Above, the connection of conformal transformations with coordinates including the radius of spheres within Lie sphere geometry was mentioned. The special case λ λ = 1 corresponds to a sphere transformation given by Edmond Laguerre (1880-1885), who called it the "transformation by reciprocal directions" and who laid down the foundation of a geometry of oriented spheres and planes. According to Darboux and Bateman, similar relations were discussed before by Albert Ribaucour (1870) and by Lie himself (1871). Stephanos (1881) pointed out that Laguerre's geometry is indeed a special case of Lie's sphere geometry. He also represented Laguerre's oriented spheres by quaternions (1883).