Re: Babylonian root 2 (was: Waradpande seems to have destroyed PIE already)

On Dec 18, 8:01 am, "Richard Wordingham" <jrw0...@xxxxxxxxxxx> wrote:
"Franz Gnaedinger" <f...@xxxxxxxxxxx> wrote:
On Dec 16, 4:34 pm, "Richard Wordingham" <jrw0...@xxxxxxxxxxx> wrote:
As i said before, the difficult part of this is getting 8,50 out of 25
divided by 1,25 divided by 2.

It occurred to me that dividing by 1,25 is actually dividing by the estimate
of the square root of 2, so perhaps one should consider the trick of not
dividing by it but multiplying by it and then halving. (We're calculating a
correction term, so in control engineering parlance it's only a gain.)
However, that gives 8,51 (or even 8,52 if you round up at each halving),
giving *1;24,51,9, quod non erat demonstrandum. I also considered replacing
1;25 by a nearby regular number, but even considering 5-digit numbers (i.e.
4 fractional digits) the best I could come up with were 1;24,32,30 (=45/32)
and 1;25,20 (=64/45), which both give 1;24,51,15 as the next estimate.
(Exercise for reader: Derive this next estimate by mental arithmetic :-)

(I had found these two by hand before I despaired and threw a computer at
the problem.)

(Thanks for being honest)

Note that the 'Babylonian method' a.k.a. Heron's method gives the ratio
577/408 if the calculations are done exactly. That comes out as
1;24,51,10,35, compared to the quoted value of 1;24,51,10 and the actual
value of 1;24,51,10,8 (all values rounded to the number of places shown).
This suggests that the final stage may have been check and tweak, though,
depending how one rounds, using sexagesimals may give you 1;24,51,10 rather
than 1;24,51,11. The latter fact suggests the use of sexagesimals rather
than vulgar fractions.

The crucial step is to form what I call mirror values.
First formula:

If the side of a square measures 5, 10, 15 ... paces,
the diagonal measures 7, 14, 21 ... paces respectively

Mirror values:

and if the side measures 7, 14, 21 ... paces, the diagonal
measures 10, 20, 30 ... paces respectively

The improved value is 5+7 = 12 for the side, and 7+10 = 17
for the diagonal, mirror value 17 for the side, 24 for the
diagonal.

Mirror values are easily found with integers, but tricky in
Babylonian sexagesimals. 7/5 = 1;24, mirror value 10/7
= 1;25,42,51,25,42,51... Improved value 17/12 = 1;24,
mirror value 24/17 = 1;24,42,21,10...

Begin with integers, reconstruct simple additive algorithms
involving integers only (my aim in experimental math-history),
and when you master them, really master them, you may
proceed to fractions.

The lines 5 7 10 and 12 17 24 are enough to establish
the basic number column for calculating the square and
octagon. Any line: a b 2a, next line: a+b b+2a 2(a+b).
First lines 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 1970 /
2378 3363 4756 / 5741 8119 11482 / 13860 19601 ...

Now try with fractions and false additions, beginning with
the numbers 5 7 10. The ratios of side and diagonal are
7/5 and 10/7. The false addition gives 7+10 / 5+7 = 17/12,
mirror value 24/17, false addition 17+24 / 12+17 = 41/29,
mirror value 58/41, false addition 41+58 / 29+41 = 99/70,
mirror value 140/99, and so on. You get the same numbers
as in the case of the number column.

Then you may try with correct additions. Add a value and
its mirror value, and halve the sum. Begin again with 5 7 10
and the fractions 7/5 and 10/7. Their correct sum is 99/35,
and half the sum is 99/70, mirror value 140/99, new sum
19601 / 6930, half the sum 19601 / 13860 ... The resulting
numbers are again part of the number column, but now you
advance much quicker, in ever bigger steps and leaps.

And if you really know how to work with fractions, you
can proceed to continued fractions: fold up the above
number column into a continued fraction, or unfold the
continued fraction 1;2,2,2,2... - it will provide the above
numbers.
.

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