Something technical

In the last month, had been out over the weekends for campus recruitments. Though hectic, I find these trips particularly enjoyable, as it provides a good change from the routine. And, travelling with friends and colleagues is always fun, especially when you travel with people like Arun ( a friend and colleague of mine ). He is a great story teller and has immense passion for the basic sciences ( mainly Physics ). His narration of historical events/stories that has scientific relevance would make any one interested in the subject. Wish we had school teachers like him.

One of the things he told me during a recent trip, which was interesting:
Euclid, the mathematician who laid the foundation for Geometry, put in some basic axioms ( intutive truths that cannot be / need not be proved ) from which one can derive any geometrical postulate. His axioms were thus:
1. Any two points can be joined by a straight line.
2. Any line segment can be extended indefenitely in a straight line.
3. All right angles are congruent ( meaning similar ).
4. If two lines intersect a third line in such a way that the sum of interior angles on one side is less than two right angles ( 180 degrees ), then the two lines would intersect each other if extended enough on that side. On the converse, if the sum is equal to two right angles, they would never intersect.
( And there is another axiom of his, not relevant to this discussion ).
Now, people thought that the fourth axiom as extraneous. They believed that it can be proved from the other axioms of his. Mathematicians have spent a lifetime trying to prove the 4th axiom from the others, in vain. And there is a case, where a mathematician father writes to his son begging him to give up trying to do that, as he himself had wasted his lifetime on that, and there was more to life than just that :)

Then, mathematicians came up with a parallel geometry where they assumed the converse of that axiom and tried to find inconsistency in that geometry, so that they can prove that this axiom was extraneous. Which was, if two lines intersect a third line such that the sum of internal angles is 180 degrees, the two lines would still meet, and tried to find an inconsistency with that geometry. But, they could not. That geometry was still consistent. They are called non-euclidean geometry.

For the interested - http://en.wikipedia.org/wiki/Euclidean_geometry#The_parallel_postulate