[p 31 ] PROBLEM VIII. TO DRAW A SQUARE, GIVEN IN POSITION AND MAGNITUDE, IN A HORIZONTAL PLANE .

[Geometric diagram]
Fig.20.

Let ABCD, 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 AB; and M, the dividing-point of AB, nearest S.

Find V', the vanishing-point of AC; and N, the dividing-point of AC, nearest S.

[p32]
Draw the measuring-line through a, and make ab', ac', each equal to the sight-magnitude of AB.

(For since ABCD is a square, AC is equal to AB.)

Draw a V' and c'N, cutting each other in c.

Draw aV, and b'M, cutting each other in b.

Then ac, ab, are the two nearest sides of the square.

Now, clearing the figure of superfluous lines, we have ab, ac, drawn in position, as in Fig.21.

[Geometric diagram]
Fig.21.

And because ABCD is a square, CD (Fig.20.) is parallel to AB.

And all parallel lines have the same vanishing-point. (Note to ProblemIII.)

Therefore, V is the vanishing-point of CD.

Similarly, V' is the vanishing-point of BD.

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

Then abcd is the square required.

COROLLARY I.

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

[p33]
COROLLARY II.

Let abcd, Fig.22., be any square drawn in perspective. Draw the diagonals ad and bc, cutting each other in C. Then C is the center of the square. Through C, draw ef to the vanishing-point of ab, and gh to the vanishing-point of ac, and these lines will bisect the sides of the square, so that ag is the perspective representation of half the side ab; ae is half ac; ch is half cd; and bf is half bd.

[Geometric diagram]
Fig.22.

COROLLARY III.

Since ABCD, Fig.20., is a square, BAC is a right angle; and as TV is parallel to AB, and TV' to AC, V'TV must be a right angle also.

As the ground plan of most buildings is rectangular, it constantly happens in practice that their angles (as the corners of ordinary houses) throw the lines to the vanishing-points thus at right angles; and so that this law is observed, and VTV' is kept a right angle, it does not matter in general practice whether the vanishing-points are thrown a little more or a little less to the right or left of S: but it matters much that the relation of the vanishing-points should be accurate. Their position with respect to S merely causes the spectator to see a little more or less on one side or other of the house, which may be a matter of chance or choice; but their rectangular relation determines the rectangular shape of the building, which is an essential point.