The Pali canon includes several words for very large numbers, which are used to describe things such as the length of rebirth in heavens and hells. As so often, I feel there are two schools of interpretation: one is that these are real and literal, these are the exact numbers of calendar years. The other is that they are simply metaphors and ought not be taken too seriously. The problem is that both approaches impose preconceptions on the text and preclude a meaningful inquiry.
It is possible, indeed probable, that the numbers mentioned in the texts mean something, even if we are not exactly sure what that is. But we are not able to decide this, or even think meaningfully about it, so long as we do not know what the numbers actually are.
The Kokālika Sutta (Snp 3.10) offers a unique opportunity to study a set of large numbers. It concerns the corrupt monk Kokālika, who, so the Buddha says, is condemned to hell for calumny. The Sutta does not just give us the length of time he is to spend in hell, it supplies no less than three independent counts of the duration of the paduma (“Pink Lotus”) hell. More, it contains a material simile that can be calculated; and a commentarial passage that gives exact numbers. At the very least, this shows that the tradition was interested in the quantity of the numbers, and that it believed they needed exegesis (neyyattha).
The passages are as follows:
- Prose: Lengths of a series of hells culminating in the paduma on a scale multiplied by twenty.
- Verse a: Length of paduma hell in abbudas and nirabbudas.
- Verse b: Length of paduma hell in nahutas and koṭis.
- Simile: The number of sesame seeds on a cart.
- Commentary: Exact quantity of all the numbers.
Verse b is unique to the Kokālika Sutta, and has no commentary (the commentary in fact says that there is no commentary in the Old Commentary on which it is based). The other passages are found also in AN 10.89 and SN 6.10 and their commentaries. They also have a parallel in DA 30:104 ff., which is similar in most details, but lacks a parallel to verse b.
First a note on the obscure numerical terms.
The koṭi, as its name suggests, was, it seems, originally the final “point” of a series of numbers that were more or less usable and practical. It is rather consistently reckoned as 10 million and may be regarded as the old word for the number known in Hindi as a crore.
The nahuta is presumably the Sanskrit niyuta with an unusual shift from y to h. In the Rig Veda and Śatapatha Brāhmaṇa it is a “team” of horses usually of Vāyu the wind. This seems to have shifted the meaning from a “harnessed team” to “countless”. The related word ayuta appears apparently as “ten thousand” in Rig Veda 4.26.7. Later Sanskrit sources give a variety of values, but ten thousand seems dominant.
Nahuta is used canonically to express the number of people in a large crowd (Kd 1:22.8.2). There, the commentary explains it as 10,000. It also appears at Milinda 5.5.7:3.1, where 10,000 would also seem to fit. We will assume that the nahuta has the value of 10,000 for this essay.
The abbuda is derived from the name of the snake Arbuda crushed beneath Indra’s foot, where he is mentioned along with other foes such as Vritra, Namuci, etc. (Rig Veda 1.51.6, 2.11.20, 2.14.4, 8.3.19, 8.32.3, 8.32.26, 10.67.12). As Arbuda Kādraveya he is, rather curiously, said to have composed Rig Veda 10.94, where he addresses the pressing stones, who “speak as if in hundreds, in thousands”. Might this have something to do with the use of arbuda as a large number? He is also invoked in Śatapatha Brāhmaṇa 13.4.3.9.
Arbuda is also used for an embryo (also in Pali, SN 10.1:3.2), and a tumor such as cancer (metaphorically for a bandit or “pest” at SN 1.77:1.4, DN 30:1.5.5.). The common thread is “a dangerous, slug-like growing entity”. I’m not exactly clear how this was parlayed into a large number. In any case, it and the following numbers in the series may be regarded as “mythic”, rather than practical numbers used for accounting such as the koṭi and nahuta.
The dominant value of abbuda is “ten million”, and while other values exist, this works out for the current text. The commentary supplies its own reckoning of a very large abbuda, and while this seems less plausible to me, it still works out as a number in the correct series.
Nirabbuda is simply an amplification of the abbuda. Usually reckoned as a hundred million, in the prose it is said to be twenty times an abbuda, so there it is 200 million.
For more on these details, there’s a handy source for several systems of reckoning numbers in History of Hindu Mathematics: a source book, Part 1, pp. 9ff, Bibhutibhusan Datta and Avadesh Narayan Singh (1935).
method
There are two main reasons I was dissatisfied with the numbers in this text. First, they did not seem to add up to the same value, even though they are counting the same thing. And second, by many methods of reckoning, depending on the chosen values of the large numbers, you end up with rounding error problems. That is, you have a very large number and a much smaller number that are supposed to be added, but the result is that the smaller number would just be rounded away. You end up with numbers like 2.000725 × 10⁴⁸. I feel like this is bad maths, and the ancient texts would not express themselves like this. So I was looking for results that would make a reasonable number.
I reached my conclusions using a somewhat scattershot method. Without making any assumptions, I simply plugged the various options in to the various passages and calculated what came out. Only after a lot of legwork did the patterns become clear: the three methods were all aiming at the same destination (in a way).
Compare it to making your way through a jungle. You have three starting points, and you’re told that there is a single exit point. Between them there are many ways you can trace your path through the jungle. How are you to know if you’re on the right track? The only way, really, is to follow every lead and see where they end up. When you find three paths that all end up at the same place, they must be the right paths.
This is complicated by the fact that the final numbers are not literally the same. Their identity is numerological rather than mathematical. That means to say, they are not exactly the same numbers, but they are related numbers with the same meaning.
The key number is 512. This expresses a numerological meaning. Five hundred is well known in Pali as expressing a “large number”, for example of people in a crowd. Twelve as a duration of time is obviously the twelve months of the year. Together they express the meaning, “a large duration of time” or “many years”.
In one instance, we find instead the number thirty-six. This is an amplified version of twelve, expressing “twelve months past, present, future”. Thus while 12 and 36 are different mathematically, they are the same numerologically.
It is perhaps relevant to note that 36 is a conventional number for a period of time, as shown by AN 7.42 and AN 7.43. There we find a discussion of whether a bhikkhu “graduates” after a certain number of years (as in the Vedic system). The Buddha denies this, and says they graduate when they are trained, no matter if it takes 12, 24, 36, or 48 years. The numbers ascend in multiples of 12, echoing various Brahmanical standards for studentship. 36 is also the number of the “streams of craving”, i.e. the internal and external senses in past, present, and future (Dhp 339:2). Finally, in Tha Ap 69:7.3 we have the ordinal locative chattiṁsatimhi, “in the 36th (aeon)”.
These basic numbers are multiplied by powers of ten. The value of these powers are different in each case, but the basic sense is the same, to express an expansion of the basic number. So each conveys the sense, “a super-long duration of time”.
We see the same principle in the relation between the sesame simile and the series of hells. As we shall see, the numbers here do not match up. But each involves the number twenty multiplied by ten or a hundred. Here the twenty is simply a reinforced ten.
I won’t bore you with all the many byways and diversions that the numbers offer. Instead, I’ll present my conclusions, with a few examples of alternatives.
the prose
The prose passage presents us with a straightforward method of getting from the abbuda to the paduma (Pink Lotus) using a series of powers of twenty. Assuming the abbuda of 10 million, we have:
| prose, abbuda = koṭi | Scale of twenty |
|---|---|
| abbuda | 10,000,000 |
| nirabbuda | 200,000,000 |
| paduma | 5.12×10¹⁸ |
The commentary gives a method of calculating, which you can see in Ven Bodhi’s translation of the Sutta Nipāta commentary. I won’t spell it out here, but it results in a very large abbuda. I feel this is too far outside the normal reckoning of the abbuda, but in any case it results in a factor of 512.
| prose, commentarial method | Scale of twenty |
|---|---|
| abbuda | 10⁴² |
| nirabbuda | 2 × 10⁴³ |
| paduma | 5.12 × 10⁵³ |
verse b
This verse is only found in the Sutta Nipāta version, and it is probably late since it has no commentary. Nonetheless, we will tackle it first as it is fairly straightforward.
Nahutāni hi koṭiyo pañca bhavanti,
Dvādasa koṭisatāni punaññā.
For there are five nahutas of koṭis,
plus another twelve hundred koṭis.
We can use this as a handy example of the rounding error problem. Here are the results obtained by plugging in various values of the nahuta.
| verse b | paduma |
|---|---|
| nahuta = 10²⁸ | 5 × 10³⁵ |
| nahuta = 10,000 | 5.12 × 10¹¹ |
| nahuta = 100,000 | 5.012 × 10¹² |
| nahuta = 1,000,000 | 5.0012 × 10¹³ |
As you can see, only the value of 10,000 has a coherent result, and that result is 512. This shows that we’re on the right track, as we now have two paths to the same destination, differing only in the power of ten.
We can now properly translate verse b:
Nahutāni hi koṭiyo pañca bhavanti,
Dvādasa koṭisatāni punaññā.
For there are five times ten thousand, times ten million years,
plus another twelve hundred times ten million.
verse a
Verse a is tricky to parse out and to translate. The cases are straightforward enough: sataṁ, chattiṁsati, and pañca are nominative singular; sahassānaṁ nirabbudānaṁ is genitive plural; abbudāni is nominative plural. The problem is how the numbers are supposed to relate to one another.
The commentary, followed by translators such as Bodhi and Norman, connects the first line with the number 36 in the second. For example, Bodhi has:
Sataṁ sahassānaṁ nirabbudānaṁ,
chattiṁsati pañca ca abbudāni
For a hundred thousand nirabbudas
and thirty-six more, and five abbudas
Thus gives a total number of nirabbudas as 100,036, which again is a strange number.
Looking further afield, we find a similar passage on the lifespan of Baka Brahma (SN 6.4:5.3):
Sataṁ sahassānaṁ nirabbudānaṁ,
Āyuṁ pajānāmi tavāhaṁ brahme”ti.
A hundred of thousands of nirabbudas
is how I understand your lifespan, Brahmā.
Here it is clear that the first line stands alone as a number. Note that this is in the Sagāthāvagga, which is where a shorter (and perhaps earlier) version of the Kokālika Sutta is also found. We find other similar constructions there too:
SN 1.32:14.3: Sataṁ sahassānaṁ sahassayāginaṁ
SN 5.5:4.1: Sataṁ sahassānipi dhuttakānaṁ
Thus we have four lines starting with “hundred thousands” in the Sagāthāvagga collection, and in three of those instances the line definitely stands on its own.
We should probably treat our current passage the same way, and read each line as a distinct number. Thus we shall have to construe chattiṁsati pañca ca abbudāni as a coherent phrase. It does seem like an odd construction, and we might wonder why this form was chosen to express a number.
Vanarata suggests that the “thousands” of the first line should be distributed to the second line as well (Bodhi, Numerical Discourses, note 626, p. 1678). This kind of distribution of terms is common in mathematics; for an example in Pali verse, see SN 6.6:5.1. Here, however, it provides a crucial key to interpretation and translation.
But we need to go further: the singular “hundred” in the first line is distributed to the singulars 36 and 5 in the second, while the plural “thousands” is distributed to the plural abbudāni (“tens of millions”).
Thus we read the distributed line here as chattiṁsati pañcasataṁ ca (five hundred plus thirty-six) sahassānaṁ abbudāni (“of thousands, tens of millions”). These juxtaposed numbers are multiplied.
The duration of the Pink Lotus then works out as (100,000 × 100,000,000) × (536 × 1000 × 10,000,000) = 5.36 × 10²⁵ years.
Note that this does not work out if we use the scale of twenty for the nirabbuda, as we did in the prose, as this would result in 1.072 × 10²⁶.
We may now translate verse a:
Sataṁ sahassānaṁ nirabbudānaṁ,
chattiṁsati pañca ca abbudāni
A hundred thousand times a hundred million years,
times five hundred and thirty-six, times a thousand, times ten million
how many sesames fit on a cart?
We have so far been using the base numbers, especially the koṭi, as defined in the commentary, which is then multiplied to yield the abbuda. However the Sutta reckons the abbuda not from the koṭi but by a simile.
Suppose there was a Kosalan cartload of twenty bushels of sesame seed. And at the end of every hundred years someone would remove a single seed from it. By this means the Kosalan cartload of twenty bushels of sesame seed would run out faster than a single lifetime in the Abbuda hell.
Perhaps this is merely illustrative of a “very large number”. But unless we can actually calculate it, we cannot say that it is not a genuine number. Fortunately, it seems we can, in fact, work it out with some confidence.
A sesame seed weighs about 0.002 grams (“Some Physical Properties of Sesame Seed”, T.Y. Tunde-Akintunde and B.O. Akintund).
The “pack” (khāri, originally “donkey-load”) of the Arthaśāstra is about 150 kg (Olivelle King, Governance, and Law, appendix 2, pg. 458). Twenty packs would be 3 tons, which is reasonable for a “load” (vāha) drawn by a heavy bullock cart (Arthaśāsatra 2.19.33 reckons the vaha similarly at about 2 tons). This results in 1.5 billion seeds and 150 billion years.
On the other hand, the Pali khāri is carried by a single person and must be closer to 20 kg (sn3.11:2.1, sn7.9:14.1, dn3:2.4.7, ud6.2:2.1). That would make a “Kosalan load” of 400 kg, which is reasonable for a load drawn by a single horse. This results in 250 million seeds and 25 billion years. This number is interesting because it is similar to the modern cosmological estimate of the life of the Universe.
Neither of these results, however, agrees very well with the other calculations in this Sutta.
It seems that the correspondence is not in the result but the simple numbers: twenty packs each hundred years correspond to the twenty times ten of the hells.
conclusion
Our final numbers for the duration of rebirth in the Pink Lotus hell are:
| passage | assumptions | paduma |
|---|---|---|
| prose | abbuda = koṭi (× 20 scale) | 5.12 × 10¹⁸ |
| verse a | abbuda = koṭi (× 10 scale), distributed and multiplied | 5.36 × 10²⁵ |
| verse b | nahuta = 10,000 | 5.12 × 10¹¹ |
We’ve tamed the apparently random and indecipherable numbers into something that, if not mathematically exact, follows a simple and culturally appropriate pattern:
- Powers of ten don’t matter too much.
- Twelve can be amplified to thirty-six.
- Use standard koṭi = abbuda (but commentarial method also works here).
- Use scale of twenties in prose, scale of tens in verse for nirabbuda.
But there’s more! What is the relationship between these figures? Turns out, it’s a very simple one. Each is derived by a multiple of ten million, namely a koṭi/abbuda. The use of this as a multiple of very large numbers is quite characteristic, as Indian numbers prefer the sequence 100,000 (lak) to 10,000,000 (crore) rather than the Western “million”.
- verse b paduma (5.12 × 10¹¹) times koṭi/abbuda (10⁷) = prose paduma (5.12 × 10¹⁸)
- prose paduma (5.12 × 10¹⁸) times koṭi/abbuda (10⁷) = verse a paduma (5.36 × 10²⁵)
The Sutta relies throughout on the symbolic number of 512. It can be multiplied or strengthened by factors of ten, two or three, but it keeps the internal coherence. In general numerological terms, these numbers express a “super-expansive duration of time”.
. Still have questions…