Quadrature series of lunes and circular portions

An ordered series of lunes, each worth the previous

Leonardo lays out a numbered progression of curved figures along the median fold: the 'first' with parts a b c d equal, the 'second' worth the first, the 'third' worth the second, the 'fourth' worth the third and the 'fifth' worth the fourth, continued to six, seven and eight. Each step equates a reshaped lune or portion to its predecessor. The chain shows curved pieces conserved in value as they are rearranged.

Sixty portions and the decuple hexagon

A circular ring of two-angles inscribed in a square carries sixty portions. Dividing them by six gives ten, so the smaller circle enters the greater ten times, and ten of the smaller portions are worth one greatest portion. The greatest hexagon is thus decuple to the smallest hexagon.

A lune worth a figure, circles in quadruple proportion

In one double figure a is worth b with the circles double to one another, and d is the square of b. In another, a lune is worth b where the circles stand in quadruple proportion. A curvilinear triangle's upper portion is shown filling its two lower concavities to become a straight-sided triangle.

Rosettes and star figures in squares and circles

The upper sheet carries several finished decorative figures: an eight-petalled rosette in a square, a star tessellation within a circle, and small florettes in an octagon and circles. Drawn as quadrature exercises, they present as symmetric applied-art motifs. Much of the lower sheet is blank with only faint offset traces.

Facciata destra.Stella bisangolare entro corona circolare:

Facciata sinistra, lungo la piega mediana. Porzione divisa, con:

Due ' triangoli curvilinei ', con:

Mezza porzione raddoppiata, con:

La precedente colle due metà capovolte:

Bisangolo vacuo, con:

Tre figure successive con:

Margine superiore:

Figura con:

Doppia figura con:

Di fianco alle precedenti:

Figura con:

Segmento circolare a due basi sopra un cerchio:

In basso. Corona circolare con bisangoli, iscritta in un quadrato:

Figura analoga entro un quadrato:

Colonna sinistra dall'alto.

Triangolo curvilineo con:

Figura sinuosa:

Falcate invertite:

Quadrilateri con ' eliche ' :

Right face. Two-angled star within a circular ring:

Left face, along the median fold. Divided portion, with:

Two 'curvilinear triangles', with:

Doubled half-portion, with:

The preceding with the two halves inverted:

Empty two-angle, with:

Three successive figures with:

Upper margin:

Figure with:

Double figure with:

Beside the preceding:

Figure with:

Circular segment on two bases upon a circle:

Below. Circular ring with two-angles, inscribed in a square:

Analogous figure within a square:

Left column from the top.

Curvilinear triangle with:

Sinuous figure:

Inverted crescents:

Quadrilaterals with 'helices':

On this page

An ordered series of lunes, each worth the previous

Sixty portions and the decuple hexagon

A lune worth a figure, circles in quadruple proportion

Rosettes and star figures in squares and circles