A synthesis of the history and main ideas of a great intellectual revolution: the non-Euclidean geometries. The book has three parts: the first is a general introduction to the history of mathematics; The second, a recount of the origins and developments of the new geometries from Euclid, through Saccheri, Gauss, Lobachevsky, Bolyai, to Riemann and Klein. Finally, they include models and visual representations, practical and suggestive of these geometries.