Chapter 1.1

 Geometry and Its
 Basic Terms

We take as the basis of every geometry the set of undefined elements (point, line, plane) which constitute space, the set of undefined relations (incidence, intermediacy, congruence) and the set of basic apriori assertions: axioms (postulates). All other elements and relations are defined by means of these primitive concepts, while all other assertions (theorems) are derived as deductive consequences of primitive propositions (axioms). So that, the character of space (this is, its geometry) is determined by the choice of the initial elements and their mutual relations expressed by axioms. The axioms of the usual approaches to geometry can be divided into a number of groups: axioms of incidence, axioms of order, axioms of continuity, axioms of congruence and axioms of parallelism. Geometry based on the first three groups of axioms is called "ordered geometry ", while geometry based on the first four groups of axioms is called "absolute geometry "; to the latter corresponds the n-dimensional absolute space denoted by Sn.

With respect to congruence, we distinguish the analytic procedure with the introduction of space metric and the synthetic procedure, also called non-metric. The justification for the name äbsolute geometry" is derived from the fact that the system of axioms introduced makes possible a branching out into the geometry of Euclid and that of Lobachevsky (hyperbolic geometry). This is achieved by adding the axiom of parallelism. By accepting the 5th postulate of Euclid (or its equivalent, Playfair's axiom of parallelism: "For each point A and line a there exists in the plane (aA) at most one line p which is incident with A and disjoint from a", where line p is said to be parallel to a) we come to Euclidean geometry. By accepting Lobachevsky's axiom of parallelism, which demands presence of at least two such lines, we come to non-Euclidean hyperbolic geometry, i.e. the geometry of Lobachevsky and that of space Ln. In particular, for n = 0 all these spaces are reduced to a point, and for n = 1 to a line; their specific characteristics come to full expression for n = 2, and we distinguish the absolute (S2), the Euclidean (E2), and the hyperbolic plane (L2). If there is no special remark, then the terms "plane" and "space" refer to the Euclidean spaces E2 and E3 respectively. By a similar extension of the set of axioms, ordered geometry supplemented with two axioms of parallelism becomes affine geometry (H.S.M. Coxeter, 1969).