The illustrious Hippocrates of Chios ( c. 470 – c. 410 BCE) worked on a great many interesting problems, one of which was squaring the circle. It was not until 1882 that Carl Louis Ferdinand von Lindemann (April 12, 1852 – March 6, 1939) provided a proof that pi was a transcendental number and that the circle could not be squared using compasses and a straight edge.
Tragically, almost all of Hippocrates’ work was lost, but at least one very interesting result survives. He found that in the image below the area of the shaded figure, known as a lune due to its resemblance to a lunar crescent, was equal to the area of the yellow triangle.
It turns out that there are five constructible lunes and that all are concave. Three were discovered by Hippocrates and two more were reported by Thomas Clausen in 1840. However, those two had already been discovered by Johan Martin Wallenius (March 7, 1731 Paimio – October 22, 1773 Turku) in 1766. I am not clear if Clausen was aware of Wallenius’ result.
More on lunes to follow.