Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself
used only the first four postulates (``Absolute Geometry'') for the first 28 propositions of the Elements,
but was forced to invoke the Parallel Postulate on the 29th. In 1823, Janos Bolyai and
Nicolai Lobachevsky independently realized that entirely self-consistent ``Non-Euclidean
Geometries'' could be created in which the parallel postulate did not hold. (Gauß
had also discovered but suppressed the existence of non-Euclidean geometries.)
- 1. A straight Line Segment can be drawn joining any two points.
- 2. Any straight Line Segment can be extended indefinitely in a straight Line.
- 3. Given any straight Line Segment, a Circle can be drawn having the segment as Radius and one
endpoint as center.
- 4. All Right Angles are congruent.
- 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less
than two Right Angles, then the two lines inevitably must intersect each other on that side if extended
far enough. This postulate is equivalent to what is known as the Parallel Postulate.
See also Absolute Geometry, Circle, Elements, Line Segment, Non-Euclidean Geometry,
Parallel Postulate, Pasch's Theorem, Right Angle
Hofstadter, D. R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 88-92, 1989.
© 1996-9 Eric W. Weisstein