Author	Laubenbacher, Reinhard. author
Title	Mathematical Expeditions [electronic resource] : Chronicles by the Explorers / by Reinhard Laubenbacher, David Pengelley
Imprint	New York, NY : Springer New York : Imprint: Springer, 1999
Connect to	http://dx.doi.org/10.1007/978-1-4612-0523-4
Descript	X, 278 p. online resource

This 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

1 Geometry: The Parallel Postulate -- 1.1 Introduction -- 1.2 Euclidโs Parallel Postulate -- 1.3 Legendreโs Attempts to Prove the Parallel Postulate -- 1.4 Lobachevskian Geometry -- 1.5 Poincarรฉโs Euclidean Model for Non-Euclidean Geometry. -- 2 Set Theory: Taming the Infinite -- 2.1 Introduction -- 2.2 Bolzanoโs Paradoxes of the Infinite -- 2.3 Cantorโs Infinite Numbers -- 2.4 Zermeloโs Axiomatization -- 3 Analysis: Calculating Areas and Volumes -- 3.1 Introduction -- 3.2 Archimedesโ Quadrature of the Parabola -- 3.3 Archimedesโ Method -- 3.4 Cavalieri Calculates Areas of Higher Parabolas -- 3.5 Leibnizโs Fundamental Theorem of Calculus -- 3.6 Cauchyโs Rigorization of Calculus -- 3.7 Robinson Resurrects Infinitesimals -- 3.8 Appendix on Infinite Series -- 4 Number Theory: Fermatโs Last Theorem -- 4.1 Introduction -- 4.2 Euclidโs Classification of Pythagorean Triples -- 4.3 Eulerโs Solution for Exponent Four -- 4.4 Germainโs General Approach -- 4.5 Kummer and the Dawn of Algebraic Number Theory -- 4.6 Appendix on Congruences -- 5 Algebra: The Search for an Elusive Formula -- 5.1 Introduction -- 5.2 Euclidโs Application of Areas and Quadratic Equations -- 5.3 Cardanoโs Solution of the Cubic -- 5.4 Lagrangeโs Theory of Equations -- 5.5 Galois Ends the Story -- References -- Credits

1. Mathematics
2. Mathematics
3. Mathematics
4. general