Common numbers and the Lalitavistara Sutra

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Common numbers and the Lalitavistara Sutra

Postby PRR » Sun May 19, 2013 2:01 pm

I just came across a passage in the Lalitavistara (http://iteror.org/big/Source/buddhism/L ... -ch12.html) that records a story of the Buddha counting incredibly high numbers. My question is--were these numbers in actual use by other astronomers/mathematicians, or was this something that the Buddha himself (or the author of this story) made up?

The site I linked to suggests that the highest order mentioned, uttaraparamanurajahpravesa, is 10 to the 421st power. Pretty high, when you think that googol is 10 to the 100th power.

Are there ancient passages that actually use these mathematical terms/numerations? Were they numbers that were actually in use (even if only by mathematicians or astronomers)? Or were these terms non-sense terms, like a modern story might use terms like "gazillions" or "oodles"? or.....?
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Re: Common numbers and the Lalitavistara Sutra

Postby cdpatton » Mon May 20, 2013 1:59 am

PRR wrote:I just came across a passage in the Lalitavistara (http://iteror.org/big/Source/buddhism/L ... -ch12.html) that records a story of the Buddha counting incredibly high numbers. My question is--were these numbers in actual use by other astronomers/mathematicians, or was this something that the Buddha himself (or the author of this story) made up?

The site I linked to suggests that the highest order mentioned, uttaraparamanurajahpravesa, is 10 to the 421st power. Pretty high, when you think that googol is 10 to the 100th power.

Are there ancient passages that actually use these mathematical terms/numerations? Were they numbers that were actually in use (even if only by mathematicians or astronomers)? Or were these terms non-sense terms, like a modern story might use terms like "gazillions" or "oodles"? or.....?


I think the extremely large numerals probably vary from one source to another, or may be specific to Buddhist texts, but it is an actual number-naming system. It's fairly standard up to a nayuta in most texts, but after that, it seems murky, arcane, and obscure. The Gandhavyuha Sutra has a similar episode, which goes up to 10^122. It would be nice if anyone can find a good source for decoding such passages. I've seen a few lists gleaned from non-Buddhist sources that are different. Here is a non-Buddhist example: http://veda.wikidot.com/sanskrit-numbers
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Re: Common numbers and the Lalitavistara Sutra

Postby PRR » Mon May 20, 2013 3:05 am

Thanks, cdpaton. Yeah nayuta isn't the highest number, but if khoti is the Sanskirt/Indian crore (which your Sanksrit numeral page suggests), that would still be a pretty high number--100 billion (100 kotis making an ayuta, and a 100 ayutas making a nayuta--calculating a koti as 10,000,000). Unfortunately, as you've suggested, the higher numbers seem to be different. Of course who knows how many centuries and provinces have been crossed, between those two systems of numbering.

Do you have a specific reference (even a book/chapter number) for the Gandhavyuha Sutra?
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Re: Common numbers and the Lalitavistara Sutra

Postby cdpatton » Mon May 20, 2013 3:29 am

PRR wrote:Thanks, cdpaton. Yeah nayuta isn't the highest number, but if khoti is the Sanskirt/Indian crore (which your Sanksrit numeral page suggests), that would still be a pretty high number--100 billion (100 kotis making an ayuta, and a 100 ayutas making a nayuta--calculating a koti as 10,000,000). Unfortunately, as you've suggested, the higher numbers seem to be different. Of course who knows how many centuries and provinces have been crossed, between those two systems of numbering.

Do you have a specific reference (even a book/chapter number) for the Gandhavyuha Sutra?


I'm only familiar with Chinese sources, myself. The Chinese translations of the Gandhavyuha are in the Avatamsaka Sutra, which has that sutra appended to the end of it. Those translations, however, abbreviated the full list of numerals.

The only full translation of the list (in Chinese) is found in Kumarajiva's Mahaprajnaparamita Commentary, which quotes the passage in full - but using obscure transliterations for much of it. It's a list of 115 numerals after an "ayuta" (billion). Lamotte's French translation of that Commentary does not attempt to translate the entire list either - some of it is probably impossible to decipher. I've worked on it some in the past, but haven't figured all of it out yet. It's the reason I've done some research on this topic.

An English translation of Lamotte's passage reads:

"The bodhisattva-mahāsattvas do not produce the thought of supreme perfect enlightenment to save just one man alone (na khalv ekasaṃdhāraṇatayā bodhisattvānāṃ mahāsattvānām anuttarāyāṃ samyaksaṃbodhau cittam utpadyate). Nor to save just two, three, etc., up to ten. Nor to save just 100 (po = śata), 1,000 (ts'ien = sahasra), 10,000 (wan = prabheda), 100,000 (che wan = lakṣha), 1,000,000 (po wan = atilakṣa), 10,000,000 (yi = koṭi), 100,000,000 (che yi = madhya), 1,000,000,000 (po yi = ayuta), 10,000,000,000 (ts'ien yi = mahāyuta), 100,000,000,000 (wan yi = nayuta)... [and so on, each term ten times as great as the preceding, up to the 122nd term of the series 1, 10, 100, 1,000... called anabhilāpya-anabhilāpya and equal to 1 followed by 121 zeros].


(In Cleary's translation of the Avatamsaka Sutra, the passage begins on p.1211 - but there the original Chinese skips over the list as Lamotte does.)
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Re: Common numbers and the Lalitavistara Sutra

Postby Huifeng » Mon May 20, 2013 4:56 am

It's just the Indian math of the time, nothing specifically Buddhist or Buddha about it.
It would be something like claiming that a modern Buddha could recall pi to n decimal places (or the like).

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Re: Common numbers and the Lalitavistara Sutra

Postby PRR » Mon May 20, 2013 5:14 am

Yes, I'm beginning to sense that this was more an Indian thing than a particularly Buddhist thing. The key (for me) was simply that these were real "numbers" that were being referred to--part of teh common parlance of the language of the time, as opposed to terms like "gazillion" or simply terms that were made up by the composer of that sutra.

Thanks to you two for your helpful responses.

Huifeng, you have an interesting blog. I've already downloaded your piece on the historical background to Buddhism. Looks interesting.
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Re: Common numbers and the Lalitavistara Sutra

Postby Huifeng » Mon May 20, 2013 9:10 am

PRR wrote:Huifeng, you have an interesting blog. I've already downloaded your piece on the historical background to Buddhism. Looks interesting.


Thanks. Most of the documents are fairly old though. A lot of old MA essays.

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Re: Common numbers and the Lalitavistara Sutra

Postby Wayfarer » Mon May 20, 2013 10:34 am

Apropos of numbers and Buddhism, I recall reading some years ago that Buddhist mathematicians were responsible to some degree for the invention of the mathematical concept of zero - without which progress in mathematics would have been impossible. This is connected with the Buddhist conception of sunyata. I seem to remember that the symbol for zero came from the hole in the middle seat of a dhow (boat) where the mast is put. Can't remember the exact source - it might have been a New Scientist article.
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Re: Common numbers and the Lalitavistara Sutra

Postby Aemilius » Mon May 20, 2013 1:40 pm

cdpatton wrote:(In Cleary's translation of the Avatamsaka Sutra, the passage begins on p.1211 - but there the original Chinese skips over the list as Lamotte does.)


In my edition of Flower Ornament Scripture the passage is on page 1229 (Thomas Cleary translation), under the heading Indriyeshvara. I have, years ago, put the numbers there in a numerical form, and thus you get much larger figures than merely 10 to the exponent of 122. Thus we have for example: Vibhaja is 10 to the power of 1835 008, Vijangha is 10 to the power of 3670 006, Vishoda is 10 to the power of 7340 032, etc... , Ilana is 10 to the power of 174 400 000, Avana is 10 to the power of 2348 800 000, etc... , Ela is 10 to the power of 10 to the power of 61573 000 000 000, Dumela is 10 to the power of 10 to the power of 123 150 000 000 000, etc... , Khelu is 10 to the power of 10 to the power of 32 282 000 000 000 000 000, Nelu is 10 to the power of 10 to the power of 64 564 000 000 000 000 000, etc..., etc...

There are large figures also earlier in the Sutra. In my edition they are on page 889, Book Thirty, chapter: The Incalculable
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Re: Common numbers and the Lalitavistara Sutra

Postby Aemilius » Mon May 20, 2013 1:56 pm

The large figures are essential for the effect of the mahayana sutras. For example, You have to purchase the old H. Kern translation of the Lotus sutra (Saddharmapundarika or the Lotus of the True Law), because there you have the numerical metaphors in full, unabridged. E.g. the Chapter XIV. Issuing of the Boddhisattvas from the Gaps of Earth. When reading the abridged versions of the Lotus sutra, I feel disappointed, something important is missing. The effect is not the same.
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Re: Common numbers and the Lalitavistara Sutra

Postby cdpatton » Mon May 20, 2013 5:51 pm

Aemilius wrote:
cdpatton wrote:(In Cleary's translation of the Avatamsaka Sutra, the passage begins on p.1211 - but there the original Chinese skips over the list as Lamotte does.)


In my edition of Flower Ornament Scripture the passage is on page 1229 (Thomas Cleary translation), under the heading Indriyeshvara. I have, years ago, put the numbers there in a numerical form, and thus you get much larger figures than merely 10 to the exponent of 122. Thus we have for example: Vibhaja is 10 to the power of 1835 008, Vijangha is 10 to the power of 3670 006, Vishoda is 10 to the power of 7340 032, etc... , Ilana is 10 to the power of 174 400 000, Avana is 10 to the power of 2348 800 000, etc... , Ela is 10 to the power of 10 to the power of 61573 000 000 000, Dumela is 10 to the power of 10 to the power of 123 150 000 000 000, etc... , Khelu is 10 to the power of 10 to the power of 32 282 000 000 000 000 000, Nelu is 10 to the power of 10 to the power of 64 564 000 000 000 000 000, etc..., etc...

There are large figures also earlier in the Sutra. In my edition they are on page 889, Book Thirty, chapter: The Incalculable


In the first place, that wasn't the passage I was referencing. I was reference the one that begins at the bottom of page 1211.

The second place, Cleary's "math" in the Incalculable chapter is his own magic, not something that is in the text. For example, he begins, "Ten to the tenth power times ten to the tenth power equals ten to the twentieth power ..." Siksananda says, "One hundred laksas is a koti." A laksa is 100,000. 100 x 100,000 = 100,000,000 = a koti. Not 100,000,000,000,000,000,000! One is much better off listening to a source such as Lamotte than he is listening to one such as Cleary.
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Re: Common numbers and the Lalitavistara Sutra

Postby PRR » Mon May 20, 2013 6:44 pm

[url]I looked that up. http://www.sacred-texts.com/bud/lotus/lot14.htm[/url]Thanks for the reference. Yeah those large numbers provide a lot of impact, when reading the passage discussing the number of Bhuddas that have arisen.


Aemilius wrote:The large figures are essential for the effect of the mahayana sutras. For example, You have to purchase the old H. Kern translation of the Lotus sutra (Saddharmapundarika or the Lotus of the True Law), because there you have the numerical metaphors in full, unabridged. E.g. the Chapter XIV. Issuing of the Boddhisattvas from the Gaps of Earth. When reading the abridged versions of the Lotus sutra, I feel disappointed, something important is missing. The effect is not the same.
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Re: Common numbers and the Lalitavistara Sutra

Postby Kim O'Hara » Mon May 20, 2013 10:37 pm

PRR wrote:[url]I looked that up. http://www.sacred-texts.com/bud/lotus/lot14.htm[/url]Thanks for the reference. Yeah those large numbers provide a lot of impact, when reading the passage discussing the number of Bhuddas that have arisen.


Aemilius wrote:The large figures are essential for the effect of the mahayana sutras. For example, You have to purchase the old H. Kern translation of the Lotus sutra (Saddharmapundarika or the Lotus of the True Law), because there you have the numerical metaphors in full, unabridged. E.g. the Chapter XIV. Issuing of the Boddhisattvas from the Gaps of Earth. When reading the abridged versions of the Lotus sutra, I feel disappointed, something important is missing. The effect is not the same.

You could say, in fact, that they are used poetically.
:thinking:
So their real numerical value doesn't matter.
:thinking:

:namaste:
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Re: Common numbers and the Lalitavistara Sutra

Postby cdpatton » Tue May 21, 2013 3:47 am

Kim O'Hara wrote:
PRR wrote:[url]I looked that up. http://www.sacred-texts.com/bud/lotus/lot14.htm[/url]Thanks for the reference. Yeah those large numbers provide a lot of impact, when reading the passage discussing the number of Bhuddas that have arisen.


Aemilius wrote:The large figures are essential for the effect of the mahayana sutras. For example, You have to purchase the old H. Kern translation of the Lotus sutra (Saddharmapundarika or the Lotus of the True Law), because there you have the numerical metaphors in full, unabridged. E.g. the Chapter XIV. Issuing of the Boddhisattvas from the Gaps of Earth. When reading the abridged versions of the Lotus sutra, I feel disappointed, something important is missing. The effect is not the same.

You could say, in fact, that they are used poetically.
:thinking:
So their real numerical value doesn't matter.
:thinking:

:namaste:
Kim


Kern doesn't abbreviate because he was translating the Sanskrit - Kumarajiva abbreviates those passages. That's a case of the English translators being faithful to their sources. It's just a little nuts to me how a text like Cleary's translation can be riddled with errors, but only those of us who read the original know. It would have been quite an undertaking to edit such a huge manuscript ... but it's being sold for $100 a copy too! The BTTS version is at least better edited.

And, yes, it is poetry with numbers. And I think there must be a little tongue-in-cheek. I've always thought of these passages as poking fun at math altogether. Infinity doesn't have a "number" but let's try counting to infinity for the mathematician's sake! That is the joke I always hear far off in the distance when I read these passages.

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Re: Common numbers and the Lalitavistara Sutra

Postby Huifeng » Tue May 21, 2013 4:52 am

Kumarajiva seldom abbreviates. Usually, the so-called "original Sanskrit" is actually a lot newer than whatever Kumarajiva was working with. He just has an older version, before all the pericopes get expanded out over the next few centuries.

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Re: Common numbers and the Lalitavistara Sutra

Postby cdpatton » Tue May 21, 2013 6:28 am

Huifeng wrote:Kumarajiva seldom abbreviates. Usually, the so-called "original Sanskrit" is actually a lot newer than whatever Kumarajiva was working with. He just has an older version, before all the pericopes get expanded out over the next few centuries.

~~ Huifeng


Yes - the temporal dimension of these things. I wasn't actually thinking Kumarajiva had himself abbreviated - just that his text was abbreviated. Things did tend to inflate into further and further fully enunciation of lists over time, didn't they?

Now, all of this made me revisit my researches of the TCTL passage I had scratched my head over last year for a few days. I've started a comparison table to ferret all of this out and I noticed the reason Cleary came up with the scheme he did in the Incalculable chapter. (Well, I have no idea how he arrives at the start with 10^10 from a hundred laksas - but - ) The text seems to suggest a squaring of each numeral - for instance is says "a koti-koti is an ayuta". But what is actually happening (assuming the text isn't completely changing what kotis, ayutas, and nayutas are valued at) is that each number is 100 times (10^2) higher than the last. So - those numbers don't get nearly so high as Cleary calculates. There are 124 numerals, so it must end somewhere around 10^255. (Yikes!)

Lamotte seems to have underestimated the list I was talking about a little, since the numbers between 10^8 and 10^13 are mentioned as a single item - he counts 122 numerals, they increase by tens, so says it ends at 10^122. But probably he missed the 10^5 in the intermission and it ought to be 10^127 (er ... something like that ...)

Haha, I am a nerd.

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Re: Common numbers and the Lalitavistara Sutra

Postby Huifeng » Tue May 21, 2013 10:57 am

cdpatton wrote:
Huifeng wrote:Kumarajiva seldom abbreviates. Usually, the so-called "original Sanskrit" is actually a lot newer than whatever Kumarajiva was working with. He just has an older version, before all the pericopes get expanded out over the next few centuries.

~~ Huifeng


Yes - the temporal dimension of these things. I wasn't actually thinking Kumarajiva had himself abbreviated - just that his text was abbreviated. Things did tend to inflate into further and further fully enunciation of lists over time, didn't they?

Charlie.


Yes. So, I'd tend to argue that the later versions expanded, rather than even saying that his text was abbreviated. ;)

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Re: Common numbers and the Lalitavistara Sutra

Postby Aemilius » Tue May 21, 2013 11:49 am

cdpatton wrote:
Aemilius wrote:
cdpatton wrote:(In Cleary's translation of the Avatamsaka Sutra, the passage begins on p.1211 - but there the original Chinese skips over the list as Lamotte does.)


In my edition of Flower Ornament Scripture the passage is on page 1229 (Thomas Cleary translation), under the heading Indriyeshvara. I have, years ago, put the numbers there in a numerical form, and thus you get much larger figures than merely 10 to the exponent of 122. Thus we have for example: Vibhaja is 10 to the power of 1835 008, Vijangha is 10 to the power of 3670 006, Vishoda is 10 to the power of 7340 032, etc... , Ilana is 10 to the power of 174 400 000, Avana is 10 to the power of 2348 800 000, etc... , Ela is 10 to the power of 10 to the power of 61573 000 000 000, Dumela is 10 to the power of 10 to the power of 123 150 000 000 000, etc... , Khelu is 10 to the power of 10 to the power of 32 282 000 000 000 000 000, Nelu is 10 to the power of 10 to the power of 64 564 000 000 000 000 000, etc..., etc...

There are large figures also earlier in the Sutra. In my edition they are on page 889, Book Thirty, chapter: The Incalculable


In the first place, that wasn't the passage I was referencing. I was reference the one that begins at the bottom of page 1211.

The second place, Cleary's "math" in the Incalculable chapter is his own magic, not something that is in the text. For example, he begins, "Ten to the tenth power times ten to the tenth power equals ten to the twentieth power ..." Siksananda says, "One hundred laksas is a koti." A laksa is 100,000. 100 x 100,000 = 100,000,000 = a koti. Not 100,000,000,000,000,000,000! One is much better off listening to a source such as Lamotte than he is listening to one such as Cleary.


There is in Har Dayal's Bodhisattva Doctrine in Buddhist Sanskrit Literature a little chapter about time in the context of the career of a bodhisattva. It is very useful, Har Dayal makes it clear that the mathematical figures were larger in India, than the ones they had afterwards in China. Har Dayal gives in his book different interpretations of the length of Kalpas, Dayal quotes Poussin who gives one interpretation of the length Kalpas that is similar to Thomas Cleary's.
There are problems with the interpretation of ancient mathematics, like theTrichiliocosm, which properly means one thousand to the power of three, i.e. 1 000 000 000. It is explained in the commentaries and in the Abhiddharma, and yet it often gets mistranslated as "three thousand worlds".
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Re: Common numbers and the Lalitavistara Sutra

Postby Rasko » Thu May 23, 2013 5:03 pm

PRR wrote:Are there ancient passages that actually use these mathematical terms/numerations? Were they numbers that were actually in use (even if only by mathematicians or astronomers)? Or were these terms non-sense terms, like a modern story might use terms like "gazillions" or "oodles"? or.....?

Not ancient Indian math, but an interesting parallel from ancient Greece:
"There are some, king Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its magnitude.

And it is clear that they who hold this view, if they imagined a mass made up of sand in other respects as large as the mass of the Earth, including in it all the seas and the hollows of the Earth filled up to a height equal to that of the highest of the mountains, would be many times further still from recognizing that any number could be expressed which exceeded the multitude of the sand so taken.

But I will try to show you by means of geometrical proofs, which you will be able to follow, that, of the numbers named by me and given in the work which I sent to Zeuxippus, some exceed not only the number of the mass of sand equal in magnitude to the Earth filled up in the way described, but also that of the mass equal in magnitude to the universe."

—Archimedes: The Sand Reckoner
http://en.wikipedia.org/wiki/The_Sand_Reckoner
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Re: Common numbers and the Lalitavistara Sutra

Postby cdpatton » Sun May 26, 2013 9:56 pm

Aemilius wrote:
cdpatton wrote:
Aemilius wrote:
In my edition of Flower Ornament Scripture the passage is on page 1229 (Thomas Cleary translation), under the heading Indriyeshvara. I have, years ago, put the numbers there in a numerical form, and thus you get much larger figures than merely 10 to the exponent of 122. Thus we have for example: Vibhaja is 10 to the power of 1835 008, Vijangha is 10 to the power of 3670 006, Vishoda is 10 to the power of 7340 032, etc... , Ilana is 10 to the power of 174 400 000, Avana is 10 to the power of 2348 800 000, etc... , Ela is 10 to the power of 10 to the power of 61573 000 000 000, Dumela is 10 to the power of 10 to the power of 123 150 000 000 000, etc... , Khelu is 10 to the power of 10 to the power of 32 282 000 000 000 000 000, Nelu is 10 to the power of 10 to the power of 64 564 000 000 000 000 000, etc..., etc...

There are large figures also earlier in the Sutra. In my edition they are on page 889, Book Thirty, chapter: The Incalculable


In the first place, that wasn't the passage I was referencing. I was reference the one that begins at the bottom of page 1211.

The second place, Cleary's "math" in the Incalculable chapter is his own magic, not something that is in the text. For example, he begins, "Ten to the tenth power times ten to the tenth power equals ten to the twentieth power ..." Siksananda says, "One hundred laksas is a koti." A laksa is 100,000. 100 x 100,000 = 100,000,000 = a koti. Not 100,000,000,000,000,000,000! One is much better off listening to a source such as Lamotte than he is listening to one such as Cleary.


There is in Har Dayal's Bodhisattva Doctrine in Buddhist Sanskrit Literature a little chapter about time in the context of the career of a bodhisattva. It is very useful, Har Dayal makes it clear that the mathematical figures were larger in India, than the ones they had afterwards in China. Har Dayal gives in his book different interpretations of the length of Kalpas, Dayal quotes Poussin who gives one interpretation of the length Kalpas that is similar to Thomas Cleary's.
There are problems with the interpretation of ancient mathematics, like theTrichiliocosm, which properly means one thousand to the power of three, i.e. 1 000 000 000. It is explained in the commentaries and in the Abhiddharma, and yet it often gets mistranslated as "three thousand worlds".


Cleary is not translating the Chinese accurately in this case, though. "One hundred laksas is a koti" =/= "Ten to the tenth power times ten to the tenth power equals ten to the twentieth power." That's rewriting a text, not translating it. As I noted in the later comment, it is a little confusing in the wording - it seems like squaring of numbers, but the answers are not squares - they are multiples of 100. The passage you had pointed out in the Gandavyuha is actually identical to the Chinese of the Incalculable chapter - except that it is abbreviated by Siksananda - Cleary decided to translate out the entire list of transliterations that he converted to his idea of the numbers in the Incalculable chapter. Anyway, I'll never be done with Cleary - it is the result of retranslating the Siksananda text and using Cleary as a reference. It really is an unedited first draft that really should have gotten edited and republished - much like Yamamoto's Mahaparinirvana Sutra. Not that I fault anyone for not having the time or for publishing a first attempt - but for a major religion to be stuck with these translations and just stop there ... It's a little ridiculous at a certain point.

As for the numbers being bigger in India as opposed to China: Keep in mind that the Chinese translations are much earlier texts compared to the Indian texts that we have overall. Sanskrit texts got bigger and more exaggerated over the centuries - it is documented well with the texts that were translated at different points of their evolution in Chinese. The old prejudice against Chinese sources is backwards if we want to know what early Indian scriptures looked like. Chinese is the only source that exists for that. If later scriptures is what one is interested in, then existing Sanskrit and Tibetan is the way to go.

The "tri-chilio-mega-chiliocosm" (to use a Lamotte-ism) was usually translated to Chinese quite literally ("Triple thousand great thousand world"). They didn't multiply the figure out to a numeral. Otherwise, I'm not sure what you mean in that case. I do remember seeing once, when comparing a Dharmaraksa text vs a later Chinese translation that in Dharmaraksa three thousand worlds turned into a trichiliocosm in the later version - but one must not simply assume something was mistranslated because it was Chinese. The original texts the two translators were working from may well have been different. There's no way to know with any certainty either way. I personally tend to assume the originals were different since those kinds of changes tended to happen over time.

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