 The new geometry posed a radical challenge to Euclidean geometry, because it denied traditional geometry its best claim to certainty, to wit, that it was the only logical system for discussing geometry at all. It also exploited the tension known to experts between the concepts of straightest and shortest. But in other ways it was conventional. It offered no new definitions of familiar concepts such as straightness or distance, it agreed with Euclidean geometry over angles, it merely offered a different behaviour of parallel lines based on a difference in the distant behaviour of straight lines. Its proponents did not offer a sceptical conclusion. Bolyai and Lobachevskii did not say: “See, there are two logical but incompatible geometries, so we can never know what is true.” Instead, they held out the hope that experiments and observations would decide. The epistemological price people would have to pay if astronomical observations had come down in favour of the new geometry would, in a sense, have been slight: it would have been necessary to say that straight lines have an unexpected property after all, but one only detectable at very long distances. To be sure, many of the theorems of geometry would then have to be reworked, and their familiar Euclidean counterparts would appear only as very good approximations. But that is broadly comparable to the situation Newtonian mechanics found itself in after the advent of special relativity.