Squaring Lunes and Circles: Geometric Games

Right side of the sheet, upper right:

* Of the wind that increases the flame.
* Of the excessive wind that destroys the flame.
* Of the wind born in the middle of the flame, rising from below upward along its axis.
Why the small flame turns into great smoke.
* The birth of the flame consumes the great smoke.
* The flame in the cold makes a more rapid and more defined surface than in the heat.

Large figure of a square inscribed in the circle, with:

ieba-gc-K-Lf-h-m

Small scored circle, with:

m -h -n

* c is worth m n, which m n is squared by adding to it the greatest portion h, which h is worth the 8 small half-portions; these, added to n m, make 2 rectilinear squares.

Scored semicircle, with:

b c a - e d

a b is worth the black square field, filling the black field with the two portions a b.

Another scored semicircle, with:

m n - p o

If I remove the 8 outer portions, the triangle remains; and if I remove the 8 inner portions, there remain the two white portions and the four triangles n m o p, equal to the greatest triangle.

Below, crescents in a scored semicircle, with:

b a

Give me a square equal to this perforated surface.
a b are worth the two greatest portions.

Large circle, with:

a g f - h - e d c

The f semicircle is worth the crescent g, and likewise the greater semicircle f g h is worth the crescent a b c d e, the greater crescent.

Crescents, with:

a -d c -b

a is worth b, because a d c is worth d b.

Portions of a circle, with:

ca-o-di-b
a b o i is worth c d, because I removed c d from c d o i and rendered it into a b.

Semicircle with lune and sector, with:

The two portions a b make as much when accompanied with c as when accompanied with d. Therefore d is worth c. But the d with S makes this one (figure of a curvilinear triangle) and the c makes this other one (figure of a rectilinear triangle).

Left side of the sheet. Right column.

First. - The c g h d lune is worth the circle e f d surrounded by such a lune.
The a c d b crescents are worth the same circle they contain, that is the circle a f.

Figure, with:

(Second) - First - d e b a. a b is worth b c. The third covers the second, that is, their crescents.

Figure, with:

(Third) - Second - d a - e b. a d is worth b c.

Central column.

The outermost crescents a b are worth the portion below, g d. It is proved because the greater circle is worth the greatest semicircle, the two crescents are supplements of the semicircle, and the portion is the residue of the greatest circle.
* f is worth d b and is worth d h. Take d from d h and h remains; take d from d b and b remains, equal to h; and if, by the contrary hypothesis, d were smaller ...
The smaller circle e f is worth half of the greater circle c e f d g h, and is worth a quarter of the greatest circle, that is the quarter circle f d b. Now it must be proved that c d equals the crescent a b; but first one must know how the crescents a c b d are worth the smaller circle e f, and also how the portion g h is worth the crescent d b; and it is proved thus, saying that since the middle circle equals the greatest semicircle, their excesses are equal, that is the crescents a b are worth the portion g h. The third figure has the crescents a d inside the semicircle and b c outside it, and this b c is worth a d; but the second figure lacks the part outside the semicircle and holds entirely within a d the value of b c of the third figure, which is worth a d. The excess of the third is lacking in the second; and because the circle exceeding that third is double the circle of the second, it puts its excess into the second, diminishing by half within the second and increasing by half the space within the second that it had in the first. Therefore whatever the third has outside the lower semicircle, the second has drawn into its body, widening the two crescents as much as the first was lacking below. The great lune above is worth the circle it encloses, and is worth the two great crescents of the second, and is worth the circle of that second.

Left column.

* The f semicircle is worth the crescent h d and the crescent d b.
* h is worth b; h d is worth the semicircle; d b (a double crescent) is worth the semicircle.
* d h is worth (the semicircle).
* d b is worth (the semicircle).
* h and b are equal to each other and are worth the semicircle.
* g h is worth f (semicircle).
* g d is worth f (semicircle).
* d b is worth f (semicircle).
* h b is worth f (semicircle).

Two semicircles, with:

n; the m, n, m are equal, and equal are their partitions.

Two semicircles, with: m;

the o, m, o are equal in their wholes and in their parts.

Circle within a circle:

c-edb-n(a)a
a n d e b c are parts equal among themselves.

Three small figures of circles and semicircles, with:

a - b; c d; d -c

Two circles and a semicircle superimposed, with:

t e S - d c b - a

If the portion a is worth half the smaller circle and half the middle circle, then, since the middle circle is double the smaller, a is worth the whole smaller circle; and it follows that the parallelogram b c d is worth the portion a, and is worth 2 crescents.
a is worth the parallelogram made within its own semicircle. It is proved so because a is worth half of the smaller circle, and the same a is worth half of the lune a b S d t; and because the smaller circle is worth half of the circle that surrounds it, which half is similar to a.

Circle divided horizontally, with:

b - c - d - a

Observe in this circle divided into 4 equal parts that each is worth half of a circle of half this size, which is proved above where the portion a is equal to half of that smaller circle. So our proposition is concluded: for if you remove a b, portions of the circle a b c d, you have removed the value of a whole smaller circle; and because the circle a b c d is worth two of those smaller circles, removing one by removing a b, the remainder e d is worth the other circle. Therefore the circle is divided into 4.

To the right of the aforesaid circle:

Make a large one: b lends two portions to c; therefore b was greater than half the semicircle by those two whole portions, and c was less by those 2 portions.

Figure, with:

(First) - Third - b d f e c a - 2 l - 2 h g
The circle e f, surrounded by the lune, is surrounded by 4 parts equal among themselves, that is c g h d, and is also surrounded by another 4 equal parts, that is a e d b.

In the central column. Horizontal semicircle, with:

n - o - m

The first found in this rule.
On the 3rd of March 1516.

Vertical semicircle, with:

a-b-f-f-c-d

If from equal things equal things are removed, the remainder stays equal.
b c f is worth 1/4 of the circle n m o, and their edges are arcs of equal curvature. I will remove the portion f from b c f, and the square b c will remain; and I will remove the portion o from the sector n m o, and the quadrangle n m with one curved side will remain.
After we have found the quantity n m, a quadrangle with one curved side, equal to the quantity b c of four straight sides, we will remove the curved part by removing the dotted half-portion, and the remainder of that square n m we will subtract from the square b c; the excess of b c will remain a rectilinear quantity equal to the aforesaid half-portion. And if you wish to square the said half-portion, make it as large as you can, and you will carry out your operation with greater certainty.

Right column, second half. Figure of two circles on a semicircle, with:

r o n a-S b-tc -v
The portion a b c of the greatest semicircle is divided into three parts, which are double one to another.

Small figures, with:

a; b. a is half of its own field, and b is half of its own.

Vertical semicircle, with:

Double circles.

Facciata destra. In alto a destra.

Grande figura di quadrato iscritto nel cerchio, con:

Piccolo cerchio biffato, con:

Semicerchio biffato con:

Altro semicerchio biffato, con:

In basso, falcate in un semicerchio biffato, con:

Grande cerchio con:

Falcate con:

Porzioni di cerchio, con:

Semicerchio con lunula e settore, con:

Facciata sinistra. Colonna destra.

Figura con:

Figura con:

Colonna centrale.

Colonna sinistra.

Due semicerchi, con:

Due semicerchi, con: m;

Cerchio entro cerchio:

Tre piccole figure di cerchi e semicerchi, con:

Due cerchi e semicerchio sovrapposti, con:

Cerchio diviso orizzontalmente, con:

A destra del cerchio predetto:

Figura con:

Nella colonna centrale. Semicerchio orizzontale, con:

Semicerchio verticale, con:

Colonna destra, seconda metà. Figura di due cerchi su un semicerchio, con:

Piccole figure con:

Semicerchio verticale, con:

Right side of the sheet, upper right:

Large figure of a square inscribed in the circle, with:

Small scored circle, with:

Scored semicircle, with:

Another scored semicircle, with:

Below, crescents in a scored semicircle, with:

Large circle, with:

Crescents, with:

Portions of a circle, with:

Semicircle with lune and sector, with:

Left side of the sheet. Right column.

Figure, with:

Figure, with:

Central column.

Left column.

Two semicircles, with:

Two semicircles, with: m;

Circle within a circle:

Three small figures of circles and semicircles, with:

Two circles and a semicircle superimposed, with:

Circle divided horizontally, with:

To the right of the aforesaid circle:

Figure, with:

In the central column. Horizontal semicircle, with:

Vertical semicircle, with:

Right column, second half. Figure of two circles on a semicircle, with:

Small figures, with:

Vertical semicircle, with:

On this page

'Give me a square equal to this perforated surface'

Lunes and crescents equated to circles and semicircles

Dividing a circle into four equal lune-parts

Squaring a quadrangle with one curved side

Notes on flame, wind, and smoke

Dated working note: 3 March 1516