99 
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, one type, or improper, the mediates 
being in regard to it two origins, two obverses, or an origin 
and an obverse ; three 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 11, 13, 83 from 
the ultimate, six types ; or else it may be three mediates dis- 
tant 111, 113, 133,333 from the ultimate; four types. 
19. An eight-fold 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 obverses 
may be evenly distant, when the remaining marks form a 
group either proper, with its origin, obverse and pair of 
mediates, or two pair of mediates, or else improper, 3 types; 
or oddly distant, when the remainder form one of the 6 pure 
four-fold statements enumerated art. 6. Two marks distant 
2 from each other may be distant 11, 33 or 13, 13 from the 
pair of obverses which is oddly distant from them ; thus 
each of the 6 four-fold statements gives 8 types of eight-fold 
statement, except the third, which gives 4; in all 19. In 
the second case the three may be a triad, or may contain a 
pair of obverses. If it is a triad, the five are mediates to 
one origin and its obverse, and we may add three proxi- 
mates distant 113 or 133 or two proximates distant 11, 13 
or 33 with an ultimate distant respectively 1, 1 or 3, 3, from 
the odd mediate, 6 types. If the three contain a pair of ob- 
verses, the five make a proper groupwith obverse of one mark; 
to this we may add the origin and obverse of the group with 
mediate distant 1 or 3 from the ultimate, or a pair of obverse 
mediates with a mediate distant 1 or 3 as before, 4 types. In 
the third case the six must be an origin and five mediates, and 
