Mathematics in Ancient and Medieval India c4.s4: 4. Late Medieval Period—A Period of Commentaries and Some New Developments (c. 1200 A.D. to 1800 A.D.)

Mathematics in Ancient and Medieval India c4.s4

Language: English

c4.s4.t1 Nārāyaṇa Paṇḍita (c. 1356 A.D.) — 6 parts
c4.s4.t2 Mādhava (c. 1400 A.D.) — 10 parts
c4.s4.t3 Prarameśvara (c. 1430 A.D.) — 15 parts
c4.s4.t4 Nīlakaṇṭha Somasutvan (c. 1443 to 1543 A.D.) — 12 parts
c4.s4.t5 Jñānarāja (c. 1503 A.D.) — 5 parts
c4.s4.t6 Citrabhānu (c. 1475 to 1550 A.D.) — 7 parts
c4.s4.t7 Śaṃkara Vāriyar of Tṛkkuṭaveli (c. 1500-60 A.D.) — 7 parts
c4.s4.t8 Jyeṣṭhadeva (c. 1500 to 1600 A.D.) — 15 parts
c4.s4.t9 Karaṇapaddhati (1596 A.D) — 7 parts
c4.s4.t10 Acyuta Piṣāraṭi (c. 1550 to 1621 A.D.) — 6 parts
c4.s4.t11 Nārāyaṇa (c. 1588 A D.) — 9 parts
c4.s4.t12 Munīśvara (1603 A.D.) — 4 parts
c4.s4.t13 Kamalākara (c. 1616 to 1700 A.D.) — 5 parts
c4.s4.t14 Paṇḍit Jagannāth (c. 1657 to 1750 A.D.) — 5 parts
c4.s4.t15 Śaṃkaravarman (c. 1800-38 A.D.) — 3 parts

4. Late Medieval Period—A Period of Commentaries and Some New Developments (c. 1200 A.D. to 1800 A.D.)

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In this period, Bhāskara II's works became so popular that the scholars of North India were perhaps content with the preservation and transmission of the knowledge from generation to generation. Several hundreds of commentaries were written on the *Līlāvatī*, *Bījagaṇita*, and *Siddhāntaśiromaṇi*. Copies of large number of these manuscripts are now available in India and abroad. For giving a picture, to what extent Bhāskara II's works gained popularity, a list of commentaries on the *Līlāvatī* is appended here as an illustration.

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| Serial No. | Name of the commentators on the *Līlāvatī* of Bhāskarācārya (c. 1150 A D.) | Approximate time of these commentators | Name of the commentaries |
| 30. | Vṛndavana Śukla | c. 1825 A.D. | *Līlāvatīṭīkā* |
| 31. | Mādhava | — | *Karmapradīpikā* |
| 32. | Anonymous | — | *Kūpikā* |
| 33. | " | — | *Laghuvimarṣiṇī* |
| 34. | " | — | *Līlāvatīkaṭakṣa* |
| 35. | " | — | *Līlāvatīṭīkā* |
| 36. | " | — | *Līlavatī udāharaṇa* |
| 37. | " | — | *Līlāvatīvilāsa* |
| 38. | " | — | *Līlāvatīvivaraṇa* |
| 39. | " | — | *Līlāvatīvivṛti* |
| 40. | " | — | *Līlāvatīvyākhyā* |
| 41. | " | — | *Pāṭīlīlāvatīvṛtti* |

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On the above commentaries of the *Līlāvatī*, there are altogether 143 copies of this manuscript now available ( vide our *Bibliography*, p. 23-25 ).

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To estimate the progress made by these commentators, some belonging to a family of astronomers,¹ details of the nature of their works need be given here.

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¹ 1

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1. *Geneology :* ( a ) *Jambusāranagara :* {l} Divākara / (c. 1375 A.D.) --- {l} Govardhana / (c. 1400) --- {l} Gaṅgādhara / (c. 1420) ( b ) *Dadhigrāma :* {l} Rāma / (c. 1425) --- {l} Trimalla / (c. 1450) + Gopirāja {l} Trimalla / (c. 1450) --- Vallāla {l} Vallāla / (c. 1495) --- {l} Rāma + Kṛṣṇa + Govinda + Raṅganātha / (c. 1520) (c. 1548) \hspace{4em} + Mahādeva {l} Govinda / (c. 1568) --- {l} Nārāyaṇa / (1588) Raṅganātha --- {l} Munīśvara / (1603) ( c ) *Nandigrāma :* {l} Kamalākara Daivajña / (c. 1466 A. D.) --- {l} Keśava / (1496) --- {l} Gaṇeśa / (1507) {l} Keśava / (1496) --- {l} Rāma / (1522) --- {l} Nṛsiṁha / (1548) ( d ) *Pārthapura :* {l} Nāganātha / (c. 1480) --- {l} Jñānarāja / (c. 1503) --- {l} Sūryadāsa / (c. 1541) ( e ) *Golagrāma :* {l} Divākara Daivajña --- Viṣṇu + Kṛṣṇa + Mallāri + Keśava / (c. 1500 A. D.) (c. 1556) (1565) (1571) / \hspace{14em} + Viśvanātha / \hspace{15em} (1580) {l} Kṛṣṇa / (1565) --- {l} Nṛsiṁha + Śiva / (1586) {l} Nṛsiṁha --- Divākara + Kamalākara + Gopinātha + / (1586) (c. 1606) (c. 1616) / \hspace{15em} Raṅganātha / \hspace{16em} (1643)

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Gangādhara ( c. 1420 A.D. ) was the son of Govardhana and grandson of Divākara, an inhabitant of Jambusāranagara. His commentary, *Gaṇitāmṛtasāgarī* or *Amṛtasāgarī* though confined to the *Līlāvatī* contains a chapter on the *Bījagaṇita*. This work appears almost verbatim of the original work.

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Gaṇeśa ( c. 1507 A.D. ) was the son of Keśava Daivajña, a distinguished astronomer and a native of Nandīgrāma near Devagiri. His *Buddhivilāsinī* commentary on the *Līlāvatī* was written in 1545 A.D. It gives a copious exposition of the text with proper demonstration of the rules. It can be considered as one of the best among available commentaries.

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Sūryadāsa ( c. 1541 A.D. ), was the son of Jñānarāja and native of Pārthapura, a village near the confluence of the Godavari and the Vidarbha rivers. His gloss on the *Līlāvatī* and the *Bījagaṇita* are known as *Gaṇitāmṛtakūpikā* ( or *Amṛtakūpikā* ) and *Sūryaprakāśa Bījavākhyā* respectively. Both contain excellent clear interpretation of the text with a concise explanation of the principles of the rules.

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Kṛṣṇa ( c. 1548 A.D. ) of Dadhigrāma was son of Vallāla and disciple of Viṣṇu. His *Bījapallava* ( or *Kalpalatāvatāra* ), a commentary on the *Bījagaṇita* of Bhāskarācārya contains a clear and copious exposition of the sense with ample demonstration of the rules much in the manner of Gaṇeśa.

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Ranganātha ( c. 1643 A.D. ) of Golagrām was the son of Nṛsiṃha Daivajña and grandson of Kṛṣṇa. His commentary on the *Līlāvatī*, entitled *Mitabhāṣiṇī* demonstrates the original passages with many demonstrations.

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Rāmakṛṣṇa was the son of Lakṣmīdāsa and grandson of Nṛsiṃha. His *Bījaprabodha*, commentary on the *Bījagaṇita* of Bhāskara II and *Gaṇitāmṛtalaharī*, a commentary on the *Līlāvatī* of Bhāskara II are similar to Ranganātha's work.

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Rāmakṛṣṇa Deva, was the son of Sadāśiva Āpadeva. His *Manorañjanā*, a commentary on the *Līlāvatī* of Bhāskara II is a copious exposition of the text with demonstrations here and there and so on.

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Though it has not become possible to turn up all the manuscripts yet our first hand checking of the contents suggest that no notable progress of arithmetic, algebra and geometry were made by these commentators.

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Such works gave a new impetus to the studies in Kerala and from fourteenth century onwards, there has been an unbroken tradition for about five hundred years (*vide*, p. 45). The mathematicians of Kerala were staunch followers of Āryabhaṭa I's school, but they understood the significance and limitations of the previous scholars.

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*Kaṭapayādi system*¹.—In this connection, it would not be quite out of place if we discuss a system of numerical notation namely, *Kaṭapayādi*, a system known in South India from 6th century A.D. from the time of Haridatta. This had begun to be utilised widely after Bhāskara II in South Indian treatises to express the time of the authors, *ahargaṇas*, eclipses and other related matters. The system² is almost self explanatory : *Kaṭapayādi* means starting with ‘k, ṭ, p and y’. The letters *k* to *jh* indicate 1 to 9 respectively; so also *ṭ* to *dh*; *p* to *m* stand for 1 to 5 and *y* to *h* represent 1 to 8 respectively; *ñ* and *n* and vowels not preceded by a consonant stand for zero. In case of conjunct consonants, it is the last consonant that has value. The vowels following a consonant have no special value. The letter *ḻ* peculiar to the Drabidian pronunciation of Sanskrit is given the value 9. In this system a right to left arrangement is employed. The system is beautifully explained in the *Sadratnamālā*.

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¹ Another system known as *bhūtasamkhyā* was current in South India. Here the numbers were indicated by well-known objects. For examples : ‘eyes’ or ‘hands’ indicate 2; ‘senses’ or ‘elements’ indicate 5. For details, *vide* Datta & Singh, *Hist. of Hindu Maths.*, 1, p. 69.

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² Bag, A. K. ‘Trigonometrical Series in the *Karaṇapaddhati* and the date of the text’, *Indian Journal of Hist. of Science*, 1, No. 2, p. 98-106, 1966; K. Kunjunni Raja, *Adyar Library Bulletin*, 27, p. 122-131, 1963.

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The activities of the scholars of this period who attained an independent distinction have been described here separately. Many of their manuscripts still unpublished are yet to be published and studied.

Nārāyaṇa Paṇḍita (c. 1356 A.D.)

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Nārāyaṇa Daivajña or Nārāyaṇa Paṇḍita, as he was popularly called, was the son of Nṛsiṃha Daivajña. He is solely a mathematician and the author of two works namely (i) *Gaṇitakaumudī*¹, a mathematical treatise in 14 chapters and (ii) *Bījagaṇitāvataṃsa*², a work on algebra.

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¹ Ed. by Padmākara Dvivedi, two parts, The Prince of Wales Saraswati Bhavan Text No. 57, Benaras, 1936 and 1942.

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² See Algebra of Nārāyaṇa by B. Datta, *Isis*, *19*, 472-85, 1933; Ed. by K. S. Shukla, Lucknow, 1970.

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The date of composition of the *Gaṇitakaumadī*³ is given in the work itself as *gaja-naga-ravi-mite śake* or 1278 śaka corresponding to 1356 A.D.

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³ *Ibid.*, Part II, p. 411; vide also the preface of *Gaṇitakaumadī* (Part I), *Saraswati Bhavan Text* no. 57, 1936.

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In this work he dealt, in addition to the eight fundamental operations, the problems on series and a general formula for the sum of any order of triangular numbers. Nārāyaṇa's method of summation of any order triangular number or *vārasaṃkalita* shortly, has established a way for the development of integration series in connection with quadrature of the circle and allied problems. In his *Bījagaṇita*, besides usual topics on algebra, the method of finding the approximate values of the square-root of a non-square number is noteworthy.

Mādhava (c. 1400 A.D.)

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Mādhava, a Kerala astronomer is well-known for his *Veṇvāroha*¹. He was regarded as the authority on spherical astronomy and mathematics and is often referred to by later writers as a *golavid*² (expert on spherical mathematics). Mādhava belonged to Saṃgamagrāma², a village of the name Saṃgama³. Mādhava gives also his family title as *bakula vihāram*⁴ in his *Veṇvāroha* (verse 13).

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¹ Ed. by K. V. Sarma with the Malayalam commentary of Acyuta Piṣāraṭi, *Ravivarma Sanskṛta* Series No. 7, Trippunithura, 1956.

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² (a) *tacca saṃgamagrāmajena golatattvavida mādhavena pradarśitam* (*Āryabhaṭīyabhāṣya* of Nīlakaṇṭha, TSS. 185, p. 75)
(b) *vande golavidaśca mādhavamukhān* (*Sphuṭanirṇaya* by Acyuta Piṣārati, Oriental Library, Madras, Ms. R. 3799 (b)).

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³ *Irriññālakkuṭa* in Cochin, famous for the temple of Saṃgameśvara.

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⁴ Bakulam=Iraññi, Viharam=palli. So the local form of the word is *Iranni-ninna-palli*. This house exists even now in that place, though the present descendents have failed to preserve their scholarly tradition.

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Mādhava's date of writing *Veṇvāroha* is fixed up to be 1400 A D. from an epoch he used in this work⁵. This date can also be supported by other evidences. Pārameśvara, one of his disciple wrote his *Dṛggaṇita* in 1431, A.D. (vide infra). Hence it is quite likely that Mādhava wrote thirty years before that of his disciple. Therefore Mādhava may be considered to have flourished during the end of fourteenth century.

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⁵ The epoch was used as 1502008 (*dīnanamrānuśāsya*) Kali days together with 5180 (*ādikurma*) cycles of moon. The first numerical figure gives 1010 years and the second gives 390 years. Two together amount to 1400 A. D.

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Besides *Veṇvāroha* which deals with an accurate method of calculating the exact positions of the moon Mādhava is well-known for his work¹ on trigonometry and on infinite series with correct determination of the value of π. Nīlakaṇṭha quotes profusely Mādhava's name and says that he has used Mādhava's theorems in his (*Āryabhaṭīya*) *bhāṣya*. In one place he quotes a verse attributing it to Mādhava and admits that he has heard it from an astrologer while travelling in North Kerala². This suggests that Mādhava besides *Veṇvāroha* might have composed other works which though extinct at present became popular throughout Kerala. Mādhava is largely quoted in later Malayalam works namely, the *Yukti-bhāṣā* of Jyeṣṭhadeva, *Kriyakramakarī*, a commentary on the *Līlāvatī* etc.

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¹ Vide. Ramavarma Maru Tampuran, '*vṛttaparidhiyiluṭe*' *Mangalodayam*, Vol. 20, p. 252-6; the verse giving the value of, *vyāse vāridhinihate rupahate......*is quoted as Mādhava's in the *Kriyākramakarī Comm.* on the *Līlāvatī* (Madras, R. 2754, p. 362). Vide also. K. V. Sarma's article on Mādhava, *Quarterly Journal of the Mythic Society*, 49, No. 3, p. 183-86, 1958.

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² *tadviṣayamapyanuṣṭubantaraṃ mādhavoktaṃ mūṣika deśajād daivajñāt paryaṭatā śrutaṃ mayā* (*muṣikadeśa* is Kolattunāḍ in North Kerala) *Āryabhaṭīyabhāṣya* of Nīlakaṇṭha, Vol. II, p. 47.

Prarameśvara (c. 1430 A.D.)

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Parameśvara, the founder of the *dṛggaṇita* system of astronomy in Kerala though mainly an astronomer was held with respect in the field of mathematics.

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He wrote commentaries on all the popular classical works on astronomy and mathematics besides some original works on astronomy¹. His commentaries are: the *Bhaṭadīpikā*² on the *Āryabhaṭīya*, the *Parameśvarī*³ on the *Laghubhāskarīya* of Bhāskara I, the *Siddhāntadīpikā*⁴ (or *Karmadīpikā*), a *bhāṣya* on the *Mahābhāskarīya*, the *Parameśvara*⁵ on Muñjāla’s *Laghumānasa*, the *Vivaraṇa*⁶ on the *Sūryasiddhānta*, and the *Vyākhyā*⁷ on the *Līlāvatī* etc.

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¹ *Dṛgganita* (Ed. K. V. Sarma, Hoshiarpur, 1963), *Goladīpikā* I (*TSS.* 49), *Goladīpikā* II (Ed. Adyar Library Pamphlet Series no. 32), etc. Vide. K. V. Sarma's article on Parameśvara, *Journal of the Oriental Research Institute, Madras*, 28, Pts. i-iv, p. 47, 1961; *Adyar Library Bulletin*, 27, 136-143, 1963; our *Bibliography*, National Institute of Sciences of India, p. 167-70, 1966.

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² Ed. by H. Kern, Leiden, 1874.

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³ *Ānandāśrama Sanskrit Series*, no. 128. Another work of the same title is referred to by V. Rājarāja Varma, op. cit., 2, p. 502.

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⁴ *Madras Govt. Oriental Ms. Library*, No. 130.

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⁵ *Ānandāśrama Sanskrit Series* No. 123.

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⁶ Ed. by Kripa Shankar Shukla, Department of Mathematics and Astronomy, Lucknow University, 1957.

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⁷ Vide. Our *Bibliography*, p. 169.

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Various information about Parameśvara is available from Nīlakaṇṭha's *Āryabhaṭīyabhāṣya* and from later works also. He belonged to the Alattūr village (Sanskritised as *aśvathagrāma*) and that his house was situated on the northern bank of the river Nīlā⁸. The name of his house Vaṭaśśeri (Sanskritised as *vaṭaśreṇi*¹ ) occurs in his name as usual in the Southern India. A meagre information is available about his parents. In his own work it is given that his grandfather was the pupil of Govinda (different from the author of *Mahābhāskarīyabhāṣya* who lived much earlier), the famous author of *Muhūrtaratna*. He came of the Bhṛgugotra and belonged to the Āśvalāyana school.

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⁸ *nīlāyāḥ sāgarasyāpi tīrasthaḥ parameśvaraḥ* | *vyākhyānamasmai bālāya līlāvatyāḥ karomyaham* || (Introductory verse of Parameśvara's Comm. on *Līlāvatī*, *Adyar Library* Ms. 68524). See also his Comm. on the *Sūryasiddhānta*. He has been referred sometimes as *Paramādīśvara*, but his name Parameśvara is also given frequently.

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¹ *Vide. Adyar Library Bulletin*, 27, p. 137, 1963.

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Parameśvara flourished during the first half of the fifteenth century as recorded in his own works *Dṛggaṇita*² (1431 A.D.) and *Goladīpikā*³ (1443 A.D.).

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² *evaṃ dṛggaṇitaṃ śake, triṣuviśvamite* (1353) *kṛtam* (*Dṛggaṇita*, Ch. 2, v. 46).

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³ *śake akṣaṣaṭtricandra* (1365) *mite parameśvaranāmneyam vada-nabhuvāgoladīpikā racitā* (*Goladīpikā* II, Ch. 4, v. 91-92).

Nīlakaṇṭha Somasutvan (c. 1443 to 1543 A.D.)

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Nīlakaṇṭha⁴ is mainly an astronomer but his *Āryabhaṭīyabhāṣya*⁵ and *Tantrasaṃgraha*⁶ contribute much to mathematics.

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⁴ Vide. S. Sambasive Sastri, *Introduction to Āryabhaṭīya with the bhāṣya of Nīlakaṇṭha* (TSS 101); Ulloor Parameśvara Aiyer, *Keralisāhitya Caritra*, 2, p. 117; K.V. Sarma ‘Gargya Kerala Nīlakaṇṭha Somayājin’, *Journal of the Oriental Research Institute Madras*, 26, p. 24-39, 1958.

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⁵ *Āryabhaṭīya with Bhāṣya of Nīlakaṇṭha Somasutvan*, Parts I & II (*Gaṇitapāda and Kālakriyā*), ed. by K. Sambhasiva Sastri, Trivandrum Sanskrit Series (TSS) nos. 101 & 110, 1931-32. Part III (*Golapāda*), ed. by Suranad Kunjan Pillai, Trivendrum Sanskrit Series No. 185, 1957.

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⁶ The *Tantrasaṃgraha with commentary Laghu-vivṛti* by Saṅkara Vārior. Published by Suranad Kunjan Pillai, Trivendrum Sanskrit Series No. 188, Trivendrum, 1958.

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He was a Nambutiri Brāhmin of Kellalure family of Tṛ-k-kaṇṭiyur near Tirur, South Malabar. He was a Bhaṭṭa and belonged to Gārgagotra and the Āśvalāyana school. His favourite deity was Lord Śiva. His father was Jātadeva and had a younger brother named Śaṃkara. He was one of the pupil of Parameśvara's son, Dāmodara.¹ From him, he learnt mainly mathematics and astronomy. Nīlakaṇṭha often refers to Parameśvara as his *paramaguru* and considers him as a great authority. Parameśvara, the great astronomer was still alive when Nīlakaṇṭha went to the house of his teacher Dāmodara; and the young Nīlakaṇṭha might have possibly received some instructions from him.²

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¹ *iti kuṇḍagrāmajena gārgyagotreṇa āśvalāyanena bhāṭṭena kerala-sadgrāmagṛhasthena śrīśchetāraṇya nāthaparameśvarakaruṇādhika-raṇabhūtavigrahena jātavedahputreṇa śaṃkarāgrajena jātavedomā-tulena dṛggaṇita nirmapakaparameśvara putraśrīdāmodarāttajyo-tiṣāmoyanena ravita āttavedāntaśāstreṇa subrahmaṇyasahṛdayena nīlakaṇṭhena somasutāviracitavividhagaṇitagranthena iṣṭa vāhupapa-ttinā………* ( Āryabhaṭīyabhīṣya of Nīlakaṇṭha, Gaṇitapāda Colophon, TSS. 101, p. 180 ).

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² *tad eva paramācāryo mamāha parameśvaraḥ |* ( *Āryabhaṭīyabhāṣya*, Golapāda v. 48 ); also compare with : *ata evoktam asmadācāryena goladīpikāyām* (*Goladīpikā*, Ch. 3, v. 35).

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The scholars³ are more or less unanimous in assig-ning Nīlakaṇṭha's time in the period 1443 A.D. to 1543 A.D. This is supported by the fact that he was a pupil of Dāmodara, son of Parameśvara.

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³ Vide. *Journal of the Bombay Branch of the Royal Asiatic Society*, 20, ( n.s. ) p. 75, 1944.

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Another work *Tantrasaṃgraha*, possibly written by him, which is still in manuscript form, deals with mathematical series, problems of algebra and geometry.⁴ Nīlakaṇṭha is the author of many other astronomical works namely, *Golasāra*, *Candracchāyāgaṇita*, *Candracchāyāgaṇitaṭīkā* etc.¹

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⁴ Vide. Mss. of *Tantrasaṃgraha* in the Trippunittura Sanskrit College Library and Adyar Library. C/O. Whish's article, *Transactions of the Royal Asiatic Society of Great Britain and Ireland*, 3, 509-23, 1835; K. Balagangadharan's article, *Journal of the Bombay Branch of the Royal Asiatic Society*, New Series, 20, 77-82, 1944.

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¹ Vide our *Bibliography*, p. 155-57.

Jñānarāja (c. 1503 A.D.)

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Jñānarāja, the author of *Siddhātasundara*² and *siddhāntasundarabīja*³ was the son of Naganātha. Both the works still remain unpublished. The *Siddhāntasundara* in the Sphuṭādhyāya chapter contains sine table and trigonometrical relations.⁴ This was written in 1503 A.D.⁵ His *siddhāntasundarbīja* is a separate work on algebra which was written as a sequel to Bhāskara II's *Bījagaṇita*. This *Bījagaṇita* has been repeatedly cited by his son Sūryadāsa.

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² For Mss. vide our *Bibliography*, p. 93-94.

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³ One good copy is deposited in the *Sans. College Library*, *Benares*. Vide Catalogue of the Library, printed by the order of the Government, Allahabad, Vol. 2, p. 257.

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⁴ Vide. p. 17-18 of Ms. deposited in *Asiatic Society Bengal* (No. 8210).

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⁵ Vide. Sudhakara's *Gaṇakataraṅginī*. Entry on Jñānarāja.

Citrabhānu (c. 1475 to 1550 A.D.)

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Citrabhānu, one of the student of Gārgya Nīlakaṇṭha⁶, is the author of *Karaṇāmṛta*⁷, a work on mathematics in four chapters. It deals with mathematical calculations following the *dṛggaṇita* system. Śaṁkara Vāriyar, the author of *Kriyākramakarī* commentary on the *Līlāvatī* was a student of Citrabhānu.

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⁶ Beginning verse of the *Karaṇāmṛta*.

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⁷ Mss. at the *Curator's Office Library*, Trivandrum (TC), 663A and B.

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Śaṃkara himself admitted that he learnt the methods of proof from his teacher, Citrabhānu.¹

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¹ *ekaviṃśatidhā kāryamityupadiṣṭaṃ citrabhānunāmnā gaṇitagolayuktividagresareṇa bhūsurottamena | tatra diṅmātramasmābhistadupadeśavaśādiha likhyate |* ( C O. K. Kunjunni Rāja’s paper, *Adyar Library Bulletin*, 27, p. 154, 1963 ). “The methods of twenty-one types have been advised by Citrabhānu the pioneer of the scholars who are well versed in *Gaṇita*, *Gola* and *Yukti*. A few of these are being written in brief by me as an advice.”

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The date of composition of his work is given as 1530 A.D. in the commentary on his work.²

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² *buddhyonmathyoddhṛtaṃ yat* ( 1691513 days i.e. c. 1530 years).

Śaṃkara Vāriyar of Tṛkkuṭaveli (c. 1500-60 A.D.)

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Śaṃkara Vāriyar was a student of Nīlakaṇṭha³ and Citrabhānu ⁴ He wrote the *Laghuvivṛti*⁵ comentary on the *Tantrasaṃgraha* in 1556 A.D. He wrote another work, the *Kriyākramakarī*,⁶ an elaborate commentary on the *Līlāvatī* of Bhāskara II—giving the rationale and proof of the theorems and formulae. The work is more or less identical with *Karmapradīpaka* of Nārāyaṇa,⁷ only difference is that Śaṃkara's work is more elaborate. The introductory verses of of both the works are same.⁸

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³ Vide, beginning of the *Karaṇāmṛta* of Citrabhānu.

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⁴ See. under Citrabhānu, fn. 3.

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⁵ *Trivandrum Sans. Series* ( TSS ). 188.

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⁶ Oriental Library. Madras, Ms. No. R2754; vide also *Bulletin of the National institute of Sciences of India*, 21, p. 320, 1963.

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⁷ Ulloor S. Parameśvara Aiyer, *Kerala Sāhitya Caritram*, 2, p. 121. For. Mss. of *Karmapradīpaka*, vide our *Bibliography*, p. 150.

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⁸ *nārāyaṇaṃ jagadanugrahajāgarūkaṃ* *śrīnīlakaṇṭhamapi sarvavidaṃ praṇamya |* “I bow to Nārāyaṇa, a great benefactor of the world and Śrī Nīlakaṇṭha, a scholar well versed in all śāstras”.

Jyeṣṭhadeva (c. 1500 to 1600 A.D.)

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Jyeṣṭhadeva¹ is another South Indian scholar on mathematics and astronomy. His *Yuktibhāṣā*² is a Malayalam work on mathematics and astronomy. The work is divided into two parts, the first comprising arithmetic, geometry and algebra and the second exclusively devoted to astronomy and allied subjects and spherical trigonometry. The contribution of this unique work is that it gives the rational or derivation of all theorems and formulae then in use among astronomers. The work is mainly based on the *Tantrasaṃgraha* of Nīlakaṇṭha.

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¹ Vide Sarma, K.V. “Jyeṣṭhadeva and his identification as the author of *Yuktibhāṣā*”, *Adyar Library Bulletin*, 22, p. 35-40, 1958; Also see, 27, p. 156-58, 1963.

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² Ed. Ramvarma Maru Thampuran and Akhileswara Aiyer, Trichur, 1947.

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A Sanskrit work *Gaṇita Yuktibhāṣā* is available at present, which seems to agree closely with the Malayalam version of the *Yuktibhāṣā*. It is somewhat difficult to decide which is the earlier version and which is the translation.

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Jyeṣṭhadeva was a Nambūtiri Brāhman of Paṟaṅṅoṭṭu family in Ālatture village. He was one of the students of Dāmodara. Jyeṣṭhadeva was the teacher of Acyuta Piṣāraṭi of Tṟkkaṇṭiyūr who refers to him in his *Uparāgakriyākrama* (vide infra).

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On the date of *Yuktibhāṣa*, there remains some controversy.

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Whish³ and the editors of the published *Yuktibhāṣā* suggested on the basis of a verse available at the end of a manuscript that the author of the work was Brahmadatta and the date of the work was 1750 A.D. Sarma¹ has shown that the passage contains a word *alekhi* which does not mean 'composed' but 'transcribed' or 'copied' Hence Brahmadatta is a transcriber and the time of his doing so was 1750 A.D.

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³ *Transactions of the Royal Asiatic Society of Great Britain and Ireland*, 3, p. 509-23, 1835.

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¹ *Adyar Library Bulletin*, 22, p. 35-40, 1958.

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A manuscript (No. 755) named *Gaṇitayuktibhāṣā* of the Kerala University Mss. Library, Trivandrum contains a fragmentary passage dealing with precision of equinoxes (*ayanacalana*) where a verse (beginning on folio 34 b) refers Jyeṣṭhadeva as the author of *Yuktibhāṣā*² The mention of Jeṣṭhadeva is more explicitly mentioned in an old palm-leaf manuscript (No. 9886) of the Oriental Institute, Baroda. It is a Malayalam commentary on the *Sūryasiddhānta* which contains also a chronological list of Kerala astronomers. This list³ contains the following names:

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² *jukriyādike pāte svarṇam tatsādhane vidhau |* *ityuktā kṣepacalanasyānitis 'tantrasaṃgrahe'* *jyeṣṭhadevo'pi 'bhāṣāyām nādhikaṃ kiṃciduktavān |* The *Tantrasaṃgraha* referred to here is the well-known work of that name by Nīlakaṇṭha Somayājin and the *Bhāṣā* referred to in continuation is obviously the *Yuktibhāṣā*, for it is expressly stated in the beginning of the *Yuktibhāṣā* that it was composed as an exposition of the *Tantrasaṃgraha*.

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³ Vide also *Adyar Library Bulletin*, 22, p. 35-40, 1958.

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Parameśvara (1360-1455 A.D.)—son, Dāmodara—pupils, Nīlakaṇṭha Somasutvan (1443-1543 A.D.) and Jyeṣṭhadeva—pupil, Acyta Piṣāraṭi (1550-1621 A.D.)—pupil, Nārāyaṇa Bhaṭṭātiri (1587 A.D.).

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Furthermore, in a Malayalam commentary⁴, the date of completion of the *Yuktibhāṣā* in number chronogram of the *kaṭapayādi* system, is given as 1, 714, 262 corresponding to 1592 A.D. This shows that Jyeṣṭhadeva must have flourished during the later part of the sixteenth century.

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⁴ Ulloor S. Parameśvara Aiyer, *Kerala Sāhitya Caritram*, 2, p. 323.

Karaṇapaddhati (1596 A.D)

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The *Karaṇapaddhati* is an important astronomical work in ten chapters by an unknown Kerala astronomer of uncertain date. Only this much is known about the author that he was a Brāhmin who took his abode in the village Śivapura.¹ The text in Devanāgarī script is published by K. Sambaśiva Śāstrī², but this edition misses the opening verse.³ Apart from the usual elements and formulae characteristic of Hindu astronomy, the work gives in the sixth chapter, trigonometrical π, *sine cosine* and *tan* series. The mathematical importance of not only *Karaṇapaddhati* but three other texts namely *Tantrasaṃgraha*, *Yuktibhāṣā* and *Sadratnamālā* was first pointed out by C.M. Whish⁴ who discussed π and tan functions given in the texts and attempted to fix their dates.

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¹ The last verse of the tenth chapter reads : *iti śivapuranāma grāmajaḥ ko'pi yajvā*. The author has been identified as Putumana Samayaji, vide *All India 18th Oriental Conference Bulletin*, p. 562-64.

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² Trivandrum Sanskrit Series ( TSS ) No. 126.

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³ Vide *Madras Triennial Catalogue*, Vol. II, Part I, Sans. A. Call No. *1310*.

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⁴ ‘On the Hindu quadrature and the infinite series of the proportion of the circumference to the diameter exhibited in the four śāstras : *Tantrasaṃgraha*, *Yuktibhāṣā*, *Karaṇapaddhati* and *Sadratnamālā’*, *Transaction of the Royal Asiatic Society* ( TRAS ) of Great Britain and Ireland, 3, part III, 509-523, 1835.

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As regarding the date of *Karaṇapaddhati*, there is some controversy.¹ Recently it is traced that the verse 11 of the chapter 10 of the *Karaṇapaddhati* contains sufficeint hints as to the date of writing of the text in 1596 A.D. The number 554754 (*viṣṇosuvṛśamanu*) when lessened by 596 (*taddhāma*), gives the date of the text in Śaka days i.e., 554158 Śaka days=1518 Śaka year=1596 A.D.

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¹ Vide, Bag, A. K. 'Trigonometrical Series in the Karaṇapaddhati and the probable date of the text', *Indian Journal of History of Science*, *1*, No. 2, p. 98-106, 1966.

Acyuta Piṣāraṭi (c. 1550 to 1621 A.D.)

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Acyuta Piṣāraṭi, a non-Brāhmin astronomer mathematician of Kerala was a student of Jyeṣṭhadeva. His *Karaṇottama*² is a manual of mathematics in five chapters containing 109 verses. In a verse³ of his astronomical work, *Uparāgakriyākrama*,⁴ he expressed that at the time of composition of this work, Jyeṣṭhadeva had become very old, but was still alive. He belonged to Piṣāraṭi community and referred to as Vaiṣṇava whose function was to look after the external affairs of the temple. His pupil, Nārāyaṇabhaṭṭa gives the date of his death as 1621 A.D. from the study of a chronogram.⁵

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² Vide Mss. at the *Curator's Office Library, Trivandrum* (TC) 697B.

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³ *prokta pravayaso dhyānājjeṣṭhadevasya sadguroḥ vicyutāśayadoṣeṇetyacyutena kriyākramaḥ* ||

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⁴ TC. 655B.

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⁵ K. Kunjunni Rāja, *Adyar Library Bulletin*, *27*, p. 160, 1963.

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“The *Kriyākrama* is being written by Acyuta to modify the faults of analogies of the work of my beloved aged guru Jyeṣṭhadeva”.

Nārāyaṇa (c. 1588 A D.)

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Nārāyaṇa was the son of Govinda and teacher of Munīśvara.¹ The manuscript of his *Khaṇḍitabījagaṇita* or shortly *Bīja*, a work on algebra is lying in Sanskrit College Library, Banaras.² He is different from his predecessor Nārāyaṇa, the author of *Gaṇitakaumudī*. All the sūtras of the *Bīja* are composed in *āryā* metre. The Ms. is incomplete at the end. His time is still not known, but he can be tentatively fixed 15 years earlier than that of Munīśvara (c. 1603 A.D.), since Munīśvara himself admitted that Nārāyaṇa was his *guru*.

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¹ 1.

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² For Ms. reference vide our *Bibliography*, p. 150.

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*Geneology*

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Rāma (Dadhigrāma, Devarāta gotra)

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Gopirāja

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Trimalla

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Vallāla

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Rāma Kṛṣṇa Govinda Raṅganātha Mahādeva (Bījāṅkurakāra) | | Nārāyaṇa Munīśvara (c. 1588 A. D.) (c. 1603 A. D.)

Munīśvara (1603 A.D.)

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Munīśvara, son of Raṅganātha, was the main astronomer in the Court of King Sā̆jāhān. His parents belonged to Devarāta gotra and lived in the Dadhigrāma of Alicpur Province near the river Payaṣṇi. He was born in 1603 A.D.¹ He wrote *Niṣṭsṭārthaduti*, a commentary on the *Līlāvatī* of Bhāskara II and *Pāṭīsāra*, another work on mathematics besides several works on astronomy². Both the works of mathematics still remain unpublished. His *Siddhāntasārvabhauma* and his own commentary on the *Siddhāntasārvabhauma* were written in 1646 A. D. and 1650 A. D. respectively³. Munīśvara was called by another name, Viśvarūpa. He had a controversy with Kamalākara and tried to refute some of the latter’s contribution.

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¹ Vide. Preface of the *Siddhāntasārvabhauma*, Princess of Wales Saraswati Bhavan Text no. 41 ( Part I ).

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² Vide. Sudhākara's *Gaṇakataraṅgiṇī*—Muniśvara; For other works, vide our *Bibliography*, p. 145-46.

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³ Vide. Sudhākara’s *Gaṇakataraṅgiṇī*; also Preface of the *Siddhāntasārvabhauma* mentioned above.

Kamalākara (c. 1616 to 1700 A.D.)

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Kamalākara, the son of Nṛsiṃha Daivajña was the younger brother as well as disciple of Divākara Daivajña. He was a Mahārāṣṭrīya Brāhmaṇa. He composed his work, *Siddhāntatattvaviveka*⁴ in 1580

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⁴ Ed. with notes by Sudhākara Dvivedi and Muralidhara Jhā, Benares Sanskrit Series, Benares, 1885; Ed. with a comm. by Gaṅgādhara Miśra Śarmā, Lucknow, 1929, etc.

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Śaka era i. e., 1658 A.D. He was a follower of *Sūrya-siddhānta* and tried to refute some of the views of Bhāskara II and Munīśvara. Though *Siddhāntatattva-viveka* is mainly a work on astronomy, he made contribution to trigonometry by giving several correct relations between chords and its corresponding arcs in his *jyotpatti* section of Spaṣṭādhyāya. His other works are : *Grahagolatattva*, *Grahasāraṇī*, *Kairāśyudāharaṇa*, a commentary on the Bhāskarācārya's *Līlāvatī*, *Manoramā*, a commentary on the *Grahalāghava* of Gaṇeśa Daivajña, *Sauravasanā*, a commentary on the *Sūryasiddhānta*, and *Śeṣavasanā*, a supplement to the author's *Siddhāntatattvaviveka*¹

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¹ For details of editions vide p. 49.

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In the concluding verse of his *Siddhāntatattvaviveka*, he admitted that his ancestors lived in the village Golagrāma near the river Godāvarī.

Paṇḍit Jagannāth (c. 1657 to 1750 A.D.)

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Paṇḍit Jagannāth worked under the patronage of king, Jayasiṁha, who was well known for his interest in the study of astronomy and mathematics. He came to Jayasiṁha at the age of twenty. He was a scholar in Sanskrit and acquired a great proficiency in both Persian and Arabic languages in a very short time². He translated Euclid's *Elements* from an Arabian version, *Taḥrīr-u-Uqlidas* by Nasīra'd-Dīn aṭ-ṭūsī ( 1201 A.D. ) under the name *Rekhāgaṇita*³.

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² vide *Rekhāgaṇita*, edited by Kamalāśaṅkara; notes on Jagannāth in the Preface.

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³ The *Rekhāgaṇita*, two volumes. Ed. with a critical preface, introduction and notes in English by Kamalāśaṅkara Trivedi, *Bombay Sanskrit Series* nos. 61-62, Bombay 1901-2.

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His other works are: *Siddhāntasamrāṭ*¹, a Sanskrit version of Ptolemy's *Almagest*, and *Siddhāntasārvabhau-ma* containing partly *Samrāṭsiddhānta* and partly Hindu astronomy.

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¹ Edited by Ramswarup Sarma, Indian Institute of Astronomical and Sanskrit Research, 3 vols, New Delhi, 1967.

Śaṃkaravarman (c. 1800-38 A.D.)

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Śaṃkarvarman, the author of *Sadratnamālā*² flourished in the first half of the nineteenth century A.D. He was born in 1800 A.D. He wrote his *Sadratnamālā* under the patronage of Rāmavarman, an heir-apparent of King Udayavarman in North Kerala³. The *Sadratnamālā* is a short work in five chapters and gives various results of the Kerala mathematicians without deductions. The date of composition of the work has been determined from the chronogram: *lokāmbasiddhasevye* i.e., 1,797,313 corresponding to 1823 A.D. He wrote also a Malayalam commentary on the work. He died in 1838 A.D.

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² Published in Malayalam script with the commentary of K. Kunjunni Rāja. Also see Adyar Library Ms. 67735 and Madras Oriental Mss. Library, Ms. No. R. 4448.

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³ Introductory verse; also see. *Journal of the Bombay Branch of the Royal Asiatic Society*, 20, p. 74 fn.

Linked texts (28)

Tantrasaṅgraha — Nīlakaṇṭha Somayājī 1500, Sanskrit
Līlāvaī — Kouhala, Māhārāṣṭrī Prakrit
Sūryasiddhānta — Maya, Sanskrit
Bījagaṇita — Bhāskara II, Sanskrit
Āryabhaṭīya — Āryabhaṭa, Sanskrit
Goladīpikā — Parameśvara of Vaṭasseri, Sanskrit
Dhātusutta — Pāli
Viyāhapannatti — Sudharman, Ardhamāgadhī Prakrit
Samrāṭsiddhānta — Jagannātha Samrāj 1707, Sanskrit
Siddhāntatattvaviveka — Kamalākarabhaṭṭa 1633, Sanskrit
Mahābhāskarīya — Bhāskara I, Sanskrit
Muhūrtaratna (Īśvaradāsa) — Īśvaradāsa 1638, Sanskrit
Yuktibhāṣā (Gaṇitayuktibhāṣā) — Jyeṣṭhadeva 1520, Malayalam
Laghubhāskarīya — Bhāskara I, Sanskrit
Rekhāgaṇita (Jagannātha Samrāṭ) — Jagannātha Samrāṭ 1720, Sanskrit
Amṛtasāgara — Pratāpasiṁha 1778, Sanskrit
Grahalāghava — Gaṇeśa Daivajña 1520, Sanskrit
Karaṇapaddhati — Putumana Somayājī 1660, Sanskrit
Sphuṭanirṇaya — Acyuta Piṣāraṭi 1570, Sanskrit
Siddhāntaśiromaṇi — Bhāskara II, Sanskrit
Dṛggaṇita — Parameśvara of Vaṭasseri, Sanskrit
Bījagaṇitāvataṃsa — Nārāyaṇa Paṇḍita (mathematician), Sanskrit
Karaṇāmṛta — Citrabhānu 1507, Sanskrit
Karaṇottama — Acyuta Piṣāraṭi 1593, Sanskrit
Golasāra — Nīlakaṇṭha Somayājī, Sanskrit
Laghumānasa — Muñjāla, Sanskrit
Candracchāyāgaṇita — Nīlakaṇṭha Somayājī, Sanskrit
Gaṇitakaumudī — Nārāyaṇa Paṇḍita (mathematician), Sanskrit