The Two Halves of the Compound Derivative Shadow
How compound shadow forms between simple shadow and light, its lighter and darker halves defined
This all-text page argues that because compound shadow is nothing but a diminution of light, beginning and ending with the luminous body, it is generated in the middle between the simple shadow and the simple light. With luminous body a b c and shadowy body d e, the simple derivative shadow d e f sees no part of the light, while the compound f e k always sees some. The proof then splits the compound shadow e f k into a lighter half i k, which sees half the light a c, and a darker half f i, which sees c b. Chapter [564] concludes that simple and compound shadow are proportioned the same way in primitive shadows joined to bodies as in the derivative shadows separated from them.
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Compound shadow lies between simple shadow and light
Because compound shadow is nothing but a diminution of light, beginning at the start of the luminous body and ending with its end, it is generated in the middle between the simple shadow and the simple light. With luminous body a b c and shadowy body d e, the simple derivative d e f sees no part of the light, whereas the compound derivative f e k always sees some.
The lighter and darker halves defined
The compound shadow e f k is split by its thickness: the half i k sees half the luminous body, a c, and is the lighter part; the darker half f i sees the second half c b. Thus the two parts of the compound derivative shadow are determined, one lighter or less dark than the other.
Primitive and derivative shadows share the same proportion
Simple and compound shadow are proportioned among themselves in the primitive shadows joined to bodies just as in the derivative shadows separated from them. This holds because the derivative simple and compound shadows are joined without interruption, exactly as are the primitive shadows from which they originate.