泰勒斯疑难The Thales puzzle

31° The Thales puzzle. It has been reported that Thales resided for a time in Egypt, and there evoked admiration by calculating the height of a pyramid by means of shadows. Two versions of the story have been given. The earlier account, furnished by Hieronymus, a pupil of Aristotle, says that Thales noted the length of the shadow of the pyramid at the moment when his own shadow was the same length as himself. The later version, given by Plutarch, says that Thales set up a stick and employed the fact that “the [height of the] pyramid was to the [length of the] stick as the shadow of the pyramid to the shadow of the stick.” Both of these versions fail to mention the very real difficulty, in either case, of obtaining the length of the shadow of the pyramid (that is, the distance from the apex of the shadow to the center of the base of the pyramid).

The above unaccounted-for difficulty has given rise to what might be called the Thales puzzle: Devise a method, based on shadow observations and similar triangles and independent of latitude and specific time of day or year, for determining the height of a pyramid. Figure 4 suggests a solution—that the reader may care to complete—requiring two shadow observations spaced a few hours apart.