Fig. 48.

As in order to fix the position of a line two points in it must be given, so in order to fix the position of a plane, two lines in it must be given.

1 Read the Article on this problem in the Appendix, p. 300, before investigating the problem itself.

Let the two lines be A B and c d, Fig. 48.

As they are given in position, the relative horizontals a E and c f must be given.

Then by Problem XVI. the vanishing-point of a b is v, and of c d, v'.

Join v v' and produce it to cut the sight-line in x.

Then v x is the sight-line of the inclined plane.

Like the horizontal sight-line, it is of indefinite length; and may be produced in either direction as occasion requires, crossing the horizontal line of sight, if the plane continues downward in that direction.

x is the vanishing-point of all horizontal lines in the inclined plane.

## 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.)

Next, draw a semicircle on the piece of pasteboard; draw its diameter, a B, Fig. 74., and a vertical line from its centre,

Fig. 74.

C D ; and draw some other lines, C E, C F, c, from the centre 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 A B. Then the line C D will be the most steeply inclined of all the lines drawn to the circumference of the circle; G C and H C will be less steep ; and E C and F c less steep still. The nearer the lines to c D, the steeper they will be; and the nearer to A B, the more nearly horizontal.

When, therefore, the line A B is horizontal (or marks the water surface), its direction is the direction of the inclined plane, and the inclination of the line D c is the inclination of have others concentrically inscribed within them, as often in Byzantine mosaics, c, the use of the vanishing-points will become essential.

the inclined plane. In beds of rock geologists call the inclination of the line D C their dip.

To fix the position of an inclined plane, therefore, is to determine the direction of any two lines in the plane, A B and C D, of which one shall be horizontal and the other at right angles to it. Then any lines drawn in the inclined plane, parallel to A B, will be horizontal; and lines drawn parallel to C D will be as steep as c D, and are spoken of in the text as the steepest lines in the plane.

But farther, whatever the direction of a plane may be, if it be extended indefinitely, it will be terminated, to the eye of the observer, by a boundary line, which, in a horizontal plane, is horizontal (coinciding nearly with the visible horizon); - in a vertical plane, is vertical; - and, in an inclined plane, is inclined.

This line is properly, in each case, called the sight-line of such plane; but it is only properly called the horizon in the case of a horizontal plane: and I have preferred using always the term sight-line, not only because more comprehensive, but more accurate; for though the curvature of the earth's surface is so slight that practically its visible limit always coincides with the sight-line of a horizontal plane, it does not mathematically coincide with it, and the two lines ought not to be considered as theoretically identical, though they are so in practice.

It is evident that all vanishing-points of lines in any plane must be found on its sight-line, and, therefore, that the sight-line of any plane may be found by joining any two of such vanishing-points. Hence the construction of Problem XVIII.

## Problem XIX. To Find The Vanishing-Point Of Steepest Lines In An Inclined Plane Whose Sight-Line Is Given

Let v x, Fig. 49., be the given sight-line. Produce it to cut the horizontal sight-line in x.

Fig.. 49.

Therefore x is the vanishing point or horizontal lines in the given inclined plane. (Problem XVIII.)

Join T x, and draw T Y at right angles to T x.

Therefore Y is the rectangular vanishing-point corresponding to x.1

From y erect the vertical Y P, cutting the sight-line of the inclined plane in p.

1 That is to say, the vanishing-point of horizontal lines drawn at right angles to the lines whose vanishing-point is x.

Then p is the vanishing-point of steepest lines in the plane.

All lines drawn to it, as Q p, r p, n p, c, are the steepest possible in the plane; and all lines drawn to x, as Q x, o x, c, are horizontal, and at right angles to the lines p Q, P R, c.