1. The following description of the projective geometry of a finite number of points in 2-space is almost certainly known to those acquainted with projective geometry or with modern algebra. The object of this brief account is to show how certain finite systems can be presented in a form easily understood by students, and how they provide simple but instructive examples of fundamental ideas and “constructions.” The fact that these examples belong to a geometry which is essentially non-Euclidean has great teaching value to those students who are apt to confuse projective geometry with the “method of projection” in Euclidean geometry. The underlying algebra is described briefly in § 4, but an understanding of this is not essential to the geometry. This algebraic work may, however, be of interest to those to whom Galois fields are fairly new.