IX

= c are parallel.

The group of loci which we next come upon are sufficiently important to deserve a chapter to themselves. But before going on to them we will dwell a little longer on the main ideas of the subject.

The position of any point $P$ is determined by arbitrarily choosing an origin, $O$ , two axes,

$O X$ and $O Y$ , at right-angles, and then by noting its coordinates $x$ and $y$ , i.e. $O M$ and $P M$ (cf. [fig.]13). Also, as we have seen in the last chapter, $P$ can be determined by the "vector" $O P$ , where the idea of the vector includes a determinate direction as well as a determinate length. From an abstract mathematical point of view the idea of an arbitrary origin may appear artificial and clumsy, and similarly for the arbitrarily drawn axes, $O X$ and $O Y$ . But in relation to the application of mathematics to the event of the Universe we are here symbolizing with direct simplicity the most fundamental fact respecting the outlook on the world afforded to us by our senses. We each of us refer our sensible perceptions of things to an origin which we call "here": our location in a particular part of space round which we group the whole Universe is the essential fact of our bodily existence. We can imagine beings who observe all phenomena in all space with an equal eye, unbiassed in favour of any part. With us it is otherwise, a cat at our

feet claims more attention than an earthquake at Cape Horn, or than the destruction of a world in the Milky Way. It is true that in making a common stock of our knowledge with our fellowmen, we have to waive something of the strict egoism of our own individual "here." We substitute "nearly here" for "here"; thus we measure miles from the town hall of the nearest town, or from the capital of the country. In measuring the earth, men of science will put the origin at the earth's centre; astronomers

even rise to the extreme altruism of putting their origin inside the sun. But, far as this last origin may be, and even if we go further to some convenient point amid the nearer fixed stars, yet, compared to the immeasurable infinities of space, it remains true that our first procedure in exploring the Universe is to fix upon an origin "nearly here."