ABSTRACT
Pseudosphere Eugenio Beltrami (1835-1900) came from an artistic family-his father was a painter who painted miniatures, and his grandfather engraved precious stones. Eugenio found that for him, music was more important, but in his career path he pursued his mathematical talent. Infl uenced by the works of Gauss, Lobachevsky, and Riemann, Beltrami considered the problem of geodesics on different surfacesthat is, what will be straight on different surfaces? He discovered that the surfaces with constant curvature will have geodesics that will be analogous to the behavior of straight lines in the plane. In 1868, he published a paper in which he showed the fi rst concrete realization of Lobachevsky geometry: he showed that this geometry holds locally on the pseudosphere , the surface generated by the revolution of a tractrix about its asymptote.
