What is Euclid's Axiom?

Answer 1

Euclid posited five axioms, statements whose truth supposedly does not require a proof, as the foundation of his work, the Elements. These still hold for plane geometry, but do not hold in the higher non-euclidean systems. The five axioms Euclid proposed are;

• Any two points can be connected by one, and only one, straight line.
• Any line segment can be extended infinitely
  
• For any point, and a line emerging from it, a circle can be drawn where the point is the centre and the line is the radius.
• All right angles are equal
• Given a line, and a point not on the line, there is only line that goes through the point that does not meet the other line. (basically, there is only one parallel to any given line)
  

This last point is controversial as it has been argued effectively that this is not in fact self evident. In fact, ignoring the fifth axiom was the starting point for many Non-Euclidean geometries. For this reason, it is probably this which is best known as Euclid's Axiom.