 To conclude: a reasonable view at the time would have been that metrical geometry needed to put its house in order, and it probably could not do so by grafting the concept of distance onto a structure modelled on Euclid’s Elements. This is an awkward position for traditional geometry to be in, and it may have opened people’s minds to the possibilities of alternatives. Certainly, two were to be produced. One, projective geometry, amplified and improved the synthetic side of geometry. The other, non-Euclidean geometry, was a new and challenging metrical geometry. But before we look at them, we turn to contemporary philosophical discussions of geometry.