The History of Mathematics: A Brief Course

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. Proto-mathematics 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. Fifth-Century 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 Four-Line 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 ibn-Qurra 302

27.3. Al-Biruni: Trigonometry 304

27.4. Al-Kuhi 305

27.5. Al-Haytham and Ibn-Sahl 305

27.6. Omar Khayyam 307

27.7. Nasir al-Din al-Tusi 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. Sixteenth-Century 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. Non-Euclidean 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 Non-Euclidean 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