stems of bases that occur in developed mathematics. But the
inertia to be overcome in the recognition of the base idea is nowhere
more obvious than in the retention by the comparatively developed
Babylonian system of a second base of 60 to supplement the decimal one
for smaller numbers. Among the American Indians (Eells, _loc. cit._) the
system of bases used varies from the cumbersome binary scale, that
exercised such a fascination over Leibniz (_Opera_, _III_, p. 346),
through the rare ternary, and the more common quarternary to the
"natural" quinary, decimal, and vigesimal systems derived from the
use of the fingers and toes in counting. The achievement of a number
base and number words, however, does not always open the way to
further mathematical development. Only too often a complexity of
expression is involved that almost immediately cuts off further
progress. Thus the Youcos of the Amazon cannot get beyond the number
three, for the simplest expression for the idea in their language is
"pzettarrarorincoaroac" (Conant, _loc. cit._, pp. 145, 83, 53). Such
names as "99, tongo solo manani nun solo manani" (i.e., 10, understood,
5 plus 4 times, and 5 plus 4) of the Soussous of Sierra Leone; "399,
caxtolli onnauh poalli ipan caxtolli onnaui" (15 plus 4 times 20 plus 15
plus 4) of the Aztec; "29, wick a chimen ne nompah sam pah nep e chu
wink a" (Sioux), make it easy to understand the proverb of the Yorubas
of Abeokuta, "You may be very clever, but you can't tell 9 times 9."

Almost contemporaneously with the beginnings of counting various
auxiliary devices were introduced to help out the difficult task. In
place of many men, notched sticks, knotted strings, pebbles, or finger
pantomime were used. In the best form, these devices resulted in the
abacus; indeed, it was not until after the introduction of arabic
numerals and well into the Renaissance period that instrumental
arithmetic gave way to graphical in Europe (D. E. Smith, _Rara
Arithmetica_, under "Counters"). "In eastern Europe," say Smith and
Mikami (_Japanese Mathematics_, pp. 18-19), "it"--the abacus--"has never
been replaced, for the tschotue is used everywhere in Russia to-day, and
when one passes over into Persia the same type of abacus is common in
all the bazaars. In China the swan-pan is universally used for the
purposes of computation, and in Japan the soroban is as strongly
entrenched as it was before the invasion of western ideas."

Given, then, the i