Why the Shadowed Sphere Is Darkest Where It Sees Most Earth
Point c, seeing the least sky and the most earth, is deepest in shadow; the cap a b stays brightest
Continuing the analysis of a sphere lit by the hemisphere of the sky, Leonardo gives a second demonstration: with the shadowed sphere a b c under the lit hemisphere d f e and the earth d e casting shadow, the cap a n b is free of shadow because the earth's darkness never sees it, while the rest is shaded in proportion to how much earth-darkness versus hemisphere-light each part sees. The point c is darkest of all because it sees the least of the hemisphere (only r d e s) and the whole earth d e, whereas a b is brightest, seeing only the earth's extremities. A further proposition states that the illuminated portion of any sphere shrinks as the part of the luminous body seeing it shrinks. Two semicircular diagrams of the sphere within the sky-hemisphere illustrate the reasoning.
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The unshadowed cap and the graded shadow
With shadowed sphere a b c, lit hemisphere d f e, and earth d e causing shadow, the cap a n b is deprived of shadow because the earth's darkness never sees it. The rest of the surface is shaded with more or less darkness according as more earth-darkness combines with less hemisphere-light.
Point c the darkest, a b the brightest
The point c is darker than any other part of the shadow because it sees the least of the hemisphere, only r d e s, while seeing the whole earth d e. The brightest is a b, because it sees only the extremities of the earth d e.
Less of the luminous seen, smaller the lit part
A proposition states that the illuminated part of any sphere is as much smaller as the part of the luminous body that sees it is smaller. Its proof takes a h as the shadowed body and c i e as the hemisphere, so that a is less lit for being seen by a smaller part of the day.