IX

Again the relation of the coordinates $O M$ and $M P$ (i.e. $x$ and $y$ ) to the vector $O P$ is an instance of the famous parallelogram law, as

can easily be seen (cf. [fig.]8) by completing the parallelogram $O M P N$ . The idea of the "vector" $O P$ , that is, of a directed magnitude, is the root-idea of physical science. Any moving body has a certain magnitude of velocity in a certain direction, that is to say, its velocity is a directed magnitude, a vector. Again a force has a certain magnitude

and has a definite direction. Thus, when in analytical geometry the ideas of the "origin," of "coordinates," and of "vectors" are introduced, we are studying the abstract conceptions which correspond to the fundamental facts of the physical world.