The ancient Greeks were very interested in the problem of “squaring the circle” or “the quadrature of the circle”, that is, drawing a square whose area is the same as a given circle, making use of only a straight edge and compasses. It was only in 1882 that Lindemann proved this to be impossible.
The Greeks also investigated a few related problems in their attempt to square the circle, and one of these was the quadrature of the lune: can one construct a square whose area is the same as that of a lune?
The Greek mathematician Hippocrates (about 470-410 BCE) wrote a treatise in which he was able to achieve the quadrature of the lune in three cases, the first of which is question 1 in this resource.
It is very likely that he was also aware of the combination of two lunes in question 2, but we do not have any record of this.
It was long thought that Hippocrates’ three types of (single) lunes were the only ones which could be squared, until two further examples were discovered in the 18th century. It is now known that there are no other types beyond these five which can be squared.1
Heath, Thomas L., A History of Greek Mathematics, Oxford 1921 (reprinted by Dover, 1981), vol. 1, p. 200.↩
