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

IMAGINARY NUMBERS 99
to OX; then evidently the triangles OQMi
and PRS are in all respects equal. Hence
MIM'=PS=xi, and RS=QM1; and therefore
OM' =OM+MM' =x+xi,
RMB' SM' +RS =y +yi.
Thus OR represents the ordered couple as
required. This figure can also be drawn
with OP and OQ in other quadrants.
It is at once obvious that we have here
come back to the parallelogram law, which
was mentioned in Chapter VI, on the laws of
motion, as applying to velocities and forces.
It will be remembered that, if OP and OQ
represent two velocities, a particle is said to
be moving with a velocity equal to the two
velocities added together if it be moving with
the velocity OR. In other words OR is said
to be the resultant of the two velocities OP
and OQ. Again forces acting at a point of
a body can be represented by lines just as
velocities can be; and the same parallelogram
law holds, namely, that the resultant of the
two forces OP and OQ is the force represented
by the diagonal OR. It follows that we can
look on an ordered couple as representing a
velocity or a force, and the rule which we
have just given for the addition of ordered
couples then represents the fundamental laws
of mechanics for the addition of forces and