Table of Contents
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
