Mettiamo ordine fra le geometrie: il programma di Erlangen e altri approcci

Since the end of the 19th century, particularly after the works by F. Klein and D. Hilbert, we know that we have to consider several geometries. In this paper we compare affine geometry, similar geometry and metric geometry from different point of views: the projective framework, Klein's Erlangen program, linear spaces. The approach via transformations and the axiomatic one are not equivalent to each other. In fact, plane transformations seem not to depend on the axioms. We also try to clarify the distinction between similar and metric geometry.