semi-civilized and civilized peoples, the same processes are retained, and
form a part of the daily life of almost every person who has to do with
counting, reckoning, or keeping tally in any manner whatever. They are no
longer necessary, but they are so convenient and so useful that
civilization can never dispense with them. The use of the abacus, in the
form of the ordinary numeral frame, has increased greatly within the past
few years; and the time may come when the abacus in its proper form will
again find in civilized countries a use as common as that of five centuries
ago.

In the elaborate calculating machines of the present, such as are used by
life insurance actuaries and others having difficult computations to make,
we have the extreme of development in the direction of artificial aid to
reckoning. But instead of appearing merely as an extraneous aid to a
defective intelligence, it now presents itself as a machine so complex that
a high degree of intellectual power is required for the mere grasp of its
construction and method of working.





CHAPTER II.

NUMBER SYSTEM LIMITS.


With respect to the limits to which the number systems of the various
uncivilized races of the earth extend, recent anthropological research has
developed many interesting facts. In the case of the Chiquitos and a few