which is identical to the stereographic projection of the ξ ξ -plane on a spherical surface already given by Klein in 1884. Since the substitutions ξ ξ , ξ ξ ′ are Möbius transformations (German : Kreisverwandtschaften) in the ξ ξ -plane or upon the ξ ξ -sphere, they concluded that by carrying out an arbitrary motion of hyperbolic space in itself, the ξ ξ -sphere undergoes a Möbius transformation, that the entire group of hyperbolic motions gives all direct Möbius transformations, and finally that any direct Möbius transformation corresponds to a motion of hyperbolic space.