100 
we may add two proximates distant 11, 13, 33 from, the odd 
mediate, or a proximate and an ultimate, or two ultimates, 
distant as before, 9 types. 
20. In an eight-fold statement with three pairs of obverses 
these may be either all evenly distant, or two of them 
evenly distant and the other oddly distant from both. In 
the first case they are mediates to a certain origin and its 
obverse, and we may add the origin with a proximate or 
ultimate, two proximates, or a proximate and ultimate, 4 
types. In the second case take the oddly distant pair for 
origin and obverse, then these are associated with two proxi- 
mates and their obverse ultimates, and we may add the two 
other proximates, a proximate and an ultimate, a proxi- 
mate and a mediate (distant 11, 13, 31, 33 from this proxi- 
mate and the remaining one) or two mediates distant 11, 33 
or 13, 13 from the two proximates, 8 types. 
Lastly, in an eight-fold statement with four pair of ob- 
verses they may be all evenly distant, or the statement may 
subdivide into six and two, or into four and four ; in the 
latter case there are two types. 
21. To obtain the whole number of types, we observe 
that for every less-than- eight-fold type there is a comple- 
mentary more-than- eight-fold type (art. 2); so tliat we must 
add the number of eight-fold types, 78, to twice the number 
of less-than-eight-fold tjpes. 159 ; the result is 396. 
Art. 
4 
5 
6 
TABLE. 
1- fold 
2- fold, distauce 1, 2, 3, 4 
3- fold, pure, distauce 112, 222, 123, 233 4 
1 pair obv., dist. 134, 224 2 
4-fold, pure, two and two- 
1 I 1 
1 I 1 
1 I 1 
1 I 3 
1|1 
3 I 3 
i_IJ 
3 1 1 
three and one 
group, proper or improper. 
113 
3 i 3 
_ C t 
3J^ 
3 ! 3 
4 
2 
12 
6 
12 
5 
2 
19 
7 
1 pair obv 
2 pair obv., dist. odd or even i 
19 
