Problem X.
To Draw a Pyramid, Given in Position and Magnitude, on a Square Base in a Horizontal Plane
Let A B, 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, C D.19
Draw a square pillar, A B G E, Fig. 26., on the square base of the pyramid, and make the height of the pillar A F equal to the vertical height of the pyramid C D (Problem IX.). Draw the diagonals G F, H I, on the top of the square pillar, cutting each other in C. Therefore C is the center of the square F G H I. (Prob. VIII. Cor. II.)
Join C E, C A, C B.
Then A B C E 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.
Notes
19 If, instead of the vertical height, the length of A D is given, the vertical must be deduced from it. See the Exercises on this Problem in the Appendix, p. 79.