Lalitavistara Sutra - the mathematical test

Story of how the young Buddha passes his maths test

King Suddhodana then asked the Bodhisattva: “Can you, my son, rival the skill of the great mathematician Arjuna in the knowledge of mathematics?”
“Sire, I can,” he replied. So the Bodhisattva was told to show his ability.

§1. The Bodhisattva knows up to ten numerations

The great mathematician Arjuna asked the Bodhisattva: “Young man, do you know the procedure of numeration called kotisatottara, more than a hundred kotis?”
The Bodhisattva answered: “I do.” [usually a koti is ten million or 10^7]
“Well then, how must one proceed to enumerate more than a hundred kotis?”

The Bodhisattva replied: “A hundred kotis is called ayuta; a hundred ayutas is called niyuta; a hundred niyutas called kankara; a hundred kankaras is called vivara; and a hundred vivaras is called aksobhya; a hundred aksobhyas is called vivaha; a hundred vivahas is called utsanga; a hundred utsangas is called bahula; a hundred bahulas is called nagabala; a hundred nagabalas is called titila; a hundred titilas is called vyavasthanaprajnapti; a hundred vyavasthanaprajnaptis is called hetuhila; and a hundred hetuhilas is called karahu; a hundred karahus is called hetvindriya; a hundred hetvindriyas is a samaptalambha; a hundred samaptalambhas is known as gananagati; a hundred gananagatis is called niravaravadya; a hundred niravaravadyas is called mudrabala; a hundred mudrabalas is called sarvabala; and a hundred sarvabalas is called visamjnagati; a hundred visamjnagatis is a sarvasamjna; a hundred sarvasamjnas is a vibhutangama; and a hundred vibhutangamas is called tallaksana. [then 100^23 kotis would mark 10^53]

“Now with the numeration called tallaksana one could take even Meru, the king of mountains, as a subject of calculation and measure it. And next is the numeration called dvajagravati; with the help of this numeration one could take all the sands of the river Ganges as a subject of calculation and measure them.
Above this is the numeration called dvajagranisamani; and above this is the numeration of vahanaprajnapti; next comes the numeration called inga; above this is the numeration of kuruta.
Again above this is the numeration called sarvaniksepa, with the help of which one could take the sands of ten Ganges rivers as a subject for calculation and measure them all.
And again above this is the numeration called agrasara, with the help of which one could take the sands of a hundred kotis of Ganges rivers as a subject of calculation measure them all. [these sands of the Gangeses would be too few for their respective numerations]

“And again above this is the highest numeration called uttaraparamanurajahpravesa, which is said to penetrate the most subtle atoms. Except for a Tathagata, or a Bodhisattva who has reached the purest essence of Enlightenment, or a Bodhisattva who has been initiated into all the Dharma, there is no being who knows this numeration, except myself or a Bodhisattva like me, who has arrived at his last existence, but has not yet left home.”

§2. Counting the atoms in a yojana and the Earth's mass

Arjuna said: “Young man, how must one proceed in the numeration which penetrates the dust of the most subtle atoms?” [counting back this list of lengths an atom would measure between 1 and 1000 picometer, in reality its diameter is 60 to 600 pm]

The Bodhisattva said: “Seven subtle atoms make a fine particle; seven fine particles make a small particle; seven small particles make a particle called vatayanaraja; and seven particles of vatayanaraja make a particle called sasaraja; seven particles of sasaraja make a particle called edakaraja; seven particles of edakaraja make a particle of goraja; seven particles of goraja make a liksaraja; seven liksaraja make a sarsapa; seven sarsapas make an adyava; seven adyavas make an anguli; twelve anguli make a parva; two parva make a hasta; four hastas make a dhanu; a thousand dhanu make a krosa of the country of Magadha; four krosas make a yojana. [a yojana measures a day's march of a royal army in distance, here covering about 108*10^12 atoms]
And now who among you knows the mass of one yojana, and how many of these subtle atoms it contains?”

Arjuna said: “I myself am even more astonished than others of lesser knowledge. Let the young prince show us the mass of a yojana, and explain how many subtle particles are found in it.”

The Bodhisattva replied: “In the mass of a yojana there are a complete niyuta of aksobhyas plus thirty hundred thousand of niyutas of kotis plus sixty hundreds of kotis plus thirty-two kotis and five times a hundred thousand and twelve thousand. [a 'mass' of 10003000000000000060320512000 atoms?] Such is the calculation of subtle particles in the mass of a yojana.
By this procedure, there are here in the land of Jambu seven thousand yojanas; in the land of Aparagodana, eight thousand yojanas; in the land Purvavideha, nine thousand yojanas; in the land of Uttarakuru, ten thousand yojanas. [Earth 34000 yojanas ~ måss 3.4E32 atoms]

§3. Three thousand great thousandfold world in essence incalculable

“Continuing with this method, beginning with the worlds composed of four continents, there are a hundred kotis of worlds with four continents and a hundred kotis of great oceans; there are the hundred kotis of Cakravalas and of Mahacakravalas; the hundred kotis of Sumerus, kings of mountains; the hundred kotis of realms of the Four Great Kings; the hundred kotis of realms of the Thirty-three gods; the hundred kotis of realms of the Yama gods; the hundred kotis of Tusita realms; the hundred kotis of Nirmanarata realms; and the hundred kotis of Parinirmita vasavartin realms. There are the hundred kotis of Brahma realms; the hundred kotis of Brahmapurohita realms; the hundred kotis of Brahmaparsadya realms; the hundred kotis of Mahabrahma realms; the hundred kotis of Parittabha realms; the hundred kotis of Apramanabha realms; the hundred kotis of Abhasvarana realms; the hundred kotis of Parittasubha realms; the hundred kotis of Apramanasubha realms; the hundred kotis of Subhakrtsna realms; the hundred kotis of Anabhraka realms; the hundred kotis of Punyaprasava realms; the hundred kotis of Brhatphala realms; the hundred kotis of Asangisattva realms; the hundred kotis of Abrha realms; the hundred kotis of Atapa realms; the hundred kotis of Sudrsa realms; the hundred kotis of Sudarsana realms; and the hundred kotis of Akanistha realms. [3000 kotis of worlds in total, maybe 3*10^10, see commentary box]

“All together these are said to be the whole of the three thousand great thousands of worlds, spread out and developed. All the calculations of the essence of the yojana includes the many hundreds of yojanas of subtle particles in this mass of three thousand great thousands of worlds, the many thousands of yojanas, the many kotis of yojanas, and the many niyutas of yojanas.
And how many subtle particles are there? It passes beyond calculation, it is incalculable. There are an incalculable number of subtle atoms in the mass of the three thousand great thousands of worlds.”

§4. Admiration of this mathematical lesson

While this lesson on enumeration was being taught by the Bodhisattva, the great mathematician Arjuna and the multitude of Sakyas listened with pleasure, joy, and happiness. Everyone there was filled with great admiration, and each of them presented the Bodhisattva with garments and ornaments. The great mathematician Arjuna then uttered these two verses:

“The hundreds of kotis and the ayutas,
the nayutas and the niyutas,
the procession of the kankaras, the vivahas,
and the aksobhyas as well:
this supreme knowledge I do not have – he is above me.
One with such knowledge of numbers is incomparable!

“And doubtless, O Sakyas, he could calculate
the dust of the three thousand worlds,
as well as all the herbs, the woods, the medicinal plants,
and even the drops of water,
in the time it takes to say ‘Hum’.
How could these five hundred Sakyas
do anything more wonderful?”

Then gods and men by the hundreds of thousands uttered cries of admiration and joy.
And the devaputras in the expanse of the sky recited this verse:

“The concepts and the ideas,
the reasonings good or bad, small or great,
the workings of the minds of all the beings
of the three times: all this he knows perfectly
through a single movement of his mind.”

Thus, O monks, the Bodhisattva distinguished himself by his superiority over all the other young Sakyas. And as they continued their contests – in jumping, in swimming, in running and all the rest – the Bodhisattva again and again demonstrated his superiority...

“Does it all add up in the sutras?”
by Asamkhyeya dasa

A commentary on the main points in the Lalitavistara Sutra's mathematical test of the young Buddha.

About the initial koti. It is doubtful if this number has the conventional value of 1E7, when we take into account the traditional number of worlds in a three thousand great thousandfold world which is 1000^3 = 1E9. Here the list of worlds goes beyond the earthly and heavenly realms of desire (places of rebirth) and also covers the two spheres of form (or creation) and formlessness (or extinction), to add up to 3000 kotis of worlds. This only makes sense if we put the koti value at 1E6.
If so the first numeration from a hundred kotis at 1E8 ends with a tallakshana at 1E52, a factor ten less than usual. Now the tallakshana may be the first value of the next enumeration (also called tallakshana) or the next numeration may start at a value after that. It's not clear if the multiplicative steps again should be 100 and if the number of steps stays equal in all numerations, namely 22 or 23. The upcoming comparisons with the sands of the Ganges seem to suggest these parameters become much smaller, but this can be brushed aside to keep the argument going at a reasonable pace. Most probably the Sutra writers didn't have a clue what they were doing, but their approach reminds us of the Big number algorithm of Archimedes and may be a flawed attempt to transplant his system to Indian buddhist soil.
Suppose the next 8 numerations starting from 1E52 all have 22 multiplicative steps of 100, then the last number expressed, the uttaraparamânurajahpraveša, is 10^(8+44*9) = 10^404 or 100^202 or 10000^101 in which form it would have had great appeal to the Indian eye. Suddenly we have a myriad times a myriad and this repeated a hundred times, with a final myriad added on top, beautiful! The right point to change over from mere mathematical numbers to the physical world – because curiously in the upcoming numeration of distances, this last number is made to express the smallest length of an atom. It is as if we were dealing with quantum-information that fills the vacuum here.
The 10th numeration of lengths goes from types of particles, to types of dust stirred up by animals of increasing size, to plant seeds, to human bodily measures, a mile, and then ending with the definition of a joyana, the measure of distance most common in buddhist sutras. The joyana is a length of about 108*1E12 of atoms in total, 108470495616000 atoms to be precise, which can be traced back to a mathematical number (sic!) of about 1E418.
The intention in the Lalitavistara was obviously to construct ever greater systems of numbers, building one on top of the other. That's why we can't make sense of the transition to measures of mass, where these have no connection with those of lengths. Specially because the value of what is called the 'mass' of a joyana – with a koti of a million this måss equals 100030000000000006032512000 atoms – cannot even cover the thinnest layer of atoms in the area of a square joyana (if a koti is 1E7 it nearly does). But maybe we cannot believe the exact expression as given above, which certainly doesn't look like a serious number. Perhaps a few additional digits have been dropped during the translation or copying, perhaps the original writers were just fools trying to impress, or perhaps it was meant as a riddle. For now we faithfully trace this måss back to a number of about 1E430.
The lands in the South, West, East and North are measured at 34000 yojanas, to form an earth world of four continents, or more generally the unit world. Such a world's måss again can be traced back to 3.4E30 atoms and 3.4E434 in pure number, which is what we have at the start of the enumeration of worlds that follows. There 100 kotis of earth worlds (with a koti of 10^6) form a måss of about 3E38 atoms or 3E442 in pure number. In all 30 types of realms are listed, each with a 100 kotis of worlds, constituting the three thousand great thousands of worlds. This seems to suggest their total can be traced back to a måss of 1E40 atoms or 1E444 in pure number, but there's a problem…
The problem is that about a third of the realms listed are of the formless type (meditative heavens, without material substance) and about a third are of the type of formation (preceding the samsâra worlds of rebirth). Even in our universe of desire there are the heavenly planets of the devas, not likely composed of the earth-like particles described above as rabbit dust (šašaraja) and such. If these higher realms do consist of atoms, their type and number is bound to disagree with that of the lowly realm of earth. This being common buddhist knowledge it is mistake to think we can derive with certainty any total number from this list of worlds on top of its first entry: the earth worlds.

We think the coming passage in the Lotus Sutra addresses just the problem left by the improbable definition of the three thousand great thousands of worlds as a måss of atoms. If this is the case, the Lotus Sutra can be dated after the Lalitavistara Sutra.
The issue here is the early evolution of the concept of a three thousand great thousandfold world, culminating centuries later in its subdivision in the Abhidharma Koša, a commentary on buddhist doctrine attributed to Vasubhandu (5th century AD). There a 1000 worlds form a small chiliocosm, there's a wall, 1000 small chiliocosms form a middle chiliocosm, then again a wall, and this 1000 times repeated to constitute the large chiliocosm. Apart from the convention to allow a total of a billion worlds, the distance between two worlds is settled there at 1203450 joyana (about a light minute), a minor travesty.

The Parable of the Magic City chapter in the Lotus Sutra starts with the Buddha explaining the time elapsed after the death of a former Buddha (called Universal Surpassing Wisdom). The argument is metaphorical, and builds further upon concepts we've met in the Lalitavistara above: the three thousand great thousandfold world, the particles of dust, the country to the East, the reference to mathematicians.
Precision is not what is aimed at here, the new quantities are just hinted at for their power to dazzle the mind, but actually the numbers do not become larger than in the Lalitavistara, as calculated within brackets. Important in this passage is that only the earth element is rightly qualified to be ground to ink.

Given the data from the Lalitavistara Sutra we solve the mathematical riddles in the Lotus Sutra as follows.

Assume the koti value is 1E7 (the usual ten million) throughout, as we aim for maximal estimates.
When there are 100 kotis of earth worlds in a three thousand great thousandfold world, the Buddha can have at most 1E9 (a billion) worlds filled with earth.
A yojana counts a length of L_y~1.08E14 atoms, and a måss of M_y~1E28 atoms.
To translate length L to måss M a possible formula is M~L^1.995 (in buddhism so it seems).
A particle of vâtâyanaraja (windy road dust), smallest dust in the list, has a diameter L_v=7^3 times the size of an atom. Then road dust can have a måss of M_v~7^6 times an atom's måss.
The måss of a world with four continents counts 34000 yojana måsses or M_w~3.4E32 atoms. Then we have M_w/M_v~3E27 particles of road dust on earth, and d~3E36 in a billion worlds: the ink drops.
The big land in the East called Pûrvavideha has a måss of 9000 yojanas, or M_p~9E31 atoms. Then there can be M_p/M_v~8E26 particles of road dust in such a land.
Because d*1000 such lands are reduced to dust this translates to k~2.35E66 kalpas at best.

And so a challenge set to mathematicians two millennia ago has finally been answered. Buddhists may be glib talkers when it comes to numbers, but we do know the end and bounds of all their lands!
Furthermore, what strikes us in this passage from the Lotus Sutra is that a higher type of recursion – a repetition of numerations as we find it rising to a power tower in the Avatamsaka Sutra – seems to hide just around the corner, waiting to be discovered.

Den Haag, 6 april 2011