The history of mathematics: a brief course
Gespeichert in:
1. Verfasser: | |
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Format: | Buch |
Sprache: | English |
Veröffentlicht: |
Hoboken, NJ
Wiley-Interscience
2005
|
Ausgabe: | 2. ed. |
Schlagworte: | |
Online-Zugang: | Inhaltsverzeichnis |
Beschreibung: | Includes bibliographical references and indexes |
Beschreibung: | XVIII, 607 S. Ill. |
ISBN: | 0471444596 |
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adam_text | Contents
Preface xv
Part 1. The World of Mathematics and the Mathematics of the
World 1
Chapter 1. The Origin and Prehistory of Mathematics 3
1. Numbers 4
1.1. Animals use of numbers 5
1.2. Young children s use of numbers 5
1.3. Archaeological evidence of counting 6
2. Continuous magnitudes 6
2.1. Perception of shape by animals 7
2.2. Children s concepts of space 8
2.3. Geometry in arts and crafts 9
3. Symbols 9
4. Mathematical inference 12
4.1. Visual reasoning 12
4.2. Chance and probability 13
Questions and problems 14
Chapter 2. Mathematical Cultures I 19
1. The motives for creating mathematics 19
1.1. Pure versus applied mathematics 19
2. India 21
2.1. The Sulva Sutras 22
2.2. Buddhist and Jaina mathematics 23
2.3. The Bakshali Manuscript 23
2.4. The siddhantas 23
2.5. Aryabhata I 24
2.6. Brahmagupta 25
2.7. Bhaskara II 25
2.8. Muslim India 26
2.9. Indian mathematics in the colonial period and after 26
3. China 27
3.1. Works and authors 29
3.2. China s encounter with Western mathematics 32
4. Ancient Egypt 34
5. Mesopotamia 35
6. The Maya 37
6.1. The Dresden Codex 37
V
vi CONTENTS
Questions and problems 38
Chapter 3. Mathematical Cultures II 41
1. Greek and Roman mathematics 41
1.1. Sources 42
1.2. General features of Greek mathematics 43
1.3. Works and authors 45
2. Japan 50
2.1. Chinese influence and calculating devices 51
2.2. Japanese mathematicians and their works 51
3. The Muslims 54
3.1. Islamic science in general 54
3.2. Some Muslim mathematicians and their works 56
4. Europe 58
4.1. Monasteries, schools, and universities 58
4.2. The high Middle Ages 59
4.3. Authors and works 59
5. North America 62
5.1. The United States and Canada before 1867 63
5.2. The Canadian Federation and post Civil War United States 66
5.3. Mexico 69
6. Australia and New Zealand 70
6.1. Colonial mathematics 70
7. The modern era 72
7.1. Educational institutions 72
7.2. Mathematical societies 73
7.3. Journals 73
Questions and problems 73
Chapter 4. Women Mathematicians 75
1. Individual achievements and obstacles to achievement 76
1.1. Obstacles to mathematical careers for women 76
2. Ancient women mathematicians 80
3. Modern European women 81
3.1. Continental mathematicians 82
3.2. Nineteenth century British women 85
3.3. Four modern pioneers 88
4. American women 100
5. The situation today 104
Questions and problems 105
Part 2. Numbers 109
Chapter 5. Counting 111
1. Number words 111
2. Bases for counting 113
2.1. Decimal systems 113
2.2. Nondecimal systems 114
3. Counting around the world 116
3.1. Egypt 116
CONTENTS vii
3.2. Mesopotamia 116
3.3. India 118
3.4. China 118
3.5. Greece and Rome 119
3.6. The Maya 121
4. What was counted? 122
4.1. Calendars 122
4.2. Weeks 125
Questions and problems 127
Chapter 6. Calculation 129
1. Egypt 129
1.1. Multiplication and division 130
1.2. Parts 131
1.3. Practical problems 134
2. China 135
2.1. Fractions and roots 136
2.2. The Jiu Zhang Suanshu 138
3. India 139
4. Mesopotamia 140
5. The ancient Greeks 142
6. The Islamic world 143
7. Europe 143
8. The value of calculation 145
9. Mechanical methods of computation 146
9.1. Software: prosthaphaeresis and logarithms 146
9.2. Hardware: slide rules and calculating machines 149
9.3. The effects of computing power 153
Questions and problems 154
Chapter 7. Ancient Number Theory 159
1. Plimpton 322 159
2. Ancient Greek number theory 164
2.1. The Arithmetica of Nicomachus 165
2.2. Euclid s number theory 168
2.3. The Arithmetica of Diophantus 170
3. China 172
4. India 175
4.1. Varahamihira s mystical square 175
4.2. Aryabhatal 175
4.3. Brahmagupta 175
4.4. Bhaskara II 178
5. The Muslims 179
6. Japan 180
7. Medieval Europe 181
Questions and problems 182
Chapter 8. Numbers and Number Theory in Modern Mathematics 187
1. Modern number theory 187
1.1. Fermat 187
viii CONTENTS
1.2. Euler 188
1.3. Lagrange 190
1.4. Legendre 191
1.5. Gauss 192
1.6. Dirichlet 193
1.7. Riemann 194
1.8. Fermat s last theorem 195
1.9. The prime number theorem 196
2. Number systems 197
2.1. Negative numbers and zero 197
2.2. Irrational and imaginary numbers 199
2.3. Imaginary and complex numbers 206
2.4. Infinite numbers 209
3. Combinatorics 210
3.1. Summation rules 210
Questions and problems 217
Part 3. Color Plates 221
Part 4. Space 231
Chapter 9. Measurement 233
1. Egypt 234
1.1. Areas 235
1.2. Volumes 239
2. Mesopotamia 241
2.1. The Pythagorean theorem 242
2.2. Plane figures 243
2.3. Volumes 244
3. China 244
3.1. The Zhou Bi Suan Jing 244
3.2. The Jiu Zhang Suanshu 247
3.3. The Sun Zi Suan Jing 248
3.4. Liu Hui 249
3.5. Zu Chongzhi 250
4. Japan 252
4.1. The challenge problems 252
4.2. Beginnings of the calculus in Japan 253
5. India 257
5.1. Aryabhata I 257
5.2. Brahmagupta 262
Questions and problems 264
Chapter 10. Euclidean Geometry 269
1. The earliest Greek geometry 269
1.1. Thales 270
1.2. Pythagoras and the Pythagoreans 271
1.3. Pythagorean geometry 272
1.4. Challenges to Pythagoreanism: unsolved problems 274
CONTENTS ix
1.5. Challenges to Pythagoreanism: the paradoxes of Zeno of Elea 283
1.6. Challenges to Pythagoreanism: incommensurables 284
1.7. The influence of Plato 285
1.8. Eudoxan geometry 287
1.9. Aristotle 293
2. Euclid 296
2.1. The Elements 296
2.2. The Data 299
3. Archimedes 299
3.1. The area of a sphere 301
3.2. The Method 302
4. Apollonius 304
4.1. History of the Conies 305
4.2. Contents of the Conies 305
4.3. Apollonius definition of the conic sections 306
4.4. Foci and the three and four line locus 308
Questions and problems 310
Chapter 11. Post Euclidean Geometry 317
1. Hellenistic geometry 318
1.1. Zenodorus 318
1.2. The parallel postulate 319
1.3. Heron 320
1.4. Pappus 322
2. Roman geometry 325
2.1. Roman civil engineering 327
3. Medieval geometry 328
3.1. Late Medieval and Renaissance geometry 330
4. Geometry in the Muslim world 332
4.1. The parallel postulate 333
4.2. Thabit ibn Qurra 333
4.3. Al Kuhi 335
4.4. Al Haytham 335
4.5. Omar Khayyam 336
4.6. Nasir al Din al Tusi 337
5. Non Euclidean geometry 338
5.1. Girolamo Saccheri 339
5.2. Lambert and Legendre 341
5.3. Gauss 342
5.4. Lobachevskii and Janos Bolyai 343
5.5. The reception of non Euclidean geometry 346
5.6. Foundations of geometry 348
6. Questions and problems 348
Chapter 12. Modern Geometries 351
1. Analytic and algebraic geometry 351
1.1. Fermat 351
1.2. Descartes 352
1.3. Newton s classification of curves 355
x CONTENTS
1.4. Algebraic geometry 355
2. Projective and descriptive geometry 356
2.1. Projective properties 356
2.2. The Renaissance artists 357
2.3. Girard Desargues 360
2.4. Blaise Pascal 364
2.5. Newton s degree preserving mappings 364
2.6. Charles Brianchon 365
2.7. Monge and his school 366
2.8. Jacob Steiner 367
2.9. August Ferdinand Mobius 368
2.10. Julius Pliicker 369
2.11. Arthur Cayley 371
3. Differential geometry 371
3.1. Huygens 371
3.2. Newton 373
3.3. Leibniz 374
3.4. The eighteenth century 375
3.5. Gauss 376
3.6. The French and British geometers 379
3.7. Riemann 380
3.8. The Italian geometers 383
4. Topology 385
4.1. Early combinatorial topology 386
4.2. Riemann 386
4.3. Mobius 388
4.4. Poincare s Analysis situs 388
4.5. Point set topology 390
Questions and problems 393
Part 5. Algebra 397
Chapter 13. Problems Leading to Algebra 399
1. Egypt 399
2. Mesopotamia 401
2.1. Linear and quadratic problems 401
2.2. Higher degree problems 403
3. India 404
3.1. Jaina algebra 404
3.2. The Bakshali Manuscript 404
4. China 405
4.1. The Jiu Zhang Suanshu 405
4.2. The Suanshu Shu 405
4.3. The Sun Zi Suan Jing 406
4.4. Zhang Qiujian 406
Questions and problems 407
Chapter 14. Equations and Algorithms 409
1. The Arithmetica of Diophantus 409
CONTENTS xi
1.1. Diophantine equations 410
1.2. General characteristics of the Arithmetica 410
1.3. Determinate problems 411
1.4. The significance of the Arithmetica 412
1.5. The view of Jacob Klein 412
2. China 413
2.1. Linear equations 413
2.2. Quadratic equations 413
2.3. Cubic equations 414
2.4. The numerical solution of equations 415
3. Japan 417
3.1. Seki Kowa 418
4. Hindu algebra 420
4.1. Brahmagupta 420
4.2. Bhaskara II 421
5. The Muslims 422
5.1. Al Khwarizmi 423
5.2. Abu Kamil 425
5.3. Omar Khayyam 425
5.4. Sharaf al Din al Muzaffar al Tusi 426
6. Europe 427
6.1. Leonardo of Pisa (Fibonacci) 428
6.2. Jordanus Nemorarius 429
6.3. The fourteenth and fifteenth centuries 429
6.4. Chuquet 430
6.5. Solution of cubic and quartic equations 431
6.6. Consolidation 432
Questions and problems 434
Chapter 15. Modern Algebra 437
1. Theory of equations 437
1.1. Albert Girard 437
1.2. Tschirnhaus transformations 438
1.3. Newton, Leibniz, and the Bernoullis 440
1.4. Euler, d Alembert, and Lagrange 440
1.5. Gauss and the fundamental theorem of algebra 443
1.6. Ruffini 444
1.7. Cauchy 444
1.8. Abel 446
1.9. Galois 447
2. Algebraic structures 451
2.1. Fields, rings, and algebras 451
2.2. Abstract groups 454
2.3. Number systems 458
Questions and problems 459
Part 6. Analysis 461
Chapter 16. The Calculus 463
xii CONTENTS
1. Prelude to the calculus 463
1.1. Tangent and maximum problems 464
1.2. Lengths, areas, and volumes 465
1.3. The relation between tangents and areas 467
1.4. Infinite series and products 467
2. Newton and Leibniz 468
2.1. Isaac Newton 468
2.2. Gottfried Wilhelm von Leibniz 470
2.3. The disciples of Newton and Leibniz 472
3. Branches and roots of the calculus 475
3.1. Ordinary differential equations 475
3.2. Partial differential equations 477
3.3. Calculus of variations 478
3.4. Foundations of the calculus 483
Questions and problems 487
Chapter 17. Real and Complex Analysis 489
1. Complex analysis 489
1.1. Algebraic integrals 490
1.2. Cauchy 493
1.3. Riemann 494
1.4. Weierstrass 495
2. Real analysis 496
2.1. Fourier series, functions, and integrals 496
2.2. Completeness of the real numbers 502
2.3. Uniform convergence and continuity 503
2.4. General integrals and discontinuous functions 503
2.5. The abstract and the concrete 504
2.6. Discontinuity as a positive property 506
Questions and problems 507
Part 7. Mathematical Inferences 509
Chapter 18. Probability and Statistics 511
1. Probability 511
1.1. Cardano 512
1.2. Fermat and Pascal 513
1.3. Huygens 514
1.4. Leibniz 515
1.5. The Ars Conjectandi of Jakob Bernoulli 515
1.6. De Moivre 517
1.7. Laplace 520
1.8. Legendre 521
1.9. Gauss 521
1.10. Philosophical issues 522
1.11. Large numbers and limit theorems 523
2. Statistics 524
2.1. Quetelet 525
2.2. Statistics in physics 525
CONTENTS xiii
2.3. The metaphysics of probability and statistics 527
2.4. Correlations and statistical inference 528
Questions and problems 531
Chapter 19. Logic and Set Theory 535
1. Logic 535
1.1. From algebra to logic 535
1.2. Symbolic calculus 539
1.3. Boole s Mathematical Analysis of Logic 539
1.4. Boole s Laws of Thought 541
1.5. Venn 542
1.6. Jevons 544
2. Set theory 544
2.1. Technical background 544
2.2. Cantor s work on trigonometric series 545
2.3. The reception of set theory 547
2.4. Existence and the axiom of choice 548
2.5. Doubts about set theory 551
3. Philosophies of mathematics 552
3.1. Paradoxes 553
3.2. Formalism 554
3.3. Intuitionism 555
3.4. Mathematical practice 556
Questions and problems 557
Literature 561
Subject Index 581
Name Index 599
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title | The history of mathematics a brief course |
title_auth | The history of mathematics a brief course |
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