Quadrature Studies: Circles, Lunes and Portions

Upper right corner. A portion divided, with:

d b a c a b, 4 times, f a c a b d.

Two semicircles with: First — e m — n a;

The sixth a is worth b of the sixth figure, and n is worth m.

Another six semicircles with: Second;

Third; Fourth; Fifth; Sixth; Seventh.

Between the sign of a hand and the figure of two portions, with:

b a; c; Double

The similar parts of circles have between them the same proportions as that of their wholes.

a b is worth c.

Around the semicircle, with:

m h x n - b e a S - L - t p of
m b p is worth the right-angled triangle b t p.
b p is worth ...
n S a is worth a S f.
f S n x are equal.
Take away the straight lines a b of the square c, and the remainder of that c will be worth n t, n o p.

A group of lunes:

* b L p triangle ...

Let the triangle L p be drawn from the square e, and note the remainder of that square. Then let p m be drawn from that square, once b has been taken from it (a corresponding figure follows), and note that they will be equal, because p is the standard (champion).

Lunes superimposed on a square, with:

b a - d c

The lunes a b enter into the space c d; they will make one whole circle.

A figure like the preceding, halved, with:

o — p

o placed in p makes a semicircle.

Lower right corner.

The quarter of 8 is 2, and the half of 8 is 4. Said of L p, which is 1/4 of the square c e; adding b to it, it is the half of e. Therefore L p and b are 2, which comes to be 4, which is the half of 8.

To the left of the preceding.

Discourse on the semicircle that contains within itself the whole circle.

Join together L p, and you will make 1/4 of the square c e; if you add b to it, you will make half of that square c. Therefore b parallel is worth 1/4 of the square e, and is worth L p. And if from b you take away L and give back m, an equal value, you will have p b m equal to the half of the square c; and if from b you take away m and give back the half of t, you will have the half of the square c.

L p 1/4 of the square, b 1/4 of the square. Give to L p, the quarter of the square, half of b, and you will have 3/4 of the 1/4 square. Give to L p the whole of b and you will have 1/2 the square c. Take away L from p b and give back m; you will have half the square c. Take away m and give back half of t to p b, and you will have the half of the square c. Therefore the triangle p remains the standard of the whole. Add to b, h L, or if you wish t; it will be doubtfully equal to m p. Therefore, since we are not yet resolved concerning the value of p, it is necessary to seek some other device.

Above, central column. A half-portion divided, with:

a - Eighth - Eighth
Method of dividing a portion of a circle into infinitely many equal parts merely by dividing one of the two upper eighths; that is, divide only the eighth a in two, twice over, as is shown here below.

Another half-portion divided, with:

a b e – m n – f - d

The cuts a b c all direct themselves to the centre d of the circle, and the transverse cuts all direct themselves to the point f, that is, of the right-angled triangle n m f.

Another half-portion divided, with:

a b e – m n – f - d

The cuts a b c all direct themselves to the centre d of the circle, and the transverse cuts all direct themselves to the point f, that is, of the right-angled triangle n m f.

Above, to the left of the preceding. Two semicircles, with:

m -b b -p n; e d -f c -o b -r a

The portion m is worth the quadrangles c d e f; n p is worth a b o r.

A circular half-ring, with:

n b l - o curved - x t d h S
b n v t is a square that has one curved side and 3 straight, which square is equal to the quarter of the portion a b c d. It is proved, because its origin is caused by the square b n o d, which is the half of the half-portion b n c d, since in that is included half the lesser portion b n o c and the half of the triangle b c d, that is b o d; and thus our intent is proved.
Therefore the 4 squares a S, l h, b d, n t, c x are equal to the portion a b c d. It follows that, placing one surface upon the other, the excesses of the one are equal to the excesses of the other, and the common contacts are similar and equal.

Left margin. A series of circles.

The first figure is worth the second. The second is worth the third. The third is worth the fourth.

Centre of the sheet. Circles in quadruple proportion:

Of circles of quadruple proportion, the quarter of the one is worth the whole of the other. It is proved ...
Of quadruple circles, the quarter of the greater is worth the whole of the lesser.

Lunes superimposed on a circle, with:

b o c a - m n

a b is worth c o. It is proved because a b is worth n m and n m is worth c o, and when two things are equal to a third, they are equal to each other.

A crossed-out figure of a square, portions and lunes, with:

d c b a - e - f

a b c d e f is worth a circle made on the periphery of the 2 portions b c.
The ...
Of circles of quadruple proportion, the quarter of the one is worth the whole of the other.

A circle and a quadrant of a circle, with:

c a - d p - n - m b

The portions a b are worth the lunes e d; the curvilinear triangle n m is worth the rectilinear triangle p.

Two figures, a circle and a quadrant of a circle, with:

a; b
a is worth b.
The whole into the half. The half into the quarter. The quarter into the eighth. The eighth into the sixteenth.

Three sectors of a circle superimposed, with:

d c - b a

a b is worth c d.

A quadrant over a circle, with:

c o b a - d e

a c is worth e d, and because e d is similar to b o, a c is worth b o.

A quadrant over a circle, with:

c o b a - d e

a c is worth e d, and because e d is similar to b o, a c is worth b o.

Angolo superiore destro. Porzione divisa con:

Due semicerchi con: Primo -e m -n a;

Altri sei semicerchi con: Secondo;

Tra il segnale di una mano e la figura di due porzioni con:

Attorno al semicerchio con:

Gruppo di falcate:

Falcate sovrapposte a quadrato, con:

Figura come la precedente dimezzata, con:

Angolo inferiore destro.

A sinistra dei precedenti.

In alto, colonna centrale. Semiporzione divisa con:

Altra semiporzione divisa, con:

Altra semiporzione divisa, con:

In alto a sinistra dei precedenti. Due semicerchi, con :

Semicorona circolare con:

Margine sinistro. Serie di cerchi.

Centro del foglio. Cerchi in quadrupla proporzione:

Falcate sovrapposte a cerchio, con:

Figura biffata di quadrato, porzioni e falcate, con:

Cerchio e quadrante di cerchio, con:

Due figure, cerchio e quadrante di cerchio, con

Tre settori di cerchio sovrapposti, con:

Quadrante sopra cerchio, con:

Quadrante sopra cerchio, con:

Upper right corner. A portion divided, with:

Two semicircles with: First — e m — n a;

Another six semicircles with: Second;

Between the sign of a hand and the figure of two portions, with:

Around the semicircle, with:

A group of lunes:

Lunes superimposed on a square, with:

A figure like the preceding, halved, with:

Lower right corner.

To the left of the preceding.

Above, central column. A half-portion divided, with:

Another half-portion divided, with:

Another half-portion divided, with:

Above, to the left of the preceding. Two semicircles, with:

A circular half-ring, with:

Left margin. A series of circles.

Centre of the sheet. Circles in quadruple proportion:

Lunes superimposed on a circle, with:

A crossed-out figure of a square, portions and lunes, with:

A circle and a quadrant of a circle, with:

Two figures, a circle and a quadrant of a circle, with:

Three sectors of a circle superimposed, with:

A quadrant over a circle, with:

A quadrant over a circle, with:

On this page

Dividing a circular portion into infinite equal parts

Squaring a lune with one curved and three straight sides

Circles in quadruple proportion

A quarter of 8 is 2, half of 8 is 4

Discourse on the semicircle containing a whole circle

Lunes entering a square make a whole circle