Quadrature of the Circle: Lunes and Squarable Figures

Right face, from right to left. Circle with inscribed octagon:

First - b - c a - d. a b c d are worth the four greatest portions of this circle, and its whole periphery is distributed over its eight sides. Therefore divide the periphery of this circle into eight equal parts and take away eight equal portions, and with these fill the lateral concavities of the four greatest portions a b c d; and thus you will have made the curvilinear cross of straight sides, and the circle will remain rectilinear, that is in octagonal figure; and so it is squarable.

Figure with eight squares inscribed in nine circles, with:

Second - b a - g f e - K i h - n m L - d c. The greatest square a b c d is worth this part of the surface that is not hatched. It is proved because each of the nine circles lacks four portions and their remainder is worth one of the nine squares, as is shown in circle e, which lacks its four portions; the same happens with f and with g and with h and with i, and so with the rest. And it is further proved that the greatest square is worth all the part of the circles that are not hatched, by putting back the eight portions of the four circles that exceed the greatest square into the eight hatched portions that lie within the said greatest square, so that then such square will remain without any perforation. Each of the nine circles, taking away its four portions, remains curve-sided, but of squarable surface.

Double circle with lunes and a two-angled star, with:

Third - d - a - b - e c. The hatched region is worth the greatest square of the greatest circle, and the lune a is worth the curvilinear triangle b c; and a b c in itself is squarable, and squarable is the lune joined with the triangle b c; the crescent d is worth the portion b and is worth the two-angled figure e.

Circle with inscribed square and eight two-angled figures:

Fourth - b m n a - f - p o - g e - d h c. e f g h are worth half of the square that surrounds them, and a b c d are worth the remainder of those squares, as is proved in its place. Again, the crescents a b fill of themselves the concave portions n m o p, and the same is done below, and the sum is worth the greatest square of the greatest circle.

Circle with lune and two-angled figures, with:

Fifth - a - b - e c d - m n. The lune a is worth the curvilinear triangle b c, and is worth the rectilinear triangle d c e, and the portion b is worth the two-angled figure d n, or if you like the two portions n m.

Four lunes and two-angled figures within a circle, with:

Sixth - m n - b a - d c - p o. The four two-angled figures n m o p are worth the four greatest portions of the greatest circle; the remainder is squarable, and from the squarable quantity, taking away the lunes a b c d, the remainder stays squarable as a square half of the greatest square of the greatest circle, as the left-hand surface shows. Therefore, having removed the four two-angled figures, the remainder can be divided into two equal squares, and of these one is generated, or composed, from the four lunes, and the other is worth the remainder that is not hatched, that is the square of the seventh figure.

Four circles within a circle, with:

Seventh. This surface is the remainder of the right-hand surface, from which the four lunes have been removed, and its square is worth half of the greatest square drawn from the greatest circle. If from known things one removes known things, the remainder is known. The seventh figure is squared by restoring to it the four lunes that you took away. It is proved because it returns to the condition of the sixth, which is squarable.

Four circles, eight two-angled figures in a circle, with:

Eighteenth - f - b a - g i e - d c - h. The eight two-angled figures of this figure are worth the greatest portions of the greatest circle, and the remainder is squarable; and half of the greatest square are the four triangles a b c d, and the other half is the remainder of the circle e f g h i. Take away therefore the four curvilinear triangles a b c d; the remainder is equal to what was removed; a quarter of the remainder is worth a quarter of what was removed.

Two circles against a crescent, with:

Nineteenth - f - i. The surface f i is the same as the f i of the eighteenth above and is worth one of the triangles a b c d, and in itself is not squarable.

Circle within a quadrant of a circle, with:

Twentieth - b a - c. The surface a b c is worth the preceding nineteenth; and it arises from the eighteenth above and in itself is not squarable.

Below the preceding, a circle diminished by two two-angled figures, with:

a. These three figures are equal among themselves in value.

Four circles in a circle, with:

Twenty-first. This twenty-first is worth half of the greatest square of its greatest circle and derives from the eighteenth above.

On the left, four unnumbered figures. In the third column, circle with two-angled star:

It is worth the greatest square of the greatest circle.

Below the preceding, circle with:

m n. Here there are lacking, besides the four greatest portions, the two squarable curvilinear triangles, which are dotted at n m.

In the fourth column, circle inscribed in a square, with:

b - c a - d. Here one removes from outside the four greatest portions and removes from inside the four curve-sided triangles a b c d; the remainder is worth the greatest square of the circle a b c d.

Below the preceding, two circles in a circle:

It is worth the greatest square of the greatest circle.

Left face. Four circles and a two-angled star in a circle, with:

First - b a - f e - h g - d c. This first figure is composed of three sorts of circles, each double the other, and the four greatest portions of the middle one are worth the eight greatest portions of the greatest circle. First, the crescents a b c d fill of themselves the four two-angled figures e f g h.

Four crescents around a square, with:

Second - b - f e - c a - h g - d. This is worth the preceding, and the crescents a b c d fill the four empty two-angled figures e f g h. The four two-angled figures e f g h are worth the four greatest portions.

Two-angled star in a circle, with:

Third - b - c a - d. Third. The four convexities of the triangles fill the eight concavities of the sides of those triangles, and from curved they become straight-sided, and the four two-angled figures are worth the four greatest portions of the greatest circle.

Two squares within two circles, with:

Fourth - m n - f e - c f a - h g - p o - d. Here there are three sorts of circles, each double the other, of which the four greatest portions of the middle one are worth the eight of the smallest; and the angles n m o p, together with the crescents a b c d, are worth the square f, and all together they are worth the greatest square of the greatest circle.

Circular crown and four two-angled figures, with:

Fifth - n - b - c m a - d. a b c d are worth the greatest portions of the greatest circle, and n m is worth the greatest square of the greatest circle.

Lunes and squares in a double circle, with:

Sixth - a. a is worth the four lunes, because the four empty portions together with the four empty crescents are worth the four greatest portions of the greatest circle, and the four lunes with the square a are worth the greatest square of the greatest circle.

Figure analogous to the preceding, with:

Seventh - f e - b a - d c - h g. The four middle two-angled figures a b c d are worth the four outer crescents e f g h, and all together are worth the four greatest portions of the greatest circle. Therefore the remainder of such circle is worth the greatest square of the greatest circle.

Four circles and a two-angled star in a circle, with:

Eighth - b - c a - d. The four curvilinear triangles are worth the greatest square of the greatest circle, and are in themselves squarable; and it derives from the seventh, and the parts removed from the seventh and from the eighth are equal and similar one to another.

Circles and lunes in a circle, with:

Ninth - f e - b c a - d - h g. The four crescents e f g h are worth the four portions a b c d, and all joined together are worth the four greatest portions of the greatest circle.

Portions, crescents and two-angled figures in a circle, with:

Tenth - d c b a. a b c d are of equal value and are equal to the four greatest portions of the greatest circle.

Four curvilinear quadrilaterals in a circle, with:

Eleventh - b a - d c. a b c d are worth the four greatest portions; the remainder of the circle is worth the greatest square of the greatest circle, and the way of squaring it is in the twelfth that follows this. The twelfth and thirteenth teach the squaring of this in the second manner.

Figure analogous to the preceding, superimposed on a square, with:

Twelfth - f - p - e - K i - b a - q o - d c - m L h r g. Let the middle quadrangle a b c d then be extracted and placed in four parts outside the circle at e f g h; the remainder o p q r is set with their convexities in contact with the concavities of the parallels i K L m; and thus it will remain a square.

Cross and circles within a square, with:

Thirteenth - b a - d c. Since in the preceding twelfth of this it was concluded how the parallels i K L m receive within their concavity the convexity of the sectors o p q r, therefore in these four hexagons of the thirteenth the curvilinear parallels a b c d are made into two that are rectilinear, that is c d, which are of the value of the two a b.

Circle with square in a double circle, with:

Fourteenth - m n - a - p o - r b q. a is worth b, and the enclosing lune is worth the enclosed circle; and the two greatest portions of the greatest circle are worth the four greatest portions of the smallest circle.

Figure analogous to the preceding, with:

Fifteenth - a - d c - f b e. a is worth b; the enclosing circle is worth the enclosed circle; c d is worth e f.

Circle with two-angled figure within a circle with portions, with:

Sixteenth - d b m s a - p o. a b is worth c d; the enclosing circle is worth the enclosed circle with its excess, and n m is worth o p.

Large two-angled figure within a circle, with:

Seventeenth - b d c a. The two lunes are worth the greatest square of the greatest circle. It is proved, because I remove from it two portions c d that are worth the four greatest portions of the greatest circle, and by the second proposition of this treatise our proof is valid.

Circle inscribed in an inscribed square, with:

Eighteenth - f a - g e - d h c. a b c d e f g h is worth the four greatest portions of the smaller circle, the same is in the left circle, and the four forks e f g h are worth the four forks of the left one n m o p. And the squaring of the eighteenth is worth the greatest square of the smallest circle.

Square with 'forks' within a circle, with:

Nineteenth - m - o n - p. This nineteenth is of the nature of the eighteenth, but they differ only in the position of their four forks e f g h n m o p.

Figure analogous to the preceding, with:

Twentieth. This twentieth is of the same nature as the eighteenth and nineteenth.

Facciata destra, da destra a sinistra. Cerchio con ottagono iscritto:

Figura con otto quadrati iscritti in nove cerchi, con:

Doppio cerchio con lunule, stella bisangolare, con:

Cerchio con quadrato iscritto e 8 bisangoli:

Cerchio con lunula e bisangoli, con:

Quattro lunule e bisangoli entro un cerchio, con:

Quattro cerchi entro un cerchio, con:

Quattro cerchi, otto bisangoli in un cerchio, con:

Due cerchi contro una falcata, con:

Cerchio entro quadrante di cerchio, con:

Sotto la precederne, cerchio diminuito di due bisangoli, con:

Quattro cerchi in un cerchio, con:

A sinistra, quattro figure non numerate. Nella terza colonna, cerchio con stella hisangolare:

Sotto la precedente, cerchio con:

Nella quarta colonna, cerchio iscritto in un quadrato, con:

Sotto la precedente, due cerchi in un cerchio:

Facciata sinistra. Quattro cerchi e stella bisangolare in un cerchio, con:

Quattro falcate attorno a un quadrato, con:

Stella bisangolare in un cerchio, con:

Due quadrati entro due cerchi, con:

Corona circolare e quattro bisangoli, con:

Lunule, quadrati in un doppio cerchio, con:

Figura analoga alla precedente, con:

Quattro cerchi e stella bisangolare in un cerchio, con:

Cerchi e lunule in un cerchio, con:

Porzioni, falcate e bisangoli in un cerchio, con:

Quattro quadrilateri curvilinei in un cerchio, con:

Figura analoga alla precedente sovrimposta a un quadrato, con:

Croce e cerchi entro quadrato, con:

Cerchio con quadrato in cerchio doppio, con:

Figura analoga alla precedente, con:

Cerchio con bisangolo entro cerchio con porzioni, con:

Grande bisangolo entro un cerchio, con:

Cerchio iscritto in quadrato iscritto, con:

Quadrato con 'forcule' entro un cerchio, con:

Figura analoga alle precedenti, con:

Right face, from right to left. Circle with inscribed octagon:

Figure with eight squares inscribed in nine circles, with:

Double circle with lunes and a two-angled star, with:

Circle with inscribed square and eight two-angled figures:

Circle with lune and two-angled figures, with:

Four lunes and two-angled figures within a circle, with:

Four circles within a circle, with:

Four circles, eight two-angled figures in a circle, with:

Two circles against a crescent, with:

Circle within a quadrant of a circle, with:

Below the preceding, a circle diminished by two two-angled figures, with:

Four circles in a circle, with:

On the left, four unnumbered figures. In the third column, circle with two-angled star:

Below the preceding, circle with:

In the fourth column, circle inscribed in a square, with:

Below the preceding, two circles in a circle:

Left face. Four circles and a two-angled star in a circle, with:

Four crescents around a square, with:

Two-angled star in a circle, with:

Two squares within two circles, with:

Circular crown and four two-angled figures, with:

Lunes and squares in a double circle, with:

Figure analogous to the preceding, with:

Four circles and a two-angled star in a circle, with:

Circles and lunes in a circle, with:

Portions, crescents and two-angled figures in a circle, with:

Four curvilinear quadrilaterals in a circle, with:

Figure analogous to the preceding, superimposed on a square, with:

Cross and circles within a square, with:

Circle with square in a double circle, with:

Figure analogous to the preceding, with:

Circle with two-angled figure within a circle with portions, with:

Large two-angled figure within a circle, with:

Circle inscribed in an inscribed square, with:

Square with 'forks' within a circle, with:

Figure analogous to the preceding, with:

On this page

Squaring the circle through octagonal division

Nine circles reduced to nine squares

A lune equal to a curvilinear triangle

Areas worth half the greatest square

Repeating rosette and interlace-like patterns