The dotted lines in 3 and 4, fig. 50, denote that it is not known whether or no any objects exist, corresponding to the space of which the dotted line forms one delimiting boundary; thus, in mood I we do not know if there are any M’S which are not P, we only know some M’S are P. [Illustration: Fig. 51.] Representing the first premiss in its various moods by regions marked by vertical lines to the right of PQ, we have in fig. 51, running up from the four letters AEIO, four columns, each of which indicates that the major premiss is in the mood denoted by the respective letter. In the first column to the right of PQ is the mood A. Now above the line RS let there be marked off four regions corresponding to the four moods of the minor premiss. Thus, in the first row above RS all the region between RS and the first horizontal line above it denotes that the minor premiss is in the mood A. The letters E, I, O, in the same way show the mood characterising the minor premiss in the rows opposite these letters. We have still to exhibit the conclusion. To do this we must consider the conclusion as a third variable, characterised in its different varieties by four moods—this being the syllogistic classification. The introduction of a third variable involves a change in our system of representation. Before we started with the regions to the right of a certain line as representing successively the major premiss in its moods; now we must start with the regions to the right of a certain plane. Let LMNR be the plane face of a cube, fig. 52, and let the cube be divided into four parts by vertical sections parallel to LMNR. The variable, the major premiss, is represented by the successive regions which occur to the right of the plane LMNR—that region to which A stands opposite, that slice of the cube, is significative of the mood A. This whole quarter-part of the cube represents that for every part of it the major premiss is in the mood A. [Illustration: Fig. 52.] In a similar manner the next section, the second with the letter E opposite it, represents that for every one of the sixteen small cubic spaces in it, the major premiss is in the mood E. The third and fourth compartments made by the vertical sections denote the major premiss in the moods I and O. But the cube can be divided in other ways by other planes. Let the divisions, of which four stretch from the front face, correspond to the minor premiss. The first wall of sixteen cubes, facing the observer, has as its characteristic that in each of the small cubes, whatever else may be the case, the minor premiss is in the mood A. The variable—the minor premiss—varies through the phases A, E, I, O, away from the front face of the cube, or the front plane of which the front face is a part.