Date: Wed, 3 Feb 1999 14:41:28 -0600 (CST)
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.
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