Brahmagupta

(b. 598; d. after 665)

astronomy.

Brahmagupta was the son of Ji??ugupta; the names compounded with –gupta may indicate membership in the Vai?ya caste. At the age of thirty, he composed the Br?hmasphu?asiddh?nta; the reigning king was Vy?ghramukha of the C?pavamsa of the Gurjaras, and we know from the account of the Chinese pilgrim Hiuan–tsang (641) that the capital of the Gurjaras was Bhillam?la (modern Bhinmal, near Mt. Abu in Rajasthan). In the colophon of chapter 24 of the Br?hmasphu?asiddh?nta and in P?th?dakasv?min’s commentary on the Kha??akh?dyaka, Brahmagupta’s second work, he is called Bhillam?l?c?rya—the teacher from Bhillam?la. The Kha??akh?dyaka uses as its epoch Sunday, 15 March 665. Both of these treatises were studied primarily in Rajasthan, Gujarat, Madhya Pradesh, Uttat Pradesh, Bihar, Nepal, Panjab, and Kashmir.

The Br?hmasphu?asiddh?nta, whose planetary parameters are mainly derived from the Pait?mahasiddh?nta of the Vi??udharamottarapur??a (see Essays IV and XII), contains twenty–four chapters, with the twnety–fifth appended in some manuscripts.

1. On the mean longitudes of the planets.
2. On the true longitudes of the planets.
3. On the three problems relating to diurnal motion.
4. on lunar eclipses.
5. On solar eclipses.
6. On heliacal risings and settings.
7. On the lunar crescent.
8. On the lunar “shadow”.
9. On planetary conjunctions.
10. On conjunctions of the planets with stars.(These ten chapters form the Da??dhy?y?, which sometimes is found independently in manuscripts.)
11. Examination of previous treatises on astronomy.
12. On mathematics.
13. Additions to chapter 1.
14. Additions to chapter 2.
15. Additions to chapter 3.
16. Additions to chapters 4 and 5.
17. Additions to chapter 7.
18. On algebra.
19. On the gnomon.
20. On meters.
21. On the sphere.
22. On instruments.
23. On measurements.
24. Summary of contents.
25. Versified tables.

A commentary (V?san?bh??ya) on the Br?hmasphu?asiddh?nta was written by P?th?dakasv?min of Kuruk?etra (fl. 864); unfortunately, it is but imperfectly preserved in the few surviving manuscripts. There are also an anonymous commentary (??k?) on the Da??dhy?y? in Vishveshvarananda Vedic Research Institute 2363, and another (??k?) on chapter 18 in two manuscripts (one in Calcutta, one in London). A translation of chapters 12 and 18 was presented by H. T. Colebrooke in Algebra, With Arithmetic and Mensuration: From the Sanscrit of Brahmegupta and Bháscara (London, 1817); the preface is reprinted in Colebrooke’s Miscellaneous Essays, II (Madras, 1872), 417–531. The whole text, with his own Sanskrit comentary, was edited by Sudh?kara Dvivedin (Benares, 1902; reprinted from The Pandit, n.s. 23 [1901] and 24 [1902]). It was published again by Ram Swarup Sharma in four volumes (New Delhi, 1966): Volume I contains a sometimes useful introduction by Satya prakash (pp. I-344) and the text based on five of some fifteen known manuscripts; Volume II, chapters 1–9 with fragments of P?th?dakasv?min’s commentary on chapters 1–3, excerpts from Sudh?kara Dvivedin’s commentary, a new Sanskrit commentary (Vijn?ñabh??ya), and a Hindi explanation; Volume III, chapters 10–16 with the same commentaries, save that, in this volume, the available portions of P?th?dakasv?min’s V?san?bh?sya are strangely missing; and Volume IV, chapter 17–25 as in Volume III, but with P?th?dakasv?min’s commentary on chapter 21 added as an appendix.

The Kha??akh?dyaka is the best–known treatise on the ?rddha?trika system (see Essay VI). It contains eight chapters:

1. On computing the tithis and nak?atras.
2. On the longitudes of the planets.
3. On the three problems relating to diurnal motion.
4. On lunar eclipses.
5. On solar eclipses.
6. On heliacal risings and settings.
7. On the lunar crescent.
8. On planetary conjunctions.

There also exists an appendix (UttaraKhandakh?dyaka). If one follows Utpala (fl. 966), it contained three chapters:

1. On the conjunctions of the planets with the stars.
2. On the P?tas of the sun and moon.
3. On the projection of eclipses.

But the (incomplete) manuscript tradition of P?th?dakasv?min’s version provides only one chapter in the UttaraKha??akh?dyaka—one in which new methods of approximation are presented; and ?mar?ja (fl. 1180) cites various verses from the Uttara which are found in neither P?th?dakasv?min nor Utpala. The whole problem of the extent and authorship of the Uttarakha??akh?dyaka needs to be extensively investigated.

Commentaries have been written on the Kha??akh?dyaka by the following scholars: Lalla; P?th?dakasv?min (Vivara?a, 864); Utpala (Viv?ti, 966); Some?vara (1040); ?mar?ja (V?san?bh?sya, 1180); ?r?datta, the son of Nage?vara Mi?ra mahop?dhy?ya (??k?); and Y?mata (??k?); as well as a number of anonymous ??k?s and ud?haranas. No manuscripts of the commentaries of Lalla and of Some?vara have yet been identified. Diksita also refers to a ??k? by one Varu?a. The version and commentary of ?mar?ja was edited by Babua Mi?ra (Calcutta, 1925). The recension and commentary of p?th?dakasv?min were edited by P.C. Sengupta (Calcutta, 1941); an appendix contains additional verses from the version of Utpala. Sengupta has also translated P?th?dakasv?min’s version into English (Calcutta, 1934).

BIBLIOGRAPHY

Virtually every paper on every aspect of Indian astronomy discusses Brahmagupta. Listed here are only those which deal especially with his works. For the others, the reader is referred to David Pingree, A Census of the Exact Sciences in Sanskrit (Philadelphia, 1969). It should also be noted that the early ?Abb?sid Zij al–Arkand and Z?j al–Sindhind have been said to be versions, respectively, of the Kha??akh?dyaka and of the Br?hmasphu?asiddh?nia:but in neither case is a direct influence discernible. See David Pingree, “The Fragments of the Works of Ya?q?b ibn ??riq,” in Journal of Near Eastern Studies, 27 (1968), 97–125; and E. S. Kennedy and David Pingree, The Kitab ?ilal al–z?j?t of al–H?shim?. Both of Brahmagupta’s works were known in Sanskrit to al–B?r?n?; he cites them extensively, especially in India and in Al–Q?n?n al–Mas??d?. In his catalog of his own works, D. J. Boilot, ed., in Mélanges de l’Institut Dominicain d’études Orientales du Caire, 2 (1955), 161–256, al–B?r?n? includes Translation of What Is in the Brahmasiddh?nta of the Method of Calculation (RG 40).

Additional literature on Brahmagupta includes B. Datta, “On the Supposed Indebtedness of Brahmagupta to Chiuchang Suan–shu,” in Bulletin of the Calcutta Mathematical Society, 22 (1930), 39–51; ?. B. D?ksita, Bh?rat?ya Jyoti???stra (Poona, 1931; repr. of Poona, 1986), pp. 216–227; Sudh?kara Dvivedin, Ga?akatarañgi?? (Benares, 1933; repr. from The Pandit, n.s. 14 [1892]), pp. 18–19; P. C. Sengupta, “Brahmagupta on Interpolation,” in Bulletin of the Calcutta Mathematical Society, 23 (1931), 125–128; M. Simon, “Zu Brahmegupta diophantischen Gleichungen zweiten Grades,” in Archiv der Mathematik und Physik, 20 (1913), 280–281; G. Thibaut, Astronomie, Astrologie und Mathematik, Grundriss der indo–arischen Philologie und Altertumskunde, III, pt. 9 (Strasbourg, 1899), 58–59; and H. Weissenborn, “Das Trapez bei Euclid, Heron und Brahmagupta,” in Abhandlungen zur Geschichte der Mathematik, 12 (1879), 167–184.

Reference should also be made to the solar and lunar tables based on the Br?hmasphu?asiddh?nta that were published by R. Sewell in Epigraphia Indica, 17 (1923–1924), 123–187, 205–290.

David Pingree