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. 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. 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). After Lindemann's impossibility proof, the problem was considered to be settled by professional mathematicians, and its subsequent mathematical history is dominated by pseudomathematical attempts at circle-squaring constructions, largely by amateurs, and by the debunking of these efforts. It was not until 1882 that Ferdinand von Lindemann proved the transcendence of π {\displaystyle \pi } and so showed the impossibility of this construction.

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. 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. 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 } .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 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. 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. Having taken their lead from this problem, I believe, the ancients also sought the quadrature of the circle.

Although much more precise numerical approximations to π {\displaystyle \pi } were already known, Kochański's construction has the advantage of being quite simple. There exist in the hyperbolic plane ( countably) infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area, which, however, are constructed simultaneously.

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. 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.

color {red}640\;\ldots },} where φ {\displaystyle \varphi } is the golden ratio, φ = ( 1 + 5 ) / 2 {\displaystyle \varphi =(1+{\sqrt {5}})/2} . This value is accurate to six decimal places and has been known in China since the 5th century as Milü, and in Europe since the 17th century. Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of π {\displaystyle \pi } . If the circle could be squared using only compass and straightedge, then π {\displaystyle \pi } would have to be an algebraic number.The hyperbolic plane does not contain squares (quadrilaterals with four right angles and four equal sides), but instead it contains regular quadrilaterals, shapes with four equal sides and four equal angles sharper than right angles. 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.