Were mathematicians studying non-euclidean geometries before Hilbert axiomatization? Or do we need the rigor of Hilbert's axioms to have non-euclidean geometry?
If non-euclidean geometry did predate Hilbert planes, how define points and lines? Did we use Euclid's axioms/definitions? It seems like that is dubious because Euclid's axioms are fairly weak (i.e. intersections of circles and lines) and alternate geometries are somewhat unintuitive.
The defining characteristic of a non-Euclidean geometry is that it abandons his parallel axiom for a different one. Although spherical geometry goes back as far as the realisation that the world was round, it was not recognised as anything revolutionary. The discovery of non-Euclidean geometry as such belongs to the independent discovery of hyperbolic geometry by Gauss, Bolyai and Lobachevski in the first part of the 19th century. When elliptic geometry was added to the non-Euclidean set a little later on, it was realised that spherical geometry had been elliptic all along. Klein began to systematise these discoveries in the second half of the century using group theory, in his Erlangen program. Hilbert is remembered for systematizing their axioms in the closing years of the century. Leonard Mlodinow's The Story of Geometry from Parallel Lines to Hyperspace (Allen Lane 2001) provides a useful historical account, although he is light on the developments between Gauss and Einstein.