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Mathematics in Ancient and Medieval India c4.s1: 1. Vedic Period—Origin of Mathematical Knowledge (c. 1500 B.C. to 200 B.C.)

Mathematics in Ancient and Medieval India c4.s1

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1. Vedic Period—Origin of Mathematical Knowledge (c. 1500 B.C. to 200 B.C.)

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About two thousand years before the Christian era, the Indus Valley was invaded by an Aryan race. Following this, at about 1500 B.C., a crude civilisation known as the Vedic civilisation began to emerge in India.

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The Vedic civilisation has four stages of development viz., ( 1 ) the *Saṃhita* ( the four *Saṃhitas*, namely *Ṛk*, *Sāma*, *Yajus* and *Atharvan* ), ( 2 ) the next stage of the *Brāhmaṇa* ( theological and ritual treatises ), ( 3 ) then the stage of the *Āraṇyakas* ( metaphysical appendices of the *Brāhmaṇas* ) and *Upaniṣads* ( philosophical texts regarding the absolute one ), and ( 4 ) the last stage of the *Vedāṅgas*. Information related to mathematical ideas in the literatures of the first three stages of the Vedic age is meagre. The *Vedāṅga* literatures are of six types namely : (i) *śikṣā* ( the science of the pronunciation of letters, accents ......), ( ii ) *kalpa* ( containing rules for ceremonial and sacrificial ceremony ), ( iii ) *vyākaraṇa* ( grammar ), ( iv ) *nirukta* ( etymology ), ( v ) *chanda* ( the science of metres ) and ( vi ) *jyautiṣa* ( astronomy ). The literatures of the *Vedāṅgas* are known as a whole Sūtra literatures, because they are written in forms of short rule, a peculiar style of composition characterised by utmost brevity and rigid systematization to preserve the cultural heritages of the Brāhmaṇas in a manageable form.

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*Sūtra Period, an age of specialisation*—It may be noted in this connection that the study of mathematics started with the sūtra period. At first, the study was strictly subservient to the primary needs, and education meant only the transmission of traditions from the teacher to the pupil and the committing to memory the sacred texts. In course of time, however, the contents of this education began to widen out and each one of the several *aṅgas* of the Veda began to develop. It is in this connection with the construction of sacrificial altars of proper size and shape that the problems of geometry and perhaps also of arithmetic and algebra were evolved. The study of astronomy arose out of the necessity for fixing the proper time for sacrifices.

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*Vedic Sources on Mathematics*—The critical mathematical knowledge of Sūtra literature led us to assume that there were mathematical works of even earlier age but they are lost. The names of seven śulbakāras are known. They are: Baudhāyana, Āpastamba, Kātyāyana, Mānava, Maitrāyana, Vārāha and Hiraṇyakeśī. The śulba works by these authors give methods and solutions of various problems of construction of altars required for Vedic sacrifices. The authors may be regarded as geometrician of the Vedic times. The *śulbasūtras* (connected to Śrauta section of the Kalpasūtra) of Baudhāyana, Āpastamba, Kātyāyana, Mānava, Maitrāyana, Vārāha and Hiraṇyakeśī are the main sources of our Vedic mathematics¹.

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¹ Beside these śulbas, we get the names of two other śulbas, namely *Vādhula* and *Māsaka*. Vide. B.Datta. *The Science of Śulba*, p.2, Calcutta University, 1932.

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There is some controversy about the birth place of Baudhāyana, the oldest *śulbakār*, some locating it in the Andhra country in the South while others in the Āryāvarta, the region between the river Gaṅgā and Yamunā². His *Baudhāyanaśulbasūtra*³ is complete in three chapters. Baudhāyana never claimed that he was the first to discover the principles of geometry or applied them to the problems of construction of altars. He gave a general enunciation of the Pythagoras’ theorem, an approximate value of √2 correct to five places of decimal, construction of a square equal to sum or difference of two squares, various methods of transformation of one figure to another etc. The native place of next *śulbakār* Āpastamba has been located similarly in a controversial manner either in the Andhra country or in Kurupāncāla¹. His *Āpastambaśulbasūtra*² is complete in six chapters. It gives in more detail the similar constructions, geometrical propositions and rules applicable for other constructions. The *śulbasūtras* of Kātyāyana³ and Mānava⁴ having similar contents are published, and the remaining śulbas are still in manuscript form. The *śulbasūtras* of Baudhāyana, Āpastamba and Kātyāyana belong to the former class and these give us a picture of development of early geometry and mathematics in India before the rise and advent of Jaina sect ( c. 500-300 B.C. ). The śulbasūtras of Mānava, Maitrāyana, Vārāha and Hiraṇyakeśī which belong to the latter class add very little to the knowledge of information available in this respect.

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² Ramgopal. *India of Vedic Kalpasūtras*. National Publishing House, Delhi, p. 100, 1959.

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³ Ed. with English translation and notes by G. Thibaut with the commentary of Dvārakānātha Yajvā, *Pandit*, Old Series, *9* and *10*, 1874-75; n.s. *1*, Benaras, 1877; Ed. by W. Caland, Vol.3, Calcutta, 1913.

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¹ Ramgopal, *ibid.*, p.96.

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² Ed. with German translation and notes by A. Burk. *Zeitschrift d. Deutsch. Morgenlandischen Gesellschaften*, 55, p.543-91, 1901; 56, p. 327-91, 1902; Ed. by D. Śrinivāsan Char and V. S.Narasimhachar, Mysore Sanskrit Series no. 73, 1931, etc.

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³ Ed. with English translation ( a portion only ) by G. Thibaut, *Pandit*, n.s. 4, 1882; Ed. with the commentary of Karkācārya, vide *Kātyāyana Śrautasūtra*, Ed. by M.M. Pāṭhaka, Benaras, 2 Vols. 1900; Published in the *Kāśī* Sanskrit Series no. 120 with Karka's *Bhāṣya* and Mahīdhara's commentary, Benaras, 1936, etc.

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⁴ Ed. with English translation by J.M.Van Gelder, New Delhi; ( Śatapiṭaka series no. 17 ), 1961.

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From the view point of mathematical contribution, the *Vedāṅga jyautiṣa*⁵, one of the Vedāṅgas dealing with astronomy, in its three recensions namely, *Ārca jyautiṣa*, *Yājuṣa jyautiṣa* and *Atharva jyautiṣa* may also be regarded as one of the sources of Vedic mathematics. A great deal of controversy prevailed about the time of the *Vedāṅga jyautiṣa*, but the modern scholars are more or less unanimous and fix it in 200 B.C.

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⁵ Ed. by A. Weber, Berlin, 1862; Ed. ( a portion ) by G. Thibaut with notes, *JASB*, 46, 1877; Ed. by Sudhākara Dvivedi with *Bhāṣya* of Somākara Śeṣa and S. Dvivedi and Muralidhara's explanatory notes, Bombay, 1908; Ed. with English translation by R. Shyama Sastri, Mysore, 1936; Ed. by A. Weber with a preface by Jagannath Tripathi, Allahabad, 1960 etc.

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