In its earliest days, geometry was empirical and inductive. Ancient measurers noticed repetitions and patterns in observed facts. They used these observations of fact as foundations for establishing geometric rules. For example, the ancient Egyptians noticed the pattern that every triangle they observed with sides of the proportional lengths of 3, 4, and 5 seemed to form a right angle. Based on these observations, they accepted that every 3-4-5 triangle, as a rule, formed a right angle. The professional Egyptian “rope stretchers” used this rule found by observation and induction to accurately survey parcels of land.
The true father of deductive geometry (which, conversely, uses rules to establish facts, as well as other rules) was probably Thales of Miletus (635-543 BC), who by the way also fathered natural philosophy (he and the the goddess of wisdom seemed to have a very intimate and fruitful relationship). Thales is said to have traveled to Babylon (another cradle of geometry) and Egypt, and in those travels may have learned inductive geometry from priests and scribes. He subsequently applied his genius to placing the science upon a more secure deductive footing.
Thales allegedly used his understanding of geometry to measure the height of the great pyramids with reference to his own shadow and that of the pyramid. To do this, he must have started with the premise that light always travels in straight lines. He deduced from this premise, and his understanding of triangles, that the ratio between the heights of two nearby bodies and their respective shadows should be equivalent. (This wouldn't be true if light happened to curve for some bodies, making their shadows proportionally longer or shorter than the shadows of other bodies.) So, Thales waited until the length of his own shadow was equal to his height. Assuming his premises were true, at this exact time, the height of the pyramid's shadow should be equal to the height of the pyramid itself. So then it was a simple matter of measuring the pyramid's shadow to determine its height (which mercifully involves no climbing).
Next in this series: The Worldview of the Theologos.