Squaring Lunes and Rings; Circular Sectors

Recombining a ring without losing quantity

n m is put back into o p and the fitting arcs are made without diminishing the quantity. Leonardo restates his governing constraint: curved figures may be reshaped freely so long as their total area is preserved.

Decomposing a ring into rectangles

Leonardo removes a d from the parallel a b c d to join it to e f; because the upper arc of a d is less curved than the lower arc of b c, he removes a lune from a and one from b, as shown at n, so that n can be united under b and f under c.

A false ring with arcs struck from one point

The arcs a b c d are unsuitable because c b is one same curvature, so Leonardo is forced to make in the space a d two arcs like those in the large space between o r of the second demonstration, where all the arcs are struck about one same point S.

Squaring the lune as the excess of the ring

Square the lune a b c, which is the excess of the larger parallel—or, if you wish, of the smaller, for it amounts to the same. This reduces the width of an annular ring to a squarable crescent.

Two overlapping sectors squared by proportion

Take 2 sectors in the same proportion within their circles as the circles bear to each other, differentiated by the difference of arc a b from arc a c. Drawing the triangle c p n from the triangle a p o, the remainder of a p o will be rectilinear and equal to the lune a b c.

Account jottings of quantities

Brief memoranda record a lot of a hundred, another hundred, and another of seventy of some commodity. These stray account lines are unrelated to the geometry filling the rest of the sheet.

Item one auto in una partita cento da tosoliere

Item in una altra setanta.

Da destra a sinistra. Parti di settori di cerchio, con:

Settore di cerchio, con:

Due figure con:

Margine destro, figure mutilate, con:

Parte di settore con:

Figura analoga, con:

Due figure scomposte in rettangoli, con:

Sei figure colle lettere:

Margine destro, tre figure con:

Al centro, cerchio con cinque settori sporgenti entro un cerchio, con:

Falcate con:

Figura:

A sinistra, doppia falcata:

Figura analoga con:

A destra, falsa corona circolare, con:

Settore suddiviso con:

A sinistra, falsa corona circolare, con:

Due settori sovrapposti, con:

Somma aritmetica 2):

Item: one had, in one lot, a hundred from tosoliere

Item: in another, seventy.

Right to left. Parts of circle sectors, with:

Circle sector, with:

Two figures, with:

Right margin, mutilated figures, with:

Part of a sector, with:

Analogous figure, with:

Two figures decomposed into rectangles, with:

Six figures with the letters:

Right margin, three figures with:

At the centre, a circle with five sectors projecting within a circle, with:

Lunes, with:

Figure:

On the left, a double lune:

Analogous figure, with:

On the right, a false circular ring, with:

Subdivided sector, with:

On the left, a false circular ring, with:

Two overlapping sectors, with:

Arithmetical sum: