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 MC gee] THE BEGINNING OF MA THEM A TICS 649

that phase of scriptorial culture marked by conventional calen- dric 'and numerical systems ; hardly less useful are several African peoples representing various stages of development; equally significant, too, are the Australasian tribes of culture so low that numerical knowledge is inchoate only ; while useful suggestions as to the origin of numerical concepts may be obtained from various subhuman animals. True, the lines of growth maturing in mathematical systems must vary with en- vironmental conditions, and doubtless with hereditary traits persistently reflecting both ancestral and proto-environmental factors ; yet, if knowledge be not an extra-natural 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 sufficiently 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 sufficiently accurate to meet practical needs.

II

In tracing lines of activity maturing in civilization and enlight- enment, it is needful to note certain habits of mind characteristic of all primitive men ; and for present purposes (as for practically all others), it suffices to define primitive peoples as those who have not yet acquired and assimilated the art of writing — i. e., as those who remain in prescriptorial culture :

1. Primitive men are mystics. Believers in extra-natural po- tencies, skilless observers, and inconstant reasoners, their faith counterfeits realities, and clothes its own figments with all man- ner of attributes, both appropriate and incongruous. In their simple (and presumptively primeval) aspect, the fear-born fig- ments are grotesque shadows or fantastic duplicates of actual things, moved by caprice or malice like unto that of human kind ; grown more complex, the figments are incarnated chiefly in

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