 Klein’s insight, following von Staudt, was that an exactly similar argument involving quadruples of collinear points can be used to define cross-ratio in projective geometry. The projective group preserves straight lines, and any ordered triple of collinear points can be mapped to any ordered triple of collinear points, and the map that sends a given ordered triple of distinct points to another ordered triple of distinct points is unique, but there is no transformation in the group that can map a quadruple of four collinear points onto an arbitrary such quadruple. Any arbitrary but fixed collinear quadruple can therefore be taken as unit of ‘size’, and a complicated but not difficult argument allows one to produce arbitrary multiples and sub-multiples of it that can be used to measure cross-ratios, by arranging as one would a ruler. Rather than give the details, it is better to give this suggestive illustration of why this can be done. Let the cross-ratio of the four collinear points \(P\), \(Q\), \(R\), \(S\) be measured by mapping the points onto the points \(A\), \(B\), \(C\), \(D\) on the real line, where \(A\) is at the origin, \(C\) at \(\infty\), and \(D\) at 1, so it is the position of \(B\) that determines the cross-ratio. This is uniquely determined, and if the length of \(AB\) is \(x\), we find that \(AB{.}CD \mathbin{/} AD{.}CB = x\).