Table of Contents of Mathematical Expeditions

Table of Contents

1 Geometry: The Parallel Postulate 1

1.1 Introduction 1

1.2 Euclid's Parallel Postulate 18

1.3 Legendre's Attempts to Prove the Parallel Postulate 24

1.4 Lobachevskian Geometry 31

1.5 Poincaré's Euclidean Model for Non-Euclidean Geometry 43

2 Set Theory: Taming the Infinite 54

2.1 Introduction 54

2.2 Bolzano's Paradoxes of the Infinite 69

2.3 Cantor's Infinite Numbers 74

2.4 Zermelo's Axiomatization 89

3 Analysis: Calculating Areas and Volumes 95

3.1 Introduction 95

3.2 Archimedes' Quadrature of the Parabola 108

3.3 Archimedes' Method 118

3.4 Cavalieri Calculates Areas of Higher Parabolas 123

3.5 Leibniz's Fundamental Theorem of Calculus 129

3.6 Cauchy's Rigorization of Calculus 138

3.7 Robinson Resurrects Infinitesimals 150

3.8 Appendix on Infinite Series 154

4 Number Theory: Fermat's Last Theorem 156

4.1 Introduction 156

4.2 Euclid's Classification of Pythagorean Triples 172

4.3 Euler's Solution for Exponent Four 179

4.4 Germain's General Approach 185

4.5 Kummer and the Dawn of Algebraic Number Theory 193

4.6 Appendix on Congruences 199

5 Algebra: The Search for an Elusive Formula 204

5.1 Introduction 204

5.2 Euclid's Application of Areas and Quadratic Equations 219

5.3 Cardano's Solution of the Cubic 224

5.4 Lagrange's Theory of Equations 233

5.5 Galois Ends the Story 247

References 259
Credits 269
Index 271