Mathematics and Its History

by Stillwell, John

• ISBN13: 9781441960528
• ISBN10: 144196052X
• Edition: 3rd
• Format: Hardcover
• Copyright: 2010-07-01
• Publisher: Springer Verlag

Summary

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 Edition	p. vii
Preface to the Second Edition	p. ix
Preface to the First Edition	p. xi
The Theorem of Pythagoras	p. 1
Arithmetic and Geometry	p. 2
Pythagorean Triples	p. 4
Rational Points on the Circle	p. 6
Right-Angled Triangles	p. 9
Irrational Numbers	p. 11
The Definition of Distance	p. 13
Biographical Notes: Pythagoras	p. 15
Greek Geometry	p. 17
The Deductive Method	p. 18
The Regular Polyhedra	p. 20
Ruler and Compass Constructions	p. 25
Conic Sections	p. 28
Higher-Degree Curves	p. 31
Biographical Notes: Euclid	p. 35
Greek Number Theory	p. 37
The Role of Number Theory	p. 38
Polygonal, Prime, and Perfect Numbers	p. 38
The Euclidean Algorithm	p. 41
Pell's Equation	p. 44
The Chord and Tangent Methods	p. 48
Biographical Notes: Diophantus	p. 50
Infinity in Greek Mathematics	p. 53
Fear of Infinity	p. 54
Eudoxus's Theory of Proportions	p. 56
The Method of Exhaustion	p. 58
The Area of a Parabolic Segment	p. 63
Biographical Notes: Archimedes	p. 66
Number Theory in Asia	p. 69
The Euclidean Algorithm	p. 70
The Chinese Remainder Theorem	p. 71
Linear Diophantine Equations	p. 74
Pell's Equation in Brahmagupta	p. 75
Pell's Equation in Bhâskara II	p. 78
Rational Triangles	p. 81
Biographical Notes: Brahmagupta and Bhâskara	p. 84
Polynomial Equations	p. 87
Algebra	p. 88
Linear Equations and Elimination	p. 89
Quadratic Equations	p. 92
Quadratic Irrationals	p. 95
The Solution of the Cubic	p. 97
Angle Division	p. 99
Higher-Degree Equations	p. 101
Biographical Notes: Tartaglia, Cardano, and Viète	p. 103
Analytic Geometry	p. 109
Steps Toward Analytic Geometry	p. 110
Fermat and Descartes	p. 111
Algebraic Curves	p. 112
Newton's Classification of Cubics	p. 115
Construction of Equations, Bézout's Theorem	p. 118
The Arithmetization of Geometry	p. 120
Biographical Notes: Descartes	p. 122
Projective Geometry	p. 127
Perspective	p. 128
Anamorphosis	p. 131
Desargues's Projective Geometry	p. 132
The Projective View of Curves	p. 136
The Projective Plane	p. 141
The Projective Line	p. 144
Homogeneous Coordinates	p. 147
Pascal's Theorem	p. 150
Biographical Notes: Desargues and Pascal	p. 153
Calculus	p. 157
What Is Calculus?	p. 158
Early Results on Areas and Volumes	p. 159
Maxima, Minima, and Tangents	p. 162
The Arithmetica Infinitorum of Wallis	p. 164
Newton's Calculus of Series	p. 167
The Calculus of Leibniz	p. 170
Biographical Notes: Wallis, Newton, and Leibniz	p. 172
Infinite Series	p. 181
Early Results	p. 182
Power Series	p. 185
An Interpolation on Interpolation	p. 188
Summation of Series	p. 189
Fractional Power Series	p. 191
Generating Functions	p. 192
The Zeta Function	p. 195
Biographical Notes: Gregory and Euler	p. 197
The Number Theory Revival	p. 203
Between Diophantus and Fermat	p. 204
Fermat's Little Theorem	p. 207
Fermat's Last Theorem	p. 210
Rational Right-Angled Triangles	p. 211
Rational Points on Cubics of Genus 0	p. 215
Rational Points on Cubics of Genus 1	p. 218
Biographical Notes: Fermat	p. 222
Elliptic Functions	p. 225
Elliptic and Circular Functions	p. 226
Parameterization of Cubic Curves	p. 226
Elliptic Integrals	p. 228
Doubling the Arc of the Lemniscate	p. 230
General Addition Theorems	p. 232
Elliptic Functions	p. 234
A Postscript on the Lemniscate	p. 236
Biographical Notes: Abel and Jacobi	p. 237
Mechanics	p. 243
Mechanics Before Calculus	p. 244
The Fundamental Theorem of Motion	p. 246
Kepler's Laws and the Inverse Square Law	p. 249
Celestial Mechanics	p. 253
Mechanical Curves	p. 255
The Vibrating String	p. 261
Hydrodynamics	p. 265
Biographical Notes: The Bernoullis	p. 267
Complex Numbers in Algebra	p. 275
Impossible Numbers	p. 276
Quadratic Equations	p. 276
Cubic Equations	p. 277
Wallis's Attempt at Geometric Representation	p. 279
Angle Division	p. 281
The Fundamental Theorem of Algebra	p. 285
The Proofs of d' Alembert and Gauss	p. 287
Biographical Notes: d' Alembert	p. 291
Complex Numbers and Curves	p. 295
Roots and Intersections	p. 296
The Complex Projective Line	p. 298
Branch Points	p. 301
Topology of Complex Projective Curves	p. 304
Biographical Notes: Riemann	p. 308
Complex Numbers and Functions	p. 313
Complex Functions	p. 314
Conformal Mapping	p. 318
Cauchy's Theorem	p. 319
Double Periodicity of Elliptic Functions	p. 322
Elliptic Curves	p. 325
Uniformization	p. 329
Biographical Notes: Lagrange and Cauchy	p. 331
Differential Geometry	p. 335
Transcendental Curves	p. 336
Curvature of Plane Curves	p. 340
Curvature of Surfaces	p. 343
Surfaces of Constant Curvature	p. 344
Geodesies	p. 346
The Gauss-Bonnet Theorem	p. 348
Biographical Notes: Harriot and Gauss	p. 352
Non-Euclidean Geometry	p. 359
The Parallel Axiom	p. 360
Spherical Geometry	p. 363
Geometry of Bolyai and Lobachevsky	p. 365
Beltrami's Projective Model	p. 366
Beltrami's Conformal Models	p. 369
The Complex Interpretations	p. 374
Biographical Notes: Bolyai and Lobachevsky	p. 378
Group Theory	p. 383
The Group Concept	p. 384
Subgroups and Quotients	p. 387
Permutations and Theory of Equations	p. 389
Permutation Groups	p. 393
Polyhedral Groups	p. 395
Groups and Geometries	p. 398
Combinatorial Group Theory	p. 401
Finite Simple Groups	p. 404
Biographical Notes: Galois	p. 409
Hypercomplex Numbers	p. 415
Complex Numbers in Hindsight	p. 416
The Arithmetic of Pairs	p. 417
Properties of + and x	p. 419
Arithmetic of Triples and Quadruples	p. 421
Quaternions, Geometry, and Physics	p. 424
Octonions	p. 428
Why C, H, and O Are Special	p. 430
Biographical Notes: Hamilton	p. 433
Algebraic Number Theory	p. 439
Algebraic Numbers	p. 440
Gaussian Integers	p. 442
Algebraic Integers	p. 445
Ideals	p. 448
Ideal Factorization	p. 452
Sums of Squares Revisited	p. 454
Rings and Fields	p. 457
Biographical Notes: Dedekind, Hilbert, and Noether	p. 459
Topology	p. 467
Geometry and Topology	p. 468
Polyhedron Formulas of Descartes and Euler	p. 469
The Classification of Surfaces	p. 471
Descartes and Gauss-Bonnet	p. 474
Euler Characteristic and Curvature	p. 477
Surfaces and Planes	p. 479
The Fundamental Group	p. 484
The Poincaré Conjecture	p. 486
Biographical Notes: Poincaré	p. 492
Simple Groups	p. 495
Finite Simple Groups and Finite Fields	p. 496
The Mathieu Groups	p. 498
Continuous Groups	p. 501
Simplicity of SO(3)	p. 505
Simple Lie Groups and Lie Algebras	p. 509
Finite Simple Groups Revisited	p. 513
The Monster	p. 515
Biographical Notes: Lie, Killing, and Cartan	p. 518
Sets, Logic, and Computation	p. 525
Sets	p. 526
Ordinals	p. 528
Measure	p. 531
Axiom of Choice and Large Cardinals	p. 534
The Diagonal Argument	p. 536
Computability	p. 538
Logic and Gödel's Theorem	p. 541
Provability and Truth	p. 546
Biographical Notes: Gödel	p. 549
Combinatorics	p. 553
What Is Combinatorics?	p. 554
The Pigeonhole Principle	p. 557
Analysis and Combinatorics	p. 560
Graph Theory	p. 563
Nonplanar Graphs	p. 567
The Konig Infinity Lemma	p. 571
Ramsey Theory	p. 575
Hard Theorems of Combinatorics	p. 580
Biographical Notes: Erdos	p. 584
Bibliography	p. 589
Index	p. 629
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