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

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    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

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    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

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    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

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    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

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    References 259
    Credits 269
    Index 271 



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