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Mathematics in Ancient and Medieval India c6.s8: Nature of Later Works

Mathematics in Ancient and Medieval India c6.s8

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Nature of Later Works

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From 300 B. C. to 300 A. D. we get a picture of early Jaina geometry. No single Jaina treatise of this period dealing with geometry and mathematics in general is known to us. Though geometry was admired as "lotus of mathematics"², yet we get only scrapy references here and there. In the *Sthānāṅgasūtra*³ (c. 300 B. C.), an early Jaina work, *rajju*, an instrument of measurement and *rāśi*, a solid quantity meaning measurement of solid bodies are the two topics mentioned under geometry¹. Though the term *rajju* does not appear in later works, there will be no difficulty in recognising it as referring to plane geometry ( *kṣetra* ). It is synonymous with the term *Śulba* of the Vedic period. A commentator of the Jaina work, *Gaṇitasārasaṃgraha* has rightly identified it ( *Śulba* ) with the *Kṣetragaṇita*. The term *rāśi* appears in later works where it means measurements of quantity of grains. In the later Hindu treatises, the topic on *rāśī* forms a part of *Khāta*² (cubic figures). Some technical terms of geometrical importance are used in the *Sūrajprajñapati*, *Bhāgavatisūtra*, *Tattvārthādhigamasūtra* of Umāsvati (c. 150 B. C.) etc. Probably these Jaina authors had more detailed knowledge, but the treatises written by them are lost. Several relations regarding circumference of a circle and their diameter are available in the *Tattvārthādhigamasūtra*, *Kṣetrasamāsa*, *Trilokasāra* and *Gommaṭasāra* and other Jaina works.³

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² *Sūtrakṛtāṅgasūtra*, 2nd Śrutaskanda, ch. 1, v. 154.

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³ Sūtra 747.

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¹ *Kṣetragaṇita* is the name for geometry appearing in the *Gaṇitasārasaṃgraha* (See ch. 7).

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² It deals with the content of the well whose length, breadth and depth are known.

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³ Vide Preface of the *Gaṇitatilaka*, dE. by H. Kapadia.

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The *Āryabhaṭīya* (c. 499 A. D.) of Āryabhaṭa I consists of four sections of which, the section of *Gaṇitapāda* (or mathematics) deals with geometry. It gives some formulae for calculating the area of a triangle, circle and trapezium and the volume of a sphere and pyramid. It also states the theorem of square on the diagonal. Brahmagupta (c. 628 A. D.)

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treated the subject ( *vide Brāhmasphuṭasiddhānta*, ch. 12 ) in the section on : (i) *Kṣetra* ( plane figures ), (ii) *Khāta* ( cubic figures ), (iii) *Citi* ( piles of bricks–solid figures ), (iv) *Krakaca* ( saw-measurement of height of a solid like wood by cutting it into pieces with a saw ) and (v) *Chāyā* ( shadow–plane figures ). Brahmagupta's rules consist mainly of certain formulae and solution of certain plain rectilinear figures such as triangles and quadrilaterals. The classification and terminology of the Jaina author Mahāvīra ( c. 850 A. D. ) are more accurate and they clearly express the relation of those subjects to the main body of the *Gaṇita*. He calls first the *Kṣetragaṇita* and rest by the name *Khātagaṇita* or simply *Khāta*. Śrīdhara ( c. 900 A. D. ) and Bhāskara II ( c. 1150 A. D. ) included *rāśi* ( solid figures ) in addition to topics discussed by Brahmagupta. In addition to some formulae, Bhāskara II gave a new proof to theorem of square on the diagonal. But on the whole, geometrical knowledge of these authors is mostly based on the consideration of practical needs. There were no difinitions, no postulates, no axioms. Thereafter no original works on mathematics discussing geometry came out. But in the *Gaṇitakaumudī* of *Nārāyaṇa* ( c. 1356 A. D. ) and *Pātīsāra* of Muniśvara ( c. 1603 A. D. ), We fiind some independent solutions of geometrical problems previously worked out by Brahmagupta and Bhāskara II, though a quite a large number of commentaries¹ were written on the famous work *Līlāvatī* by Bhāskara II during this period. Euclid’s Elements came to be known to Indian scholars in the mid-fourteenth century though it was actually translated into Sanskrit first by Jagannātha (c. 1718 A. D.) (Vide p. 50–51).

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¹ Vide *A Bibliography of Sanskrit Works on Astronomy and Mathematics*, Part I, by S. N. Sen, A. K. Bag and S. R. Sarma, National Institute of Science of India, New Delhi, 1966, p. 25.

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