Lotusland of Mathematical Dreams

1. There is a unique straight line segment connecting any two points

2. Unlimited (continuous) extendability of any straight line segment .

3. Existence of a circle with any center and any value for radius .

4. Equality of right angles.

5. If two straight line segments, a and b in a plane intersect another straight line c such that the sum of the interior angles on the same side of c is less than two right angles, then a and b, extended far enough on that side of c, will intersect somewhere.

Playfair’s Axiom is an alternate variation of the 5th Euclidean postulate (parallel postulate). It states that for any line and any point that is not on the line, there is a unique straight line through the point which is parallel to the line.

Parallel postulate has a lot of story attached to it. History says of a lot of people who tried to make the fifth postulate a theorem, derivable from the rest four. But finally Beltrami proved the independence of 5th postulate from others.

Remarkable is the effort by Saccheri, which speaks of the origins of elliptic geometry and hyperbolic geometry.