Contribution of Ancient Jaina Scholars To Modern Mathematics

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Dr. Anupam Jain
(Prof.-Mathematics & Director-Ganini Gyanmati Shodhpheeth)

Jainism is one of the oldest living religion of the world. Its literature is very vast and varied. Jainåcåryas contributed in the field of science too.

Mathematics is an integral part of Jaina literature. According to subjectwise classification scheme entire Jaina literature can be classified in four sections which are known as anuyoga

(According to Digambara tradition)

• I. Prathamånuyoga
• II. Karanånuyoga
• III. Caranånuyoga
• Iv. Dravyånuyoga

(According to Svetåmbara tradition)

• I Dharmakathånuyoga
• II Karana- Caranånuyoga
• III GanîtånuyogaIV Dravyånuyoga

Out of these four literature of Karanånuyoga and Ganîtånuyoga group are full of Mathematics. Apart from it some exclusively mathematical texts were also written by Jaina scholars to facilitate and train the students of Jainology. A detailed list of all such books will be given in the present paper.

Many such formulae and concepts are available in these texts which are still in use in the mathematical world. Some of them are following.

• I. General formula for combination
  nCr = n! / r!(n-r)!
• II. General formula for Permutation nPr = n! / (n-r)!
• III. Area of some specific geometrical figures like ellipse
• IV. Volume of some specific geometrical figures
• V. Concept of logarithms and related formulae like

log m.n = log m + log n
log m/n = log m- log n
log mn = n log m
log log mn = log n + log log m etc.

• VI. Concept of set, type of sets and operations.
• VII. Treatment of series and sequences.
• VIII. Treatment of continued fraction and unit fraction.

In the present paper original sa´skâta/pråkâta verses related with above formulae will be quoted with translation from Jaina texts and then deduction of the formulae will be made The necessary reference from the renowned books on History of Mathematics will be taken to support the originality, historical importance and uniqueness of the work of Jaina Scholars.

In the salutation of his famous Indian Mathematical text Ganita-såra sa´graha (GSS) of 9th century great Jainåcårya Mahåvîra (814-877 A.D.) says-

Alaghya´ trijagatsåra´ yasyånantacatu²ayam,
namastasmai jinendråya Mahåvîråya tåyine. (1)

Sakhyåjñånapradîpena jainendrena mahåtviå,
prakå¹ita´ jagatsarva´ yena ta´ pranamåmyaham1.(2)

‘I bow to Lord Mahåvîra who are unsurpassable in all the three world and acquired four infinite attributes. I bow to that highly glorious Lord Jina by whom as forming the shining lamp of the knowledge of numbers, the whole of the Universe has been made to shine1.

In this mangalåcarana Mahåvîråcårya refers Lord Mahåvîra as an illuminator of knowledge of numbers.
Another great mathematician Åcårya ©rîdhara of 8th C.A.D. writes in the salutation (mangalåcarana) of Tri¹atikå that-

Natvå Jina´ svavircita påtya ganitasya såra muddhratya lokavyavhåråya pravaksayati Sridharåcarya2.
©rîdharåcårya bowing to god Jina tells the substance of Mathematics as extracted from the Påtî composed by him for the use of people.

This slightly indicates the purpose of compiling the completely mathematical texts by Jainåcåryas. Of course it does not give complete picture; even then, it is important. We discuss it again after some time.
Now question arise what is Jaina School of Mathematics or Jaina Mathematics.

The Mathematics developed by Jainåcåryas are the Mathematics found in Jaina Literature is called Jaina Mathematics. Modern mathematical world was completely unaware with this school before the publication of Ganitasårasa sa´graha (GSS) in 1912 with English translation by M. Rangåcårya. The first information about GSS was given by Prof.

David Eugen Smith in April 1908 in Fourth International Congress on Mathematics in Rome.3 After this a detailed survey article under the title ‘The Jaina School of Mathematics’ appeared in Bulletin of Calcutta Mathematical Society, 21 (1929) by Prof. B.B. Dutta4.

In the ©vetåmbara tradition, in connection of 72 arts, we find the quotation-
Lehaiyao Ganiyappahanao5

i.e. Script etc. but full of mathematics. In other words we can say that they accept that all the arts are full of mathematics. Role of Mathematics in Jainism is very high.

It can be observed in the following words of a famous Jaina commentator, ¢odarmala (1740-1767 A.D.) of Jaipur. He writes-

Bahuri je jîva sanskâtådi ke jñåna sahita hai kintu ganitådika ke jñåna ke abhåva te mýla grantha yå tîkå vi,ai prave¹a nå karahu tina bhavya Jîvana kåje ina granthana kî racanå karî hai3.

For those people, who have the knowledge of Sa´skrita etc., but due to lake of knowledge of Mathematics they can’t understand the origin texts, these texts have been prepared. This indicate the utility of Mathematics in understanding Jaina philiosophy6.

Not only it but in 9th century a famous Jaina Mathematician, Åcårya Mahåvîra writes that-

Bahubhirvi Pralåpaiha Ki´ Trailokye Sacaråcare
Yatkincidvastu Tatsarva Ganitena Binå nahi 7.

What is good of saying much in vain ? Whatever there is in all three worlds, which are proposed of moving and non-moving being all that indeed cannot exists as apart from Mathematics.7

These references are enough to show the place of Mathematics among Jainas. literature have a lot of Mathematics. It is vast and varied too. The technical word used for mathematics in ©vetåmbara tradition is Sa´khyåna In Vyåkhya prajñåpti (Bhagavatî Sýtra) knowledge of Sa´khyåna is essential for all the Jaina Monks (Sådhus). In the lane of twelve a´gas, third one is Sthånånga, which is also known as Thana´. Here we find an important verse (Ch. 10, No. 100) related to the topics of Sa´khyåna.

Dasavidhe sa´khåne pannatte ta´ jahå
parikamma´ vavahåro rajju råsî kalåsavanne ya 1

Jåvantåvati Vaggo, ghano yataha vaggavaggo vikappo 1 ya.8 Types of Sa´khyåna are 10, which are following-

No.  Term	Old Interpretation by Abhaidevasýri	New Interpretation
1. Parikarma	Addition etc.	Fundamental Operations
2. Vyavahåra	Series etc.	Applications of fundamental Operations
3. Rajju	Plane Geometry	Para worldly Mathema-tics related to Simile Measure etc
4. Rå¹i	Heap of Grains	Set Theory
5. Kalåsavarna	Fractions	Mathematics of Fractions
6. Jåvata Tåvata	Multiplication of Natural Numbers	Simple Equations
7. Varga	Square	Quadratic Equations
8. Ghana	Cube	Cubic Equations
9. Vargavarga	Fourth Power	Higher Order Equations
10. Kalpa or Krakacikå	Vyavahåra	Combinations and Permutations Vikalpa or Mathematics related to Cutting of Saw

Attempt to explain these ten types has been made by Abhaidevasýri, B.B. Dutta, H.R. Kapadia and L.C. Jain. But the explanations made by L.C. Jain are more close to the reality.

Here I want to mention that the mathematics found in the Dhavalå and Gomma²asåra are comparatively more advanced and these are not available before Abhaidevasýri, B.B. Dutta, H.R. Kapadia, hence could’t imagine these advanced topics.

In many commentaries of Thåna´, we find this verse with minor changes. The interpretation made by Abhaidevasýri is misleading. The same type of interpretation is given by Åcårya Tulsî and other commentator in the editions of Thånam. This topic has been discussed in detail in author’s Hindi article. Ten Topics of Mathematical discussion in Jainågamas10.

During past two decades I have tried to see and collect the information about Vast Jaina Mathematical Literature, but we can do only a small part of it. More than 100 mathematical manuscripts written by different Jaina scholars are still remains unexplored, unidentified and we are still unknown about the mathematical knowledge contained in it. Even then, available information is very huge and attracts attention.

More information about it is available in the following two papers written by the author in Hindi-

• 1- Some Unknown Jaina Mathematical Text, Ganita Bhåratî (Delhi), 4 (1-2), 1982, PP 61-71.
• 2- Jaina Mathematical Literature, Arhat Vacana (Indore), 1 (1), Sep. 1988, PP 19-40

Of course above two articles are very exhaustive, but here I would like to give some information from these. We can classify all the available mathematical texts of Jaina School in the following 6 groups.

”’Group I”’ – In this category, we include those mathematical texts, which are well known and its critical edition has been published.

• 1. Ganita Såra Sa´graha of Mahåvîråcårya (850 A.D.).
• 2. Ganita Tilaka, commentory on Påtiganita of ©rîpati, written by Si´ha Tilaka Sýri (1275 A.D.)10.

”’Group II”’ – In this category, we include only those texts whose original texts has been published but its critical edition has not been published so far.

• 1. Tri¹atikå (Påtîganitasåra) of ©rîdhara. (8th c. A. D.)
• 2. Ganita Såra Kaumudî of Thakkara Pheru (1265-1330).
• 3. Angula Saptati.
• 4. Lilåvatî of poet Lålachand.
• 5. A´ka Praståra.

Out of these Tri¹atikå is very important. Its mangalåcarana has been changed. Really he was earlier ©aiva hindu but later he became Jaina. At the time of copying, some one changed the mangalåcarana. Original mangalåcarana was-

Natvå Jina´ svavircita påtyå ganitasya såra muddhratya

And letter Jina´ word was changed to ©iva´.
We have internal references in support of it 11.

”’Group III”’-In this group we include those texts whose manuscripts are lying with me but so far unpublished.

• 1. a²trin¹ikå by Mådhavacandra Traividya (11th C.A.D.)
• 2. Ganita Såra by Hemaråja (1673 A.D.)
• 3. Istå´kapancavi´¹atikå by Tejasingh Sýri (1686)

The specimen Pages are shown in appendix of this paper.

”’Group IV”’-In this group we include the name of those texts which are still preserved in different libraries of India. In the catalogues of different bhandårs we have the information about these texts. They should be preserved immediately. This list includes-

• 1. Ganita Såthasau – Mahimodaya
• 2. Ganita Såra – Ånanda Kavi
• 3. Ganividyå Pannatti
• 4. Ganita Sa´graha- Yallåcårya
• 5. Kshetra Ganita – Nemichand
• 6. Kshetra Samåsa – Somatilaka Sýri
• 7. Kshetra Samåsa Prakarana- ©rîcandra Sýri
• 8. Vâahat Kshetra Samåsa Vâatti- Siddha Sýri
• 9. Laghu Kshetra Samåsa Vâatti- Haribhadra Sýri
• 10. Kshetra Samåsa – Ratna ©ekhara Sýri
• 11. Kshetra Samåsa – Si´hatilak Sýri
• 12. Uttara Chattîsî Tîkå – ©rîdhara & Sumatakirti
• 13. Ganita åstra – Rajåditya
• 14. Ganita åstra – Gunabhadra
• 15. Ganita Vilåsa – Candram
• 16. Ganita Sa´graha – Råjåditya
• 17. Ganita Vilåsa – Rajåditya
• 18. Ganita Ko,thaka
• 19. Pudgala Bha´ga Vâatti

This list may be extended by making extensive survey. More detailed information about these MSS are available in my article Jaina Mathematical Literature (1988).

Group V-In this group we list out those texts whose names are found in other texts but at present there is no information about the availability of these texts. They should be searched.

• 1. Vâahada Dhårå Parikarma (In Trilokasåra) 12
• 2. Siddhabhýpaddhati Tikå (In Uttara puråna) 13
• 3. Karana Sýtra (Yativâ,abha)
• 4. Karana Bhåvanå (Anantapåla)
• 5. Påtîganita (Anantapåla)
• 6. Chattîsa Pýrva Prati Uttara Pratisaha (Mahåvîråcårya)
• 7. Kshetra Ganita (Mahåvîråcårya)
• 8. Aloukika ganita (Ratna ©ekhara Sýri)
• 9. Ganita Sýtra (Ratna ©ekhara Sýri)
• 10. Tri¹ati (Ratna ©ekhara Sýri)
• 11. Kshetra Vicåranå (Ratna ©ekhara Sýri)
• 12. Kshetra Samåsa Bålåvabodha
• 13. Lilåvati Bhå,å Caupai.

Group VI-There are so many old mathematical works which were composed in soura¹enî Pråkâta prose and poetry and in Ardhamågadhî. The quotations of these works found in later works but the original texts are not available. The names of the texts containing such quotations are following-

• 1. Tatvårthådhigama Sýtra Bhå,ya of Umåsvåti
• 2. Commentary of Åryabhatîya by Bhåskara-I
• 3. Dhavalå Commentary by Vîrasena
• 4. Commentary of Anuyogadvåra Sýtra by ©îlånka5. Lîlåvatî of Bhåskara-II

The well known method of solving quadratic equation given by ©rîdhara is available in Bhåskara’s Lîlåvatî14.

More detailed information is available in my article “Jaina Mathematical literature (1988)” in Hindi.
Now I am giving the list of the Åcåryas/Scholars, who are related with Jaina School and whose works have the material of mathematician’s interest. Unless we identify all the available manuscripts and search out the mathematical treatise mentioned by later mathematicians, we can’t claim about the completeness of it. Even then I giving brief idea.

No. Name	Period	Text of Mathematical Interest
1. Gunadhara	1st c.B.C.	Ka,åya Påhuda
2. Kundakunda	1st c.B.C.	Pancåstikåya etc
3. Dharasena, Pupadantaand Bhýtabali	1st c.A.D.	Satakhanðågama &Mahåbandha
4. Umåsvåmî	2nd c.A.D.	Tatvårthasýtra
5. Umåsvåti	2nd-4th c.A.D.	Tatvårthådhigamasýtrabhå,ya
6. Yativâ,abha	176-609 A.D.	Tiloyapannattî
7. Pýjyapåda(Devanandi)	539 A.D.	Sarvåtha Siddhi
8. Jinabhadragani (Bhå,yakåra)	609 A.D.	Vi¹e,åvå¹yaka Bhå,ya
9. Akalanka	620-680 A.D.	Tatvårtha Råjavårtikå
10. Vidyånanda	775-840 A.D.	Tatvårtha ©lokavårtikå
11. Rîdharåcårya	8th c.A.D.	Påtiganita, Tri¹atikå Jyotirñånavidhi,Bîjaganita (not available) etc.
12. Vîrasena	816 A.D.	Dhavalå Commentary
13. Jinasena	9th c.A.D.	Jayadhavalå Commentary
14. Mahåvîra or	850 A.D.	Ganita-såra-sa´graha etc. Mahåvîråcårya
15. Kumudendu	860-880 A.D.	Siribhývalaya
16. Ilånka	9th c.A.D.	Commentaries of ågams
17. Nemicandra Siddhåntacakravarti	10-11th c.A.D.	Gommatasåra, Trilokasåra, Labdhisåra, K,apanasåra
18. Mådhavåcandra Traividya	10-11th c.A.D.	attrin¹ikå, Commentaries of Gomma²asåra, Trilokasåra etc.
19. Padmanandi- I	977-1043 A.D.	Jambýddîvapannatti sa´gaho
20. Amitagati- II	11th c.A.D.	Candraprajñapti, Sårdhadvaya prajñapti, Vyåkhyå prajñapti
21. Abhaideva Sýri	1015-1078 A.D.	Commentaries of 9 ågamas
22. Hemcandra Sýri	1107 A.D.	Anuyogadvåra Vratti,Vi¹e,avåsyaka Bhå,yavâatti
23. Malayagiri	1080-1172 A.D.	Commentaries of sýrya prajñapti Candra prajñapti, Jambýdvipa prajñapti etc.
24. Råjåditya	1120 A.D.	Vyavahåra Ganita, Kshetra Ganita Vyavahå Ratna, Jaina Ganita Sýtrodåharana,Citrahasuge, Lîlåvatî.
25. Si´hatilaka Sýri	13th c.A.D.	Ganita Tilaka Tîkå
26. Thakkara Pheru	1265-1330 A.D.	Ganitasåra Kaumudî
27. Ratna¹ekhara Sýri	1440 A.D.	Laghu Kshetra Samåsa
28. Mahimodaya	1665 A.D.	Ganita Såthasau
29. Hemaråja	1673 A.D.	Ganitasåra
30. Tejasi´ha	17th c.A.D.	Itånka Pancavin¹atikå
31. Coaaramala	1740-1767 A.D.	Samyakajñåna Candrikå Tîkå on Gommatasåra, Trilokasåra etc.

 The names of the mathematicians whose works included in Group IV should also be added after getting the copies of the MSS from bhandårs. The above list of authors may be extended by adding the names Haribhadra Sýri, Padmaprabha Sýri, Candrama, Siddhasena, Mahendra Sýri, Malayendu Sýri, Bulåkîcandra, Bulåkîdåsa etc.

The books written by these authors include some material of mathematician’s interest.

Apart from it in Jaina Canonical Text of ©vetåmbara tradition, which is known as A´ga & Upa´ga, we find enough interesting material. These a´gas contains valuable information regarding Number System, Theory of Infinity, Theory of Indices, Combination etc. Sthånångasýtra, Bhagavaýsýtra.

Anuyogadvårasýtra, Uttarådhyayanasýtra, Jambýdvîpap-rajñapti are more important from this point of view. All these references are collectively available in Ganitånuyoga compiled by Muni Kanhaiyalal ‘Kamal’15.

A detailed study of the Mathematical content available in Arddhamågadhi Jaina canon is made by author under a minor research project of J.V.B.I. Ladnun16

I would like to mention that ©rîdhara. Mahåvîra, Si´hatilakasýri & Thakkarpheru are purely mathematician. While Yativâ,abha, Vîrasena, Nemicandra & Toðarmala who are basically philosophers, but they contributed a lot to the development of Mathematics.

A deep study of the works of these philosopher mathematicians is urgently needed. I am happy to note that Prof. Padmavathamma, Dr. Pragati Jain, Mr. Dipak Jadhav, Mr. N. Shivkumar are doing such studies.
Now I am mentioning few points which were earlier discovered by Jainåcåryas but still in the existing books on History of Mathematics we find wrong information.

1. Process of finding perpendiculars & base for fixed C is available in GSS, 17

but credit is given to Fibonacci (1202 A.D.) and Vieta (1580 A.D.) 18 It is also clear by the name also.

Actually credit should be given to Mahåvîråcårya (850 A.D.)

2. The General formula for combination nCr = n!/r!(n-r)! is available in GSS by Mahåvîra (850 A.D.) 19, but credit is given to Herigon (1634 A.D.) 20. The theory of combination & permutation is available in many Jaina texts by the name Vikalpa or Bha´ga. Bhagavatî Sutra is important regarding it.

3. The general formula for Permutation nPr = n!/(n-r)! is given in the commentary of Anuyogadvåra sýtra by Hemcandra21, While it is mentioned in the book of Smith that it is invented in Europe in 14-15th Century.22

4. The concept and formula for logarithms is available in the Tiloyapannatti23
(2-7 c.A.D.) under the name arddhaccheda etc. and in the Dhavalå commentary of Sha²akhanðågama written by Vîrasena (816 A.D.)24 The formulae which are available-

log m.n = log m + log n
log m/n = log m – log n
log mn = n log m

Not only it, but the concept of log log and log log log is also available in the Dhavalå commentory. More details are available in the article of Prof. L.C. Jain ‘On Some Mathematical Topics of Dhavalå text or in the book ‘Exact Sciences from Jaina Sources, Vol.-1, Basic Mathematics’.25

But credit for the invention of logarithm is given to John Napier (1550-1617) and Just Burgi (1552-1632) which is not proper.26

Of course it is true that in Dhavalå all the rules are discussed with base 2,3,4, but in the modern mathematics base is ’10’ and ‘e’ are more popular.

The concept of antiardhaccheda and antivarga¹alaka is also available in the commentories written by Mådhvacandra Traividya. 27

5. In the same Dhavalå. There exist concepts and illustrations of set theory. In fact, the word ‘råsi’ is used for sets. Other synonomical words are Ogha, Punja, Sampåta, Bhavya Jîva Råsî, Mithyå dri,ti Jiva Råsi, Vanaspati Kåyika Jiva Råsi, all are well defined hence they are sets.

The concept of finite & infinite set, Singleton set, Null set, Sub set, Super set, etc. are available in Dhavalå in detail but the credit for invention and development of set theory goes to george Cantor (17th c.A.D.). 28 Of course it may be true that concept of set was developed by Cantor independently but on this ground we can not neglect the contributions of Vîrasena.

6. The continued fractions are available in Dhavalå, Vol-3, in 9th century, while credit goes to Antonio Cataldi (1540-1620 A.D.).29

7. Concept of Probability is available in the Aptamimånsa commentory written by Samantabhadra (2nd c.A.D.) 30 while credit goes to Galileo (1564-1642), Fermat (1601-1625), Pascal (1623-1662), Bernouli (1654-1705).31 In Jainology it is available in the name of Avaktavya.

8. Famous book Ganita Såra Sa´graha of Mahåvîråcårya contains the rule adding fraction of unequal denominator by the name niruddha.32

This rule was invented in Europe in 15th C.A.D. and came in use about 17th C.A.D. 33

9. The use of unit fraction is an unique contribution of Mahåvîråcårya. No other Indian Mathematician discusses it.34 Seven different type of cases are available in GSS, ch-3. 35

10. The rule for finding the area & circumference of ellipse is available in GSS-7/21 & 7/63. The name of ellipse is here Åyatavâatta in the Tri¹atika of ©rîdhara. It is discussed under the name Yavåkåra. Both are not available in any contemporary or prior book. Many other unique geometrical figures are discussed in the 7th chapter of G.S.S. A detailed study is available in author’s Hindi Book Mahåvîråcårya – A critical study.36

11. The development of different rules related with A.P., G.P. and mixed progressions are found in G.S.S. of Mahåvîråcårya and they are all illustrated in my above mentioned book.37

12. So many other concept related with evaluation of surds, classification of numbers are available in ardhamågadhî Jain ågams and they are all discussed in my minor research project of J.V.B.I, Ladnun.38 We are not taken here due to shortage of space.

A deeper study of Jaina School of Mathematics is urgently required.

References-

1. Ganita Såra Sa´graha, Åcårya Mahåvîra, English Translation by M. Rangacharya, Madras, 1912, Hindi Translation by L.C. Jain, Sholapur, 1963, Kannada Translation by Padmavathamma with English Translation of M. Rangacharya, Hombuj, 2000, ch.-1, verse-1,2.

2. Manuscript Tri¹atikå, Lucknow University, verse-1.

3. Gommatasåra (Jîvakånda), Åcårya Nemicandra Siddhåntacakravarti, with commentory of Todarmal, Jaina Siddhånta Prakå¹inî Sansthå, Kolkata, Pýrva Pîthîkå, p. 58.

4. Preface of the GSS (Kannada Translation), 2000

5. B.B. Dutta, the Jaina School of Mathematics, B.C.M.S. (Calcutta), 21 (1929), pp. 115-143.

6. Bhagavatî Sýtra, 90.

7. Ganita såra sa´graha, Åcårya Mahåvîra (850 A.D.), ch-1, ¹loka-16, (1/16).

8. Thana´, ch. 10. Verse-100 (747). Ladnun, 1976, p-926, 992-994

9. Tulsî Prajñå (Ladnun), 13(1987), pp 57-64.

10. Ganita Tilaka, Gaikwada Oriental Series, Baroda, 1935.

11. Mamta Agrawal, Åcårya ©rîdhara and his Mathematical Contributions, Ch. Charanasingh University, Meerut, Ph.D. Thesis, 2001, p. 169.

12. Trilokasåra, Gåthå 91.

13. Uttarapuråna Pra¹asti, Bhartiya Jñånapitha, Delhi

14. See, Ref. 11

15. Ganitånuyoga, Muni Kanhaiyalal ‘Kamal’, Agam Anuyoga Prakåshan Samiti, Sanderao, 1970.

16. Mathematical Content of Ardhamågadhî Jaina Canon (in Hindi) Minor Research Project, Jain Vishva Bharati Ladnun, 2006.

17. G.S.S. Ch.-7, Verse 95, 97 & 122½

18. L.E. Dickson, History of Theory of Numbers, Vol-II Washington, 1923, p. 167.

19. G.S.S. 6/218, p. 246.

20. D.E. Smith, History of Mathematics, Dower Publication, New York, Vol-2, p. 527.

21. Commentary of Anuyogadvåra sýtra by Hemchandra, Biyavar

22. Smith, History, Vol-2, p. 524-528.

23. Tiloyapannattî, Yatîvâ,abha, Hindi commentary by Åryikå Vi¹uddhamatî, 3 Vols, Dig. Jaina Mahåsabhå, Kota, 1984-89.

24. atakhanðågama with Dhavalå Commentary, Vol 1-16, 1st Ed. Amravati, Vidisha, 1939, 2nd edition, Sholapur, Specially book 3

25. L.C. Jain, On Some Mathematical Topics of Dhavalå Texts, I.J.H.S., 11 (2), pp. 85-111, 1976.
L.C. Jain, Exact Sciences from Jaina Sources, Vol I & II, Råjasthåna Pråkrita Bhårtî, Jaipur, 1982-83.

26. Smith, History, Vol II, p. 419.

27. Dipak Jadhav, the laws of logarithms in INDIA, Historia Scientairum, 11(3), p-261-267, 2002.

28. L.C. Jain, Set Theory in Jaina School of Mathematics, I.J.H.S.(Calcutta), 8(1) P 1-27, 1973

29. Dhavalå Vol 3 PP 45-46 & Smith, History of Mathematics Vol 2 P 419

30. Åpta Mimån,å by Samantabhadra, Ch.-7, Verse 15,16.

31. Ramesh Chand Jain, Syådvåda ke Sapta Ba´ga and Modern Mathematics Proceedings of International Seminar on Jaina Mathematics & Cosmology, D.J.I.C.R. (Hastinapur), 1985, 95-99.

32. Ganita Såra Sa´graha, of Mahåvîråcårya, 3/56, p. 49.

33. B.L. Upadhyaya, Ancient Indian Mathematics, (Hindi), Delhi 1971, p. 169.

34. Brijmohan, History of Mathematics, Lucknow p. 81.

35. Ganita Såra Sa´graha, Ch.-3, verse 75-84.

36. Anupam Jain & S.C. Agrawal, Mahåvîråcårya- A critical study (in Hindi) D.J.I.C.R., Hastinapur1985

37. See, Ref. 36

38. See, Ref. 16 Sa²tri¹ika of Mådhavacandra Traividya (11th c.A.D.)Ganitasåra of Hemaråja (17th c.A.D.)I,tånkapancavinsasatika of Tejasingh