Squaring the circle: sectors, lunes and inscribed squares

The sheet rotated 90 degrees to the right.

Right half. Six figures, a portion and a sector accompanied by a hand with an extended index finger and the letters:

a - b

Double figure with:

a; b

Sector of a circle divided, with:

a - d c - f b e - n

a c d the maximum portion is double the minimum portion b; and add to that maximum portion the two triangles e f and you will have divided the whole sector into two equal parts, that is b n and the remainder.

Two circle figures with inscribed squares, with:

First - b e a - d c; Second - g f - h
Let the 4 portions be removed from the first round, and let the same be done to the second circle. Then give back to the first the value of one of those 4 portions, which value shall be the 4 lesser portions. Do the like with the second circle by giving it back the value of one of those portions, which value is found in the two half portions. Then subtract the second with the rectilinear part of the whole first rectilinear; the remainder, rectilinear, of the first will be worth one of the whole and greater portions of the second circle.

To the right of the preceding, a figure with a pointing finger and a little cross:

This wavy surface is squarable and is made of unequal curvature on each of its two sides.

Below the preceding:

The deception ...

Near the middle fold, a figure with:

a - b

a is worth b.

Sector of a circle with portions:

c b - a

a is 3/4 of b. Remove the quarter of b, which is c, and b will remain equal to a.

Circle with a ring and inscribed square:

First -

m - n - o

The curvilinear parallel m is worth the portion n. It is proved thus: the portion n is worth 3/4 of the portion o ...

Circle with inscribed squares:

Second - o - p
Here 3 portions are removed and a square remains. And because the first circle is worth 3/4 of the second, the 4 portions of the first are worth the 3 of the second; therefore the 4 of the first are worth 4 of the second.

Figure with a pointing finger:

Whole and half of double circles.

Numbers:

3 4 2
2 2 8
3
If
4 3 2
6
4 3 2
6
4 3

Left half. Near the centre, a group of portions with:

m n p - f-o; c-n-b
Remove p n, m f is squarable. Put p back into o and the greatest lune regains its first quantity. Remove n m, o f is a square.
I remove b from the second figure. n c is a square, because it is similar to f o.
Remove the lune p and put it in o, its value, and you will have the surface n m f o equal to the lune p n m f and equal to the portion n b c. Then remove n m from f o, and that f o triangle remains. Besides this, remove b, similar to n m; there will remain n c equal to the o f triangle, as you see [Figure].

A large circle divided into biangles with the letters and numbers:

1 1 - b 1 1 a - 1 1 - e - 1 1 - 1 1 - 4 4 h g 4 4 - 1 1 - 1 1 - f - 1 1 - d 1 1 c - 1 1
Here from the square made of the surface b [Figure b] the two squares o r [Figure o r] are removed, and the remainder of that square [Figure i S] S is divided into equal parts, if there be such, etc. But there remain to us a b c d, lunes equal to the said portions, to which a curvilinear quadrangle will be lent [Figure] and another square remade, which too will be subtracted from the greater square. The remainder will be divided by the 12 curved triangles [Figure] and by the 4 lent curved triangles [Figure], so that such remainder will thus be divided into sixteen equal parts; and from this quantity remove the four lent triangles, and the remainder will be the capital.

Returning to the centre, two circles within a circle, with:

a - b - n

the lune a b is worth a quarter of the portion b n.

Towards the right:

These remainders are equal.

Two circles and squares within a circle:

The whole perforated figure is squarable.

To the left of the preceding, two semicircles with:

Double - a - c b; double - a - c b - Quadruple circles

Below the preceding, a figure with:

The whole.

To the right of the preceding, a triangle inscribed in and circumscribed about circles:

The circle made outside the triangle is quadruple that which is made inside. The circle made outside the triangle is worth 4 times that made inside.
It is proved because the diameter of the greater circle is double the diameter of the lesser circle.

To the right of the preceding, a circle with:

First — c a — b - e d – f – h g
This is whole like the one below.

Analogous figure with:

the one below.
Analogous figure with: Second - Whole and it is worth the circular parallel.
In the first figure the greater circle is worth 4 of the lesser. Therefore four portions similar to one another and similar to the greater are worth one of the greater portions. It follows that, removing from the greater circle its 4 greater portions and removing from the two lesser circles sixteen of their lesser portions, the remainder a c d e g h of the greater circle is equal to the remainder b f of the lesser circle.

To the right of the 'first' figure, a circle with:

b - a

Draw a, parallel of b, a square; the remainder of that b is worth the remainder of a.

At the bottom:

He who does not fear is often full of harms, and often repents.

On the lower margin, from right to left, five portions with:

e b a – S t; m; m - o; n; o a
* From the portion the lesser portions m o are removed, and from the lune is removed e m, which is worth the two said portions m o.
The whole greatest portion is worth the half of the great portion, but of the two smallest portions removed from the greatest, there is the black one which is common to the lune o and to the greatest portion; and the lune o is worth the black portion, since the black portion, together with the lune, composes the middle portion, which middle portion is worth the two lesser portions a and the black one.

The sheet turned upside down, a semicircle:

First - Double proportion.

To the left, circles and squares inscribed and circumscribed, with:

Second - Triple proportion.

Two circles, with:

Third - Third

On the lower margin; a quadrant and a circle, with:

b; a
a is worth b.

Ruotato il foglio di 90° verso destra.

Metà destra. Sei figure, porzione e settore accompagnati da una mano con indice teso e le lettere:

Doppia figura con:

Settore di cerchio diviso, con:

Due figure di cerchio con quadrati iscritti, con:

A destra del precedente, figura con indice ' e crocetta:

Sotto il precedente:

Presso la piega mediana, figura con:

Settore di cerchio con porzioni:

Cerchio con corona e quadrato iscritto:

Cerchio con quadrati iscritti:

Figura con ' indice ':

Numeri:

Metà sinistra. Presso il centro, gruppo di porzioni con:

Grande cerchio diviso in bisangoli colle lettere e i numeri:

Tornando al centro, due cerchi entro un cerchio, con:

Verso destra:

Due cerchi e quadrati entro un cerchio:

A sinistra del precedente, due semicerchi con:

Sotto il precedente, figura con:

A destra dei precedenti, triangolo iscritto e circoscritto a cerchi:

A destra del precedente, cerchio con:

Analoga figura con:

A destra della figura ' prima', cerchio con:

In basso:

Sul margine inferiore da destra a sinistra, cinque porzioni con:

Capovolto il foglio, semicerchio:

A sinistra cerchi e quadrati iscritti e circoscritti, con:

Due cerchi, con:

Sul margine inferiore; quadrante e cerchio, con: