a curved surface, in which case a similar
method to that explained by Leonardo da Vinci has to be adopted.

In Chapter CCCI he shows us how to draw a figure twenty-four braccia
high upon a wall twelve braccia high. (The braccia is 1 ft. 10-7/8 in.).
He first draws the figure upright, then from the various points draws
lines to a point _F_ on the floor of the building, marking their
intersections on the profile of the wall somewhat in the manner we have
indicated, which serve as guides in making the outline to be traced.

[Illustration: Fig. 67.

'Draw upon part of wall _MN_ half the figure you mean to represent, and
the other half upon the cove above (_MR_).' Leonardo da Vinci's
_Treatise on Painting_.]

XXI

INTERIORS

[Illustration: Fig. 68. Interior by de Hoogh.]

To draw the interior of a cube we must suppose the side facing us to be
removed or transparent. Indeed, in all our figures which represent
solids we suppose that we can see through them, and in most cases we
mark the hidden portions with dotted lines. So also with all those
imaginary lines which conduct the eye to the various vanishing points,
and which the old writers called 'occult'.

[Illustration: Fig. 69.]

When the cube is placed below the horizon (as in Fig. 59), we see the
top of it; when on the horizon, as in the above (Fig. 69), if the side
facing us is removed we see both top and bottom of it, or if a room, we
see floor and ceiling, but otherwise we should see but one side (that
facing us), or at most two sides. When the cube is above the horizon we
see underneath it.

We shall find this simple cube of great use to us in architectural
subjects, such as towers, houses, roofs, interiors of rooms, &c.

In this little picture by de Hoogh we have the application of the
perspective of the cube and other foregoing problems.

XXII

THE SQUARE AT AN ANGLE OF 45 DEG.

When the square is at an angle of 45 deg to the base line, then its sides
are drawn respectively to the points of distance, _DD_, and one of its
diagonals which is at right angles to the base is drawn to the point of
sight _S_, and the other _ab_, is parallel to that base or ground line.

[Illustration: Fig. 70.]

To draw a pavement with its squares at this angle is but an
amplification of the above figure. Mark off on base equal distances, 1,
2, 3, &c., representing the diagonals of required squares, and from each
of these points draw lines to poi