Secret_History

484 The Secret History of the World
The Sumerian numbering system often required excessive repetitions of
identical marks, placing symbols side by side to represent addition of their values.
The number 3,599 required a total of twenty-six symbols. For this reason, the
Sumerians would often use a “subtractive convention” with a little symbol that
meant “take this number away from that number to get the number that is being
indicated”.
In the pre-Sargon era, certain irregularities started to appear in the cuneiform
representations of numbers. In addition to the subtractive convention, entirely new
symbols were being created for multiples of 36,000. This means that instead of
repeating 36,000 however many times it was to be indicated, the numbers 72,000,
108,000, 144,000, 180,000 and 216,000 had their own symbols assigned to them.
In all of human history, the Sumerians are the only ones we know of who
invented and used a sexagesimal system. This can be seen as a “triumph” of their
civilization, and a great mystery as well. Many people have tried to understand
why they did this and numerous hypotheses were offered from Theon of
Alexandria to Otto Neugebauer. These hypotheses range from “It was the easiest
to use” and the “lowest of numbers that had the greatest number of divisors”, to “it
was natural” because the number of days in a Solar year rounded down to 360, and
so on. Daniel Boorstin suggested that the Sumerians used base 60 because they
multiplied the number of planets known to them (5) times the number of months in
the year. It was pointed out by the Assyriologist, G. Kewitsch in 1904 that neither
astronomy nor geometry can really explain the origin of a number system,
presupposing that abstract considerations preceded concrete applications.
Kewitsch speculated that the sexagesimal system actually resulted from the fusion
of two civilisations, one of which used a decimal number-system, and the other
used base 6 derived from a special form of finger-counting. This was not easily
accepted since there is no historical record of a base 6 numbering system
anywhere in the world.
However, duodecimal systems, or base 12 numbering systems ARE widely
attested, especially in Western Europe. It is still used for counting eggs and
oysters. We regularly use the words “dozen” and “gross” and measurements based
on 12 were used in France right up to the Revolution, and are still used in Britain
and the U.S.
The Romans had a unit of weight, money and arithmetic called the as, divided
into 12 ounces. One of the monetary units of pre-Revolutionary France was the
sol, divided into 12 deniers. The Sumerians, Assyrians, and Babylonians used base
12 and its multiples and divisors vary widely as well. The Mesopotamian day was
divided into twelve equal parts, and they divided the circle, the ecliptic, and the
zodiac into twleve equal sectors of 30 degrees. This means that base 12 could very
well have played a major part in shaping the Sumerian number system.
The major role of 10 in the base 60 system is well attested as well, since it was
used as an auxiliary unit to circumvent the main difficulty of the sexagesimal
system. This leads us to an important clue: the Sumerian word for “ten” also
means “fingers” suggesting an earlier counting system.
Taking this back to a variation on Kewitsch’s hypothesis, Georges Ifrah
proposes that base 60 was a “learned solution” to the union between two peoples,
one of which used a decimal system derived from a vigesimal system and the other