Squaring the circle - 'Thinking the unthinkable'

Methods to approximate the area of a given circle with a square were known already to Babylonian mathematicians. The Rhind papyrus in 1800BC gives the area of a circle as 64 / 81d2, where d is the diameter of the circle. Indian mathematicians also found an approximate method, though less accurate, documented in the Sulba Sutras. Indian mathematicians also gave an approximate solution to the problem of circling the square.

The first person to be associated with the problem in Greece was Anaxagoras, who worked on it while in prison. Hippocrates of Chios squared certain lunes, thought to be in the hope it will lead to a solution. Antiphon the Sophist believed that inscribing regular polygons within a circle and doubling the number of sides will eventually fill up the area of the circle, and since a polygon can be squared, it means the circle can be squared. Even then there were skeptics - Eudemus argued that magnitudes cannot be divided up without limit, so the area of the circle will never be used up. The problem was even mentioned in Aristophenes's play Birds.