By fixing a conic κ and ranging point pairs over the region bounded by κ, the cross-ratio becomes a function of point pairs alone.

Can metric properties be fixed in this way? Traditionally one defines the distance between two points (x1, … ,xn) and (y1, … ,yn) of a numerical manifold as the positive square root of (x1 − y1) 2 + … + (xn − y n)2. The group of isometries consists of the transformations that preserve this function. However, this is just a convention, adopted to ensure that the geometry is Euclidean. Using projective geometry, Klein thought of something better. No real-valued function of point pairs, defined on