The parallel postulate was Euclid's famous, and sometimes infamous, fifth postulate. Euclid's five postulates, or axioms, as they are sometimes called, appear in his masterpiece the Elements, one of the most influential mathematics books ever written. Euclid's program in theElements was to build a deductive geometry starting with the minimum number of postulates necessary. Since postulates are essentially assumptions stated without proof and since a deductive system is primarily about proving theorems, mathematicians from Euclid (c. 300 BC) to the present have desired to assume as little as necessary and prove as much as possible. Euclid thought that five, but no more than five, postulates were necessary to build up his deductive geometry. The first four of Euclid's postulates caused no controversy at all. They were four simple assumptions that nobody could doubt; and those are the properties, simplicity and certainty, for which one looks in...