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, ▲ B, Fig. 74., and a vertical line from its centre, o D

and draw some other lines, o E, O 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 ine OD will be the most steeply inclined of all the lines drawn tc the circumference of the circle; G C and H o 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 ▲ B, the more nearly borizontal.

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 Do is the inclination of the inclined plane. In beds of rock geologists call the inclination of the line D o 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 O 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 OD will be as steep as o 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 considerea 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 sightline of any plane may be found by joining any two of such vanishing-points. Hence the construction of Problem XVIII.

IN Fig. 8. omit the lines o D, o' D', and D s; and, as here in Fig 75., from a draw a d parallel to A B, cutting в T in d; and from d draw d e parallel to в C.

Now because the triangles a c v, bc' v, are similar

acbc:: av: b v;

and because the triangles d e T, bc' r are similar

de: b c d r : br.

But a c is equal to de

.. av: bv:: dr: h

..the two triangles a b d, b r v, are similar, and their angles are

alternate:

.. Tv is parallel to a d.

But a d is parallel to ▲ B —

.. T v is parallel to ▲ a