Fig. 50.

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 G F upwards and downwards.

From A set off any portion of the relative horizontal A c, and on A c describe a semicircle in a vertical plane, a d c, 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 A 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 vanishing-point 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 its surface.

The preceding series of constructions, with the examples in the first Article of the Appendix, put it in the power of the student to draw any form, however complicated l, 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 as in algebraic science, much depends, in complicated perspective, 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.

sight-point, and the sight-point is opposite the eye, the sight-line is always on a level with the eye. Above and below the sight-line, the eye comprehends, 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, sometimes 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.

There is no reason, in the laws of perspective, why the picture should not be moved to the right or left of the sight-point, as well as up or down; but there is this practical reason. The moment the spectator sees the horizon in a picture high, he tries to hold his head high, that is, in its right place. When he sees the horizon in a picture low, he similarly tries to put his head low. But, if the sight-point is thrown to the left hand or right hand, he does not understand that he is to step a little to the right or left; and if he places himself, as usual, in the middle, all the perspective is distorted. Hence it is generally inadvisable to remove the sight-point laterally, from the centre of the picture. The Dutch painters, however, fearlessly take the license of placing it to the right or left; and often with good effect.

The rectilinear limitation of the sides, top, and base of the picture is of course quite arbitrary, as the space of a landscape would be which was seen through a window; less or more being seen at the spectator's pleasure, as he retires or advances.

The distance of the station-point is not so arbitrary. In ordinary cases it should not be less than the intended greatest dimension (height or breadth) of the picture. In most works by the great masters it is more; they not only calculate on their pictures being seen at considerable distances, but they like breadth of mass in buildings, and dislike the sharp angles which always result from station-points at short distances.1