This new geometry is important because it generalizes and unites in the same time all together: Euclid, Lobachevsky/Bolyai/Gauss, and Riemann geometries. And separates them as well. It is based on the first four Euclid's postulates, but the fifth one is replaced so that there exist various straight lines and points exterior to them in such a way that none, one, more, and infinitely many parallels can be drawn through the points in this mixted smarandacheian space.