Studies on squaring the circle: lunes and nested circles

Reducing figures by removing equal lunes ('first, second, third')

The two removals of the two circles are equal, each worth half its circle. Because the eight circles of the third figure share the value of the first and second but cannot fit within the second, the central circle of the second is halved and remade to match one of the first, from which eight portions are taken with the four bi-angles of the third, leaving the third figure worth the greatest square of the sub-double circle.

Circle-to-circle as square-to-square of the diameters

a b is worth half of the circle a b c d. The proportion from one circle to another is the same as from one square to another, formed by multiplying the diameters of those circles. This ratio governs the reductions throughout the sheet.

Nested circles in continuous double proportion (d, c, a, b)

Circle d is double circle c, c is double a, and a is double b, because the greatest circle is quadruple the smallest. As the periphery of the greatest circle d is quadruple that of the smallest a, so the smaller curvilinear triangle's greatest periphery is quadruple its least, making periphery d octuple periphery h. The fourth figure is thereby reduced to a circle sub-double to the greatest.

If from a squareable figure you remove a squareable figure

If from a squareable figure you take away a squareable figure, the remainder is squareable. This principle underlies removing half a circle by taking away the two inner circles, so that the remainder equals the value of a circle sub-double to the greatest.

Three sizes of circle nested by diameter (greatest, middle, least)

The middle circle enters four times into the greatest, and each least circle enters four times into the middle, so the greatest is to the middle as the middle is to the least. Since the diameters are in double proportion, the circles follow the square-of-diameter ratio; the eight portions of the middle circle within its four bi-angles equal the four greatest portions of a circle sub-double to the greatest.

Dated note: Ascension Day at Amboise, 1517

Along the upper right margin Leonardo dates the sheet: the day of the Ascension at Amboise, in May 1517, at Clou[x]. The note anchors these late geometrical studies to his final French years.

Facciata destra, presso il margine superiore:

Colonna a destra; tre figure con:

Colonna sinistra, figura con:

Figura con:

Figura con:

Tre figure unite con tratti di penna, colle lettere:

La seconda figura con:

Facciata sinistra, dall'alto:

Figura:

Figura:

Figura:

Figura:

Figura di quadrato iscritto e circoscritto:

A matita colla relativa figura:

Con riferimento alla maggior figura sottostante:

A destra e a sinistra della figura con:

Sotto la figura suddetta:

A sinistra:

Right side, near the upper margin:

Right-hand column; three figures with:

Left-hand column, figure with:

Figure with:

Figure with:

Three figures joined by pen strokes, with the letters:

The second figure with:

Left side, from the top:

Figure:

Figure:

Figure:

Figure:

Figure of a square inscribed and circumscribed:

In pencil, with the related figure:

With reference to the larger figure below:

To the right and left of the figure, with:

Beneath the above figure:

On the left:

On this page

Reducing figures by removing equal lunes ('first, second, third')

Circle-to-circle as square-to-square of the diameters

Nested circles in continuous double proportion (d, c, a, b)

If from a squareable figure you remove a squareable figure

Three sizes of circle nested by diameter (greatest, middle, least)

Dated note: Ascension Day at Amboise, 1517