## Mathematical Expeditions: Chronicles by the ExplorersThis book contains the stories of five mathematical journeys into new realms, told through the writings of the explorers themselves. Some were guided by mere curiosity and the thrill of adventure, while others had more practical motives. In each case the outcome was a vast expansion of the known mathematical world and the realization that still greater vistas remained to be explored. The authors tell these stories by guiding the reader through the very words of the mathematicians at the heart of these events, and thereby provide insight into the art of approaching mathematical problems. The book can be used in a variety of ways. The five chapters are completely independent, each with varying levels of mathematical sophistication. The book will be enticing to students, to instructors, and to the intellectually curious reader. By working through some of the original sources and supplemental exercises, which discuss and solve - or attempt to solve - a great problem, this book helps the reader discover the roots of modern problems, ideas, and concepts, even whole subjects. Students will also see the obstacles that earlier thinkers had to clear in order to make their respective contributions to five central themes in the evolution of mathematics. |

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### Contents

Geometry The Parallel Postulate | 1 |

12 Euclids Parallel Postulate | 18 |

13 Legendres Attempts to Prove the Parallel Postulate | 24 |

14 Lobachevskian Geometry | 31 |

15 Poincares Euclidean Model for NonEuclidean Geometry | 43 |

Set Theory Taming the Infinite | 54 |

22 Bolzanos Paradoxes of the Infinite | 69 |

23 Cantors Infinite Numbers | 74 |

38 Appendix on Infinite Series | 154 |

Number Theory Fermats Last Theorem | 156 |

42 Euclids Classification of Pythagorean Triples | 172 |

43 Eulers Solution for Exponent Four | 179 |

44 Germains General Approach | 185 |

45 Kummer and the Dawn of Algebraic Number Theory | 193 |

46 Appendix on Congruences | 199 |

Algebra The Search for an Elusive Formula | 204 |

24 Zermelos Axiomatization | 89 |

Analysis Calculating Areas and Volumes | 95 |

32 Archimedes Quadrature of the Parabola | 108 |

33 Archimedes Method | 118 |

34 Cavalieri Calculates Areas of Higher Parabolas | 123 |

35 Leibnizs Fundamental Theorem of Calculus | 129 |

36 Cauchys Rigorization of Calculus | 138 |

37 Robinson Resurrects Infinitesimals | 150 |

52 Euclids Application of Areas and Quadratic Equations | 219 |

53 Cardanos Solution of the Cubic | 224 |

54 Lagranges Theory of Equations | 233 |

55 Galois Ends the Story | 247 |

259 | |

Credits | 269 |

271 | |

### Common terms and phrases

aggregate algebraic analysis angle sum Archimedes arithmetic Axiom Axiom of Choice Bolzano called Cantor Cardano cardinal number Cauchy Cauchy's Cavalieri's century chapter coefficients complex numbers contains Continuum Hypothesis cube curve definition divisor elements equal equations of degree equivalent Euclid Euclid's Euclid's Elements Euclidean Euclidean geometry Euler Exercise exponent factors Fermat equation Fermat's Last Theorem FIGURE finite follows formula functions Fundamental Theorem Galois Gauss Germain given Greek hyperbolic geometry Hypothesis indivisibles infinite sets infinitesimal Lagrange Legendre Leibniz Lemma linear Lobachevsky's mathematicians mathematics method natural numbers non-Euclidean non-Euclidean geometry number theory one-to-one correspondence parabola parallel postulate perpendicular PHOTO polynomial positive integer prime numbers problem proof proposed equation Proposition prove Pythagorean triples Quadrature rational numbers real numbers reduced equation relatively prime result right angles roots segment set theory sides solution solve square straight line tangent triangle FDC values variable