We must, in conclusion, investigate the perspective of inclined lines, beginning with a single one given in position. For the sake of completeness of system, I give in Appendix II. Article III. the development of this problem from the second. But, in practice, the position of an inclined line may be most conveniently defined by considering it as the diagonal of a rectangle, as _AB_ in Fig. 39., and I shall therefore, though at some sacrifice of system, examine it here under that condition. If the sides of the rectangle _AC_ and _AD_ are given, the slope of the line _AB_ is determined; and then its position will depend on that of the rectangle. If, as in Fig. 39., the rectangle is parallel to the picture plane, the line _AB_ must be so also. If, as in Fig. 40., the rectangle is inclined to the picture plane, the line _AB_ will be so also. So that, to fix the position of _AB_, the line _AC_ must be given in position and magnitude, and the height _AD_. [Illustration: Fig. 41.] If these are given, and it is only required to draw the single line _AB_ in perspective, the construction is entirely simple; thus:— Draw the line _AC_ by Problem I. Let _AC_, Fig. 41., be the line so drawn. From _a_ and _c_ raise the vertical lines _ad_, _cb_. Make _ad_ equal to the sight-magnitude of _AD_. From _d_ draw _db_ to the vanishing-point of _ac_, cutting _bc_ in _b_. Join _ab_. Then _ab_ is the inclined line required. [Illustration: Fig. 42.] If the line is inclined in the opposite direction, as _DC_ in Fig. 42., we have only to join _dc_ instead of _ab_ in Fig. 41., and _dc_ will be the line required. I shall hereafter call the line _AC_, when used to define the position of an inclined line _AB_ (Fig. 40.), the “relative horizontal” of the line _AB_. OBSERVATION. [Illustration: Fig. 43.] In general, inclined lines are most needed for gable roofs, in which, when the conditions are properly stated, the vertical height of the gable, _XY_, Fig. 43., is given, and the base line, _AC_, in position. When these are given, draw _AC_; raise vertical _AD_; make _AD_ equal to sight-magnitude of _XY_; complete the perspective-rectangle _ADBC_; join _AB_ and _DC_ (as by dotted lines in figure); and through the intersection of the dotted lines draw vertical _XY_, cutting _DB_ in _Y_. Join _AY_, _CY_; and these lines are the sides of the gable. If the length of the roof _AA′_ is also given, draw in perspective the complete parallelopiped _A′D′BC_, and from _Y_ draw _YY′_ to the vanishing-point of _AA′_, cutting _D′B′_ in _Y′_. Join _A′Y_, and you have the slope of the farther side of the roof. [Illustration: Fig. 44.] The construction above the eye is as in Fig. 44.; the roof is reversed in direction merely to familiarize the student with the different aspects of its lines. PROBLEM XVI. TO FIND THE VANISHING-POINT OF A GIVEN INCLINED LINE. If, in Fig. 43. or Fig. 44., the lines _AY_ and _A′Y′_ be produced, the student will find that they meet.