DESCRIPTIVE GEOMETRY. ol 
The preceding construction, however satisfactorily it illustrates the prin- 
ciples of perspective, is yet too complicated in practice ; simpler construc- 
tions therefore become necessary. 
Fig. 58 exhibits a simplified construction for representing the square just 
mentioned. Let zy again be the basis of the plane of the picture, lying 
then above this line; the space below zy is the ground plane upon which 
the square abcd is described, and which rests immediately against the basis. 
. We assume for the first case that the point of sight A is opposite to the 
middle of the square, and that DD’ is the horizontal line. The distance 
of the point A from the plane of the picture or the perspective plane must 
be given in numbers or otherwise, and laid off right and left from the visual 
point on the horizon; D and D are then the two points of distance. From 
all points of the square draw perpendiculars to the base, meeting it in a and 
b, from which points, rays, as aA, DA, are to be drawn to the visual point. 
The lines aD and 0D are also to be drawn from the points a and 6 to the 
points of distance D and D opposite to them; they will intersect those first 
-drawn ine and f. By connecting the points of intersection thus obtained 
by straight lines, we shall obtain the figure abef as the perspective of the 
square abcd, for the situation of the visual point at A. If the visual point 
be not in the middle, but at A’ for instance, the points of distance will lie 
at D’ and D’: visual rays from a and b to A’ willi ntersect the lines aD’ and 
bD’, and thus determine abgh as the perspective of the square. It must be 
remarked that two points of distance are not always necessary, one being 
sufficient in most cases, as will be shown in the next example. 
If the square abcd does not lie immediately against the basis, as in fig. 60, 
the process is somewhat different, as the distance from the basis is to be 
taken into account. In this case let D be the point of sight, and A the 
point of distance, and draw perpendiculars to the basis from the four cor- 
ners. These lie here in two lines, as the square is parallel to the basis. 
From the points where these perpendiculars meet the basis, draw lines to 
the point of sight D. Take off the distance of the corners from the basis, 
in the direction opposite to the point of distance D; for the point @ we ob- 
tain a'; for 6, b’; for c, b'; and for d, d'. Drawing lines from a’, b’, and 
d' to the point of distance-A, they will intersect those drawn to the point 
of sight in a’, b,c’, d?; connecting the four corners @’, b’, c’, d* by straight 
lines, we shall have the perspective picture of the square at its proper 
distance from the basis. 
From this figure we perceive that all lines in the object which run 
parallel to the basis, must be parallel to it in the perspective represen- 
tation. 
Pl. 4, fig. 59, exhibits a complicated rectilineal figure, with the construc- 
tion of its perspective representation. The mode of operation is precisely 
the same as in the instance just explained. 
Fig.61 shows how a curve is:to be represented in perspective. The 
curve is here acircle, ab ; Ais the point of sight, and D the point of distance. 
Here it is necessary to determine the perspective of several points, through 
51 
