PROBLEM XX.

TO FIND THE VANISHING-POINT OF LINES PERPENDICULAR TO THE SURFACE OF A GIVEN INCLINED PLANE.

As the inclined plane is given, one of its steepest lines must be given, or may be ascertained.

Let A B, Fig. 50., be a portion of a steepest line in the given plane, and v the vanishing-point of its relative horizontal.

Through v draw the vertical GF upwards and downwards.

From A set off any portion of the relative horizontal ▲ c, and on a c describe a semicircle in a vertical plane, ADC, cutting A B in E.

Join E c, and produce it to cut G F in F.

Then F is the vanishing-point required.

For, because A E C is an angle in a semicircle, it is

a right angle; and therefore the line E F is at right

angles to the line ▲ B; and similarly all lines drawn to F, and therefore parallel to E F, are at right angles

with any line which cuts them, drawn to the vanishingpoint of a B.

And because the semicircle A D C is in a vertical plane, and its diameter a c is at right angles to the horizontal lines traversing the surface of the inclined plane, the line E C, being in this semicircle, is also at right angles to such traversing lines. And therefore the line E C, being at right angles to the steepest lines in the plane, and to the horizontal lines in it, is perpendicular to it: surface.

THE preceding series of constructions, with the exam ples in the first Article of the Appendix, put it in the power of the student to draw any form, however complicated, which does not involve intersection of curved surfaces. I shall not proceed to the analysis of any of these more complex problems, as they are entirely useless in the ordinary practice of artists. For a few words only I must ask the reader's further patience, respecting the general placing and scale of the picture.

As the horizontal sight-line is drawn through the sight-point, and the sight-point is opposite the eye, the sight-line is always on a level with the eye.

* As in algebraic science, much depends, in complicated perspec tive, on the student's ready invention of expedients, and on his quick sight of the shortest way in which the solution may be accomplished when there are several ways.

Above and below the sight-line, the eye compre hends, as it is raised or depressed while the head is held upright, about an equal space; and, on each side of the sight-point, about the same space is easily seen without turning the head; so that if a picture represented the true field of easy vision, it ought to be circular, and have the sight-point in its centre. But because some parts of any given view are usually more interesting than others, either the uninteresting parts are left out, or somewhat more than would generally be seen of the interesting parts is included, by moving the field of the picture a little upwards or downwards, so as to throw the sight-point low or high. The operation will be understood in a moment by cutting an aperture in a piece of pasteboard, and moving it up and down in front of the eye, without moving the eye. It will be seen to embrace sometimes the low, some times the high objects, without altering their perspective, only the eye will be opposite the lower part of the aperture when it sees the higher objects, and vice versa.