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NUMBER-SYSTEMS AND NUMERALS 9
local value, such as we have in the notation now in use.
Having missed this principle, the ancients had no use for a
symbol to represent zero, and were indeed very far removed
from an ideal notation. In this matter even the Greeks and
Romans failed to achieve what a remote nation in Asia, little
known to Europeans before the present century, accomplished
most admirably. But before we speak of the Hindus, we
must speak of an ancient Babylonian notation, which, strange
to say, is not based on the scale 5, 10, or 20, and which,
moreover, came very near a full embodiment of the ideal
principle found wanting in other ancient systems. We refer
to the sexagesimal notation.
The Babylonians used this chiefly in the construction of
weights and measures. The systematic development of the
sexagesimal scale, both for integers and fractions, reveals a
high degree of mathematical insight on the part of the early
Sumerians. The notation has been found on two Babylonian
tablets. One of them, probably dating from 1600 or 2300 b.c,
contains a list of square numbers up to 601 The first seven
are 1, 4, 9, 16, 25, 36, 49. We have next 1,4 = 8^ 1.21 = 9^
1.40 = 10^, 2.1 = 11^, etc. This remains, unintelligible, unless
we assume the scale of sixty, which makes 1.4 = 60 -|- 4,
1.21 = 60 -f- 21, etc. The second tablet records the magnitude
of the illuminated portion of the moon's disc for every day
from new to full moon, the whole disc being assumed to consist
of 240 parts. The illuminated parts during the first five
days are the series 5, 10, 20, 40, 1.20(= 80). This reveals
ag^n the sexagesimal scale and also some knowledge of
geometrical progressions. Erom here on the series becomes an
arithmetical progression, the numbers from the fifth to the
fifteenth day being respectively, 1.20, 1.36, 1.52, 2.8, 2.24, 2.40,
2.56, 3.12, 3.28, 3.44, 4. In this sexagesimal notation we have,
then, the principle of local value. Thus, in 1.4 (= 64), the 1 is