MCGEF] SENARY-SEPTENARY NOTATION 841 
toward the left), and accordingly that this form would be normal if 
the form itself were normal to advanced culture; but that since the 
symbol pertains in all essential respects to the lowly culture charac- 
terized by centripetal hand-movement, the counter-clockwise form 
and it is 

would seem to be more properly considered the norm: 
drawn herein. 
While the concept of the senary-septenary system is much more 

complex than that of the quaternary-quinary system, the law of aug- 
mentation is similar; and it is significant that the similarity accom- 
panies (and presumptively results from) analogous efforts at graphic 
representation. Commonly the concept is directional, as in that form 
of the Cult of the Quarters in which zenith and nadir are reckoned as 
cardinal points; and the mechanical symbol is complicated, and event- 
ually modified, through the difficulty of depicting tridimensional rela- 
tions on the bidimensional surface. Among the pueblo peoples this 
difficulty is overcome by bisecting two of the quadrants in a simple 
cruciform symbol in such manner as to produce the asymmetric figure 
Me; but the ever-acting mechanical tendency operates to produce the 
regular figure 44 as the applications of the systems are extended. In 
either case, augmentation is effected by doubling or further increas- 
ing the peripheral extremities in such manner as to produce simple 
hexagrams, at first irregular, 4g, and eventually regular, &: or &. 
The value of successive augmentations is expressed by the figures 
6+1, 12+1, 18+1, ete., i. e., by successive additions (mechanical or 
mental) to a once-reckoned Middle. 
Now, comparison of these two number systems, especially as 
illumined by the Pueblo method of depicting the fifth and sixth direc- 
tions, indicates that the higher is produced from the lower simply by 
the superposition of a binary system on the quaternary system; and 
the inference, coupled with the ‘patent fact that the higher base is the 
measure of increased intellectual capacity, seems to define the course 
of development of both systems. True, it is difficult for the arith- 
metical thinker to see how the mathematical pioneer missed the now- 
plain road from the indefinite quaternary-quinary notion to the defi- 
nite quinary concept; but the fact can not be gainsaid that the road 
was missed by many primitive tribes of especially mystical cast of 
mind, and that it was found and followed only by the ancestors of 
the practical Arabs with their decimal system, the barefoot Mexicans 
with their vigesimal system, and a few other peoples of exceptionally 
vigorous mind. The failure to find so plain a way may be ascribed 
largely to the complete domination of primitive thought by mystical 
concepts; and it would seem to repeat the demonstration by other 
facts that throughout much of preseriptorial culture little if any use 

