34 MATHEMATICS, 
Fig. 60 represents the environs of the town of Greitz, and fig. 59, the plan 
and profile of a mountain top, drawn according to the declivities of the 
surface. 
2. DESCRIPTIVE GEOMETRY. 
A. Prosecrion. 
a. Projection in vertical and horizontal planes. 
By the theory of projection is understood in general, a combination of all 
those propositions by whose application we are enabled to represent an 
object as it appears to us in a certain direction, and from a certain distance. 
If we suppose lines to be drawn from our eyes to all points of the object, 
representing lines of sight, a pyramid of rays will be formed, whose base is 
the surface of the object, whose sides are the rays of sight, whose apex is the 
eye, and whose altitude is the perpendicular distance of the object from the 
eye. If we suppose a plane to be passed through this pyramid, parallel to 
its base, according to the principles of similar figures and the laws of Stere- 
ometry, we will have an image in the plane of intersection which is similar 
to the body in question, and which is smaller as the distance of the plane 
from the eye is less. If we suppose the object to be at an infinite distance, the 
pyramid produced will be of great altitude, and the angle made by the sight rays 
with the base of the pyramid will be obtuse ; if the section be made tolerably 
near the base, we may assume the portions of the sight rays thus cut off as 
parallel to each other, and perpendicular to the base of the pyramid. The 
intersecting plane is called the plane of projection, and EES it the image of 
the object is supposed to be represented. 
According to the preceding principles; we can find the projection of a 
point, by drawing a perpendicular from it to the plane of projection; the 
intersection of the line with the plane will be the projection of the. point. 
Nevertheless, as the distance from the plane at which the point is situated _ 
is not determined, we cannot ascertain its actual position. from this projec- 
tion. This will be possible, however, if we employ a second plane, upon 
which we may suppose the distance of the projection from the point itself 
to be described. The first plane is called the plane of elevation, or the 
vertical plane ; the second, the ground, or horizontal plane. Both planes 
may be considered as perpendicular to, each other, and the position of a 
point in space may be accurately determined by the intersection of. the two 
perpendiculars erected from the projection of the point on the two planes. 
In pi. 4, fig. 20, AB is the vertical plane, and BC the horizontal plane; a’6' 
is the vertical, and ab” the horizontal projection of a line. If. we suppose 
lies to be drawn from the four points, perpendicular to their respective 
planes, a’ and a’ will intersect each other in ab’ and b’' in 6, and the posi- 
tion of the points, a,b, in space will thus be determined. Now whenever 
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