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Bhāskara II

Bhāskara II

(b. 1115)

astronomy, mathematics.

Bhāskara II has been one of the most impressive Indian astronomers and mathematicians, not only to modern students of the history of science but also to his, contemporaries and immediate successors. An important inscription discovered at Pātnā, near Chalisgaon in East Khandesh, Mahārāstra, by Bhāu Dājī, and reedited by F. Kielhorn (Epigraphia Indica, 1 [1892], 338–346), records the endowment, by Soïdeva the Nikumbha, on 9 August 1207, of an educational institution (matha) for the study of Bhāskara’s works, beginning with the Siddhāntaśiromaṇi. There is further reference in this inscription to Soïdeva’s brother and successor, Hemādideva, who was a feudatory of the Yādava king of Devagiri, Sin̄ghana, whose rule began in 1209/1210. The following genealogy is given in the inscription.

Trivikrama belonged to the Śāndilya gotra—which indicates that he and his descendants were Brāhmanas. His son was Bhāskarabhatta, who was given the title of Vidyāpati by Bhojarāja (the Paramāra king of Dhārā from ca. 995 to ca. 1056). The next four generations were respectively Govinda, Prabhākara, Manoratha, and Maheśvara; the last was the father of Bhāskara II. Bhāskara’s son, Laksmīdhara, was made chief of the Panditas by Sin̄ghana’s predecessor, Jaitrapāla (1191–1209); and Laksmīdhara’s son, Can̄gadeva, was the chief astrologer to Singhana himself. It is confirmed in Bhāskara’s works—e.g., in the concluding verses of the Siddhāntaśiromaṇi—that his father was Maheśvara of the Śāndilya gotra; it is further added that he came from the city Vijjadavida (Bijāpur in Mysore), which was probably named after the Kalacūri king Vijjala II (1156–1175). If this identification is correct—since the Siddhāntaśiromaṇi was written in 1150—Bhaskara II must have been in Vijjala’s capital while the latter was still dandanāyaka of the Cālukya kings, Jagadekamalla II (1138–1150) and Taila III (1150–1156). We further know from Trivikrama’s Damayantikathā that he was the son of Nemāditya (Devāditya?) and the grandson of Śrīdhara; and there exists a popular astrological work by Maheśvara, Bhāskara II’s father, entitled Vrttaśataka

Bhāskara II is the author of at least six works, andpossibly of a seventh as well:

  1. Līlāvatī (see Essay XII).
  2. Bījaganita (see Essay XII).
  3. Siddhāntaśiromaṇi(see Essay IV).
  4. Vāsanābhaṣya on the Siddhāntaśiromaṇi (see Essay. IV).
  5. Karaṇakutūhala (see Essay IV).
  6. Vivarana on the Śisyadhīvraddhidatantra of Lalla (see Essay V).
  7. Bījopanaya (see Essay IV).

The Līlavatī and the Bījaganita are sometimes taken to be parts of the Siddhāntaśiromaṇi;the ascription of the Bījopanaya to Bhāskara II is questionable.

1. The Līlāvatī is a work on mathematics addressed by Bhaskara II to a lady (his daughter or wife?)named Līlāavatī. It contains thirteen chapters:

  1. Definitions of terms.
  2. Arithmetical operations.
  3. Miscellaneous rules.
  4. Interest and the like.
  5. Arithmetical and geometrical progressions.
  6. Plane geometry.

7–10 Solid geometry.

11. On the shadow of a gnomon.

12. Algebra: the pulverizer (kuṭṭaka). This is the same as chapter 5 of the Bīaganita

13. Combinations of digits.

The Līlāvatī has been commented on many times:

  1. Karmapradīpikā of Nārāyana (fl. 1356).
  2. Vyākhyā of Paraśurāma Misra (1356).
  3. Vyākhyā of Parameśvara (fl. 1400–1450).
  4. Ganitāmrtasāgari of Gangādhara (ca. 1420).
  5. Vyākhyā of Laksmīdāsa (fl. 1501).
  6. Ganitāmrtakūpikā of Sūryadāsa (1541). See K. Madhava Krishna Sarma, Siddha-Bhāratī, part 2 (Hoshiarpur, 1950), 222–225.
  7. Buddhivillāsinī of Ganeśa (1545). Published. See below, Sanskrit text of the Līlāvalī no. 14.
  8. Kriyākramakarī of Śan̄kara (fl. 1556).
  9. Vivarana of Mahīdhara, alias Mahīdāsa (1587). Published. See below, Sanskrit text of the Līlāvatī no. 14.
  10. Mitabhāsinī of Ran̄ganātha (1630).
  11. Nisrstārthadūtī of Munīśvara, alias Viśvarūpa (1635).
  12. Ganitāmrialalharī of Rāmakrsna (1687). See P. K. Gode, “Date of Ganitārtalaharś of Rāmakrsna:’ in Annals Of the Bhandarkar OrientalResearch 1nstitute, 11 (1930), 94–95
  13. Sarvabodhinī of Śrīdhara (1717).
  14. Udāharana of Nīlāmbara Jhā (fl. 1823).
  15. Tīkā in Kannada of Alasīngārya, alias Aliśin̄garāia
  16. Vyākhyā of Bhaveśa.
  17. Udāharano of Candraśekhara Patanāyaka.
  18. Tīkā of Dāmodara(?).
  19. Vilāsa of Devīsahāya
  20. Bhūsana of Dhaneśvara. Refers to Sūryādāsa (1541).
  21. Tīkā (in vernacular) of Giridhara.
  22. Vyākhyā of Keśava.
  23. Tippana of Mukunda.
  24. Vrtti of Mosadeva.
  25. Subodhinī of Rāghava.
  26. Ganakabhūsana of Rāmacandra, son of Ṣoṣaṇabhaṭṭa.
  27. Kautukalīlāvatī of Rāmacandra, son of Vidyādhara.
  28. Ṭippana of Rāmadatta (?).
  29. Manoranjana of Rāmakrsnadeva.
  30. Ṭīkā of Rāmeśvara.
  31. Ṭīkā of Śrīkantha.
  32. Ganitāmrtavanī of Sūryamani.
  33. Udāharana of Vīreśvara. Refers to Laksmīdāsa (1501).
  34. Udāharana of Viīveśvara.
  35. Ṭīkā of Vrndāvana (?).

In addition to these and a number of anonymous commentaries, there are others in Marāhī and Gujarātī. A modern Sanskrit commentary (aside from those which accompany some of the editions listed below) was published by candra Śekhara Jhā under the title Vyaktavilāsa (Benares, 1924).

There are also numerous editions of the Sanskrit text of the Līlāvatī

  1. Calcutta, 1832.
  2. Tārānātha Śarman, ed. (Calcutta, 1846).
  3. Baptist Mission Press (Calcutta, 1846; 2nd ed., Calcutta, 1876).
  4. With the Vivarana of Mahīdhara and a Telugu commentary by Tadakamalla Ven̄kata Kṛṣṇarāva, Vāvilla Rāmasvāmin Śāstrin, ed. (Madras, 1863).
  5. Jīvānanda Vidyāsāgara, ed. (Calcutta, 1816).
  6. Sudhākara Dvivedin, ed. (Benares, 1878).
  7. Edited, with his own Sanskrit commentary, by Bāpūdeva Śāstrin (Benares, 1883).
  8. Bhuvanacandra Basak, ed. (Calcutta, 1885).
  9. Edited as an appendix to Banerji’s edition of Colebrooke’s translation (Calcutta, 1892; 2nd ed., Calcutta, 1927).
  10. Edited, with a Marāthī commentary, by Vināyaka Pānduraga Khānāpūrkar (Poona, 1897).
  11. Sudhākara Dvivedin, ed., Benares Sanskrit Series, no. 153 (Benares, 1912).
  12. Rādāvallnbha, ed. (Calcutta, 1914)
  13. Edited, with his own Sanskrit commentary, by Muralīdhara Thākura, as Śrī Harikrsna Nibandha Manimālā Series, no. 3 (Benares, 1928; 2nd ed., Benares. 1938).
  14. With Buddhivilāsinī of Ganeśa and Vivarana of Mahīdhara, Daāitreya Āpte, ed., Ānandāśrama Sanskrit Series, no. 107, 2 vols. (Poona, 1937).
  15. With Sanskrit commentary, edited by Dāmodara Miśra and Payānatha Jhā, as PrācīnācaryaGranthāvalā Series, no. 8 (Durbhanga, 1959).
  16. With Sanskrit and Hindī commentaries of Lasana Lāla Jhā edited by Śuresa Śarman, as Vidyābhavana SamaskrtaGranthamālā Series no 62 (Benares, 1961).

There are also many translations of the Līlāvatī. A Kannada version is supposed to have been made by Bhāskara Il’s contemporary Rājāditya, who flourished, apparently, under the Hoysala king Visnuvardhana (111–1141). There also exists a Hindī translation, and the various commentaries in Gujarātī, Marāthī, and Telugu have already been referred to. Three Persian translations are known. That made by Abū al-Fayd Faydī at the request of Akbar in 1587 was published at Calcutta in 1827; another was done by Dharma Narāyan ibn KalyānmalKāyath ca. 1663 (H. J. J. Winter and A. Mirza, in Journal of the Asiatic Society of Science, 18 [1952], 1–10); and the third was made in 1678 by Muhammad Amīn ibn Shaykh Muhammad Saʿīd. There are also two English translations. That by J. Taylor was published at Bombay in 1816, and that by H. T. Colebrooke in his Algebra, With Arithmetic and Mensuration: From the Sanscrit of Brahmegupta and Bhāscara (London, 1817). The latter was republished by Haran Chandra Banerji as Colebrooke’s Translation of the Lílāvati (Calcutta,)892; 2nd ed., Calcutta, 1927).

2 The Bījagnita, on algebra, contains twelve chapters:

  1. On positive and negative numbers.
  2. On zero.
  3. On the unknown.
  4. On surds.
  5. On the pulverizer (kuṭṭaka)
  6. On indeterminate quadratic equations.
  7. On simple equations.
  8. On quadratic equations.
  9. On equations having more than one unknown.
  10. On quadratic equations having more than one unknown.
  11. On operations with products of several unknowns.
  12. On the author and his work.

The commentaries on the Bījaganita are all relatively late, and they are far fewer in number than those on the Līlāvatī

  1. Sūryaprakāūa of Sūryadāsa (1538). See K. Madhava Krishna Sarma, in Poona Orientalist, 11 (1946), 54–66, and his article in Siddha-Bhāratī, part 2 (Hoshiarpur, 1950), 222–225.
  2. Navān̄ura (or Bījapallava, or Bījāvatamsa, or Kalpalatāvarāara) of Krsna (1602). See M. M. Patkar, in Poona Orientalist, 3 (1938), 169. Published. See below, Sanskrit texts nos. 13 and 16.
  3. Bījaprabodha of Rāmakrsna (1687). See P. K. Gode in Annals of the Bhandarkar Oriertal Research lnstitute, 10 (1919), 160–161, and 11 (1930), 94–95.
  4. Bālabodhinī of K′rpārāma (1792).
  5. Vāsanābhaṣya of Haridāsa.
  6. Bījāvala of Nijānanda.
  7. Kalpalatā of Paramaśukla (most likely Krsna’s work?).
  8. Bījavivarana of Vīresvara (?).

The Sanskrit text has been frequently published:

  1. Calcutta, 1834; rev ed., Calcutta, 1834.
  2. Calcutta, 1846.
  3. Partial edition with a German tranislation by H. Brockhaus,“Üerdie Algebra des Bhāskara,” in Berichte über die verhandlungen der Königlich Säch-sischen Gesellschaft der Wissenschaften Zu Leipzig, Philosophisch-historische Klasse, 4 (Leipzig, 1852), 1–46.
  4. Calcutta, 1853.
  5. Gopinātha Pāthka, ed. (Benares 1864).
  6. Bāpūdeva Sāstrin, ed., 2 parts (Calcutta [?], 1875)
  7. Jīvīnanda Vidyāsāgara, ed. (Calcutta, 1878).
  8. Edited, with his own Sanskrit commentary, by Jīvāniitha Śarman (Benares, 1885).
  9. Edited, with his own Sanskrit Commentary, by Sudhākara Dvivedin (Benares, 1888).
  10. Edited, with a Marāthī translation and commentary, by Vināyaka Pānduranga Khānāpūrkar (Poona, 1913)
  11. Edited, with his own Sanskrit commentary, by Rādhāvallabha (Calcutta, 1917).
  12. Edited, with Sudhākara Dvivedin’s Sanskrit commentary and one of his own, by Muralīdhara Jhā, as Benares Sanskrit Series, no. 154 (Benares, 1927).
  13. Edited with the Navān̄kura of Kṛṣṇa, by DattāHreya Āpte, as Anandānśma Sanskrit Series, no. 99 (Poona, 1930).
  14. Edited, With his own Sanskrit and Hindī commentaries, by Durgāprasāda Dvivedin (3rd. ed., Lakṣmaṇapura, 1941; the preface is dated Jayapura, 1916).
  15. Edited, with Jīvanātha Śarman’s Sanskrit commentary and with his own in Sanskrit and Hindī by Acyurānanda Jhā, as Kāśī Sanskrit Series, no. 148, (Benares, 1949).
  16. Edited, with the Bījapallava of Kṛṣṇa, by T. V. Rādhākrsna Sāstriri, as Tanjore Sarasvati Mahal Series, no. 78 (Tanjore, 1958).

There are two Persian translations of the Bījaganita, one anonymous and the other by ʿAtā allāh Rashīdīibn A′hmad Nādir for Shah Jahan in 1634/1635. An English translation of the latter by E. Strachey, with notes by S. Davis, was published at London in 1813. It was also translated into English directly from the Sanskrit by H. T. Colebrooke in Algebra, With Arithmetic and Mensuratign… (London, 1817).

3. The Siddhāntaśīromani, which was written in 1150, consists of two parts—the Grahaganitādhyāya (or Gaitādhyāya) and the Golādhyāya—which are sometimes preserved singly in the manuscripts. The first part, on mathematical astronomy, contains twelve chapters:

  1. On the mean longitudes of the planets.
  2. On the true longitudes of the planets.
  3. On the three problems involving diurnal motion.
  4. On the syzygies.
  5. On lunar eclipses.
  6. On solar eclipses.
  7. On planetary latitudes.
  8. On the heliacal risings and settings of the planets.
  9. On the lunar crescent.
  10. On planetary conjunctions.
  11. On conjunctions of the planets with thestars.
  12. On the pātas of the sun and moon.

The second part, on the sphere, contains thirteen chapters:

  1. Praise of (the study of)the sphere.
  2. On the nature of the sphere.
  3. On cosmography and geography.
  4. Principles of planetarymean motion.
  5. On the eccentric-epicyclic model of the planets.
  6. On the construction of an armillary sphere.
  7. Principles of spherical trigonometry.
  8. Principles of eclipse calculations.
  9. Principles of the calculation of the first and last visibilities of the planets.
  10. Principles of the calculation of the lunar crescent
  11. On astronomical instruments.
  12. Descriptions of the seasons.
  13. On problems of astronomical computations.

The chapter on the sine function is placed differently in different editions. The Golādhyāya, then, is to a large extent an expansion and explanation of the Ganitādhyāya.

The following commentaries on the Siddhāntaśiromaṇi are known (besides various anonymous ones):

  1. Mitāksarā (or Vāsanābhāya) of Bhāskara II himself (see 4 , below). Published. See below, under Sanskrit texts.
  2. Ganitattvacintāmani of Laksmīdāsa (1501).
  3. Śiromaniprakāśa of Ganeśa (b. 1507). Published in part. See below. Sanskrit text of Grahaganitādhyāya, no. 4.
  4. Marīci of Munīsvara, alias Viśvarūpa(b. 1603). Published. See below, under Sanskrit texts.
  5. Tīkā of Rāmakrsna (fl. 1687).
  6. Tīkā of Cakracūdāmani (?).
  7. Vyākhyā;of Dhaneśvara.
  8. Vyākhyā of Harihara (?).
  9. Tīkā of Jayalaksmana (?).
  10. Laksmīnāthī of Laksmīnātha Miśra (?).
  11. Bhāsya of Maheśvara (?).
  12. Vāsanā of Mohanadāsa (?).
  13. Vyākhyā of Ran̄ganātha.
  14. Tīkā of Vācaspati Miśra (?).

The Tippanīvivarana of Buddhinātha Jhā was published at Benares in 1912.

The list of editions of the text is arranged under three headings: Siddhāntaśiromaṇi, Grahaganitādhyāya, and Golādhyāya.

Siddhāntaśiromaṇi.

  1. Siddhāntaśiromaniprakāśa (of Ganeśa?), with a Marāthí translation (Bombay, 1837).
  2. Siddhāntaśiromani, with the Prakāśa (of Gaṇeśa?), Rāmacandra, ed. (Madras, 1837).
  3. Edited, with the Vāsanābhasya, by Bāpūdeva Śastrin (Benares, 1866); revised by Candradeva (Benares, 1891); revised by Gaṇapatideva Śāstrin, as Kāså Sanskrit Series, no. 72 (Benares, 1929).
  4. Edited with the Vāsanābhaṣya, the Vāsanāvārttika of Nrsimha, and the Marīci of Munīśvara, by Muralīdhara Jhā, in The Pandit, n.s. 30–38 (1905–1916)—incomplete; the first chapter of the Grahaganitādhyāya was reprinted at Benares in 1917.
  5. Edited, with a Sanskrit commentary, by Girijāprasāda Dvivedin (Ahmadabad, 1936).

Grahaganitādhyāya.

  1. Edited, with the Mitāksarā, by L. Wilkinson (Calcutta, 1842).
  2. Edited, with the Mitāksarā by Jīvānanda Vidyāsāgara (Calcutta, 1881).
  3. Edited, with a Marāthi translation and commentary, by Vināyaka Pānduran̄ga Khānāpūrkar (Poona, 1913).
  4. Edited, with the Vāsanābhaṣya and the Śiromaniprkāśa of Ganeśa, by Dattātreya Āpte, as Ānandāaśrama Sanskrit Series, no. 110, 2 vols, (Poona, 1939–1941).
  5. Edited, with the Vāsanābhaṣya and his own Sanskrit commentary, by Muralīdhara Thakkura, as Kāśi Sanskrit Series, no. 149 (Benares, 1950)—the first two chapters only.
  6. Edited, with the Vāsanābhaṣya, the Marīci of Munīśvara, and his own Sanskrit and Hindi commentaries, by Kedāradatta Jośī, 3 vols. (Benares, 1961–1964); this edition does not include the Marīci on chapter 1.

Golādhyāya.

  1. Edited, with the Mitāksarā, by L. Wilkinson (Calcutta, 1842).
  2. Calcutta, 1856.
  3. Edited, with the Vāsanābhaṣya, by Jīvānanda Vidyāsāgara (Calcutta, 1880).
  4. Edited, with the Vāsanābhaṣya and a Bengali translation, by Rasikamohana Chattopādhyāya (Calcutta, 1887).
  5. Edited, with the Vāsanābhaṣya and a Bengali translation, in Arunodaya, 1 (1890), part 6.
  6. Edited, with a Marāthī translation and commentary, by Vināyaka Pānduran̄ga Khānāpūrkar (Bombay, 1911)—chapters 1–8 only.
  7. Edited, with the Vāsanābhaṣya and a Hindī commentary, by Girijāprasāda Dvivedin (Lucknow, 1911).
  8. Edited, with the Vāsanābhaṣya and a Bengali translation, by Rādhāvallabha (Calcutta, 1921).
  9. Edited, with the Vāsanābhaṣya and the Marīci of Munīśvara, by Dattātreya Āple, as Ānandāśrama Sanskrit Series, no. 122, 2 vols. (Poona, 1943–1952).

Aside from the translations into the vernacular mentioned above, I know only of the following two: a Latin translation of the Grahaganitādhyāya publishedby E. Roer in Journal of the Royal AsiaticSociety of Bengal, 13 (1844), 53–66, and an English translation of the Golādhyāya L. Wilkinson, revised by Bāpūdeva Śāstrin, as Bibliotheca Indica, no. 32 (Calcutta, 1861), with the Pandit’s translation of the Sūryasiddhānta. See also L. Wilkinson, “On the Use of the Siddhāntas in the Work of Native Education,” in Journal of the Royal Asiatic Society of Bengal, 3 (1834), 504–519.

4. The Vāsanābhaṣya or Mitāksarā is Bhāṣkara II’s own commentary on the Siddhāntaśiromaṇi A commentary on it, the Vāsanāvārttika, was written by Nrsimha of Golagrāma in 1621. Editions of both these works have been listed in the preceding material on the Siddhāntaśiromaṇi.

5. The Karanakutūhala, which is also known as the Brahmatulya, the Grahāgamakutūhala, and the Vidagdhabuddhivallablla, was written in 1183; it gives simpler rules for solving astronomical problems than does the Siddhāntśiromaṇi. There are ten sections:

  1. On the mean longitudes of the planets.
  2. On the true longitudes of the planets
  3. On the three problems involving diurnalmotion.
  4. On lunar eclipses.
  5. On solar eclipses.
  6. On heliacal risings and settings,
  7. On the lunar crescent.
  8. On planetary conjunctions.
  9. On the pātas of the sun and moon.
  10. On the syzygies.

There are, aside from the usual quantity of anonymous commentaries on the Karaṇakutūhala, eight whose authors’ names are known:

  1. Bhāsya of Ekanātha (ca. 1370).
  2. Nārmadī of Padmanābha (ca. 1575).
  3. Udāhana of Viśvanātha (1612).
  4. Ganakakumudakaumudī of Sumatiharṣa Gaṇi (1622). Published. See below.
  5. Tīkā of Caṇḍīdāsa.
  6. Brahmatulyasāra of Keśavārka (?).
  7. Tīkā of San̄kara.
  8. Tīkā of Soḍhala.

For a set of tables based on the Karaṇakutūhala, see David Pingree, “Sanskrit Astronomical Tables in the United States,” in Transactions of the American, Philosophical Society, n.s. 58 , no. 3 (1968). 36–37.

The Karanakutūhala has twice been edited: by Sudhākara Dvivedin, with his own Sanskrit commentary (Benares, 1881); and, with the Ganakakumadakaumudī of Sumatiharṣa Gani, by Mādhava Sāstri Purohita (Bombay, 1902).

6. Bhāskara II’s Vivaraṇa on the Śiṣyadhśvṛddhidatantra of Lalla has not been studied or published. There are three manuscripts: in Benares, in Bikaner, and in Ujjain.

7. A short text of fifty-nine verses entitled Bījiopanaya is attributed to Bhāskara II, The author claims to be that scholar and to have written this work in 1151. A Tithinirnayadārikā; published with it is the only other Sanskrit work to mention it; the author of this text claims to be Śrīnivāsa Yajvan, who flourished in Mysore in the second half of the thirteenth century and wrote a Śuddjhidśpikā and a commentary on the Karanaprakāśa of Brahmadeva. Both works, despite their acceptance by Mukhopadhyaya and sengupta, are evidently late forgeries.

Kuppanna Sastri has shown that the Bījopanaya, which gives rules for computing a correction to the moon’s equation of the center and variation, was most probably forged in south India in the early 1810’s to buttress the position of the partisans of the drk system against those of the Vākya system. His argument is based on three main points:

(1) The first correction is astronomically invalid and would have appeared so to the author of the Siddhāntaśiromanī.

(2) The style is completely at variance with Bhāskara’s normal method of exposition.

(3) There are oblique references in the Vāsanābhāṣya, a commentary accompanying the Bījopanaya, which is also alleged to be by Bhāskara II, to Ran̄ganātha’s commentary on the Sūryasiddhānta, which was written in 1602 and was publishedin 1859.

These arguments seem to this writer quite convincing.

The Bījopanaya has been published twice: by Cintāmaṇi Raghunāthācārya and Taḍhakamalla Ven̄kaṭakṛṣṇa Rāya at Madras in 1876; and by Ekendranāth Ghosh at Lahore in 1926.

BIBLIOGRAPHY

The following bibliography generally excludes articles that deal only in part with Bhāskara II. It is divided into five sections: General, Līlāvatī, Bījaganita, Siddhāntaīiromaṇi, and Bījopanaya. All entries are listed in chronological order.

I. General. The following deal with Bhāskara II and his works in general: Bhāu Dājī, “Brief Notes on the Age and Authenticity of the Works of Aryabhata, Varāhamihira, Brahmagupta. Bhaṭṭotpala, and Bhāskarāchārya,” in Journal of the Royal Asiatic Society (1865), 392–418, esp. 410–418: Janārdana Bālājī Modaka, Bhāskara Āchārya and His Astronomical System (n.p., 1887); Sudhākara Dvivedin, Ganakataran̄ginī (Benares, 1933; repr. from The Pandit, n.s. 14 [1892]), pp. 34–42; Bāpūdeva Śāstrin, “A Brief Account of Bhāskara, and of the Works Written, and Discoveries Made, by Him.” in Journal of the Asiatic Society of Bengal, 62 (1893), 223–229; S. B. Dīksita, Bhāratīya Jyotihś āstra (Poona, 1931; repr. of Poona, 1896), pp. 246–254; G. Thibaut, Astronomie, Astrologie und Mathematik, Grundriss der indo-arschen Philologie und Altertumskunde, III, pt. 9 (Strasbourg, 1899), 60; S. K. Ganguly, “Bhāskarācbārya’s References to Previous Teachers; “in Bulletin of the Calcutta Mathematical Society, 18 (1927), 65–76; B. Datta. “The Two Bhāskaras,” in Indian Historical Quarterly, 6 (1930), 727–736; and Brij Mohan. “The Terminology of Bhāskara,” in Journal of the Oriental Institute, Baroda, 9 (1959/1960), 17–22.

II. LĪlĀvatĪ. The Līlāvatī is discussed in E. Strachey, Observations on the Mathematical Science of the Hindoos. With Extracts From Persian Translations of the Leelawuttee and Beej Gunnit (Calcutta, 1805); H. Suter, “Über die Vielecksformel in Bhāskara,” in Verhandlungen des 3.Mathematikerkongresses in Heidelberg (Leipzig, 1905), pp. 556–561; Sarada Kanta Ganguly, “Bhāskarācārya and Simultaneous Indeterminate Equations of the First Degree,” in Bulletin of the Calcutta Mathematical Society, 17 (1926), 89–98; M. G. Inamdar, “A Long Forgotten Method,” in Annals of the Bhandarkar Oriental Research Institute, 9 (1927–1928), 304–308; A. A. Krishnaswami Ayyangar, “Bhaskara and Samclishta Kuttaka,” in Journalof the Indian Mathematical Society18 (1929), 1–7; Saradakanta Ganguli, “Bhāskara and Simultaneous Indeterminate Equations of the First Degree,” ibid., 19 (1931/1932), 6–9; A. S. Bhandarkar, “’Method of False Assumption’of Pacioli, an Italian Mathematician,” in Indian culture, 8 (1941/1942), 256–257; K. S. Nagarajan, “Bhaskara’s Leelavathi,” in The Mathematics Student, 18 (1950), 1–8; and Brij Mohan, “The Terminology of Līlāvāti,” in Journal of the Oriental Institute, Baroda, 8 (1958/1959), 159–168.

III. BĪjaganlta. The Bījagaṇita is dealt with in Reuben Burrow, “A Proof That the Hindoos Had the BinomialTheorem," in Asiatick Researches2 (1790), 487–497; A. A. Krishnaswami Ayyangar, “New Light on Bhaskara’s Chakravala or Cyclic Method of Solving Indeterminate Equations of the Second Degree in Two Variables,” in Journal of the Indian Mathematical Society, 18 (1929), 225–248; K. J. Sanjana, “A Brief Analysis of Bhaskara’s Bijaganita With Historical and Critical Notes,” ibid., 176–188; and D. H. Potts, “Solution of a Diophantine System Proposed by Bhaskara,” in Bulletion of the Calcutta Mathematical Society, 38 (1964), 21–24.

IV. SlddhĀntaŚirdmaṄi Works discussing the Siddhāntaśiromaṇi are Bapudeva Sastri, “Bhāskara’s Knowledge of the Differential Calculus,” in Journal of the Asiatic Society of Bengal, 27 (1858), 213–216; W. Spotiswoode, “Note on the Supposed Discovery of the Principle of the Differential Calculus by an Indian Astronomer,” in Jounal of the Royal Asiatic Society (1860), 221–222; H. Suter, “Eine indische Methode der Berechnung der Kugeloberfläche,” in Bibliotheca mathematica, 3rd ser., 9 (1908/1909), 196–199; R. Sewell, “The Siddhanta-siromai,” in Epigraphic Indica, 15 (1919/1920), 159–245; M. G. Inamdar, “A Formula of Bhaskara for the Chord of a Circle Leading to a Formula for Evaluating Sin α°,” in The Mathematics Student, 18 (1950), 9–11; and A. A. Krishnaswami Ayyangar, “Remarks on Bhaskara’s Approximation to the Sine of an Angle,” ibid., 12.

V. BĪjopainaya. Further discussion of the Bījopanayacan be found in Dhirendranath Mukhopadhyaya. “The Evection and the Variation of the Moon in Hindu Astronomy,” in Bulletin of the Calcutta Mathematical society, 22 (1930), 121–132; P. C. Sengupta, “Hindu Luni-solar Astronomy,” ibid., 24 (1932), 1–18; and T. S. Kuppanna Sastri, “The Bījopanaya: Is It a Work of Bhāskarācārya?,” in Journal of the Oriental Institute, Baroda, 8 (1958/1959), 399–409.

Davld Pingree

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