THE ELEMENTS OF PERSPECTIVE

cated; another triangle being drawn on the line ▲

or B C.

COROLLARY.

It is evident that by this application of Problem VI. any given rectilinear figure whatever in a hori zontal plane may be drawn, since any such figure may be divided into a number of triangles and the triangles then drawn in succession.

More convenient methods may, however, be gene rally found, according to the form of the figure required, by the use of succeeding problems; and for the quadrilateral figure which occurs most frequently in practice, namely, the square, the following construction is more convenient than that used in the present problem.

he line

PROBLEM VIII.

TO DRAW A SQUARE, GIVEN IN POSITION AND MAGNITUDE,
IN A HORIZONTAL PLANE.

LET A B C D (Fig. 20.) be the square.

As it is given in position and magnitude, the position

and magnitude of all its sides are given.

Fix the position of the point a in a.

Find v, the vanishing-point of A B; and м, the dividing-point of AB, nearest s.

Find v', the vanishing-point of A c; and N, the dividing-point of A c, nearest s.

Draw the measuring-line through a, and make a b′,
a c', each equal to the sight-magnitude of a B.
(For since A B C D is a square, A c is equal to ▲ B.)
Draw a v' and c' N, cutting each other in c.
Draw a v, and b' м, cutting each other in b.

Then a c, a b, are the two nearest sides of the square.
Now, clearing the figure of superfluous lines, we have

a b, a c, drawn in position, as in Fig. 21.

And because A B C D is a square, CD (Fig. 20.) is

parallel to ▲ B.

And all parallel lines have the same vanishing-point.

(Note to Problem III.)

Therefore, v is the vanishing-point of c D.

Similarly, v' is the vanishing-point of B D.

Therefore, from b and c (Fig. 22.) draw b v', cv, cutting each other in d.

Then a b c d is the square required.

COROLLARY I

It is obvious that any rectangle in a horizontal plane may be drawn by this problem, merely making a b', on the measuring-line, Fig. 20., equal to the sight-magnitude of one of its sides, and a c' the sigaagui de of the other.

COROLLARY II.

Let a bed, Fig. 22., be any square drawn in perspective. Draw the diagonals a dando c, cutting each other in c. Then c is the centre of the square. Through c, draw e f to the vanishing

point of a b, and g h to the vanishing-point of a c and these lines will bisect the sides of the square, so that a g is the perspective representation of half the side a b; a e is half a c; c h is half cd; and b ƒ 18 half b d.