The Codex Atlanticus (Atlantic Codex) is a twelve-volume, bound set of drawings and writings by Leonardo da Vinci, the largest such set; its name indicates its atlas-like breadth. It comprises 1,119 leaves dating from 1478 to 1519, the contents covering a great variety of subjects, from flight to weaponry to musical instruments and from mathematics to botany. This codex was gathered by the sculptor Pompeo Leoni, son of Leone Leoni, in the late 16th century, who dismembered some of Leonardo's notebooks in its formation. It is currently preserved at the Biblioteca Ambrosiana in Milan.
https://www.discoveringdavinci.com/codexes
To explore it
https://codex-atlanticus.ambrosiana.it/#/
Objective: the same number π for perimeter and area?
The number π is defined as the ratio between the perimeter of a circle and its diameter: this ratio is constant, and does not depend on the circle chosen. In his treatise De la mesure du cercle, the illustrious scientist Archimedes of Syracuse (287 to 212 BC) shows that the same constant is also involved when relating the area of a disk to the square of its radius.
Below we present a geometric method for understanding the relationship between the area of a disk and the perimeter of a circle. Method: sharing a pizza
Let's consider a disk-shaped pizza with radius R. Its perimeter is 2×R×π. Let's now justify that its area is expressed using the same constant π, and is π×R×R.
We first cut the pizza in half along a diameter, then divide each half-pizza into a large number of equally sized slices.
Each pizza slice is almost triangular in shape: it's a geometric figure whose two sides are segments whose length is the radius of the pizza, and the last side is a small arc of a circle. The larger the number of slices, the more triangular the slices become, as the arc of the circle becomes a small segment of a straight line.
We now rearrange the pieces as follows.
then
The result is a new pizza shape, which almost coincides with a rectangle. The width of this rectangle is the radius R of the initial pizza, and its length is the half-perimeter π×R. Its area is therefore π×R×R=πR^2^. Now the new pizza has kept the same area as the original disk. The area of this disk is therefore also πR^2^.
We don't know who first came up with the idea for the cut-out shown above, but it goes back at least as far as Leonardo da Vinci, as this detail from the Codex Atlanticus dating from 1513 shows.
Translated from https://sorciersdesalem.math.cnrs.fr/Autour_de_pi/Pizza_Archimede.html
