Quadrature of Lunules and the Flight of Birds

A circle of four lunules, a worth bcde

A circle carries four lunules lettered so that a is worth bcde. Leonardo notes that the circles are double, so that four halves are worth as much as one whole, as demonstrated in the figures below.

On the transmutation of surfaces

A programmatic list of quadrature problems: transmuting rectilinear surfaces into curvilinear ones and back, turning curved surfaces into other varied curved surfaces, and dividing curvilinear surfaces into equal parts that are themselves still curved. It reads as the plan of a treatise on squaring curved figures.

The bird making the shorter turn

The bird that makes the shorter revolution extends its wings with less obliquity, so the tighter the turn the more curved its circulation. Leonardo adds that a bird never moves backwards, because its open wingtips are never further forward than its centre of gravity, and cites the way a bird always turns its beak to the wind when it alights.

Steering with the horns of the tail

If the left horn of the tail rises as far above the central lines of motion and weight as the right horn drops below them, the bird's course must run straight, the two horns balancing one another. The left horn above bends the flight to the right exactly as much as the right horn below bends it to the left.

The bird sinks toward its heavier side

A lettered bird figure shows that a bird always inclines toward the side where it weighs more, which is the side of least breadth, so it descends along the line a b and not b a. Leonardo records having watched the sparrow and the lark climb in a straight line while held level, the raised wing rising on acquired impetus that renews as the feathers close one beneath the other.

A tree turning its branches to the wind

A sketch of a tree carries the observation that the underside of the branches of any plant always shows itself to the wind that strikes them, and that each branch leans upon another. It pairs the behaviour of vegetation in wind with the flight notes alongside.

Prima colonna. Cerchio con quattro lunule e le lettere:

La stessa figura iscritta in un cerchio, colle lettere:

Seconda colonna. Cerchio con due lunule e le lettere:

Due cerchi concentrici, con:

Sotto le precedenti colonne con una serie di lunule e di falcate:

Terza colonna. Figura d'uccello in virata:

Quarta colonna. Uccello in discesa:

Figura d'uccello colle lettere:

Figura d'uccello in discesa:

Uccello che atterra:

Figura d'albero:

First column. Circle with four lunules and the letters:

The same figure inscribed in a circle, with the letters:

Second column. Circle with two lunules and the letters:

Two concentric circles, with:

Below the preceding columns, with a series of lunules and crescents:

Third column. Figure of a bird in a turn:

Fourth column. Bird in descent:

Figure of a bird with the letters:

Figure of a bird in descent:

Bird alighting:

Figure of a tree: