842 PRIMITIVE NUMBERS [ETH. ANN.19 
was made of nature’s abacus, the ever present human hand—for a 
habit of finger-counting could hardly fail to fix the quinary system in 
the minds of. counters able to grasp so high a number as five without 
aid of extraneous symbols. 
The growth of the senary-septenary system out of the quaternary- 
quinary arrangement forcibly suggests the genesis of the latter; for 
just as the hexagram of the higher system represents the swastika of 
the lower system plus a trigram of the binary-ternary system super- 
posed by almacabalic augmentation, so the swastika itself merely 
represents two superposed trigrams. This view of the growth of the 
three systems in the order of passage from the simple to the complex 
is supported by all that is known of the relative intellectual capacity 
ot their users; and it would seem to be established by the occasional 
advances from the binary-ternary system to the quaternary-quinary 
plane by some of the Australian numerations, as well as by various 
vestiges of the binary-ternary system along various culture lines, 
notably the Mongolian and Aryan. 
The presumptively primeval system apparently arose spontaneously 
(perhaps along lines noted later) and became fixed through habitual 
mental effort shaped less by purpose-wrought symbols than by per- 
sonal or subjective associations. Analogy with the higher systems 
would indicate that the number-concept outlined vaguely through the 
dull mentation of the Australian blackfellows might be symbolized 
by any regular trigram uniting the perceived pair of objects and the 
unapperceived Ego, i. e., connecting the objective impression with its 
subjective reflex; but the inequality of all social pairs in the tribal 
organization, the ever-varying relative potencies of the good and evil 
mysteries, the unequal rank of the two ghostly Doppel-ichen, and 
divers other indications, would suggest that a better figure for the 
concept would be an irregular trigram. Yet howsoever the system 
be represented graphically by the student (for apparently the black- 
fellow had no notion of notation), the law of augmentation common to 
the two higher systems prevailed, as is shown both by certain of the 
Australian number-terms and by the Mongolian vestiges—i. e., the 
augmentation proceeded by successive additions to a once-reckoned 
middle, yielding the values 2-+-1, 4+1, 6-+-1. 
It is questionable whether any enlightened student will ever enter 
sufliciently into the prescriptorial thought represented by any consid- 
erable number of distinct primitive peoples to grasp and record all 
the stages and substages in the growth of number systems; yet the 
records already extant would seem to indicate the lines of growth in 
fairly adequate fashion. ‘The records are consistent in indicating that 
primitive peoples used integral numbers rather as symbols of extra- 
natural potencies than as tokens for natural values; that they com- 
bined the symbols through mechanical devices by aid of a simple rule 
