One of the greatest engineering achievements of ancient times is a water tunnel, 1,036 meters (4,000 feet) long, excavated through a mountain on the Greek island of Samos in the sixth century B. C. It was dug through solid limestone by two separate teams advancing in a straight line from both ends, using only picks, hammers, and chisels. This was a prodigious feat of manual labor. The intellectual feat of determining the direction of tunneling was equally impressive. How did they do this?
No one knows for sure, because no written records exist. When the tunnel was dug, the Greeks had no magnetic compass, no surveying nstruments, no topographic maps, nor even much written mathematics at their disposal. Euclid’s Elements, the first major compendium of ancient mathematics, was written some 200 years later. There are, however, some convincing explana-tions, the oldest of which is based on a theoretical method devised by Hero of Alexandria five centuries after the tunnel was completed.
It calls for a series of right-angled traverses around the mountain beginning at one entrance of the proposed tunnel and ending at the other, main-taining a constant elevation, as suggested by the diagram below left. By measuring the net istance traveled in each of two perpendicular directions, the lengths of two legs of a right triangle are determined, and the hypotenuse of the triangle is the proposed line of the tunnel. By laying out smaller similar right triangles at each entrance, markers can be used by each crew to determine the direction for tunneling.
Later in this article I will apply Hero’s method to the terrain on Samos. Hero’s plan was widely accepted for nearly 2,000 years as the method used on Samos until two British historians of science visited the site in 1958, saw that the terrain would have made this method unfeasible, and suggested an alternative f their own. In 1993, I visited Samos myself to investigate the pros and cons of these two methods for a Project MATHEMATICS! ideo program, and realized that the engineering problem actually to be determined at the same elevation above sea level; and second, the direction for tunneling between these points must be established. I will describe possible solutions for each part; but first, some historical background. Samos, Just off the coast of Turkey in the Aegean Sea, is the eighth largest Greek island, with an area of less than 200 square miles. Separated from Asia Minor by the narrow Strait f Mycale, it is a colorful island with lush vegeta-tion, beautiful bays and beaches, and an abun-dance of good spring water.
Samos flourished in the sixth century B. C. during the reign of the tyrant Polycrates (570-522 B. C. ), whose court attracted poets, artists, musicians, philosophers, and mathematicians from all over the Greek world. His capital city, also named Samos, was situated on the slopes of a mountain, later called Mount Castro, dominating a natural harbor and the narrow strip of sea between Samos and Asia Minor. The historian Herodotus, who lived in Samos in 457 B. C. , described it as the most famous city of its time.