the heyday of synthetic geometry can be considered to have been the 19th century, when analytic methods based on coordinates and calculus were ignored by some geometers such as jakob steiner, in favor of a purely 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. projective geometry has in fact the simplest and most elegant synthetic expression of any geometry.