At heart, a geometry consists of a set S of objects, called the points of the geometry.
The figures are subsets of S. Usually, a particular collection of subsets are selected as
the lines of the geometry.
The geometry is defined by a set of axioms. Different geometries have different axioms.
Note that this is quite different from group theory where we have four fixed axioms.
A group is any system which obeys these axioms.
claimed to derive all of euclidean geometry from just
In fact, he made many other assumptions in the course of his proofs.
In 1899, Hilbert
gave a complete description of euclidean geometry using
Five of Hilbert's axioms are for 3-dimensional geometry, so
plane geometry needs 16.
The most important new axioms involved
Euclid regarded it as intuitive that, if three distinct points lie on a line, then
exactly one lies between the other two. Hilbert made this an axiom, thus
defining the concept of betweenness. Some other axioms are needed to
complete the characterization.
Euclid did not actually define congruence. He often considered the conditions sufficient
to ensure that two figures had identical geometrical properties. The first example is the
SAS condition for triangles (Proposition 1.4)
This was one of the few proofs where he used The Principle of Superposition.
Again this is undefined, and there are no axioms to govern its use. Various formulations
have been suggested based on the words "applied to". For our purposes, we take it as
Two figures are congruent if and only if there is a rigid transformation mapping one to the other.
By congruence, we mean having identical geometrical properties.
By a rigid transformation, we mean a map which preserves geometrical properties.
Of course, the "if" part is an obvious consequence of our definitions.
The "only if" is vital - it allows us to move, say,
a line segment
to lie on top of any congruent segment (i.e. one of equal length).
Hilbert's resolution was to introduce axioms to define congruence.
Instead of moving figures, he postulated the ability to construct an exact copy of a
figure at any place on the plane, rotated through any angle, and oriented either way.
The SAS condition is one of the axioms.
In his Erlangen lecture in 1872,
Klein adopted a much more radical approach.
Inspired by the Principle of Superposition, he began with the notion of transformations.
From this he defined congruence and the notion of a geometrical property and then
derived the theorems of geometry without the need for axioms!
This is known as the klein view