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Mathematics in Ancient and Medieval India c4.s3.t2: Āryabhaṭa I (c. 476 A.D.)

Mathematics in Ancient and Medieval India c4.s3.t2

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Āryabhaṭa I (c. 476 A.D.)

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Āryabhaṭa I perhaps completed his education at the school of Kusumpura near modern Patna¹. His *Āryabhaṭīya*² or *Āryasiddhānta* (*laghu*) which is complete in four chapters, namely, *Daśagītikā* (the ten Gīti stanzas), *Gaṇitapāda* (Mathematics), *Kālakriyā* (reckoning of time), and *Gola* (sphere), deals with arithmetic and astronomy. The work was written in 499 A.D. when he was 23 years old³. Thirtyfour verses of the *Āryasiddhānta* on astronomical instruments have recently been traced by K.S. Shukla in a commentery of the *Sūryasiddhānta* by Rāmkṛṣṇa Ārādhya (1472 A.D.). Since these passages and Āryabhaṭa's midnight astromony do not occur in the *Āryasiddhānta* (*laghu*) of Āryabhaṭa, it is strongly presumed that he wrote another bigger work under the title, *Āryasiddhānta* (*mahā*). Though this work is not available at present, the glimpses of his midnight astronomy is still available in Bhāskara's *Mahābhāskarīya*, *Laghubhāskarīya* and in others. Some scholars believe that Āryabhaṭa I's works were based on the old *Sūryasiddhānta* for he was a worshipper of Sun, the promulgator of original *Sūryasiddhānta* (*vide* Introductory verse, Kern's edition )

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¹ *Āryabhaṭīya*, introductory verse, Gaṇitapāda.

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² Ed. by H. Kern with commentary *Bhaṭadīpikā* of Parameśvara, Leiden, 1874; by Rodet, L. *Journal Asiatique*, 13, p. 393-434, 1879; Tr. into English by P. C. Sengupta, *JDL, CU*, 16, p. 1-56, 1927; Tr. by W. E. Clark into English, Chicago, 1930; Ed. with commentary of Nīlakaṇṭha Somasutvan, *Trivandrum Sanskrit Series* (*TSS*), nos. 101, 110 and 185, Trivandrum, 1931, 1932, 1957 respectively.

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³ *ṣaṣṭhyabdānāṃ ṣaṣṭhir yadā vyatītāstrayaśca yugapādāḥ/* *tryadhikā viṃśatirabdāstadeha mama janmano'tītāḥ//* (*Āryabhaṭīya*, Kālakriyā v. 10) began with the beginning of Kaliyuga *i. e.* at 3102 B.C. Evidently in 499 A.D. (for 3600-3101=499), 23 years of Āryabhaṭa's life has passed.

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The *Āryabhaṭīya* is essentially a systematisation of the results contained in the *Siddhāntas* and it is of particular value because of the picture it gives of the state of mathematical knowledge of the period, no less than for the impulse it gives to the study of the subject. In its pages are to be found alphabet-numeral system of notation, rules for the usual arithmetical operations including involution and evolution. He gave methods of solution of simple and quadratic equations, indeterminate equations of first degree. Āryabhaṭa I devoted considerable attention to trigonometry and his introduction of the sines and versed sines was a notable improvement upon the clumpsy half chords of Ptolemy. He hit upon a remarkably close approximations to the ratio of the circumference of a circle to its diameter ( i.e., π=3.1416 ), which is

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“when three Yugapādas and sixty times sixty years (3600 years) have elapsed then twenty-three years of my life have passed”—(Clark). The fourth Yugapāda of Āryabhaṭa I

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undoubtedly an achievement over the mathematicians of the world. He also gave correct generalised rules for computing the sum of natural numbers and of their squares, and cubes.

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