The fourth dimension

A FOUR-DIMENSIONAL FIGURE 123

The rule will be made more evident by a simple application.

Let us consider one dimension and one position. I will call the axis 2, and the position 0.

—_—__S

o

Here the figure is the position o on the line 4. Take

now two dimensions and two positions on each. We have the two positions 0; 1 on ¢, and the two positions 0, 1 on J, fig. 63. These give

the J vise to a certain complexity. I will

4 let the two lines 4 and 7 meet in the

; position I call o on each, and I will

v une consider @ asa direction starting equally ig. 63.

from every position on j, and 7 as starting equally from every position on 7. We thus obtain the following figure :—a is both o7 and oj, B is 1% and oj, and so on as shown in fig. 63b. The positions on Ac are all 07 positions. They are, if we like to consider it in that way, points at no distance in the 4 direction from the line ac. We can call the line ac the o# line. Similarly the points on AB are those no distance

Fig. 630. from 4B in the j direction, and we can call them oj points and the line AB the oj line. Again, the line op can be called the lj line because the points on it are at a distance, 1 in they direction.

We have then four positions or points named as shown, and, considering directions and positions as “ kinds,” we have the combination of two kinds with two kinds. Now, selecting every one of one kind with every other of every other kind will mean that we take 1 of the kind 4 and