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.
When, therefore, the line _AB_ is horizontal (or marks the water
surface), its direction is the direction of the inclined plane, and
the inclination of the line _DC_ is the inclination of the inclined
plane. In beds of rock geologists call the inclination of the line
_DC_ their “dip.”
To fix the position of an inclined plane, therefore, is to determine
the direction of any two lines in the plane, _AB_ and _CD_, of which
one shall be horizontal and the other at right angles to it. Then any
lines drawn in the inclined plane, parallel to _AB_, will be
horizontal; and lines drawn parallel to _CD_ will be as steep as _CD_,
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.