^Special  methods  are  such  as  are  founded  on  the  distinguishing
peculiarities  of  individual  problems,  or  groups  of  problems.  They
are  therefore  as  numerous  as  those  peculiarities,  and  hence  afford
the  natural  field  for  the  exercise  of  ingenuity  in  their  discovery
and  application.
Direct  methods  are  those  which  immediately  and  obviously
conform  to  the  usual  definition  of  a  shade  or  a  shadow  ;  as  when
a  point  of  shade  is  found  as  the  point  of  contact  of  a  given  ray
with  a  given  surface.
Indirect  methods  are  those  in  which  auxiliary  magnitudes  are
employed,  to  avoid  the  laborious  constructions  which  direct
methods  may  sometimes  occasion ;  as  when  we  find  a  point  of
shadow  on  some  irregular  surface,  by  noting  where  the  easily
found  shadow  upon  some  secant  plane  meets  the  intersection  of
that  plane  with  the  surface.
SHADES  AND   SHADOWS.
SERIES  I.
SHADES    AND    SHADOWS    FORMED    BY   PARALLEL    RAYS

DIVISION  I.
SHADES  AND  SHADOWS  ON  RULED  SURFACES.
23.  The  line  of  shade  on  any  ruled  surface  is  obtained  by  find-
ing, either  by  inspection,  or  by  construction — by  means  of  tangent
rays  or  planes  of  rays — the  edges,  elements,  or  points  which
constitute  that  line  of  shade.
24.  A  shadow  cast  by  a  body  upon  any  ruled  surface,  is  deter-
mined by  finding  an  element  of  that  surface,  and  a  point  of  the  line
of  shade  of  the  body,  both  of  which  shall  be  in  the  same  plane
of  rays.     A  ray  through  the  point  of  shade  will  then  intersect
the  element  of  the  ruled  surface  in  a  point  of  the  required  shadow.
CLASS  I.
SHADES  AND  SHADOWS  ON  PLANE  SURFACES.
§  I. — Shades.
