 It is not just that there are formulae, but that they hint at an alternative formulation of geometry, one in which the geometry described in Euclid’s Elements might prove to be but a special case. If there could be another way to define geometry, one that would lead to these formulae in various cases, the way would be open to rethink all of the questions about geometry that critical examination had opened. The person best placed to do this in the 1830s and 1840s was Gauss: he knew very well what Bolyai and Lobachevskii had done, and his differential geometry gave him the means to proceed, but, curiously, he did not do so. In the early 1840s he wrote some notes that show he could connect the new two-dimensional geometry with geometry on a surface of constant negative curvature, but he did nothing with this observation.