 122
12. Athenian Mathematics II: Plato and Aristotle 128
12.1. The Influence of Plato 128
12.2. Eudoxan Geometry 130
12.3. Aristotle 134
13. Euclid of Alexandria 140
13.1. The Elements 140
13.2. The Data 144
14. Archimedes of Syracuse 148
14.1. The Works of Archimedes 149
14.2. The Surface of a Sphere 150
14.3. The Archimedes Palimpsest 153
14.4. Quadrature of the Parabola 155
15. Apollonius of Perga 160
15.1. History of the Conics 161
15.2. Contents of the Conics 162
15.3. Foci and the Three and FourLine Locus 165
16. Hellenistic and Roman Geometry 169
16.1. Zenodorus 169
16.2. The Parallel Postulate 171
16.3. Heron 172
16.4. Roman Civil Engineering 174
17. Ptolemy’s Geography and Astronomy 177
17.1. Geography 177
17.2. Astronomy 180
17.3. The Almagest 184
18. Pappus and the Later Commentators 190
18.1. The Collection of Pappus 190
18.2. The Later Commentators: Theon and Hypatia 196
PART IV. INDIA, CHINA, AND JAPAN 500 BCE–1700 CE
Contents of Part IV 201
19. Overview of Mathematics in India 203
19.1. The Sulva Sutras 205
19.2. Buddhist and Jain Mathematics 206
19.3. The Bakshali Manuscript 206
19.4. The Siddhantas 206
19.5. Hindu–Arabic Numerals 206
19.6. Aryabhata I 207
19.7. Brahmagupta 208
19.8. Bhaskara II 209
19.9. Muslim India 210
19.10. Indian Mathematics in the Colonial Period and After 210
20. From the Vedas to Aryabhata I 213
20.1. Problems from the Sulva Sutras 213
20.2. Aryabhata I: Geometry and Trigonometry 219
21. Brahmagupta, the Kuttaka, and Bhaskara II 227
21.1. Brahmagupta’s Plane and Solid Geometry 227
21.2. Brahmagupta’s Number Theory and Algebra 228
21.3. The Kuttaka 230
21.4. Algebra in the Works of Bhaskara II 233
21.5. Geometry in the Works of Bhaskara II 235
22. Early Classics of Chinese Mathematics 239
22.1. Works and Authors 240
22.2. China’s Encounter with Western Mathematics 243
22.3. The Chinese Number System 244
22.4. Algebra 246
22.5. Contents of the Jiu Zhang Suan Shu 247
22.6. Early Chinese Geometry 249
23. Later Chinese Algebra and Geometry 255
23.1. Algebra 255
23.2. Later Chinese Geometry 262
24. Traditional Japanese Mathematics 267
24.1. Chinese Influence and Calculating Devices 267
24.2. Japanese Mathematicians and Their Works 268
24.3. Japanese Geometry and Algebra 270
24.4. Sangaku 277
PART V. ISLAMIC MATHEMATICS, 800–1500
Contents of Part V 281
25. Overview of Islamic Mathematics 283
25.1. A Brief Sketch of the Islamic Civilization 283
25.2. Islamic Science in General 285
25.3. Some Muslim Mathematicians and Their Works 287
26. Islamic Number Theory and Algebra 292
26.1. Number Theory 292
26.2. Algebra 294
27. Islamic Geometry 302
27.1. The Parallel Postulate 302
27.2. Thabit ibnQurra 302
27.3. AlBiruni: Trigonometry 304
27.4. AlKuhi 305
27.5. AlHaytham and IbnSahl 305
27.6. Omar Khayyam 307
27.7. Nasir alDin alTusi 308
PART VI. EUROPEAN MATHEMATICS, 500–1900
Contents of Part VI 311
28. Medieval and Early Modern Europe 313
28.1. From the Fall of Rome to the Year 1200 313
28.2. The High Middle Ages 318
28.3. The Early Modern Period 321
28.4. Northern European Advances 322
29. European Mathematics: 1200–1500 324
29.1. Leonardo of Pisa (Fibonacci) 324
29.2. Hindu–Arabic Numerals 328
29.3. Jordanus Nemorarius 329
29.4. Nicole d’Oresme 330
29.5. Trigonometry: Regiomontanus and Pitiscus 331
29.6. A Mathematical Skill: Prosthaphæresis 333
29.7. Algebra: Pacioli and Chuquet 335
30. SixteenthCentury Algebra 338
30.1. Solution of Cubic and Quartic Equations 338
30.2. Consolidation 340
30.3. Logarithms 343
30.4. Hardware: Slide Rules and Calculating Machines 345
31. Renaissance Art and Geometry 348
31.1. The Greek Foundations 348
31.2. The Renaissance Artists and Geometers 349
31.3. Projective Properties 350
32. The Calculus Before Newton and Leibniz 358
32.1. Analytic Geometry 358
32.2. Components of the Calculus 363
33. Newton and Leibniz 373
33.1. Isaac Newton 373
33.2. Gottfried Wilhelm von Leibniz 375
33.3. The Disciples of Newton and Leibniz 379
33.4. Philosophical Issues 379
33.5. The Priority Dispute 381
33.6. Early Textbooks on Calculus 382
34. Consolidation of the Calculus 386
34.1. Ordinary Differential Equations 387
34.2. Partial Differential Equations 390
34.3. Calculus of Variations 391
34.4. Foundations of the Calculus 397
PART VII. SPECIAL TOPICS
Contents of Part VII 404
35. Women Mathematicians 405
35.1. Sof’ya Kovalevskaya 406
35.2. Grace Chisholm Young 408
35.3. Emmy Noether 411
36. Probability 417
36.1. Cardano 418
36.2. Fermat and Pascal 419
36.3. Huygens 420
36.4. Leibniz 420
36.5. The Ars Conjectandi of James Bernoulli 421
36.6. De Moivre 423
36.7. The Petersburg Paradox 424
36.8. Laplace 425
36.9. Legendre 426
36.10. Gauss 426
36.11. Philosophical Issues 427
36.12. Large Numbers and Limit Theorems 428
37. Algebra from 1600 to 1850 433
37.1. Theory of Equations 433
37.2. Euler, D’Alembert, and Lagrange 437
37.3. The Fundamental Theorem of Algebra and Solution by Radicals 439
38. Projective and Algebraic Geometry and Topology 448
38.1. Projective Geometry 448
38.2. Algebraic Geometry 453
38.3. Topology 456
39. Differential Geometry 464
39.1. Plane Curves 464
39.2. The Eighteenth Century: Surfaces 468
39.3. Space Curves: The French Geometers 469
39.4. Gauss: Geodesics and Developable Surfaces 469
39.5. The French and British Geometers 473
39.6. Grassmann and Riemann: Manifolds 473
39.7. Differential Geometry and Physics 476
39.8. The Italian Geometers 477
40. NonEuclidean Geometry 481
40.1. Saccheri 482
40.2. Lambert and Legendre 484
40.3. Gauss 485
40.4. The First Treatises 486
40.5. Lobachevskii’s Geometry 487
40.6. J´anos B´olyai 489
40.7. The Reception of NonEuclidean Geometry 489
40.8. Foundations of Geometry 491
41. Complex Analysis 495
41.1. Imaginary and Complex Numbers 495
41.2. Analytic Function Theory 500
41.3. Comparison of the Three Approaches 508
42. Real Numbers, Series, and Integrals 511
42.1. Fourier Series, Functions, and Integrals 512
42.2. Fourier Series 514
42.3. Fourier Integrals 516
42.4. General Trigonometric Series 518
43. Foundations of Real Analysis 521
43.1. What is a Real Number? 521
43.2. Completeness of the Real Numbers 525
43.3. Uniform Convergence and Continuity 525
43.4. General Integrals and Discontinuous Functions 526
43.5. The Abstract and the Concrete 527
43.6. Discontinuity as a Positive Property 529
44. Set Theory 532
44.1. Technical Background 532
44.2. Cantor's Work on Trigonometric Series 533
44.3. The Reception of Set Theory 536
44.4. Existence and the Axiom of Choice 537
45. Logic 542
45.1. From Algebra to Logic 542
45.2. Symbolic Calculus 545
45.3. Boole’s Mathematical Analysis of Logic 546
45.4. Boole’s Laws of Thought 547
45.5. Jevons 548
45.6. Philosophies of Mathematics 548
45.7. Doubts About Formalized Mathematics: Gödel’s Theorems 554
Literature 559
Name Index 575
Subject Index
The History of Mathematics: A Brief Course, 3rd Edition (US $129.00)
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