TYPES OF COMPOUND STATEMENT. 85 



distances are i or 3 ; and it will be easily seen that the fol- 

 lowing are all the possible cases : — 



111 Lii 111 111 Lll 111 

 11' 'i I 3* "3 I 3" *3 I i" '3 I 3" '3 13 



In these figures the dots indicate the four marks, the cross 

 lines indicate distance 2, and the other figures the distances 

 between the marks on either side of them. Next, from the 

 pairs of marks at distance 2 let one of the others at least 

 be evenly distant, i. e. at distance 2. Then we have three 

 marks which are all at distance 2 from one another ; and it 

 is easy to show that they are all proximates of a certain 

 other mark. For, select one of them as origin ; then the 

 other two are mediates which are not obverse, and which 

 consequently are at distance i from some one proximate. 

 With this proximate as origin, therefore, all three are proxi- 

 mates. We have therefore only to inquire what different 

 relations the fourth mark can bear to these three. It may 

 be the origin, its obverse, the remaining proximate, its ob- 

 verse, or one of two kinds of mediates, viz. at distance i or 

 3 from the remaining proximate. Thus we have 6 types, 

 in which the distances of the fourth mark from the triad are 

 respectively i i i, 3 3 3, 2 2 2, 2 2 2, i 3 3, i 13. The third 

 and fourth of these are especially interesting, as being di- 

 stinct types with the same set of distances ; I call them 

 proper and improper groups respectively : viz., a proper group 

 is the four proximates of any origin ; an improper group is 

 three proximates with the obverse of the fourth. On the 

 whole we get 1 2 types of pure fourfold statement 



7. In a fourfold statement with one pair of obverses, 

 take one of them for origin ; the remaining two marks must 

 then be either a pair of proximates or ultimates, a proxi- 

 mate and an ultimate, a pair of mediates, or a proximate or 

 ultimate with one of two kinds of mediate — in all, 5 types. 



