Then _GB_ and _GH_ are the true magnitudes of _GB_ and _GH_ in Fig. 56. If _GB_ is given, and not the vertical altitude, with center _B_, and distance _GB_, describe circle cutting _FG_ in _G_, and _FG_ is the vertical altitude. If _GH_ is given, describe the circle from _H_, with distance _GH_, and it will similarly cut _FG_ in _G_. It is especially necessary for the student to examine this construction thoroughly, because in many complicated forms of ornaments, capitals of columns, etc., the lines _BG_ and _GH_ become the limits or bases of curves, which are elongated on the longer (or angle) profile _GH_, and shortened on the shorter (or lateral) profile _BG_. We will take a simple instance, but must previously note another construction. It is often necessary, when pyramids are the roots of some ornamental form, to divide them horizontally at a given vertical height. The shortest way of doing so is in general the following. [Illustration: Fig. 58.] Let _AEC_, Fig. 58., be any pyramid on a square base _ABC_, and _ADC_ the square pillar used in its construction. Then by construction (Problem X.) _BD_ and _AF_ are both of the vertical height of the pyramid. Of the diagonals, _FE_, _DE_, choose the shortest (in this case _DE_), and produce it to cut the sight-line in _V_. Therefore _V_ is the vanishing-point of _DE_. Divide _DB_, as may be required, into the sight-magnitudes of the given vertical heights at which the pyramid is to be divided. [Illustration: Fig. 59.] [Illustration: Fig. 60.] From the points of division, 1, 2, 3, etc., draw to the vanishing-point _V_. The lines so drawn cut the angle line of the pyramid, _BE_, at the required elevations. Thus, in the figure, it is required to draw a horizontal black band on the pyramid at three fifths of its height, and in breadth one twentieth of its height. The line _BD_ is divided into five parts, of which three are counted from _B_ upwards. Then the line drawn to _V_ marks the base of the black band. Then one fourth of one of the five parts is measured, which similarly gives the breadth of the band. The terminal lines of the band are then drawn on the sides of the pyramid parallel to _AB_ (or to its vanishing-point if it has one), and to the vanishing-point of _BC_. If it happens that the vanishing-points of the diagonals are awkwardly placed for use, bisect the nearest base line of the pyramid in _B_, as in Fig. 59. Erect the vertical _DB_ and join _GB_ and _DG_ (_G_ being the apex of pyramid). Find the vanishing-point of _DG_, and use _DB_ for division, carrying the measurements to the line _GB_. In Fig. 59., if we join _AD_ and _DC_, _ADC_ is the vertical profile of the whole pyramid, and _BDC_ of the half pyramid, corresponding to _FGB_ in Fig. 57. [Illustration: Fig. 61.] We may now proceed to an architectural example.