The Ruler and Compass

Through his oracle at Delos, Apollo informed the Delians that if they wanted to be rid of the plague they must construct a new cubical altar that exactly doubled the volume of the existing one. The Delian Problem then was to construct a cube having a side $$ \sqrt[3]{2} $$ times as long as a side of the original cube. This problem also has the somewhat misleading name The Duplication of the Cube. According to another legend, Eratosthenes reported that the problem was sent to the geometers at Plato’s Academy in Athens. Plato is reported to have said that the god had assigned the task to shame the Greeks for their neglect of mathematics and their contempt for geometry. It was not that the Greeks could not construct segments of the required length by various methods but that they could not do so using only the ruler and compass. That was their task. There is little doubt that the Greeks soon suspected the problem had no solution. However, they lacked the algebra to prove this fact. Our task is to prove the ancient Greeks necessarily failed because they were asking for the impossible. To do this, we must formulate our problems in the language of algebra.