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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 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. 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.

As well, several later mathematicians including Srinivasa Ramanujan developed compass and straightedge constructions that approximate the problem accurately in few steps. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge. displaystyle \left(9 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. 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.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. Ancient Indian mathematics, as recorded in the Shatapatha Brahmana and Shulba Sutras, used several different approximations to π {\displaystyle \pi } . 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 the same work, Kochański also derived a sequence of increasingly accurate rational approximations for π {\displaystyle \pi } .

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. If the circle could be squared using only compass and straightedge, then π {\displaystyle \pi } would have to be an algebraic number. The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible. Squaring the circle: the areas of this square and this circle are both equal to π {\displaystyle \pi } .

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. 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). The more general goal of carrying out all geometric constructions using only a compass and straightedge has often been attributed to Oenopides, but the evidence for this is circumstantial. 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. Contemporaneously with Antiphon, Bryson of Heraclea argued that, since larger and smaller circles both exist, there must be a circle of equal area; this principle can be seen as a form of the modern intermediate value theorem.

In 1914, Indian mathematician Srinivasa Ramanujan gave another geometric construction for the same approximation. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found.Although much more precise numerical approximations to π {\displaystyle \pi } were already known, Kochański's construction has the advantage of being quite simple.

Hippocrates of Chios attacked the problem by finding a shape bounded by circular arcs, the lune of Hippocrates, that could be squared. Now imagine that instead of the pattern growing, we start with a square and the pattern continues inwards - with the circles and squares becoming smaller and smaller.The problem of constructing a square whose area is exactly that of a circle, rather than an approximation to it, comes from Greek mathematics. 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. In contrast, Eudemus argued that magnitudes cannot be divided up without limit, so the area of the circle would never be used up. This identity immediately shows that π {\displaystyle \pi } is an irrational number, because a rational power of a transcendental number remains transcendental.