 Beyond the eye-catching and thought-provoking idea of a physical geometry alternative to Euclid’s system, which had stood for two thousand years, there was the panoply of metrical geometries hinted at in work on differential geometry by Gauss and above all by Riemann. Here it finally proved possible to explain the relationship between straightest and shortest in a suitably general setting. It also became possible to discuss geometry as a body of ideas that grew out of naive ideas of length, angle, shape, and size, and to do so in a sophisticated and rigorous way. What had been regarded as inescapable axiomatic truths, necessary truths, now became hypothetical assumptions that could be modified in many ways; the old notion of an axiom yielded gradually to the modern one. It was also possible to contemplate all kinds of theoretical structures as ‘spaces’, whether or not they were presented axiomatically or intended as distillations of intelligible experience. In this way, it became possible to apply geometrical ideas in novel settings and in novel ways. Geometrisation has been ubiquitous in maths and science ever since.