Quadrature of lunes and curvilinear surfaces

Squaring the curvilinear surface a c n g

A lettered figure asserts that the square a b c d is worth the curvilinear surface a c n g. It is one of the sheet's opening statements that a curved region can be given an equal square.

Lunes are squarable; a square remains

The two circles of a lune are each double the other, so the whole lune is worth half the greatest circle. Removing from it the two greatest portions leaves a square. In these surfaces the lunes are squarable and their fields are made square.

Hexagon in a circle and its squarable remainder

A first hexagon of white area equals the largest hexagon that fits in the greatest circle, and its remainder is squarable in itself. A second, 'empty' hexagon is shown to be worth the first, matching white to white and dark to dark, thereby squaring a curvilinear surface. Because the field is rectilinear, removing a squarable known part leaves a squarable remainder.

Doubled circles: 6 portions of the greater equal 12 of the smaller

For circles each double the other, the 6 portions of the larger are worth the 12 portions of the smaller. The rule underlies the star and rosette figures of the third column.

The 'vasicola' surface, squared by its diameter

A surface Leonardo names the 'vasicola' (little vessel) is said to be squarable in itself, and is squared by multiplying its diameter by itself. A related bi-angular surface a b c is labelled below.

In alto, prima colonna. Figura colle lettere:

Figura di bisangolo entro un rombo:

Figura sotto la precedente:

Figura analoga alla precedente:

A sinistra della precedente:

Seconda colonna. Figura:

Figura:

Terza colonna. Cerchio con stella bisangolare: 5)

Sopra l'analoga figura contenente la lettera

Sotto la precedente, figura colla lettera:

Quarta colonna. Figura colla lettera:

Altra figura colla lettera:

Sopra la successiva figura colle lettere:

Sotto la stessa:

Sotto la figura successiva:

Top, first column. Figure with letters:

Figure of a bi-angle (lens) within a rhombus:

Figure below the previous:

Figure similar to the previous:

To the left of the previous:

Second column. Figure:

Figure:

Third column. Circle with a bi-angular star:

Above the similar figure containing the letter a:

Below the previous, figure with the letter:

Fourth column. Figure with the letter:

Another figure with the letter:

Above the following figure, with letters:

Below the same:

Below the following figure: