Circle Squaring
Construct a Square equal in Area to a Circle using only a Straightedge and Compass. This was one of the three Geometric Problems of Antiquity, and was perhaps first attempted by Anaxagoras. It wasfinally proved to be an impossible problem when Pi was proven to be Transcendental by Lindemann 
in 1882.
However, approximations to circle squaring are given by constructing lengths close to 
.Ramanujan (1913-14), Olds (1963), and Gardner (1966, pp. 92-93) give geometric constructions for
. Dixon (1991) gives constructions for 
and
.
While the circle cannot be squared in Euclidean Space, it can in Gauss-Bolyai-Lobachevsky Space(Gray 1989).
See also Geometric Construction, Quadrature, Squaring
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 190-191, 1996. Dixon, R. Mathographics. New York: Dover, pp. 44-49 and 52-53, 1991. Dunham, W. ``Hippocrates' Quadrature of the Lune.'' Ch. 1 in Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 20-26, 1990. Gardner, M. ``The Transcendental Number Pi.'' Ch. 8 in Martin Gardner's New Mathematical Diversions from Scientific American. New York: Simon and Schuster, 1966. Gray, J. Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, 2nd ed. Oxford, England: Oxford University Press, 1989. Meyers, L. F. ``Update on William Wernick's `Triangle Constructions with Three Located Points.''' Math. Mag. 69, 46-49, 1996. Olds, C. D. Continued Fractions. New York: Random House, pp. 59-60, 1963. Ramanujan, S. ``Modular Equations and Approximations to 
.'' Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.
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