1.2.2.3 Unintended Consequences
The Greeks studied the 45 - 45 -90 degree triangle shown below.
They then came upon a major problem.
They realised that even if you drew the sides AB and BC exactly to be one inch
the length of AC could not be calculated ,.
This is because Pythagorras' Theorem determines it to be 
and try as they might
no-one could write down precisely what was.
This is impossible .
One can go fro a to B exactly,
one can go from B to C exactly
but one could not go from A to C exactly????
The Greeks thought that this was nonsense and so abandoned all interest in numbers.
Thus Greek mathematics concentrated almost exclusively on Geometry
and the world had to wait another 500 years
before Algebra was discovered.