ANALYTICAL GEOMETRY 
OF 
THREE DIMENSIONS. 
SECTION I. 
ON THE PLANE, AND STRAIGHT LINE. 
1. In order to determine the position of a point in space, 
some fixed point is taken for the origin of co-ordinates, and 
through it are drawn three fixed planes, called the co-ordinate 
planes, at right angles to one another, and intersecting one 
another two and two in straight lines, which are also at right 
angles to one another, and are called the axes of the co 
ordinates. Then, if the perpendicular distances of a point 
from each of the co-ordinate planes be given, its position will 
be completely determined. 
Let O (fig. 1) be the origin of the co-ordinates, and yOz, 
zOx, xOy the co-ordinate planes; M any point, and ME, 
MF, MN the perpendiculars let fall from it upon the co 
ordinate planes; these perpendiculars are called the rectan 
gular co-ordinates of M, and, as their values change for the 
different points of space, are denoted by the variables x, y, z. 
The point M will be determined in position, if we know the 
values of its three co-ordinates; that is, if we know that for 
that point 
x = a, y = b, z = c. 
For, if along Ox we measure OA = a, and through A draw 
an indefinite plane parallel to y Oz, this plane will contain all 
points whose distance from yOz is «, or for which x = a, 
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