
Re: Ariadne's thread
Posted:
Nov 24, 2011 2:29 AM


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

