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

300 BRAHMAGUPTA'S ASTRONOMY : ITS HIGHLIGHTS +13 (Sun's mean longitude) This expression may be rearranged to get the mean longitude of the Sun from the mean longitude of the Moon¹. Mean longitude of the Sun = 1/13 [mean longitude of the Moon

• ((intercalary months in a yuga) × ahargaṇa / civil days in a yuga) revolutions] Calculating the Mean Longitudes of the Sun and the Moon without using Ahargaṇa Bhāskara I follows the method of Āryabhaṭa I and the same method more or less has been adopted by Brahmagupta in calculating the mean longitudes of the Moon and the Sun without the use of ahargaṇa. The method may be described thus : Reduce the years elapsed since the beginning of kali- yuga to months and add to them elapsed months of the current year. Then multiply the sum by 30 and add the product to the number of lunar days elapsed since the beginning of the current month. Multiply that sum by the number of intercalary months in a yuga and divide by the number of solar months in a yuga reduced to days; the quotient denotes the number of intercalary months elapsed. The remainder is the adhimāsaśeṣa. Multiply the complete intercalary months thus obtained by 30 and to the product add the number of solar days elapsed since the beginning of kaliyuga² : then multiply that sum by the number of omitted lunar days in a yuga and divide by the number of lunar days in a yuga ; the remainder obtained is the avamaśeṣa called āhnika. Then multiply the avamaśeṣa

1. कुमुदतीनां सुहृदोऽधवाऽऽगतं विशोध्य शेषस्य लवस्त्रयोदशः । स मध्यमार्को गणकैर्निरूप्यते गुरुप्रसादात्प्रति बुद्ध बुद्धिभिः ॥ MBh. I. 11-12
2. By the number of solar days here is meant the number obtained above by reducing the years elapsed since the beginning of kaliyuga to months, then adding to them the number of months elapsed since the beginning of the current year, then multiplying the sum by 30, and then adding to the product thus obtained the number of lunar days elapsed of the current month.

MEAN LONGITUDES OF THE SUN AND THE MOON 301 (also called āhnika) by the number of intercalary months in a yuga and divide by the number of civil days (in a yuga). Add the resulting quotient to the adhimāsaśeṣa and divide the sum by the number of lunar months in a yuga : this gives degrees etc. (This is the total adhi- māsaśeṣa). Next multiply again the avamaśeṣa called āhanika by 60 and divide by the number of civil days in a yuga : the result is in minutes, seconds, thirds etc. The number of months elapsed (since the beginning of Caitra) are to be taken as signs and the number of lunar days elapsed of the current month as degrees. The sum of these signs and degrees and the minutes, seconds etc. corresponding to the avamaśeṣa is the grahatanu. From thirteen times and from one time that grahatanu severally subtract the degrees, minutes etc. corresponding to the total adhimāsaśeṣa : the remain- ders thus obtained are the mean longitudes of the Moon and the Sun respectively¹.

1. गुणिताद्यु गाधिमासैर्यु गभूदिवसैर्ह तादवमशेषात् । फलयुक्तमधिकमासकशेषं मध्यावतोऽर्केन्दू ॥ अधिमासावमशेषे युगशशि भूदिनहृते पृथग्लब्धेः । मासदिनाद्ये स्थाप्ये गतमासदिनानि चैत्रादेः ।। अवमशेषलब्ध्या सहितानि पृथक् त्रयोदश गुणानि । अधिमास शेषलब्ध्या हीनानि पृथक् रविशशांकौ । —BrSpSi. XIII. 20-22 विनाद्यु राशेरपि चन्द्रभास्करौ प्रकुर्व्वतो वा विधिरेष क थ्यते । समास मासिकृतविग्रहासु ये ह्यतीतमासा विनियोज्य तान् पुनः ।। खरामनिघ्नान् दिवसेषु योजयेद् गतेषु मासस्य ततोऽधिमासकैः । निहत्य सर्व विभजेत् सर्वदा युगार्कमासैर्दिवसत्वमागतैः ॥ भवन्ति लब्धास्त्वधिमासकाः पुनस्ततोऽपनीयाशु च भागहारकम् । भजेत शेषं शशिमास संख्यया ततोऽशलिप्ता विकलाः स तत्पराः ।। ततोऽधिमासान् प्रणिहत्य खाग्निभिर्नियोज्य सम्यग्गतवासरैः क्रमात् । युगावमघ्नाञ्छशिवासरैर्हरेत् तमत्र शेषं प्रवदन्ति चान्हिकम् ॥ हत्वाऽधिमासैरवमस्य शेषं छित्वा धराया दिवसैः प्रलब्धम् । संयोज्य नित्यं त्वधिमासशेषे कार्यं पुनस्तत् करणैर्यथोक्तम् ॥ युगप्रसिद्धैर्धरणी दिनैर्हरेन्निहत्य षष्ट्यावमशेषमान्हिकम् । कलाविलिप्ताः क्रमशस्तत्परास्ततोऽवमास्ता दिवसा गृहांशकाः ॥ त्रयोदशघ्नादपि रूपतादिताद्विशोधयेत्त्वधिमासशेषजम् । निशाकरार्कौ गणकैः प्रकीर्त्तितौ भट प्रणीताविति बुद्धिमत्तमैः । —MBh. I. 13-19 (Here verse 17 should follow verse 15—K.S. Shukla)

302 BRAHMAGUPTA'S ASTRONOMY : ITS HIGHLIGHTS K. S. Shukla has provided the following rationale to the rule cited above : The fraction of the intercalary month (obtained in the rule) = (adhimāsaśeṣa) / (solar days in a yuga), in mean lunar months. = (adhimāsaśeṣa) / (lunar days in a yuga), in mean solar months. (i) The fraction of the omitted lunar day (obtained in the rule) = (avamaśeṣa or āhnika) / (lunar days in a yuga), in mean civil days. = (avamaśeṣa) / (civil days in a yuga), in mean lunar days. = (avmaśeṣa × 60) / (civil days in a yuga), in mean lunar ghaṭīs. (ii) The fraction of the intercalary month corresponding to the above fraction of the omitted lunar day = [(intercalary months in a yuga) × (avamaśeṣa)] / [(lunar days in a yuga) × (civil days in a yuga)] in mean solar months. (iii) Adding (i) and (iii) and multiplying by 30, the total fraction of the intercalary month = { (adhimāsaśeṣa) / (lunar months in a yuga)

• [(intercalary months in a yuga) × (avamaśeṣa)] / [(lunar months in a yuga) × (civil days in a yuga)] } in mean solar days. (iv) Suppose that m lunar months and d lunar days have elapsed since the beginning of Caitra. Then, treating them as mean lunar months and mean lunar days, m months and d days denote the time elapsed since the beginning of mean Caitra up to the beginning of the current lunar day (treated as mean lunar day). As (ii) is the interval, in mean lunar ghaṭīs, between the beginning of the current lunar day and the mean sunrise on that day, therefore m months + d days + (ii) denotes the time in mean lunar months, days, ghaṭīs¹ elapsed

1. 1 hour = 2½ ghaṭīs; 1 ghaṭī = 60 vighaṭīs; 1 vighaṭī = 60 pravighaṭīs.

CONCORDANCE OF WORKING RULES 303 since the beginning of mean Caitra up to the mean sunrise on the current lunar day. Like wise m months + d days + (ii) - (iv) denotes the time in mean solar months, days, ghaṭīs etc. elapsed since the beginning of the current mean solar year up to the mean sunrise on the current lunar day¹. Let M, D, G, V, and P denote respectively the mean solar months, mean solar days, mean solar ghaṭīs mean solar vighaṭīs and mean solar pravighaṭīs elapsed since the beginning of the current mean solar year up to the mean sunrise on the current lunar day. Then evidently mean longitude of the Sun = M signs, D degrees, G minutes, V seconds and P thirds. = (m signs and d degrees) + [minutes, seconds etc. corresponding to (ii)] - [degrees, minutes etc. corresponding to (iv)]. and mean longitude of the Moon = 13 [m signs and d degrees + (minutes, seconds, etc. corresponding to (ii)] - [degrees, minutes etc. corresponding to (iv)] because [(1/12) mean longitude of the Moon - mean longitude of the Sun. = m signs + d degrees + (minutes, seconds etc. corresponding to (ii)] (This equality is based on the fact that the left hand side denotes the mean lunar date also known as madhyama tithi). A similar rule of these calculations of the mean longitude

1. Because (iv) is equal to fraction of a lunar month between the beginning of Caitra and the beginning of the current mean solar year fraction of an intercalary month corresponding to the tithis elapsed up to the beginn- ing of the current mean lunar day since the beginning of Caitra fraction of an intercalary month corresponding to the avamaśeṣa, i.e., the lunar portion between the beginning of the current lunar date and the follow- ing sunrise.

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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