Light and shadow on spheres, with sphere geometry and a comet

Laws of light on illuminated spheres

A sequence of propositions states that the illuminated part of any spherical body is the smaller the smaller the portion of the luminous body that sees it, and that its brightness is more intense where fewer shadowy species accompany it. A shadow resting on the earth in contact with it is darker than one surrounded by air, because it also receives the shadow the body casts on the ground.

Definitions of shadow and darkness

Beside a lettered figure Leonardo defines the terms: darkness is the privation of light, shadow is the diminishing of light, and shadow is a mixture of darkness with light, of greater or lesser darkness according to the power of the light mixed into it.

The tail of a firebrand in moving air

Figures of a firebrand (stizzo) analyse how its glowing tail (chioma) is shaped by the wind: let a be the firebrand, b the wind, c the tail of the fire. A firebrand held still in moving air makes as long a tail as a moving brand does in still air, and if fire and air move uniformly together the fire has no tail at all.

The comet's ray, the air, and the sky

A short note reasons that if the air moves while the sky stands still, the top of the air leaves the ray of the comet turned toward the east; and if the air stands still and the sky moves, the comet's ray turns toward the west.

To make a crucifix appear in a room

On the left half, beside a small box and a quadrilateral marked with a cross, a note reads 'to make a crucifix appear in a room', apparently a projected-image or optical trick using a darkened chamber.

Trisecting a line and equal circular sectors

Triangle and sector diagrams carry constructions: a line a b is divided into three equal parts by stepping a fixed compass opening three times along a line and drawing to a point q. A further sector figure proves spaces equal by the rule that two things equal to a third are equal to one another.

Metà destra, prima colonna.

Cerchio in semicerchio, colle lettere:

Figura analoga, con:

Figura analoga, con:

Margine inferiore. Figura analoga, con:

Figura analoga, con:

Seconda colonna.

Figure di 'stizzo del foco', con:

Terza colonna.

Zona sferica con:

Figura analoga con:

Due calotte sferiche, con:

Parte di zona sferica, con :

Metà sinistra. In alto, cassetta e quadrilatero con croce:

Calotta sferica, con:

Sotto la figura col putto, settore di cerchio, con:

Triangolo con:

Triangolo con:

Settore di cerchio, con:

Right half, first column.

Circle within a semicircle, with the letters:

Analogous figure, with:

Analogous figure, with:

Lower margin. Analogous figure, with:

Analogous figure, with:

Second column.

Figures of the 'firebrand', with:

Third column.

Spherical zone, with:

Analogous figure, with:

Two spherical caps, with:

Part of a spherical zone, with:

Left half. Above, a small box and a quadrilateral with a cross:

Spherical cap, with:

Below the figure with the putto, a sector of a circle, with:

Triangle, with:

Triangle, with:

Sector of a circle, with:

On this page

Laws of light on illuminated spheres

Definitions of shadow and darkness

The tail of a firebrand in moving air

The comet's ray, the air, and the sky

To make a crucifix appear in a room

Trisecting a line and equal circular sectors