Around 430 BC (the same time Sparta was invading) a plague hit Athens. (Using modern forensics, scholars now think it was typhus.) According to legend, the Athenians consulted the oracle at the island of Delos, who told them they needed to double the size of their altar. They doubled the length of each side, but this multiplied the volume of the altar by 8, not 2. Even knowing this was the problem, it was difficult for the Greeks to say precisely how much each side needed to be enlarged so the volume would be exactly doubled.
In 350 BC (unfortunately long after the plague had already passed) Menaechmus resolved the doubling the cube problem by solving the equation . To do this he split the equation into two: and .
Why is solving the system of two equations equivalent to solving ?