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Mathematics in Ancient and Medieval India c4.s3: 3. Early Medieval Period—Period of Revival and Maximum Activity of Indian Mathematics with Systematisation (c. 400-1200 A.D.)

Mathematics in Ancient and Medieval India c4.s3

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English

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3. Early Medieval Period—Period of Revival and Maximum Activity of Indian Mathematics with Systematisation (c. 400-1200 A.D.)

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Several modern historians of mathematics namely, Kaye¹, Cajori², Smith³, etc. are almost unanimous in presuming that the revival in Hindu mathematics is due to the Greek contact. From a thorough discussion of all those matters, Burgess⁴, concluded that the transmitting knowledge was pre-Ptolemian Greek astronomy of Hipparchus and it took place in the 5th and 6th centuries of the Christian era. Biot followed by Thibaut⁵ is of opinion that the Greek astronomy was not transmitted to India through scientific treatises but through manuals used by Greek astrologers and calendar makers. P.C. Sengupta has thoroughly re-examined the whole subject and rejected the hypothesis that Indian astronomy was derived from the Greek sources. He says, "if Indian astronomy is to be held indebted to any foreign system of astronomy, that system of astronomy was the Babylonian astronomy—the fountain head from which both the systems took their rise"¹. The specific instances of the influence of the Babylonian astronomy on the Indian have been given by Neugebauer² and Sen³. My intention in referring to this complex situation is not to dispute the hypothesis of the Greek contact with Hindu astronomy. Indeed, there is incontrovertible evidence of foreign contact. In the opinion of Datta⁴ however the renaissance in the Hindu astronomy is entirely indigenous.

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¹ Kaye, G. R. *Indian Mathematics*, Calcutta, p. 9, 1915.

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² Cajori, F. *Hist. of Math.*, p.84, also p.83. Revised ed.

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³ Smith, D. E. *Hist. of Math.*, 1, p. 145, 2nd ed., 1923.

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⁴ *Journal of the American Oriental Society*, 6, p. 471, 1857.

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⁵ Thibaut and Dvivedi, *Pañcasiddhāntikā*, Introduction p. 19.

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¹ Sengupta, P. C. "Āryabhaṭa–the father of Indian Epicyclic Astronomy" *Journal of the Department of Letters* (*JDL/CU*), *18*, p. 55, 1929; vide also S. K. Das "Alleged Greek Influence on Hindu Astronomy; *Indian Historical Quarterly* (*IHQ*), p. 68-77, 1928.

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² Neugebauer, O. *The Exact Sciences in Antiquity*, Princeton, 1952.

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³ Sen, S. N. "Transmission of Scientific ideas . . . . . " *Bulletin of the National Institute of Sciences of India*, *21*, p. 13, 1963.

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⁴ Datta, B. "The Scope and Development of Hindu Gaṇita", *Indian Historical Quarterly* (*IHQ*), 5, p. 486, foot note, 1929.

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Whatever be the causes of the revival, their effect on the scope of the science of the *gaṇita* in this period was great. Astronomy, which had been a branch of Mathematics, separated out and began to be regarded by the name *jyotiṣa*. Geometry which belonged to a separate group of sciences viz. the *Kalpasūtra*, came to be regarded as an integral part of the *gaṇita*. Thus the readjusted science of the *gaṇita* consisted mainly of arithmetic, algebra and geometry⁵. Trigonometry was probably discovered by that time and came within the scope of *jyotiṣa*. Amongst the Jainas and the Buddhists the *gaṇita* was known by the name *saṃkhyāna* (or science of numbers).

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⁵ Vide, the works of Āryabhaṭa I, Brahmagupta, Bhāskara II and others.

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*Schools of Ujjain and Mysore* : In this period, the mathematical activity centered mainly round three schools namely, Kusumpura, Ujjain and Mysore. Āryabhaṭa I took his lesson in the Kusumpura school Varāhamihira, Brahmagupta and Bhāskara II (Bhāskarācārya) belonged to the Ujjain school while Mahāvīra may be said to represent the Southern school of Mysore.

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The relation of the two schools of Ujjain and and Mysore with Kusumpura is very interesting. A terrible famine is said to have raged for twelve years and devasted completely the Magadha country. As a result, one section of the Jaina community of Magadha headed by their priest Bhadrabāhu emigrated to South India (Mysore). On his way he passed through Ujjain and halted there for some time. This story is supported by the local tradition, several inscriptions (of 650 A.D.) and literature.¹ The connection between these three important schools of Hindu mathematics might thus be said to have been established in a very early times.

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¹ *BCMS*, 21, p.128, 1929.

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Some scholars² have claimed Āryabhaṭa I and Bhāskara I as residents of Kerala on the grounds that practically all the important astronomical and mathematical works produced in Kerala follow the Āryabhaṭa school, and works of these ancient authorities have been very popular in that part of India as indicated by the existence of a large number of their manuscripts and commentaries there on. Bhāskara I's reference to *Āryasiddhānta* as *Asmaka-* *tantra* also testifies that the work was written in Asmakadeśa in Kerala. But from Āryabhaṭa I's own statement, *kusumpure abhyarcitam jñānam* (*Āryabhaṭīya*, Gaṇita, v. 1) it is not unlikely to presume that after his lesson was over at the school of Kusumpura, Āryabhaṭa returned to his native place at Kerala where he established a school and wrote his mathematical works. Bhāskara I was more popular in the South India than in the North (vide Bhāskara I) but there is no mention of any place in Kerala as the seat of his activities.

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² Shukla, K. S. English translation of the *Mahābhāskarīya*, p.2, Lucknow Univ., 1960; Srinivasiengar, C. N. *The History of Ancient Indian Mathematics*, p. 41, World Press, Calcutta, 1967.

Bakshālī Manuscript (c. 400 A.D.)

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This manuscript¹ written in Sāradā script was first discovered, in course of excavation by a farmer at Bakshālī, a village near Peshwar in 1881 A.D. It attracted the attention of H. Hoernlé², G. R. Kaye³ and Datta⁴, who examined the manuscript from various aspects but failed to come to a satisfactory conclusion in ascertaining its age. Hoernlé considered it as 3rd to 4th century A.D. manuscript, which has been accepted as fair by Bühler, Cantor and Cajori. But Kaye placed it in the 12th century A.D. Datta opines that since the direct evidence is absent, there will be no better evidence than to depend on historical grounds. In this respect, the mathematical principles, symbols and terminology employed in the work would be better guides. Accordingly he examined the manuscript thoroughly and came to the conclusion that it was a work of 3rd to 4th century A.D.

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¹ Edited with Introduction by G. R. Kaye, Part I & II, New Imperical Series no. 43, Calcutta, 1927.

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² 'Notices under History of Asiatic Society for 1882', *Indian Antiquary*, 12, p. 89-90, 1883; 'The Bakshali Ms', *Indian Antiquary*, 17, p. 33-48, p. 275-79, 1888.

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³ p. 74-77 (vide his edition); *JASB*, 8, p. 349-61, 1912.

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⁴ Datta, B. "The Bakshali Mathematics", *Bulletin of the Calcutta Mathematical Society* (*BCMS*), 21, p.1-60, 1929.

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The work contains rules, with illustrative examples and solutions of many arithmetical, algebraic and geometrical (including mensuration) problems. The major portion deals with arithmetic e.g. fractions, square roots, progressions, income and expenditure, profit and loss, computation of gold, interest, rule of three and summation of complex series. The discussion of algebraic operations includes simple equations, simultaneous linear equations, quadratic equations, problems relating to mensuration and miscellaneous subjects.

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.

Varāhamihira (c. 505–587 A.D.)

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Varāhamihira was a native of Avanti and pupil of his astronomer father Ādityadāsa¹. The statement of Utpala that he was a Magadha Brāhmaṇa must be understood in this sense that his family derived its origin from Magadha². At the enquiry of Hunter³, Hindu astronomers of Ujjain furnished a list assigning the date of Varāhamihira to be 427 Śaka era corresponding to 505 A.D. Alberuni⁴ about eight centuries earlier from now gave the date which gives a corroborating evidence, though we do not know the exact period of his life. The date of Varāhamihira's death has been ascertained by Bhau Dāji⁵ in a comm.

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¹ *ādityadāsatanayastadavāptabodhaḥ kāpitthake* *savitṛlabdhavaraprasādaḥ |* *avantiko munimatānyavalokya samyag ghorāṃ* *varāhāmihiro rucirāṃ cakāra ||* ( *Bṛhajjātaka* ch. 26, v. 5) “Varāhamihira, a native of Avanti, was the son of Ādityadāsa and instructed by him. Having obtained the gracious favour of the sun, at Kapitthaka, composed his elegant work on horoscopy, after making himself duly acquainted with the doctrines of the ancient sages”.

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² c/o. Colebrooke, Algebra, p. XLV, fn.

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³ Published by Colebrooke, vide his *Miscellaneous Essays*, 2, p. 461, London, 1837. Also vide Bhaudaji's article, *Journal of the Royal Asiatic Society* (*JRAS*, ) (n.s. ), 1, p. 407, 1865.

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⁴ *Journal Asiatique*, tome IV, p. 285, 1844.

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⁵ “Varāhamihira went to heaven in the 509 in (year) of Śaka era i. e., 587 A.D.” *JRAS*, 1, (n.s.), p. 407, 1865.

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of Āmarāja on the *Khaṇḍakhādyaka* of Brahmagupta to be 587 A.D.

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Varāhamihira's *Pañcasiddhāntikā*¹, among other works², is considered important in the history of astronomy, as it gives the description of the five *siddhāntas* namely, Paulīśa, Romaka, Vaśiṣṭa, Saura and Paitāmaha of earlier dates. In the history of mathematics also, the work has a high place for its amount of trigonometrical information. It gives different relations among three functions, *jyā*, *kojyā* and *utkramajyā* and values of different *jyās* in a quadrant, drawn at a fixed interval (sine table) besides many other information of mathematical importance.

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¹ Ed. with English translation by G. Thibaut and M. M. Sudhākara Dvivedī, Benaras, 1889; Reprinted by Motilal Banarsidas, Benaras, 1930; vide *Jyautiṣasiddhāntasaṃgraha* Ed. by V. P. Dvivedi, Benaras, 1917; Ed. and tran. into English by Neugebaner and Pingree, Kopenhagen 1970

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² *Bṛhatsaṃhitā*, *Bṛhajjātaka*, and some astrological works.

Bhāskara I (c. 600 A.D.)

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We do not get sufficient evidence to say anything definitely about the place of his birth. In his *Āryabhaṭīyabhāṣya*, while giving problems on time and distance, he used the name of places like, Valabhi, Harukaccha and Sivabhagapura. He gave the equinoxial shadow of Sthāneśvara. It partly indicates that he belonged to the part of India near Valabhi. It is known that he imbibed his knowledge of astronomy from his father. Modern researches have shown that Bhāskara I was not a direct pupil of Āryabhaṭa I, but was undoubtedly the most competent exponent of Āryabhaṭa I's school of Astronomy³. Kuppanna Sastri¹ has concluded from external sources that Bhāskara I flourished in c. 600 A.D.

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³ *IHQ*, 6, p. 927-36, 1930.

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¹ Vide his edition of *Mahābhāskarīya*, Introduction, p. XII.

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Three works were written by him in the following chronological order: ( i ) *Mahābhāskarīya*², an astronomical work complete in eight chapters, ( ii ) *Āryabhaṭīyabhāṣya*³, a commentary on the *Āryabhaṭīya* of Āryabhaṭa and ( iii ) *Laghubhāskarīya*⁴, an abridged and simplified version of *Mahābhāskarīya* in eight chapters. Bhāskara I was mainly an astronomer but made commendable progress in the solution of the indeterminate equation of the first degree, the method of whose solution is given in his *Mahābhāskarīya*⁵ for use in the solution of astronomical problems⁶. Brahmagupta and others simply adopted the methods of Āryabhaṭa I and Bhāskara I. The method was however improved at a later period by Āryabhaṭa II (c. 950 A.D.) by abridging the operations in certain cases.

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² Ed. by Balavantaraya Apte with the commentary *Karmadīpikā* by Parameśvara, *Ānandāśrama Sanskrit Series* no. 126, Poona, 1945; Ed. with introduction and appendices by T. S. Kupanna Sastri, *Madras Govt. Oriental Series* no. 130, Madras 1957; Ed. with English translation and notes by Kripa Shankar Shukla, Lucknow University, 1960.

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³ Ms; For Mss. vide *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, p. 19, 1966. From now on, this bibliography will be abbreviated as our *Bibliography*.

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⁴ Ed. with the commentary *Vivaraṇa* of Śaṅkaranārāyaṇa, Trivandrum Sanskrit Series no. 162, Trivandrum, 1949; Ed. and translated into English by Kripa Shankar Shukla, Lucknow Univ. 1963.

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⁵ Ch. 1, verses 41-52.

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⁶ Examples are given in his *Mahābhāskarīya* (Ch. 8, verses 13, 19, 20, 23), *Āryabhaṭīyabhāṣya* (Ch. 2, verses 32-33), and *Laghubhāṣkrīya* (Ch. 8, verse 17).

Brahmagupta (c. 628 A.D.)

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The most prominent of Hindu mathematicians belonging to the school of Ujjain was Brahmagupta. His father’s name was Jiṣṇu. He wrote in 628 A.D. his *Brāhmasphuṭasiddhānta* at the age of thirty¹ and the other astronomical work, *Khaṇḍakhādyaka* in 655 A.D.² He lived during the reign of Śrīvyāghramukha, the greatest king of the Cāpa dynasty.

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¹ *śrīcāpavaṁśatilake śrīvyāghramukhe nṛpe śakanṛpālāt/* *pañcāśatsaṁyaktairvarṣaśataiḥ pañcabhiratītaiḥ//* *brāhmaḥsphuṭasiddhāntaḥ sajjanagaṇitagolavit prītyai/* *triṁśadvarṣeṇa kṛto jiṣṇusutabrahmaguptena//* (*Brāhmasphuṭasiddhānta*, *Ch.* 24, *v.* 7-9) “In the reign of Śrī Vyāghramukha, of the Śrī Cāpa dynasty, five hundred and fifty years after the Śaka king (i.e., 628 A.D. having passed, Brahmagupta, the son of Jiṣṇu, at the age of thirty, composed the *Brāhmasphuṭasiddhānta*, for the gratification of mathematicians and astronomers”.

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² Vide. P. C. Sengupta’s tr. of the *Khaṇḍakhādyaka*, Introduction, p. XXVII, University of Calcutta, 1934.

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The *Brāhmasphuṭasiddhānta*³ which is complete in twentyfour chapters includes two chapters on mathematics namely; *Gaṇitādhyāya* (chapter on mathematics) and *Kuṭṭakādhyāya* (chapter on indeterminate equations). Many other details of mathematical interests have been found in other chapters of this work. The *Gaṇitādhyāya* deals with cyclic triangle and quadrilateral, rules for arithmetical operations involving zero, negative numbers; quadratic equations. The *Kuṭṭakādhyāya*, contains solutions of the indeterminate equations of both first and second degree. The *Spaṣṭādhikāra* chapter contains, however, trigonometrical notations including sine table. In chapter IX of his *Khaṇḍakhādyaka*, Brahmagupta gave a method of obtaining from the given table of sines, the sines of intermediate angles.

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³ Ed. with commentary of Sudhākara Dvivedi, Benaras, 1902; Published originally in the *Pandit*, (n.s.) 23-24, 1901-2; tr. into English (a portion only) by H. T. Colebrooke, vide his *Algebra with Arithmetic and Mensuration from the Sanskrit of Brahmagupte and Bhascara*, London, 1817; Ed. with *Vāsanābhāṣya* by Ram Swarup Sarma and a board of editors and with notes by Satya Prakash. 4 vols. Indian Institute of Astronomical and Sanskrit Research, New Delhi, 1966.

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Brahmagupta was a bitter opponent of Āryabhaṭa I, though in his *Khaṇḍakhādyaka*, in his old days he seems to have recognised Āryabhaṭa's merit and adopted one of the system of Āryabhaṭa's astronomy. Brahmagupta holds a place in the history of Indian Civilisation. His works which were translated into Arabic seem to have influenced Arabian astronomy and mathematics in the 8th century A.D. possibly before Arabian scholars came to know about Ptolemy's work.¹

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¹ Arabian *Sindhind* and *Alarkand* were probably translations of *Brāhmasphuṭasiddhānta* and *Khaṇḍakhādyaka*. According to some scholars, *Sindhind* may be a translation of the *Sūryasiddhānta*, or some other work bearing the title *siddhānta* ( Smith, D. E. History of Math., *1*, p. 167-8. Dover Publication ).

Lalla (c. 768 A.D.)

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² Sengupta, P. C. *Khaṇḍakhādyaka*, Introduction, p. XXVI, Calcutta, 1934.

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Lalla was the grandson of Śāmba and son of Bhaṭṭa Trivikrama. His *Śiṣyadhīvṛddhida*³ in one thousand ślokas is fully devoted to astronomy. The work contains some important information on trigonometry. He was associated with the school of Kusumpura and his work was based on the *Āryabhaṭīya* of Āryabhaṭa I. Lalla has written two other works namely, *Pāṭīgaṇita* and *Siddhāntatilaka* both of them are now lost. Only references are available in later works.¹ The *Pāṭīgaṇita* dealt exclusively with mathematics, whereas the *Siddhāntatilaka* which was similar to the *Brāhmasphuṭasiddhānta* contained chapters on arithmetic and algebra.

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³ Ed. by Sudhākara Dvivedi, Benares, 1886.

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¹ Vide *Pāṭīgaṇita* of Śrīdharācārya. Ed. by K. S. Shukla, Introduction, p. IX, Lucknow Univ. 1957.

Govindasvāmin (c. 800-850 A.D.)

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Govindasvāmin who wrote a *bhāṣya* on the *Mahābhāskarīya*² of Bhāskara I seems to have flourished in Kerala in the first half of the ninth century A.D. Nīlakaṇṭha refers to the tradition that Govindasvāmin was the teacher of Śaṅkaranārāyaṇa who flourished at Mahodayapura. His *Govindakṛti* which was written as a sequel to the *Āryabhaṭīya* of Āryabhaṭa I appears to be lost. It was also similar to the *Brāhmasphuṭasiddhānta* of Brahmagupta and contained chapters on arithmetic and mensuration. Quotations from this work are found to appear in the commentaries of the astronomers, Śaṅkaranārāyaṇa (869 A.D.) and Udayadivākara (1073 A.D.) on the *Laghubhāskarīya*³ of Bhāskara I and in the *Āryabhaṭīyabhāṣya* of Nīlakaṇṭha.

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² Ed. by T. S. Kuppanna Śāstri with the *Bhāṣya* of Govindasvāmin and *Siddhāntadīpikā* of Parameśvara, Madras, 1957; vide. our *Bibliography*, p. 78.

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³ Vide the *Laghubhāskarīya* with the commentary *Vivaraṇa* of Śaṅkaranārāyaṇa, Trivandrum Sanskrit Series no. 162, Trivandrum, 1949; For Mss. of Udayadivākara vide *Bibliography*, p. 230.

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*SKANDASENA* (c. 9th century beginning)

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Pṛthūdakasvāmi (860 A.D.) has referred to this mathematician three times in his commentary on ch. 12 of the *Brāhmasphuṭasiddhānta* of Brahmagupta (v.2, 9 and 55). This reference to Skandasena is of great historical importance, as they show that the methods of multiplication known as *kapāṭasandhi*, *tatstha* and *khaṇḍa* as well as the mixed fractions called *bhāgamātā* occurred also in early works on Hindu *pāṭīgaṇita*. We also understand that the geometrical interpretation of an arithmetic series with the help of a figure which has been explained in detail in Śrīdhara's *Pāṭīgaṇita* (vide Rules 79-89) is due to Skandasena or some earlier mathematician and not to Śrīdhara. The works of Skandasena are now lost and nothing more can be said definitely (vide Shukla's Introduction of the *Pāṭīgaṇita* of *Śrīdhara*, Lucknow University, 1959).

Pṛthudaka Svāmi (c. 850 A.D.)

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Pṛthudakasvāmi wrote in 860 A.D. an important commentary¹ on the *Brāhmasphuṭasiddhānta*, which contains many examples to illustrate the mathematical statement and results contained in the *Brāhmasphuṭasiddhānta*. A considerable difference of opinion prevails whether these examples are those of Brahmagupta or of Pṛthudakasvāmi. The examples are not found in any text of the *Brāhmasphuṭasiddhānta*. The writing materials were scarce in those days and the students learnt at the feet of their masters and passed that knowledge in turn to their students. It may be that in this way some or all the examples might have come down to Pṛthudakasvāmi.

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¹ Vide. Colebrooke's translation with comm. of Pṛthudakaswāmi in his *Algebra with Arithmatic........from the Sanskrit of Brahmagupta and Bhāscara*, London, 1817.

Mahāvīra (c. 850 A.D.)

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The fame of Mahāvīra, a Jaina mathematician rests on his brilliant work known as the *Gaṇitasārasaṃgraha*¹. This was written in 850 A.D. From the comparison of his work and that of Brahmagupta, it seems that Mahāvīra was familiar with latter's work but he has improved upon it a lot. He had associations with the school of Mysore and lived perhaps at the court of Amoghavarṣa Nṛpatuṅga, one of the Rāṣṭrakuṭa monarchs in Mysore.

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¹ Translated into Telugu by Pabuturi, 11 th century A. D., Edited and tr. into English with notes by M. Raṅgācārya, Madras, 1912.

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The work in nine chapters is exclusively written on arithmetic, geometry and algebra. He has dealt with almost all the problems of his predecessors. He made the classification of the arithmetical operations simpler and gave a number of examples to elucidate the rules. His treatment of the work appears quite lucid and elaborate. His most noteworthy work however is the treatment of fractions. Mahāvīra shows ability to handle geometrical as well as arithmetical series. He gives a general rule of the combination of *n* different things taken *r* at a time. He deals with different types of quadratic equations and with the right angled triangle whose sides are rational. He continued on the line of his predecessor in dealing with indeterminate equation and fundamental arithmetical operations.

Śrīdhara (c. 850-950 A.D.)

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Śrīdharācārya's *Pāṭīgaṇita*² is a work on arithmetic and mensuration. He dealt with multiplication, division, square, cube, square-root, cube-root, fraction, rule of three and the areas of plane figures. Eight rules are given for operation involving zero but division by zero is not one of them. His method of summation of different A.P. and G.P. and solution of the problems pertaining to mixtures of things are interesting. He for the first time gave a rule to extract the root of ax²+bx=c, which is known usually as Śrīdhara’s formula.

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² Ed. by Sudhākara Dvivedi, Nirnayasagar Press, 1879; Ed. with English translation and notes by Kripa Shankar Shukla, Lucknow University, 1959.

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There remains a great deal of controversy about his time and place of origin. Some scholar opined that he was a Ācārya Brāhmin of Bengal and a Saivite Hindu, and others assigned him to the South India. Neither of these views are considered tenable by Shukla.¹ From a comparative study of the works of Śrīdharācārya with those of Mahāvīra ( 850 A.D. ) and Āryabhaṭa II ( 950 A.D. ), he came to the conclusion that Śrīdharācārya flourished between c. 850 A.D. to 950 A.D.

Āryabhaṭa II (c. 950 A.D.)

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The *Mahābhāskarīya*², an astronomical work in eighteen chapters of Āryabhaṭa II dealt various problems of mathematical interest, besides preliminary operations. The work was written in 950 A.D. Āryabhaṭa II mentions separately about three branches of mathematics viz. *pāṭī*, *kuṭṭaka* and *bīja* in his *Mahāsiddhānta* ( ch. 1,v.¹ ). Chapter XV of his treatise contains the *pāṭīgaṇita* and chapter XVIII deals with *kuṭṭaka*. In solving the quadratic, he followed the method of Āryabhaṭa I and Brahmagupta. He advised some corrections in the treatment of solution of simultaneous indeterminate equations of first degree. The high estimation of the subject of *kuṭṭaka* continued so distinctly after Āryabhaṭa II that the subject received special treatment at the hands of Hindu workers after Āryabhaṭa II onwards.

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¹ Vide his edition of *Pāṭīgaṇita* of Śrīdhara, Introduction, p. XXXV-XLIII.

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² Ed. with his own commentary by Sudhākara Dvivedi, *Benares Sanskrit Series* no. 36, Benares, 1910.

Śrīpati (c. 1039 A.D.)

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Śrīpati was a Jaina astronomer cum mathematician. He wrote *Gaṇitatilaka*¹, *Siddhāntaśekhara*² ( in 1039 A D. ) and *Bījagaṇita* besides five other works on astronomy and astrology³. The *Gaṇitatilaka* is devoted exclusively to arithmetic. The *Siddhāntaśekhara*, a work mainly on astronomy in twenty chapters, deals with algebra in two chapters namely *vyaktagaṇitādhyāya* ( ch. 13 ) and *avyaktagaṇitādhyāya* ( ch. 14 ). But his *Bījagaṇita* is now lost. Though he is the author of so many works, very little is known about his life. Probably he was the son of Nāgadeva and was a native of Rohinikhaṇḍa⁴. Like his predecessors, he discussed in his *Gaṇitatilaka* the eight fundamental operations and dealt with permutation, combination, proportion, notational places and many other things of relative interst.

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¹ Ed. with introduction by H. R. Kapadia with comm. of Siṃhatilaka Suri. *Gaekwad Oriental Series* no. 78, Baroda, 1937.

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² Ed. by Babūaji Miśra, 2 vols. Calcutta University, 1936, and 1947.

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³ *Dhikoṭikaraṇa*, *Jyautiṣaratnamālā*, *Dhruvamānasa*, *Jātakapaddhati* and *Daivajñavallabha*.

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⁴ Vide. *Gaṇitatilaka* introduction, by H. R. Kapadia.

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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