mc gee] THE BEGINNING OF MA THEM A TICS 663
this difficulty is overcome by bisecting two of the quadrants in a simple cruciform symbol in such manner as to produce the asymmetric figure ^r ; but the ever-acting mechanical ten- dency operates to produce the regular figure >jc as the applica- tions of the systems are extended. In either case, augmentation is effected by doubling or further increasing the peripheral ^extremities in such manner as to produce simple hexagrams, at first irregular, ^, and eventually regular, $) , or $. The value of successive augmentations is expressed by the figures 6+1, 1 2+1, 1 8+ 1 1 etc., 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 directions, 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 sys- tems. True, it is difficult for the arithmetical thinker to see how the mathematical pioneer missed the now-plain road from the indefinite quaternary-quinary notion to the definite quinary con- cept ; but the fact cannot 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 prac- tical Arabs with their decimal system, the barefoot Mexicans with their vigesimal system, and a few other peoples of excep- tionally 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 prescripto- rial culture little if any use 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.
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