other two are mediates which are not obverse, and which 
consequently are at distance 1 from some one proximate. 
With this proximate as origin, therefore, all three are proxi- 
mates. We have therefore only to enquire 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 1 or 3 
from the remaining proximate. Thus we liave 6 types, in 
which the distances of the fourth mark from the triad are 
respectively 111, 333, 222, 222, 133, 113. The third and 
fourth of these are especially interesting, as being distinct 
types with the same set of distances; I call them pToper 
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 12 types of pure four-fold statement. 
7. In a four-fold 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, 
with the distances 18^13 ; 13',31; 22^22 13^,22; 13^22. 
With two pair of obverses, they must be either at odd or 
even distances from one another; two types. Altogether 
12-l-5-f-2=:19 four-fold types. 
8. In a pure five^fold statement there is always a triad 
of marks at distance 2 from one another. For there is a 
pair evenly distant ; if there is not another mark evenly 
distant from these, the remaining three are aU oddly dis- 
tant, and therefore evenly distant from one another. First 
then let the remaining two marks be both oddly distant 
from the triad. In regard to the origin of which these are 
proximates, the two to be added must be either two 
mediates, like (of two kinds) or unlike, or a mediate of 
either kind with the origin or the obverse ; 7 t}q)es, Next, 
