Squaring the circle: lunes, portions and the inscribed square

Taking four greatest portions leaves an inscribed square

Removing the 4 greatest portions from the circle leaves a square whose four corners touch the surrounding circle. The square drawn outside the same circle, tangent along its sides, is double that inner square, so any figure made in the circle equals the smaller square and is half the greater.

The ring, the field m and the sixteen squares

In the large circular ring the field m equals the field of the four squares a b c d, and the solid a b c d equals the solid e f g h. The space enclosed by the smaller circumference equals its circumscribing parallelogram, which holds 16 squares plus four portions like m.

The doubling axiom on two circles in a 2:1 ratio

The two lower circles are double one to the other; taking double parts leaves a double remainder, as when 4 and 2 taken from 16 and 8 leave 12 and 6. Removing 8 bi-angles from one circle and 16 from the other keeps the remainder double and reduces it to squares.

The nine-pointed star and eight sectors

A greatest portion is divided into 8 sectors, together worth the 4 greatest portions. Taking 16 bi-angles from the double circle and 8 from the half circle leaves a double remainder, tallied in the column 16, 8, 4, 2, 1.

In alto, a destra.

Figura con:

Analoga figura con:

Grande corona circolare con quadrato iscritto, con:

A sinistra dall'alto. Figura con:

Analoga figura con:

Analoga figura con:

Analoga figura con:

Analoga figura con:

Analoga figura con:

Analoga figura con:

Figura a destra del maggior cerchio, con:

Tra i due grandi cerchi:

Sotto la figura:

A sinistra delle due figure:

Da destra a sinistra. Quadrato iscritto con porzioni:

Stella a nove punte entro un cerchio:

Upper right.

Figure with:

Analogous figure with:

Large circular ring with inscribed square, with:

On the left, from the top. Figure with:

Analogous figure with:

Analogous figure with:

Analogous figure with:

Analogous figure with:

Analogous figure with:

Analogous figure with:

Figure to the right of the larger circle, with:

Between the two large circles:

Below the figure:

To the left of the two figures:

From right to left. Inscribed square with portions:

Nine-pointed star within a circle:

Squaring the circle: lunes, portions and the inscribed square

On this page

Taking four greatest portions leaves an inscribed square

The ring, the field m and the sixteen squares

The doubling axiom on two circles in a 2:1 ratio

The nine-pointed star and eight sectors