The mathematics contained in Euclid's Elements are based on five axioms and five postulates. Euclid regarded these ten statements as fundamental truths that had to be accepted without proof. Later mathematicians asked whether all of these statements really had to be accepted on faith or whether one or more could be stated as a theorem which could be proved. Among the first to pursue this question vigorously was the Italian mathematician Girolamo Saccheri (1667-1733). In the early 1700s, Saccheri chose to study Euclid's postulate about parallel lines, namely his assumption that through any given point not on a line, one and only one line can be drawn parallel to the given line. Saccheri's approach was to assume that the postulate is not true, and then to look for some contradiction that would result from that assumption. If he could do so, he would be able to...