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The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible. 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.

Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of π {\displaystyle \pi } . In Chinese mathematics, in the third century CE, Liu Hui found even more accurate approximations using a method similar to that of Archimedes, and in the fifth century Zu Chongzhi found π ≈ 355 / 113 ≈ 3.

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. However, they have a different character than squaring the circle, in that their solution involves the root of a cubic equation, rather than being transcendental. 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. displaystyle \left(9 Although the circle cannot be squared in Euclidean space, it sometimes can be in hyperbolic geometry under suitable interpretations of the terms.

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. In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi ( π {\displaystyle \pi } ) is a transcendental number.To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. 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. 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. For if a parallelogram is found equal to any rectilinear figure, it is worthy of investigation whether one can prove that rectilinear figures are equal to figures bound by circular arcs.

Greek mathematicians found compass and straightedge constructions to convert any polygon into a square of equivalent area. Although squaring the circle exactly with compass and straightedge is impossible, approximations to squaring the circle can be given by constructing lengths close to π {\displaystyle \pi } . Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i. In contrast, Eudemus argued that magnitudes cannot be divided up without limit, so the area of the circle would never be used up. 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.Ancient Indian mathematics, as recorded in the Shatapatha Brahmana and Shulba Sutras, used several different approximations to π {\displaystyle \pi } . This identity immediately shows that π {\displaystyle \pi } is an irrational number, because a rational power of a transcendental number remains transcendental. If π {\displaystyle {\sqrt {\pi }}} were a constructible number, it would follow from standard compass and straightedge constructions that π {\displaystyle \pi } would also be constructible.

The solution of the problem of squaring the circle by compass and straightedge requires the construction of the number π {\displaystyle {\sqrt {\pi }}} , the length of the side of a square whose area equals that of a unit circle. Two other classical problems of antiquity, famed for their impossibility, were doubling the cube and trisecting the angle. 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. Hippocrates of Chios attacked the problem by finding a shape bounded by circular arcs, the lune of Hippocrates, that could be squared.In 1914, Indian mathematician Srinivasa Ramanujan gave another geometric construction for the same approximation. Although much more precise numerical approximations to π {\displaystyle \pi } were already known, Kochański's construction has the advantage of being quite simple.