Squaring the circle: lunes reduced to triangles and squares

A semicircle's portions reduced to a triangle

The triangle labelled m f n is called the equivalent of all the workings of the semicircles of the 'first book'. From the semicircle its two greatest portions n m are drawn off, and what remains is left equal to the triangle f.

A circle's four portions reduced to a square

The inscribed square is presented as the equivalent of all the workings of the circles of the 'second book'. From the circle its four greatest portions n m o p are drawn off, and the remainder is left equal to the square g.

A lune made equal to a semicircle

In the central column a lune (falcata) a is set equal to a semicircle b, so that a equals b. A companion figure of internally tangent circles repeats that the lune a equals b.

Four equal parts of a lune

The four small figures n, a, b, c are declared equal, each worth a quarter of the lune a of the first figure and a quarter of that figure's half-circle b, as demonstrated above on the left.

Circles in quadruple proportion

For circles quadruple one another, a quarter of the larger equals the whole smaller circle, and four smaller circles equal the larger. Since the wholes stand in that ratio, like parts must stand in the same ratio as whole to whole.

Colonna destra. Semicerchio con:

Quadrato iscritto nel cerchio, con:

Cerchi internamente tangenti, con:

Quadrato in due cerchi concentrici, con:

Due cerchi concentrici, porzioni depennate, con

Due circoli concentrici, porzioni depennate, con

Rettangolo e triangoli fra porzioni depennate, con:

Colonna centrale. Figura di cerchi entro cerchio, con:

Sotto la precedente, cerchi tangenti internamente, con:

Cerchi concentrici, quadrati iscritti, con:

Quattro piccole figure di parti di cerchio, con:

A sinistra, cerchio con:

Sotto una linea orizzontale:

Sotto un'altra linea orizzontale:

Right column. Semicircle with:

Square inscribed in the circle, with:

Internally tangent circles, with:

Square in two concentric circles, with:

Two concentric circles, portions crossed out, with

Two concentric circles, portions crossed out, with

Rectangle and triangles among crossed-out portions, with:

Central column. Figure of circles within a circle, with:

Below the previous, internally tangent circles, with:

Concentric circles, inscribed squares, with:

Four small figures of parts of a circle, with:

On the left, circle with:

Beneath a horizontal line:

Beneath another horizontal line: