Euclidean Geometry is the attempt to build geometry out of the rules
of logic combined with some ``evident truths'' or axioms. The axioms
of Euclidean Geometry were *not* correctly written down by Euclid,
though no doubt, he did his best. There are now a number of different
ways of giving the formal basis for the *same* geometry. These
are

- The ``High School Geometry'' text book approach.
- Hilbert's ``Foundations of Geometry'' approach.
- Through Projective Geometry as in Coxeters' ``Non-Euclidean Geometry''.
- Trough the study of the Euclidean group as done by Sophus Lie.

The method that (to my mind) comes closest to the original approach is
that of Hilbert's Foundations of Geometry. Unlike the
``High School Geometry'' text books, this makes no reference to the
``Ruler Placement Postulate'' or a ``Protractor Placement Postulate'',
both of which are anti-thetical to a purely geometric approach. The
arithmetic aspects of geometry should grow out of it rather than be
imposed from outside. Another difference is that the notion of a line
is not *as* a set of points in Euclid's approach; points, lines
and planes are distinct notions in Hilbert's approach too.

Without much more ado then let us examine Hilbert's axioms for
Euclidean geometry. The fundamental notions are points (denoted by
*A*, *B*, *C*, ...), lines (denoted by *a*, *b*, *c*, ...)
and planes (denoted by , , , ...). The
mutual relations between these are those of Incidence (``contains'' or
``lies on''), Order (``is between'') and Congruence. The axioms
characterise the ``evident'' or fundamental properties of these
relations. We divide the axioms into four classes, Incidence, Order,
Parallels, Continuity, Congruence.

- The Axioms of Incidence
- Axioms of Order
- Axiom of Parallels
- Models
- Putting co-ordinates
- Suggested Reading