These bodies standing apart in a room lighted by a single window
will have derivative shadows more or less short according as they
are more or less opposite to the window. Among the shadows cast by
bodies of equal mass but at unequal distances from the opening by
which they are illuminated, that shadow will be the longest of the
body which is least in the light. And in proportion as one body is
better illuminated than another its shadow will be shorter than
another. The proportion _n m_ and _e v k_ bear to _r t_ and _v x_
corresponds with that of the shadow _x_ to 4 and _y_.

The reason why those bodies which are placed most in front of the
middle of the window throw shorter shadows than those obliquely
situated is:--That the window appears in its proper form and to the
obliquely placed ones it appears foreshortened; to those in the
middle, the window shows its full size, to the oblique ones it
appears smaller; the one in the middle faces the whole hemisphere
that is _e f_ and those on the side have only a strip; that is _q r_
faces _a b_; and _m n_ faces _c d_; the body in the middle having a
larger quantity of light than those at the sides is lighted from a
point much below its centre, and thus the shadow is shorter. And the
pyramid _g_ 4 goes into _l y_ exactly as often as _a b_ goes into _e
f_. The axis of every derivative shadow passes through 6 1/2
[Footnote 31: _passa per_ 6 1/2 (passes through 6 1/2). The meaning
of these words is probably this: Each of the three axes of the
derived shadow intersects the centre (_mezzo_) of the primary shadow
(_ombra originale_) and, by prolongation upwards crosses six lines.

This is self evident only in the middle diagram; but it is equally
true of the side figures if we conceive of the lines 4 _f_, _x n v
m_, _y l k v_, and 4 _e_, as prolonged beyond the semicircle of the
horizon.] and is in a straight line with the centre of the primary
shadow, with the centre of the body casting it and of the derivative
light and with the centre of the window and, finally, with the
centre of that portion of the source of light which is the celestial
hemisphere, _y h_ is the centre of the derived shade, _l h_ of the
primary shadow, _l_ of the body throwing it, _l k_ of the derived
light, _v_ is the centre of the window, _e_ is the final centre of
the original light afforded by that portion of the hemisphere of the
sky which illuminates the solid body.

[Footnote: Compare the diagram on Pl. IV, No. 3. In the original
this drawing is placed between lines 3 and 22; the rest, from line 4
to line 21, is written on the left hand margin.]