A Note concerning Veblen ' S Axioms for Geometry

Veblen has given a set of axioms for geometry.!. The first eight of these are as follows : % Axiom I. There exist at least two distinct points. Axiom II. If points A, B, C are in the order ABC, they are in the order CBA. Axiom III. If points A, B, C are in the order ABC, they are not in the order BCA. Axiom IV. If points A, B, C are in the order ABC, then A is distinct from C. Axiom V. If A and B are any two distinct points, there exists a point C such that A, B, C are in the order ABC. Def. I. The line AB (A =4= B) consists of A and B and all points X in one of the possible orders ABX, AXB, XAB. The points X in the order AXB constitute the segment AB. A and B are the end-points of the segment. Axiom VI. If points C and D ( C 4s D ) lie on the line AB, then A lies on the line CD. Axiom VII. If there exist three distinct points, there exist three distinct points A, B, C not in any of the orders ABC, BCA, or CAB. Axiom VIII. If three distinct points A, B and C do not lie on the same line, and D and E are two points in the orders BCD and CEA, then a point F exists in the order AFB and such that D, E, F lie on the same line. Axiom II may be divided into two parts which I will call Axiom Hi and Axiom II2. Axiom IL.. If A, B, C are three distinct points in the order ABC, they are in the order CBA . Axiom II2. If the points A, B, C are not all distinct, then, if they are in the order ABC, they are in the order CBA . Though Veblen's Axioms I-XII, as they stand, are mutually independent,