> (I wrote:)
>> On this hot summer, which is a cold winter to some, let me go off on a
>> tangent to Luigi Borzacchini's theme.
>>
>> In a message of July 12th, you made, in passing, the following remark
>>
>> "the numerical 'fact' of some incommensurable results (for example side
>> and diagonal of the square) was already known to the Babylonians".
>>
>> Can you elaborate on that? What is the evidence?
>>
>> Avinoam Mann
>>
> It is raining. Even in a Mediterranean July, it may happen,
It's damned hot, unpleasant. Rain in July in Israel is extremely rare.
> I thought to two clay tablets analyzed by Neugebauer in his "The exact
> sciences in Antiquity". I have not the volume now available, but if you
> need I can give you the exact references in the near future.
The discussion starts on p.34 (2nd ed.). On p.33 he refers to the
statements "7 does not divide", "11 does not divide", that you mention below.
> The first tablet gives an approximate value or root(2) near a picture of a
> square with the two diagonal.
> The second tablet gives a series of values which are substantially
> equivalent to Pythagorean triples relative to triangles listed according to
> their (increasing) angles. The last is almost 45 degrees, but the isosceles
> triangle is absent from the list.
> It reminds me other clay-tablets of inverses, where 7 was missing ("it does
> not divide").
>
> Interpretations of Babylonian Mathematics are always risky, but I think it
> is not too hazardous to think that they realized, even if just by trials,
> that in the isosceles triangle the value of the diagonal was not exactly
> computable (in a reasonable time) from the value of the side, roughly as in
> the case of the inverse of 7. Obviously this last "numerical fact" depends
> on the sexagesimal system, whereas the incommensurability is invariant for
> the numerical system, but they could scarcely understand the difference.
>
> Luigi Borzacchini
>
It would be easy for the Babylonian mathematicians to decide that "7 does
not divide" etc. 1/7 has period 3 in the sexagesimal system (6 in the
decimal one), 1/11 has period 5, 1/13 period 4. Thus only few easy
divisions were necessary, and then they realized the computation enters
into an infinite loop. So they gave approximations instead. One wonders
how long they worked on the square root of 2 before giving up. Did the
distinction between repeating and non-repeating positional fractions even
occur to them? If it did not, one can hardly speak about
incommensurability in that connection. Another intriguing question is:
since base 60 was used for scientific calculations, but base 10 for
everyday writing of numbers, e.g. in dates, did they realize that some
fracions, say 1/3, are finite sexagesimal fractions, but that "3 does not
divide" in decimal notation?
Avinoam Mann