Historically, projective geometry was introduced in order to make the propositions of incidence true (without exceptions such as are caused by parallels). From the point of view of synthetic geometry it was considered that projective geometry *should be* developed using such propositions as axioms. This turns out only to make a major difference only for the projective plane (for reasons to do with Desargues' theorem).

The modern approach is to define projective space starting from linear algebra and homogeneous coordinates. Then the propositions of incidence are derived from the following basic result on vector spaces: given subspaces U and V of a vector space W, the dimension of their intersection is at least dim U + dim V - dim W. Bearing in mind that the dimension of the projective space **P**(W) associated to W is dim W - 1, but that we require an intersection of subspaces of dimension at least 1 to register in projective space (the subspace {0} being common to all subspaces of W), we get the basic proposition of incidence in this from: linear subspaces **L** and **M** of projective space **P** meet provided dim **L** + dim **M** is at least dim **P**.

See also: Incidence matrix, incidence algebra, angle of incidence