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Mathematics in Ancient and Medieval India c6.s7: Śulbasūtras and the Later Mathematics

Mathematics in Ancient and Medieval India c6.s7

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Śulbasūtras and the Later Mathematics

c6.s7.1

Certain scholars think that later Mathematics bears no noticeable relation with the *Śulbasūtras*. Kaye¹ pointed out that later mathematicians completely ignored the mathematical contents of the *Śulbasūtras*. He goes even further by saying that no Indian writer earlier than the 19th century is known to have referred to the *Śulbasūtras* as having any mathematical value. This has been supported by Cajori¹ and smith.²

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¹ Kaye, G. R. *Indian Mathematics*, p. 3, Calcutta, 1918; Vide also p. 9.

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¹ Cajori, F. *A History of Mathematics*, 2nd revised and enlarged edition, p. 84–86.

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² Smith, D. E. *History of Mathematics*, 1, p. 145, Dover Publication.

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The real fact has not been reproduced by these scholars. Before discussing the extent to which mathematics was cultivated after the *Śulba* period, some attempts must be made to understand the mental background of the people of that time. When the Vedic tradition was on the wane, a current of revolutionary thought began to interplay. Basically this current of thought was intellectual – an upsurge of a desire to solve the mental problems. By this time, Buddha and Mahāvīra appeared in forefront with their great personalities. Owing to the rise of Buddhism and Jainism which preached against the Vedic sacrifices for several centuries, the Vedic sacrifices were on the wane. As a result, the occasions for the construction of altars requiring high skill and ingenuity on the part of the constructors were few and far between. On the other hand, it should be observed that the Hindu geometry in the Śulba-period was of extremely practical nature and it was guided by religious motives whereas the later mathematicians were generally guided by academic interests. Nevertheless, it is not correct to say that *Śulbasūtras* became obsolete and were overlooked by all in India. The following items will readily show that the later mathematicians were not unaware of the *Śulbasūtras*.

c6.s7.6

From the table of the technical terms supplied, one can easily see that the following terms of the *Śulbasūtras* were used almost in the same sense in the later mathematical works viz. *caturasra*, *karṇa*, *karaṇī*, *kṣetra*, *vṛtta*, *parināha*, *rekhā*, *viṣama-caturasra*, *viṣkambha*, *vyāsa*, etc. Āryabhaṭa I (c. 476 A. D. ), used the terms *citighana*, *vargacitighana* and *ghanacitighana* for sum of triangular numbers, the sum of the squares of the natural numbers and the sum of the cubes of the natural numbers respectively. *Citi* means a pile or altar ( of bricks ) and *ghana* means cubic contents. These terms may be explained only if we assume that Āryabhaṭa I studied these series in relation to construction of altars. And this is not improbable, since we have shown that the mathematics first developed in India in connection with the construction of Vedic altars.

c6.s7.7

Various rules of construction and transformation occurring in the *Śulbasūtras* are : 1) the so-called Pyathagorian theorem, 2) the properties of similar figures, 3) the circle squaring, 4) construction of right triangle, 5) calculation of areas, through different division and transformation of figures. The first two of these topics enter largely in the later Hindu Mathematics and astronomy which will be seen in our subsequent discussion on geometry of Āryabhaṭa I, Brāhmagupta, Bhāskara II etc. That the later Hindu mathematicians did not lose their interest as regards circle squaring can be found from our discussion on the method of construction of a right triangle. It is mainly found in the works of Brahmagupta, Mahāvīra and others. The item 5) does not appear in the later Hindu works on mathematics in the particular manner in which they occur in the *Śulbasūtras*. But that the later mathematicians were interested in finding the area of triangular, tetragonal and circular plots and the volume of a right prism, sphere, can also be seen from our discussion of geometry of Āryabhaṭa I, Brahmagupta, Mahāvīra and others. The history of development of arithmetic and algebra of the post-Vedic mathematics also shows that they were not unaware of the work of the *Śulbasūtras*. But in this connection, it must be clear that the first two items were used more significantly than others. Whitney's remark may be noted in this connection, “The two main principles, by aid of which the greater portion of all Hindu calculations are made, are, on one hand, the equality of the square of the hypotenuse in a right angled triangle to the sum of the square of the other two sides, and on the other hand, the proportional relation to the corresponding parts of similar triangles.”²

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² Vide Whitney's Note under *Sūryasiddhānta*, ch. 2, v. 27. Proportional relation to the corresponding parts of the similar triangles is not used in the *Śulbasūtras*, but it may be that this relation is a development to the relation used in the type of similar figures having different areas.

c6.s7.9

The *Śulbas* continued to remain classics with the priestly class having academic interest and were followed in practice wherever, there were occasions during the succeeding centuries. That people interested in them did never disappear will be sufficiently realised from the necessity that arose in the middle ages from writing commentaries on the *Śulbasūtras*. Two commentaries are now available on the *Śulbasūtras* of Baūdha-yana. They are *Śulbadīpikā* of Dvārakānātha Yajvā (between 5th to 8th centuries), *Śulbamīmāṁsā* of Vyaṅkaṭeśvara Diksita. On the *Śulbasūtra* of Āpastamba, five commentaries are available viz., *Vyākhyā* by Karavindasvāmī (between 5th to 8th century A. D.), *Śulbasūtra-bhāṣya* by Kapardisvāmī (before 12th century A. D.), *Vyākhyā* by Sundararāja (c. 1575 A. D.), *Āpastamba Śulba-bhāṣya* by Sundarasūri, *Vyākhyā* by Gopāla Yajvā. Similarly four commentaries are written by Gaṅgādhara, Karka, Mahīdhara and Rāma Candra on the *Kātyāyana-Śulbasūtra* etc. These commentaries, apart from interpretations, have collated many-verses on the *Śulbasūtras* which show that these texts became popular in later times to suit different purposes. For manuscripts and other references, our *Bibliography*¹ may be consulted.

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¹ *Āryabhaṭīya* (*gaṇitapāda*), v. 21–22.

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¹ *A Bibliography of Sanskrit Works on Astronomy and Mathematics*, by S. N. Sen, A. K. Bag and S. R. Sarma, National Institute of Sciences of India, New Delhi, 1966.

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