The curves most required in architectural drawing, after the circle, are those of pointed arches; in which, however, all that will be generally needed is to fix the apex, and two points in the sides. Thus if we have to draw a range of pointed arches, such as _APB_, Fig. 63., draw the measured arch to its sight-magnitude first neatly in a rectangle, _ABCD_; then draw the diagonals _AD_ and _BC_; where they cut the curve draw a horizontal line (as at the level _E_ in the figure), and carry it along the range to the vanishing-point, fixing the points where the arches cut their diagonals all along. If the arch is cusped, a line should be drawn, at _F_ to mark the height of the cusps, and verticals raised at _G_ and _H_, to determine the interval between them. Any other points may be similarly determined, but these will usually be enough. Figure 63. shows the perspective construction of a square niche of good Veronese Gothic, with an uncusped arch of similar size and curve beyond. [Illustration: Fig. 64.] In Fig. 64. the more distant arch only is lettered, as the construction of the nearest explains itself more clearly to the eye without letters. The more distant arch shows the general construction for all arches seen underneath, as of bridges, cathedral aisles, etc. The rectangle _ABCD_ is first drawn to contain the outside arch; then the depth of the arch, _Aa_, is determined by the measuring-line, and the rectangle, _abcd_, drawn for the inner arch. _Aa_, _Bb_, etc., go to one vanishing-point; _AB_, _ab_, etc., to the opposite one. In the nearer arch another narrow rectangle is drawn to determine the cusp. The parts which would actually come into sight are slightly shaded. PROBLEM XIV. Several exercises will be required on this important problem. I. It is required to draw a circular flat-bottomed dish narrower at the bottom than the top; the vertical depth being given, and the diameter at the top and bottom. [Illustration: Fig. 65.] Let _ab_, Fig. 65., be the diameter of the bottom, _ac_ the diameter of the top, and _ad_ its vertical depth. Take _AD_ in position equal to _ac_. On _AD_ draw the square _ABCD_, and inscribe in it a circle. Therefore, the circle so inscribed has the diameter of the top of the dish. From _A_ and _D_ let fall verticals, _AE_, _DH_, each equal to _ad_. Join _EH_, and describe square _EFGH_, which accordingly will be equal to the square _ABCD_, and be at the depth _ad_ beneath it. Within the square _EFGH_ describe a square _IK_, whose diameter shall be equal to _ab_. Describe a circle within the square _IK_. Therefore the circle so inscribed has its diameter equal to _ab_; and it is in the center of the square _EFGH_, which is vertically beneath the square _ABCD_. Therefore the circle in the square _IK_ represents the bottom of the dish. Now the two circles thus drawn will either intersect one another, or they will not.