38 MATHEMATICS. 
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surface of the cone, only in different planes above the base; and it is neces- 
sary to find the projections of these planes in both views of the cone. In 
the vertical projection, these planes appear as straight lines; in the horizontal 
plane, as circles. When we pass planes through D and E, in the vertical 
projection, the points D and EH, of the curve, will be situated in them; 
if, then, the length DE be divided into any number of equal parts, and 
horizontal planes be passed through the points of division, there will be 
two points of the ellipse in each plane, which will be situated in that part 
where the line DE cuts these planes in succession. To obtain the form of — 
the ellipse in the horizontal projection, we draw in it the diameter, A’B’, and 
let fall upon it perpendiculars from the points 1.2....DE: the points 
1’, 2’, &c., will answer to the horizontal projection of those points, and as 
the horizontal projections of the surfaces projected as straight lines in eleva- 
tion must be circles, these can readily be determined, knowing their radii, 
C’, I’, &c., and their common centre,C. These circles are cut successively 
by the ellipse. By drawing the perpendiculars, DD’ and EE’, we obtain 
the projection of the extremities, since the axis of the ellipse lies parallel to 
the plane of projection. Drawing a perpendicular from the point where the. 
ellipse cuts the plane marked 6, until it cuts the circle 6’ in the horizontal 
projection, we shall obtain one point of the horizontal projection of the 
ellipse, or two as the ellipse is symmetrical. By a repetition of the process, 
a number of points in the horizontal projection will be obtained, through 
which the ellipse itself may be passed. The figure standing near fig. 27, 
exhibits the actual view, or the orthographic projection. of the ellipse. It is: 
obtained by taking off the axis, DE, of the ellipse from the vertical projec-: 
tion, with its planes of intersection, which would here be represented as. 
straight lines. From the horizontal projection, we obtain.the true breadth, 
and if these be described one after the other upon the corresponding planes’ 
on each side of the axis, we shall obtain the points through which the ellipse 
is to pass. aid ; | . 
Pl. 4, fig. 28, exhibits the vertical and horizontal. projection of a right 
cone, with a parabolic intersection. DE is the projection ,of the parabola, 
which, for the vertical plane, is a straight line. The horizontal projection is 
obtained precisely as in the case of the ellipse. Thus, 1, 2, 3, ——hori- 
zontal planes are passed through that part of the front view of. the cone 
traversed by the parabola, at equal distances from each other ;: these appear 
as straight lines: they are circles in the horizontal view of the cone. . In fig. 
28, semicircles only are drawn. From the points, 1’, 2',. &c., where the 
planes passed through the elevation cut the parabola, draw perpendiculars 
to the horizontal projections of these planes; the perpendicular, DD’D”, will 
form the foot of the parabola, EE’ its vertex, and. perpendiculars from 
1’, 2’, ——— let fall upon the cirles :1%, 2’, 3°, ——-—wwill form inter- 
sections, all lying in one arm of the parabola, the other being easily 
constructed, as shown in fig. 28. The orthographic projection of the 
parabola is shown in the figure near fig. 28. It is obtained by erecting a 
perpendicular from the middle of DD’, and marking successively upon this 
the height D1’, D2’, D3’, &c., and drawing throngh these points, parallels to 
38 
