Squaring the circle took two thousand years to prove impossible
Build a square with exactly the area of a given circle, using only compass and straightedge, in a finite number of steps. Greek geometers tried. So did prisoners, Jesuits and legions of amateurs. In 1882 a proof about the nature of pi closed the question, and the phrase became shorthand for attempting the impossible.
Long before anyone framed the exact puzzle, cultures estimated a circle's area, which amounts to estimating pi. Babylonian mathematicians around 2000 BCE used 25 over 8, while Egyptians at roughly the same time used a fraction close to 3.16. Archimedes pinned pi between about 3.141 and 3.143, and in fifth-century China Zu Chongzhi reached 355 over 113, accurate to six decimal places.
The Greeks posed the exact version because they already knew how to turn any polygon into a square of equal area and wanted to compare curved shapes the same way. Anaxagoras reportedly worked on it in prison. Hippocrates of Chios found a moon-shaped region bounded by arcs, now called his lune, that could be squared. Antiphon the Sophist argued that polygons with ever more sides would eventually fill the circle, while Bryson of Heraclea reasoned that since bigger and smaller circles exist, one must match the square, an early glimpse of the intermediate value theorem.
Failed attempts had side benefits. Grégoire de Saint-Vincent published a flawed squaring in 1647 but, while tackling the hyperbola, helped develop the natural logarithm. The decisive steps came later. Lambert proved pi irrational in 1761. Pierre Wantzel showed in 1837 that constructible lengths must be algebraic numbers, roots of polynomials with rational coefficients. Building on Charles Hermite's 1873 proof about Euler's number, Ferdinand von Lindemann proved in 1882 that pi is transcendental, so the required side length can never be constructed.
The problem differs from its famous siblings, doubling the cube and trisecting an angle, which involve cubic equations and yield to paper folding or neusis. Circle squaring does not, although extra curves such as the quadratrix of Hippias make it possible. Mathematicians including Ramanujan and the Polish Jesuit Adam Adamandy Kochański, in 1685, devised short constructions that come remarkably close.
Source: Squaring the circle
