790] GEOMETRY. 567
of a point in the plane may be regarded as the coordinates of the point; or, if
instead of a single point they determine a system of two or more points, then as the
coordinates of the system of points. But, as noticed under Curve, [785], there are also
line-coordinates serving to determine the position of a line; the ordinary case is when
the line is determined by means of the ratios of three quantities f, y, £ (correlative to
the trilinear coordinates x, y, z). A linear equation ci^ + by + c^ — 0 represents then the
system of lines such that the coordinates of each of them satisfy this relation, in fact,
all the lines which pass through a given point; and it is thus regarded as the line-
equation of this point; and generally a homogeneous equation (*]££, y, £) n = 0 represents
the curve which is the envelope of all the lines the coordinates of which satisfy this
equation, and it is thus regarded as the line-equation of this curve.
II. Solid Analytical Geometry (§§ 26—40).
26. We are here concerned with points in space,—the position of a point being
determined by its three coordinates x, y, z. We consider three coordinate planes, at
right angles to each other, dividing the whole of space into eight portions called
octants, the coordinates of a point being the perpendicular distances of the point from
the three planes respectively, each distance being considered as positive or negative
according as it lies on the one or the other side of the plane. Thus the coordinates
in the eight octants have respectively the signs
X, y, z
+ + +
+ - +
+ +
- - +
+ + -
+
+
Fig. 16.
z
The positive parts of the axes are usually drawn as in fig. 16, which represents
a point P, the coordinates of which have the positive values OM, MN, NP.