This paper was originally planned as an exposition of finite geometry in a form in which it could be used to illustrate all the work covered in the normal school geometry course. The beginning of it, however, was anticipated by an article for the Gazette by Mr. H. Martyn Cundy (1), which he has kindly allowed me to read before publication. This present paper shows how a coordinate system can be constructed for the geometry described in his article by using the field of residue classes (mod. 5); that is, the Galois field GF(5). A definition of distance is introduced which extends that adopted by Cundy to a form suited to the discussion of a wider range of theorems.