 The epistemological character of Klein’s Erlangen Programme becomes clearer when one looks at how it resolved the well-known nagging doubt about the definition of cross-ratio in projective geometry. Klein’s answer proceeded by analogy with lengths in Euclidean or non-Euclidean geometry. In those geometries, the corresponding group preserves straight lines, and any point can be mapped to any other point, but there is no transformation in the group that can map a line segment onto a proper sub-segment of itself. Any arbitrary but fixed line segment (determined by two points) can therefore be taken as unit of length and used to measure line segments, by constructing arbitrary multiples and sub-multiples of it and arranging them as one would a ruler. Now to measure the length of a segment \(AB\), one simply lays the point \(A\) at one end of the ruler and sees where the point \(B\) falls on the ruler.