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Plofker, 2009a, p. 162). Rājāditya 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 Varaballāla II, who ruled from 1173 CE to 1220 CE, whom he referred to as Visnunrpāla (Padmavathamma et al., 2013, pp. xxiv-xxv). Thakkara Pherū 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 Tattvārthādhigama Sūtra Bhāsya ("Commentary on the aphorisms of the learning and meaning of the fundamental principles") of Umāsvāti (some period between 150 BCE and 219 CE), B B Datta is, about the mathematical formulae quoted in the Tattvārthādhigama Sūtra Bhāsya by Umāsvāti, of the following opinion. "Umāsvāti'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 Tattvārthādhigama Sūtra Bhāsya 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 Śrīdhara's time. This is why we do not find measure to have been classified into laukika and lokottara in the Anuyogadvāra Sūtra (3rd century CE) (Madhukara 1987, sūtra, 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. Mahāvīra seems to have worked at the court of the famous and benevolent ninth-century Rāstrakūta king Amoghavarca Nrpatunga who ruled at Mānyakheta 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-sāra-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
