84 PROF. W. K. CLIFFORD ON THE 



It will l)e seen from the above table that the' four proxi- 

 mates are respectively obverse to the four ultimates, and 

 that the mediates form three pairs of obverses. Every 

 proximate or ultimate is distant i and 3 respectively from 

 such a pair of mediates. Thus each proximate or ultimate 

 divides the mediates into two classes ; three of them are at 

 distance i from it, and three at distance 3. Two mediates 

 which are not obverse are at distance 2. Two proximates 

 or two ultimates, or an ultimate and a proximate which are 

 not obverse, are also at distance 2. 



This view of the mutual relations of the marks is the 

 basis of the following enumeration of types. 



4. There is clearly only one type of simple statement. 

 But of twofold statements there are four types; viz. the 

 distance may be i , 1, 3, or 4 ; and so, in general, with n 

 classes there are n types of twofold statement. 



5. A compound statement containing no pair of obverses 

 is called pure. In a threefold statement there are three 

 distances ; one of these must be not less than either of the 

 others. If this be i, the remaining mark must be at odd 

 distance from both of these or at even distance from both ; 

 thus we get the types i, \,i, and 1, 1, 2. If the not-less 

 distance be 3, the remaining distances must be one even and 

 the other odd ; the even distance must be i, the odd one 

 either i or 3 ; and the types are i, 2, 3; 1, 3, 3. Thus there 

 are 4 pure threefold types. With a pair of obverses, the re- 

 maining mark must be at odd or even distance from them ; 

 I, 3, 4; 2, 2, 4. In all six threefold types observe that 

 there is necessarily one even distance. 



6. A fortiori, in a fourfold statement there must be one 

 even distance. In a pure fourfold statement this distance 

 is 2. From this pair of marks let both the others be oddly 

 distant ; then they must be evenly distant from one another 

 i. e. at distance 2, obverses being excluded. The odd 



