fives.
20 = two tens, one man, two feet.[165]

CHAPTER V.

MISCELLANEOUS NUMBER BASES.

In the development and extension of any series of numbers into a systematic
arrangement to which the term _system_ may be applied, the first and most
indispensable step is the selection of some number which is to serve as a
base. When the savage begins the process of counting he invents, one after
another, names with which to designate the successive steps of his
numerical journey. At first there is no attempt at definiteness in the
description he gives of any considerable number. If he cannot show what he
means by the use of his fingers, or perhaps by the fingers of a single
hand, he unhesitatingly passes it by, calling it many, heap, innumerable,
as many as the leaves on the trees, or something else equally expressive
and equally indefinite. But the time comes at last when a greater degree of
exactness 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 ba