The quadrature of the circle, better known as squaring the circle, is a classical problem of mathematics, or more specifically, of geometry.
Construction
The problem is to construct, using only ruler-and-compass constructions, a square with the same area as a given circle. The problem dates back to the invention of geometry, and occupied mathematicians for centuries. It was not until 1882 that the impossibility was proven rigorously, though even the ancient geometers had a very good practical and intuitive grasp of its intractability. It should be noted that it is the limitation to just compass and straightedge that makes the problem difficult. If other simple instruments, for example something which can draw an Archimedean spiral, are allowed then it is not difficult to draw a square and circle of equal area.
A solution demands construction of the number 
, and the impossibility of this undertaking follows from the fact that π is a transcendental number, i.e. it is non-algebraic, and therefore a non-constructible number. The transcendence of π was proven by Ferdinand von Lindemann in 1882. If you solve the problem of the quadrature of the circle, this means you have also found an algebraic value of π — this is impossible. This does not imply that it is impossible to construct a square with an area very close to that of a given circle.
"Squaring the circle" as a metaphor
The mathematical proof that the quadrature of the circle is impossible has not hindered many "free spirits" to invest years in this problem anyway. The futility of undertaking exercises aimed at finding the quadrature of the circle has brought this term into use in totally unrelated contexts, where it is simply used to mean a hopeless, meaningless, or vain undertaking.
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