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More About This Textbook
Overview
This third edition includes new chapters on simple groups and combinatorics, and new sections on several topics, including the Poincaré conjecture. The book has also been enriched by added exercises.
Editorial Reviews
Booknews
A beautiful little book, certain to be treasured by several generations of mathematics lovers, by students and teachers so enlightened as to think of mathematics not as a forest of technical details but as the beautiful coherent creation of a richly diverse population of extraordinary people. In twenty chapters which, though densely crosslinked, can be read in virtually any order the author reviews the broad sweep of the issues in the principal topical sub fields of mathematics. His writing is so luminous as to engage the interest of utter novices, yet so dense with particulars as to stimulate the imagination of professionals even, on occasion, to instruct. Takes a broader view and is more biographical than either What is mathematics? Courant & Robbins, 1941 or Geometry and the imagination Hilbert & CohnVossen, 1932, and is technically more specific that E.T. Bell's Men of mathematics 1937. Very attractively produced, with nice figures and an excellent bibliography. NW Annotation c. Book News, Inc., Portland, OR booknews.comFrom the Publisher
From the reviews of the third edition:"The author’s goal for Mathematics and its History is to provide a “bird’seye view of undergraduate mathematics.” (p. vii) In that regard it succeeds admirably. ... Mathematics and its History is a joy to read. The writing is clear, concise and inviting. The style is very different from a traditional text. ... The author has done a wonderful job of tying together the dominant themes of undergraduate mathematics. ... While Stillwell does a wonderful job of tying together seemingly unrelated areas of mathematics, it is possible to read each chapter independently. I would recommend this fine book for anyone who has an interest in the history of mathematics. For those who teach mathematics, it provides lots of information which could easily be used to enrich an opening lecture in most any undergraduate course. It would be an ideal gift for a department’s outstanding major or for the math club president. Pick it up at your peril — it is hard to put down!"
(Richard Wilders, MAA Reviews)
“I appreciate and recommend Stillwell’s presentation of mathematics and history written in a lively style. The author’s concept (history mostly as the means of approaching mathematics) remains a matter of interest for both the mathematician and the historian … .” (Rüdiger Thiele, Zentralblatt MATH, Vol. 1207, 2011)
From the reviews of the second edition:
"This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations. The fact that the topics are introduced in their historical contexts will enable students to better appreciate and understand the mathematical ideas involved...If one constructs a list of topics central to a history course, then they would closely resemble those chosen here."
(David Parrott, Australian Mathematical Society)
"The book...is presented in a lively style without unnecessary detail. It is very stimulating and will be appreciated not only by students. Much attention is paid to problems and to the development of mathematics before the end of the nineteenth century... This book brings to the nonspecialist interested in mathematics many interesting results. It can be recommended for seminars and will be enjoyed by the broad mathematical community."
(European Mathematical Society)
"Since Stillwell treats many topics, most mathematicians will learn a lot from this book as well as they will find pleasant and rather clear expositions of custom materials. The book is accessible to students that have already experienced calculus, algebra and geometry and will give them a good account of how the different branches of mathematics interact."
(Denis Bonheure, Bulletin of the Belgian Society)
Product Details
Meet the Author
John Stillwell is a professor of mathematics at the University of San Francisco. He is also an accomplished author, having published several books with Springer, including The Four Pillars of Geometry; Elements of Algebra; Numbers and Geometry; and many more.
Table of Contents
Preface to the Third Edition vii
Preface to the Second Edition ix
Preface to the First Edition xi
1 The Theorem of Pythagoras 1
1.1 Arithmetic and Geometry 2
1.2 Pythagorean Triples 4
1.3 Rational Points on the Circle 6
1.4 RightAngled Triangles 9
1.5 Irrational Numbers 11
1.6 The Definition of Distance 13
1.7 Biographical Notes: Pythagoras 15
2 Greek Geometry 17
2.1 The Deductive Method 18
2.2 The Regular Polyhedra 20
2.3 Ruler and Compass Constructions 25
2.4 Conic Sections 28
2.5 HigherDegree Curves 31
2.6 Biographical Notes: Euclid 35
3 Greek Number Theory 37
3.1 The Role of Number Theory 38
3.2 Polygonal, Prime, and Perfect Numbers 38
3.3 The Euclidean Algorithm 41
3.4 Pell's Equation 44
3.5 The Chord and Tangent Methods 48
3.6 Biographical Notes: Diophantus 50
4 Infinity in Greek Mathematics 53
4.1 Fear of Infinity 54
4.2 Eudoxus's Theory of Proportions 56
4.3 The Method of Exhaustion 58
4.4 The Area of a Parabolic Segment 63
4.5 Biographical Notes: Archimedes 66
5 Number Theory in Asia 69
5.1 The Euclidean Algorithm 70
5.2 The Chinese Remainder Theorem 71
5.3 Linear Diophantine Equations 74
5.4 Pell's Equation in Brahmagupta 75
5.5 Pell's Equation in Bhâskara II 78
5.6 Rational Triangles 81
5.7 Biographical Notes: Brahmagupta and Bhâskara 84
6 Polynomial Equations 87
6.1 Algebra 88
6.2 Linear Equations and Elimination 89
6.3 Quadratic Equations 92
6.4 Quadratic Irrationals 95
6.5 The Solution of the Cubic 97
6.6 Angle Division 99
6.7 HigherDegree Equations 101
6.8 Biographical Notes: Tartaglia, Cardano, and Viète 103
7 Analytic Geometry 109
7.1 Steps Toward Analytic Geometry 110
7.2 Fermat and Descartes 111
7.3 Algebraic Curves 112
7.4 Newton's Classification of Cubics 115
7.5 Construction of Equations, Bézout's Theorem 118
7.6 The Arithmetization of Geometry 120
7.7 Biographical Notes: Descartes 122
8 Projective Geometry 127
8.1 Perspective 128
8.2 Anamorphosis 131
8.3 Desargues's Projective Geometry 132
8.4 The Projective View of Curves 136
8.5 The Projective Plane 141
8.6 The Projective Line 144
8.7 Homogeneous Coordinates 147
8.8 Pascal's Theorem 150
8.9 Biographical Notes: Desargues and Pascal 153
9 Calculus 157
9.1 What Is Calculus? 158
9.2 Early Results on Areas and Volumes 159
9.3 Maxima, Minima, and Tangents 162
9.4 The Arithmetica Infinitorum of Wallis 164
9.5 Newton's Calculus of Series 167
9.6 The Calculus of Leibniz 170
9.7 Biographical Notes: Wallis, Newton, and Leibniz 172
10 Infinite Series 181
10.1 Early Results 182
10.2 Power Series 185
10.3 An Interpolation on Interpolation 188
10.4 Summation of Series 189
10.5 Fractional Power Series 191
10.6 Generating Functions 192
10.7 The Zeta Function 195
10.8 Biographical Notes: Gregory and Euler 197
11 The Number Theory Revival 203
11.1 Between Diophantus and Fermat 204
11.2 Fermat's Little Theorem 207
11.3 Fermat's Last Theorem 210
11.4 Rational RightAngled Triangles 211
11.5 Rational Points on Cubics of Genus 0 215
11.6 Rational Points on Cubics of Genus 1 218
11.7 Biographical Notes: Fermat 222
12 Elliptic Functions 225
12.1 Elliptic and Circular Functions 226
12.2 Parameterization of Cubic Curves 226
12.3 Elliptic Integrals 228
12.4 Doubling the Arc of the Lemniscate 230
12.5 General Addition Theorems 232
12.6 Elliptic Functions 234
12.7 A Postscript on the Lemniscate 236
12.8 Biographical Notes: Abel and Jacobi 237
13 Mechanics 243
13.1 Mechanics Before Calculus 244
13.2 The Fundamental Theorem of Motion 246
13.3 Kepler's Laws and the Inverse Square Law 249
13.4 Celestial Mechanics 253
13.5 Mechanical Curves 255
13.6 The Vibrating String 261
13.7 Hydrodynamics 265
13.8 Biographical Notes: The Bernoullis 267
14 Complex Numbers in Algebra 275
14.1 Impossible Numbers 276
14.2 Quadratic Equations 276
14.3 Cubic Equations 277
14.4 Wallis's Attempt at Geometric Representation 279
14.5 Angle Division 281
14.6 The Fundamental Theorem of Algebra 285
14.7 The Proofs of d' Alembert and Gauss 287
14.8 Biographical Notes: d' Alembert 291
15 Complex Numbers and Curves 295
15.1 Roots and Intersections 296
15.2 The Complex Projective Line 298
15.3 Branch Points 301
15.4 Topology of Complex Projective Curves 304
15.5 Biographical Notes: Riemann 308
16 Complex Numbers and Functions 313
16.1 Complex Functions 314
16.2 Conformal Mapping 318
16.3 Cauchy's Theorem 319
16.4 Double Periodicity of Elliptic Functions 322
16.5 Elliptic Curves 325
16.6 Uniformization 329
16.7 Biographical Notes: Lagrange and Cauchy 331
17 Differential Geometry 335
17.1 Transcendental Curves 336
17.2 Curvature of Plane Curves 340
17.3 Curvature of Surfaces 343
17.4 Surfaces of Constant Curvature 344
17.5 Geodesies 346
17.6 The GaussBonnet Theorem 348
17.7 Biographical Notes: Harriot and Gauss 352
18 NonEuclidean Geometry 359
18.1 The Parallel Axiom 360
18.2 Spherical Geometry 363
18.3 Geometry of Bolyai and Lobachevsky 365
18.4 Beltrami's Projective Model 366
18.5 Beltrami's Conformal Models 369
18.6 The Complex Interpretations 374
18.7 Biographical Notes: Bolyai and Lobachevsky 378
19 Group Theory 383
19.1 The Group Concept 384
19.2 Subgroups and Quotients 387
19.3 Permutations and Theory of Equations 389
19.4 Permutation Groups 393
19.5 Polyhedral Groups 395
19.6 Groups and Geometries 398
19.7 Combinatorial Group Theory 401
19.8 Finite Simple Groups 404
19.9 Biographical Notes: Galois 409
20 Hypercomplex Numbers 415
20.1 Complex Numbers in Hindsight 416
20.2 The Arithmetic of Pairs 417
20.3 Properties of + and x 419
20.4 Arithmetic of Triples and Quadruples 421
20.5 Quaternions, Geometry, and Physics 424
20.6 Octonions 428
20.7 Why C, H, and O Are Special 430
20.8 Biographical Notes: Hamilton 433
21 Algebraic Number Theory 439
21.1 Algebraic Numbers 440
21.2 Gaussian Integers 442
21.3 Algebraic Integers 445
21.4 Ideals 448
21.5 Ideal Factorization 452
21.6 Sums of Squares Revisited 454
21.7 Rings and Fields 457
21.8 Biographical Notes: Dedekind, Hilbert, and Noether 459
22 Topology 467
22.1 Geometry and Topology 468
22.2 Polyhedron Formulas of Descartes and Euler 469
22.3 The Classification of Surfaces 471
22.4 Descartes and GaussBonnet 474
22.5 Euler Characteristic and Curvature 477
22.6 Surfaces and Planes 479
22.7 The Fundamental Group 484
22.8 The Poincaré Conjecture 486
22.9 Biographical Notes: Poincaré 492
23 Simple Groups 495
23.1 Finite Simple Groups and Finite Fields 496
23.2 The Mathieu Groups 498
23.3 Continuous Groups 501
23.4 Simplicity of SO(3) 505
23.5 Simple Lie Groups and Lie Algebras 509
23.6 Finite Simple Groups Revisited 513
23.7 The Monster 515
23.8 Biographical Notes: Lie, Killing, and Cartan 518
24 Sets, Logic, and Computation 525
24.1 Sets 526
24.2 Ordinals 528
24.3 Measure 531
24.4 Axiom of Choice and Large Cardinals 534
24.5 The Diagonal Argument 536
24.6 Computability 538
24.7 Logic and Gödel's Theorem 541
24.8 Provability and Truth 546
24.9 Biographical Notes: Gödel 549
25 Combinatorics 553
25.1 What Is Combinatorics? 554
25.2 The Pigeonhole Principle 557
25.3 Analysis and Combinatorics 560
25.4 Graph Theory 563
25.5 Nonplanar Graphs 567
25.6 The Konig Infinity Lemma 571
25.7 Ramsey Theory 575
25.8 Hard Theorems of Combinatorics 580
25.9 Biographical Notes: Erdos 584
Bibliography 589
Index 629