Quadrature of the circle

The problem of constructing a square of equal area as the given circle; one of the classical Ancient problems on constructions with a ruler and compass. The side of a square equal in area to a circle of radius $r$ has length $r\sqrt\pi$. Thus the problem of the quadrature of the circle reduces to the following: To construct a line of length $\sqrt\pi$. Such a construction cannot be realized with a ruler and compass since $\pi$ is a transcendental number, as was proved in 1882 by F. Lindemann. However, the problem of the quadrature of a circle is solvable if one extends the means of construction, for example, by using certain transcendental curves, called quadratrices (cf. Quadratrix).

References

[1]	Yu.I. Manin, "Ueber die Lösbarkeit von Konstruktionsaufgaben mit Zirkel und Lineal" , Enzyklopaedie der Elementarmathematik , 4. Geometrie , Deutsch. Verlag Wissenschaft. (1969) pp. 205–230 (Translated from Russian)

Comments

The Ancient problem of squaring the circle led to a problem in measure theory which asks if a disc and a square of equal area are equi-decomposable, i.e. if the disc can be decomposed into a finite number of disjoint subsets which can be reassembled to form a square, [a6]. Cf. Tarski problem for results on this problem.

References

[a1]	L. Bieberbach, "Theorie der geometrischen Konstruktionen" , Birkhäuser (1952)
[a2]	F. Klein, et al., "Famous problems and other monographs" , Chelsea, reprint (1962) (Translated from German)
[a3]	I. Stewart, "Galois theory" , Chapman & Hall (1973) pp. Chapt. 5
[a4]	B.L. van der Waerden, "Science awakening" , 1 , Noordhoff (1975) (Translated from Dutch)
[a5]	U. Dudley, "A budget of trisections" , Springer (1987)
[a6]	S. Wagon, "Circle squaring in the twentieth century" Math. Intelligencer , 3 : 4 (1981) pp. 176–181
[a7]	E.W. Hobson, "Squaring the circle" , Squaring the circle and other monographs , Chelsea, reprint (1953)
[a8]	O. Perron, "Irrationalzahlen" , de Gruyter (1960)
[a9]	W.W.R. Ball, H.S.M. Coxeter, "Mathematical recreations and essays" , Dover, reprint (1987) pp. 347–359