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 1_i_ 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 1_j_ line because the points on it are at a distance, 1 in the _j_ direction. [Illustration: Fig. 63_b_.] 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_. 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 O_i_, 1_j_. [Illustration: Fig. 64.] 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. [Illustration: Fig. 65.] In the same way, measuring the distances from the face ADC, we see that every position in the face ADC is an _oi_ position, and the whole plane of the face may be called an _oi_ plane. Thus we see that with the introduction of a new dimension the signification of a compound symbol, such as “_oi_,” alters. In the plane it meant the line AC. In space it means the whole plane ACD.