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

54 INTRODUCTION TO MATHEMATICS
Consider the vector AC in figure 6 as representative of the changed position of a body
from A to C: we will call this the vector of
transportation. It will be noted that, if the
reduction of physical phenomena to mere
changes in positions, as explained above, is
correct, all other types of physical vectors are
really reducible in some way or other to this
single type. Now the final transportation
from A to C is equally well effected by a
transportation from A to B and a transportation from B to C, or, completing the parallelogram ABCD, by a transportation from A to
D and a transportation from D to C. These
transportations as thus successively applied
are said to be added together. This is simply
a definition of what we mean by the addition
of transportations. Note further that, considering parallel lines as being lines drawn in
the same direction, the transportations B to
C and A to D may be conceived as the same
transportation applied to bodies in the two
initial positions B and A. With this conception we may talk of the transportation
A to D as applied to a body in any position,
for example at B. Thus we may say that
the transportation A to C can be conceived
as the sum of the two transportations A to
B and A to D applied in any order. Here
we have the parallelogram law for the addition of transportations: namely, if the