Lighting a Shadowed Sphere; Colour of Painted Shadows
Treatise on Painting, Part Five: tangent-line proofs and how a surface takes its object's colour
This folio of the Treatise's Fifth Part opens by finishing a proof that a shadowed sphere, divided along the diameter a b, has one half wholly in shadow and the other wholly lit, the two regions bounded by parallel lines e f s t tangent to its front. A chapter [697] then argues that a light larger than a shadowed body can never illuminate exactly half of it, because the extremities of two unequal spheres cannot be tangent to the same two parallel lines. A new chapter [698], 'On the various darknesses of the shadows of bodies counterfeited in painting', states the governing rule that the surface of every opaque body partakes of the colour of its object, more or less according to distance. Small circle-and-tangent diagrams accompany the geometric arguments in the text column.
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[697] Can a light illuminate only half of a smaller shadowed body?
The chapter asks whether at some distance a luminous body can illuminate only part of a shadowed body smaller than itself, and answers that a light larger than the shadowed body can never illuminate exactly half of it. The reasoning treats the boundary between lit and shadowed regions as set by tangent illumination on the spherical surface.
Proof by parallel tangent lines to unequal spheres
The diameter a b of the shadowed sphere passes through its centre, so the sphere is divided into equal halves, one wholly shadowed and one wholly luminous, with the parallels e f s t tangent to its front. Between equidistant lines only spheres of that diameter are enclosed, so the extremities of two unequal spheres cannot both be tangent to the same two parallels.
[698] The darknesses of shadows counterfeited in painting
A new chapter states that the surface of every opaque body partakes of the colour of its object, and so much the more or less as the object is nearer or more remote. The proof is begun by letting a b c be the surface of an opaque body taken to be white.
Marginal diagrams of shadowed spheres
Two small pen diagrams set into the text column show circles with tangent lines meeting at a point, illustrating the parallel-tangent argument for the boundary between the lit and shadowed halves of a sphere. They are drawn, not transcribed, and support the geometric proofs of the facing text.