Leonardo da Vinci re-discovered the "Lunes of Alhazen" while working
with right triangles.
Lune-easy continued
THE LUNES OF ALHAZEN
Leonardo starts with an isosceles right triangle shown
on the left.
The areas of the 2 semi-circles on the legs of this isosceles rt. triangle,
added together, equal the area of the large semi-circle in blue. This is
follow from the Pythagorean Theorem.
Leonardo applies his lune-easy technique to the above isoceles
rt. triangle.
He folds over the large blue semi-circle which creates 2 lunes in orange.
He subtracts the areas in white from all 3 semi-circles. The areas left
over show that the 2 orange lunes equal the area of the rt. triangle in
red stripes. Hippocrates discovered this in 460 B.C.
Next, Leonardo tries the same analysis to any right triangle.
And the results are the same: The areas of the 2 lunes in orange equal the
area of the right triangle. This was first
discovered by Alhazen in around 1000 A.D. However, his work was not available
in Europe until 1899. So Leonardo
re-discovered the Lunes of Alhazen on his own.