transport the
length _L'f'_ to the opposite side and draw the vertical downwards from
the horizon, as at _FL_, we can then suppose point _L_ to be exactly
opposite the sun, and if we make that the vanishing point for the sun's
rays we shall find that we obtain precisely the same result. As in Fig.
277, if we wish to find the length of _C_, which we may suppose to be
the shadow of _P_, we can either draw a line from _A_ through _O_ to
_B_, or from _B_ through _O_ to _A_, for the result is the same. And as
we cannot make use of a point that is behind us and out of the picture,
we have to resort to this very ingenious device.

[Illustration: Fig. 277.]

In Fig. 276 we draw lines L1, L2, L3 from the luminary to the top of the
object to meet those drawn from the foot _F_, namely F1, F2, F3, in the
same way as in the figures we have already drawn.

[Illustration: Fig. 278.]

Fig. 278 gives further illustration of this problem.

CLV

THE SHADOW OF AN INCLINED PLANE

The two portions of this inclined plane which cast the shadow are first
the side _fbd_, and second the farther end _abcd_. The points we have to
find are the shadows of _a_ and _b_. From luminary _L_ draw _La_, _Lb_,
and from _F_, the foot, draw _Fc_, _Fd_. The intersection of these lines
will be at _a'b'_. If we join _fb'_ and _db'_ we have the shadow of the
side _fbd_, and if we join _ca'_ and _a'b'_ we have the shadow of
_abcd_, which together form that of the figure.

[Illustration: Fig. 279.]

CLVI

SHADOW ON A ROOF OR INCLINED PLANE

To draw the shadow of the figure _M_ on the inclined plane _K_ (or a
chimney on a roof). First find the vanishing point _P_ of the inclined
plane and draw horizontal _PF_ to meet vertical raised from _L_, the
luminary. Then _F_ will be the vanishing point of the shadow. From _L_
draw L1, L2, L3 to top of figure _M_, and from the base of _M_ draw
1F, 2F, 3F to _F_, the vanishing point of the shadow. The
intersections of these lines at 1, 2, 3 on _K_ will determine the
length and form of the shadow.

[Illustration: Fig. 280.]

CLVII

TO FIND THE SHADOW OF A PROJECTION OR BALCONY ON A WALL

To find the shadow of the object _K_ on the wall _W_, drop verticals
_OO_ till they meet the base line _B'B'_ of the wall. Then from the
point of sight _S_ draw lines through _OO_, also drop verticals _Dd'_,
_Cc'_, to meet these lines in _d'c'_; draw _c'F_ and _d'F_ to foot of
luminary. From th