We have the two positions o; 1 on i, and the two positions o, 1 on j, [fig. 63]. These give rise to a certain complexity. I will let the two lines i and j meet in the position I call o on each, and I will consider i as a direction starting equally from every position on j, and j as starting equally from every position on i. We thus obtain the following figure:—A is both oi and oj, B is 1i and oj, and so on as shown in [fig. 63]b. The positions on AC are all oi positions. They are, if we like to consider it in that way, points at no distance in the i direction from the line AC. We can call the line AC the oi line. Similarly the points on AB are those no distance from AB in the j direction, and we can call them oj points and the line AB the oj line. Again, the line CD can be called the 1j line because the points on it are at a distance, 1 in the j direction.
Fig. 63b.
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 i and with it o of the kind j; and then, that we take o of the kind i and with it 1 of the kind j.
Fig. 64.
Thus we get a pair of positions lying in the straight line BC, [fig. 64]. We can call this pair 10 and 01 if we adopt the plan of mentally, adding an i to the first and a j to the second of the symbols written thus—01 is a short expression for Oi, 1j.
Fig. 65.
Coming now to our space, we have three dimensions, so we take three positions on each. These positions I will suppose to be at equal distances along each axis. The three axes and the three positions on each are shown in the accompanying diagrams, [fig. 65], of which the first represents a cube with the front faces visible, the second the rear faces of the same cube; the positions I will call 0, 1, 2; the axes, i, j, k. I take the base ABC as the starting place, from which to determine distances in the k direction, and hence every point in the base ABC will be an ok position, and the base ABC can be called an ok plane.