Squaring the circle
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.
Contents
Description
More abstractly and more precisely, it may be taken to ask whether specified axioms of Euclidean geometry concerning the existence of lines and circles entail 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, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients.
It had been known for some decades before then that the construction would be impossible if pi were transcendental, but pi was not proven transcendental until 1882.
Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a finite number of steps, since there are rational numbers arbitrarily close to pi.
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 to mean the same thing as squaring the circle, but it may also refer to approximate or numerical methods for finding the area of a circle.
History
Methods to approximate the area of a given circle with a square were known already to Babylonian mathematicians. The Egyptian Rhind papyrus of 1800 BC gives the area of a circle as (64/81) d 2, where d is the diameter of the circle, and pi approximated to 256/81, a number that appears in the older Moscow Mathematical Papyrus and used for volume approximations (i.e. hekat).
Indian mathematicians also found an approximate method, though less accurate, documented in the Sulba Sutras.
Archimedes showed that the value of pi lay between 3 + 1/7 (approximately 3.1429) and 3 + 10/71 (approximately 3.1408). See Numerical approximations of π for more on the history.
The first known Greek to be associated with the problem was Anaxagoras, who worked on it while in prison. Hippocrates of Chios squared certain lunes, in the hope that it would lead to a solution — see Lune of Hippocrates.
Antiphon the Sophist believed that inscribing regular polygons within a circle and doubling the number of sides will eventually fill up the area of the circle, and since a polygon can be squared, it means the circle can be squared. Even then there were skeptics—Eudemus argued that magnitudes cannot be divided up without limit, so the area of the circle will never be used up.
The problem was even mentioned in Aristophanes's play The Birds.
It is believed that Oenopides was the first Greek who required a plane solution (that is, using only a compass and straightedge).
James Gregory attempted a proof of its impossibility in Vera Circuli et Hyperbolae Quadratura (The True Squaring of the Circle and of the Hyperbola) in 1667. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of pi.
It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility.
See also
External links
- Squaring the circle @ Wikipedia
