Lunes, Sectors and the Squaring of Curved Surfaces

Semicircles and sectors set equal

The figure cdb is a semicircle less one portion, while the sector fab is worth that same semicircle. If ab is removed, the remainder is stated to be equal to cd. These equalities let a curved figure be exchanged for another of known area.

A circle worth four smaller circles, proven by lunes

The large circle is declared worth 4 of the smaller ones, and nm worth the remainder of the circle. Leonardo grounds this on the claim that the two irregular lunes db and Ki are worth the circle they contain, since lunes plus circle decompose into four equal parts: two semicircles and two sectors equal to them.

Sixteen biangles equal to eight triangles

In the double square at the right, sixteen biangles are said to be worth the eight portions drawn from the greatest triangle. In the three-square construction below, the removed part equals the remainder, so a square can be assigned to each; the 8 biangles carry the value of the 16 portions.

On the transmutation of surfaces

A short heading at the left announces the theme of the whole sheet: the transmutation of surfaces, that is, converting one figure into another of equal area. It frames the surrounding diagrams as exercises in exchanging curved and straight-sided shapes.

Columns of figures

Two small stacks of numbers sit in the lower left and lower right of the sheet, running through sums such as 740, 200 and 940. They appear to be working calculations set beside the geometrical proofs rather than a finished result.

Da destra a sinistra lungo la linea mediana:

Tre figure con:

Semicerchio con:

A destra, doppio quadrato con sedici bisangoli:

Due figure di cerchi con semicerchi e settori, e le lettere:

Settore di cerchio sovrapposto a semicerchio, con:

Figura analoga con:

A sinistra:

Tre quadrati con cerchi e bisangoli iscritti, e con:

Operazioni aritmetiche. In alto:

In basso:

From right to left along the median line:

Three figures, with:

Semicircle, with:

At the right, a double square with sixteen biangles:

Two figures of circles with semicircles and sectors, and the letters:

A circle sector superimposed on a semicircle, with:

An analogous figure, with:

At the left:

Three squares with inscribed circles and biangles, and with:

Arithmetic operations. Above:

Below: