ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

304 BRAHMAGUPTA'S ASTRONOMY : ITS HIGHLIGHTS of the Sun and the Moon without basing on ahargaṇa has also been given by Brahmagupta in the Khaṇḍakhādyaka¹. Concordance of Working Rules There has been a good deal of agreement on various rules of astronomical constants from the time of Āryabhaṭa I (499 A. D.) to the Bhāskara II (1150 A. D.) or even later to the days of Munīśvara (1620 A, D). Earliest concepts were formulated during the days of the Vedāṅga-Jyautiṣa and the Siddhāntas of Indian and the western origin, for example of Brahma, Vasiṣṭha, Pitāmaha, Romaka and Puliśa. We in this section are giving some important concordances which we find common in the writings of Brahmagupta and his predecessors, contemporaries and successors as listed below. The list is not exhaustive. Only a few illustrations have been cited.

1. Ārya.—Āryabhaṭīya, Āryabhaṭa I, 499 A. D.
2. BrSpSi. —Brāhmasphuṭasiddhānta, Brahmagupta, 628 A. D.
3. K. K.—Khaṇḍakhādyaka, Brahmagupta, 628 A. D.
4. KKu.—Karaṇa-kutūhala, Bhāskara II, 1150 A. D.
5. LBh.—Laghu-Bhāskarīya, Bhāskara I, 522 A. D. MBh.—Mahā-Bhāskarīya, Bhāskara I, 522 A. D. MSi.—Mahā-siddhānta, Āryabhaṭa II, 950 A. D. PSi.—Pañcasiddhāntikā, Varāhamihira, 505 A. D. ŚiDVṛ.—Śiṣyadhīvṛddhida, Lalla, 598 A. D. SiSā.—Siddhāntasārvabhauma, Munīśvara, 1620 A. D. SiŚe—Siddhāntaśekhara, Śrīpati, 1039 A. D. SiŚi.—Siddhānta-Śiromaṇi, Bhāskara II, 1150 A. D. SuSi.—Sūryasiddhānta, Modern, 6th or 7th Century.
6. Rule for finding the mean longitudes of the Sun, Mercury and Venus : BrSpSi. I. 44. Also MBh. 1. 31. MSi I. 26 ; SiŚe II. 42, 43 ; SiŚi. I i. (d). 15 ; SiSā I. 105 ; KKu I. 7.
7. Rule for finding the mean longitude of the Moon's ascending node : BrSpSi, XXV. 35.
8. दिनदलभक्तमवमावशेषभाप्त दिनादि तत्सहितान् । अधिमासशेषाकाञ्च त्रिंशद् गुणितादृ‌तुखदिग्भिः ॥ मासदिन प्रथमैक्यं पृथक् त्रयोदशगुणं द्वितीयोनौ । द्वाक्येवं भृव्यौ राश्याद्यवार्कचन्द्रौ वा ॥ =KK. I. 11–12

CONCORDANCE OF WORKING RULES 305 Also MBh. I. 33 ; ŚiDVṛ. I. i. 52 (ii) 3. Rule for finding the mean longitude of the Śīghrocca of Venus and also giving the additives for the Śīghrocca of Mercury and Moon : BrSpSi. XXV. 36. Also MBh. I. 35 ; ŚiDVṛ. I. i. 57 (ii) 4. Rule for finding the mean longitude of the Śīghrocca of Mercury : BrSpSi. XXV. 34. Also MBh. I. 36 ; ŚiDVṛ. I. i. 50 (ii) 5. Rule for finding the mean longitude of Saturn : BrSpSi. XXV. 35. Also ŚiDVṛ. I. i. 52 (i) ; MBh. I. 37. 6. Rule for finding the mean longitude of Mars : BrSpSi. XXV. 33. Also ŚiDVṛ I. i. 50 (i) MBh. I. 38. 7. Rule for finding the mean longitude of Jupiter: BrSpSi. XXX. 35. Also MBh. I. 39 ; ŚiDVṛ. I. i. 51 (i). 8. Rule for finding the distance of a place from the prime meridian : BrSpSi. I. 36. Also MBh. II. 3-4 ; LBh. I. 25-26 ; ŚiDVṛ. I. 57-58 (i) ; SiŚā. I. 143-144. 9. Rule for finding the directions : BrSpSi. III. 1. Also MBh. III. 2 ; SūSi. III. 1-4 ; LBh. III. 1; ŚiDVṛ. I. iii. 1 ; MSi. IV. 1-2 ; SiŚe. IV 1-3 ; SiŚi. I. iii. 8-9. Alternative rule : BrSpSi. III. 2. Also MBh. III. 3 ; PSi. XIV. 14-16 ; ŚiDVṛ. I. iii. 2 ; SiŚe. IV. 4. 10. Rule for finding the latitude and colatitude and the zenith distance and altitude of the Sun : BrSpSi. III. 10. Also MBh. III 5; SūSi III. 13-14; LBh. III 2-3 ŚiDVṛ I. iii. 4-5; SiŚe. IV. 7; SiŚi I. iii. 18. 11. Rule for determining the declination, day—radius, earth sine and ascensional difference (for the Sun or a point on the ecliptic) : BrSpSi. II. 55. Also SūSi II. 28 ; LBh. II. 16; . I. ii ŚiDVṛ. 17; SiŚe. III.

306 BRAHMAGUPTA'S ASTRONOMY : ITS HIGHLIGHTS 63-64 ; SiŚi. I. ii. 47 (ii) (For RSine of the Declination). BrSpSi. II. 56 ; also Ārya. IV 24 ; MBh. III, 6; LBh. II. 17 ; ŚiDVṛ. I. ii. 18 ; SiŚe. III. 66 ; . SiŚi I. ii. 48 (For day- radius). BrSpSi. II. 57-58 ; also MBh. III. 7 ; LBh. II. 18 ; SūSi .II. 61; ŚiDVṛ. I. ii. 18; SiŚe III. 67 ; SiŚi. I. ii. 49 (i) (For the ascensional difference). 12. For finding the times of rising of the sāyana signs at the equator : BrSpSi. III. 15. Also MBh. III. 9 ; SūSi. III. 42–43 : ŚiDVṛ. I. iii. 8 ; SiŚe. IV. 15 ; SiŚi I. 11. 51. 13. Rule for finding the ascensional differences of the Sāyana signs Aries. Taurus and Gemini : KK. 1. 21. Also MBh. III. 8 ; PSi. III. 10 ; ŚiDVṛ. 1. XIII. 9 ; SiŚi. I. ii. 50-51. 14. Rule for the determination of the meridian zenith distance and meridian altitude of the Sun with the help of the Sun’s declination and the latitude of the place ; BrSpSi. III, 47. Also MBh. III. 11 ; LBh. III. 27 ; ŚiDVṛ. I. iii. 16 ;SiŚe. IV. 42. 15. Rule for determination of the latitude with the help of the Sun’s meridian zenith distance and declination : BrSpSi. : III. 13. Also MBh. III. 17: LBh. III. 34; SūSi. III. 15-16; SiŚe. IV. 51, 16. Rule for finding out the Rsine of the Sun’s altitude for the given time in ghaṭīs : BrSpSi. III. 25-26 Also Ārya IV. 28; MBh. III. 18-20; LBh. III. 7-10: ŚiDVṛ. I. iii. 24-25; SiŚe. IV. 32,34; SiŚi. I. iii. 53-54. 17. Rule for finding out the Sun’s altitude : RSin α= M×day radius gnomon ───────────── × ─────────────────────────────────────── R hypotenuse of equinoctial midday shadow where M=R Sin (given ghaṭīs∓asc. diff.) Rsin (asc. diff.), the upper or lower sign being taken according as the Sun is the nor- thern or southern hemisphere, α is the Sun’s altitude. BrSpSi. III. 27 (i).

CONCORDANCE OF WORKING RULES 307 Also MBh. III 24; ŚiDVṛ. I. iii. 27; SiŚe IV. 37 18. Rule for finding the Sun's altitude when the Sun's ascensional difference is greater than the given time : BrSpSi. III. 33 : Also MBh. III 25, LBh. III. 11; ŚiDVṛ. I. iii. 29 SiŚe. IV. 41. 19. Rule for finding the Sun's altitude in the night : BrSpSi. III. 63. Also MBh. III. 26; LBh. III. 11; SiŚe. IV. 89. The Sun's altitude for the night has been called Pātāla Śaṅku by Brahmagupta (BrSpSi. XV.9) 20. Rule for finding the longitude of the rising point of the ecliptic with the help of (i) the instantaneous sāyana longi- tude of the Sun and (ii) the civil time measured since sunrise, or with the help of (i) the Sun's sāyana longitude at sunrise and (ii) the sidereal time elapsed since sunrise : BrSpSi III. 18-20. Also MBh. III. 30-32: LBh. III. 17-19; SūSi. III. 46-48; ŚiDVṛ. I. iii. 11-12: SiŚe IV. 18-19 (i) : SiŚi I. iii. 2-4. 21. Rule for obtaining the civil time measured since sunrise with the help of (i) the Sun's instantaneous sāyana longi- tude and (ii) the sāyana longitude of the rising point of the ecliptic, or the sidereal time elapsed since sunrise with the help of (i) the Sun's sāyana longitude at sunrise and (ii) the sāyana longitude of the rising point of the ecliptic: BrSpSi III. 21-23. Also SūSi. III.50-51; MBh. III. 34-36; LBh. III. 20; ŚiDVṛ. I. iii. 13; SiŚe IV. 19 (ii)—22 (i); SiŚi I. iii. 5–7. (i). 22. Rule for determining the R Sines of the Sun's prime vertical altitude : BrSpSi III. 52. Also Ārya. IV; MBh. III.37-38; LBh. III. 52. (An error created by Āryabhaṭa has been criticised by Brahmagupta.) 23. Construction of the locus of the end of the shadow of a gnomon : BrSpSi. III. 2-3. Also MBh. III 52; ŚiDVṛ. I. iii. 3; SiŚe, IV. 5; 24. Rule for finding the Sun's mean anomaly: BrSpSi, II. 12 (i).

308 BRAHMAGUPTA' ASTRONOMY : ITS HIGHLIGHTS Also MBh. IV. 1; SūSi. II. 29 ; ŚiDVṛ I. ii. 10; SiŚe. iii. 12; SiŚi. I. ii. 18-19 (i). 25. Rule for finding the RSine (Reversed sine) of an arc (<90°) : BrSpSi II. 10. Also SūSi. II. 31-32 ; MBh. IV. 3-14; LBh. II, 2 (11)-3 (i); ŚiDVṛ. I. ii. 12; SiŚe. III. 15; SiŚi. I, ii. 10 (ii)-11. (We shall discuss it separately in the light of Brahmagupta formula.) 26. Rule for finding the Sun's equation of the centre: BrSpSi.. II. 15 (ii); Also MBh. IV. 4 (ii); . III SiŚe. 27 27. Rule for determining the Sun's true longitude: BrSpSi. XIV. 17-18. Also MBh. IV. 21-23; SiŚe. III. 52. 28. Rule for finding the Sun's bhujāntara correction under the eccentric theory : BrSpSi. XIV. 19. Also MBh.-IV. 24. 29. Rule for determining the cara-saṁskāra or cara correction : KK. I. 22. 30. Rule for finding the semi-durations of the day and night : BrSpSi. II. 60 ; KK. I. 23. Also SūSi. II. 62-63 ; ŚiDVṛ. I. ii. 20-21 ; SiŚe. III. 70 ; SiŚi. I. ii. 52. 31. Rule for calculating the tithi : BrSpSi. II. 62 ; KK. I. 25. Also SūSi. II. 66 ; ŚiDVṛ I. ii. 22 ; SiŚe. III. 71 ; SiŚi. I. ii. 66. 32. Rule for calculating the Karaṇa : KK. I. 27. Also. ŚiDVṛ. I. ii. 24 ; SiŚe. III. 77 ; SiŚi. I. ii. 66. 33. Rule for calculating nakṣatra : BrSpSi. II. 62 ; KK. I. 24. Also SūSi. II. 64 ; ŚiDVṛ. I. ii. 23 (i) ; SiŚe. III. 75; SiŚi. I. ii. 67. 34. Rule pertaining to direct and retrograde motions of a planet : BrSpSi. II. 50-51.

CONCORDANCE OF WORKING RULES 309 Also MBh. IV. 56-57 ; SiŚe. III. 59 ; ŚiDVṛ. I. ii. 42 ; 35. A rule for converting true distances known in min- utes into true distances into yojanas : for example : Sun's true distance in yojanas Sun's mean distance in yojanas × Sun's true dist. in minutes —————————————————————————————————————————— Radius BrSpSi. XXI. 31 (ii). Also MBh. V. 3 ; ŚiDVṛ. I. iv. 5 (i) ; LBh. IV. 3 ; SiŚe. V. 4 (ii) ; SiŚi I.v. 5 (i) ; 36. Rule for finding angular diameters of the Sun and the Moon : BrSpSi. XXI. 34 (ii) ; Also MBh. V. 5 : ŚiDVṛ. I. iv. 8 ; SiŚe. V. 6 ; SiŚi. I. v. 7, 37. Formulae for the true (i. e. angular) diameters of the Sun, the Moon and the shadow in terms of the true daily motions of the Sun and the Moon (Here by shadow is meant the section of the cone of the Earth's shadow at the Moon's distance.) : BrSpSi. IV. 6 (i) : KK. IV. 2 (i). Also MBh. V. 6-7 ; ŚiDVṛ. I. iv. 9 ; MSi. V. 5 (ii) ; SiŚe V. 9 ; SiŚi. I. v. 8-9 ; 38. Rule for finding the spaṣṭa-valana (resultant valana) for the circle drawn with half the sum of the diameters of the eclipsed and eclipsing bodies as radius : BrSpS.. IV. 18 (i). Also MBh. V. 46 ; ŚiDVṛ. I. iv. 26. 39. Method for calculating the phase of the eclipse for the given time : BrSpSi. IV. 11-12. Also MBh. V. 62-63 ; ŚiDVṛ. I. iv. 19-20 ; SiŚe. V. 14. 40. Rule for the determination of the diameter of the shadow i.e., the diameter of the Section of the Earth's shadow where the Moon crosses it : BrSpSi. XXIII. 8-9. Also MBh. V. 71-73 ; Āryc. IV. 39-40 ; ŚiDVṛ. I. iv. 6 (ii)-7. 41. Process of successive approximations in connection with calculations of a lunar eclipse : BrSpSi. IV. 8-9. Also MBh. V. 75-76 ; LBh. IV. 10-12 ; ŚiDVṛ. I. iv. 14-16 SiŚe. V. 12-13; SiŚi. I, v. 12-13.

310 BRAHMAGUPTA'S ASTRONOMY : ITS HIGHLIGHTS 42. Rule relating to the visibility-correction known as akṣa-dṛkkarma : BrSpSi. VI. 4. Also MBh. VI. 1-2; ŚiDVṛ. I. vii. 3 (ii); MSi. VII. 4; SiŚe. IX. 7. 43. Rule relating to the visibility correction known as ayanadṛkkarma : BrSpSi. VI. 3; X. 17. Slightly modified in MBh. VI. 2 (ii)-3; ŚiDVṛ. I. vii. 2-3 (i) · SiŚe. IX. 4-5; similar in MSi. VII. 2-3; more accurate in SiŚi. I. viii. 4-5. 44. Rule relating to the visibilily of moon : BrSpSi. VI. 6 ; X. 32. Also MBh; VI. 4-5 (i) PSi. V. 3 : ŚiDVṛ. I. vii. 5; SiŚe. IX. 8 (i). 13. 45. Rule for calculating the phase of the Moon : BrSpSi. VII. 11 (ii)-12. Also MBh. VI. 5 (ii)-7; ŚiDVṛ. I. ix. 12. 46. Rule for the determination of the Moon's true declination (i.e. the declination of the centre of the Moon's disc) : BrSpSi. VII. 5. Also MBh. VI. 8; ŚiDVṛ. 1. viii. 2: SiŚe. X. 7. (these are approximate rules; a more accurate rule occurs in SiŚi. I. vii. 3 and 13). 47. Graphical representation of the elevation of the lunar horns in the first quarter of the month at sunset : BrSpSi. VII. 7-10. Also MBh. VI. 13-17; ŚiDV. I. ix; SiŚi. I. ix. 48. Minimum distances of the planets from the Sun when they are visible : BrSpSi. vi 6 ; X. 32. Also MBh. VI. 44; ŚiDVṛ. I. vii. 5 (i); SiŚe. IX. 8 (i). 12. 49. Rule relating to the determination of the time and the common longitude of two planets when they are in conjunc- tion in longitude : BrSpSi. IX. 5-6. Also MBh. VI. 49-51; ŚiDVṛ. I. x. 7-9 (i); SiŚe. XI. 12-12. 50. Rule relating to the distance between two planets

CONCORDANCE OF WORKING RULES 311 which are in conjunction in longitude : BrSpSi. IX. 11. Also MBh. VI. 54 ; ŚiDVṛ. I. x. 11 ; SiŚe. XI. 10. 51. Rule For finding the Bhujaphala and Koṭiphala etc. without the use of the RSine-difference table : BrSpSi. XIV. 23-24. Also MBh. VII. 17-19 ; SiŚe. III. 17. 52. To obtain the Sun's mean true longitude derived from the midday shadow of the gnomon : BrSpSi. XIV. 28; III. 61-62. Also MBh. VIII. 5; SiŚi I. ii. 45. 53. Rule to find the arc corresponding to a given RSine : BrSpSi II. 11. Also MBh. VIII. 6; SūSi. II. 33, ŚiDVṛ. I. ii. 13, SiŚe. III. 16, SiŚi. I. ii. 11 (ii)—12 (i). For the concordance given here we express our indebted- ness to the work of K.S. Shukla on the Mahābhāskarīya. Tables of Constants Some of the Tables of Constants have been given in an earlier chapter. We here give a few more tables which would indicate how far Brahmagupta introduced new concepts in eva- luating these constants of greater accuracy and refinement. TABLE I Position of Planets for the Beginning of kaliyuga In this Table are given the positions of the planets, includ- ing the Moon's apogee and ascending node, for the beginning of Kaliyuga. The calculations of Brahmagupta are different from those of the Sūryasiddhānta and of Āryabhaṭa I,

312 BRAHMAGUPTA'S ASTRONOMY : ITS HIGHLIGHTS

SIDEREAL REVOLUTION OF THE NODES 313 TABLE III Sidereal Revolutions of the Apogees of the Planets in a Kalpa

314 BRAHMAGUPTA'S ASTRONOMY : ITS HIGHLIGHTS TABLE V Peripheries of the Epicycles of the Planets

1. In the middle of quadrants. it is 237°

LATITUDES OF JUNCTION STARS 315 TABLE VI Mean Diameters of the Planets

316 BRAHMAGUPTA'S ASTRONOMY : ITS HIGHLIGHTS TABLE VIII Longitudes of the Junction Stars according to Different Authorities

LATITUDES OF JUNCTION STARS 317

318 BRAHMAGUPTA'S A 1 | 2 | 3 Hasta | 11°S | 11°S Citrā | 1°45'S | 2°S Svāti | 37°N | 37°N Viśākhā | 1°23'S | 1°30'S Anurādhā | 1°44'S | 3°S Jyeṣṭhā | 3°30'S | 4°S Mūla | 8°30'S | 9°S P-Āṣāḍhā | 5°20'S | 5°30'S U-Āṣāḍhā | 5°S | 5°S Śravaṇa | 30°N | 30°N Dhaniṣṭhā | 36°N | 36°N Śatabhiṣak | 18'S | 30'S P-Bhādra | 24°N | 24°N U-Bhādra | 26°N | 26°N Revatī | 0 | 0

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320 BRAHMAGUPTA A GREAT CRITIC backing to the west) in one day. It is sufficient to postulate the existence of only one Sun, one Moon and twenty-seven nakṣatras to explain the astronomical phenomena.¹ 3. Brahmagupta differs from Āryabhaṭa I in the length of the four yugas. Ārvabhaṭa regards all the four yugas of equal lengths, i. e. 1,080,000 years; the caturyuga being of 4,320,000 years. Brahmagupta regards Kaliyuga to be of 432,000 years, Dvāpara to be twice of it, Tretā to be thrice of it and Kṛtayuga of four times of the length of the Kaliyuga. Both have the caturyuga of the same length.² 4. Āryabhaṭa was not clear with respect to the number of civil days (sāvana dina) in a yuga; in one of his treatises he gives this number to be 1,577,917,800 and in the other 1,577,917, 500 with a difference of 300 days. though in both the treatises, Ārya- bhaṭa I regards the number of solar years to be 4,320,000 in a Caturyuga or Mahāyuga. Why is this difference ? asks Brahmagupta.³ 5. Āryabhaṭa regards mandocca (the apogee) and pāta (the ascending node of the orbit on the ecliptic) as constant or stationary; then how could he propound a sphuṭa-yuga or the concept of true yuga with the concurrence of year, month and day on the Caitra Śukla Pratipadā (the first day of bright half of the month Caitra) at the same time as indicated by Āryabhaṭa in his Laghvāryabhaṭīya Tantra.⁴ Āryabhaṭa was not clear in respect to the variance in the pāta. In the Āryāṣṭa-śata (in the Aryabhaṭīya which has 108 Āryā verses), Aryabhaṭa states that the pāta of all the planets

1. भानिचतुष्पञ्चाशद्द्वा द्वावकैर्द्वौ जिनोक्तं यत् । ध्रुवमत्स्यस्यावर्त्तो भवति यतोऽन्हा ततस्तदसत् ॥ BrSpSi. XI. 3
2. आर्यभटोयुगपादांस्त्रीन् यातानाह कलियुगादौ यत् । तस्यकृतान्त्यस्मात् स्वयुगान्तौ न तत् तस्मात् ॥ —BrSpSi. XI. 4
3. युगरविभगणाः स्युर्धृति यत्प्रोक्तं तत् तयोर्युगं स्पष्टम् । त्रिशती स्युदयानां तदन्तरं हेतुना केन ॥ —BrSpSi. XI. 5
4. कुजवर्षादीन् वदतांचैत्रसितादेः समं प्रवृत्तान् यत् । उदसत् यतः स्फुटयुगं तत् स्थैर्यान्मन्दपातानाम् ॥ —BrSpSi. XI. 6

BRAHMAGUPTA A GREAT CRITIC 321 show variance of movement, but in the Daśagītikā (a chapter of ten Āryā verses), he states that with the exception of the pāta of the Moon, the pāta's of all other planets are stationary or cons- tant. Brahmagupta points out this anomaly in the concept of Āryabhaṭa.¹ 6. Brahmagupta points out a self-contradiction in Ārya- bhaṭa. At one place he says that the Moon covers the Sun during the solar eclipse and similarly the shadow of the Earth covers the Moon during the lunar eclipse (and he does not mention Rāhu) in this connection (Āryabhaṭīya, Gola. 37). At the same time it is said that Āryabhaṭa was familar with the movement of "eight" planets, and thus postulating the presence of Rāhu; in fact, the pātas of planets are responsible for their eclipse, and the eighth planet Rāhu is not present.² 7. Āryabhaṭa gives measures to Manus, Yugas and Kalpas different from what have been given in the recognised Smṛtis.³ 8. Āryabhaṭa regards Guruvāra or Thursday to be the first day of the Kalpa, and not Sunday, which is wrong according to Brahmagupta.⁴ 9. Brahmagupta unnecessarily criticises Āryabhaṭa in respect to the order of days. The motion of the planets decreases in the following order : Moon, Budha (Mercury), Śukra (Venus), Sun, Kuja (Mars), Guru (Jupiter) and Śani (Saturn). Āryabhaṭa in one of his verses states that starting from the Sun and pro- ceeding in the increasing order every fourth is the dinapati or the "lord of the day" (Āryabhaṭīya, Kāla. 16); this gives the order : Sun, Moon, Mars, Mercury, Venus and Saturn and hence the order of days as Ravivāra (Sunday); Candravāra (Monday,)

1. आर्य्याष्टशते पाता भ्रमन्ति दशगीतिके स्थिराः पाताः । मुक्तेन्दु पातमपमण्डले भ्रमन्ति स्थिरा नान्ये ॥ —BrSpSi. XI. 8
2. आर्य्यभटो जानाति ग्रहाष्टगतिं यदुक्तवांस्तदसत् । राहुकृतं न ग्रहणं तत्पातो नाष्टमो राहुः ॥ —BrSpSi. XI. 9
3. समामनुयुगकल्पाः कल्पादिगतं कृतादियातं च । स्मृत्युक्तं र्रार्य्यभटो नातो जानाति मन्वादिम् ॥ —BrSpSi. XI. 10
4. ओङ्कारो दिनवारो गुरुरौदयिकोऽस्य भवति कल्पादौ । न भव्यर्को यस्मादोङ्कारो विस्तरस्तस्मात् ॥ —BrSpSi. XI. 11

322 BRAHMAGUPTA A GREAT CRITIC Maṅgalavāra (Tuesday), (Budhavāra) (Wednesday), Guruvāra (Thursday) Śukravāra (Friday) and Śanivāra (Saturday). Thus Āryabhaṭa gives the same order of days as other authorities and there is no reason why he should be criticised. Certainly what is Sunday for Laṅkā, may not be Sunday at the same time for Siddhapura; and Āryabhaṭa should have emphasised that the day (Monday, Tuesday etc.) is not cons- tant for all the places.¹ 10. Brahmagupta expresses surprise why Āryabhaṭa, in two of his treatises propounds two different systems of reckoning one from the Sunrise in Laṅkā and the other from the Midnight in Laṅkā.² This causes a difference of one-fourth of the daily- motion in the two reckonings of the motion of planets.³ 11. Brahmagupta criticises Āryabhaṭa on the point of dia- meter of the Earth. In the Gītikāpāda, 5 and 6, Āryabhaṭa states that one yojana = 8,000 × puruṣa, and 1 puruṣa = 4 hasta, thus 1 yojana = 32,000 puruṣas, and the diameter of the Earth is 1050 yojanas. Brahmagupta further says that an error in the diameter of the Earth would cause an error in deśāntara or longitude and thus also in the true tithi and consequently in the calculation of eclipses also.⁴ 12. Āryabhaṭa has rightly stated that the Earth is in motion and the Bhagaṇas are stationary. Brahmagupta's objec- tion is that if the Earth is in motion, birds would not be able to return to their nests, and if the Earth's motion is upside-down,

1. सूर्याद्यश्चतुर्थी दिनवारा यदुवाच तदसदार्यभटः । लङ्कोदयो यतोऽर्कं स्यास्तमयं प्राह सिद्धपुरे ॥ — BrSpSi. XI. 12
2. अधिकैः शतैश्चतुर्भिर्वर्षसहस्रै रचतुर्दशभिरेकः । युगपाद्दिनवारान्तर मौदयिकार्ध रात्रिकयोः ॥ — BrSpSi. XI. 13.
3. औदयिकाद्दिनमुक्त गत्यंशो नार्ध रात्रिको भवत्यूनः । कतरं स्फुटं न निश्चितमनयोः स्फुटमेकमपि नातः ॥ — BrSpSi. XI. 14
4. षोडशगणितोजनरिधिं प्रतिभूव्यासं पुलावदतः । आत्मानं र या.पितमनिश्चयस्तनि कृतकन्द्यात् ।। भूव्यासस्याज्ञानाद् व्यर्थ देशान्तरं तदज्ञानात् । स्फुटतिथ्यज्ञानात् तिथिनाशाद् ग्रहणयोर्नाशः ॥ — BrSpSi. XI. 15-16

BRAHMAGUPTA A GREAT CRITIC 323 then the roof and hills would come down, which is contrary to our observation.¹ Obviously Brahmagupta is not justified in his criticism. 13. Brahmagupta points out to the differences in his calcu- lations and the calculations of Āryabhaṭa in the peripheries of the manda and śīghra epicycles of planets in the odd and even quadrants. (This difference we have shown in Table V, p. 313)² 14. Āryabhaṭa I and Bhāskara I have both given a rule for the determination of the dṛkkṣepajyās of the Sun and the Moon : Take the product of the Sun's or Moon's own madhya- jyā and udayajyā, then divide the product by the radius and then take the square of the quotient. Sub- tract that from the square of the own madhyajyā ; the square-root of that difference is known as the Sun's or Moon's dṛkkṣepajyā.³ The Sun's dṛkkṣepajyā is the Rsine of the zenith distance of that point of the ecliptic which is at the shortest distance from the zenith (this point is called nonagesimal or the central ecliptic point). The Moon's dṛkkṣepajyā is the Rsine of the zenith distance of that point of the Moon's orbit which is at the short- est distance from the zenith. The rule given above is only approximate and has been criticised by Brahmagupta.⁴ 15. In the Āryabhaṭīya, there is a rule in the Golapāda for finding the Rsine of the agrā of the true Sun; and also for the

1. प्राणेनैति कलां भूर्यदि तर्हि कुतो ब्रजेत् कमध्वानम् । आवर्त्तनमुर्व्याश्चेन्न पतन्ति समुच्छ्रयाः कस्मात् ॥ —BrSpSi. XI. 17
2. औदयिको यः परिधिर्विषमेऽन्योमेऽन्यः समे भुजस्य गुणः । तदसद्विषमान्तफलं यतो न युग्मादि फलतुल्यम् ॥ विषमेऽन्योऽन्यो युग्मे परिधिर्गुणकः क्रमोत्क्रमज्यानाम् । चक्रार्धे फलनाशो न भवति यस्मादसत् तदपि । —BrSpSi. XI. 18-19
3. स्वमध्यज्योदयाभ्यासं विष्कम्भार्धाप्तवर्गितम् ॥ मध्यज्यावर्गंतोऽपास्य स्वदृक् क्षेपं पदं विदुः ॥ —Mbh. V. 19
4. वित्रिभलग्ने दृक्क्षेपमण्डलं तदपमण्डलयुतौ ज्या । मध्याद्दक्क्षेपज्या नार्थभटोक्ताऽनया तुल्या ॥ दृक्क्षेपज्याऽतोऽसव तन्नाशादवनतेर्नांशः । अवनतिनाशात् ग्रासस्योनाधिकता रविग्रहणे ॥ —BrSpSi. XI. 29-30