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. 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