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Therefore, to draw any line in a plane parallel to that of the picture, we have only to fix the position of one of its extremities, a or b, and then to draw from a or b a line parallel to the given line, bearing the proportion to it that r s bears to T D.

COROLLARY II.

If the line A B is in a horizontal plane, the vertical distance of one of its extremities must be the same as

that of the other.

Therefore, in that case, ▲ C equals в c' (Fig. 6.).

And the construction is as in Fig. 8.

In Fig. 8. produce a b to the sight-line, cutting the

sight-line in v; the point v, thus determined, is called the VANISHING-POINT of the line A B.

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Join r v. Then the student will find experimentally that T v is parallel to ▲ B.*

COROLLARY III.

If the line A B produced would pass through some point beneath or above the station-point, C D is to D T as c'D' is to D' T; in which case the point c coincides with the point c', and the line ab is vertical.

Therefore every vertical line in a picture is, or may be, the perspective representation of a horizontal one which, produced, would pass beneath the feet or above the head of the spectator.f

The demonstration is in Appendix II. Article I.

The reflection in water of any luminous point or isolated object (such as the sun or moon) is therefore, in perspective, a vertical line; since such reflection, if produced, would pass under the feet of the spectator. Many artists (Claude among the rest) knowing something of optics, but nothing of perspective, have been led occasionally to draw such reflections towards a point at the centre of the base of the picture.

PROBLEM IIL

TO FIND THE VANISHING-POINT OF A GIVEN HORIZONTA

LET A B, Fig. 9., be the given line.

From T, the station-point, draw T v parallel to ▲ B cutting the sight-line in v.

v is the Vanishing-point required.*

The student will observe, in practice, that, his paper lying flat on the table, he has only to draw the line TV on its horizontal sur. face, parallel to the given horizontal line A B. In theory, the paper

COROLLARY I

As, if the point b is first found, v may be determined by it, so, if the point v is first found, b may be deter

should be vertical, but the station-line s T horizontal (see its definition above, page 10.); in which case T v, being drawn parallel to a B, will be horizontal also, and still cut the sight-line in v.

The construction will be seen to be founded on the second Corol. lary of the preceding problem.

It is evident that if any other line, as M N in Fig. 9., parallel to A B, occurs in the picture, the line T v, drawn from т, parallel to м N, to find the vanishing-point of M N, will coincide with the line drawn from T, parallel to A B, to find the vanishing-point of A B.

Therefore A B and M N will have the same vanishing-point.

Therefore all parallel horizontal lines have the same vanishingpoint.

It will be shown hereafter that all parallel inclined lines also have the same vanishing-point; the student may here accept the general conclusion "All parallel lines have the same vanishing. point."

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It is also evident that if A B is parallel to the plane of the picture, Tv must be drawn parallel to G H, and will therefore never cut & H. The line A B has in that case no vanishing-point: it is to be drawn by the construction given in Fig. 7.

It is also evident that if A B is at right angles with the plane of the picture, T v will coincide with r s, and the vanishing-point of ▲ B will be the sight-point.