Lune of Hippocrates

This afforded some hope of solving the circle-squaring problem, since the lune is bounded only by arcs of circles.

[3] Hippocrates' proof was preserved through the History of Geometry compiled by Eudemus of Rhodes, which has also not survived, but which was excerpted by Simplicius of Cilicia in his commentary on Aristotle's Physics.

[2][4] Not until 1882, with Ferdinand von Lindemann's proof of the transcendence of π, was squaring the circle proved to be impossible.

[5] Hippocrates' result can be proved as follows: The center of the circle on which the arc AEB lies is the point D, which is the midpoint of the hypotenuse of the isosceles right triangle ABO.

In the mid-20th century, two Russian mathematicians, Nikolai Chebotaryov and his student Anatoly Dorodnov, completely classified the lunes that are constructible by compass and straightedge and that have equal area to a given square.