The history of mathematics

Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Contents......Page 6
Preface......Page 11
A Sense of Number......Page 14
Notches as Tally Marks......Page 15
The Peruvian Quipus: Knots as Numbers......Page 19
The History of Herodotus......Page 22
Hieroglyphic Representation of Numbers......Page 24
Egyptian Hieratic Numeration......Page 28
The Greek Alphabetic Numeral System......Page 29
Babylonian Cuneiform Script......Page 33
Deciphering Cuneiform: Grotefend and Rawlinson......Page 34
The Babylonian Positional Number System......Page 36
Writing in Ancient China......Page 39
Egyptian Mathematical Papyri......Page 46
A Key to Deciphering: The Rosetta Stone......Page 48
Early Egyptian Multiplication......Page 50
The Unit Fraction Table......Page 53
Representing Rational Numbers......Page 56
The Method of False Position......Page 59
A Curious Problem......Page 62
Egyptian Mathematics as Applied Arithmetic......Page 63
Approximating the Area of a Circle......Page 66
The Volume of a Truncated Pyramid......Page 69
Speculations About the Great Pyramid......Page 70
A Tablet of Reciprocals......Page 75
The Babylonian Treatment of Quadratic Equations......Page 77
Two Characteristic Babylonian Problems......Page 82
A Tablet Concerning Number Triples......Page 85
Babylonian Use of the Pythagorean Theorem......Page 89
The Cairo Mathematical Papyrus......Page 90
Greece and the Aegean Area......Page 96
The Dawn of Demonstrative Geometry: Thales of Miletos......Page 99
Measurements Using Geometry......Page 100
Pythagoras and His Followers......Page 103
Nicomachus’s Introductio Arithmeticae......Page 107
The Theory of Figurative Numbers......Page 110
Zeno’s Paradox......Page 114
Geometric Proofs of the Pythagorean Theorem......Page 118
Early Solutions of the Pythagorean Equation......Page 120
The Crisis of Incommensurable Quantities......Page 122
Theon’s Side and Diagonal Numbers......Page 124
Eudoxus of Cnidos......Page 129
Hippocrates and the Quadrature of the Circle......Page 133
The Duplication of the Cube......Page 137
The Trisection of an Angle......Page 139
Rise of the Sophists......Page 143
Hippias of Elis......Page 144
The Grove of Academia: Plato’s Academy......Page 147
A Center of Learning: The Museum......Page 154
Euclid’s Life and Writings......Page 156
Euclid’s Foundation for Geometry......Page 157
Common Notions......Page 159
Book I of the Elements......Page 161
Euclid’s Proof of the Pythagorean Theorem......Page 169
Book II on Geometric Algebra......Page 172
Construction of the Regular Pentagon......Page 178
Euclidean Divisibility Properties......Page 183
The Algorithm of Euclid......Page 186
The Fundamental Theorem of Arithmetic......Page 190
An Infinity of Primes......Page 193
The Sieve of Eratosthenes......Page 196
Measurement of the Earth......Page 199
The Almagest of Claudius Ptolemy......Page 201
Ptolemy’s Geographical Dictionary......Page 203
The Ancient World’s Genius......Page 206
Estimating the Value of π......Page 210
The Sand-Reckoner......Page 215
Quadrature of a Parabolic Segment......Page 218
Apollonius of Perga: The Conics......Page 219
The Waning of the Golden Age......Page 226
The Spread of Christianity......Page 228
Diophantus’s Number Theory......Page 230
Problems from the Arithmetica......Page 233
The Cattle Problem of Archimedes......Page 236
Early Mathematics in India......Page 238
The Chinese Hundred Fowls Problem......Page 241
The Mathematical Collection of Pappus......Page 245
Hypatia, the First Woman Mathematician......Page 246
Roman Mathematics: Boethius and Cassiodorus......Page 248
The Algebra of al-Khowârizmî......Page 251
Abû Kâmil and Thâbit ibn Qurra......Page 255
Omar Khayyam......Page 260
The Astronomers al-Tûsî and al-Kashî......Page 262
The Ancient Chinese Nine Chapters......Page 264
Later Chinese Mathematical Works......Page 272
The Carolingian Pre-Renaissance......Page 282
Transmission of Arabic Learning to the West......Page 285
The Pioneer Translators: Gerard and Adelard......Page 287
The Hindu-Arabic Numerals......Page 290
Fibonacci’s Liber Quadratorum......Page 293
The Works of Jordanus de Nemore......Page 296
The Liber Abaci’s Rabbit Problem......Page 300
Some Properties of Fibonacci Numbers......Page 302
Pythagorean Number Triples......Page 306
Fibonacci’s Tournament Problem......Page 310
The Italian Renaissance......Page 314
Artificial Writing: The Invention of Printing......Page 316
Founding of the Great Universities......Page 319
A Thirst for Classical Learning......Page 323
Restoring the Algebraic Tradition: Robert Recorde......Page 325
The Italian Algebraists: Pacioli, del Ferro, and Tartaglia......Page 328
Cardan, A Scoundrel Mathematician......Page 332
Cardan’s Solution of the Cubic Equation......Page 333
Bombelli and Imaginary Roots of the Cubic......Page 337
The Resolvant Cubic......Page 341
The Story of the Quintic Equation: Ruffini, Abel, and Galois......Page 344
The Seventeenth Century Spread of Knowledge......Page 350
Galileo’s Telescopic Observations......Page 352
The Beginning of Modern Notation: François Viéta......Page 358
The Decimal Fractions of Simon Stevin......Page 361
Napier’s Invention of Logarithms......Page 363
The Astronomical Discoveries of Brahe and Kepler......Page 368
The Writings of Descartes......Page 375
Inventing Cartesian Geometry......Page 380
The Algebraic Aspect of La Géométrie......Page 385
Descartes’s Principia Philosophiae......Page 388
Perspective Geometry: Desargues and Poncelet......Page 390
The Textbooks of Oughtred and Harriot......Page 394
Wallis’s Arithmetica Infinitorum......Page 396
The Lucasian Professorship: Barrow and Newton......Page 399
Newton’s Golden Years......Page 405
The Laws of Motion......Page 411
Later Years: Appointment to the Mint......Page 417
The Early Work of Leibniz......Page 422
Leibniz’s Creation of the Calculus......Page 426
Newton’s Fluxional Calculus......Page 429
The Dispute over Priority......Page 437
Maria Agnesi and Emilie du Châtelet......Page 443
Graunt’s Bills of Mortality......Page 452
Games of Chance: Dice and Cards......Page 456
The Precocity of the Young Pascal......Page 459
Pascal and the Cycloid......Page 465
De Méré’s Problem of Points......Page 467
The Traité du Triangle Arithm etique......Page 469
Mathematical Induction......Page 474
Francesco Maurolico’s Use of Induction......Page 476
Christiaan Huygens’s Pamphlet on Probability......Page 481
The Bernoulli Brothers: John and James......Page 484
De Moivre’s Doctrine of Chances......Page 490
The Mathematics of Celestial Phenomena: Laplace......Page 491
Mary Fairfax Somerville......Page 495
Laplace’s Research in Probability Theory......Page 496
Daniel Bernoulli, Poisson, and Chebyshev......Page 502
Scientific Societies......Page 510
Marin Mersenne’s Mathematical Gathering......Page 512
Numbers, Perfect and Not So Perfect......Page 515
Fermat’s Arithmetica......Page 524
The Famous Last Theorem of Fermat......Page 529
The Eighteenth-Century Enlightenment......Page 533
Maclaurin’s Treatise on Fluxions......Page 537
Euler’s Life and Contributions......Page 540
The Period of the French Revolution: Lagrange, Monge, and Carnot......Page 552
Gauss’s Disquisitiones Arithmeticae......Page 559
The Legacy of Gauss: Congruence Theory......Page 564
Dirichlet and Jacobi......Page 571
The Efforts of Proclus, Playfair, and Wallis......Page 576
Saccheri Quadrilaterals......Page 579
The Accomplishments of Legendre......Page 584
Legendre’s Eléments de géométrie......Page 587
Gauss’s Attempt at a New Geometry......Page 597
The Struggle of John Bolyai......Page 601
Creation of Non-Euclidean Geometry: Lobachevsky......Page 605
Models of the New Geometry: Riemann, Beltrami, and Klein......Page 611
Grace Chisholm Young......Page 616
D’Alembert and Cauchy on Limits......Page 617
Fourier’s Series......Page 623
The Father of Modern Analysis, Weierstrass......Page 627
Sonya Kovalevsky......Page 629
The Axiomatic Movement: Pasch and Hilbert......Page 632
Babbage and the Analytical Engine......Page 639
Peacock’s Treatise on Algebra......Page 642
The Representation of Complex Numbers......Page 643
Hamilton’s Discovery of Quaternions......Page 646
Matrix Algebra: Cayley and Sylvester......Page 652
Boole’s Algebra of Logic......Page 659
Ascendency of the German Universities......Page 670
American Mathematics Takes Root: 1800–1900......Page 672
The Twentieth-Century Consolidation......Page 682
The Last Universalist: Poincaré......Page 686
Cantor’s Theory of Infinite Sets......Page 689
Kronecker’s View of Set Theory......Page 694
Countable and Uncountable Sets......Page 697
Transcendental Numbers......Page 702
The Continuum Hypothesis......Page 707
The Early Paradoxes......Page 711
Zermelo and the Axiom of Choice......Page 714
The Logistic School: Frege, Peano, and Russell......Page 717
Hilbert’s Formalistic Approach......Page 721
Brouwer’s Institutionism......Page 724
The Tripos Examination......Page 734
The Rejuvenation of English Mathematics......Page 735
A Unique Collaboration: Hardy and Littlewood......Page 738
India’s Prodigy, Ramanujan......Page 739
Frechet’s Metric Spaces......Page 742
The Neighborhood Spaces of Hausdorff......Page 744
Banach and Normed Linear Spaces......Page 746
Emmy Noether’s Theory of Rings......Page 748
Von Neumann and the Computer......Page 754
Women in Modern Mathematics......Page 757
A Few Recent Advances......Page 760
General Bibliography......Page 768
Additional Reading......Page 772
The Greek Alphabet......Page 774
Solutions to Selected Problems......Page 775
Index......Page 790