In the beginning, geometry was a collection of rules for computing lengths, areas, and volumes. Many were crude approximations arrived at by trial and error. This body of knowledge, developed and used in construction, navigation, and surveying by the Babylonians and Egyptians, was passed on to the Greeks. Blessed with an inclination toward speculative thinking and the leisure to pursue this inclination, the Greeks transformed geometry into a deductive science. About 300 b.c., Euclid of Alexandria organized some of the knowledge of his day in such an effective fashion that all geometers for the next 2000 years used his book, The Elements, as their starting point.

First he defined the terms he would use – points, lines, planes, and so on. Then he wrote down five postulates that seemed so clear that one could accept them as true without proof. From this basis he proceeded to derive almost 500 geometrical statements or theorems. The truth of these was in many cases not at all self-evident, but it was guaranteed by the fact that all the theorems had been derived strictly according to the accepted laws of logic from the original (self-evident) assertions.

Although a great breakthrough in their time, the methods of Euclid are imperfect by modern standards. To begin with, he attempted to define everything in terms of a more familiar notion, sometimes creating more confusion than he removed.