IV

By far the simplest type of parallelogram is a rectangle, and in pure mathematics it is

the relation of the single vector $AC$ to the two component vectors, $AB$ and $AD$, at right angles (cf. [fig.]7), which is continually recurring. Let $x$, $y$, and $r$ units represent the lengths of $AB$, $AD$, and $AC$, and let $m$ units of angle represent the magnitude of the angle $BAC$. Then the relations between $x$, $y$, $r$, and $m$, 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.