Mathematics and Its History / Edition 3

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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.

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Editorial Reviews

From the Publisher

From the reviews of the third edition:

"The author’s goal for Mathematics and its History is to provide a “bird’s-eye 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 non-specialist 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)

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 cross-linked, 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 & Cohn-Vossen, 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.com
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Product Details

  • ISBN-13: 9781441960528
  • Publisher: Springer New York
  • Publication date: 6/1/2010
  • Series: Undergraduate Texts in Mathematics Series
  • Edition description: 3rd ed. 2010
  • Edition number: 3
  • Pages: 662
  • Product dimensions: 6.30 (w) x 9.30 (h) x 1.70 (d)

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.
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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 Right-Angled 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 Higher-Degree 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 Higher-Degree 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 Right-Angled 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 Gauss-Bonnet Theorem 348

17.7 Biographical Notes: Harriot and Gauss 352

18 Non-Euclidean 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 Gauss-Bonnet 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

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