Dr. Sarah's Notes on Gauss

In about 300 BC Euclid wrote The Elements, a book which was to become one of the most famous books ever written. Euclid stated five postulates on which he based all his theorems:

  1. To draw a straight line from any point to any other.
  2. To produce a finite straight line continuously in a straight line.
  3. To describe a circle with any centre and distance.
  4. That all right angles are equal to each other.
  5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

It is clear that Euclid's fifth postulate (#5 above) is different from the other four as it is hard to understand and much more complicated. Its lack of simplicity did not satisfy Euclid and he tried to avoid its use as long as possible - in fact the first 28 propositions of The Elements are proved without using it. Another comment worth making at this point is that Euclid, and many that were to follow him, assumed that straight lines were infinite.

Playfair tried to replace Euclid's 5th postulate with a simpler formulation (which is the form that we use in high school today) Playfair's Axiom:- Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.

A non-Euclidean geometry is a geometry that does not satisfy Euclid's postulates.

If Playfair's postulate is false (and it is not always possible to draw exactly one line through the given point parallel to the line) then there are 2 possibilities:

1) There is some point and line so that there are no parallels (spherical geometry)

2) There is some point and line so that there are at least 2 parallels (hyperbolic geometry).

People debated about the validaty and necessity of Euclid's 5th postulate for thousands of years. While the geometries that arise from the negation of Euclid's 5th postulate (1) and 2) above) are certainly different than Euclidean geometry (Euclid's geometry) and also seem impossible, these new geometries do exist and are important. They represent a change in world view and require thinking that is "outside of the box".