[Illustration: Fig. 37.] As it is given in magnitude, its diameters must be given at the base and summit, _AB_ and _CD_; and its vertical height, _CE_.[24] And as it is given in position, the center of its base must be given. [Illustration: Fig. 38.] Draw in position, about this center,[25] the square pillar _afd_, Fig. 38., making its height, _bg_, equal to _CE_; and its side, _ab_, equal to _AB_. In the square of its base, _abcd_, inscribe a circle, which therefore is of the diameter of the base of the cone, _AB_. In the square of its top, _efgh_, inscribe concentrically a circle whose diameter shall equal _CD_. (Coroll. Prob. XIII.) Join the extremities of the circles by the right lines _kl_, _nm_. Then _klnm_ is the portion of cone required. COROLLARY I. If similar polygons be inscribed in similar positions in the circles _kn_ and _lm_ (Coroll. Prob. XII.), and the corresponding angles of the polygons joined by right lines, the resulting figure will be a portion of a polygonal pyramid. (The dotted lines in Fig. 38., connecting the extremities of two diameters and one diagonal in the respective circles, occupy the position of the three nearest angles of a regular octagonal pyramid, having its angles set on the diagonals and diameters of the square _ad_, inclosing its base.) If the cone or polygonal pyramid is not truncated, its apex will be the center of the upper square, as in Fig. 26. COROLLARY II. If equal circles, or equal and similar polygons, be inscribed in the upper and lower squares in Fig. 38., the resulting figure will be a vertical cylinder, or a vertical polygonal pillar, of given height and diameter, drawn in position. COROLLARY III. If the circles in Fig. 38., instead of being inscribed in the squares _bc_ and _fg_, be inscribed in the sides of the solid figure _be_ and _df_, those sides being made square, and the line _bd_ of any given length, the resulting figure will be, according to the constructions employed, a cone, polygonal pyramid, cylinder, or polygonal pillar, drawn in position about a horizontal axis parallel to _bd_. Similarly, if the circles are drawn in the sides _gd_ and _ec_, the resulting figures will be described about a horizontal axis parallel to _ab_. [24] Or if the length of its side, _AC_, is given instead, take _ae_, Fig. 37., equal to half the excess of _AB_ over _CD_; from the point _e_ raise the perpendicular _ce_. With center _a_, and distance _AC_, describe a circle cutting _ce_ in _c_. Then _ce_ is the vertical height of the portion of cone required, or _CE_. [25] The direction of the side of the square will of course be regulated by convenience. PROBLEM XV. TO DRAW AN INCLINED LINE, GIVEN IN POSITION AND MAGNITUDE. We have hitherto been examining the conditions of horizontal and vertical lines only, or of curves inclosed in rectangles. [Illustration: Fig. 39.] [Illustration: Fig. 40.]