TO DRAW A SQUARE PILLAR, GIVEN IN POSITION AND MAGNITUDE, ITS BASE AND
TOP BEING IN HORIZONTAL PLANES.
Let _AH_, Fig. 23., be the square pillar.
Then, as it is given in position and magnitude, the position and
magnitude of the square it stands upon must be given (that is, the
line _AB_ or _AC_ in position), and the height of its side _AE_.
[Illustration: Fig. 23.] [Illustration: Fig. 24.]
Find the sight-magnitudes of _AB_ and _AE_. Draw the two sides _ab_,
_ac_, of the square of the base, by Problem VIII., as in Fig. 24. From
the points _a_, _b_, and _c_, raise vertical lines _ae_, _cf_, _bg_.
Make _ae_ equal to the sight-magnitude of _AE_.
Now because the top and base of the pillar are in horizontal planes,
the square of its top, _FG_, is parallel to the square of its base,
_BC_.
Therefore the line _EF_ is parallel to _AC_, and _EG_ to _AB_.
Therefore _EF_ has the same vanishing-point as _AC_, and _EG_ the same
vanishing-point as _AB_.
From _e_ draw _ef_ to the vanishing-point of _ac_, cutting _cf_ in
_f_.
Similarly draw _eg_ to the vanishing-point of _ab_, cutting _bg_ in
_g_.
Complete the square _gf_ in _h_, by drawing _gh_ to the
vanishing-point of _ef_, and _fh_ to the vanishing-point of _eg_,
cutting each other in _h_. Then _aghf_ is the square pillar required.