On The Sum of An A. P. With Optional Number in Ancient India

सारांश

प्रस्तुत आलेख में विद्वान लेखक ने आचार्य यतिवृषभ (४७३-६०९ ई.) द्वारा प्रणीत तिलोयपण्णत्ती ग्रंथ में समान्तर श्रेणी के ह पदों के योग निकालने के एक विशिष्ट सूत्र के महत्व एवं उपयोगिता पर सोदाहरण प्रकाश डाला गया है।

The formula for finding the sum `S’ of an arithmetic progression (abbreviated as A. P.) is given by

S = [(n – 1) d + 2 a] n/2 [1]

where `a’ is the first term, `d’ is the common difference and `n’ is the number of terms in the A. P.
The formula can be written as

S = [ nd – id + id – d + 2 a ] n /2 [2]or S = [ ( n – i ) d + ( i – 1 ) d + 2 a ] n / 2

where `i’ is an optional number.

This is what is contained in the Tiloyappannati (abbreviated as TP) as follows:

चय—हममिच्छूण—पदं रुवूणिच्छाए गुणिद—चय—जुत्तं।
दुगुणिद—वदणेण जुदं, पद—दल—गणिदं हवेदि संकलिद।।

The number of terms (pada, n) as subtracted by an optional number (iccha, i) is multiplied by the common difference (caya, d) and then joined with the product of the optional number as subtracted by unity and the common difference. To this sum is added twice the first term (vadana, vadana in Sanskrit, a) and then the result is multiplied by half the number of terms. This gives the sum (sanikalida, sanikalita in Sanskrit, S).

The TP is a text on cosmology, cosmogony and cosmography composed by Yativrsabha in 9 chapters in Sauraseni Prakrit verses. Not much is known about the life of Yativrsabha for except that he was a Digambara Jaina ascetic, and disciple of the ascetics Aryamanksu and Nagahasti. Some date between 473 A.D. and 609 A. D. is assigned to him.

The formula [1] can be written also as

S = [nd + id – id – d + 2a ] n/2
or S = [ (n +i) d – ( i + 1) d + 2a / n/2 [3]

where `i’ is an optional number.

This is what he sets forth in the verse of the TP as below :

चय—हदमिट्ठाधिय—पढमेक्काधिय—इट्ठ गुणिद—चय—हीणं।
दुगुणिद—वदणेण जुदं, पद—दल—गुणिदम्मि होदि संकलिद।।

`The number of terms (pada, n) as added by an optional number (ittha, ista in Sanskrit, i) is multiplied by the common difference (caya, d) and then subtracted by the common difference as multiplied by the optional number increased by unity. To this difference is added twice the first term (vadana, vadana in Sanskrit, a) and the whole result is multiplied by half the number of terms. This gives the sum (sankalida, sanikalita in Sanskrit, S) .”

The general formula [1] seems to have been well known to him as it is explicit from his own illustration given below.

अट्ठत्तालं दलिदं, गुणिदं अट्ठेहि पंच—रूव—जुदं।
उणवण्णए पहदं, सव्वधणं होई पुढवीणं।।

“Half of forty eight is multiplied by eight and then added by 5. The result thus obtained is multiplied by 49. Thus, the sum (savvadhana, Sarvadhana in Sanskrit, S) for the earths (prthavis) is obtained.

i.e.S = [ 48/2 x + 5] 49
or S = [(49 -1 ) x 8 + 2 x 5 ] 49/2

which implies a = 5,
d = 8 and n = 49 ;

( and S comes to be 9653 ; vide table A for the reference of the seven earths).

It thus becomes clear that he superflously inducted i into the formula [1] to obtain the formulae [2] and [3].

Why should i have been inducted at all ? It is a question which T A. Sarasvati Amma posed in her article appeared in 1962. The present article may be considered as a sequel to her question.

Yativrsabha seems to me to have done so for some purpose. What is the purpose ? To know that one we have to see the context of the application of these two formulae. Below is described that context.
There are 8400000 cavaties (bilas) distributed all over the seven earths (prthavis) from Ratnaprabha to Mahatamahprabha, which serve as residence of hellish souls (narkiya jwas).

These are of three types : central cavaties (indrakas), cavaties arranged in rows (srenibaddhas bilas), and scattered cavaties (prakirnakas) : vide Table A. The first two form a large arithmetic progression having the common difference minus eight within which they again form six small arithmetic progressions having the same common difference except for the last earth where is only one term.

A : Account of hellish cavaties in the seven earths

Oridinal number of the earth

(1) Neme of the earth 

Number of the central cavaties

Number of the cavaties arranged

The sum of the central cavaties in rows

The sum of the scattered cavaties in nows and the Gaties auared in Nows.

in the first earth there are 13 central cavaties. This number goes on decreasing by 2 in each earth, so that the last earth has only one. Thus, the total number of central cavaties in the seven earths is 49; vide Table A.

The cavatives arranged in rows are attached to the central cavatives like the spokes of wheel, with 4 rows in the directions and 4 rows in the subdirections attached to each central cavaty. All these arrays take the shape of discs (padalas). Thus the total number of discs in the seven earths is 49, being 13, 11, 9, 7, 5, 3, 1 respectively.

The first disc has 49 cavaties arranged in rows in each of the directions and 48 in each of the subdirections; vide Fig. 1. The second disc has 48 in each direction and 47 in each subdirections while the last disc has 4 in the directions only; vide Fig. 2 and Fig. 3 respectively. Thus the total number of cavaties arranged in rows to each central cavaty goes on decreasing by 8.

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He has given the two formulae [2] and [3] . of them any one can be used, for finding out the sum of all the central cavaties and the cavaties arranged in rows in any one of the six earths from the first to the sixth, for which the sum of the central cavaties and the cavaties arranged in rows in the last disc of these earths are given, respectively by 293, 205, 133, 77, 37, 13 being called the first term `a’, 13, 11, 9, 7, 5, 3 are, respectively, taken as the number of terms `n’, and the common difference is 8 everywhere. Below are the illustrations in which the formula

[2] is being applied.

For the first earth i.e. i = 1

S = [(13 – 1) 8 + ( 1 – 1) 8 + 2 x 293] 13/2
or S = 4433

For the second earth i.e. i = 2
S = [(11 – 2 ) 8 + (2 – 1 ) 8 + 2 x 205] 11/2
or S = 2695

For the third earth i.e. i = 3
S = [(9 – 3 ) 8 + ( 3 – 1 ) 8 + 2 x 133 ] 9 /2
or S = 1485

For the fourth earth i.e. i = 4
S = [(7 – 4 ) 8 + ( 4 – 1) 8 + 2 x 77] 7/2
or S = 707

For the fifth earth i.e. i = 707
S = [(5 – 5) 8 + ( 5 – 1) 8 + 2 x 37] 5/2
S = 265

For the sixth earth i.e. i = 6
S = [(3-6) 8 + (6-1) 8 + 2 x 13 ] 3/2
or S = 63

vide Table A for comparing S with the sum of the central vavaties and the cavaties arranged in rows.
Thus, we see that the optional number `i’ has been employed only to associate a reckoner with the first term `a’ and the number of terms `n’ of the ith small A.P. within the large A.P.

ACKNOWLEDGEMENT

The author expresses his gratitude to Dr. Anupam Jain (Indore) for providing facilities from Kundakunda Jnanapitha, Indore for the preparation of this paper.

Reference

A.Jadhav, Dipak [2004] “Theories of A.P. and G.P. in Nemicandra’s Works”,Arhat Vacana, 16 (2), 35-40, pp. 35 and 39.

B.Patni, C.P. [Editor] Tiloyapannatti of Yativrsabha (with Aryika Visuddhamati’s Hindi Commentary), Part-1, Second Edition, Sri 1008 Candraprabha Digambara Jain Atisayaksetra, Dehra-Tijara, 1997, pp. 140-172.

C.Sarsvati Amma, T.A. [1961-62] “The Mathematics of the First Four Mahadhikaras of the Trilokaprajnapti, Journal of the Ganganatha Jha Research Institute, 18, 27-51, pp. 34-38.

1.Tiloyapannatti (T.P., v. 2.64, p. 158.   

 2.Tiloyapannatti (T.P., v. 2.70, p. 161   

3.Tiloyapannatti (T.P., v. 2.71. p. 162.

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