o2 MATHEMATICS. 
which the curve is then to be passed. Divide the circle into any number 
of equal parts, the number being greater with the size of the circle and the 
degree of accuracy required. From the points of division draw perpendicu- 
lars to the basis. As two of these points le exactly behind two others, we 
shall have only five points on the basis, from which lines are to be drawn te 
the visual point, A. Each one of the three middle lines determines two 
points ; the two external lines, only one each; these last points being the 
extremities of a diameter parallel to the basis. The distances of the five 
points are now to be laid off on the basis, in a direction opposite to the point 
of distance, and through the points thus determined, lines drawn to the point 
of distance: these, by their intersections with the visual rays, will determine 
five points in the curve. The remark made with reference to fig. 60, that 
all natural lines parallel to the basis, are parallel to it and to each other in 
perspective, enables us to obtain the remaining three points. Through the 
three points of division to the left of ab, parallels to the basis are supposed to 
be drawn, which must then meet the division points of the circle, opposite to 
them. Drawing, in the perspective view, lines parallel to the basis, from the 
three perspective points obtained for those points, these will intersect the 
visual rays corresponding to the opposite division points. The three 
deficient points of the curve will then be obtained, with which the 
perspective view of the circle can be readily completed, as shown in 
jig. 61. 
We have hitherto spoken of figures in a plane, that is, of surfaces: to deal 
with solids, we must determine the height, in addition to the length and 
breadth. It is to be observed that while the depth of the figure, speaking 
with reference to the plane of the picture, has its vanishing point in the 
point of distance, and the breadth in the point of sight, the height must like- 
wise have its vanishing point in the horizontal line. If, then, a certain 
height be applied to the base, and lines be drawn from its top and bottom to 
a point in the horizontal line—generally the point of sight—the top and 
bottom of all bodies which have the height supposed, will lie in these 
lines. 
Fig. 62 exhibits the method of determining the perspective height. Let 
a four-sided prism be here drawn, whose plane is given, and whose height is 
yz. Ais the point of sight, D the point of distance, and xy the basis of the 
plane of the picture. After the base, abcd, of the object has been perspec- 
tively represented at a*d*, by the methods already given, apply the height, 
yz, to the basis, and draw the lines yA and zA; these will contain the top 
and bottom of the prism for the different planes in the plane of the picture. 
To ascertain, for instance, the perspective height of the prism at the point @’, 
draw a line parallel to the basis, meeting the line yA in 2. A perpendicular 
line, 2,2’, cutting the line zA, determines at 2’ the perspective shortening of 
the height yz, for the plane through d@’, and so on for any other point. Let 
perpendiculars be next erected at the four corners of the perspective ground 
plane, and parallels be drawn to the basis, intersecting the line Ay, and per- 
pendiculars again from these points intersecting the line Az; finally, draw 
52 
