An introduction to mathematics, by A. N. Whitehead.

124 INTRODUCTION TO MATHEMATICS
Consider y-x=O; here the a, b, and c, of
the general form have been replaced by 1, -1,
and O respectively. This line passes through
the "origin," O, in the diagram and bisects
the angle XOY. It is the line L'OL of the
diagram. The fact that it passes through the
origin, O, is easily seen by observing that the
equation is satisfied by putting x= O and
y =O simultaneously, but O and O are the coordinates of 0. In fact it is easy to generalize
and to see by the same method that the
equation of any line through the origin is of
the form ax+by=0. The locus of equation
y+x=O also passes through the origin and
bisects the angle X'OY: it is the line L1OL'1
of the diagram.
Consider y-x = 1: the corresponding locus
does not pass through the origin. We therefore seek where it cuts the axes. It must cut
the axis of x at some point of coordinates
x and 0. But putting y =O in the equation,
we get x= -1; so the coordinates of this
point (A) are -1 and 0. Similarly the point
(B) where the line cuts the axis OY are O and
1. The locus is the line AB in the figure and
is parallel to LOL'. Similarly y +x = 1 is the
equation of line A1B of the figure; and the
locus is parallel to LIOL'. It is easy to prove
the general theorem that two lines represented
by equations of the forms ax +by- O and
ax + by = c are parallel.