If, in Fig. 43. or Fig. 44., the lines A y and a' y' be produced, the student will find that they meet.

Let P, Fig. 45., be the point at which they meet.

From P let fall the vertical p v on the sight-line, cutting the sight-line in v.

Then the student will find experimentally that v is the vanishing-point of the line A c.1

Complete the rectangle of the base A c', by drawing a'c' to v, and c c' to the vanishing-point of A a'.

Join y' c',

Now if Y c and y' c' be produced downwards, the student will find that they meet.

Let them be produced, and meet in p'.

Produce p v, and it will be found to pass through the point p'.

Therefore if A Y (or c y), Fig. 45., be any inclined line drawn in perspective by Problem XV., and A c the relative horizontal (a c in Figs. 39, 40.), also drawn in perspective;

Through v, the vanishing-point of A c, draw the vertical p p' upwards and downwards.

Produce a y (or c y), cutting p p' in p (or p').

Then p is the vanishing-point of A Y (or p' of c y).

The student will observe that, in order to find the point P by this method, it is necessary first to draw a portion of the given inclined line by Problem XV. Practically, it is always necessary to do so, and, therefore, I give the problem in this form.

Theoretically, as will be shown in the analysis of the problem, the point P should be found by drawing a line from the station-point parallel to the given inclined line: but there is no practical means of drawing such a line; so that in whatever terms the

1 The demonstration is in Appendix II. Article III problem may be given, a portion of the inclined line

Fig. 45.

(a y or c y) must always be drawn in perspective before p can be found.

## Problem XVI

It is often possible to shorten other perspective operations considerably, by finding the vanishing-points of the inclined lines of the object. Thus, in drawing the gabled roof in Fig. 43., if the gable A Y C be drawn in perspective, and the vanishing-point of A V determined, it is not necessary to draw the two sides of the rectangle, a' d' and D' b', in order to determine the point Y'; but merely to draw Y Y' to the vanishing-point of A a' and a' Y' to the vanishing-point of A Y, meeting in Y', the point required.

Again, if there be a series of gables, or other figures produced by parallel inclined lines, and retiring to the point v, as in Fig. 72.1, it is not necessary to draw each separately, but merely to determine their breadths on the line A v, and draw the slopes of each to their vanishing-points, as shown in Fig. 72. Or if the gables are equal in height, and a line be drawn from Y to v, the construction resolves itself into a zigzag drawn alternately to P and Q, between the lines Y V and A v.

The student must be very cautious, in finding the vanishing-points of inclined lines, to notice their relations to the horizontals beneath them, else he may easily mistake the horizontal to which they belong.

Thus, let A B c D, Fig. 73., be a rectangular inclined plane, and let it be required to find the vanishing-point of its diagonal B D.

Find v, the vanishing-point of A D and B C

Draw a E to the opposite vanishing-point, so that D A E may represent a right angle.

Let fall from B the vertical B E, cutting A E in E.

Join E D, and produce it to cut the sight-line in V'.

1 The diagram is inaccurately cut. Y Y should be a right line.

Fig. 72.

Then, since the point E is vertically under the point B, the horizontal line E D is vertically under the inclined line B D.

Fig. 73.

So that if we now let fall the vertical v' p from v', and produce B D to cut v' P in P, the point P will be the vanishing-point of B D, and of all lines parallel to it 1

1 The student may perhaps understand this construction better by completing the rectangle a d f e, drawing D F to the vanishing-point of A E, and E F to V. The whole figure, b f, may then be conceived as representing half the gable roof of a house, A F the rectangle of its base, and A c the rectangle of its sloping side.

In nearly all picturesque buildings, especially on the Continent, the slopes of gables are much varied (frequently unequal on the two sides), and the vanishing-points of their inclined lines become very important, if accuracy is required in the intersections of tiling, sides of dormer windows, c.

Obviously, also, irregular triangles and polygons in vertical planes may be more easily constructed by finding the vanishing-points of their sides, than by the construction given in the corollary to Problem IX.; and if such triangles or polygons

## Problem XVII. To Find The Dividing-Points Of A Given Inclined Line

Let p, Fig. 46., be the vanishing-point of the inclined line, and v the vanishing-point of the relative horizontal.

Find the dividing-points of the relative horizontal, D and d'.

Through p draw the horizontal line x y.

Fig. 46.

With centre p and distance D p describe the two arcs D x and d' y, cutting the line x Y in x and Y.

Then x and y are the dividing-points of the inclined line.1

Obs. The dividing-points found by the above rule, used with the ordinary measuring-line, will lay off distances on the retiring inclined line, as the ordinary dividing-points lay them off on the retiring horizontal line.

Another dividing-point, peculiar in its application, is sometimes useful, and is to be found as follows:

1 The demonstration is in Appendix II., p. 303.

Let A b, Fig. 47., be the given inclined line drawn in perspective, and A c the relative horizontal.

Find the vanishing-points, v and E, of A c and A B; D, the dividing-point of A c; and the sight-magnitude of A c on the measuring-line, or A c.

From d erect the perpendicular d f.

Join C b, and produce it to cut d f in f.Join e f.

Fig. 47.

Then, by similar triangles, d f is equal to E v, and e f is parallel to d v.

Hence it follows that if from D, the dividing-point of A c, we raise a perpendicular and make D F equal to e v, a line c F, drawn from any point c on the measuring-line to F, will mark the distance A B on the inclined line, A B being the portion of the given inclined line which forms the diagonal of the vertical rectangle of which a c is the base.