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Concept
Squaring the circle
Summary
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.
In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi () is a transcendental number.
That is, is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found.
Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i.e. the work of mathematical cranks). The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible. The term quadrature of the circle is sometimes used as a synonym for squaring the circle, but it may also refer to approximate or numerical methods for finding the area of a circle.
Methods to calculate the approximate area of a given circle, which can be thought of as a precursor problem to squaring the circle, were known already in many ancient cultures. These methods can be summarized by stating the approximation to pi that they produce. In around 2000 BCE, the Babylonian mathematicians used the approximation , and at approximately the same time the ancient Egyptian mathematicians used . Over 1000 years later, the Old Testament Books of Kings used the simpler approximation . Ancient Indian mathematics, as recorded in the Shatapatha Brahmana and Shulba Sutras, used several different approximations to . Archimedes proved a formula for the area of a circle, according to which .
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