| Squaring The Circle |
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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 Axiom s 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 fact that Pi (π) is a Transcendental , rather than 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 ''if'' π ''were'' transcendental then the construction would be impossible, but that π ''is'' transcendental was not proven until 1882. Approximate squaring to any given non-perfect accuracy, on the other hand, is possible in a finite number of steps, as a consequence of 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 , St Andrews University .
:A text found at Saquara dating to c 3000 BC or 5000 years BP has a picture of a curve and under the curve the dimensions given in fingers to the right of the circle as reconstructed to the right. It was presumed the horizontal spacing was based on a royal cubit but the results if based on an ordinary cubit are different. It appears the value for PI being used is 3 '8 '64 '1024. which at 3.141601563 is slightly better than the Rhind value. :The circumference of the circle is 1200 fingers and the diameter of the circle is 191 x 2 = 382 :3 '8 '64 '1024 x 382 ~= 1200.0 :The side of the square is 12 royal cubits and its area is 434 square feet. :The area of the circle is 191^2 x 3.141601563. :The algorithm suggests working with coordinates and numerical analysis to define a curve. :1 :1 1 :1 2 1 :1 3 3 1 :1 4 6 4 1 :1 5 10 10 5 1 :3 :3 + 1/2y^3 is 3 '8, = 3.125 :3 + 1/2y^3 + 1/2y^6 is 3 '8 '64,= 3.140625 :3 + 1/2y^3 + 1/2y^'''6''' + 1/2y^'''10''' is 3 '8 '64 '1024 = '''3.141601563''' :For purposes of comparison(3 '7 = 3.142857143) of attempts at the problem.Florian Cajori, ''A History of Mathematics'', second edition, p.143, New York: The Macmillan Company, 1919.]] 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 up1. 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 Space s also makes squaring the circle possible. For example, although the circle cannot be squared in Euclidean space, it can in Gauss-Bolyai-Lobachevsky Space (hyperbolic geometric 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 Benjamin 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 : giving a remarkable 8 decimal places of π. In 1991, Robert Dixon gave constructions for
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