Quadrature of Lunes and Circular Sectors

Right half. On the right, two figures with the letters:

a; b. a b are equal.

From right to left, four figures of sectors with:

Double a; b; c; f. a b c d are 4 surfaces equal among themselves.

On the left:

Give me a parallelogram equal to a segment, and a sector equal to that segment, and a lune equal to that segment, and a [...]

Two superimposed sectors:

1/4 – 1/16 of a circle double the first.

Two more superimposed sectors:

Double.

A lune and a sector with the letters:

a; b. a b are equal.

Two sectors with the letters:

a – a; b – b. a b are equal.

Hexagon inscribed and circumscribed:

Double – Half.

Moving toward the right. Two small squares with:

6; double; 3. Per arcibra 5. 2. 3 6.

A figure with:

a n.

Another figure with:

n – n.

Left half. Right column. Part of a circular ring (annulus) with:

a b. Knowing the quadrature of the parallelogram b a c, I will remove from it the lune a, which always has as its lower side one of the sides of the parallelogram and whatever thickness one wishes; for by the rule below I shall always have its quadrature. From this it will follow that I shall have true knowledge of the remainder b e of that parallelogram b a c. See how I shall always have the quadrature of every lune with irrational sides.

Figure of the lunes of Alhazen, with:

b – n a. Here the two semicircles a b, joined together, are always equal to the third one in which the right triangle is made. And if equal parts are taken from equal things, the remainders will be equal. I know, therefore, that when the crossed-out area (which is double) is removed from a and removed from b, the two lunes remain; and then, removing the crossed-out area from the greater semicircle n, which equals the two aforesaid, it follows that the right triangle n remains equal to the two lunes a b. It remains for me to assign to those two lunes the part of the right triangle that is squarable, which will be done in the angle of proportions.

To the left of the preceding passage, a triangle divided from apex to base in the proportion 1/3 : 2/3, with the letters:

c b a – f e d – f. Angle of proportions. a b and b c are the bases of the lower semicircles; so I shall draw the conclusive line b f, which cuts the triangle a c f in the same proportion as that of the two diameters of the semicircles. Then I shall take the diameter of the greater lower circle and set it at d e f, and thus it will be cut by the conclusive line b f at point e; and likewise below I shall make the cut S o in the right triangle, and the triangle b o s remains equal to the lune b n S.

Figure with a double construction of the lunes of Alhazen, with:

rima – d – e – S – n – a – f – c o b. Here, when the angle of the right triangle is at the middle of the circumference of its semicircle, the lune a d e c equals the right triangle b a c. This is the greatest lune and the greatest supplement and right triangle that can be. The like occurs in the figure you see; and the right triangle never dies without the lune equal to it, together with the other lune its supplement, dying at the same moment, nor is one ever born without the lune and the supplement being born. This proof you shall make by squaring the lunes upon themselves; but first you must know the proportions of the circles from which such lunes are drawn, then you can square them with certainty by taking and restoring the value of the parts removed on the opposite side of such lunes, as is shown below. No straight line within a semicircle can be equal to the diameter of its circle. If two equal surfaces are partly superimposed one upon the other, the part of their common contact will be equal in quantity and in shape, and the part of the one [...]

A figure like the preceding one with greatly reduced lunes and the letters:

b – e – a – c d. a d c equals a b e c.

Upper margin.

On the squaring of the greater lune. Because the greater lune has its portion in a body whose circumference is 2/3 of the circumference of the greater circle, it will be said that the portion n is 2/3 of the portion g. Because the outer circumference of the greater lune is divided into 3 portions, and the greater circle in which the right triangle is enclosed does likewise—of which the greater lune takes 2 on its inner side—it follows that one outer portion of that lune is worth two thirds of an inner portion of that lune. There being, then, 3 outer portions worth 6/3, and 2 inner ones rendered, likewise worth 6/3 at 3/3 each, of which one of the outer ones, as said, is 2/3, therefore the square f o l g equals the lune i o l m n; and within the smaller lune the 2 equal portions are worth the 8 equal outer ones, because the proportion is quadruple.

In the lower part, above the figure of the lunes of Alhazen with half-hexagons inscribed in the semicircles, with the letters:

m o – l f a – n g K d b – h c. Here the part stands to the part proportioned to it, according to the proportion of the whole to the whole, as the whole stands to the whole;

From the top, down the left margin,

which in this arrangement is in quadruple proportion. Therefore, if the whole is quadruple to the whole, the half is also quadruple to the half, and the third to the third; nor does the fourth to the fourth deviate from the first proportion. Example. You have two numbers, namely 48 and 12, which are in quadruple proportion, and each term has a half, a third and a fourth; of which, the whole being in quadruple proportion, the half of the greater number, which is 24, is quadruple to the half of the lesser number, which is 6; the third of the greater is 16, which is quadruple to the third of the lesser, which is 4; and likewise the fourth of the greater is 12, which is quadruple to the fourth of the lesser, which is 3; and continuous quantities of irrational terms would do the same, without departing from the same proportion. Therefore it follows that the part of the irrational term which is e is quadruple to the lesser term c; and thus is proved the quadrature of that lesser lune, which, drawn from the right triangle, leaves the remainder of that right triangle equal to the greater lune, which by the seventh of the second is also in itself squarable. The lesser semicircle is worth a quarter of the greater semicircle, because its semidiameter is worth half of the greater semidiameter. The portion e forms a side of the third of the greater circumference, and the portion c is worth 1/3 of the lesser circumference; and if I had taken a quarter of the greater circumference, it would equal the 4 circumferences of the lesser semicircle. But because I have taken the third of the greater circumference, I shall also take the third of the lesser circumference, which will be in the same proportion to the whole and to the part of its remainder as the greater circumference is to its parts and to the whole. And if you divide the rectilinear field into 4 triangles in the lesser circle, each of those 4 will be one quarter of one of the 4 triangles of the greater field.

To the right of the figure, two small figures with the letters:

a; b. I take from you the quarter a. a b [...] equals [...] known.

Lower left corner, figures with the letters:

a – b. c e d. To make lunes double and triple to one another and in every proportion.

Metà destra. A destra due figure colle lettere:

Da destra a sinistra, quattro figure di settori con:

A sinistra:

Due settori sovrapposti:

Altri due settori sovrapposti:

Falcata e settore colle lettere:

Due settori colle lettere:

Esagono iscritto e circoscritto:

Andando verso destra. Due quadratini con:

Figura con:

Altra figura con:

Metà sinistra. Colonna destra. Parte di corona circolare con:

Figura di lunule di Alhazen, con:

A sinistra del brano precedente, triangolo diviso dal vertice alla base, secondo la proporzione 1/3 : 2/3, colle lettere:

Figura con duplice costruzione delle lunole di Alhazen, con:

Figura analoga alla precedente con lunole ridottissime e le lettere:

Margine superiore.

Nella parte inferiore, sopra la figura delle lunole di Alhazen con semiesagoni iscritti nei semicerchi, colle lettere:

Dall'alto lungo il margine sinistro,

A destra della figura due piccole figure colle lettere:

Angolo inferiore sinistro figure colle lettere: