The geometry of Euclid was indeed synthetic, though not all of the books covered topics of *pure geometry*. The heyday of synthetic geometry can be considered to have been the nineteenth century; when methods based on coordinates and calculus were ignored by some geometers such as Jakob Steiner, in favour of a synthetic development of projective geometry.

For example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory (with more models) than is found by starting with a vector space of dimension three. The close axiomatic study of Euclidean geometry led to the discovery of non-Euclidean geometry. The question is whether this is success or failure.

If the axiom set is not *categorical* (so that there is more than one model) one has the geometry/geometries debate to settle. That's not a serious issue for a modern axiomatic mathematician, since the implication of axiom is now *starting point for theory* rather than *self-evident plank in platform based on intuition*. And since the Erlangen programme of Klein the geometrical nature of *a* geometry has been seen as the connection of symmetry and the content of propositions, rather than the style of development.