## 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 *S*^{n}.

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 (*a*, *A*) 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 *L*^{n}. 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 (*S*^{2}), the Euclidean
(*E*^{2}), and the hyperbolic plane (*L*^{2}). If there is no special
remark, then the terms "plane" and "space" refer to the Euclidean
spaces *E*^{2} and *E*^{3} 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).