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:

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]]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.

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]]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

190 x = 100000

x = 10000/19

2x = 20000/19

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

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 |

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.

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

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:

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.

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]]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]]

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,

OD^{2} = AD^{2} + OA^{2}

AD^{2} = OD^{2} – OA^{2}

AD = √

(*OD*2 –*OA*2)

2AD = 2 √

(*OD*2 –*OA*2)

CD = 2 √

(*OD*2 –*OA*2)

CD = 2 √

(*OD*2 -(*OB – AB*2)

CD = 2 √

(*OD*2 –*OB*2 – *AB*2 + 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.

Tags:
Jain Mathematics