Quadrature of lunes, cylinder geometry and a bitter letter to a brother

Squaring the lune from tangent quarter-circles

Two quarter-circles, one double the other (a b c d e f g h for the larger, d e f g h for the smaller), are compared so their excess equals the smaller quarter. By subtracting sector c, transposing part f to point i beneath f, and adding part b, the lune a b is shown equal to surface g d.

Companion figure of equalities

A second analogous figure lists a chain of equalities among its parts: a b is worth c d e f, a c is worth a b, c is worth b, b e is worth e f and d f, b d is worth e f, and b e f is worth d e f.

Cylinder, right triangle and uniform curvature

Cylinder studies relate geometry to the solid: the sides a b c of a right triangle equal half the circumference containing the cylinder's circle together with its height, and a line of uniform curvature drawn from the top circumference down to the base is said to be worth 'in power' the whole base circumference plus the height.

Draft letter to a brother

In fine script Leonardo writes to his 'most beloved brother,' having learned of the birth of an heir. With heavy irony he declares that his brother, thinking himself prudent, has rejoiced at creating a solicitous enemy who will crave the liberty that comes only with the father's death.

Siege engines (drawn, untranscribed)

The upper left carries small pen sketches of catapult- and trebuchet-like siege engines. These drawings are not covered by the transcribed text.

Domed building with columns (drawn, untranscribed)

At upper right stands a small sketch of a centrally planned domed building flanked by columns. It is not described in the transcribed notes.

Metà superiore, a destra.

Figura con:

Muovendo verso sinistra.

Figura analoga, con:

In scrittura più minuta:

Sotto la precedente:

Metà inferiore, girato il foglio sul lato destro . Numeri e operazioni:

Con riferimento alla figura sottostante:

Due figure di cilindro e cilindro con rettangolo, con:

Cilindro sezionato a metà, con:

Cilindro, con:

Upper half, at right.

With a figure:

Moving toward the left.

Analogous figure, with:

In finer writing:

Below the preceding:

Lower half, the sheet turned on its right side. Numbers and operations:

With reference to the figure below:

Two figures of a cylinder, and a cylinder with a rectangle, with:

Cylinder cut in half, with:

Cylinder, with: