# Mathematical Background Of The Karma Theory In The Gomma¢as†ra As Exposed In The JTP And SJC Commentaries

A few research and exposition articles have appeared on the mathematics of the Gomma†asāra and its commentaries, the JTP and The SJC.1 It may be noted that out of print edition of the Gomma†asāra [GJK (o) and GKK (o)] were published in Calcutta round about 1919, along with JTP (Sanskrit), MPB and SJC commentaries, by Gandhi Hari Bhai Devakarana, and edited by Pundits G.L. Jain and S.L. Jain Now we have at our disposal, the Gommatasara in four volumes published by Bharatiya Jnana Pitha, New Delhi (1978-1981). These contain JTP (Kar¿å²avâtti), JTP (Sanskrit) and a Hindi commentary on the lives of ¢oðaramala’s SJC without his symbolic ArthaSa´dâ¼²i-Chapters which explain various results and symbolism of various important topics, serving as guide live for research workers. ¢oðaramala’s ArthaSa´dâ¼²i Chapters of the Gomma†asāra have been given in English as Predude to Gauge-Symbolism of the Labdhisåra (PGL) by me, separately in this research project on the Labdhisåra of Nemicandra Siddhånta cakravartî, so that the work Labdhisåra may be understood easily.

Summation Formulae

Denomination working Symbol

middle term (madhyamadhana)

m

or

initial sum (adi-dhana) na

common-difference sum

(caya dhana)

post-sum (uttra-dhana)

Formulas:

1. Sum (sarva-dhana) is equal to the sum of the initial sum (ådi-dhana) and post-sum (uttara-dhana) :

S = na + .

2. Sum is half the total of first term (ådi) and as term (antadhana) as multiplied by the number of terms (pada)

S = n ()

3. Number of terms (gaccha or pada) is equal to the ratio of difference of the last term (anta-dhana) and first term

(ådi) to the common difference (caya), and then added by unily

n = .

4. Half the common-difference (caya) is obtained by first dividing the sum (sarva dhana) by the number of term

(gaccha) and reducing the quotient by the first term (ådi) and then dividing the result by number of terms

(gaccha) as reduced by unity:

.

One more formula in GJK (o), 49/123 is ÷ numerate.12

Verse 49. The duration of the [low tended-operation (adha³-pravâtta-kara¿a)] is one inter muhurta (antar-muhýrta,

the transforms (pari¿åmas) in it are innumerate ties the innumerate spare-points in the universe (loka); and in the

upper levels they increase in purity through similar increment.13

Note: In the JTP kar¿å²a and Sanskrit commentaries certain rules for dealing with progression relevant to verse

49 are given from p. 81 to 112. We have explained this through ancient symbols elsewhere. The verses have also

been translated in the collection of all verses in the appendix.

Hence we summarize the process of manifoulation in the commentaries as follows; used for the three types of

operations:

Denomination Working Symbol

sum (sarva-dhana S

or pada-dhana)

number of terms n

(pada or gaccha)

common difference d

(caya or vi¹e¼a)

first term (mukha- a

ådi or prabhava

last term (antadhana l

or bhými)

5. Half the common difference is also obtained by first subtracting from the sum (sarvadhana) the initial-sum

(ådidhana) [first term (ådi) as multiplied by number of terms (gaccha)], and then dividing the remainder by the

product of the number of terms (gaccha) and the number of terms (gacchas) as reduced by unity :

6. First term is obtained by first subtracting the post sum (uttara dhana) from the sum (sarva-dhana), 6. First

term is obtained by first subtracting the post sum (uttara dhana) from the sum (sarva-dhana), and then dividing

the remainder by the number of terms (gaccha) :

7. The last term is obtained as the sum of the first term and the common difference sum (caya-dhana)

l = a + (n-1)d

8. The sum is also obtained by multiplying the middle term (madhyama dhana) by the number of terms (gaccha)

9. Example there of about the low-tended operation (adha³-pravâtta kara¿a) through numerical symbolism :

S = 3072, n = 16 instants, d = 4,

then numerate (samkhyåta) =

a = Hence the series is 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222.

It may be noted that mathematical manipulation of the theory of karma in the Digambara Jaina School of

mathematics has been through sequences and series, progressions and regressions, as the series or

sequences take up several types of structures in course of time duration, lapses, progressions, as nisusus

(ni¼ekas), or cell-like structures with various life-times inn successive instants (samayas). The Digambara Jaina

School developed their studies in their own way, and it may be seen that their seem to be original attempts in

developing their own techniques for the study of their functional system theory (karma-siddhånta).14

In the JTP (kar¿å²aka vâtti and Sanskrit ²ikå), there is given a detailed description of the following topics under

the non-universal mathematics (alaukika ga¿ita). The non-universal measure (alaukika pramå¿a or måna) is of

four types : fluent measure (dravya måna), quarter measure (k¼etra måna), time measure (kåla måna), and

phase measure (bhåva måna). The fluent measure is of two types : number measure (samkhyåmåna) and simile

measure (upamå måna). The number measure is of three types numerate (samkhyåta), innumerate (asa´khyåta),

infinite (ananta), and so on. These are construction sets, constructed through a set process. They assess the

number measure of an existentialset. Then fourteen sequences, divergent in character are mentioned, describing

in details the dyadic-square-sequence (dvirýpa varga dhårå), dyadic-cube-sequence (dvirýpa ghana-dhårå)

applicable in the topic. The method of spread, distribute and multiply (viralana-deya-gu¿ana) is also given.

Logarithms is also applied. Defining various types of sets, existent, the commentator proceeds to describe the

eight types of simile-measure (upamå-pramå¿a or måna). Description is through quotations from earlier texts,

specially the Tiloyapa¿¿attî. Logarithms and logarithms of logarithms to base two of certain simile sets have

been extracted, as in the TLS15

Note: Let the numerate be denoted by s, intermuhýrta may be denoted by M;, which ranges from the trail (åvali)

as increased by unity instant (samaya) and denoted by A instant set, to the maximum value of instant set

contained in 48 minutes as decreased by one instant (samaya).

1. The development of assimilation takes time A, for completion

2. The development of the body then takes time as instant-set instant-set

3………………………………..senses……………………………………………………..

4………………………………..respiration………………………………………………..

5. ……………………………..speech……………………………………………………

6. ……………………………..mind……………………………………………………….

Each one of the above six periods from A to is an inter-muhýrta which is a variable as already told. And the total

is also an intermuhýrta, given by as we shall see by making use of the value of s.

Let S be taken as minimal numerate, ie, 2, and let A be the minimal inter-muhýrta, ie, one trail(avali) + 1 instants

(samayas). Then the total is equal to trails (avalis) + in stants (samayas). This is the minimal period of

development (paryåpti) for developable rational bios. The total time of the first two development is

instant (samayas). Summation of a geometric finite progression is worthy of note here.

Verse 153. Hellish souls or bios in all [are equal to the set of the space point in] universe line as multiplied by

the second square root [i.e. the fourth root] of one cube-finger. [Hellish bios] in the second and other [ie. the

third, fourth, fifth, sixth and the seventh hells are in number equal to the quotient of te space point set of

Universe-line as divided by its own twelfth, tenth, eighth, sixth, third and second root [respectively].17

Note: Let the universe line be denoted by L and cube of finger- space-point set be denoted by F3 . Then the total

number of all the hellish bios = L [F3]1/4 .

Those of the second hell = L ÷ [L]1/12 .

Those of the third hell = L ÷ [L]1/10 ,

Those of the fourth hell = L ÷ [L]1/8 ,

Those of the fifth hell = L ÷ [L]1/6 ,

Those of the sixth hell = L ÷ [L]1/3 ,

Those of the seventh hell = L ÷ [L]1/2 ,

Those of the first hell = – [the bios in the second hell to the seventh hell].

Verse 157. [If we] divide the universe line [space-point-set] by the square root of a linear-finger [space-point set],

and divide the quotient set so obtained by the third root [of the linear-finger space-point set], [and then] subtract

one [therefrom], [we obtain] the total number of all human beings [in the universe]. [The number of] developable

[human beings alone] is equal to the cube of 2 squared times, [expressed as follows] :18

Verse158. The number of the developable human bios is19 79, 22, 81, 62, 51, 42, 64, 33, 75, 93, 54, 39, 50, 336.

Verse 170. The range of ocular vision is one lac minus three hundred and sixty [yojanas squared], multiplied by

ten and then reduced to it square root, and then multiplied by nine as divided by sixty, would give the range of

sight.20 Note: This works out as

For other senses maximal range of activities cf. vv. 168, and 169.21

Verse 213. [The quotient of] the pit (palya) divided by innumerate part of a trail (avali) subtracted [Once, twice,

thrice, fourth and fifth times] from a sea (sagara) is the number of the logarithm to base two respectively of gross

fire bodied, non host individuals, host individuals, earth bodies, water bodies [bios] and [the number of logarithm

to base two of gross air-bodied [bios] [is] the last, ie. full sea (sagara).22

Note: For definition of sagara, cf. Jaini, J.L. (1918, BB), appendix D.

Verse 215. Divide the logarithm to base two of result of data by the logarithm to base two of the given figure in

data.

This will give the index number of the data. Divide the index of the ‘desired’ by the index of the data, writing the

result of the data as many times as there are units in the last quotient, and multiply them all into each other.

This is the desired result.21

Note: We shall explain this numerically.

If 2 is raised to power 16, we get (2)16 = 65536 or pa¿¿a²²hî

If 2 is raised to power 64, we get (2)64 = ? or eka²²hî

Thus the measure (pra¿¿a) set (rå¹i)

16 is the spread set (viralana rå¹i)

2 is the distribution set (deya rå¹i)

and (2)16 = 65536 = acquisition set (phala rå¹i), (or labdha rasi)

In requisition set (icchå rå¹i)

64 is the spread set (viralana rå¹i)

2 is the distribution set (deya rå¹i)

and (2)64 is the acquisition set (labdha rå¹i)

Formula is : = spread set in requisition set

or

[acquisition set in measure set] = acquisition in requisition set.

Applying the above formula to the numerical symbolism we have

= (65536)4 = 18, 44, 67, 44, 07, 37, 09, 55, 16, 16, which is called eka²²hî. In the dyadic sequences,

pa¿¿a²²hî = 65536 =

vådåla =

and eka²²hî =

A verse accurs quoted in the GJK, I. p. 352 :

“viralana rasido puna jettiya mettani hinaruvani /

tesim annonnahade haro uppannarasissa //”22

Verse 254. At every instant (samaya), only one unit, instant effective bond (samaya prabaddha) is bound and

comes into operation, or rises. At the last [instant of the duration of any instant effective bond (samaya

prabaddha)] the number of state (sattva), existence of functional ultimate particles16 is the spread set (viralana

rå¹i) 2 is the distribution set (deya rå¹i) and (2)16 = 65536 = acquisition set (phala rå¹i), (or labdha rasi)

In requisition set (icchå rå¹i)

64 is the spread set (viralana rå¹i)

2 is the distribution set (deya rå¹i)

and (2)64 is the acquision set (labdha rå¹i)

Formula is : = spread set in requisition set

or

[acquisition set in measure set] = acquisition in requisition set.

Applying the above formula to the numerical symbolism we have

= (65536)4 = 18, 44, 67, 44, 07, 37, 09, 55, 16, 16, which is called eka²²hî. In the dyadic sequences,

pa¿¿a²²hî = 65536 =

vådåla =

and eka²²hî =

A verse accurs quoted in the GJK, I. p. 352 :

“viralana rasido puna jettiya mettani hinaruvani /

tesim annonnahade haro uppannarasissa //”22

Verse 254. At every instant (samaya), only one unit, instant effective bond (samaya prabaddha) is bound and

comes into operation, or rises. At the last [instant of the duration of any instant effective bond (samaya

prabaddha)] the number of state (sattva), existence of functional ultimate particles

Basic Scientific IdeasS

From what has been described earlier, the mathematical contents bear testimony to evolution of a scientific spirit

from the period of the source material (c. 1st century A.D.) and even from still remoter period, in India,

specifically in the Jaina School of Mathematics, where the Karma theory became predominant, flourishing in

various schools of thought.

The Digambara Jaina School took the lead in its mathematical exposition, not only through semantics but also

through symbolism.

According to Russell, “Most sciences, at their inception, have been connected with some form of false belief,

which gave them a fictitious velue. Astronomy was connected with astrology, chemistry with alchemy. Mathematics

was associated with a more refined type of error. Mathematical knowledge appeared to be certain, exact, and

applicable to the real world; moreover it was obtained by mere thinking, without the need of observation……….

This form of philosophy begins with Pythagoras.”72

At this threshold, we pause to give a thought to the evolution of the theory of Karma as a naive scientific thought

in India with Vardhamåna Mahåvîra. We wish to go through, in brief, the underlying concepts, methods and

procedures in the vast literature available specifically with the Digambara Jaina School. No doubt, these might

have been intermingled with astrology, alchemy, metaphysics, and so on, through various passages, yet the

precision in the Digambara Jaina School draws special attention owing to the following descriptions which may

be compared with their analogous set up in modern science.