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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
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