 By the opening years of the 20th century a variety of Riemannian differential geometries had been proposed, which made rigorous sense of non-Euclidean geometry. There were also significant advances in the domain of abstract geometries, such as those proposed by David Hilbert. It follows that the terms ‘geometry’ and ‘physical space’ do not have simple meanings in the 19th century, and changing conceptions of these terms do not follow a simple pattern of refinement. Their inter-relations therefore also have a complicated history.