 Credit for the first fully mathematical descriptions of space in terms other than Euclid’s must go to János Bolyai in Hungary and Nicolai Ivanovich Lobachevskii in Russia independently. Bolyai in his “Appendix scientiam spatii absolute veram exhibens” (1832) and Lobachevskii in his Neue Anfangsgründe der Geometrie (1835) and again in his Geometrische Untersuchungen (1840) replaced the parallel postulate with the assumption that given a line L and a point not on that line, there are many lines through the point that lie in the plane defined by L and the given point, but that do not meet the line L. Of these, as they then showed, one line in each direction is asymptotic to L, and these asymptotic lines divide the family of all the other lines in the given plane and through the given point into two families: those that meet L, and those that do not. Much work then followed, famously similar in each case, in particular to show that in the three-dimensional space described by their assumptions there is a surface upon which Euclidean geometry holds, and to deduce the existence of trigonometrical formulae describing triangles in the plane. These formulae resemble the corresponding formulae for triangles on the sphere.