DESCRIPTIVE GEOMETRY. 37 
zy, as, accoraing to the assumption, the axis of the pyramid remains parallel 
to the plane of elevation: the line gg’ is the projection of the circular arc 
described by the apex g, during the supposed change of position about the 
corner 6°. The same condition holds good for all the other points of the 
pyramid; and by drawing perpendiculars from points a@*, b*\——f”, their in- 
tersections with lines parallel to zy, drawn from a, b, c——f, will give the 
points a’*, b°——”, determining the projection of the obliquely situated base : 
connecting these points with each other, and with the apex g’, we obtain 
the horizontal projection of the oblique pyramid. In a similar manner we 
obtain the projection of the plane of intersection, h°/’, exhibited in the ele- 
vation as a straight line: this projection is h’, [’ ae 
Let us now suppose the pyramid to be rotated upon the corner 0’, still at 
the same inclinaticn to the base zy; the axis of the pyramid will no longer 
be parallel to the plane of elevation. It is evident that all points of the 
pyramid must describe horizontal arcs during this rotation, whose centres 
will le in a perpendicular, supposed to be erected from the point 6°. Their 
perpendicular distance from the base must, consequently, remain the same 
as before. As, however, the inclination to the base remains the same, the 
projection in the ground plane needs to be changed only with respect of the 
direction of the edge g*b’ to the base zy. PI. 4, fig. 26, represents the upper 
view seen in fig. 25, at the same angle with the basis zy. The preceding 
explanations have taught us that we can draw horizontal lines from all points 
of the elevation, in which the new projections of these points, for the new 
position, must lie. The points are absolutely defined, by drawing perpen- 
diculars from the corresponding portions of the plan to these horizontal 
lines. Thus, to obtain the position of the point g* in the new projection, 
we draw the horizontal lines g*g*, and the perpendicular, g’g*. In like man- 
ner we obtain the projection of the base a*, b'——f*, and consequently the 
projection of the entire pyramid, by uniting g* with these corners. As this 
pyramid is no longer parallel to the base, the plane of intersection, h*/*( fig. 25), 
can no longer appear as a straight line in this last position of the pyramid. 
Its projection, h*, /’, k'——n', is obtained by the preceding methods. 
As an additional illustration, we give the projection of the three principal 
conic sections. If we imagine a plane to be passed through a cone, which 
is parallel neither to the axis nor to one of the sides, we shall obtain a 
regular, symmetrical, curved line, termed an ellipse ; if the plane be passed 
parallel to one side of the cone, a parabola will be produced; and when 
parallel to the axis, a hyperbola. The development and properties of these 
three curves are cases of the higher Geometry and Analysis (see pages 24, 
25)... In‘this place we have to de only with their projections. 
Pl. 4, fig. 27, is the projection of a right cone in the horizontal and 
vertical planes. The circle, A’B’, and the straight line, AB, are the projec- 
tions on the vertical plane of the base, C is the apex, and DE the intersecting 
plane, appearing in elevation as a straight line, and whose intersection is to 
form the ellipse, whose shape in horizontal projection is to be obtained. The 
question reduces itself to finding the breadth of the ellipse for the different 
points of the circumference. These points lie symmetrically upon the 
37 
