The Quest to go beyond Euclid

This describes the history of geometry from Euclid in about 300 BC to Europe in the eighteenth century. Euclid laid down five axioms, and geometers writing in Arabic, and later in Latin, made repeated attempts to show that the axiom on parallel lines was a consequence of the other four. I give a convincing but false proof of this. At the end we see why it is wrong, and how in the eighteenth century three mathematicians independently created a plane geometry where the parallel postulate fails. Here is a sample.