ON A SYSTEM OF AXIOMS FOR GEOMETRY ' BY

In these Transactions, vol. 3 (1902), p. 142, E. H. Moore has given a set of projective axioms for geometry. In further development of the standpoint of Moore a system of axioms for three dimensional Euclidian geometry has been constructed by Veblen f by means of the " betweenness " relation. J A feature of Veblen's system is that a planar axiom (axiom VIII, 1. c.) is necessary to establish the existence of an infinitude of points and to prove the theorem, "To any four distinct points of a line the notation A, B, C, D may always be assigned so that they are in the order AB CD." In this note I give a set of independent axioms in terms of sameness of sense of dyads, abstractly expressed by aßKy8§, which implies Veblen's axioms I-VIII, but which does not require a planar axiom for the proof of the preceding properties. Of the two relations, sameness of sense and betweenness, one is definable in terms of the other; but the two definitions are not equally simple, as will appear below. §2. The axioms and definitions in terms of the relation K are as follows : ||