Page:IntroductionToMathematicsWhitehead.pdf/58

Consider the vector ${\displaystyle AC}$ in figure 6 as representative of the changed position of a body from ${\displaystyle A}$ to ${\displaystyle C}$: 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 ${\displaystyle A}$ to ${\displaystyle C}$ is equally well effected by a transportation from ${\displaystyle A}$ to ${\displaystyle B}$ and a transportation from ${\displaystyle B}$ to ${\displaystyle C}$, or, completing the parallelogram ${\displaystyle ABCD}$, by a transportation from ${\displaystyle A}$ to ${\displaystyle D}$ and a transportation from ${\displaystyle D}$ to ${\displaystyle C}$. 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 ${\displaystyle B}$ to ${\displaystyle C}$ and ${\displaystyle A}$ to ${\displaystyle D}$ may be conceived as the same transportation applied to bodies in the two initial positions ${\displaystyle B}$ and ${\displaystyle A}$. With this conception we may talk of the transportation ${\displaystyle A}$ to ${\displaystyle D}$ as applied to a body in any position, for example at ${\displaystyle B}$. Thus we may say that the transportation ${\displaystyle A}$ to ${\displaystyle C}$ can be conceived as the sum of the two transportations ${\displaystyle A}$ to ${\displaystyle B}$ and ${\displaystyle A}$ to ${\displaystyle D}$ applied in any order. Here we have the parallelogram law for the addition of transportations: namely, if the