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Mathematics in Ancient and Medieval India c4.s3.t13: Bhāskara II (c. 1114 A.D. to 1200 A.D.)

Mathematics in Ancient and Medieval India c4.s3.t13

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English

Bhāskara II (c. 1114 A.D. to 1200 A.D.)

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The general appreciation and fame of Bhāskara II or Bhāskarācārya, as he was popularly called as a mathematician, is based on his *Līlāvatī*¹, *Bījagaṇita*² and *Siddhāntaśiromaṇi*³. He is the author of two other works namely *Vāsanābhāaṣya*, his own commentary on the *Siddhāntaśiromaṇi*, and *Karaṇakutūhala*⁴, a treatise on planetary motion. As to the time of his activity, we gather the following statement from his own writing, “In the year 1036 of the Śaka kings I was born, and at the age of 36 I have written the *Siddhāntaśiromaṇi*.”¹ The year of his birth i.e., 1036 of the Śaka kings corresponds to 1114 A.D. and the *Siddhāntaśiromaṇi* was written in 1150 A.D. The *Karaṇakutūhala* was written in 1183 A.D. Of his parentage and native place, he has said that he was born of a renowned Brāhmaṇa scholar and astronomer Maheśvara at a city called Bijjaḷabiḍa.² He had association with the schools of Ujjain and received his education from his father. All these have been confirmed by an inscription discovered by Bhau Daji.³

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¹ Tr. by J. Taylor, Bombay, 1817; Tr. by H. T. Colebrooke (a portion only) vide his *Algebra with Arithmetic* . . . . . , London, 1817; Ed. by Bāpudeva Śāstrī, Benares, 1883; Ed. by Sudhākara Dvivedi, Benares, 1912; Colebrooke's translation with notes and texts by Haran Candra Banerjee, Calcutta 1893; second edition, Calcutta, 1927; Ed. by Dattatreya Apte with *Buddhivilāsinī* comm. of Gaṇeśa Daivajña and *Līlāvatīvilāsa* of Mahīdhara, Ānandāśram Sanskrit Series no. 107, Poona, 1937; Ed. by Payanātha Jhā with commentary *Vāsanā* by Dāmodara Miśra, Mithila Institute of Post Graduate Studies in Sans. Learning, Durbhanga, 1959 etc.

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² Tr. by E. Strachey, London, 1813; By H. T. Colebrooke (a portion only) vide his *Algebra with Arithmetic*........., London, 1817; Ed. by Bāpudeva Śāstrī, 2 parts, 1875; Ed. with comm. by Sudhākara Dvivedi, Benares, 1888; Ed. with commentary *Navāṅkura* of Kṛṣṇa Daivajña, Ānandāśrama Sanskrit Series no. 99, Poona, 1930; Ed. by T. V. Rādhākṛṣṇa Śāstri with comm. *Vijapallava* of Kṛṣṇa Daivajña, Tanjore Saraswati Mahal Series no. 98, Tanjore, 1958 etc.

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³ Ed. and trans into Eng. (Golādhyāya only) by L. Wilkinson with the *Vāsanābhāṣya*, Calcutta, 1842; Revised edition by Bāpudeva Śāstri, Calcutta, 1861; Ed. by Jīvānanda Bhattācārya with *Mitākṣarā* of Raṅganātha, Calcutta, 1881; Ed. by Muralidhara Jhā with *Vāsanābhāṣya*, *Vāsanāvārttika* of Nṛsiṃha and *Marīci* of Viśvarūpa, Benares, 1917; Ed. by Bāpudeva Śāstrī with *Vāsannbhāṣya*, revised by Ganapati Śāstrī, Kāśī Sans. Series no. 72, Benares, 1929, etc.

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⁴ Ed. by Sudhākara Dvivedi with comm. *Vāsanā*, Benares, 1881; Ed. with comm. of Sumatiharṣa, Bombay, 1901.

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¹ *rasaguṇapūrṇamahīsamaśakanṛpasamaye’bhavanmamotpattiḥ|* *rasaguṇavarṣena siddhāntaśiromaṇi racitāḥ||* (*vide* Bāpudeva Śāstrī’s ed. of the *Siddhāntaśiromaṇi*, *Gola*, XIII, 58)

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² Ibid. *Gola*, XIII, 61-2. Compare also the concluding verse of the *Bījagaṇita*. In 1150 A.D. when Bhāskara wrote his *Siddhāntaśiromaṇi*, the town of Biḍa was under the rule of Prince Bijjala, a vassal of the Western Cālukya King Tailapa II and hence it has been called Bijjala-Biḍa.

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³ *JRAS*, 1, ( n.s. ), p. 414-418, 1865.

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The *Līlāvatī* which is based on Brahmagupta’s *Brāhmasphuṭasiddhānta*, Śrīdhara’s *Pāṭīgaṇita* and Āryabhaṭa II’s *Mahāsiddhānta* exhibits a profound system of arithmetic and also contains many useful propositions in geometry and arithmetic. In addition to fundamental operations his rule of three, five, seven, nine, and eleven, permutation and combination, treatment of zero prove his deep knowledge in the subject. The work was translated into Persian by Fyzi in 1587 A.D. by the direction of the emperor Akbar. A curious account of the occasion of writing the *Līlāvatī* is current which is given by Fyzi¹ in the preface of his translation. His *Bījagaṇita* contains problems of unknown quantities, surds, pulverizer (or *kuṭṭaka*), simple and quadratic equations including general rule which went beyond that of Śrīdhara, solutions of several indeterminate equations of second degree and solutions of certain equations of the third and fourth degree. The full solution of the equation and of its more general form ax^2+bx+c=y^2 was given by Bhāskara II. About his ‘Cyclic method’ Hankel² wrote, “It is beyond all praise: it is certainly the finest thing achieved in the theory of numbers before Lagrange.” H.T. Colebrooke³ points out that the method is exactly the same which Lord Brouncker devised to answer a question proposed by way of challenge by Fermat in 1657. In his *Siddhāntaśiromaṇi*, we get evidence of his knowledge of trigonometry including sine table and different relations among the three functions known as *jyā*, *kojyā* and *utkramajyā*. The idea of differentiation of the earlier workers was finally developed by him. He was acquainted with the principle of infinitesimal calculus and is often given credit for originating the idea of integration long before Newton and Leibniz.

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¹ It is said that *Līlāvatī* was the name of Bhāskara II's daughter. Bhāskara II came to know by the power of his astrological knowledge that she should never be wed. The father, however, ascertained a lucky hour for contracting her in marriage. He placed a cup with a small hole at the bottom on a vessel of water. It was so arranged that the cup would sink at the end of an hour. When everything was ready and the cup was placed on the vessel of water, Līlāvatī suddenly out of her childish curiosity came near to the cup to observe the water coming into the cup when, by chance, a pearl being separated from her bridal dress, fell into the cup and chanced to stop the influx. The lucky hour passed without the sinking of the cup. Bhāskara II, out of great dejection wrote a book and named it *Līlāvatī* after her name.

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² Hankel, H. *Zur Geschichte der Mathematik*, p. 202, Leipzig, 1874.

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³ *Bījagaṇita*, Rules 80-81 (vide his *Algebra with Arithmetic*... London, 1817.

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He earned so great a distinction that his manuscripts have been copied and commented widely in later years. An ancient temple inscription referred to him in the following terms : “Triumphant is the illustrious Bhāskarācārya whose feet are revered by the wise, eminently learned……, a poet,…… endowed with good fame and religious merit……………”¹

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¹ Kaye, G. R. *Indian Mathematics*, p. 37, Calcutta, 1915.

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