Jambudiva-pannatti-samgaho in Modern Mathematical Aspects

Jambūdīva-pannatti-samgaho in Modern Mathematical Aspects
Jitendra Sharma  & Anupam Jain

ABSTRACT '''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. "

F.R.Bradley

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:

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:



Fig 1: Concept of Universe 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. 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. 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. There are seven regions in Jambūdvīpa namely Bharata, Haimavata, Harivarșa, Videha, Ramyaka,Hairaņyavata and Airāvata

Calculation  of  the  width   of  different    sections
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 190

The above description can be understood with the following figure:



Fig 2: Proportionate view of Jambudvipa

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:

Table-2 Width of the sections of Jambūdvīpa

Calculation of the Arrow
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  have

arrow = diameter of Jambhudvipa/190 * proportionate part



Fig 3: Arrow of the Sections 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:

Table-3 Arrow of the sections of Jambudvipa 

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.

Calculation of the Chord
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.

chord = &radic; (diameter - arrow) * arrow * 4



Fig.4 Representation     of  chord 

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

chord = &radic; (diameter - arrow) * arrow * 4

chord = &radic; 100000 - 70000/19 * 70000/19 * 4

chord = &radic; 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 concept 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.

<Div align=center >Fig. 5 Determination of chord by Pythagoras theorem 

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 &Delta;OAD,

OD2 = AD2 + OA2

AD2 = OD2 - OA2

AD = &radic; (OD<span style="font-size: 10px;vertical-align:+17%;">2 - OA<span style="font-size: 10px;vertical-align:+17%;">2 )

2AD = 2 &radic; (OD<span style="font-size: 10px;vertical-align:+17%;">2 - OA<span style="font-size: 10px;vertical-align:+17%;">2 )

CD = 2 &radic; (OD<span style="font-size: 10px;vertical-align:+17%;">2 - OA<span style="font-size: 10px;vertical-align:+17%;">2 )

CD = 2 &radic; (OD<span style="font-size: 10px;vertical-align:+17%;">2 -( OB - AB<span style="font-size: 10px;vertical-align:+17%;">2 )

CD = 2 &radic; (OD<span style="font-size: 10px;vertical-align:+17%;">2 - OB<span style="font-size: 10px;vertical-align:+17%;">2 - AB<span style="font-size: 10px;vertical-align:+17%;">2 + 2OB.AB)

CD = 2 &radic; AB(20B - AB)

CD = &radic; 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.