structural:expansion, information:classification
Synthetic vs. analytic geometry

Since the introduction of analytic geometry by René Descartes in the 17th century, there was two approaches to geometry, the old one called synthetic geometry, and the new one, where everything is specified in terms of real numbers called coordinates. On the other hand, in classical synthetic geometry, numbers are defined as ratios of distances, or, equivalently, as distances measured in terms of a fixed length unit.

Mathematicians did not worried much about this circular definition before the mid-nineteenth century, where there was "an acrimonious controversy between the proponents of synthetic and analytic methods in projective geometry, the two sides accusing each other of mixing projective and metric concepts". Indeed, there is no concept of distance in a projective space, and the cross-ratio, which is a number, is a basic concept of synthetic projective geometry.

Karl von Staudt developed a purely geometric approach to this problem by introducing "throws" that form what is presently called a field, in which the cross ratio can be expressed.

Apparently, the problem of the equivalence between analytic and synthetic approach was completely solved only with Emil Artin 's book Geometric Algebra published in 1957. It was well known that, given a field k, one may define affine and projective spaces over k in terms of k - vector spaces. In these spaces, the Pappus hexagon theorem holds. Conversely, if the Pappus hexagon theorem is included in the axioms of a plane geometry, then one can define a field k such that the geometry is the same as the affine or projective geometry over k.