Bh?skara II

(b. 1115)

astronomy, mathematics.

Bh?skara 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 P?tn?, near Chalisgaon in East Khandesh, Mah?r?stra, by Bh?u D?j?, 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 Bh?skara’s works, beginning with the Siddh?nta?iroma?i. There is further reference in this inscription to Soïdeva’s brother and successor, Hem?dideva, who was a feudatory of the Y?dava king of Devagiri, Sin?ghana, whose rule began in 1209/1210. The following genealogy is given in the inscription.

Trivikrama belonged to the ??ndilya gotra—which indicates that he and his descendants were Br?hmanas. His son was Bh?skarabhatta, who was given the title of Vidy?pati by Bhojar?ja (the Param?ra king of Dh?r? from ca. 995 to ca. 1056). The next four generations were respectively Govinda, Prabh?kara, Manoratha, and Mahe?vara; the last was the father of Bh?skara II. Bh?skara’s son, Laksm?dhara, was made chief of the Panditas by Sin?ghana’s predecessor, Jaitrap?la (1191–1209); and Laksm?dhara’s son, Can?gadeva, was the chief astrologer to Singhana himself. It is confirmed in Bh?skara’s works—e.g., in the concluding verses of the Siddh?nta?iroma?i—that his father was Mahe?vara of the ??ndilya gotra; it is further added that he came from the city Vijjadavida (Bij?pur in Mysore), which was probably named after the Kalac?ri king Vijjala II (1156–1175). If this identification is correct—since the Siddh?nta?iroma?i was written in 1150—Bhaskara II must have been in Vijjala’s capital while the latter was still dandan?yaka of the C?lukya kings, Jagadekamalla II (1138–1150) and Taila III (1150–1156). We further know from Trivikrama’s Damayantikath? that he was the son of Nem?ditya (Dev?ditya?) and the grandson of ?r?dhara; and there exists a popular astrological work by Mahe?vara, Bh?skara II’s father, entitled Vrtta?ataka

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

1. L?l?vat? (see Essay XII).
2. B?jaganita (see Essay XII).
3. Siddh?nta?iroma?i(see Essay IV).
4. V?san?bha?ya on the Siddh?nta?iroma?i (see Essay. IV).
5. Kara?akut?hala (see Essay IV).
6. Vivarana on the ?isyadh?vraddhidatantra of Lalla (see Essay V).
7. B?jopanaya (see Essay IV).

The L?lavat? and the B?jaganita are sometimes taken to be parts of the Siddh?nta?iroma?i;the ascription of the B?jopanaya to Bh?skara II is questionable.

1. The L?l?vat? is a work on mathematics addressed by Bhaskara II to a lady (his daughter or wife?)named L?l?avat?. 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 (ku??aka). This is the same as chapter 5 of the B?aganita

13. Combinations of digits.

The L?l?vat? has been commented on many times:

1. Karmaprad?pik? of N?r?yana (fl. 1356).
2. Vy?khy? of Para?ur?ma Misra (1356).
3. Vy?khy? of Parame?vara (fl. 1400–1450).
4. Ganit?mrtas?gari of Gang?dhara (ca. 1420).
5. Vy?khy? of Laksm?d?sa (fl. 1501).
6. Ganit?mrtak?pik? of S?ryad?sa (1541). See K. Madhava Krishna Sarma, Siddha-Bh?rat?, part 2 (Hoshiarpur, 1950), 222–225.
7. Buddhivill?sin? of Gane?a (1545). Published. See below, Sanskrit text of the L?l?val? no. 14.
8. Kriy?kramakar? of ?an?kara (fl. 1556).
9. Vivarana of Mah?dhara, alias Mah?d?sa (1587). Published. See below, Sanskrit text of the L?l?vat? no. 14.
10. Mitabh?sin? of Ran?gan?tha (1630).
11. Nisrst?rthad?t? of Mun??vara, alias Vi?var?pa (1635).
12. Ganit?mrialalhar? of R?makrsna (1687). See P. K. Gode, “Date of Ganit?rtalahar? of R?makrsna:’ in Annals Of the Bhandarkar OrientalResearch 1nstitute, 11 (1930), 94–95
13. Sarvabodhin? of ?r?dhara (1717).
14. Ud?harana of N?l?mbara Jh? (fl. 1823).
15. T?k? in Kannada of Alas?ng?rya, alias Ali?in?gar?ia
16. Vy?khy? of Bhave?a.
17. Ud?harano of Candra?ekhara Patan?yaka.
18. T?k? of D?modara(?).
19. Vil?sa of Dev?sah?ya
20. Bh?sana of Dhane?vara. Refers to S?ry?d?sa (1541).
21. T?k? (in vernacular) of Giridhara.
22. Vy?khy? of Ke?ava.
23. Tippana of Mukunda.
24. Vrtti of Mosadeva.
25. Subodhin? of R?ghava.
26. Ganakabh?sana of R?macandra, son of ?o?a?abha??a.
27. Kautukal?l?vat? of R?macandra, son of Vidy?dhara.
28. ?ippana of R?madatta (?).
29. Manoranjana of R?makrsnadeva.
30. ??k? of R?me?vara.
31. ??k? of ?r?kantha.
32. Ganit?mrtavan? of S?ryamani.
33. Ud?harana of V?re?vara. Refers to Laksm?d?sa (1501).
34. Ud?harana of Vi?ve?vara.
35. ??k? of Vrnd?vana (?).

In addition to these and a number of anonymous commentaries, there are others in Mar?h? and Gujar?t?. A modern Sanskrit commentary (aside from those which accompany some of the editions listed below) was published by candra ?ekhara Jh? under the title Vyaktavil?sa (Benares, 1924).

There are also numerous editions of the Sanskrit text of the L?l?vat?

1. Calcutta, 1832.
2. T?r?n?tha ?arman, ed. (Calcutta, 1846).
3. Baptist Mission Press (Calcutta, 1846; 2nd ed., Calcutta, 1876).
4. With the Vivarana of Mah?dhara and a Telugu commentary by Tadakamalla Ven?kata K???ar?va, V?villa R?masv?min ??strin, ed. (Madras, 1863).
5. J?v?nanda Vidy?s?gara, ed. (Calcutta, 1816).
6. Sudh?kara Dvivedin, ed. (Benares, 1878).
7. Edited, with his own Sanskrit commentary, by B?p?deva ??strin (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 Mar?th? commentary, by Vin?yaka P?nduraga Kh?n?p?rkar (Poona, 1897).
11. Sudh?kara Dvivedin, ed., Benares Sanskrit Series, no. 153 (Benares, 1912).
12. R?d?vallnbha, ed. (Calcutta, 1914)
13. Edited, with his own Sanskrit commentary, by Mural?dhara Th?kura, as ?r? Harikrsna Nibandha Manim?l? Series, no. 3 (Benares, 1928; 2nd ed., Benares. 1938).
14. With Buddhivil?sin? of Gane?a and Vivarana of Mah?dhara, Da?itreya ?pte, ed., ?nand??rama Sanskrit Series, no. 107, 2 vols. (Poona, 1937).
15. With Sanskrit commentary, edited by D?modara Mi?ra and Pay?natha Jh?, as Pr?c?n?caryaGranth?val? Series, no. 8 (Durbhanga, 1959).
16. With Sanskrit and Hind? commentaries of Lasana L?la Jh? edited by ?uresa ?arman, as Vidy?bhavana SamaskrtaGrantham?l? Series no 62 (Benares, 1961).

There are also many translations of the L?l?vat?. A Kannada version is supposed to have been made by Bh?skara Il’s contemporary R?j?ditya, who flourished, apparently, under the Hoysala king Visnuvardhana (111–1141). There also exists a Hind? translation, and the various commentaries in Gujar?t?, Mar?th?, and Telugu have already been referred to. Three Persian translations are known. That made by Ab? al-Fayd Fayd? at the request of Akbar in 1587 was published at Calcutta in 1827; another was done by Dharma Nar?yan ibn Kaly?nmalK?yath 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 Am?n ibn Shaykh Muhammad Sa??d. 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 Bh?scara (London, 1817). The latter was republished by Haran Chandra Banerji as Colebrooke’s Translation of the Líl?vati (Calcutta,)892; 2nd ed., Calcutta, 1927).

2 The B?jagnita, 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 (ku??aka)
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 B?jaganita are all relatively late, and they are far fewer in number than those on the L?l?vat?

1. S?ryaprak??a of S?ryad?sa (1538). See K. Madhava Krishna Sarma, in Poona Orientalist, 11 (1946), 54–66, and his article in Siddha-Bh?rat?, part 2 (Hoshiarpur, 1950), 222–225.
2. Nav?n?ura (or B?japallava, or B?j?vatamsa, or Kalpalat?var?ara) of Krsna (1602). See M. M. Patkar, in Poona Orientalist, 3 (1938), 169. Published. See below, Sanskrit texts nos. 13 and 16.
3. B?japrabodha of R?makrsna (1687). See P. K. Gode in Annals of the Bhandarkar Oriertal Research lnstitute, 10 (1919), 160–161, and 11 (1930), 94–95.
4. B?labodhin? of K?rp?r?ma (1792).
5. V?san?bha?ya of Harid?sa.
6. B?j?vala of Nij?nanda.
7. Kalpalat? of Parama?ukla (most likely Krsna’s work?).
8. B?javivarana of V?resvara (?).

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 Bh?skara,” 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. Gopin?tha P?thka, ed. (Benares 1864).
6. B?p?deva S?strin, ed., 2 parts (Calcutta [?], 1875)
7. J?v?nanda Vidy?s?gara, ed. (Calcutta, 1878).
8. Edited, with his own Sanskrit commentary, by J?v?niitha ?arman (Benares, 1885).
9. Edited, with his own Sanskrit Commentary, by Sudh?kara Dvivedin (Benares, 1888).
10. Edited, with a Mar?th? translation and commentary, by Vin?yaka P?nduranga Kh?n?p?rkar (Poona, 1913)
11. Edited, with his own Sanskrit commentary, by R?dh?vallabha (Calcutta, 1917).
12. Edited, with Sudh?kara Dvivedin’s Sanskrit commentary and one of his own, by Mural?dhara Jh?, as Benares Sanskrit Series, no. 154 (Benares, 1927).
13. Edited with the Nav?n?kura of K???a, by Datt?Hreya ?pte, as Anand?n?ma Sanskrit Series, no. 99 (Poona, 1930).
14. Edited, With his own Sanskrit and Hind? commentaries, by Durg?pras?da Dvivedin (3rd. ed., Lak?ma?apura, 1941; the preface is dated Jayapura, 1916).
15. Edited, with J?van?tha ?arman’s Sanskrit commentary and with his own in Sanskrit and Hind? by Acyur?nanda Jh?, as K??? Sanskrit Series, no. 148, (Benares, 1949).
16. Edited, with the B?japallava of K???a, by T. V. R?dh?krsna S?striri, as Tanjore Sarasvati Mahal Series, no. 78 (Tanjore, 1958).

There are two Persian translations of the B?jaganita, one anonymous and the other by ?At? all?h Rash?d?ibn A?hmad N?dir 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 Siddh?nta??romani, which was written in 1150, consists of two parts—the Grahaganit?dhy?ya (or Gait?dhy?ya) and the Gol?dhy?ya—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 p?tas 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 Gol?dhy?ya, then, is to a large extent an expansion and explanation of the Ganit?dhy?ya.

The following commentaries on the Siddh?nta?iroma?i are known (besides various anonymous ones):

1. Mit?ksar? (or V?san?bh?ya) of Bh?skara II himself (see 4 , below). Published. See below, under Sanskrit texts.
2. Ganitattvacint?mani of Laksm?d?sa (1501).
3. ?iromaniprak??a of Gane?a (b. 1507). Published in part. See below. Sanskrit text of Grahaganit?dhy?ya, no. 4.
4. Mar?ci of Mun?svara, alias Vi?var?pa(b. 1603). Published. See below, under Sanskrit texts.
5. T?k? of R?makrsna (fl. 1687).
6. T?k? of Cakrac?d?mani (?).
7. Vy?khy?;of Dhane?vara.
8. Vy?khy? of Harihara (?).
9. T?k? of Jayalaksmana (?).
10. Laksm?n?th? of Laksm?n?tha Mi?ra (?).
11. Bh?sya of Mahe?vara (?).
12. V?san? of Mohanad?sa (?).
13. Vy?khy? of Ran?gan?tha.
14. T?k? of V?caspati Mi?ra (?).

The Tippan?vivarana of Buddhin?tha Jh? was published at Benares in 1912.

The list of editions of the text is arranged under three headings: Siddh?nta?iroma?i, Grahaganit?dhy?ya, and Gol?dhy?ya.

Siddh?nta?iroma?i.

1. Siddh?nta?iromaniprak??a (of Gane?a?), with a Mar?thí translation (Bombay, 1837).
2. Siddh?nta?iromani, with the Prak??a (of Ga?e?a?), R?macandra, ed. (Madras, 1837).
3. Edited, with the V?san?bhasya, by B?p?deva ?astrin (Benares, 1866); revised by Candradeva (Benares, 1891); revised by Ga?apatideva ??strin, as K?så Sanskrit Series, no. 72 (Benares, 1929).
4. Edited with the V?san?bha?ya, the V?san?v?rttika of Nrsimha, and the Mar?ci of Mun??vara, by Mural?dhara Jh?, in The Pandit, n.s. 30–38 (1905–1916)—incomplete; the first chapter of the Grahaganit?dhy?ya was reprinted at Benares in 1917.
5. Edited, with a Sanskrit commentary, by Girij?pras?da Dvivedin (Ahmadabad, 1936).

Grahaganit?dhy?ya.

1. Edited, with the Mit?ksar?, by L. Wilkinson (Calcutta, 1842).
2. Edited, with the Mit?ksar? by J?v?nanda Vidy?s?gara (Calcutta, 1881).
3. Edited, with a Mar?thi translation and commentary, by Vin?yaka P?nduran?ga Kh?n?p?rkar (Poona, 1913).
4. Edited, with the V?san?bha?ya and the ?iromaniprk??a of Gane?a, by Datt?treya ?pte, as ?nand?a?rama Sanskrit Series, no. 110, 2 vols, (Poona, 1939–1941).
5. Edited, with the V?san?bha?ya and his own Sanskrit commentary, by Mural?dhara Thakkura, as K??i Sanskrit Series, no. 149 (Benares, 1950)—the first two chapters only.
6. Edited, with the V?san?bha?ya, the Mar?ci of Mun??vara, and his own Sanskrit and Hindi commentaries, by Ked?radatta Jo??, 3 vols. (Benares, 1961–1964); this edition does not include the Mar?ci on chapter 1.

Gol?dhy?ya.

1. Edited, with the Mit?ksar?, by L. Wilkinson (Calcutta, 1842).
2. Calcutta, 1856.
3. Edited, with the V?san?bha?ya, by J?v?nanda Vidy?s?gara (Calcutta, 1880).
4. Edited, with the V?san?bha?ya and a Bengali translation, by Rasikamohana Chattop?dhy?ya (Calcutta, 1887).
5. Edited, with the V?san?bha?ya and a Bengali translation, in Arunodaya, 1 (1890), part 6.
6. Edited, with a Mar?th? translation and commentary, by Vin?yaka P?nduran?ga Kh?n?p?rkar (Bombay, 1911)—chapters 1–8 only.
7. Edited, with the V?san?bha?ya and a Hind? commentary, by Girij?pras?da Dvivedin (Lucknow, 1911).
8. Edited, with the V?san?bha?ya and a Bengali translation, by R?dh?vallabha (Calcutta, 1921).
9. Edited, with the V?san?bha?ya and the Mar?ci of Mun??vara, by Datt?treya ?ple, as ?nand??rama 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 Grahaganit?dhy?ya publishedby E. Roer in Journal of the Royal AsiaticSociety of Bengal, 13 (1844), 53–66, and an English translation of the Gol?dhy?ya L. Wilkinson, revised by B?p?deva ??strin, as Bibliotheca Indica, no. 32 (Calcutta, 1861), with the Pandit’s translation of the S?ryasiddh?nta. See also L. Wilkinson, “On the Use of the Siddh?ntas in the Work of Native Education,” in Journal of the Royal Asiatic Society of Bengal, 3 (1834), 504–519.

4. The V?san?bha?ya or Mit?ksar? is Bh??kara II’s own commentary on the Siddh?nta?iroma?i A commentary on it, the V?san?v?rttika, was written by Nrsimha of Golagr?ma in 1621. Editions of both these works have been listed in the preceding material on the Siddh?nta?iroma?i.

5. The Karanakut?hala, which is also known as the Brahmatulya, the Grah?gamakut?hala, and the Vidagdhabuddhivallablla, was written in 1183; it gives simpler rules for solving astronomical problems than does the Siddh?nt?iroma?i. 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 p?tas of the sun and moon.
10. On the syzygies.

There are, aside from the usual quantity of anonymous commentaries on the Kara?akut?hala, eight whose authors’ names are known:

1. Bh?sya of Ekan?tha (ca. 1370).
2. N?rmad? of Padman?bha (ca. 1575).
3. Ud?hana of Vi?van?tha (1612).
4. Ganakakumudakaumud? of Sumatihar?a Ga?i (1622). Published. See below.
5. T?k? of Ca???d?sa.
6. Brahmatulyas?ra of Ke?av?rka (?).
7. T?k? of San?kara.
8. T?k? of So?hala.

For a set of tables based on the Kara?akut?hala, 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 Karanakut?hala has twice been edited: by Sudh?kara Dvivedin, with his own Sanskrit commentary (Benares, 1881); and, with the Ganakakumadakaumud? of Sumatihar?a Gani, by M?dhava S?stri Purohita (Bombay, 1902).

6. Bh?skara II’s Vivara?a on the ?i?yadh?v?ddhidatantra 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 B?jiopanaya is attributed to Bh?skara II, The author claims to be that scholar and to have written this work in 1151. A Tithinirnayad?rik?; published with it is the only other Sanskrit work to mention it; the author of this text claims to be ?r?niv?sa Yajvan, who flourished in Mysore in the second half of the thirteenth century and wrote a ?uddjhid?pik? and a commentary on the Karanaprak??a of Brahmadeva. Both works, despite their acceptance by Mukhopadhyaya and sengupta, are evidently late forgeries.

Kuppanna Sastri has shown that the B?jopanaya, 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 V?kya 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 Siddh?nta?iroman?.

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

(3) There are oblique references in the V?san?bh??ya, a commentary accompanying the B?jopanaya, which is also alleged to be by Bh?skara II, to Ran?gan?tha’s commentary on the S?ryasiddh?nta, which was written in 1602 and was publishedin 1859.

These arguments seem to this writer quite convincing.

The B?jopanaya has been published twice: by Cint?ma?i Raghun?th?c?rya and Ta?hakamalla Ven?ka?ak???a R?ya at Madras in 1876; and by Ekendran?th Ghosh at Lahore in 1926.

BIBLIOGRAPHY

The following bibliography generally excludes articles that deal only in part with Bh?skara II. It is divided into five sections: General, L?l?vat?, B?jaganita, Siddh?nta?iroma?i, and B?jopanaya. All entries are listed in chronological order.

I. General. The following deal with Bh?skara II and his works in general: Bh?u D?j?, “Brief Notes on the Age and Authenticity of the Works of Aryabhata, Var?hamihira, Brahmagupta. Bha??otpala, and Bh?skar?ch?rya,” in Journal of the Royal Asiatic Society (1865), 392–418, esp. 410–418: Jan?rdana B?l?j? Modaka, Bh?skara ?ch?rya and His Astronomical System (n.p., 1887); Sudh?kara Dvivedin, Ganakataran?gin? (Benares, 1933; repr. from The Pandit, n.s. 14 [1892]), pp. 34–42; B?p?deva ??strin, “A Brief Account of Bh?skara, and of the Works Written, and Discoveries Made, by Him.” in Journal of the Asiatic Society of Bengal, 62 (1893), 223–229; S. B. D?ksita, Bh?rat?ya Jyotih? ?stra (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, “Bh?skar?cb?rya’s References to Previous Teachers; “in Bulletin of the Calcutta Mathematical Society, 18 (1927), 65–76; B. Datta. “The Two Bh?skaras,” in Indian Historical Quarterly, 6 (1930), 727–736; and Brij Mohan. “The Terminology of Bh?skara,” in Journal of the Oriental Institute, Baroda, 9 (1959/1960), 17–22.

II. L?l?vat?. The L?l?vat? 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 Bh?skara,” in Verhandlungen des 3.Mathematikerkongresses in Heidelberg (Leipzig, 1905), pp. 556–561; Sarada Kanta Ganguly, “Bh?skar?c?rya 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, “Bh?skara 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 L?l?v?ti,” in Journal of the Oriental Institute, Baroda, 8 (1958/1959), 159–168.

III. B?jaganlta. The B?jaga?ita 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. Slddh?nta?irdma?i Works discussing the Siddh?nta?iroma?i are Bapudeva Sastri, “Bh?skara’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 ?°,” 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. B?jopainaya. Further discussion of the B?jopanayacan 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 B?jopanaya: Is It a Work of Bh?skar?c?rya?,” in Journal of the Oriental Institute, Baroda, 8 (1958/1959), 399–409.

Davld Pingree