Then, since the point _E_ is vertically under the point _B_, the horizontal line _ED_ is vertically under the inclined line _BD_. [Illustration: Fig. 73.] So that if we now let fall the vertical _V′P_ from _V′_, and produce _BD_ to cut _V′P_ in _P_, the point _P_ will be the vanishing-point of _BD_, and of all lines parallel to it.[35] [34] The diagram is inaccurately cut. _YV_ should be a right line. [35] The student may perhaps understand this construction better by completing the rectangle _ADFE_, drawing _DF_ to the vanishing-point of _AE_, and _EF_ to _V_. The whole figure, _BF_, may then be conceived as representing half the gable roof of a house, _AF_ the rectangle of its base, and _AC_ 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, etc. 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 have others concentrically inscribed within them, as often in Byzantine mosaics, etc., the use of the vanishing-points will become essential. PROBLEM XVIII. Before examining the last three problems it is necessary that you should understand accurately what is meant by the position of an inclined plane. Cut a piece of strong white pasteboard into any irregular shape, and dip it in a sloped position into water. However you hold it, the edge of the water, of course, will always draw a horizontal line across its surface. The direction of this horizontal line is the direction of the inclined plane. (In beds of rock geologists call it their “strike.”) [Illustration: Fig. 74.] Next, draw a semicircle on the piece of pasteboard; draw its diameter, _AB_, Fig. 74., and a vertical line from its center, _CD_; and draw some other lines, _CE_, _CF_, etc., from the center to any points in the circumference. Now dip the piece of pasteboard again into water, and, holding it at any inclination and in any direction you choose, bring the surface of the water to the line _AB_. Then the line _CD_ will be the most steeply inclined of all the lines drawn to the circumference of the circle; _GC_ and _HC_ will be less steep; and _EC_ and _FC_ less steep still. The nearer the lines to _CD_, the steeper they will be; and the nearer to _AB_, the more nearly horizontal.