The Shadow of a Sphere and When the Eye Can See It
Whether an eye set behind the light can perceive a spherical body's shadow
This page of the Treatise on Painting reasons geometrically about the shadow cast by a spherical body and whether an observing eye can perceive it. It argues that when the luminous body equals the sphere, an eye placed behind the light can never see any part of the shadow, because the bounding rays run parallel and never converge to a point; but when the light is smaller than the sphere, some viewing distance can always be found from which the shadow becomes visible. A small spindle-shaped diagram at mid-page labels the light and the shadowed body o, p, b. The section closes by turning to the shadow of an opaque sphere set in the open air.
On this page
An eye behind a light equal to the sphere sees no shadow
When the luminous body equals the shadowed sphere, an eye l placed behind the light a b, at whatever distance, can never see any part of the shadow. The argument turns on the shadow of the sphere c f d and the sight-lines that reach the eye.
Parallel bounding rays never meet in a point
The proof invokes the rule that parallels never converge to a point (cited as the 7th of the ninth book). Because a c and b d are set parallel and just embrace half the sphere, and the lines n, m converge at l, that point can never see half the sphere across its diameter c d.
When the light is smaller, some distance reveals the shadow
If the luminous body a b is smaller than the shadowed body o p e f, some distance can always be found from which the eye n, standing behind the light, sees a shadowed part of the sphere's shadow, as the straightness of the sight-lines shows.
Diagram of the light and the shadowed body
A small elongated, spindle-like figure runs across the middle of the page, marking the light source and the shadowed body with the letters o, p, b, illustrating the geometry of the rays discussed in the text.