"Lobachevsky believed that another form of geometry existed, a non-Euclidean geometry, and this 1840 treatise is his argument on its behalf. Line by line in this classic work he carefully presents a new and revolutionary theory of parallels, one that allows for all of Euclid s axioms, except for the last. This 1891 translation includes a bibliography and translator George B. Halsted s essay on elliptic geometry. Russian mathematician NICHOLAS LOBACHEVSKY (1792 1856) is best remembered as the founder (along with Janos Bolyai) of non-Euclidean geometry. He is also the author of New Foundations of Geometry (1835 1838) and Pangeometry (1855)."