The pyramidal shadow produced by a columnar body will be narrower
than the body itself in proportion as the simple derived shadow is
intersected farther from the body which casts it.
[Footnote 166: Compare the first diagram to No. 161. If we here
conceive of the outlines of the pyramid of shadow on the ground as
prolonged beyond its apex this gives rise to a second pyramid; this
is what is spoken of at the beginning of No. 166.]
167.
The cast shadow will be longest when the light is lowest.
The cast shadow will be shortest when the light is highest.
168.
Both the primary and derived shadow will be larger when caused by
the light of a candle than by diffused light. The difference between
the larger and smaller shadows will be in inverse proportion to the
larger and smaller lights causing them.
[Footnote: In the diagrams _A_ stands for _celo_ (sky), _B_ for
_cadela_ (candle).]
169.
ALL BODIES, IN PROPORTION AS THEY ARE NEARER TO, OR FARTHER FROM THE
SOURCE OF LIGHT, WILL PRODUCE LONGER OR SHORTER DERIVED SHADOWS.
Among bodies of equal size, that one which is illuminated by the
largest light will have the shortest shadow. Experiment confirms
this proposition. Thus the body _m_ _n_ is surrounded by a larger
amount of light than the body _p q_, as is shown above. Let us say
that _v c a b d x_ is the sky, the source of light, and that _s t_
is a window by which the luminous rays enter, and so _m n_ and _p q_
are bodies in light and shade as exposed to this light; _m n_ will
have a small derived shadow, because its original shadow will be
small; and the derivative light will be large, again, because the
original light _c d_ will be large and _p q_ will have more derived
shadow because its original shadow will be larger, and its derived
light will be smaller than that of the body _m n_ because that
portion of the hemisphere _a b_ which illuminates it is smaller than
the hemisphere _c d_ which illuminates the body _m n_.
[Footnote: The diagram, given on Pl. IV, No. 2, stands in the
original between lines 2 and 7, while the text of lines 3 to 6 is
written on its left side. In the reproduction of this diagram the
letter _v_ at the outer right-hand end has been omitted.]
170.
The shadow _m_ bears the same proportion to the shadow _n_ as the
line _b c_ to the line _f c_.
171.