Quadrature of the Lune

Ancient Greek mathematicians were enchanted by the beauty, symmetry, and order of geometry.Succumbing to this passion, Greek mathematician Hippocrates of Chios demonstrated how to construct a square with an area equal to a particular lune. A lune is a crescent-shaped area, bounded by two concave circular arcs, and this Quadrature of the Lune is one of the earliest-known proofs in mathematics In other words, Hippocrates demonstrated that the areas of these lunes could be expressed exactly asa rectilinear area, or "quadrature."
For the ancient Greeks, finding the quadrature meant using a straightedge and compass to construct a square equal in area to a given shape. If such construction is possible, the shape is said to be "quadrable" ( or "squarable"). The Greeks had accomplished the quadrature of polygons, but curved forms were more difficult. In fact, it must have seemed unlikely, at first, that curved objects could be quadrable at all.
Hippocrates is also famous for compiling the first-known organized work on geometry, nearly a century before Euclid. Euclid may have used some of Hippocrates' ideas in his own work, Elements. Hippocrates' writings were significant because they provided a common framework upon which other mathematicians could build.
Hippocrates' lune quest was actually part of a research effort to achieve the "quadrature of the circle" - that is, to construct a square with the same area as the circle. Mathematicians have tried to solve the problem of "squaring the circle" for more than 2,000 years, until Ferdinand von Lindemman in 1882 proved that it is impossible. Today, we know that only five types of lunes are quadrable. Three of these were discovered by Hippocrates, and two more were found in the mid-1700s.