A cross ratio in continuous geometry

A. Fuhrmann generalizes Sperner's definition of the cross ratio of four collinear points with coordinates in a division ring to apply to four linear varieties over a division ring. The form of, his cross ratio is still very classical. In Part 2 of this a cross ratio is defined for a configuration of subspaces of a continuous or discrete geometry. Although this cross ratio couldn't be further removed in appearance from the classical form we give a simple proof (Section 4) to show that, for the case of four points on a line, it does in fact agree with the usual cross ratio. Moreover, it will follow from the results of Sections 5,6 that for the case of finite dimensional (i.e discrete) geometries, Fuhrmann's cross ratio is essentially the same as the one introduced here. The cross ratio has the desired property of invariance under collineations ( Theorem 3).