Squaring the Circle: Lunes, Squares and Circle-Portions

A squarable part taken from a quadrilateral

If from the quadrilateral you take away a squarable part, the remainder is squarable, not by itself with its own parts, but in that a square equal to it can be given. The principle opens the sheet's reduction of curved figures to squares.

The lune d worth four circle-portions e f g h

In the first figure, a b joined together are worth the greatest square of the greatest circle; and since the circles are double one to the other, the lune-shaped surface d is worth the four portions e, f, g and h.

Fitting a convex portion into a concave one

In the lower margin Leonardo instructs: place the convex portion a o n into the concave portion p r q. There remains a surface K, shown below, equal to the square. It is a cut-and-fit proof that a lune equals a rectilinear square.

Segments each a third of a semicircle

The two segments a and b are each by themselves a third part of a semicircle, and joined together are worth two thirds of a semicircle; those two thirds in turn are worth the circle n drawn below.

Circles in double and quadruple proportion

The composition of circles within a rectangle is captioned n: these circles of the cross-shaped surface are of quadruple proportion, while elsewhere the circles are said to be double one to the other. The figures thus encode ratios of area, not only equalities.

Colonna destra.

Figura con:

Figura con:

Figura con:

Figura con:

Figura con:

Margine inferiore. Tre figure con:

Colonna sinistra. Cerchio con stella bisangolare:

Idem con stella semivuota:

Cerchio con corona circolare:

Quadrato curvo e semicerchio:

Cerchio con stella bisangolare:

Quadrato curvo e cerchio:

Tre settori di cerchio con quadrati curvi:

Figure composte di segmenti circolari a due basi:

Composizione di cerchi entro un rettangolo:

Croce entro cerchio:

Corona circolare e quattro settori in campo bianco:

Idem entro un grande cerchio, con:

Figura analoga con:

Right column.

Figure with:

Figure with:

Figure with:

Figure with:

Figure with:

Lower margin. Three figures with:

Left column. Circle with a lune-star:

The same, with a half-empty star:

Circle with a circular ring (annulus):

Curved square and semicircle:

Circle with a lune-star:

Curved square and circle:

Three circle-sectors with curved squares:

Figures composed of circular segments with two bases:

Composition of circles within a rectangle:

Cross within a circle:

Circular ring and four sectors on a white field:

The same within a large circle, with:

Analogous figure with:

On this page

A squarable part taken from a quadrilateral

The lune d worth four circle-portions e f g h

Fitting a convex portion into a concave one

Segments each a third of a semicircle

Circles in double and quadruple proportion