In Artin [1, pp. 51-75] the affine plane is developed and coordinatized from a system of four axioms. The present paper also develops affine plane geometry from a system of four axioms. The first three are the same as Artin's. The fourth is concerned with parallel projections from lines to lines. It leads to a construction of a skew field in a fashion analogous to Bachmann's [2, pp. 137-140] construction of a commutative field. This field can then be used to coordinatize the plane as in Levi [3, pp. 32-37], [4, pp. 50-53]. Both Artin's treatment and the present one deal with mappings rather than configurations. They differ in that Artin considers mappings of the whole plane into itself, whereas here we deal only with mappings from individual lines to lines. Each of our axioms has a natural counterpart for projective geometry, and our development of affine geometry suggests a parallel development of projective geometry. However, the field which this approach yields is necessarily commutative. To cover the general case we present a system of four axioms for the projective plane, three being counterparts as stated, and the fourth a modified counterpart. This fourth axiom, which is a weakened form of the Fundamental Theorem of Projective Geometry, can also be regarded as a form of Desargues' Theorem, in which only those instances are asserted which contain a prescribed line in a prescribed place in the configuration. The author wishes to thank H. S. M. Coxeter and Joseph Lipman for comments on an earlier version of this paper which resulted in substantial simplifications.