Page:IntroductionToMathematicsWhitehead.pdf/59

transportations are ${\displaystyle A}$ to ${\displaystyle B}$ and ${\displaystyle A}$ to ${\displaystyle D}$, complete the parallelogram ${\displaystyle ABCD}$, and then the sum of the two is the diagonal ${\displaystyle AC}$.

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 ${\displaystyle AD}$ (cf. fig. 6) and a man walks across its deck. If the steamer were still, in one minute he would arrive at ${\displaystyle B}$ but during that minute his starting point ${\displaystyle A}$ on the deck has moved to ${\displaystyle D}$, and his path on the deck has moved from ${\displaystyle AB}$ to ${\displaystyle DC}$. So that, in fact, his transportation has been from ${\displaystyle A}$ to ${\displaystyle C}$ over the surface of the sea. It is, however, presented to us analysed into the sum of two transportations, namely, one from ${\displaystyle A}$ to ${\displaystyle B}$ relatively to the steamer, and one from ${\displaystyle A}$ to ${\displaystyle D}$ 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 ${\displaystyle AC}$ represents his velocity. For if ${\displaystyle AC}$ 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 ${\displaystyle AB}$ and ${\displaystyle AD}$ 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