Quadrature of lunes and transformation of areas
Cutting a square into curvilinear parts, proving areas equal, with columns of arithmetic
A dense sheet of geometry in which Leonardo works on the quadrature of curvilinear figures: he divides squares and rectangles into small curved 'portions' (the pointed lune shapes drawn in the big strip at lower right) and reckons how many equal parts each is worth in order to restore a rectangle to its full value. A central figure resolves a lettered square r S o c t v into thirds and units, while a left-hand trapezoidal figure invokes 'one of the Elements' to prove two composed rectangles equal. Columns of arithmetical operations (600, 1200, 800, 5000) run down the upper right, and further figures repeat when the sheet is turned. The mirror-script is legible where transcribed but faint elsewhere.
On this page
A square resolved into thirds and units (r S o c t v)
r S is worth a third of the whole square r S o c t v; adding o e makes r S o c half of the square, which is 12, so 4 falls in r S, 4 in o t and 4 in c v. Taking o from t (worth 1) leaves t 3, and removing c leaves v 3; it follows that v is worth 3 and c is worth 1.
Removing 24 curvilinear parts to square the figure
From the square a b c d Leonardo removes 24 little parts like o S r (16 like o r, 8 like S) to test whether S equals the portion o. What remains is the value of eight triangles in the surface n m o p, which he restores with the surface d n o m; if the supplement is true, the rectangle a b c d is made good of its deficiency.
Two equal triangles compose equal rectangles (by the Elements)
Here are two equal triangles, a and b. Adding c to triangle b composes rectangle c b; adding the same c to triangle a makes the surface a c; and by one of the Elements the surface c b is made equal to a c.
Columns of arithmetical operations
Down the upper right run columns of figures being summed and reckoned: 600, 1200, 1200, 800, 1200, totalling 5000, alongside smaller numbers 14, 5, 70. A second inverted set (3, 4, 1 41, 4, 3) appears when the sheet is turned.