Quadrature of the circle: inscribed circles, lunes, and roots

Three inscribed circles and their curvilinear triangles

The 3 circles enclosed in the greatest circle are worth half of it. Striking away their 24 hatched portions is the same as removing them from the 3 curvilinear triangles def, leaving the perforated circle abcdef equal to the greatest square of the greatest circle; the remainder, divided by 3, gives the quadrature of each triangle def.

Proportions of circles inscribed within a circle

The circle at the top contains the 3 largest circles it can hold, together worth half of it, so each smaller circle is worth a sixth of the greater. Hence one of their 4 greatest portions is worth one-sixth of a greatest portion of the containing circle, and the 4 greatest portions of the great circle equal 24 of the small circles' portions.

Circle proportion equals the proportion of the squares of diameters

Stated as a general rule: the proportion from circle to circle is the same as that from square to square formed by multiplying the diameters of those circles by themselves.

A compass rule for the square root of a number

To find any root on the line af — here the root of two — lay out the two spaces ab and bc, add a like space cd to obtain ad, take its middle o, set the compass foot at o, and open the compass to the spaces a-o and o-d.

Angolo superiore destro:

Figura di tre cerchi entro un cerchio, con:

Sotto la figura detta:

Colonna centrale, figura con

Figura di cerchio iscritto in un quadrante di cerchio:

Quattro bisangoli in un quadrante:

Continuando sulla prima colonna, figura con:

Figura analoga con:

In basso:

Figure di quadrato e di triangolo iscritti nel cerchio :

Colonna sinistra:

Figura di quadrati, cerchi e triangoli coi numeri

Figura di triangoli e quadrati coi numeri in senso antiorario

Figura colle lettere:

Figura colle lettere:

Upper right corner:

Figure of three circles within a circle, with:

Beneath the said figure:

Central column, figure with:

Figure of a circle inscribed in a quadrant of a circle:

Four lunes in a quadrant:

Continuing in the first column, figure with:

Analogous figure with:

Below:

Figures of a square and a triangle inscribed in the circle:

Left column:

Figure of squares, circles and triangles with the numbers:

Figure of triangles and squares with the numbers, counterclockwise:

Figure with the letters: