MCGEE] CORRESPONDENCE IN THE THREE SYSTEMS 839 
of the Toba and perhaps other South American tribes, assume definite 
and harmonious shape in a binary-ternary system, in which things are 
conceived in pairs related subconsciously to an initial or central inter- 
pretative nucleus—that is, to the dominating Ego of primitive ideation. 
The three number-systems pertaining to prescriptorial culture are 
essentially distinct from modern Aryan numeration, and indeed from 
the whole of Arabic algorithm and arithmetic, in motive as well as in 
mechanism. Primarily, they are devices for divination or for con- 
necting the real world with the supernal, and it is only later or in minor 
way that they are prostituted to practical uses; yet by reason of the 
magical potency imputed to them they dominate thought and action 
in the culture-stages to which they belong and profoundly affect the 
course of intellectual development—indeed, like other figments (or pure 
abstractions, dissevered from the actualities of nature), their office is 
first to stimulate and later to enchain mentation. 
In mechanism the three systems correspond substantially, even if 
they are not actually correlative, for each rests on an exoteric base in 
the form of a small even number, and each is really controlled and per- 
fected by a half-apperceived unity, itself the reflection of the Ego, 
whereby the base is raised esoterically to the next higher odd number. 
The systems differ only in the value of the exoteric base, which is 
a measure of the intellectual capacity normal to the culture-stage to 
which it pertains. The two higher systems have graphic equivalents 
which shape and intensify their mystical potency (for the mechanical 
conditions attending graphic representation always interact with pri- 
mary concepts in primitive thought); but the lowest and presumptively 
primeval system is without known graphic symbol. 
NoraTION AND AUGMENTATION 
Resting as they do on inconstant and largely subjective bases, and per 
taining as they do to prescriptorial culture (or at the best to inchoate 
ideographic representation), the primitive number systems are not 
susceptible of algorithmic notation. Concordantly they are insuscep- 
tible of treatment by the methods of rational arithmetic; though the 
two higher systems (and probably the lowest also) lend themselves to 
combinations made in accordance with a method or law which may 
be styled augmentation—a process tending to perpetuate itself, and, 
while neither addition nor multiplication, tending to generate both. 
This curious law of augmentation is of much significance; in the first 

place, it represents a process apparently lost (along with the observa- 
tional basis of arithmetic) from the recorded history of mathematics; 
and, in the second place, it seems to explain the interrelations and evo- 
lution of the magica) number-systems; again, it would seem to con- 
stitute the germ of the fundamental arithmetic processes, and hence 
to explain the transition from magical to rational numbers; and finally 
