Squaring the circle

Solved, 1882GeometryAlgebra

Statement

Using only a compass and an unmarked straightedge, construct in finitely many steps a square whose area equals the area of a given circle.

Squaring a circle of radius 1 by compass and straightedge would require constructing a segment of length $\sqrt{\pi}$ , and Pierre Wantzel had shown in 1837 that every length constructible this way is an algebraic number obtained by a tower of square roots — in particular, algebraic of degree a power of 2. Ferdinand von Lindemann closed the problem in 1882 by proving that $\pi$ itself is transcendental (not a root of any nonzero polynomial with rational coefficients), adapting the method Charles Hermite had used in 1873 to prove $e$ transcendental. Since $\pi$ is transcendental, so is $\sqrt{\pi}$ , and no compass-and-straightedge construction can produce it.

Solved by, 1882Ferdinand von Lindemann

Wantzel's same algebraic framework simultaneously settled the other two classical problems: doubling the cube (impossible, since $\sqrt[3]{2}$ has degree 3, not a power of 2) and trisecting an arbitrary angle (impossible in general, since it would require solving a cubic). All three become possible if the tools are extended — for instance, with a marked ruler (neusis constructions) or by allowing origami folds (which can solve cubics) — and approximate quadratures of the circle accurate to many decimal places, such as those found by Ramanujan, remain popular constructions even though no exact one can exist.

References

Ferdinand von Lindemann (1882). Über die Zahl π
Pierre Laurent Wantzel (1837). Recherches sur les moyens de reconnaître si un problème de géométrie peut se résoudre avec la règle et le compas
Petr Beckmann (1971). A History of Pi

History of the problem

450 BCEGreek geometers such as Anaxagoras and Hippocrates of Chios study the problem of constructing a square equal in area to a given circle, one of the three classical construction problems of antiquity alongside doubling the cube and trisecting the angle.
1837Pierre Wantzel characterizes exactly which lengths are constructible by compass and straightedge — they must be algebraic numbers reachable by a finite tower of square roots — giving a rigorous framework for proving constructions impossible.
1873Charles Hermite proves that $e$ is transcendental, introducing analytic techniques that would soon be adapted to $\pi$ .
1882Ferdinand von Lindemann proves $\pi$ is transcendental; since squaring the circle would require constructing the transcendental length $\sqrt{\pi}$ , the construction is proved impossible.Ferdinand von Lindemann