Bird-flight notes and De ludo geometrico figures
Wing-beat versus natural descent of birds, and a numbered series of curvilinear figures each equal to a square
The upper-left corner carries two bird sketches with notes on flight: whether a bird rises or falls depends on whether its downward wing-beat is faster than its natural descent, and Leonardo resolves to dissect a bird's wing together with the chest muscles that move it, then to do the same for man. The rest of the sheet is a vast array of 'playful geometric elements' (elementi ludici geometrici) — a numbered series of figures, First through Sixteenth, each shown to equal the maximum square of the maximum circle once the 'four maxima' are removed. Curvatures of circular rings and semicircles, curvilateral quadrangles and empty squares are compared by proportion throughout.
On this page
When a wing-beat overcomes the bird's natural descent
If a bird does not beat its wings downward faster than it would fall in natural descent with the same wing-spread, its motion is downward. But if the wing-beat is faster than that natural descent, the bird rises — and rises the more swiftly, the faster the wings are driven down.
Dissect the wing and chest muscles of bird and man
A bird descends on the side where the wing-tip lies nearest its centre of gravity. Leonardo resolves to make the anatomy of a bird's wings together with the chest muscles that move them, and to do the same for man, to show that a man might sustain himself in the air by beating wings.
Elements of geometric play: 'less the four maxima'
The numbered figures each demonstrate that a curved figure equals the maximum square of the maximum circle once its 'four maxima' (sixteen or thirty-two 'minima') are taken away. The First establishes the pattern; later figures are shown to derive from and equal earlier ones.
Curvilateral quadrangles double the empty square
In the square a p o q the curvilateral quadrangle is proven double the empty square: an eighth of the curvilinear square takes the diameter while an eighth of the empty square takes the side, and the two are similar in figure. The Sixteenth figure divides four circles into sixteen sectors to conclude the remainders equal.
Figures compared as equal or double in value
Throughout, figures are weighed against one another by proportion: a b is worth the maximum square of a circle double the circle a; other pairs are declared of equal value, or one worth half or double another.