Euclid's Postulates
- 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 oneendpoint 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 lessthan two Right Angles, then the two lines inevitably must intersect each other on that side if extendedfar enough. This postulate is equivalent to what is known as the Parallel Postulate.
himselfused only the first four postulates (``Absolute Geometry'') for the first 28 propositions of the Elements,but was forced to invoke the Janos Bolyai andNicolai Lobachevsky independently realized that entirely self-consistent ``Non-EuclideanGeometries'' could be created in which the parallel postulate did not hold. (Gauß had also discovered but suppressed the existence of non-Euclidean geometries.)See also Absolute Geometry, Circle, Elements, Line Segment, Non-Euclidean Geometry,Parallel Postulate, Pasch's Theorem, Right Angle
References
Hofstadter, D. R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 88-92, 1989.
- Semicircle
- Semicolon Derivative
- Semiconvergent Series
- Semicubical Parabola
- Semiderivative
- Semidirect Product
- Semiflow
- Semigroup
- Semi-Integral
- Semilatus Rectum
- Semimagic Square
- Semimajor Axis
- Semiminor Axis
- Semiperfect Magic Cube
- Semiperfect Number
- Semiperimeter
- Semiprime
- Semiprime Ring
- Semiregular Polyhedron
- Semiring
- Andrews-Curtis Link
- Andrews-Schur Identity
- Andrew's Sine
- Andrica's Conjecture
- André's Problem