_Note._—At the beginning of the next chapter the same structures as those which follow are exhibited in more detail and a reference to them will remove any obscurity which may be found in the immediately following passages. They are there carried on to a greater multiplicity of dimensions, and the significance of the process here briefly explained becomes more apparent. [Illustration: Fig. 59.] Take three mutually rectangular axes in space 1, 2, 3 (fig. 59), and on each mark three points, the common meeting point being the first on each axis. Then by means of these three points on each axis we define 27 positions, 27 points in a cubical cluster, shown in fig. 60, the same method of co-ordination being used as has been described before. Each of these positions can be named by means of the axes and the points combined. [Illustration: Fig. 60.] Thus, for instance, the one marked by an asterisk can be called 1_c_, 2_b_, 3_c_, because it is opposite to _c_ on 1, to _b_ on 2, to _c_ on 3. Let us now treat of the states of consciousness corresponding to these positions. Each point represents a composite of posits, and the manifold of consciousness corresponding to them is of a certain complexity. Suppose now the constituents, the points on the axes, to interchange arbitrarily, any one to become any other, and also the axes 1, 2, and 3, to interchange amongst themselves, any one to become any other, and to be subject to no system or law, that is to say, that order does not exist, and that the points which run _abc_ on each axis may run _bac_, and so on. Then any one of the states of consciousness represented by the points in the cluster can become any other. We have a representation of a random consciousness of a certain degree of complexity. Now let us examine carefully one particular case of arbitrary interchange of the points, _a_, _b_, _c_; as one such case, carefully considered, makes the whole clear. [Illustration: Fig. 61.] Consider the points named in the figure 1_c_, 2_a_, 3_c_; 1_c_, 2_c_, 3_a_; 1_a_, 2_c_, 3_c_, and examine the effect on them when a change of order takes place. Let us suppose, for instance, that _a_ changes into _b_, and let us call the two sets of points we get, the one before and the one after, their change conjugates. Before the change 1_c_ 2_a_ 3_c_ 1_c_ 2_c_ 3_a_ 1_a_ 2_c_ 3_c_}Conjug- After the change 1_c_ 2_b_ 3_c_ 1_c_ 2_c_ 3_b_ 1_b_ 2_c_ 3_c_} ates. The points surrounded by rings represent the conjugate points. It is evident that as consciousness, represented first by the first set of points and afterwards by the second set of points, would have nothing in common in its two phases. It would not be capable of giving an account of itself. There would be no identity.