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目次

Foreword by Isaac Asimov xi Preface to the Third Edition xiii Preface to the Second Edition xv Preface to the First Edition xvii 1 Traces 1 Concepts and Relationships 1 Early Number Bases 3 Number Language and Counting 5 Spatial Relationships 6 2 Ancient Egypt 8 The Era and the Sources 8 Numbers and Fractions 10 Arithmetic Operations 12 “Heap” Problems 13 Geometric Problems 14 Slope Problems 18 Arithmetic Pragmatism 19 3 Mesopotamia 21 The Era and the Sources 21 Cuneiform Writing 22 Numbers and Fractions: Sexagesimals 23 Positional Numeration 23 Sexagesimal Fractions 25 Approximations 25 Tables 26 Equations 28 Measurements: Pythagorean Triads 31 Polygonal Areas 35 Geometry as Applied Arithmetic 36 4 Hellenic Traditions 40 The Era and the Sources 40 Thales and Pythagoras 42 Numeration 52 Arithmetic and Logistic 55 Fifth–Century Athens 56 Three Classical Problems 57 Quadrature of Lunes 58 Hippias of Elis 61 Philolaus and Archytas of Tarentum 63 Incommensurability 65 Paradoxes of Zeno 67 Deductive Reasoning 70 Democritus of Abdera 72 Mathematics and the Liberal Arts 74 The Academy 74 Aristotle 88 5 Euclid of Alexandria 90 Alexandria 90 Lost Works 91 Extant Works 91 The Elements 93 6 Archimedes of Syracuse 109 The Siege of Syracuse 109 On the Equilibriums of Planes 110 On Floating Bodies 111 The Sand–Reckoner 112 Measurement of the Circle 113 On Spirals 113 Quadrature of the Parabola 115 On Conoids and Spheroids 116 On the Sphere and Cylinder 118 Book of Lemmas 120 Semiregular Solids and Trigonometry 121 The Method 122 7 Apollonius of Perge 127 Works and Tradition 127 Lost Works 128 Cycles and Epicycles 129 The Conics 130 8 Crosscurrents 142 Changing Trends 142 Eratosthenes 143 Angles and Chords 144 Ptolemy’s Almagest 149 Heron of Alexandria 156 The Decline of Greek Mathematics 159 Nicomachus of Gerasa 159 Diophantus of Alexandria 160 Pappus of Alexandria 164 The End of Alexandrian Dominance 170 Proclus of Alexandria 171 Boethius 171 Athenian Fragments 172 Byzantine Mathematicians 173 9 Ancient and Medieval China 175 The Oldest Known Texts 175 The Nine Chapters 176 Rod Numerals 177 The Abacus and Decimal Fractions 178 Values of Pi 180 Thirteenth–Century Mathematics 182 10 Ancient and Medieval India 186 Early Mathematics in India 186 The Sulbasutras 187 The Siddhantas 188 Aryabhata 189 Numerals 191 Trigonometry 193 Multiplication 194 Long Division 195 Brahmagupta 197 Indeterminate Equations 199 Bhaskara 200 Madhava and the Keralese School 202 11 The Islamic Hegemony 203 Arabic Conquests 203 The House of Wisdom 205 Al–Khwarizmi 206 ‘Abd Al–Hamid ibn–Turk 212 Thabit ibn–Qurra 213 Numerals 214 Trigonometry 216 Tenth– and Eleventh–Century Highlights 216 Omar Khayyam 218 The Parallel Postulate 220 Nasir al–Din al–Tusi 220 Al–Kashi 221 12 The Latin West 223 Introduction 223 Compendia of the Dark Ages 224 Gerbert 224 The Century of Translation 226 Abacists and Algorists 227 Fibonacci 229 Jordanus Nemorarius 232 Campanus of Novara 233 Learning in the Thirteenth Century 235 Archimedes Revived 235 Medieval Kinematics 236 Thomas Bradwardine 236 Nicole Oresme 238 The Latitude of Forms 239 Infinite Series 241 Levi ben Gerson 242 Nicholas of Cusa 243 The Decline of Medieval Learning 243 13 The European Renaissance 245 Overview 245 Regiomontanus 246 Nicolas Chuquet’s Triparty 249 Luca Pacioli’s Summa 251 German Algebras and Arithmetics 253 Cardan’s Ars Magna 255 Rafael Bombelli 260 Robert Recorde 262 Trigonometry 263 Geometry 264 Renaissance Trends 271 François Viète 273 14 Early Modern Problem Solvers 282 Accessibility of Computation 282 Decimal Fractions 283 Notation 285 Logarithms 286 Mathematical Instruments 290 Infinitesimal Methods: Stevin 296 Johannes Kepler 296 15 Analysis Synthesis the Infinite and Numbers 300 Galileo’s Two New Sciences 300 Bonaventura Cavalieri 303 Evangelista Torricelli 306 Mersenne’s Communicants 308 René Descartes 309 Fermat’s Loci 320 Gregory of St. Vincent 325 The Theory of Numbers 326 Gilles Persone de Roberval 329 Girard Desargues and Projective Geometry 330 Blaise Pascal 332 Philippe de Lahire 337 Georg Mohr 338 Pietro Mengoli 338 Frans van Schooten 339 Jan de Witt 340 Johann Hudde 341 René François de Sluse 342 Christiaan Huygens 342 16 British Techniques and Continental Methods 348 John Wallis 348 James Gregory 353 Nicolaus Mercator and William Brouncker 355 Barrow’s Method of Tangents 356 Newton 358 Abraham De Moivre 372 Roger Cotes 375 James Stirling 376 Colin Maclaurin 376 Textbooks 380 Rigor and Progress 381 Leibniz 382 The Bernoulli Family 390 Tschirnhaus Transformations 398 Solid Analytic Geometry 399 Michel Rolle and Pierre Varignon 400 The Clairauts 401 Mathematics in Italy 402 The Parallel Postulate 403 Divergent Series 404 17 Euler 406 The Life of Euler 406 Notation 408 Foundation of Analysis 409 Logarithms and the Euler Identities 413 Differential Equations 414 Probability 416 The Theory of Numbers 417 Textbooks 418 Analytic Geometry 419 The Parallel Postulate: Lambert 420 18 Pre– to Postrevolutionary France 423 Men and Institutions 423 The Committee on Weights and Measures 424 D’Alembert 425 Bézout 427 Condorcet 429 Lagrange 430 Monge 433 Carnot 438 Laplace 443 Legendre 446 Aspects of Abstraction 449 Paris in the 1820s 449 Fourier 450 Cauchy 452 Diffusion 460 19 Gauss 464 Nineteenth–Century Overview 464 Gauss: Early Work 465 Number Theory 466 Reception of the Disquisitiones Arithmeticae 469 Astronomy 470 Gauss’s Middle Years 471 Differential Geometry 472 Gauss’s Later Work 473 Gauss’s Influence 474 20 Geometry 483 The School of Monge 483 Projective Geometry: Poncelet and Chasles 485 Synthetic Metric Geometry: Steiner 487 Synthetic Nonmetric Geometry: von Staudt 489 Analytic Geometry 489 Non–Euclidean Geometry 494 Riemannian Geometry 496 Spaces of Higher Dimensions 498 Felix Klein 499 Post–Riemannian Algebraic Geometry 501 21 Algebra 504 Introduction 504 British Algebra and the Operational Calculus of Functions 505 Boole and the Algebra of Logic 506 Augustus De Morgan 509 William Rowan Hamilton 510 Grassmann and Ausdehnungslehre 512 Cayley and Sylvester 515 Linear Associative Algebras 519 Algebraic Geometry 520 Algebraic and Arithmetic Integers 520 Axioms of Arithmetic 522 22 Analysis 526 Berlin and Göttingen at Midcentury 526 Riemann in Göttingen 527 Mathematical Physics in Germany 528 Mathematical Physics in English–Speaking Countries 529 Weierstrass and Students 531 The Arithmetization of Analysis 533 Dedekind 536 Cantor and Kronecker 538 Analysis in France 543 23 Twentieth–Century Legacies 548 Overview 548 Henri Poincaré 549 David Hilbert 555 Integration and Measure 564 Functional Analysis and General Topology 568 Algebra 570 Differential Geometry and Tensor Analysis 572 Probability 573 Bounds and Approximations 575 The 1930s and World War II 577 Nicolas Bourbaki 578 Homological Algebra and Category Theory 580 Algebraic Geometry 581 Logic and Computing 582 The Fields Medals 584 24 Recent Trends 586 Overview 586 The Four–Color Conjecture 587 Classification of Finite Simple Groups 591 Fermat’s Last Theorem 593 Poincaré’s Query 596 Future Outlook 599 References 601 General Bibliography 633 Index 647