Name: Kai

Question:
A prof once told me that a certain type of lune is quadrable given that the diameter is an integer. She used the construction of a right isosceles triangle within a semicircle and later constructed another semicircle on the base of the first semicircle and used area subtraction to show equality to a smaller triangle with quadrable area. What happens when the original inscribed triangle is no longer isosceles? She mentioned something about other lunes also being quadrable; but not all. What are the dimensions of other such lunes? Note: I'm not certain if I still have my hercules account; please simply post on Q&Q.
Thanks!

Hi Kai

According to William Dunham in his JOURNEY THROUGH GENIUS (p. 26) exactly 5 types of lunes are quadrable. Three were discovered by Hippocrates (around 440 BC, more than a century before Euclid), and the other two by Euler in 1771. The proof that there are no more came in the 20th century. (An elementary reference is Tobias Dantzig, THE BEQUEST OF THE GREEKS, Chap. 10, but the book is not in the U. of Regina library.) Hippocrates' three lunes are discussed in Carl B. Boyer, A HISTORY OF MATHEMATICS, pp. 73-74. In each case (and also for a related result) the treatment comes with pictures and an indication of how the quadrature works.

Cheers,
Chris