Quadrature of lunes: area equivalences of circles and squares

Lunules in a rectangle and square each 'equal the field'

At the head of the sheet a lunule enclosed in a rectangle labelled d c b a is said to equal the field, and squares carrying two lunules labelled a b likewise equal the field. Each caption asserts an equality of area between the curved figure and the surrounding rectilinear space.

Postulate: take and restore a like quantity to a surface

A 'petizione' (postulate) in the central column asks that Leonardo may take away and give back one same quantity to one same surface. This licenses the cutting-away and restoring of equal areas that the whole sheet performs.

The greatest square is double the middle square

In the lower 'first' figure the greatest square a b c d is shown to be double the middle square c f g h, so that the excess of the greater equals the middle itself and the four lesser semicircles equal the middle circle. Removing eight lesser portions and four middle portions leaves eight lune-shapes equal to two of the greatest portions.

A circular ring equal to a square surface

The 'second' figure sets a circular ring (paralello circulare) a i c L m K f h equal to the square surface a b c d m e f g. The 'third' and 'fourth' figures then prove one another equal to this ring by transforming a part i into an equal part b.

Four lunules with their enclosed square equal a doubled square

In the 'sixth' figure the four lunules together with the square they enclose equal a square double that enclosed square, and the surrounding field equals as much. Two squares touching one circle, one within and one without, are said to generate two squares each double the other, so the remainder of the greater equals the whole lesser.

Metà superiore, colonna destra. Lunula in un rettangolo, con:

Quadrato con due lunule:

Quadrato con due lunule:

Colonna centrale.

Semicerchio:

Semicerchio con:

Colonna sinistra.

Settore e grande porzione di cerchio, con:

Cerchio, con:

Quadrato entro cerchio, con:

Cerchio in semicerchio, con:

Metà inferiore. Cerchio iscritto e circoscritto a quadrati, con:

Corona circolare entro quadrato, con:

Quadrato e porzioni entro quadrato, con:

Quadrato e porzioni entro quadrato, con:

Quadrati e cerchi entro quadrato, con:

Quadrato, cerchio e lunule entro quadrato, con:

Cerchio e due quadrati entro cerchio, con:

Triangolo e quadrato entro cerchio, con:

Angolo inferiore destro. Cerchio e due quadrati entro cerchio, con:

Verso sinistra, serie di figurine accoppiate, con:

Upper half, right column. Lunule in a rectangle, with:

Square with two lunules:

Square with two lunules:

Central column.

Semicircle:

Semicircle with:

Left column.

Sector and large portion of a circle, with:

Circle, with:

Square within a circle, with:

Circle in a semicircle, with:

Lower half. Circle inscribed in and circumscribed about squares, with:

Circular ring within a square, with:

Square and portions within a square, with:

Square and portions within a square, with:

Squares and circles within a square, with:

Square, circle and lunules within a square, with:

Circle and two squares within a circle, with:

Triangle and square within a circle, with:

Lower right corner. Circle and two squares within a circle, with:

Toward the left, a series of paired little figures, with:

On this page

Lunules in a rectangle and square each 'equal the field'

Postulate: take and restore a like quantity to a surface

The greatest square is double the middle square

A circular ring equal to a square surface

Four lunules with their enclosed square equal a doubled square