Quadrature of the circle: portions and lunes

Dividing a portion of a circle into equal parts

A construction to divide a portion of a circle into many equal and similar parts, in an even number. Related figures count out the parts of a divided circle, for example 17 equal and known parts made of 9 portions and 8 lunes.

Whoever squares n squares the circle (2a - n)

Beside a figure labelled 2a - n - scempia stands the claim that whoever squares n squares the circle, reducing the quadrature of the whole circle to the squaring of a single lettered figure.

Eight triangles with one curved and two straight sides

A semicircle is cut into 8 triangles equal among themselves, each with one curved side and two straight ones. This takes away the value of 8 of the smaller portions while the other takes 4, that is half; so the remainder of the half-taking equals the remainder of the eight-taking.

Circles in octuple proportion

Of circles in octuple proportion, the eighth of the greater is worth the whole of the lesser; therefore half of the lesser is worth a sixteenth of the greater. A neighbouring note treats circles in double proportion, where the quarter of the greater's portion equals half the lesser's.

Proof by equals taken from equals (a b c)

a b is worth c, proved by the axiom that if from equal things equal things are taken away, the remainder is equal. The circle is divided into two equal semicircles, and equals removed from equals leave equal remainders.

Columns of calculations

The lower-right corner carries several columns of arithmetical operations, including repeated multiplications around the number 6516 and figures such as 1530, 13032 and 68192.

Prima colonna da destra. Terza figura colle lettere:

Sesta figura:

Settima figura:

Ottava figura:

Le due figure successive:

Seconda colonna: Dodicesima figura:

Tredicesima figura:

Quattordicesima figura:

Quindicesima figura

Terza colonna. Terza figura, cerchio con quadrato inscritto:

Quarta figura, semicerchio con mezzo quadrato:

Quinta figura, settore di cerchio:

Sesta figura, settore di cerchio:

Settima figura, settore di cerchio

Ultima figura:

Quarta colonna. Seconda figura:

Terza figura:

Quinta colonna, semicerchio:

Angolo superiore sinistro:

Dal margine sinistro, coppia di semicerchi:

A destra delle precedenti, semicerchio :

Sotto il precedente, verso destra, semicerchio:

Sotto il precedente:

A destra del precedente entro un cerchio con:

Sotto il precedente, semicerchio con:

Operazioni aritmetiche. A destra:

First column from the right. Third figure, with the letters:

Sixth figure:

Seventh figure:

Eighth figure:

The two following figures:

Second column: Twelfth figure:

Thirteenth figure:

Fourteenth figure:

Fifteenth figure

Third column. Third figure, a circle with an inscribed square:

Fourth figure, a semicircle with half a square:

Fifth figure, a sector of a circle:

Sixth figure, a sector of a circle:

Seventh figure, a sector of a circle

Last figure:

Fourth column. Second figure:

Third figure:

Fifth column, a semicircle:

Upper left corner:

From the left margin, a pair of semicircles:

To the right of the preceding, a semicircle:

Below the preceding, toward the right, a semicircle:

Below the preceding:

To the right of the preceding, within a circle with:

Below the preceding, a semicircle with:

Arithmetical operations. On the right: