Pointers to Katapayadi formula needed

[thayashi at doshisha.ac.jp]

At  5:26 AM 95.6.27 +0100, Krishna Kunchithapadam wrote:
>Anand Venkt Raman writes:
>:
>: [ questions about the katapayadi sankhya deleted ]
>:
>
>
>I too have a question regarding the sankhya.  I have come across
>a sloka which encodes the digits of the the decimal expansion of
>pi (correct to about 20+ decimal places) based on the katapayadi
>sankhya.
>
>I would like to know the source of the sloka.  Those readers with
>Web access can check out:
>
>    http://www.cs.wisc.edu/~krisna/pi.html
>
>[The reason I am giving a Web address is that the page has the
>sloka in devanagari---which might avoid some of the problems in
>transliteration]
>
>For those without Web access, the sloka approximately
>transliterates as:
>
>    gopi bhaagya madhu vrata, srngisho dadhisandhiga |
>    khalajivita khatava, galahata rasandhara ||
>
>I appreciate a pointer to the source or any other relevant
>information.  Thank you.
>
>--Krishna
> 

A few comments about the sloka mentioned by Krishna (I cannot find its source).

(1) The sloka gives the 32 digits,
3141592653589793238462643386(3?)2792(5?), in the descending order, that is,
from the highest (g=3) to the lowest (r=2) decimal places, while the
original katapayadi system of numeration would start from the lowest.  
(2) The system of the sloka seems to be the same as `a variation' mentioned
by Datta and Singh (History of Hindu Mathematics, single volume edition,
Bombay, etc.: Asia Publishing House 1962, Part 1, p. 72), according to whom
it `is knwon as the Kerala System', but no reference is given.
(3) Aryabhata II (usually assigned to the 9th to the 11th centuries, but to
ca. 1500 according to R. Billard) certainly read numbers in the descending
order of places (see his Mahaasiddhaanta), but in his system every
consonant has a numerical value (in the original katapayadi system, only
the last of a consonants cluster has a numerical value), and therefore
`gopibhaagya' would express 31431 instead of 3141.  
(4) As far as I know, the traditional Indian mathematics did not extend
their decimal place-value system to the fractional part of a number (in
astronomy they used sexagesimal place-value notation for that purpose, but
always with the names of units such as kalaa, vikalaa, liptaa, viliptaa,
etc.).  So it looks strange to me (from the view point of ganita) that the
sloka does not contain the 31st power of ten as a divisor.  See, for
example, a stanza ascribed to the great mathematician Madhava of
Sangamagrama (ca 1400):
vibudhanetragajaahihutaazanatriguNavedabhavaaraNabaahavaH /
navanikharvamite vRtivistaare paridhimaanam idaM jagadur budhaaH //
(cited by Sankara in his comm. on Bhaskara's Lilavati, VVRI edition, p. 377)
This says that the circumference of a circle is 2827433388233 (expressed in
the so-called bhutasamkhya system) when its diameter is measured by `nine
nikharva' (9 times the 11th power of ten).

Takao Hayashi
Science and Engineering Research Institute,
Doshisha University,
Kyoto, Japan.