Mathematical Contributions of Srldharacarya and Mahaviracarya : A Comparative Study
Abstract
Anupam Jain & Shaifali Jain
During the golden period of Indian Mathematics (500 – 1200 A.D.), the work of 5 mathematicians namely Aryabhata, Brahmagupta, Srldharacarya, Mahaviracarya and Bhaskaracarya has been very significant and famous. The contributions of these mathematicians not only influenced the entire mathematical fraternity of contemporary world but also has been a driving force ever since then.
Among these five, two mathematicians i.e. Sridharacarya (799 A D.) and Mahavir acarya (850 A.D.) belong to the Jaina tradition.
While the period of Mahaviracarya is fixed about 850 A.D. but the period of Srldharacarya has remained controversial for a long period.
Some place him in 10th C.A.D. (after Mahdvlra) while some others during 8th C.A.D. (before Mahdvlra). Due to it the contributions of Srldhara have not been evaluated properly yet. An attempt has been made here to make a comparative study between the works of Srldharacarya and those of Mahaviracarya.
The Jaina literature of Karananuyoga section is full of Mathematics. Actually no one can understand it without the knowledge of Mathematics. In Jaina literature, we find two types of Mathematics :
1. Loukika (Wordly) 2. Aloukika or Lokottara (Para-worldly)
Loukika is related with day to day Mathematics, i.e. Arithmetic, Algebra, Geometry etc. and Aloukika is related with Yojana, Rajju, Palya, Sagropama etc., which though useful for the understanding of the structure of the cosmos, has nothing to do with our daily life.
The works of Srldharacarya (Srldhara) and Mahaviracarya (Mahavlra) come under the category of Loukika Ganita. Following texts are undoubtedly written by Srldharacarya are available at present :
1.PatlganitaPatlganita, Srldharacarya, Traslated with introdu;:::r : *15 5-u I e 7 anment of Mathematics,Lucknow University, Lucknow, 1959
2.Patlganitasara (Ganitasara or Trisatika).Trasatika (Ganltasara or Patlganltasara), Sridharaean – S-jdbSi Dvivedi, Varanasi, 1894. Translated in Hindi with Sumedha Commentary by 5_r _rm – :lr … – _.:~rr a Samskrta Sarhsthana, N. Delhi, 2004.
3. JyotirjnanavidhiJyotirjnanavidhi published with Acarya Srldhara e1 tyrlna dhi. Anupam Jain, ISBN81-86933-44-1 Kundakunda Jnanapltha, Indore, 21 1
4. Ganitapancavimsi.T. Hayashi, Srldharas Authorship of the Mather*.2: : a Trer r 2an::2pahcavims i, HistoriaScientiarum, 4(3) 1995, P. 233-250.
Bljaganita is another famous book of Srldharacarya which is not available at present but the quotation from it can be seen in the Bljaganita of BhaskaracaryaCaturahatvargasamai rupaih paksadvayam gunaye:a Avyakta vargarupairyuktav pakso tato raulam.
Bhaskaracarya Pranlta Bijaganitam, Vas ana Sudh2 L * 1 .nChoukhamba Krisnadasa Acedemy, Varanasi, 2008, Avyakta Varga Samlkz.. The credit of Vrhatpatl and Navamsati is given to Srldhara but as they are still not available and we can’t finally admit that these :ere composed by Srldhara and whether they are in any way differs::m the existing works mentioned above.
Ganita-sara-samgrahaGanita-Sara-Samgraha, Mahavlracarya, Translates in Eng ;xxes by M. Rangacharya,
Madras, 1912, Hindi Translation with notes by L.C Jam Srrl 1-1 Kannada Translation
by Padmavathamma, Published with Kannada and English Trans : Hombuj, 2000, Telugu Translation by Tanneti, Hyderabad, 2005. is only available text of Miiaviracarya.
The credit of following four texts is also given to Mahavlracarya but they are not available nor there is there any definite proof of their composition*.For more details about the authorship of these four texis see-Shaifali Jain, Mathematical Contributions of Acarya Sri dhara and – Ir Mahavlra, A Comparative Study, Ph.D. Thesis, Shobhit University, Meerut, 2015
| 1. Sattrisatika, | 3. Chattlsapurva prati uttara pratisaha |
| 2.Ksetragar.ita. | 4.Jyotisa Patala |
The content and the area covered by Patigar (PT) and Trisatika (TS) of Sridharacarya and Ganita-Sara-Samgraha (GSS) of Mahavlracarya is almost the same. Now that it has been finally accepted that Srfdhara is prior to Mahavlra, we make a comparative study of the contributions o: these two mathematicians of Jaina Traditions.
During this comparative study, we get three types of siraations :
1. Srldhara is better than Mahavlra
In some cases like the volume of the sphere and the urea of the segment of the circle we find that results of Srldhara more accurate In fact Sridhara and Mahavlra both adopted different techniques. The technique adopted by Mahavlra yields inferior results. (See Table, Sr. No. 1 & 2)
In the Patiganita (PT) we find some results which are neither given in Trisatika (TS) of Srldhara nor in the Ganita-Sara-Samgraha GSS) of Mahavlra. Why are they not given by Srldhara in TS? Are they left by both the mathematician being complicated or because they are not much useful for the beginners which is the target group for TS and GSS? See Table, Sr. No. 3-6) Rule for area of trapezium given in Tristika is correct and still in use while it is not properly delt by Mahavlra in GSS. (See Table, Sr. No.-7
2. Srldhara and Mahavlra derived the same results
In some cases like the area of traingle and quadrilateral. Rule of three and some problems related to fractions, both mathematicians draw upon the same original source. Many texts of Karnanuyoga section which contains a lot of Material of Mathematician’s interest like in TiloyapannattI, commentaries on Satkhandagama (Which are not available at present), Lokavibhaga (Prakrta) and commentaries on Tattvarthasutra may have been those original sources from which they have borrowed.
It is clearly admitted by Mahavlra in G.S.SGanlta-Sara-Samgraha, Ch.-l, Verse 17-19.. (See Table, No. 8-13).
3. Mahavlra is better than Sridhara
In some cases, we find that the results of Mahavlra are better. It is very natural due to the development of knowledge. There are many things which were discussed by Srldhara but Mahavlra discussed them in a better way later and therefore the results given by Mahavlra are more accurate. (See Table, Sr. No. 14 to 21.)
