View source for Quadrature of the circle

{{TEX|done}}{{MSC|51M04}}

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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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)</TD></TR></table>



====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, [[#References|[a6]]]. Cf. [[Tarski problem]] for results on this problem.

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