Now in a revised and expanded new edition, this volume chronologically traces the evolution of Euclidean, non-Euclidean, and relativistic theories regarding the shape of the universe. A unique, highly readable, and entertaining account, the book assumes no special mathematical knowledge. It reviews the failed classical attempts to prove the parallel postulate and provides coverage of the role of Gauss, Lobachevskii, and Bolyai in setting the foundations of modern differential geometry, which laid the groundwork for Einstein's theories of special and general relativity. This updated account includes a new chapter on Islamic contributions to this area, as well as additional information on gravitation, the nature of space and black holes.

PART I: Early geometry; Euclidean geometry and the parallel postulate; Investigations by Islamic mathematicians. PART II: Saccheri and his Western Predecessors; J H Lambert's work; Legendre's work; Gauss' contribution; Trigonometry; the first new geometries; the discoveries of Lobachevskii and Bolyai; Curves and surfaces; Riemann on the foundations of geometry; Beltrami's ideas; New models and old arguments; Resume. PART III: Non-Euclidean mechanics; The question of absolute space; Space, time and space-time; Paradoxes of special relativity; Gravitation and non-Euclidean geometry; Speculations; Some last thoughts.