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. COROLLARY. It is obvious that if _AE_ is equal to _AC_, the whole figure will be a cube, and each side, _aefc_ and _aegb_, will be a square in a given vertical plane. And by making _AB_ or _AC_ longer or shorter in any given proportion, any form of rectangle may be given to either of the sides of the pillar. No other rule is therefore needed for drawing squares or rectangles in vertical planes. Also any triangle may be thus drawn in a vertical plane, by inclosing it in a rectangle and determining, in perspective ratio, on the sides of the rectangle, the points of their contact with the angles of the triangle. And if any triangle, then any polygon. A less complicated construction will, however, be given hereafter.[18] [18] See page 96 (note), after you have read Problem XVI. PROBLEM X. 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. PROBLEM XI. TO DRAW ANY CURVE IN A HORIZONTAL OR VERTICAL PLANE. [Illustration: Fig. 28.] Let _AB_, Fig. 28., be the curve. Inclose it in a rectangle, _CDEF_. Fix the position of the point _C_ or _D_, and draw the rectangle. (Problem VIII. Coroll. I.)[20] Let _CDEF_, Fig. 29., be the rectangle so drawn. [Illustration: Fig. 29.] If an extremity of the curve, as _A_, is in a side of the rectangle, divide the side _CE_, Fig. 29., so that _AC_ shall be (in perspective ratio) to _AE_ as _AC_ is to _AE_ in Fig. 28. (Prob. V. Cor. II.) Similarly determine the points of contact of the curve and rectangle _e_, _f_, _g_.