Squaring the circle
Using only a compass and an unmarked straightedge, can one construct in finitely many steps a square with exactly the same area as a given circle? Prove your answer.
History and references
Solved in 1882 by Ferdinand von Lindemann.
The problem goes back to ancient Greek geometry, and Anaxagoras is the first Greek known to have worked on it. In 1837 Pierre Wantzel showed that every length constructible with compass and straightedge is an algebraic number. Squaring a circle of radius 1 means constructing a segment of length , so the construction would be impossible if were transcendental. In 1882 Ferdinand von Lindemann proved exactly that, so the answer is no. Constructions that approximate the area to any desired accuracy are still possible.
References
Solutions 0
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