92 PROF. W. K. CLIFFORD ON THE 



■ may be all evenly distant, or two evenly and the other pair 

 oddly distant from each. If they are all evenly distant 

 they are the mediates to a certain origin or its obverse, and 

 the seventh mark may be the origin or a proximate, 2 types. 

 In the other case we have an origin, obverse, and pair of 

 mediates together with a proximate and its obverse ulti- 

 mate ; we may add a proximate or a mediate, 2 types. 



17. A pure eightfold statement must consist of two 

 groups, either both proper or both improper, or one of 

 each. Two proper groups may have their origins distant 

 I or 3 ; 2 types. To an improper group we may add a 

 proper group made of one origin and three obverses or of 

 three origins and one obverse, or an improper group made 

 of four origins or four obverses or two origins and two 

 obverses ; 5 types. Altogether there are 7 types of pure 

 eightfold statement. 



18. An eightfold statement with one pair of obverses 

 must subdivide into four and four, or into five and three. 

 In the former case we have a pair of obverses, viz. an origin 

 and its obverse, and two mediates ; to which we must add 

 a group formed out of the proximates and ultimates. This 

 group may be proper, i type, or improper, the mediates 

 being in regard to it two origins, two obverses, or an origin 

 and an obverse ; 3 types. In the latter case the five 

 marks must be a proper group with the obverse of one 

 mark, to which we must add a triad made out of the origin, 

 obverse, and mediates of the group. This triad may be the 

 origin or obverse together with two mediates distant i i, i 3, 

 3 3 from the ultimate, 6 types ; or else it may be three me- 

 diates distant i i i, i i 3^ i 3 3^ 3 3 3 from the ultimate, 4 

 types. 



19. An eightfold statement with two pairs of obverses 

 must subdivide into four and four, or into five and three, 

 or into six and two. In the first case the two pairs of ob- 



