• Hippocrates and his lunules:
• Description of the method reported by Alexander of Aphrodisias for squaring lunules,
• according to Simplicius, Commentary on Aristotle's Physics pp. 55 line 25 - 57 line 29
• in his discussion of the squaring of lunules by Hippocrates of Chios©
by Henry Mendell (Cal. State U., L.A.)

main diagram (lunule and rectilinear figure or inner sector or outer sector)The basic idea is to construct an outer sector A and an inner sector B, where the endpoints of their arcs coincide and satisfying the following condition.  Let RA be the radius of A and RB the radius of B, and let  be the angle of the arc of A and  the angle of the arc of B. Then let  :  = RBsqr : RAsqr = p : q (where p : q is any ratio of magnitudes, not necessarily commensurate). (diagram 2) Since  >  , cut off from  an angle equal to  to make sector A*, a part of A.  Hence A* is similar to B.  Since the radius of A* is R, by the sector theorem that similar sectors are as the squares of their radii,
A* : B = RAsqr : RBsqr = q : p
But
A : A* =  :  = p : q
(diagram 3 or inner sector or outer sector) Hence, ex aequali,
A : B = p : p
That is:  A = B.
(diagram 4) Take the common area C of the two sectors and (diagram 5) subtract it from each sector. (diagram 6) The remaining figures will be equal, the lunule and the rectilinear area.

(diagram 7) A brief note on Alexander's method of finding a circle and lunule that can be squared is also in order.  Suppose that one were to look for a lunule to be squared that is the next simplest after the lunule on the quadrant.  One might think that this is the lunule on the radius of the circle, i.e., the lunule on the side of an inscribed hexagon, where the outer arc is also a semicircle. One would quickly find that three such semicircles are equal to 3/4 the larger semicircle or that 4 such semicircles equal the larger semicircle.   Hence the three lunules will be less than the rectilinear area by one semicircle.