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Foundations of geometry

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

  1. The ``High School Geometry'' text book approach.
  2. Hilbert's ``Foundations of Geometry'' approach.
  3. Through Projective Geometry as in Coxeters' ``Non-Euclidean Geometry''.
  4. Trough the study of the Euclidean group as done by Sophus Lie.
We shall examine the middle two approaches in the following text. The first method which was learned in school should now be forgotten since we are looking at (to paraphrase Klein) ``elementary mathematics from an advanced standpoint''.

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 $ \alpha$, $ \beta$, $ \gamma$, ...). 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.

next up previous
Next: The Axioms of Incidence Up: euclid Previous: euclid
Kapil H. Paranjape 2001-01-20