828 PRIMITIVE NUMBERS [ETH. ANN. 19 
patches, while the mental growth is luxuriant and spreading exuber- 
antly from province to proyince throughout the lands of the earth. 
In both cases the interpretation in terms of growth-stages is established 
by conformity with natural law: did the grove receive extranatural 
impulse at any stage, or did knowledge arise otherwise than through 
interactions of nature, the interpretation would fail; but in the absence 
of evidence against the uniformity of nature, the equivalence of corres- 
ponding stages must be recognized alike for the figurative forests of 
ideas and the material forests of wood and leafage. 
Now the acceptance of these principles, and the recognition of the 
general course of intellectual development, afford a means of tracing 
the unrecorded history of Aryan culture and of interpreting the meager 
records of Arabia’s mathematical pioneering in terms of the culture 
of other peoples still below, or just rising above, the plane marked by 
the birth of writing—i. e., the beginning of scriptorial culture. 
Especially useful for comparison are various practically independent 
Amerind peoples, some low in prescriptorial culture, others grap- 
pling with the rudiments of definite graphic art, and still others just 
within that phase of scriptorial culture marked by conventional calen- 
dric and numeral systems; hardly less useful are several African peo- 
ples representing various early stages of development; of much 
significance, too, are the Australian tribes, of culture so low that 
numerical knowledge is inchoate only, together with different Polyne- 
sian tribes whose culture curiously reflects their distinctive environ- 
ment; while useful suggestions as to the origin of numerical concepts 
may be drawn from various subhuman animals. True, the lines of 
mental growth maturing in mathematical systems must vary with 
environmental conditions, and doubtless with hereditary traits per- 
sistently reflecting both ancestral and proto-environmental factors; 
yet, if knowledge be not an extranatural product rather than a reflex 
of nature (as brilliantly conceived by Bacon) the lines must be so far 
conformable as to render the comparisons trustworthy and sufliciently 
accurate for practical purposes—just as the retracing of the history of 
an isolated grove by comparison with the growth-lines of other groves 
must be inexact in detail, though trustworthy in general and sufli- 
ciently accurate to meet practical needs. 
CHARACTERISTICS OF PRIMITIVE THOUGHT 
In tracing the lines of intellectual growth maturing in modern 
enlightenment, it is needful to note certain habits of mind character- 
istic of all primitive men, yet measurably distinct (in degree if not in 
kind) from those common to civilized and enlightened men; and for 
present purposes, as for practically all others, it will suftice to define 
primitive peoples as those who have not yet acquired and assimilated 
