TO DRAW A PYRAMID, GIVEN IN POSITION AND MAGNITUDE, ON A SQUARE BASE
IN A HORIZONTAL PLANE.
[Illustration: Fig. 25.]
Let _AB_, Fig. 25., be the four-sided pyramid. As it is given in
position and magnitude, the square base on which it stands must be
given in position and magnitude, and its vertical height, _CD_.[19]
[Illustration: Fig. 26.]
Draw a square pillar, _ABGE_, Fig. 26., on the square base of the
pyramid, and make the height of the pillar _AF_ equal to the vertical
height of the pyramid _CD_ (Problem IX.). Draw the diagonals _GF_,
_HI_, on the top of the square pillar, cutting each other in _C_.
Therefore _C_ is the center of the square _FGHI_. (Prob. VIII.
Cor. II.)
[Illustration: Fig. 27.]
Join _CE_, _CA_, _CB_.
Then _ABCE_ is the pyramid required. If the base of the pyramid is
above the eye, as when a square spire is seen on the top of a
church-tower, the construction will be as in Fig. 27.
[19] If, instead of the vertical height, the length of _AD_ is
given, the vertical must be deduced from it. See the Exercises
on this Problem in the Appendix, p. 79.