Shadow of a sphere under star-shaped light, with quadrature figures

Apparatus for the clean edge of a shadow

To see the sharp boundary separating simple from mixed shadow, use a screen of paper greased with turpentine and oil struck by sunlight, behind it a thin board pierced with eight equal holes evenly spaced on a circle half a braccio across, and a dense spherical body set half a braccio from its centre. A thin chancery sheet between the eye and the sphere then shows the shadow exactly.

Shadow cast by an eight-rayed 'star' light

The striking figure of overlapping circles is explained as the shadow made by a spherical shadowy body lit by a star-shaped light whose rays are of equal thickness. To grade the simple shadows finely, make a star of eight rays equally broad from tip to base, turn it to the sun, and place behind it the sphere, then the paper, then the eye.

Reminder to record the qualities of shadows

Near the central fold Leonardo writes a note to himself: remember to record the qualities and quantities of the shadows. The instruction connects the optical experiments to the practical rendering of shade.

Quadrature of a circle sector

On a quadrant over a semicircle, a b are equal and b is the quadrature of a, leaving c to be squared. In the second figure the semicircle a o c equals the figure o b n m; removing o from each side leaves a e equal to b n m, and since b squares a, m n remains equal to c.

Equalities among superimposed sectors

Working two superimposed sectors, Leonardo sets a equal to b o plus b: subtracting b o from a leaves a remainder equal to b squared, or subtracting b squared leaves a equal to the un-squared b o. A neighbouring quarter-and-eighth pair shows that lending b the quantity d c makes b c d equal to a alone.

An inverted salutation

With the sheet reversed, a single word survives at the head: 'Most beloved'. It reads as the opening of a letter jotted among the diagrams.

Metà destra. Figura con:

Presso la piega mediana:

Meta sinistra.

Intersezione di cerchi:

Metà superiore, da destra a sinistra. Quarto e ottavo di cerchio sovrapposti, con:

Parte di settori sovrapposti, con:

Figura rifatta dal Leoni, con:

Metà inferiore, da destra a sinistra. Due figurine, con:

Due settori sovrapposti, con:

Angolo inferiore destro. Figurina con :

Quadrante ampliato di cerchio su semicerchio, con:

Quadrante su semicerchio, con:

Foglio capovolto.

Right half. Figure with:

Near the median fold:

Left half.

Intersection of circles:

Upper half, from right to left. Quarter and eighth of a circle superimposed, with:

Part of superimposed sectors, with:

Figure redone by Leoni, with:

Lower half, from right to left. Two small figures, with:

Two superimposed sectors, with:

Lower right corner. Small figure with:

Enlarged quadrant of a circle on a semicircle, with:

Quadrant on a semicircle, with:

Sheet turned upside down.