Jambūdīva-pannatti-samgaho in Modern Mathematical Aspects

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ABSTRACT

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In the present paper we will discuss about the geography of the universe and some facts about the earth where we live,where is it situated exactly. All these facts have been discussed in the text Jambudiva­ pannatti-samgaho (JPS).JPS is the text of Karnanyoga section of Jaina literature.

The literature related with Karnanuyoga includes Mathematics,Geography,Cosmology,Cosmogony and Karma theory.JPS of l0th-11th century composed by Padmanandi-I,is a big text of 13 Uddesa(chapters) which includes the precise details about the existence of the universe with its geography and the dimensions.

In fact,when we go through this text we find that there are so many geometrical concepts that had been used to calculate the area, circumference and other geometrical measures of Jambūdvīpa. These geometrical concepts were quite different from the concepts of modern mathematics but practically when we compare the results obtained from both the techniques,we observe that the results are approximately the same.

In the present paper we will elaborate some mathematical or more precisely geometrical concepts in context to the comparison with other mathematical texts and modern mathematics with some illustration.

“We may be asked whether, Nature is finite or infinite. If nature is infinite, we have the absurdity of something which exists and still does not exist. For actual existence is, all finite. But on the other hand, if nature is finite, then nature must have an end and this is again impossible.

 For a limit of extension must be relative to an extension beyond and to fall back on empty space will not help us at all. But we can not escape the conclusion that that nature is infinite. Every physical world is essentially and necessarily infinite. “

In the Jaina philosophy the universe is divided into three parts as mentioned below:

1. Adholoka: Its base is similar to vetrāsana(wedge).

2. Madhyaloka: It appears like the upper portion of a standing mrdanga(trumpet).

3. Ūrdhvaloka: It appears like a standing mrdanga(trumpet).

The complete loka has a height of 14 rāju and a thickness of 7 rāju throughout.Adholoka and ūrdhvaloka have a height of 7 raju each and in between them lies the madhyaloka. The Sumeru Mountain is stretched over madhyaloka with the height of 100040 yojana.

Madhyaloka comprises of 1 rāju as length and breadth and 1 lac yojana height.The number of islands and oceans in it are uncountable.The islands and oceans form alternate rings with Jambūdvīpa(the island of Jambu),the only disc of land mass at the centre.The alternate positions of islands and oceans in their proper order from the inner most islands are given as under:

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No. Island(Dvipa)  Ocean (Samundra)

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1.	Jambūdvīpa	Lavana Samudra
2.	Dtātakīkhanda Dvīpa	Kālodadhi Samudra
3.	Puskaravara Dvīpa	Puskara vara Samudra
4.	Vārunīvara Dvīpa	Vārunīvara Samudra
5.	Ksīravara Dvīpa	Ksīravara Samudra
6.	Ghrtavara Dvīpa	Ghrtavara Samudra
7.	Ksaudravara Dvīpa	Ksaudravara Samudra
8.	Nandīsvara Dvīpa	Nandīsvara Samudra
9.	Arunavara Dvīpa	Arunavara Samudra
10.	Arunābhāsa Dvīpa	Arunābhāsa Samudra
11.	Kundalavara Dvīpa	Kundalavara Samudra
12.	Śankhavara Dvīpa	Śankhavara Samudra
13.	Rucakavara Dvīpa	Rucakavara Samudra
14.	Bhujagavara Dvīpa	Bhujagavara Samudra
15.	Kuśavara Dvīpa	Kuśavara Samudra
16.	Krauncavara Dvīpa	Krauncavara Samudra
17.	Manahśila Dvīpa	Manahśila Samudra
18.	Haritāla Dvīpa	Haritāla Samudra
19.	Sindūra Dvīpa	Sindūra Samudra
20.	Śyāma Dvīpa	Śyāma Samudra
21.	Añjanavara Dvīpa	Añjanavara Samudra
22.	Hingula Dvīpa	Hingula Samudra
23.	Rūpyavara Dvīpa	Rūpyavara Samudra
24.	Kañcana Dvīpa	Kañcana Samudra
25.	Vajraavara Dvīpa	Vajravara Samudra
26.	Vaidūrya Dvīpa	Vaidūrya Samudra
27.	Nāgavara Dvīpa	Nāgavara Samudra
28.	Bhūtavara Dvīpa	Bhūtavara Samudra
29.	Yakșavara Dvīpa	Yakșavara Samudra
30.	Devavara Dvīpa	Devavara Samudra
31.	Abhīndravara Dvīpa	Abhīndravara Samudra
32.	Svayambhūramaņa Dvīpa	Svayambhūramaņa Samudra

As a general rule,an island always precedes the ocean.Thus, the innermost island is Jambūdvīpa which is surrounded by Lavana samudra and the outer most ocean is Svayambhuramana Samudra surrounding the Svayambhūramaņa Dvīpa.This concept can be understood with the help of the following figure:

[[File:Fig1_JPS.JPG|700px|center]]

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Fig 1: Concept of Universe

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The universe may be considered as concentric rings of the islands and the oceans.There are many Jaina Āgamas related to the Geography of the universe which also includes Mathematics, Cosmology, Cosmogony and Karma theory. The major related texts are Tiloypaņņattī, Trilokasāra, Jambūdīva-paņņatti­ samgaho and Loka Vibahāga. With reference to these texts the diameter of Jambūdvīpa is 1 Lac yojanas and the diameter of the Lavanasamudra is 2 lakh yojanas, double the diameter of the Jambudvipa. Likewise the diameter goes on doubling subsequently till the diameter of the last ocean Svayambhūramaņa.
There are many facts described in Jambūdīva-paņņatti-samgaho(JPS) related to the geography of Jambūdvīpa some of them are given below for the ready reference.

The Jambū island has been related as the circular solar disc in the centre of the islands and oceans and has one lakh yojanas of diameter or length. Paumnandi(Padrnanandi), Jarnbudiva-pannatti-samgaho ed. by Prof.A.N. Upadhye and Dr. Hiralal Jain, Shri Jaina Sarnskrti Sarnraksaka Sarngha, Sholapur,1958,Uddesa.I Gatha:2Q.

The diameter as multiplied by the diameter is multiplied by ten and then the square root of the product is taken out resulting in its circumference.Ibid, Uddesa: I Gatha:23.

The circumference is multiplied by one fourth part of the diameter, resulting in the area of the circular areas like the disc of the sun. Ibid, Uddesa: 1 Gathii:24.

There are seven regions in Jambūdvīpa namely Bharata, Haimavata, Harivarșa, Videha, Ramyaka,Hairaņyavata and AirāvataIbid, Uddesa. Z Giitha:2.

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Calculation of the width of different sections 

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In the Jambū island, up to videha region there are 4 regions and 3 family mountains. Thus, Jambū island is comprising of 7 division which are successively double the preceding and the successive six divisions are each such that the succeeding division is half of the preceding. Hence, the proportions of different sections are given as below:

Bharata-1	Himavāna-2
Haimavata-4	Mahāhimavāna-8
Hari-16	Nișadha-32
Videha-64	Nīla-32
Ramyka-16	Rukmī-8
Hairaņyavata-4	Śikharī-2
Airāvata-l

Total number of the proportions of the Jambū island are 190Ibid, Uddesa:2 Gatha:6-15

The above description can be understood with the following figure:

[[File:Fig2_JPS.jpg|600px|center]]

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Fig 2: Proportionate view of Jambudvipa

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Since the diameter of Jambūdvīpa is 100000 yojana & it is divided in to 190 proportionate parts. If the proportionate part of Bharata and Airāvata region is considered as x then we can write

Thus the proportionate width of each of the region can be given as below:

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Table-2 Width of the sections of Jambūdvīpa

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No.	Name of Region/Mountain	Width in yoganas
1.	Bharata Region	10000/19
2.	Himavāna Region	20000/19
3.	Haimavata Region	40000/19
4.	Mahahimavana mountain	80000/19
5.	Hari Region	160000/19
6.	Nișadha Mountain	320000/19
7.	Videha Region	640000/19
8.	Nila Mountain	320000/19
9.	Ramyka Region	160000/19
10.	Rukmī Mountain	80000/19
11.	Hairanyavata Region	40000/19
12.	Śikharī Mountain	20000/19
13.	Airāvata Region	10000/19

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Calculation of the Arrow

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The concepts given in JPS to determine the arrow of the desired segment is as follows:
When the arc as bow’s surface,chord and height of segments are divided by 19 they are obtained in the form of fractions.The arrows of Videha etc. are obtained,when the diameter of Jambūdvīpa is divided by 190 and multiplied by  ‘95,63,31,15,7,3,1’  respectively.Thus,we can haveIbid, Uddesa.Z Gatha:21-22.

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”’arrow = diameter of Jambhudvipa/190 * proportionate part”’

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[[File:Fig3 JPS.jpg|700px|center]]

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”’Fig 3: Arrow of the Sections”’

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Videha region is the middle most part of Jambūdvīpa and divided by Sitā­ Sitodā rivers in to two equal parts.Thus,the proportions below the rivers are 95.Thus the Arrow of each of the region up to Videha region can be given as below:

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Table-3 Arrow of the sections of Jambudvipa

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No.	Name of Region/Mountain	Arrow in yoganas
1.	Bharata Region	10000/19
2.	Himāvana Region	30000/19
3.	Haimavata Region	70000/19
4.	Mahāhimavāna mountain	150000/19
5.	Hari Region	310000/19
6.	Nișadha Mountain	630000/19
7.	Videha Region	950000/19

Arrow of any section can also be determined by doubling its width then subtracting the width of the Bharata region. Similarly, the arrow of the sections up to Videha region can also be determined-by adding the widths of the preceding region.

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Calculation of the Chord

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The concepts given in JPS to determine the length of the chord of the desired segment is given below.

When arrowless diameter is multiplied by arrow and multiplied by 4 the square-root of the product gives the measure of the chord.Ibid, Uddesa:2 Gatha:23.

chord = √ (diameter – arrow) * arrow * 4 

[[File:Fig4_JPS.jpg|700px|center]]

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Fig.4 Representation of chord

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With reference to the above mentioned concept the chord of Haimavata region can be determined as follows :

chord = √ (diameter – arrow) * arrow * 4 

chord = √  100000 – 70000/19 * 70000/19 * 4 

chord = √  715822/19 

chord = 37674 16/19 yogana

The concept to determine the chord given in JPS is quite similar to the concept described in the text Tiloyapannati(TP).

This conceptTilcyapannatti,Yativrasabha,Hindi Translation by Aryika Visuddharnati, Bharatavarshiya Dig.Jain Mahasabha, Kota,Uddesa.q Gatha:180. is

chord = 4[(d/2)² – (d/2 – h)²]

where,
d – diameter; & h – depth

The chord of Haimavata region by this formula comes out to be 37674.80037 yojanas which is approximately equal to the length of the chord determined earlier.
In continuation, we can also prove that the concepts described in JPS are also identical to the concepts of modern mathematics. The same formula can be determined by using the well-known Pythagoras theorem.

[[File:Fig5_JPS.jpg|700px|center]]

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Fig. 5 Determination of chord by Pythagoras theorem

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In the given figure, OB and OD are radius. AB is the depth or arrow about the chord CD.CD is the chord whose length is to be determined.

In ΔOAD,

OD² = AD² + OA²

AD² = OD² – OA²

AD = √
(OD2 –
OA2) 

2AD = 2 √
(OD2 –
OA2) 

CD = 2 √
(OD2 –
OA2) 

CD = 2 √
(OD2 -(
OB – AB2) 

CD = 2 √
(OD2 –
OB2 – AB2 + 2OB.AB) 

CD = 2 √ AB(20B – AB) 

CD = √ 4AB(20B – AB) 

Thus, the same result can be derived from the modern approach. There are many more mathematical or more precisely geometrical concepts embedded in the text JPS which are quite appropriate in context to the modern mathematics.