Leonardo da Vinci developed his Lune-easy method for quadrature problems
using a "key geometrical figure."
Leonardo's Lune-easy
Key Geometrical Figure
In the above figure of nested circles touching nested squares, the outer
circle is twice the area of the next smaller circle. It follows that the
corresponding parts are in the same proportion as the areas of the nested
circles. Leonardo used this to calculate the area "x" shown in
the figure below.
The lune or segment from the largest circle in the key geometrical figure
has 4 times the area of the smaller lune from the third circle. So that
"a" is one fourth the total area of the lune above. By subtracting
both lunes in orange, Leonardo derives that area "a" equals the
falcata area "x."
In the figure above, circles A and B are each one fourth the area of
the large circle. It follows that the striped area is also equal to the
areas A or B. It follows that one half the striped area equals a the area
of a semi-circle of A or B.
The above figure shows another way of drawing the falcata "x"and
its equivalent area lune "a."