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NUMBER-SYSTE:StS AND NUMERALS 5
two systems have been found in use among previously unknown
tribes.^ The decimal scale was used in North America by
the greater number of Indian tribes, but in South America it
was rare.
In the construction of the decimal system, 10 was suggested
by the number of fingers as the first stopping-place in counting,
and as the first higher unit. Any number between 10
and 100 was pronounced according to the plan &(10) + a(T),
a and 6 being integers less than 10. But the number 110
might be expressed in two ways, (1) as 10 x 10 -)-10, (2) as
11 X 10. The latter method would not seem unnatural.
Why not imitate eighty, ninety, and say eleventy, instead of
hundred and ten ? But upon this choice between 10 x 10 -f-10
and 11 X 10 hinges the systematic construction of the number
system.^ Good luck led all nations which developed the
decimal system to the choice of the former;' the unit 10
being here treated in a manner similar to the treatment of the
lower unit 1 in expressing numbers below 100. Any number
between 100 and 1000 was design3ted c(10)^ -|- &(10) + a,
a, b, c representing integers less than 10. Similarly for numbers
below 10,000, d(10y + c(10y + 6(10)1 _^ ^(^Qy. a,nd simi­
larly for still higher numbers.
Proceeding to describe the notations of numbers, we
begin with the Babylonian. Cuneiform writing, as also the
accompanying notation of numbers, was probably invented
^ Conant, op. cit., p. 588.
2 Hbemah:!? Hankel, Zur Geschichte der MathematiTc in Alterthum
und Mittelalter, Leipzig, 1874, p. 11. Hereafter we shall cite this brilliant
work as Hankel.
^ In this connection read also Moeitz Cantoe, Vorlesungen uber
Geschichte der MathematiTc, Vol. I., (Second Edition), Leipzig, 1894,
pp. 6 and 7. This history, by the prince of mathematical historians of
this century, wiU be in tliree volumes, when completed, and vfill be cited
hereafter as Cantor.