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Circling the Square: Cwmbwrla, Coronavirus and Community

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James Gregory attempted a proof of the impossibility of squaring the circle in Vera Circuli et Hyperbolae Quadratura (The True Squaring of the Circle and of the Hyperbola) in 1667. There is no method for starting with an arbitrary regular quadrilateral and constructing the circle of equal area. In contrast, Eudemus argued that magnitudes cannot be divided up without limit, so the area of the circle would never be used up. Having taken their lead from this problem, I believe, the ancients also sought the quadrature of the circle. color {red}640\;\ldots },} where φ {\displaystyle \varphi } is the golden ratio, φ = ( 1 + 5 ) / 2 {\displaystyle \varphi =(1+{\sqrt {5}})/2} .

Lindemann was able to extend this argument, through the Lindemann–Weierstrass theorem on linear independence of algebraic powers of e {\displaystyle e} , to show that π {\displaystyle \pi } is transcendental and therefore that squaring the circle is impossible. In modern mathematics the terms have diverged in meaning, with quadrature generally used when methods from calculus are allowed, while squaring the curve retains the idea of using only restricted geometric methods. As well, several later mathematicians including Srinivasa Ramanujan developed compass and straightedge constructions that approximate the problem accurately in few steps. It had been known for decades that the construction would be impossible if π {\displaystyle \pi } were transcendental, but that fact was not proven until 1882.It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge. In 1837, Pierre Wantzel showed that lengths that could be constructed with compass and straightedge had to be solutions of certain polynomial equations with rational coefficients. 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. After the exact problem was proven unsolvable, some mathematicians applied their ingenuity to finding approximations to squaring the circle that are particularly simple among other imaginable constructions that give similar precision.

In 1914, Indian mathematician Srinivasa Ramanujan gave another geometric construction for the same approximation. Bending the rules by introducing a supplemental tool, allowing an infinite number of compass-and-straightedge operations or by performing the operations in certain non-Euclidean geometries makes squaring the circle possible in some sense. It takes only elementary geometry to convert any given rational approximation of π {\displaystyle \pi } into a corresponding compass and straightedge construction, but such constructions tend to be very long-winded in comparison to the accuracy they achieve. It was not until 1882 that Ferdinand von Lindemann proved the transcendence of π {\displaystyle \pi } and so showed the impossibility of this construction. One of many early historical approximate compass-and-straightedge constructions is from a 1685 paper by Polish Jesuit Adam Adamandy Kochański, producing an approximation diverging from π {\displaystyle \pi } in the 5th decimal place.Antiphon the Sophist believed that inscribing regular polygons within a circle and doubling the number of sides would eventually fill up the area of the circle (this is the method of exhaustion). It was not until 1882 that Ferdinand von Lindemann succeeded in proving more strongly that π is a transcendental number, and by doing so also proved the impossibility of squaring the circle with compass and straightedge. The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible. The problem of finding the area under an arbitrary curve, now known as integration in calculus, or quadrature in numerical analysis, was known as squaring before the invention of calculus.

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