Jaina School Of Indian Mathematics

Author(s): Dipak Jadhav
Publisher: Indian Journal of History of Science
Catalog link: https://jainqq.org/explore/034015/1

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Indian Journal of History of Science, 52.3 (2017) 316-334 1. INTRODUCTION The part played by the Jainas in the development of mathematics in ancient and medieval India is very significant. In accordance with their requirements they followed their own line of adoption and development of mathematics, which is generally called the Jaina school of mathematics. It, being developed within the intelligentsia of Indian mathematics, is, hereupon and in his previous papers', termed "the Jaina school of Indian mathematics" by the present author. DOI: 10.16943/ijhs/2017/v52i3/49161 The Jaina School of Indian Mathematics* Dipak Jadhav** (Received 13 January 2016; revised 13 July 2017) Abstract The Jaina school of Indian mathematics had a considerable standing. The school, on the basis of theorization, could be divided into the canonical class and the exclusive class. The treatises of the former contain mathematics along with discussion on Jaina canons while those of the latter are composed exclusively on mathematics. The object of the former was to demonstrate canonical thoughts including on karma and cosmos using mathematics while that of the exclusive class was to provide mathematics education to the contemporary civil life. Besides laukika ganita (worldly mathematics) and lokottara ganita (post-worldly mathematics) the paper also addresses some related issues. Key words: Canonical class, Exclusive class, The Jaina school of Indian mathematics. The school is suggested to have been divided into the canonical class and the exclusive class. A large number of papers? emphasizing the canonical class and the exclusive class have Historical Notes already been reported by the present author before. He first elaborated them in his paper appeared in 2004 (Jadhav 2004, p. 37), and thereafter in his doctoral thesis (Jadhav 2013, pp. 34-48). In this paper we are going to deal with them in detail. However, there is a general impression, especially outside India, that there was not any organized school of mathematics in ancient and medieval India except the Kerala school of astronomy and mathematics. On the Jain school of Indian mathematics there is a general concern, again outside India, whether Jain mathematics should be treated as a school rather than a specific tradition, closely related to mainstream of classical Sanskrit mathematical writings, and why the mathematicians like Sridhara and Mahavira be * Invited talk, except for a few changes and additions, delivered at International Conference on Science and Jaina Philosophy, held at Indian Institute of Technology, Bombay during January 8-10, 2016. The author takes this opportunity to thank the organizers, including Prof. Samani Chaitanya Prajna, of the conference for inviting him. The paper is dedicated to the memory of Prof. L. C. Jain (1926-2015). **Lecturer in Mathematics and Principal-in-charge, Govt. School of Excellence, Barwani (M. P.) India. E-mail: dipak jadhav17@yahoo.com Jadhav 2003, p. 53; 2004, p. 37; 2006, p. 75; 2008, pp. 139 and 146-147; 2009, p. 52; 2014, pp. 260-261 and 263-265; Jadhav and Jain 2003, p. 91; Jadhav and Padmavathamma 2002, p. 31. 2 Jadhav 2001, p. 94; Jadhav and Padmavathamma 2002, pp. 51-53; Jadhav 2002b, pp. 251-252; Jadhav and Jain 2003, pp. 108, 113-117; Jadhav 2004, p. 37; 2006, p. 75; 2008, p. 139; 2013, pp. 140 and 146-147; 2014, pp. 260-261; Jadhav and Jain 2016, pp.190-204

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HISTORICAL NOTE: THE JAINA SCHOOL OF INDIAN MATHEMATICS 317 included into exclusive class rather than as members of the mainstream classical Sanskrit mathematical tradition, who simply happen to be Jainas. The present paper is mainly aimed at justifying and discussing the division of the Jaina school of Indian mathematics into the canonical class and the exclusive class. This will be done by means of theorization. In this regard, a set of factual ideas will be developed about the school in order to find some basis. By studying the way in which its treatises contain mathematics along with canonical discussion or exclusively and its mathematicians treat, we will explain how it fits into the above classes. In order to avoid any misunderstanding regarding the exclusive class against laukika ganita (worldly mathematics) and the canonical class against lokottara ganita (postworldly mathematics) an appropriate discussion containing a comparative analysis of these two different divisions has been accommodated. the essence of mathematics') of Mahavira (c. 850) published in 1912, opines that: "The answer to the questions as to the relation between the schools of India cannot yet be easily given. At first it would seem a simple matter to compare the treatises of the three or four great algebraists and to note the similarities and differences. When this is done, however, the result seems to be that the works of Brahmagupta, Mahaviracarya and Bhaskara may be described as similar in spirit but entirely different in detail. For example, all of these writers treat of the areas of polygons, but Mahaviracarya is the only one to make any point of those that are re-entrant. All of them touch upon the area of a segment of a circle, but all give different rules. The so called janya operation is akin to work found in Brahmagupta and yet none of the problems is the same. The shadow problems, primitive case of trigonometry and gnomonics, suggest a similarity among these three great writers and yet those of Mahaviracarya are much distinct than the one to be found in either Brahmagupta or Bhaskara and no questions are duplicated (Padmavathamma, 2000, p. 762)." Smith accepts as early as in 1912 in more or less clear terms that there were the schools of mathematics in ancient and medieval India. In ancient India, mathematics was not separated from astronomy. In fact, the former was developed for the service of the latter. It is now recognized that there was Bramapaksa in Indian classical mathematical astronomy or Bramagupta school of Indian astronomy after the name of Indian mathematician and astronomer Brahmagupta (628 CE) (Plofker, 2014). As far as the Jaina school of Indian mathematics is concerned, we shall see that it sustained for more than two thousand years adopting, developing, following and practicing certain kinds of mathematical thoughts in ancient and medieval India. 2. APPROVAL OF THE JAINA SCHOOL OF INDIAN MATHEMATICS Every academic discipline, from old theology to modern science and technology, has competing theories and perspectives with which it grows. Mathematics has been no exception. For example, John Napier (1550-1617 CE) and Jobst Burgi (1552-1632 CE) discovered logarithms, but through an entirely different line of approach. The former's approach was geometric while the latter's was algebraic. Long before them the Jaina school of Indian mathematics approached logarithms on the basis of the number of possible divisions of a quantity by two (Jadhav 2002a; Jadhav 2003; Jadhav 2014). The studies made on mathematical thoughts developed in ancient and medieval India and about their followers make us to appreciate that the schools of some sort did exist. David Eugene Smith, in the introduction written by him to the Ganita-sara-sangraha ('Compendium of

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318 INDIAN JOURNAL OF HISTORY OF SCIENCE Bibhutibhusan Datta appears to be the first historian of mathematics who wrote a paper of which title contains the name of the school. The paper, which he wrote in 1929, is "The Jaina school of mathematics" (Datta 1929). It was aimed at professing, although he did not claim so, an account of the mathematical achievements of the Jainas. It was based on those sources that he collected up to that time. Most of them were of the Svetambara tradition. In 1934, SK Das wrote "The Jaina school of astronomy" (Das 1934). It gave the details of the astronomical and cosmographical speculations of the Jainas. LC Jain produced four papers. The first was "On the Jaina school of mathematics" (Jain, LC, 1967). It was also aimed at professing a brief account of the mathematical achievements of the Jainas but the sources that he used were different from those used by Datta. Those sources were of the Digambara tradition. "Jaina school of mathematics" was the second one which he produced (Jain, LC, 1975). It was a study in Chinese influences and transmissions. The third one was "The Jaina schools of mathematical sciences" (Jain, LC, 1992). It describes that there have been two Jaina schools of mathematics in India, the Digambara and the Svetambara. The former held proficiency in the symbolico- mathematical theory of karma, whereas the latter seems to be more interested in astronomy and astrology. "The Jaina school of mathematical atical philosophy" was the fourth paper (Jain, LC, 2000) which discussed how the Jainas approached their philosophy all the way through mathematics. Apart from the above four papers, LC Jain wrote five more papers that contain the expression "Jaina school of mathematics" in their respective titles, namely (1) "Set theory in Jaina school of mathematics" (Jain, LC, 1973), (2) "On certain mathematical topics of the Dhavala texts (the Jaina school of mathematics)" (Jain, LC, 1976), (3) "Perspective of system-theoretic technique in Jaina school of mathematics between 1400-1800 CE" (Jain, LC 1978), (4) "System theory in Jaina school of mathematics" (Jain, LC, 1979), and (5) "System theory in Jaina school of mathematics II" (Jain, LC and Jain, Meena 1989). Despite all these publications, the expression "the Jaina school of mathematics" did not get an essential amount of exposure at international level. One of the reasons behind this situation may have been that the above papers were published in less known journals except in Indian Journal of History of Science. Another may be that it was not recognized that the mathematicians of Jaina faith shared common mathematical thoughts to a great extent mathematical thoughts irrespective of the languages they used to compose their treatises. For example, (1) the Jainas shared 1o as the value for it for the long period commencing from 500 BCE at least to the time of Todaramala (1720 CE-1767 CE) (Jadhav, 2013, pp. 502-517 and 528-538), and (2) Thakkura Pheru borrowed most of the rules into the Ganita-sarakaumudi ("Moonlight of the essence of mathematics") composed by him in Apabhramsa from the Trisatika composed by Sridhara in Sanskrit (see SAKHYA 2009) With the assessment of the situation RC Gupta approves, while writing a note on the research work done by L C Jain, the expression "the Jaina school of mathematics" in the following words. "The Jaina school of mathematics was one of the most remarkable institutions of ancient India. Its contribution in the development of scientific thought especially as part of philosophicmathematical thinking may be regarded as quite significant and is a known fact to some extent. But the paradoxical situation is that it is yet to find due place in the historical expositions of the development of mathematics in India, what to say of that in the world. Nevertheless, in the pursuit of scientific thinking the depth of Jain philosopher-mathematicians is comparable to that of Greece. For

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HISTORICAL NOTE: THE JAINA SCHOOL OF INDIAN MATHEMATICS 319 from right to left. This is communicated in the Adipurana of Jinasena as late as in 9th century (Jain, Pannalal 1993, v. 108, p. 356). It is, on the basis of this communication, not easy to corroborate that the school took, as Lord Rsabha belongs to the prehistoric period its initiation in his time. However, this is only meant to suggest the antiquity of the school. However, he is well known to the Vedic literature. Hence, according to TA Sarasvati example, they were the earliest to transcend the simplistic thesis that all infinities are equal (Gupta 1991, p. 88)." In fact, the number of truly devoted research scholars in the field of the study of the Jaina school of Indian mathematics is relatively very small. There are certain difficulties which are responsible for this situation. According to RC Gupta, some of them are difficulties of ancient languages and peculiar terminology, of historical as well as scientific methodology, and of other technicalities involved in the Jaina texts (Gupta 1992, p. xi). One more difficulty Kim Plofker, while putting a remark of appreciation on the research work done by RC Gupta, maps out in the following words. "Gupta has published several key papers on the remarkable mathematical discoveries of the Jaina tradition, this has been a yeoman service especially in the case of the many works that have been almost inaccessible to anyone not closely linked with the Jaina canon (Plofker 2009b, p. 116)." Despite the aforesaid difficulties the Jaina school of Indian mathematics is a fascinating field of ancient and medieval Indian mathematics to be explored. "it is more likely that this dissident faith Ki.e., Jainism, revolting against sacrificial killing, was quite an old rival to the Vedic faith or that it had taken root in India even before the Vedic faith. The mathematical knowledge contained in the Jaina writings should therefore have been more or less parallel to that in the Vedic literature (Sarasvati 1979, p. 61)." On the other hand, according to L C Jain, the school appears to have originated soon after Lord Mahavira (599 BCE-527 BCE), the twenty fourth and last Tirtharkara in the history of Jainism. It means that the school flourished in India around the same period in which the school of Pythagoras (572 BCE-510 BCE) flourished in Greece. It was formed mainly of some niggamtha (Skt. nirgrantha, outwardly and inwardly free from worldly ties) ascetics who left a few record of their knowledge (Jain, LC 1967, p. 265). The present author, however, believes that the Jaina school of Indian mathematics was in cradle prior to Lord Mahavira, if existed. It started to flourish from Lord Mahavira's own time and came into black and white a little long after him. And it developed along with the progress and expansion of Jainism in India. 3. THE DIVISION OF THE SCHOOL INTO THE CANONICAL CLASS AND THE EXCLUSIVE CLASS The Jaina school of Indian mathematics is said to have initiated its activity in the time of Lord Rsabha, the first and foremost Tirtharkara in the history of Jainism. He is said to have taught numbers to his daughter Sundari with his left hand * For the details regarding the example cited in this passage, see Singh, Navjyoti 1991, p. 229. 4 For the research work done by RC Gupta, see Hayashi 2011. 5 Also see Jain, Anupam 1994, p. 127 where it is also stated that there are the other texts such as the Bhagavati Sutra, the Puranasara Samgraha of Damanandi, the Satrunjaya Kavya etc that document that Sundari learnt mathematics from her father. 6 Johar 2000, pp. 46-51. Also see Jain, H. L. 2000, pp. 3-28; Kumar 1997, pp.44-45; McEvilley 1996, pp. 6-20; Ranga 2000, pp. 73-75. 7 Johar 2000, pp. 46-51. Also see Jain, H. L. 2000, pp. 3-28; Ranga 2000, pp. 73-75.

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320 INDIAN JOURNAL OF HISTORY OF SCIENCE The teachings of the last Tirthankara were systematized by their disciples into doctrinal theories called canons (agamas or siddhantas) or sacred scriptures (srutis). Agamas refer to that which have come down to us from Lord Mahavira. They are called siddhantas because they are the fundamentals of his speeches. According to the unanimous tradition of the Digambara and Svetambara Jainas, the teachings of Lord Mahavira were arranged in twelve Books called angas by his disciples and successors, each anga has been called a sutta which is sanskritized as sutra. Those twelve argas including the Thana (Skt. Sthananga) and the Vikkhapannatti or Viyahapannatti (Skt. Vyakhyaprajnapti or better known as the Bhagavati Page #6
HISTORICAL NOTE: THE JAINA SCHOOL OF INDIAN MATHEMATICS over 2000 years the Jainas adopted, developed, followed and practiced certain kinds of mathematical thoughts as a school in ancient and medieval India. It was very vast and wide. Those treatises that are on the Jaina canons and contain mathematics for their discourse are placed in the canonical class. For example, the Bhagavati Sutra refers to minimum number of points (paesas, Skt. pradesas) required to construct each of the eleven formations as it was essential for the discussion therein. Distinction has been drawn between odd and even number of points. Those numbers have been recognized as figurate numbers (Jadhav 2009, pp. 35-55). The treatises that have been composed by the authors of Jaina faith exclusively on mathematics are placed in the exclusive class. The contents in the 3.2. Exclusive class treatises of this class are cent per cent mathematical ones. For example, the well known treatise of this class is the Ganita-sara-sangraha of Mahavira (c. 850 CE). The division of the Jain school into these two classes is based on factual ideas and may be appreciated as explianed below separately: sometimes occurs in the form of rules and results in their treatises and every so often occurs in the functional form. For example, the laws of logarithms to the base two (Jadhav, 2002a) and combinatorics of tuples (Jadhav and Jain, 2016) are available in the form of rules in the Trilokasara and the Gommatasara (Jivakanda) respectively whereas system theory of its own kind is available in the functional form (Jain, LC 1979; 1989; Gupta, 1993, p. 24). 321 Some of the mathematicians of this class are listed in Table 1. This does not however profess to be a complete list of the mathematicians of this class. In fact, it is a small list that covers only the prominent mathematicians. The treatises of this class are exclusively treatises is mathematics and only mathematics. It written on mathematics. The subject matter of their usually, but not at all times, happens to be at complete course on arithmetic and mensuration to cater the needs of both students and civil activities. The authors of this class were originally mathematicians except for few ones. Some of the mathematicians of this class are listed in Table 2, however it does not profess to be a complete list of the mathematicians of this class. In fact, it covers only the prominent mathematicians. 3.1. Canonical class The authors of this class were not mathematicians alone. They were the authors of the canonical or the post-canonical works. In other words, they were the authors of philosophy or ontology or metaphysics or cosmology or Karma theory or any combination of them oriented treatises. Historians of mathematics put them into the rank of mathematicians as there is found a good deal of knowledge of mathematics in their treatises. Mathematical material found embedded in their treatises seem to have been developed or dealt by them in accordance with their need and was accurately applied on cosmological system or Karma system or used for some philosophical discussion. It is available in abundance. It 9 It is inexactly written the Bhasa Vacanika. See Bharilla 1999, pp. 101-102. Unlike in the treatises of the canonical class except the Sthananga sutra (Madhukara 1992, Chapter X, sutra 100, p. 720) that suggests the ten topics for discussion in samkhyana (computation) and the Trilokasara-Bhasatika of Todaramala (1720-1767 CE) (Sastri, Manoharalal 1918, pp. 1-22, and Bharilla 1999, p. 104) and like in those of the non-Jaina mathematicians such as in the Brahma-sphuta-siddhanta of Brahmagupta (628 CE), the subject matter in the treatises of the exclusive class is broadly divided into two sections. One is parikarma (logistics) and

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322 INDIAN JOURNAL OF HISTORY OF SCIENCE Table 1: Canonical class Mathematician Sect S. No. Major Works Written in Anonymous (c. 500 BCE) Anonymous (c. 500 BCE) Svetambara Svetambara Prakrit Prakrit 2 3 Anonymous (c. 500 BCE) Svetambara Prakrit Surya Prajnapti (Madhukara 1995) Jambudvipaprajnapti (Sastri, Chhaganlal 1994) Jivajivabhigama Sutra (Madhukara 1989, Part I and 1991, Part II) Uttaradhyayana Sutra (Madhukara 1991, Publication No. 19 ) Sthananga Sutra (Madhukara 1992) Bhagavati Sutra (Deleu 1970) Prakrit Prakrit Prakrit 6 Anonymous Svetambara (c. 300 BCE or earlier) Anonymous(c. 300 BCE) Svetambara Sudharma Svami Svetambara (c. 300 BCE or earlier) Puspadanta and Bhutabali Digambara (between 87 CE and 156 CE) Umasvati /Umasvami Svetambara/ OBCE and 219 CE) Digambara 7 Satkhandagama (Jain, H. L. et al. 1996) Prakrit 8 Sanskrit Tattvarthadhigama Sutra Bhasya (Surisvara 1994) Jambudvipa Samasa (Srisatyavijaya 1923) Anuyogadvara Sutra (Madhukara 1987) Jyotiskarandaka (Anonymous 1928) Sanskrit 9 10 Svetambara Svetambara Prakrit Prakrit 11 Digambara Tiloyapannatti (Patni 1997) Prakrit Aryaraksita(3rd century CE) Anonymous Vallabhicarya (c. 300 CE) Yativrsabha (between 176 and 609 CE) Jinabhadra Gani (609 CE) Virasena (816 CE) Nemicandra (981 CE) 12 13 14 Svetambara Digambara Digambara Prakrit Prakrit Prakrit Prakrit Prakrit Sanskrit 15 Digambara Madhavacandra Traividya (c. 982 CE) Padmanandi (1000 CE) 16 Digambara Prakrit Brhatksetrasamasa (Vijayaji 1988) Dhavala (Jain, H. L. et al. 1996) Trilokasara (Mukhtara and Patni 1975) Gommatasara (Jain, G L. and Jain, S. L. 1919) Labdhisara (Jain, G. L. and Jain, S. L. 1919) Commentary on the Trilokasara (Mukhtara and Patni 1975) Jambudvipapannattisamgaho (Upadhye and Jain 1958) Vrttis (commentaries) on the the Bhagavati Sutra, Sthananga Sutra etc (Jain, Anupam 2008, pp. 45-47) Commentaries on the Jyotiskarandaka etc (Anonymous 1928) Lokaprakasa (Vijayaji 1932) Samyakjnanacandrika (Jain, G. L. and Jain, S. L. 1919) Trilokasara-Bhasatika (Sastri, Manoharalal 1918) 17 Svetambara Sanskrit Abhayadeva Suri (1015-1078 CE) 18 Malayagiri (1080-1172 CE) Svetambara Prakrit 19 20 Vinayavijaya Gani (1639 CE) Todaramala (1720-1767 CE) Svetambara Digambara Sanskrit Dhundhari Dhundhari

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HISTORICAL NOTE: THE JAINA SCHOOL OF INDIAN MATHEMATICS the other is vyavahara (determination). The parikarma (logistics or operation) consists of fundamental arithmetic operations and other important ways of computation. The vyavahara (determination or practice or procedure) shares the topics of common concern. The topics suggested in the Sthananga Satra are parikarma, vyavahara, rajju, rasi, kalasavarna, yavat tavat, varga, ghana, vargavarga, and kalpa.10 Brahmagupta offers twenty parikarmas and eight vyavaharas. According to Prthudakasvami (c. 850 CE), a commentator of the Brahma-sphuta-siddhanta, sankalita (addition), vyavakalita (subtraction), gunana (multiplication), bhagahara (division), varga (square), vargamala (square root), ghana (cube), ghanamila (cube root), panca jati (five rules of reduction relating to the five standard forms of fractions), trairasika (the rule of three), vyasta-trairasika (the inverse rule of three). pancarasika (the rule of five), saptarasika (the rule of seven), navarasika (the rule of nine), ekadasarasika (the rule of eleven), and bhandapratibhanda (barter and exchange) are those twenty parikarmas and misraka (mixture). sredhi (progression or series), ksetra (plane figures), khata (excavation), citi (stack), krakacika (saw), rasi (mound), and chaya (shadow) are those eight vyavaharas (Datta and Singh 1935, p. 124). Sridhara (c. 799 CE) tenders twenty nine parikarmas and nine vyavaharas. He excluded ekadasarasika (the rule of eleven) from the twenty parikarmas offered by Brahmagupta and offers the other expression pratyutpanna for gunana (multiplication). The ten parikarmas added by him are the first eight parikarmas for bhinnas (fractions), one more rule of reduction relating to one more standard form of fraction, and jivavikraya (sale of living beings). One vyavahara added by him to the list of eight vyavaharas offered by Brahmagupta is sunya-tatva (principle of zero) (Shukla 1959, vv. 2-6, p. 2). 10 For their interpretations see Rajgopal 1991, pp. 1-8. 11 See Sastri, N. C. Jain 1947, p. 31. 323 The exclusive class was aimed at providing mathematics for the sake of worldly business. In other words, the object of this class was to provide mathematics education to common people according to the necessity of the contemporary civil life. This can be corroborated from the statements of the mathematicians of this class. In the beginning of the Trisatika Sridhara states that: navAjinaM svaviracitapATyA gaNitasya sAramuddhRtya / lokavyavahArAya pravakSyati zrIdharAcAryaH / / natva jinam svaviracitapatya ganitasya saramuddhrtya lokavyavaharaya pravakya Sridharacaryah "Paying homage to Jina, having excerpted the essence (sara) of mathematics (ganita) from the Pati ganita (algorithms) composed by himself, the teacher (acarya) Sridhara will state for the sake of worldly business." And the Pariganita of Sridhara as well is aimed at providing mathematics lokavyavaharartha ('for the sake of worldly business') (Shukla 1959, v.1, p.1). Rajaditya (12 century CE) claims that with a view to support all the scholars in the field of mathematics and help businessmen and common men better deal with their day to day transactions he wrote the Vyavahara-ganita (Padmavathamma et al 2013, vv. 1.12-1.14, pp. 5-6). For the sake of all people Thakkara Pheru expounded the Ganita-sarakaumudi after he had taken some material from the writings of the ancient teachers, especially from Mahavira (SaKHYa 2009, pp. xx-xxi) and Sridhara (SaKHYa 2009, pp. xxi-xxii), gained some from direct experience, and heard some from others (SaKHYA 2009, v. 1.2, pp. 9 and 45). Sridhara's faith whether he was a Saiva follower or a Jaina has been a great deal of controversy among scholars (Shukla 1959, Introduction, pp. xv and xxxv-xxxvi). The benediction referred to above is from an old palm

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324 INDIAN JOURNAL OF HISTORY OF SCIENCE Table 2: Exclusive class S. No. Mathematician Sect Major Works Written in Sridhara (c. 799 CE) Digambara Sanskrit Sanskrit Sanskrit Mahavira (c. 850 CE) Patiganita (Shukla 1959) Trisatika (Dvivedi 1899) Ganita-sara-sangraha (Padmavathamma 2000) Saftrimsika (Jain, Anupam 1982, 1988) Digambara Digambara Sanskrit Madhavacandra Traividya (c. 982 CE) Rajaditya (12th century CE) Digambara Kannada Vyavahara-ganita (Padmavathamma et al 2013) Ganita-tilaka-vrtti (Kapadia 1937) Svetambara Sanskrit Simhatilaka Suri (13th Century CE) Thakkara Pheru (c. 1265-c.1330 CE) Anonymous Svetambara Ganita-sara-kaumudi (SaKHYa 2009) Apabhramsa Prakrit Sanskrit Hemaraja (c. 1673) Digambara Tejasimha Suri (died in 1686) Svetambara Patana Mathematical Anthology (Hayashi 2006a) Ganitasara (Jain, Anupam 1988b) Istanka Pancavimsatika (Hayashi 2006) Hindi Sanskrit 9 leaf manuscript of the Trisatika, written in Kanarese script, discovered in the Jaina Library at Mudabidri in south Karnataka. It contains Jinam' whereas the other manuscript contains 'Sivam' (Dvivedi 1899, p. 1). N C Jain Sastri is of the opinion that the occurrence of the reading Sivam' is a deliberate change as such a custom of changing the benediction of a text is found in other texts too. He regards the reading Jinam' to be authentic. So he suggests that Sridhara was a Jaina (Sastri, N C Jain 1947, pp. 31-32). Anupam Jain and Jaychand Jain support N C Jain with scores of arguments (Jain, Anupam and Jain, Jaychand 1988, pp. 49-53). Mamata Agrawal has followed them (Agrawal 2001, pp. 41-43). were frequently used by L C Jain (Jain, LC 1961, pp. 222-231; 1973, p. 3; 1980, p. 43; 2007, p. 9). RC Gupta understands them in the way as follows. "The lokottara type of Jaina mathematics is somewhat of abstract and its higher level surpasses that of laukika mathematics. The laukika Jaina mathematics is mostly mensurational and is related to simpler problems of the type which we come across in ordinary life. It is covered by what we call elementary arithmetic, algebra and geometry. ... It is in the category of alaukika mathematics that the work of the Jaina School is unique. In fact, the remarkable achievement in this area clearly distinguishes the Jaina school of Mathematics from other ancient schools whether it is in India, or outside India. One is often surprised to find parallels of several modern mathematical concepts and notions in ancient Jaina texts (Gupta 1993, pp. 22-23)." The two expressions, laukika ganita and lokottara ganita, were rarely employed in the treatises of the Jaina school of Indian mathematics. 4. LAUKIKA AND LOKOTTARA GANITA The mathematics found in the treatises of the Jaina school of Indian mathematics is viewed into two categories. One is laukika ganita (worldly mathematics) and the other is alaukika ganita (non-worldly mathematics) or lokottara ganita (post-worldly mathematics). These expressions

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HISTORICAL NOTE: THE JAINA SCHOOL OF INDIAN MATHEMATICS 325 It is Todaramala who made use of the expression alaukika ganita in the Bhasatika, an introduction to the mathematics appeared in the commentary of Madhavacandra Traividya on the Trilokasara of Nemicandra (Sastri, Manoharalal 1918, pp. 122; also see Jain, L C and Trivedi, RK 1987, p. 365 and Bharilla 1999, p. 102). Long before him the expression laukikaganita was employed, in order to show how the rule for finding the meeting time of the sun and the moon can be applied on a worldly problem, by Bhaskara I (629 CE) outside the Jaina school of Indian mathematics (Shukla 1976, p. 131; also see Keller 2006, p. 127). Here it may be clearly concluded that mathematics on its own was not laukika (worldly) or lokottara (post-worldly) but because of its application it was considered to be laukika or lokottara. Bhaskara I (629 CE) seems to have been in contact with the treatises of the Jainas as he is found to have quoted five passages in Prakrit gathas (verses) in the Aryabhatiya-bhasya. Two of them state the following two formulae. (1) C 10d where is the circumference of a circle of diameter d, and (2) A: V10 ch/4) where A, C and h are the area, chord and height of a segment of a circle of diameter d. The value for at implied in these formulae is 10, which was then regarded as the subtle one. His purpose of touching upon the passages was to criticizing vio and value for it (Jadhav 2013, pp. 502-517 and 528538). The two adjectives, laukika and lokottara, do occur in the treatises of the school but with the term mana or pramana (measure). The Trilokasara refers to two kinds of mana (Skt. mana, measure), logiga (Skt. laukika, worldly) and loguttara (Skt. lokottara, post-worldly). The laukika mana is of six types while dravyamana (fluent-measure), ksetramana (space-measure), kalamana (timemeasure and bhavamana (thought-measure are the four lokottara manas (Mukhtara and Patni 1975, vv. 9-10, pp. 12-13). The Tatvarthavartika ("Explanatory of the meaning of the fundamental principles") of Akalanka (7th century) appears to be the first treatise that contains the classification of measure in this manner (Jain, Mahendra 1999, sutra 3.38, pp. 205-209). One of the six laukika manas is gani mana (or ganana mana or ganima mana, countingmeasure). One, two, three and so on are countingmeasures (Jain, Mahendra 1999, sutra 3.38, p. 205). Salaries, wages, provisions, income, expenditure, cost etc are, according to the Anuyogadvara Sutra, determined using it (Madhukara 1987, sutra 327, p. 239). On the other hand, two, three and so on are samkhyapramana (number-measure). It is, according to the Tatvarthavartika (Jain, Mahendra 1999, sutra 3.38, p. 206) and the Trilokasara (Mukhtara and Patni 1975, v. 12, p. 13), one of the two types of dravyamana. It has three divisions: samkhyata (numerate), asamkhyata innumerate) and ananta (infinite). Asamkhyata (innumerate) is further divided into three sub-classes: parita (preliminary), yukta (proper) and asamkhyata innumerate). Ananta (infinite) is also divided into three sub-classes: parita (preliminary), yukta (proper) and ananta (infinite). Each of samkhyata (numerate), three sub-classes of asamkhyata (innumerate) and three sub-classes of ananta (infinite) is again divided into jaghanya infimum), madhyama (intermediate), and utkrsta emphasizing on Aryabhata's new value value 20000 (Shukla 1976, p. lvi; also see below v. 2.10, pp. 72-73). He seems to have cited those five passages from some contemporary or earlier work. That work according to B B Datta, must have been of Jaina authorship (Datta 1936, p. 41). It may be noted that 10 continued to be used by one and all in the school as the value for at for the long period of more than two thousand years commencing from 500 BCE. The mathematicians of the canonical class except Virasena (816 CE) and Nemicandra (981 CE) never used any other

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326 INDIAN JOURNAL OF HISTORY OF SCIENCE (supremum) (Mukhtara and Patni 1975, vv. 1352, pp. 14-49). This twenty one-fold number system12 was used by the Jainas to demonstrate their Karma theory and cosmology. The other dravyamana is upamapramana (simile-measure). It is of eight kinds. They are playa, sagara, sucyangula, pratarangula, ghanangula, jagacchreni (or jagatsreni), lokapratara (or jagatpratara), and loka (Jain, Mahendra 1999, sutra 3.38, pp. 206-208; for details also see Mukhtara and Patni 1975, vv. 12 and 92-112, pp. 13 and 86-109). (number-measure) is alaukika ganita if we confine ourselves to samkhyapramana alone, and all of the four lokottara manas (post-worldly measures) are alaukika ganita if our concern is both samkhyapramana and its area of application. 5. DISCUSSION 5.1. No directive, except the literal meaning of laukika ganita and lokottara ganita, has been issued by the Jainas, which shall help us to determine what mathematical thought will be placed in the category of laukika ganita and what one in the category of lokottara ganita. This intricacy will get illustrated if the following example is paid attention. The formula, This twenty one-fold number system seems to have been said alaukika ganita by Todaramala (1720-1767 CE). Almost one century earlier than him it was called lokottara ginative), for finding the volume of a sphere (post-worldly reckoning) by Hemaraja in his Ganitasara (Jain, Anupam 1988b, v. 4, p. 56). The same seems to have been discussed in the Alaukika-ganita ("Non-worldly mathematics") of an anonymous author, of which copy is said to have been preserved in Pancayati Mandira, Delhi (Jain, Anupam 1988a, p. 25). "Gommanasara grantha mem upayogi alaukika ganita ki kucha samjnam ka khulasa (Eng. Revelation of some terms, applied in the book Gommatasara, of nonworldly mathematics)", a write-up inserted just after the foreword (pragnivedana) into the Gommatasara (Karmakanda) of Nemicandra, edited by Khubachand Jain, reads that samkhyapramana (number-measure) of twenty one-fold and upamapramana (simile-measure) of eight kinds along with ksetramana (spacemeasure) that contains pradesa (indivisible part of space (akasa)), kalamana (time-measure) that contains samaya (indivisible part of time), and bhavamana (thought-measure) that contains avibhagapraticcheda (indivisible correspondingsection of omniscience (kevalajnana)) pertain to alaukika ganita (Jain, Khubchanda 1986/1913, pp. 6-11). On the basis of the above facts, the present author is of the opinion that samkhyapramana whose diameter is d, referred to by Mahavira (c. 850 CE) in the Ganita-sara-sangraha (Padmavathamma, 2000, v. 8.28%, pp. 612-613) is also found in the Trilokasara of Nemicandra (c. 981), that too employed in the process of finding the first asamkhyata (innumerate) (Mukhtara and Patni 1975, v. 19 first hemistich, p. 25). Theorization of the school into the two classes does not put any hurdle to accept that the formula is a content of the treatises of the both classes. On the other hand, it is very difficult to explain whether the formula is laukika or lokottara if the area of its application is taken into consideration. 12 For understanding this system it is suggested to read Singh, Navjyoti 1991, pp. 209-232. Laukika ganita and lokottara ganita are the two divisions of mathematics in the school while the canonical class and the exclusive class are the two divisions of the school. The canonical class does not stand for lokottara ganita although most of the latter are the contents of the treatises belonging to the former. Similar is the case of the exclusive class and laukika ganita. 5.2. Though the list of the mathematicians of the exclusive class is smaller than that of the canonical

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HISTORICAL NOTE: THE JAINA SCHOOL OF INDIAN MATHEMATICS class, the mathematicians contained in it are very important with regard to ancient and medieval Indian mathematics. The reason behind to be the small list is that the Jaina school of Indian mathematics has generally been under the domination of its canonical class. The authors of the exclusive class pay obeisance to those of the canonical class and state that they have gleaned material from the treatises of the canonical class (Padmavathamma 2000, vv. 1.17-1.19, p. 6; v. 1.70, p. 20; v. 7.49, p. 453). This is why the status of the canonical class can be said to be upper than that of the exclusive class in the Jaina school of Indian mathematics. This claim and others, including the one regarding laukika ganita and alaukika ganita, of the present author get support from the following facts revealed and views expressed by Catherine Morice-Singh. "During the one hundred and odd years since 1912, much has been written on the Ganitasarasangraha's mathematical contents, but no attempt has been undertaken to re-examine the text established by Rangacharya<, its first editor,> nor to trace the Jaina philosophical and cosmological elements in it, in spite of the fact that Jaina Studies has developed rapidly during the 20th century. The importance given to mathematics (ganita) by the Jaina thinkers who wanted to quantify in full details the entities existing in the universe is now well known, and the technical and specialized Jaina vocabulary attached to it is also better understood (Morice-Singh 2016, p. 41)." Here it may be noted that Mathematiques et cosmologie jaina Nombres et algorithmes dans le Ganitasarasangraha et la Tiloyapannatti is her doctoral thesis. "In the Ganitasarasangraha," further writes she, "the exceptionally developed and well-written introductory chapter supplies a great amount of details about the organization of mathematical topics and many explicit references to the Jaina context. The Ganitasarasangraha's 327 first two chapters are then both fundamental, and in my thesis I have proposed a deep study of them along with a French translation. In order to reexamine Rangacharya's text and to identify his editorial choices, I examined some manuscripts available at the Government Oriental Manuscripts Library (Madras) hoping to find traces of his work and, in order to get a wider view on the elements of mathematics linked to the Jaina universe, I explored excerpts of different original texts (Dhavala, Trilokasara, etc.) but mainly of the Tiloyapannalli, a Prakrit text (-6th to 9th century) belonging to the same Digambara tradition . The study of the impressive mathematical content of these texts has led me to propose answers to the two questions about the Ganitasarasangraha's structure... To express numbers, Mahaviracarya makes an intensive use of the word-numeral (bhuta-samkhya) system, choosing often words belonging to the Jaina terminology, as for example lesya, associated to number 6. Here, Rangacharya, probably not knowing the meaning of lesya, deliberately corrected it into lekhya (v. 2.34)>>, which is incorrect... Mahaviracarya has, in every aspect of his work, managed to retain the essential and to separate "alaukikaganita" from "laukikaganita" without departing from the teachings of his tradition. For instance, the units of length in the Ganitasarasangraha ( 1.25) start with the atom (anu) which is made of an ananta quantity of ultimate particles (paramanu), and an asamkhya number of samaya is required to constitute the first unit of time, the avali ( 1.32): The distinction between ananta and asamkhyata is kept here, even if its utility doesn't appear in a mathematical text (Morice-Singh 2016, pp. 41-43)." 5.3. As far as their chronological order is concerned, the exclusive class must have appeared later than the canonical class. We do not know who all were the mathematicians of the exclusive class prior to Sridhara. However, it is certain that this class was in existence long before him.

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328 INDIAN JOURNAL OF HISTORY OF SCIENCE Plofker, 2009a, p. 162). Rajaditya flourished either around 1120 CE in the royal court of the king Visnuvardhana, who reigned from 1111 CE to 1141 CE, of the Hoyasal dynasty (Padmavathamma et al, 2013, pp. xxiii-xxiv) or around 1190 CE in the court of the king Varaballala II, who ruled from 1173 CE to 1220 CE, whom he referred to as Visnunrpala (Padmavathamma et al., 2013, pp. xxiv-xxv). Thakkara Pheru held the positions during the period of the successive Sultans Alauddin Khalji (1296-1316 CE), Shihabuddin Umar (1316 CE), Qutubddin Mubarak Shah (1316-1320 CE) and Ghiyasuddin Tughluq (1320-1325 CE) at their treasuries (SaKHYa 2009, p. xiii). 5.5. It may need a separate paper to list and discuss the achievements of the Jaina school of Indian mathematics. Some remarkable achievements of the school, which distinguish the school from other ancient schools, R C Gupta has summarized as follows. Following Siddhasena Gani, the commentator of the Tattvarthadhigama Sutra Bhasya ("Commentary on the aphorisms of the learning and meaning of the fundamental principles") of Umasvati (some period between 150 BCE and 219 CE), B B Datta is, about the mathematical formulae quoted in the Tattvarthadhigama Sutra Bhasya by Umasvati, of the following opinion. "Umasvati's name has come down to us as a great writer on the Jaina doctrines, but not as a writer on mathematics. He is not even known to have ever devoted himself to a study of this science. Hence it will have to be concluded that the mathematical formulae quoted in his Tattvarthadhigama Sutra Bhasya were taken from some other treatise on mathematics known at his time (Datta 1929, pp. 126-127)." The time when it happened seems to be the one before which the exclusive class of the school began to come forward. The period preceding the fifth century CE or preceding the time of Aryabhata I (born 476) is considered to be the darkest period of the history of Indian mathematics (Singh, A. N. 1942, p. 4). It was that period during which the exclusive class seems to have been struggling for its executive shape and it finally came in that shape before or by Sridhara's time. This is why we do not find measure to have been classified into laukika and lokottara in the Anuyogadvara Sutra (3rd century CE) (Madhukara 1987, sutra, 327, p. 239). 5.4. The exclusive class seems to have helped the society a lot by producing the treatises exclusively written on mathematics as and when its mathematician got the seat in the court of the state. Mahavira seems to have worked at the court of the famous and benevolent ninth-century Rastrakuta king Amoghavarca Nrpatunga who ruled at Manyakheta in south India, much of what is known as Karnataka today as he has praised the king in glowing terms and wished for his prosperity in the Ganita-sara-sangraha (Padmavathamma, 2000, vv. 1.3-1.8, pp. 2-3 and "Closed and open number systems both finite and transfinite were developed. The Jainas had realized the notion of actual infinity in the realm of numbers, formulated the idea of cardinality, and thus made first attempts towards the calculus of transfinite numbers. Logarithms (especially to base two) were applied and their laws of combinations were made known. Mathematics of transfinite class (called ananta) was dealt. In fact, the mathematical operations developed to handle transfinite numbers, was one of the greatest achievements of the Jainas. The Jaina Karma system has been developed, like modern system theory, on the basis of several postulates and hypothesis, and utilizing such notions as that of one-to-one correspondence. Ideas of structuralism and functionalism of system theory have been developed. System-theoretic knowledge of maxima and minima was evolved. Several settheoretic relations are found quoted in Prakrit texts. Fourteen special divergent sequences have been discussed. ... Ten

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HISTORICAL NOTE: THE JAINA SCHOOL OF INDIAN MATHEMATICS 329 types of infinities are mentioned in canonical texts (Gupta 1993, p. 24)". These achievements are, according to R C Gupta, from ancient lokottara ganita of the school (Gupta 1993, p. 24). The present author too would like to point out that these achievements belonged to only the canonical class of the school. The exclusive class, which separated from the canonical class to provide mathematics education to common people, never referred to these achievements, perhaps for the reason that it could not find the areas of their application in the contemporary civil life. 5.6. We have noticed, although in brief, above that the Jaina school of Indian mathematics lasted for more than two thousand years. It not only prolonged but also it was varied in terms of using the languages, although Prakrit along with its different forms was the most used one among them, and having different purposes including the demonstration of canonical thoughts using mathematics and providing mathematics education to the civil society. The division, whatever it may be, of the school of this sort cannot be an ideal one. This is illustrated from the following examples. Madhavacandra Traividya was an immediate disciple of Nemicandra as he himself claims to be so (Mukhtara and Patni 1975, v. 1, p. 768). He wrote a very useful commentary on the Trilokasara of Nemicandra (981 CE). The most interesting, especially for the historians of mathematics feature about his commentary is that it contains rationales to mathematical rules given in the Trilokasara. He is said to have authored the Sattrimsika ("The textbook of thirty six logistics and determinations>'). It is the refined-essence (sodhya-sara) of the >-sara-sangraha of <Page #15
330 INDIAN JOURNAL OF HISTORY OF SCIENCE "classical Sanskrit mathematical writings" only if a few treatises, for example, those of Sridhara and Mahavira, of the exclusive class are assessed at face value. Sridhara was the most distinguished mathematician of his time. His reputation spread all over India. Similarly, Mahavira was a celebrated mathematician of his time. His fame rests on the Ganita-sara-sangraha. It was used as a text-book for centuries in the whole of south India. Their treatises being composed in Sanskrit, they appear to be members of the mainstream "classical Sanskrit mathematical tradition" but when we go in detail we find that the case is not so. For example, trapezium, especially isosceles trapezium, was a household geometrical figure for the Jainas. Each of the front and backside faces of the three fold universe of the Jainas is in this shape. Sridhara gave it so importance that he considered it primary figure.14 In Jaina cosmography, the middle universe is supposed to be a flat plane divided into an innumerable number of concentric annuli which are alternatively islands and seas. Following this concept, Mahavira coined the expressions bahincakravalavrtta (outer-annuluscircle i. e., the outer circle of an annulus) and antascakravalavrtta (inner-annulus-circle i. e., the inner circle of an annulus) (Padmavathamma, 2000, v. 7.6, p. 427; Jadhav, 2013, pp. 97-98 and 558). The expressions such as "Sanskrit mathematics and astronomy" (Plofker, 2010, p. 1), "Sanskrit mathematicians" (Hoyrup, 2012, p. 2), "Sanskrit formulas" (Plofker, 2001, p. 284), "Sanskrit mental-calculation algorithms" (Plofker 2009, p. 16), "Sanskrit geometry" (Plofker, 2009a, p. 28) and so forth are mostly popular in foreign publications on the history of Indian mathematics. Why these sorts of expressions are particularly followed in those publications is not known. However, "Sanskrit mathematics" refers to mathematics contained in the treatise composed in Sanskrit. "Sanskrit mathematicians" mean to be those mathematicians whose treatises are in Sanskrit. In the same manner we shall have to interpret the remaining expressions. Sometimes we come across the expressions such as "Prakrit mathematical work" (Plofker 2009a, p. 209). Indian researchers, if not at all, rarely employ them. Those expressions, consciously or unconsciously, intend to show a linguistic division of ancient and medieval Indian mathematics, especially to a common reader. In fact, original mathematical thoughts were developed in linguistically varied India's irrespective of language although Sanskrit has been the panIndian medium of intellectual discourse. In the section 5.5 of this paper, we have already noticed the highly original mathematical thoughts of the Jainas. All of them, belonging to the canonical class of their school, were composed in Prakrit. Most of them never found any place in the treatises composed in Sanskrit. A few of them had found some place in some Sanskrit texts such as in Madhavacandra Traividya's Sanskrit commentary on the Trilokasara of Nemicandra, but they were not paid any attention by the intelligentsia of the perceived "classical Sanskrit mathematical tradition". For example, the two important mathematical concepts addhached (Skt. ardhaccheda, logarithms to the base two) (Jadhav 2002a; 2003; 2014) and vaggidasamvaggida of a (Skt. vargitasamvargita of a, 'the self-power of a' or 'raising a to its own power' where a is a positive integer) (Jadhav, 2008) always remained untouched by the others including the exclusive class. 14 Dvivedi 1899, vv. 42-43 and examples vv. 80-81, pp. 30-32; Shukla 1959, v. 115, p. 161; examples vv. 122-124, pp. 161-162; vv. 126-127, p. 165; Jadhav 2013, pp. 157-160 and 558. 15 That how poor is the status of the mapping of the mathematical literature composed in the ancient and medieval Indian regional languages can be had from: Sarma 2011, pp. 201-211; SaKHYa 2009, p. xi.

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HISTORICAL NOTE: THE JAINA SCHOOL OF INDIAN MATHEMATICS 331 6. CONCLUDING REMARKS The Jaina school of Indian mathematics is an approved school in the field of history of ancient and medieval Indian mathematics. The division of the school into the canonical class and the exclusive is based on theorization. The approaches adopted for theorization can be seen in the sections three and five; the former initiates its process while the latter establishes it by means of analysis. One should not try to form the image of the school from the works of its exclusive class alone. The image which lies with its exclusive class is the outer one. The intrinsic image of the school lies with its canonical class. Dvivedi, Sudhakara (ed.). Trisatika of Sridhara. Chandraprabha Press, Benares, 1899. Gupta, R C. Introduction. In: L. C. Jain. The Tao of Jaina Sciences. Arihant International, Delhi, 1992, pp. viii xvi. Gupta, RC. New researches in Jaina mathematics: The work of Prof. L. C. Jain. Nanasayara 9 (1993): 22-27 and 96. Published first in: Jinamanjari 3.2 (October 1991): 88-94. Hayashi, Takao. A Sanskrit mathematical anthology. SCIAMVS 7 (2006a): 175-211. Hayashi, Takao. Istankapancavimsatika of Tejasimha. Ganita Bharati 28.1-2 (2006): 129-145. Hayashi, Takao. Current bibliography of Radha Charan Gupta. Ganita Bharati 33.1-2 (2011):137-172. Hoyrup, Jens. Sanskrit Prakrit interaction in elementary mathematics as reflected in Arabic and Italian formulations of the rule of three - and something more on the rule elsewhere. Ganita Bharati 34.1-2 (2012):1 28. Jadhav, Dipak. On values for it used in the Trilokasara. Ganita Bharati 23.1-4 (2001):91-100. Jadhav. Dipak. The laws of logarithms in India. Historia Scientiarum 11.3 (2002a): 261-267. Jadhav, Dipak. Nemicandra's rule for the volume of a sphere. IJHS 37.3 (2002b): 237-254. Jadhav, Dipak. Theories of indices and logarithms in India from Jaina sources. Arhat Vacana 15.4 (2003):53-73. Jadhav, Dipak. Theories of A.P. and GP. in Nemicandra's works. Arhat Vacana 16.2 (2004): 35-40. Jadhav, Dipak. Why do I assign 981 A. D. to Nemicandra? Arhat Vacana 18.1 (2006):75-81. BIBLIOGRAPHY Agrawal, Mamata. Acarya Sridhara evam unaka ganitiya avadana. Ph. D. Thesis (Unpublished). University of Meerut, Meerut, 2001. Alsdorf, Ludwig. The Pratyayas: Indian contribution to combinatorics (Translated from the German by S. R. Sarma). IJHS 26.1(1991):17-61. Anonymous (ed.). Jyotiskarandaka of Vallabhicarya (with Malayagiri's commentary). Sri Rsabhadevaji Kesarimaletyakhya Svetambara Sanstha, Ratlam, 1928. Bharilla, Hukumchanda. Pandita Todaramala: Vyaktitatva aura Karttrtva. Pandita Todaramala Smaraka Trust, Jaipur, 1999. First Edition in 1973. Das SK. The Jaina school of astronomy. Indian Historical Quarterly 8, (1934):30-42. Datta, B B. "The Jaina school of mathematics". Bulletin of the Calcutta Mathematical Society 21 (1929): 115145. Reprinted with the title: "The mathematical achievements of the Jainas". In: Studies in the History of Science in India (ed. D. Chattopadhyaya), New Delhi, 1982, pp. 684-716. Datta, B B. A lost Jaina treatise on arithmetic. The Jaina Antiquary 2.1 (1936):38-41. Datta, B. B. and Singh, A. N. History of Hindu Mathematics, Part I. Motilal Banarsidass, Lahore, 1935. Deleu, Jozef. Vivahapannatti (Bhagavai): Introduction, Critical Analysis, Commentary and Indexes. University of Ghent, Ghent. Jadhav, Dipak. On raising a number to its own power in ancient India. Ganita Bharati 30.2 (2008):139-149. Jadhav, Dipak. On the figurate numbers from the Bhagavati Sutra. Ganita Bharati 31.1-2 (2009): 35-55. Jadhav, Dipak, Mensuration in India from Jaina Sources. Ph. D. Thesis (Unpublished). Devi Ahilya University, Indore, 2013 Jadhav, Dipak. Nemicandra's rules for computing multiplier and divisor. IJHS 49.3 (2014): 260-267. Jadhav, Dipak and Jain, Anupam. Mensuration of circle in the Trilokasara. In: R. C. Gupta. Ancient Jain Mathematics. Jain Humanities Press, Canada and USA, 2003, pp. 89-177.

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332 INDIAN JOURNAL OF HISTORY OF SCIENCE Jadhav, Dipak and Jain, Anupam. Combinatorics as found in the Gommatasara (Jivakanda) of Nemicandra. IJHS 51.2.1(2016):190-205. Jadhav, Dipak and Padmavathamma. The Mensuration of a conch in ancient India. Arhat Vacana 14.1 (2002):3154. Jain, Anupam. Katipaya ajnata Jaina ganita grantha. Ganita Bharati 4.1-2 (1982):61-71. Jain, Anupam. Jaina ganitiya sahitya. Arhat Vacana 1.1 (1988a):19-40. Jain, Anupam. Hemaraja, vyaktitva evam krtitva. Arhat Vacana 1.1 (1988b):51-63. Jain, Anupam. Madhavacandra evam unaki Sattrimsika. Arhat Vacana 1.1 (1988c):65-74. Jain, Anupam. Bhagavana Rsabhadeva ki parampara mem vikasita ganita. Arhat Vacana 6.2 (1994):125-130. Jain, Anupam. Ardhmagadhi Sahitya mem Ganita. Jain Vishva Bharati University, Ladnun, 2008. Jain, Anupam and Jain, Jaychand. Kya Sridhara Jaina the?. Arhat Vacana, 1.2 (1988):49-54. Jain, G. L. and Jain, S. L. (eds. & trs.). Gommatasara (Jivakanda and Karmakanda), Labdhisara (inclusive of Ksapanasara) of Nemicandra (with various commentaries including Samyakajnanacandrika of Todaramala). Gandhi Haribhai Devakaran Jain Series, Calcutta, circa 1919. Jain, HL. Jaina Tradition in Indian Thought (ed. D. C. Jain). Sharada Publishing House, Delhi, 2000. Jain, HL. et al. (eds. and trs.). Satkhandagama of Puspadanta and Bhutabali (with the Dhavala commentary of Virasena and with Hindi translation). 16 vols. Jaina Samskrti Samraksaka Sangha, Sholapur, 1996. First Printed, Amaraoti, 1940-50. Jain, JP. Jaina Authors and their works. The Jaina Antiquary 32.2 (1979):8-18. Jain, Khubchanda (ed.). Gommatasara (Karmakanda) of Nemicandra (with Manoharlal Sastri's Sanskrit Chaya and abridged Hindi commentary). Srimad Rajacandra Asrama, Agas, Vikrama Samvat 2042 (= 1986 CE). First Edition in Vikrama Samvat 1969 (= 1913 CE). Jain, L C. Bharatiya lokottara ganita-vijnana ke sodhapatha. In: Acarya Bhiksu Smrti Grantha, Calcutta, 1961, pp. 222-231. Jain, L C. On the Jaina school of mathematics. In: Babu Chote Lala Jaina Smrti Grantha, Kolkata, 1967, pp. 265-292. Jain, L. C. Set theory in Jaina school of mathematics. IJHS 8.1-2 (1973):1-27. Jain, L. C. "Jaina school of mathematics (A study in Chinese influences and transmissions)". In: Contribution of Jainism to Indian Culture. Delhi-Varanasi-Patna, 1975, pp. 206-220. Jain, L. C. On certain mathematical topics of the Dhavala texts (the Jaina school of mathematics). IJHS 11.2 (1976):85-111. Jain, L. C. Perspective of system-theoretic technique in Jaina school of mathematics between 1400-1800 A.D. Jain Journal (October 1978): 49-66. Jain, L. C. System theory in Jaina school of mathematics. IJHS 14 (1979):31-65. Jain, L C. Agamom mem ganitiya samagri tatha usaka mulyankana. Tulasi Prajna 6.9, (1980):35-69. Jain, L. C. The Jaina schools of mathematical sciences (The Digambara and the Svetambara schools). Arhat Vacana 4.2-3 (1992):95-101. Jain, L C. The Jaina school of mathematical philosophy. Tulsi Prajna 28/109 (2000):104-117. Jain, L. C. Antarrastriya stara para Jaina ganita aura laukika ganita mem dharmika tatha etihasika vicaradharae. Bhava Vijnana (December 2007): 6-17. Jain, L. C. and Jain, Meena. System theory in Jaina school of mathematics II. IJHS 24.3 (1989):163-180. Jain, L. C. and Trivedi, R. K. "Todaramala of Jaipur (A Jaina philosopher-mathematician)". IJHS 22.4 (1987):359-371. Jain, Mahendra Kumar (ed. & tr.). Tatvarthavartika of Akalanka. Bhartiya Jnanapitha, New Delhi, 1999. Jain, Pannalal (ed. & tr.). Adipurana of Jinasena. Bhartiya Jnanapitha, Fourth Edition, 1993, Delhi. Jaini, J. L. (ed. & tr.). Gommatasara (Jivakanda) of Nemicandra. The Central Jaina Publishing House, Lucknow, 1927. Johar, K. L. "Bhagavana Rsabhadeva evam Jainadharma". In: Anupam Jain (ed.). Akhya: Bhagavana Rsabhadeva Rastriya Kulapati Sammelana. Digambara Jaina Triloka Sodha Samsthana, Meerut, 2000, pp. 46-51. Kapadia, H. R. (ed.). Ganitatilaka of Sripati (with Simhatilaka Suri's vrtti). Gaekwad's oriental series 78. Oriental Institute, Baroda, 1937. Keller, Agathe. Expounding the Mathematical Seed. A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya. Vol. I. Birkhauser, Basel etc, 2006.

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HISTORICAL NOTE: THE JAINA SCHOOL OF INDIAN MATHEMATICS Kumar, Bhuvanendra. "Jainas and their religion in America: A social survey". Arhat Vacana 9.1 (1997):43-53. Madhukara, Misrimalaji Maharaja (ed.). Anuyogadvara Sutra of Aryaraksita. Publication No. (Granthanka) 28, Sri Agama Prakasana Samiti, Beawar (Rajsthan), VNY 2513 (1987 CE). Madhukara, Misrimalaji Maharaja (ed.). Jivajivabhigama Sutra. Sri Agama Prakasana Samiti, Part I, Publication No. 30, 1989, and Part II, Publication No. 31, Beawar, 1991. Madhukara, Misrimalaji Maharaja (ed.). Uttaradhyayana Sutra. Publication No. 19, Beawar, 1991. Madhukara, Misrimalaji Maharaja (ed.). Sthananga Sutra. Publication No. 07, Beawar, 1992. Madhukara, Misrimalaji Maharaja (ed.). Suryaprajnapti Candraprajnapti. Publication No. 29, Beawar, 1995. Morice-Singh, Catherine. Jaina mathematics and cosmology: Numbers and algorithms in the Ganitasarasangraha and the Tiloyapannatti. Centre of Jaina Studies Newsletter (School of Oriental and African Studies, University of London) 11 (March), 2016, 41-43. McEvilley, Thomas. Approaches to the question of the antiquity of Jainism. Jinamanjari 13.1 (1996): 6-21. Mukhtara, R. C. Jain and Patni, C. P. (eds. & trs.). Trilokasara of Nemicandra (with Madhavacandra Traividya's Sanskrit commentary and Aryika Visuddhamati's Hindi commentary). 1975. Padmavathamma (ed. & tr. into Kannada). Ganitasarasangraha of Mahavira (with notes and translation into English by M. Rangacharya along with the 'Introduction' written by D. E. Smith, 1912). Siddhartakirti Granthamala, Hombuja, 2000. Padmavathamma, Krishnaveni and K. G. Prakash (eds. & trs.). Sri Rajaditya's Vyavaharaganita and Lilavati. Published by Padmavathamma, Mysore, 2013. Patni, CP. (ed.). Tiloyapannatti of Yativrsabha (with Aryika Vioeuddhamati's Hindi commentary). 3 Vols. Sri 1008 Candraprabha Digambara Jaina Atisayaksetra, DehraTijara, 2523 VNY (=1997 CE.). Plofker, Kim. The 'Error' in the Indian 'Taylor series approximation' to the sine. Historia Mathematica 28 (2001):283-295. Plofker, Kim. Mathematics in India. Princeton University Press, Princeton and Oxford, 2009a. 333 Plofker, Kim. Professor R. C. Gupta receives the Kenneth O. May Prize. Ganita Bharati 31.1-2 (2009b): 115118. Reprinted in: Ganitananda (ed. K Ramasubramanian), Indian Society for History of Mathematics, Delhi, 2015, pp. 7-9. Plofker, Kim. Links between Sanskrit and Muslim science in Jaina astronomical works". International Journal of Jaina Studies (Online) 6.5 (2010):1-13. Plofker, Kim. Mathematical models and data in the Brahmapaksa school of Indian astronomy. Ganita Bharati 36.1 (2014):1-12. Rajgopal, P. The Sthananga Sutra programme in Indian mathematics. Arhat Vacana 3.2 (1991):1-8. Ranga, Munilal. Vaidika parampara ke granthom mem Bhagavana Rsabhadeva. In: Anupam Jain (ed.), Akhya: Bhagavana Rsabhadeva Rastriya Kulapati Sammelana. Digambara Jaina Triloka Sodha Samsthana, Meerut, 2000, pp. 73-75. SAKHYA (ed.). Ganita-sara-kaumudi of Thakkura Pheru (with introduction, translation, and mathematical commentary by SaKHYa). Manohar Publishers and Distributors, New Delhi, 2009. Sarasvati, T A. Geometry in Ancient and Medieval India. Motilal Banarsidass, Delhi, 1979. Sarma, S R. Mathematical literature in the regional languages of India. In B. S. Yadav & Man Mohan (eds.), Ancient Indian Leaps into Mathematics. Springer, New York, 2011, pp. 201-211. Sastri, Chhaganlal (ed. & tr.). Jambudvipaprajnapti Sutra. Publication No. 26, Beawar, VNY 2520 (= 1994 CE). Sastri, Manoharalal (ed.). Trilokasara of Nemicandra Siddhantacakravarti with Bhasa Vacanika of Pandita Todaramala. Hindi Jaina Sahitya Prasaraka Karyalaya, Bombay, 1918. Sastri, NC Jain. Sridharacarya. Jaina Siddhanta Bhaskara 14.1 (1947):31-42. Schubring, Walther. The Doctrine of the Jainas (Translated from German by Wolfgang Beurlen into English). Lala Sundarlal Jain Research Series, Vol. XV, Motilal Banarsidass, Delhi. The first Edition in 1962. Shukla, K S. (ed.). Patiganita of Sridhara. Lucknow University, Lucknow, 1959. Shukla, K S. (ed.). Aryabhatiya of Aryabhata I (with the commentaries of Bhaskara I and Somesvara). Indian National Science Academy, New Delhi, 1976.

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