We learn from this table, for instance, that two propositions of the form A = AB, B = BC, which are such as constitute the premises of the old syllogism Barbara, exclude as impossible four of the eight combinations in which three terms may be united, and that these propositions are capable of taking twenty-four variations by transpositions of the terms or the introduction of negatives. This table then presents the results of a complete analysis of all the possible logical relations arising in the case of three terms, and the old syllogism forms but one out of fifteen typical forms. Generally speaking, every form can be converted into apparently different propositions; thus the fourth type A = B, B = BC may appear in the form A = ABC, *a* = *ab*, or again in the form of three propositions A = AB, B = BC, *a*B = *a*B*c*; but all these sets of premises yield identically the same series of combinations, and are therefore of equivalent logical meaning. The fifth type, or Barbara, can also be thrown into the equivalent forms A = ABC, *a*B = *a*BC and A = AC, B = A ꖌ *a*BC. In other cases I have obtained the very same logical conditions in four modes of statements. As regards mere appearance and form of statement, the number of possible premises would be very great, and difficult to exhibit exhaustively. The most remarkable of all the types of logical condition is the fourteenth, namely, A = BC ꖌ *bc*. It is that which expresses the division of a genus into two doubly marked species, and might be illustrated by the example--“Component of the physical universe = matter, gravitating, or not-matter (ether), not-gravitating.” It is capable of only two distinct logical variations, namely, A = BC ꖌ *bc* and A = B*c* ꖌ *b*C. By transposition or negative change of the letters we can indeed obtain six different expressions of each of these propositions; but when their meanings are analysed, by working out the combinations, they are found to be logically equivalent to one or other of the above two. Thus the proposition A = BC ꖌ *bc* can be written in any of the following five other modes, *a* = *b*C ꖌ B*c*, B = CA ꖌ *ca*, *b* = *c*A ꖌ C*a*, C = AB ꖌ *ab*, *c* = *a*B ꖌ A*b*.