Looking at the shape which represents the totality of the valid conclusions, it does not present any obvious symmetry, or easily characterisable nature. A striking configuration, however, is obtained, if we project the four-dimensional figure obtained into a three-dimensional one; that is, if we take in the base cube all those cubes which have a marked space anywhere in the series of four regions which start from that cube. This corresponds to making abstraction of the figures, giving all the conclusions which are valid whatever the figure may be. [Illustration: Fig. 57.] Proceeding in this way we obtain the arrangement of marked cubes shown in fig. 57. We see that the valid conclusions are arranged almost symmetrically round one cube—the one on the top of the column starting from AAA. There is one breach of continuity however in this scheme. One cube is unmarked, which if marked would give symmetry. It is the one which would be denoted by the letters I, E, O, in the third wall to the right, the second wall away, the topmost layer. Now this combination of premisses in the mood IE, with a conclusion in the mood O, is not noticed in any book on logic with which I am familiar. Let us look at it for ourselves, as it seems that there must be something curious in connection with this break of continuity in the poiograph. [Illustration: Fig. 58.] The propositions I, E, in the various figures are the following, as shown in the accompanying scheme, fig. 58:—First figure: some M is P; no S is M. Second figure: some P is M; no S is M. Third figure: some M is P; no M is S. Fourth figure: some P is M; no M is S. Examining these figures, we see, taking the first, that if some M is P and no S is M, we have no conclusion of the form S is P in the various moods. It is quite indeterminate how the circle representing S lies with regard to the circle representing P. It may lie inside, outside, or partly inside P. The same is true in the other figures 2 and 3. But when we come to the fourth figure, since M and S lie completely outside each other, there cannot lie inside S that part of P which lies inside M. Now we know by the major premiss that some of P does lie in M. Hence S cannot contain the whole of P. In words, some P is M, no M is S, therefore S does not contain the whole of P. If we take P as the subject, this gives us a conclusion in the mood O about P. Some P is not S. But it does not give us conclusion about S in any one of the four forms recognised in the syllogism and called its moods. Hence the breach of the continuity in the poiograph has enabled us to detect a lack of completeness in the relations which are considered in the syllogism. To take an instance:—Some Americans (P) are of African stock (M); No Aryans (S) are of African stock (M); Aryans (S) do not include all of Americans (P).