IV

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 transportations are $A$ to $B$ and $A$ to $D$,

complete the parallelogram $ABCD$, and then the sum of the two is the diagonal $AC$.

All this at first sight may seem to be very artificial. But it must be observed that nature itself presents us with the idea. For example, a steamer is moving in the direction $AD$ (cf. [fig.]6) and a man walks across its deck. If the steamer were still, in one minute he would arrive at $B$; but during that minute his starting point $A$ on the deck has moved to $D$, and his path on the deck has moved from $AB$ to $DC$. So that, in fact, his transportation has been from $A$ to $C$ over the surface of the sea. It is, however, presented to us analysed into the sum of two transportations, namely, one from $A$ to $B$ relatively to the steamer, and one from $A$ to $D$ which is the transportation of the steamer.

By taking into account the element of time, namely one minute, this diagram of the man's transportation $AC$ represents his velocity. For if $AC$ represented so many feet of transportation, it now represents a transportation of so many feet per minute, that is to say, it represents the velocity of the man. Then $AB$ and $AD$ represent two velocities, namely, his velocity relatively to the steamer, and the velocity of the steamer, whose "sum" makes up his complete velocity. It is evident that diagrams and definitions concerning transportations

are turned into diagrams and definitions concerning velocities by conceiving the diagrams as representing transportations per unit time. Again, diagrams and definitions concerning velocities are turned into diagrams and definitions concerning accelerations 7 by conceiving the diagrams as representing velocities added per unit time.

Thus by the addition of vector velocities and of vector accelerations, we mean the addition according to the parallelogram law.

Also, according to the laws of motion a force is fully represented by the vector acceleration it produces in a body of given mass. Accordingly, forces will be said to be added when their joint effect is to be reckoned according to the parallelogram law.

Hence for the fundamental vectors of

science, namely transportations, velocities, and forces, the addition of any two of the same kind is the production of a "resultant" vector according to the rule of the parallelogram law.