One combination, ABCD, may be converted into another A*b*C*d* by interchanging one or more of the classes with the complementary classes. The number of such changes is called the *distance*, which in the above case is 2. In two similar compound statements the distances of the combinations denied must be the same; but it does not follow that when all the distances are the same, the statements are similar. There is, however, only one example of two dissimilar statements having the same distances. When the distance is 4, the two combinations are said to be *obverse* to one another, and the statements denying them are called *obverse statements*, as in ABCD = 0 and *abcd* = 0 or again A*b*C*d* = 0 and *a*B*c*D = 0. When any one combination is given, called the *origin*, all the others may be grouped in respect of their relations to it as follows:--Four are at distance *one* from it, and may be called *proximates*; six are at distance *two*, and may be called *mediates*; four are at distance *three*, and may be called *ultimates*; finally the obverse is at distance *four*. Origin and Six Obverse and four proximates. mediates. four ultimates. *ab*CD | *a*BCD A*bc*D | A*b*C*d* A*bcd* | \ | / | | \ | / | | \|/ | ABC*d*--ABCD--A*b*CD + *abc*D--*abcd*--*a*B*cd* | /|\ | | / | \ | | / | \ | AB*c*D *a*B*c*D | *a*BC*d* *ab*C*d*. | AB*cd* It will be seen that the four proximates are respectively obverse to the four ultimates, and that the mediates form three pairs of obverses. Every proximate or ultimate is distant 1 and 3 respectively from such a pair of mediates. Aided by this system of nomenclature Professor Clifford proceeds to an exhaustive enumeration of types, in which it is impossible to follow him. The results are as follows:-- 1-fold statements 1 type } 2 " " 4 types} 3 " " 6 " } 4 " " 19 " } 159 5 " " 27 " } 6 " " 47 " } 7 " " 55 " } 8-fold statements 78 " Now as each seven-fold or less-than-seven-fold statement is complementary to a nine-fold or more-than-nine-fold statement, it follows that the complete number of types will be 159 × 2 + 78 = 396. It appears then that the types of statement concerning four classes are only about 26 times as numerous as those concerning three classes, fifteen in number, although the number of possible combinations is 256 times as great.