Mathematics and Its History

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  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 2010-07-01
  • Publisher: Springer Verlag

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This third edition includes new chapters on simple groups and combinatorics, and new sections on several topics, including the Poincar conjecture. The book has also been enriched by added exercises. Book jacket.

Author Biography

John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer, including The Four Pillars of Geometry (2005), Elements of Number Theory (2003), Numbers and Geometry (1998) and Elements of Algebra (1994).

Table of Contents

Preface to the Third Editionp. vii
Preface to the Second Editionp. ix
Preface to the First Editionp. xi
The Theorem of Pythagorasp. 1
Arithmetic and Geometryp. 2
Pythagorean Triplesp. 4
Rational Points on the Circlep. 6
Right-Angled Trianglesp. 9
Irrational Numbersp. 11
The Definition of Distancep. 13
Biographical Notes: Pythagorasp. 15
Greek Geometryp. 17
The Deductive Methodp. 18
The Regular Polyhedrap. 20
Ruler and Compass Constructionsp. 25
Conic Sectionsp. 28
Higher-Degree Curvesp. 31
Biographical Notes: Euclidp. 35
Greek Number Theoryp. 37
The Role of Number Theoryp. 38
Polygonal, Prime, and Perfect Numbersp. 38
The Euclidean Algorithmp. 41
Pell's Equationp. 44
The Chord and Tangent Methodsp. 48
Biographical Notes: Diophantusp. 50
Infinity in Greek Mathematicsp. 53
Fear of Infinityp. 54
Eudoxus's Theory of Proportionsp. 56
The Method of Exhaustionp. 58
The Area of a Parabolic Segmentp. 63
Biographical Notes: Archimedesp. 66
Number Theory in Asiap. 69
The Euclidean Algorithmp. 70
The Chinese Remainder Theoremp. 71
Linear Diophantine Equationsp. 74
Pell's Equation in Brahmaguptap. 75
Pell's Equation in Bhâskara IIp. 78
Rational Trianglesp. 81
Biographical Notes: Brahmagupta and Bhâskarap. 84
Polynomial Equationsp. 87
Algebrap. 88
Linear Equations and Eliminationp. 89
Quadratic Equationsp. 92
Quadratic Irrationalsp. 95
The Solution of the Cubicp. 97
Angle Divisionp. 99
Higher-Degree Equationsp. 101
Biographical Notes: Tartaglia, Cardano, and Viètep. 103
Analytic Geometryp. 109
Steps Toward Analytic Geometryp. 110
Fermat and Descartesp. 111
Algebraic Curvesp. 112
Newton's Classification of Cubicsp. 115
Construction of Equations, Bézout's Theoremp. 118
The Arithmetization of Geometryp. 120
Biographical Notes: Descartesp. 122
Projective Geometryp. 127
Perspectivep. 128
Anamorphosisp. 131
Desargues's Projective Geometryp. 132
The Projective View of Curvesp. 136
The Projective Planep. 141
The Projective Linep. 144
Homogeneous Coordinatesp. 147
Pascal's Theoremp. 150
Biographical Notes: Desargues and Pascalp. 153
Calculusp. 157
What Is Calculus?p. 158
Early Results on Areas and Volumesp. 159
Maxima, Minima, and Tangentsp. 162
The Arithmetica Infinitorum of Wallisp. 164
Newton's Calculus of Seriesp. 167
The Calculus of Leibnizp. 170
Biographical Notes: Wallis, Newton, and Leibnizp. 172
Infinite Seriesp. 181
Early Resultsp. 182
Power Seriesp. 185
An Interpolation on Interpolationp. 188
Summation of Seriesp. 189
Fractional Power Seriesp. 191
Generating Functionsp. 192
The Zeta Functionp. 195
Biographical Notes: Gregory and Eulerp. 197
The Number Theory Revivalp. 203
Between Diophantus and Fermatp. 204
Fermat's Little Theoremp. 207
Fermat's Last Theoremp. 210
Rational Right-Angled Trianglesp. 211
Rational Points on Cubics of Genus 0p. 215
Rational Points on Cubics of Genus 1p. 218
Biographical Notes: Fermatp. 222
Elliptic Functionsp. 225
Elliptic and Circular Functionsp. 226
Parameterization of Cubic Curvesp. 226
Elliptic Integralsp. 228
Doubling the Arc of the Lemniscatep. 230
General Addition Theoremsp. 232
Elliptic Functionsp. 234
A Postscript on the Lemniscatep. 236
Biographical Notes: Abel and Jacobip. 237
Mechanicsp. 243
Mechanics Before Calculusp. 244
The Fundamental Theorem of Motionp. 246
Kepler's Laws and the Inverse Square Lawp. 249
Celestial Mechanicsp. 253
Mechanical Curvesp. 255
The Vibrating Stringp. 261
Hydrodynamicsp. 265
Biographical Notes: The Bernoullisp. 267
Complex Numbers in Algebrap. 275
Impossible Numbersp. 276
Quadratic Equationsp. 276
Cubic Equationsp. 277
Wallis's Attempt at Geometric Representationp. 279
Angle Divisionp. 281
The Fundamental Theorem of Algebrap. 285
The Proofs of d' Alembert and Gaussp. 287
Biographical Notes: d' Alembertp. 291
Complex Numbers and Curvesp. 295
Roots and Intersectionsp. 296
The Complex Projective Linep. 298
Branch Pointsp. 301
Topology of Complex Projective Curvesp. 304
Biographical Notes: Riemannp. 308
Complex Numbers and Functionsp. 313
Complex Functionsp. 314
Conformal Mappingp. 318
Cauchy's Theoremp. 319
Double Periodicity of Elliptic Functionsp. 322
Elliptic Curvesp. 325
Uniformizationp. 329
Biographical Notes: Lagrange and Cauchyp. 331
Differential Geometryp. 335
Transcendental Curvesp. 336
Curvature of Plane Curvesp. 340
Curvature of Surfacesp. 343
Surfaces of Constant Curvaturep. 344
Geodesiesp. 346
The Gauss-Bonnet Theoremp. 348
Biographical Notes: Harriot and Gaussp. 352
Non-Euclidean Geometryp. 359
The Parallel Axiomp. 360
Spherical Geometryp. 363
Geometry of Bolyai and Lobachevskyp. 365
Beltrami's Projective Modelp. 366
Beltrami's Conformal Modelsp. 369
The Complex Interpretationsp. 374
Biographical Notes: Bolyai and Lobachevskyp. 378
Group Theoryp. 383
The Group Conceptp. 384
Subgroups and Quotientsp. 387
Permutations and Theory of Equationsp. 389
Permutation Groupsp. 393
Polyhedral Groupsp. 395
Groups and Geometriesp. 398
Combinatorial Group Theoryp. 401
Finite Simple Groupsp. 404
Biographical Notes: Galoisp. 409
Hypercomplex Numbersp. 415
Complex Numbers in Hindsightp. 416
The Arithmetic of Pairsp. 417
Properties of + and xp. 419
Arithmetic of Triples and Quadruplesp. 421
Quaternions, Geometry, and Physicsp. 424
Octonionsp. 428
Why C, H, and O Are Specialp. 430
Biographical Notes: Hamiltonp. 433
Algebraic Number Theoryp. 439
Algebraic Numbersp. 440
Gaussian Integersp. 442
Algebraic Integersp. 445
Idealsp. 448
Ideal Factorizationp. 452
Sums of Squares Revisitedp. 454
Rings and Fieldsp. 457
Biographical Notes: Dedekind, Hilbert, and Noetherp. 459
Topologyp. 467
Geometry and Topologyp. 468
Polyhedron Formulas of Descartes and Eulerp. 469
The Classification of Surfacesp. 471
Descartes and Gauss-Bonnetp. 474
Euler Characteristic and Curvaturep. 477
Surfaces and Planesp. 479
The Fundamental Groupp. 484
The Poincaré Conjecturep. 486
Biographical Notes: Poincarép. 492
Simple Groupsp. 495
Finite Simple Groups and Finite Fieldsp. 496
The Mathieu Groupsp. 498
Continuous Groupsp. 501
Simplicity of SO(3)p. 505
Simple Lie Groups and Lie Algebrasp. 509
Finite Simple Groups Revisitedp. 513
The Monsterp. 515
Biographical Notes: Lie, Killing, and Cartanp. 518
Sets, Logic, and Computationp. 525
Setsp. 526
Ordinalsp. 528
Measurep. 531
Axiom of Choice and Large Cardinalsp. 534
The Diagonal Argumentp. 536
Computabilityp. 538
Logic and Gödel's Theoremp. 541
Provability and Truthp. 546
Biographical Notes: Gödelp. 549
Combinatoricsp. 553
What Is Combinatorics?p. 554
The Pigeonhole Principlep. 557
Analysis and Combinatoricsp. 560
Graph Theoryp. 563
Nonplanar Graphsp. 567
The Konig Infinity Lemmap. 571
Ramsey Theoryp. 575
Hard Theorems of Combinatoricsp. 580
Biographical Notes: Erdosp. 584
Bibliographyp. 589
Indexp. 629
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