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.Among other constructions given by Ramanujan in 1914 (Approximate geometrical constructions for Ï, Quarterly Journal of Mathematics XLV (1914), 350-374) was a ruler and compass construction which was equivalent to taking the strange yet remarkable approximate value for Ï to be (92+ 192/22)1/4. Now this is 3.1415926525826461253.... which differs from Ï only in the ninth decimal place (Ï = 3.1415926535897932385...). For a circle of diameter 8000 miles, the error in the length of the side of the square constructed was only a fraction of an inch.