We have hitherto been examining the conditions of horizontal and vertical lines only, or of curves inclosed in rectangles.

[Geometric diagram]
Fig.39. [Geometric diagram]
Fig.40.

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 AppendixII. ArticleIII. 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 [p51] 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.

[Geometric diagram]
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 ProblemI.

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.

[Geometric diagram]
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.

[Geometric diagram]
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 [p52] 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.

[Geometric diagram]
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.