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.

In the selection of a base,--of a number from which he makes a fresh start,
and to which he refers the next steps in his count,--the savage simply
follows nature when he chooses 10, or perhaps 5 or 20. But it is a matter