Introduction to Mathematics by Alfred North Whitehead

Author:Alfred North Whitehead
Language: eng
Format: epub
Publisher: Barnes & Noble
Published: 2017-05-11T04:00:00+00:00

Fig. 14.

Consider y-x=0; here the a, b, and c, of the general form have been replaced by 1, -1, and 0 respectively. This line passes through the “origin,” O, in the diagram and bisects the angle XOY. It is the line L‘OL of the diagram. The fact that it passes through the origin, O, is easily seen by observing that the equation is satisfied by putting x=0 and y=0 simultaneously, but 0 and 0 are the coordinates of O. In fact it is easy to generalize and to see by the same method that the equation of any line through the origin is of the form ax+by=0. The locus of equation y+x=0 also passes through the origin and bisects the angle X’OY: it is the line L1 OL’1 of the diagram.

Consider y-x=1: the corresponding locus does not pass through the origin. We therefore seek where it cuts the axes. It must cut the axis of x at some point of coordinates x and 0. But putting y =0 in the equation, we get x=-1; so the coordinates of this point (A) are -1 and 0. Similarly the point (B) where the line cuts the axis OY are 0 and 1. The locus is the line AB in the figure and is parallel to LOL′. Similarly y+x=1 is the equation of line A1B of the figure; and the locus is parallel to L1 OL’1. It is easy to prove the general theorem that two lines represented by equations of the forms ax+by=0 and ax+by=care 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, OX and OY, at right-angles, and then by noting its coordinates x and y, i.e. OM and PM. Also, as we have seen in the last chapter, P can be determined by the “vector” OP, 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, OX and OY. But in relation to the application of mathematics to the events 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.

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