Hippocrates of Chios: the quadrature of a lune 1

    Hippocrates of Chios was a merchant who fell in with a pirate ship and lost all his possessions. He came to Athens to prosecute the pirates and, staying a long time in Athens by reason of the indictment, consorted with philosophers, and reached such proficiency in geometry that he tried to affect the quadrature of the circle. He did not discover this, but having squared the lune3  he falsely thought from this that he could square the circle also. For he thought that from the quadrature of the lune the quadrature of the circle could also be calculated. 4
 
 

    Eudemus6, however, in his History of Geometry says that Hippocrates did not demonstrate the quadrature of the lune on the side of a square but generally, as one might say. 7  For every lune has an outer circumference equal to a semicircle or greater or less, and if Hippocrates squared the lune having an outer circumference equal to a semicircle and greater and less, the quadrature would appear to be proved generally. I shall set out what Eudemus wrote word for word, adding only for the sake of clearness a few things taken from Euclid's Elements on account of the summary style of Eudemus, who set out his proofs in abridged form in conformity with the ancient practice. 8  He writes thus in the second book of the History of Geometry.
 

    "The quadratures of lunes, which seemed to belong to an uncommon class of propositions by reason of the close relationship to the circle, were first investigated by Hippocrates, and seemed to be set out in correct form; therefore we shall deal with them at length and go through them. He made his starting-point, and set out as the first of the theorems useful to this purpose, that similar segments of circles have the same ratios as the squares on their bases.9  And this he proved by showing that the squares on the diameters have the same ratios as the circles. 10
    "Having first shown this he described in what way it was possible to square a lune whose outer circumference was a semicircle. He did this by circumscribing about a right-angled isosceles triangle a semicircle and about the base a segment of a circle similar to those cut off by the sides. 11  Since the segment about the base is equal to the sum of those about the sides 12 , it follows that when the part of the triangle above the segment about the base is added to both the lune will be equal to the triangle. Therefore the lune, having been proved equal to the triangle, can be squared.13  In this way, taking a semicircle as the outer circumference of the lune, Hippocrates readily squared the lune."