What did mathematicians mean by the words “space” and “geometry” in the nineteenth century? This chapter will try to answer this question, starting with an analysis of Hertz’s, Lipschitz’ and Darboux’s geometrization of mechanics and continuing with a discussion of the use of the words by mathematicians who are usually credited as the principal inventors of non-Euclidean and higher dimensional geometries. The conclusion is that most mathematicians prior to 1880 used the words to denote (intuited) physical space and the geometry describing that space. This is the background against which one should evaluate the sometimes confusing nineteenth century discussions about the existence of geometries other than Euclid’s geometry. The question was radically changed with the advent of modernist structuralist mathematics, as described in (Gray, Plato’s Ghost: the modernist transformation of mathematics. Princeton University Press, Princeton, 1994). © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.