Leonardo's Lunes


                        A lune is the geometrical figure created by the                                        overlap of two circles of different size.  

Although Lunes were known to the Greeks,  it was                                 the 10th century Arab mathematician  
(Ibn al-Haytham ) who cast them in their
most familiar form..

Looking at the main diagram below:  within the upper semicircle BC is  inscribed a triangle ABC.  Following
Thales’ rules, all triangles inscribed in a semicircle are right-angle triangles. The Pythagorean Theorem holds that constructing semicircles on each side of a right-angle triangle, demonstrates that the areas of the semicircles AB + AC equal the area of the lower semicircle BC.

What is more interesting, and caught  the attention of
Leonardo (1452-1519, ) is the claim that the areas of the lunes Y & Z are equal to the area of the triangle ABC (X.). This is easily seen (with some thought) by noting that if the areas of semicircles AB + AC equal the lower semicircle BC, they also equal the upper semicircle BC.  Thus, if the same relative amounts (green figures) are subtracted from semicircles AB & BC creating lunes Y & Z they are at the same time subtracted from the semicircle AC, yielding Y & Z = X , a proper Pythagorean result.

This interested
Leonardo, who was an
amateur geometer, because the
quadrature of the circle was still
a live issue.  And here was the graphic
illustration of a bridge between
curvilinear and rectilinear forms,
perhaps providing a path to
squaring the circle.