Synthesis of the history and main ideas of a great intellectual revolution: non-Euclidean geometries. The book is divided in three parts: First, a general introduction to the history of mathematics; second, an account of the origins and development of new geometries, from Euclid, to Saccheri, Gauss, Lobachevsky, Bolyai, Riemanna, and Klein; third, models, visual representations, and practices suggestive of such geometries.