is required.  Perhaps the number 11 is to be indicated, and indicated precisely.  A fresh mental effort is required of the ignorant child of nature; and the result is “all the fingers and one more,” “both hands and one more,” “one on another count,” or some equivalent circumlocution.  If he has an independent word for 10, the result will be simply ten-one.  When this step has been taken, the base is established.  The savage has, with entire unconsciousness, made all his subsequent progress dependent on the number 10, or, in other words, he has established 10 as the base of his number system.  The process just indicated may be gone through with at 5, or at 20, thus giving us a quinary or a vigesimal, or, more probably, a mixed system; and, in rare instances, some other number may serve as the point of departure from simple into compound numeral terms.  But the general idea is always the same, and only the details of formation are found to differ.

Without the establishment of some base any system of numbers is impossible.  The savage has no means of keeping track of his count unless he can at each step refer himself to some well-defined milestone in his course.  If, as has been pointed out in the foregoing chapters, confusion results whenever an attempt is made to count any number which carries him above 10, it must at once appear that progress beyond that point would be rendered many times more difficult if it were not for the fact that, at each new step, he has only to indicate the distance he has progressed beyond his base, and not the distance from his original starting-point.  Some idea may, perhaps, be gained of the nature of this difficulty by imagining the numbers of our ordinary scale to be represented, each one by a single symbol different from that used to denote any other number.  How long would it take the average intellect to master the first 50 even, so that each number could without hesitation be indicated by its appropriate symbol?  After the first 50 were once mastered, what of the next 50? and the next? and the next? and so on.  The acquisition of a scale for which we had no other means of expression than that just described would be a matter of the extremest difficulty, and could never, save in the most exceptional circumstances, progress beyond the attainment of a limit of a few hundred.  If the various numbers in question were designated by words instead of by symbols, the difficulty of the task would be still further increased.  Hence, the establishment of some number as a base is not only a matter of the very highest convenience, but of absolute necessity, if any save the first few numbers are ever to be used.