Introduction
Euclid's parallel postulate
Legendre's attempts to prove the parallel postulate
Lobachevskian geometry
Poincare's Euclidean model for non-Euclidean geometry
Bolzano's paradoxes of the infinite
Cantor's infinite numbers
Zermelo's axiomatization
Archimedes' quadrature of the parabola
Archimedes' method
Cavalieri calculates areas of higher parabolas
Leibniz's fundamental theorm of calculus
Cauchy's rigorization of calculus (Continued) Robinson resurrects infinitesimals
Appendix on infinite series
Euclid's classification of Pythagorean triples
Euler's solution for exponent four
Germain's general approach
Kummer and the dawn of algebraic number theory
Appendix on congruences
Euclid's application of areas and quadratic equations
Cardano's solution of the cubic
Lagrange's theory of equations
Galois ends the story

The stories of five mathematical journeys into new realms, pieced together from the writings of the explorers themselves. The authors tell these stories by guiding readers through the very words of the mathematicians at the heart of these events, providing an insightinto the art of approaching mathematical problems.