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Re: Ariadne's thread
Posted:
Nov 24, 2011 2:29 AM
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Additive number patterns were a powerful tool of
early mathematics. I found ample evidence for their
use in ancient Egypt and Mesopotamia. They allow
to work with integers and provide many values from
which you can choose the one that comes handy
in a given calculation. Number sequences generate
good values from bad and mediocre ones, and
excellent values from mediocre and good ones.
The Babylonian clay tablet YBC 7289 dates to
ca. 1650 BC. The round tablet shows a square
with its diagonals. On a side of the square is
written the number 30, and on a diagonal are
given the numbers 1;24,51,10 and 42;25,35.
The square is supposed to measure 1 by 1 cubit
or 30 by 30 fingers, while the diagonal measures
1;24,51,10 cubits or 42;25,35 fingers. Marvellous
values. How were they found? Consider the first
lines of the number column of the square and octagon:
1 1 2 / 2 3 4 / 5 7 10 / 12 17 24 / 29 41 58 / 70 99 140 /
169 239 338 / 408 577 816 / 985 1393 ... Divide 1393
by 985 and you obtain 1;24,51,10,3,2 ... Let go the
small numbers and keep 1;24,51,10. Multiply this
number by a factor of 30 and you get 42;25,35.
Now let us imagine a circle inscribed in the square
1 by 1 cubit or 30 by 30 fingers. How long is the
circumference? 3;8,29,44 cubits or 1,34;14,52 fingers,
implicit pi value 84823/27000.
4/1 (plus 3/1) 7/2 10/3 13/4 16/5 19/6 22/7 25/8
3/1 (plus 22/7) 25/8 ... 333/106 355/113
333/106 (plus 355/113) ... 84823/27000
A divisor of 27000 fits well into the Babylonian number
system. 84823 divided by 27000 equals 3;8,29,44
(circumference of the circle in cubits). 84823 divided
by 900 equals 1,34;14,52 (circumference in fingers).
Next time: a cosmogram in the city layout of Teotihuacan,
discovered by Suburo Sugiyama
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