Definition:Erlangen Program/Historical Note

Historical Note on Erlangen Program

The Erlangen program was published by Felix Christian Klein in $1872$ in his Vergleichende Betrachtungen über neuere geometrische Forschungen.

It is named after the University Erlangen-Nürnberg, where Klein worked.

By $1872$, non-Euclidean geometries had emerged.

However, nothing had been done to determine their hierarchy and relationships.

Klein's approach was as follows:

• Projective geometry was identified as the unifying frame for all other geometries considered by him. Hence the following hierarchy was developed:
  • Affine geometry was more restrictive than projective geometry
  • Euclidean geometry was more restrictive than affine geometry.

• Klein proposed that group theory was the most useful way of organizing geometrical knowledge, emerging from the discipline of Galois theory.

• Klein specificially identified the appropriate geometrical language in which to define the relevant concepts for each level of the hierarchy.