Page:IntroductionToMathematicsWhitehead.pdf/61

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.

By far the simplest type of parallelogram is a rectangle, and in pure mathematics it is the relation of the single vector ${\displaystyle AC}$ to the two component vectors, ${\displaystyle AB}$ and ${\displaystyle AD}$, at right angles (cf. fig. 7), which is continually recurring. Let ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle r}$ units represent the lengths of ${\displaystyle AB}$, ${\displaystyle AD}$, and ${\displaystyle AC}$, and let ${\displaystyle m}$ units of angle represent the magnitude of the angle ${\displaystyle BAC}$. Then the relations between ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle r}$, and ${\displaystyle m}$, in all their many aspects are the continually recurring topic of pure mathematics; and the results are of the type required for application to the fundamental vectors of mathematical physics. This diagram is the chief bridge over which the results of pure mathematics pass in order to obtain application to the facts of nature.