Mathematical Contributions of Sridharacarya and Māhavirācārya : A Comparative Study

Mathematical Contributions of Sridharacarya and Māhavirācārya : A Comparative Study
Abstract During the golden period of Indian Mathematics (500-1200 A.D.) the work of 5 mathematicians namely Aryabhata Brahmagupta, Sridhācārya, Mahāvirācārya and Bhāskarācārya 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. Sridharācārya (799 A.D.) and Mahāvirācārya (850 A.D.) belong to the Jaina tradition. While the period of Matāvirācārya is fixed about 850 A.D. but the period of Sridharācārya has remained controversial for a long period. Some place him in 10th C.A.D. (after Mahāvira) while some others during 8th C.A.D. (before Mahāvira). Due to it the contributions of Sridhara have not been evaluated properly yet. An attempt has been made here to make a comparative study between the works of Sridharācārya and those of Mahāvirācārya. The Jaina literature of Karanānuyoga 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 relates with Yojana, Rajju, Palya, Sāgropama etc. Which though useful for the understanding of the structure of the cosmos, has nothing to do with our daily life. The works of Sridharācārya (Sridhara) and Mahāvirācārya (Mahāvira) come under the category of Loukika Ganita. Following texts are undoubtedly written by Sridharācārya are available at present :

1.Pātiganita

2.Pātiganitasāra (Ganitasāra or Trisatikā)

3.Jyotirjnānavidhi 4.Ganitāpancavimsi

Bijaganita is another famous book of Sridharācārya which is not available at present but the quotation form it can be seen in the Bijaganita of Bhāskarācārya. The credit of Vrhatpāti and Navamsati is given to Sridhara but as they are still not available and we can’t finally admit that these works were composed by Sridhara and whether they are in any way different from the existing works mentioned above. Ganita-sāra-samgraha is only available text of Mahāvirācārya. The credit of following four texts is also given to Mahāvirācārya but they are not available nor there is there any definite proof of their composition 1.Sattrisatikā, 2.Ksetraganita, 3.Chattisapurvā, prati uttara pratisaha 4.Jyotisa Patala The content and the area covered by Pātiganita (PT) and Trisatikā (TS) of Sridharācārya and Ganita-Sāra-Samgraha (GSS) of Mahāvirācārya is almost the same. Now that it has been finally accepted that Sridhara is prior to Mahāvira, we make a comparative study of the contributions of these two mathematicians of Jaina Traditions. During this comparative study, we get three types of situations :

1.Sridhara is better than Mahāvira.
In some cases like the volume of the sphere and the area of the segment of the circle we find that results of Sridhara more accurate. In fact Sridhara and Mahāvira both adopted different techniques. The technique adopted by Mahāvira yields inferior results. (See Table, Sr. No. 1 & 2) In the Pātiganita (PT) we find some results which are neither given in Trisatikā (TS) of Sridhara nor in the Ganita-Sāra-Samgraha (GSS) of Mahāvira. Why are they not given by Sridhara 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 Tristikā is correct and still in use while it is not properly delt by Mahāvira in GSS. (See Table, Sr. No.-7).

2.Sridhara and Mahāvīra derived the Same results.
In some cases like the area of tringle and quadrilateral, Rule of three and some problems related to fractions, both mathematicians draw upon the same original source. Many texts of Karnānuyoga section which contains a lot of Material of Mathematician’s interest like in Tiloyapannatti, commentaries on Satkhandāgama (Which are not available at present), Lokavibhāga (Prākrta) and commentaries on Tattvārathasutra may have been those original sources from which they have borrowed. It is clearly admitted by Mahāvira in G.S.S.8 (See Table, No. 8-13).

3.Mahāvira is better than Sridhara.
In some cases, we find that the results of Mahāvira are better. It is very natural due to the development of knowledge. There are many things which were discussed by Sridhara but Mahāvira discussed them in a better way later and therefore the results given by Mahāvira are more accurate. (See Table, Sr. No. 14 to 21.)