Quadrature of Lunes and Curved Figures

The rule for squaring a curved surface

You will never square the curved surface unless you restore what you take away, either in figure or in value. And no figure of a single curved side can be squared.

Squares built on a right triangle (3 4 5)

In chalk Leonardo constructs squares on the sides of a right-angled triangle and marks them 3 4 5, the classic Pythagorean triple. The same figures recur with the letters r i, tying the square constructions to the quadrature studies alongside.

Squaring a lune by restoring inside what is cut outside (a n)

To the lune a Leonardo restores on the inside the value of what he removes from the outside, so that it remains squared and equal to the whole portion n. He notes the result matches the first and second lunes of the group of five figures.

Circles quadruple and double one another (a b, c d)

The two subdivided portions are equal in value while the circle of a is quadruple the circle of b, and c and d are of equal quantity while the circle of c is double that of d. The note reads the curved areas against the ratios of their circles.

Double and sub-double semicircles (m n o, p)

Because two semicircles are double one of the other, the greater receives the whole of the lesser, and what overhangs outside equals that lesser. Hence m n o, the parts by which the double semicircle exceeds its sub-double, equal the sub-double semicircle p.

Facciata destra. A matita, disegni di quadrati costruiti sui lati d'un ortogonio, colle lettere:

Mezza porzione divisa in due falcate e una porzione minore, colle lettere:

A matita quadrati costruiti attorno a un ortogonio, coi numeri:

A foglio capovolto, tre semicerchi divisi in settori:

Facciata sinistra. In alto. Segmento circolare a due basi e settore di cerchio con:

Colonna destra.

Quattro figure, di cui una biffata, colle lettere:

Gruppo di cinque figure di falcate colle lettere:

Due figure di porzioni suddivise colle lettere:

A sinistra del precedente, segmento circolare diviso colle lettere:

Lungo il margine sinistro, settore di cerchio con:

Segmenti circolari colle lettere:

Figura analoga alla precedente con:

Settori di cerchio e segmenti circolari a due basi sovrapposti, con:

Segmento circolare a due basi con:

Lungo il margine inferiore. Figura analoga alle due predette, e due minori figure con:

A destra del precedente, semicerchio con:

Right face. In pencil, drawings of squares constructed on the sides of a right triangle, with the letters:

A half-portion divided into two lunes and a lesser portion, with the letters:

In pencil, squares constructed about a right triangle, with the numbers:

With the sheet inverted, three semicircles divided into sectors:

Left face. Above. A circular segment with two bases and a sector of a circle with:

Right column.

Four figures, one of them crossed out, with the letters:

Group of five figures of lunes with the letters:

Two figures of subdivided portions with the letters:

To the left of the preceding, a divided circular segment with the letters:

Along the left margin, a sector of a circle with:

Circular segments with the letters:

A figure like the preceding with:

Sectors of a circle and circular segments with two bases superimposed, with:

A circular segment with two bases with:

Along the lower margin. A figure like the two aforesaid, and two smaller figures with:

To the right of the preceding, a semicircle with:

On this page

The rule for squaring a curved surface

Squares built on a right triangle (3 4 5)

Squaring a lune by restoring inside what is cut outside (a n)

Circles quadruple and double one another (a b, c d)

Double and sub-double semicircles (m n o, p)