During the years that Leonardo da Vinci worked on squaring the
circle, he also developed a method using movement, a forerunner of calculus.
Leonardo's Kinematic Technique
"The thing which moves acquires as much space as it loses
of it."
Codex Atlanticus 152 ca. 1508-1509
In the example above, Leonardo draws a circle and divides it into segments
(brown lines). Then he moves the segments in the left half of the circle
directly to the left. In doing so, he creates an ellipse! The two areas
in brown stripes are equal to the area of the semi-circle. While drawing
this ellipse, Leonardo invents an adjustable compass which led to the elliptical
compass. (from Codex Atlanticus 369 v-a, ca 1510)
In the next example above, Leonardo moves a shovel-shaped figure to
the right. In the new position, the area in orange represents the area
shared. Therefore, the area of the lune on the far right (in blue stripes)
is equal to the sum of the 3 rectangles (in red stripes) on the left side.
(from Codex Atlanticus 152 v-a, ca. 1508-1509)
In the last example, Leonardo experiments with a semi-circle. He starts
by positioning it horizontally at the top. Then he rotates it, keeping
the right end fixed. The area in blue stripes is shared by both positions.
Therefore, he can say that areas C + A = D + B. Leonardo was wrong in
also asserting that area A = B. (from manuscript G58r, ca. 1510-1515)