Playfair's postulate is false, because any two great circles intersect. Thus the sphere is a model of non-Euclidean geometry. So is the pseudo-sphere at the bottom of the figure. If straight lines are interpreted as being the shortest curves connecting any two points on the surface. For more information check out the following web pages.
History of mathematics archive An excellent starting point.
In particular, you should look at their page on non-Euclidean geometry . You should also look at their indexes of biographies to find more info about Euclid, Gauss, J. and F. Bolyai, Lobachevsky, Riemann and Beltrami. Introduction to the works of Euclid Student paper about Euclid. Euclid's Elements, Introduction Euclid, the Java way.