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SECTION III. 
ON THE PROJECTIONS OF LINES AND PLANE SURFACES, 
AND ON THE TRANSFORMATION OF CO-ORDINATES. 
The meaning of the projection of a point, and of a line, 
upon any plane has already been explained, (Arts. 2, and 13.) 
Moreover if from the extremities of a limited line we drop 
perpendiculars upon any indefinite line either in the same 
plane with it or not, the part of the latter line intercepted 
between the feet of the perpendiculars, is called the projection 
of the former line upon the latter. 
Also, if the sides of any plane surface be projected upon 
a plane, the figure bounded by these projections is called the 
projection of the given surface upon that plane. Between 
the length of a finite line and the length of its projection 
upon any plane or line, and also between the area of any 
plane surface and the area of its projection upon a plane, a 
remarkable relation exists, which we shall now exhibit. 
79. The length of the projection of a limited line upon a 
plane, is equal to the length of the line multiplied by the 
cosine of the acute angle which it forms with the plane. 
Eet CD (fig. 25) be the line produced to meet GdK, the 
plane on which it is to be projected, in H. Let DHd be 
the projecting plane, intersecting GdK in Hd ; and in the 
projecting plane draw Cc, Dd, perpendicular to Hd, and CD' 
parallel to it; then cd is the projection of CD, and Z. DHd = i 
is the inclination of CD to the plane GdK; also CD' = CD 
x cos i; 
cd - CD’ = CD cos i.