Three natures of derivative shadow; the sphere's simple shadow
Why the sphere's shadow is called simple, and the narrowing, columnar and widening cast shadows
This page of the Treatise on Painting closes the account of the sphere's shadow — it receives reflection only from the simple shadow e, so it too is called a simple shadow — and then sets out the three figures of the derivative shadow. One is wide at its birth and narrows with distance; the second keeps an infinite length at the same width; the third grows wider at every degree of distance. Leonardo relates these to the size of the light: a shadow from a body smaller than the light is pyramidal and shorter the nearer the light, the parallel does not vary, and the dilatable widens as it nears the light. A final paragraph names the three natures — dilatable, columnar, and concurrent — the last of infinite length beyond the intersection of its sides.
On this page
Why the sphere's shadow is a simple shadow (e)
The shadow of the sphere sees neither the incident light nor reflected light of any degree. It therefore receives its reflection only from the simple shadow e, and for this reason it too is called a simple shadow.
Three figures: narrowing, columnar, widening
The derivative shadow has three varieties: one wide at its birth that narrows the farther it moves from that beginning; a second that keeps an infinite length at the same width as its birth; and a third that, at every degree of distance beyond its birth-width, acquires further degrees of width.
Pyramidal, parallel and dilatable shadows compared
The shadow from a body smaller than the light that illuminates it is pyramidal, and the shorter the nearer it is to the luminous body. The parallel shadow does not vary in such a case, while the dilatable widens the more it approaches its light. Leonardo names the three natures as dilatable, columnar, and concurrent to the intersection of its sides.