The immediate concern of this science is the means of rigorous thinking--undefined terms and definitions, axioms and propositions. Its collateral concern is the things to which these may apply, the things which may be thought about rigorously--everything. But now the mathematician's domain is so vastly extended that it becomes more than ever important for him to attain the utmost generality in all his pronouncements. One barrier to such generalization is the very name "geometry," with the restricted significance which its derivation and long usage carry. The geometer therefore must have it distinctly understood that for him "geometry" means simply the process of deducing a set of propositions from a set of undefined primitive terms and axioms; and that when he speaks of "a geometry" he means some particular set of propositions so deduced, together with the axioms, etc., on which they are based. If you take a new set of axioms you get a new geometry. The geometer will, if you insist, go on calling his undefined terms by the familiar names "point," "line," "plane." But you must distinctly understand that this is a concession to usage, and that you are not for a moment to restrict the application of his statements in any way. He would much prefer, however, to be allowed new names for his elements, to say "We start with three elements of different sorts, which we assume to exist, and to which we attach the names A, B and C--or if you prefer, primary, secondary and tertiary elements--or yet again, names possessing no intrinsic significance at all, such as ching, chang and chung." He will then lay down whatever statements he requires to serve the purposes of the ancient axioms, all of these referring to some one or more of his elements. Then he is ready for the serious business of proving that, all his hypotheses being granted, his elements A, B and C, or I, II and III, or ching, chang and chung, are subject to this and that and the other propositions.