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Squaring the circle is a problem proposed by ancient geometers. It is the challenge to construct a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. 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 problem was proven to be impossible, as a consequence of the fact that that pi (π) is transcendental, not algebraic; that is, it is not the root of any polynomial with rational coefficients. That the transcendence of π would have that consequence had already been known for some decades; but the transcendence was proved in 1882. Approximate squaring to any given nonzero tolerance, on the other hand, is possible in a finite number of steps, corresponding to the fact that there are rational numbers arbitrarily close to π. The term quadrature of the circle is sometimes used synonymously, or may 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 1800BC gives the area of a circle as , where is the diameter of the circle. Indian mathematicians also found an approximate method, though less accurate, documented in the Sulba Sutras. Indian mathematicians also gave an approximate solution to the problem of circling the square. The first person to be associated with the problem in Greece 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. 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 Birds. It is believed that Oenopides was the first person 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 incorrect, it was the first paper to attempt to solve the problem using algebraic properties of π. It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility. Impossibility A solution of the problem of squaring the circle by compass and straightedge demands construction of the number , and the impossibility of this undertaking follows from the fact that π is a transcendental number—that is, it is non-algebraic and therefore a non-constructible number. If one solves the problem of the quadrature of the circle using only compass and straightedge, then one has also found an algebraic value of π, which is impossible. Johann Heinrich Lambert conjectured that π was transcendental in 1768 in the same paper he proved its irrationality, even before the existence of transcendental numbers was proved. It wasn't until 1882 that Ferdinand von Lindemann proved its transcendence. It is possible to construct a square with an area arbitrarily close to that of a given circle. If a rational number is used as an approximation of π, then squaring the circle becomes possible, depending on the values chosen. However, this is only an approximation and does not meet the constraints of the ancient rules for solving the problem. Several mathematicians have demonstrated workable procedures based on a variety of approximations. Bending the rules by allowing an infinite number of compass-and-straightedge operations or by performing the operations on certain non-Euclidean spaces also makes squaring the circle possible. For example, although the circle cannot be squared in Euclidean space, it can in Gauss-Bolyai-Lobachevsky space. Note that the transcendence of π implies the impossibility of exactly "circling" the square, as well as of squaring the circle. Modern approximative constructions Though squaring the circle is an impossible problem using only compass and straightedge, approximations to squaring the circle can be given by constructing lengths close to π. It takes only minimal knowledge of elementary geometry to convert any given rational approximation of π into a corresponding compass-and-straightedge construction, but constructions made in this way tend to be very long-winded in comparison to the accuracy they achieve. After the exact problem was proved unsolvable, some mathematicians have applied their ingenuity to finding elegant approximations to squaring the circle, defined roughly (and informally) as constructions that are particularly simple among other imaginable constructions that give similar precision. Among the modern approximate constructions was one by E. W. Hobson in 1913. This was a fairly accurate construction which was based on constructing the approximate value of 3.14164079..., which is accurate to 4 decimals. Indian mathematician Srinivasa Ramanujan in 1913, C. D. Olds in 1963, Martin Gardner in 1966, and B. Bold in 1982 all gave geometric constructions for which is accurate to 6 decimal places of π. Srinivasa Ramanujan in 1914 gave a ruler and compass construction which was equivalent to taking the approximate value for π to be ight)^ = sqrt4 = 3.1415926525826461253dots giving a remarkable 8 decimal places of π. In 1991, Robert Dixon gave constructions for and (Kochanski's approximation), though these were only accurate to 4 decimal places of π. Squaring or quadrature as integration The problem of finding the area under a curve, known as integration in calculus, or quadrature in numerical analysis, was known as squaring before the invention of calculus. Since the techniques of calculus were unknown, it was generally presumed that a squaring should be done via geometric constructions, that is, by compass and straightedge. For example Newton wrote to Oldenberg in 1676 "I believe M. Leibnitz will not dislike ye Theorem towards ye beginning of my letter of pag. 4 for squaring Curve lines Geometrically."* After Newton and Leibniz invented calculus, they still referred to this integration problem as squaring a curve. "Squaring the circle" as a metaphor The mathematical proof that the quadrature of the circle is impossible using only compass and straightedge has not proved to be a hindrance to the many people who have invested years in this problem anyway. Having squared the circle is a famous crank assertion. (See also pseudomathematics.) 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. Aleister Crowley used the metaphor in a different sense, to represent the goal of magick and mysticism. He implicitly associated his system of Thelema with π. For more information, see Abrahadabra. 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