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Mathematics in Ancient and Medieval India c4.s2: 2. Post-Vedic Period—Period of Arrested Progress and Some Foreign Contacts (c. 200 B.C. to 400 A.D.)

Mathematics in Ancient and Medieval India c4.s2

Language
English

2. Post-Vedic Period—Period of Arrested Progress and Some Foreign Contacts (c. 200 B.C. to 400 A.D.)

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During this period, no original contribution to the development of astronomy and mathematics in the strictest sense was made. But several works on astronomy and mathematics were composed on the basis of knowledge of the earlier Vedic period and there is also evidence of foreign contacts.

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( a ) *Jaina works* : Amongst the religious works of the Jainas, those that are important from the view point of mathematics are : *Sūryaprajñapti*,¹ *Sthānāṅgasūtra*, *Bhagavatīsūtra*,² *Tattvārthādhigamasūtra*³ of Umāsvātī, *Anuyogadvārasūtra*⁴, *Kṣetrasamāsa*, *Trilokasāra*⁵, etc. There were certainly other mathematical treatises by the early Jaina scholars, which are now lost. Considerable amount of search and research about the ancient works of the Jainas is necessary. We get the names of Siddhasena, Bhadrabāhu and many others who quoted mathematical formulae in connection with their doctrines but were not mathematicians by themselves.

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¹ Ed. with commentary of Malaya Giri. Agamadaya Samiti, 1918; Ed. by Amolakrisi with Hindi translation, Hyderabad; Ed. with English translation and notes by G. Thibaut, *Journal of the Asiatic Society of Bengal (JASB)*, 49, p. 107-27, p. 181-206, 1880.

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² Ed. by N. L. Vaidya with the commentary of Abhayadeva Sūri, *Śrivijayadeva Sursangha* Series no. 8, 1954.

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³ Ed. by K.P.Mody, Calcutta, 1903; By H.R.Kapadia, Bombay, 1926; By A. S. Sastri with the commentary *Sukhabodha* of Bhāskaranandi, *Govt. Oriental Library Sans. Series* No. 84, Mysore, 1944.

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⁴ Ed. with Hemacandra's commentary, *Jaina Pustakoddhāra* Series no. 37, Bombay, 1916.

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⁵ Ed. by Nathurama Premi with the commentary of Mādhava Candra, Bombay, 1919.

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The literature of the Jainas is generally classified into four groups¹ viz. ( i ) *Dharmakathanuyoga* ( or the exposition of the principle of religion ), ( ii ) *Gaṇitānuyoga* ( or exposition of the principle of mathematics ), ( iii ) *Saṃkhyāna* ( or science of numbers ) and ( iv ) *Jyotiṣa* ( or astronomy ). According to them, a child should be taught firstly writing, then arithmetic. The *arithmetic* and *jyotiṣa* had been considered as one of the main accomplishment of a Jaina saint.

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¹ Kapadia, H. L. Preface of *Gaṇitatilaka*, Ed. by H. R. Kapadia Gaekward Oriental Series no. 78, Baroda, 1937.

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*Kusumpura school*² : Umāsvātī ( 150 B.C. ), one of the greatest Jaina metaphysician of India first cites the existence of this school of mathematics. He resided in the city of Kusumpura ( ancient Pataliputra ) near Patna. Probably the school was originated long before the time of famous Jaina saint Bhadrabāhu ( 300 B.C. ) who lived at Kusumpura. The culture of mathematics and astronomy survived in this school for many centuries, where Āryabhaṭa I perhaps took his lesson in the fifth century A.D.

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² Datta, B., “The Jaina School of Mathematics”, *Bulletin of the Calcutta Mathematical Society* ( *BCMS* ), 21, p. 126-8, 1929.

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( b ) *Hindu works* : Eighteen siddhāntas³ were composed during this period. They go after the name of their authors namely Sūrya, Paitāmaha, Vyāsa, Vaśiṣṭha, Atri, Parāśara, Kāśyapa, Nārada, Garga, Marīci, Manu, Aṅgira, Lomaśa, Pauliśa, Cavana, Yavana, Bhṛgu and Śaunaka. Only five siddhāntas namely, the *Pauliśa Siddhānta*, *Romaka Siddhānta* (*Yavana Siddhānta*), *Vaśiṣṭha Siddhānta*, *Saurya Siddhānta* (i.e. *Sūrya Siddhānta*) and the *Paitāmaha Siddhānta* and a few others have survived, and the remainings are lost. These works also have gone through different recensions. It is now very difficult to assert what original matters are retained in their present redactions.

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³ *Sūryasiddhānta*, Ed. by Burgess, vide. Introduction of P. C. Sengupta, p. VIII, fn. University of Calcutta, 1935.

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The modern *Sūrya Siddhānta*¹ which is a standard astronomical treatise is of particular interest from mathematical point of view. It is divided into 14 chapters and its astronomical calculations are accepted and followed throughout India. The original *Sūrysiddhānta* which is now lost, began to take shape from about the fourth or fifth century A.D. at a time when old astronomical ideas and calculations came to be revised and placed on a scientific and mathematical basis. A rough idea of the contents of this work may be had from chs. i, ix, x, xi, xvi and xvii of the *Pañcasiddhāntikā* of Varāhamihira. The text in its present form is of much later development and result of many corrections and interpolations. There is a considerable agreement between the *Sūryasiddhān-*

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¹ Translated with notes into English by E. Burgess with introduction by P. C. Sengupta, University of Calcutta, 1935.

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*sūrya pitāmaho vyāso vaśiṣṭho'triḥ parāśaraḥ*
*kāśyapo nārado gargo marīcirmanuraṅgiraḥ* ||
*lomaśaḥ pauliśaścaiva cyavano yavano bhṛguḥ*
*śaunako'ṣṭādaśascaite jyotiḥśāstrapravartakāḥ* ||

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*ta* as described by Varāhamihira and the modern *Sūryasiddhānta*. P.C. Sengupta has shown, the old version was amended by Varāhamihira himself in accordance with the teaching of Āryabhaṭa I ( vide Āryabhaṭa's lost work, *BCMS*, 22, 1930; Introduction to Burgess' *Sūryasiddhānta*, Calcutta Univ. 1935 ). The *Sūryasiddhānta* gives in addition various information on trigonometry.

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