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

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

324 BRAHMAGUPTA A GREAT CRITIC Rsine of the Sun's prime vertical altitude.¹ Bhāskara also gives the rule in his Mahābhāskarīya : Multiply the Rsine of the (Sun's) greatest declination by the Rsine of the Sun's true (sāyana) longitude; then divide (the product) by the Rsine of the colati- tude, the result is (the Rsine of) the agrā of the true Sun. When that (agrā) is less than the latitude and when the Sun is also in the northern hemisphere, multiply (the Rsine of the Sun's agrā) by (the Rsine of) the colatitude; the result is the Rsine of the Sun's prime vertical altitude.² The condition laid down in the rule that the "Sun's agrā should be less than the latitude" is incorrect. The error was originally comitted by Āryabhaṭa and Bhāskara followed it. This error was noticed by Brahmagupta³. Bhāskara I, however, corrected the error in the Laghu-Bhāskarīya. There he gives the correct conditions. It is not the agrā that should be less than the latitude, it is the Sun's declination (or rather the Rsine of the Sun's northern declination as we have in the Laghu- Bhāskarīya), which should be less than the latitude (or rather the Rsine of the latitude). This condition is necessary for the existence of the prime vertical shadow of the gnomon. It may be pointed out that the commentators of Āryabhaṭa I have interpreted the rule given by Āryabhaṭa I as conveying the correct meaning; they say that Āryabhaṭa also meant declination when he used the term agrā.

1. परमापक्रमजीवामिष्टज्यार्हतां ततो विभजेत् । ज्यालम्बकेन लब्धार्काग्रा पूर्वापरे क्षित्तिजे ॥ सा विषुवज्ज्योना चेद् विषुवदुदग्लम्बकेन संगुणिता । विषुवज्ज्या विभक्ता लब्धः पूर्वापरे शंकुः ॥ — Ārya. V. 30-31
2. स्फुटरविभुजनिघ्नां यां परां क्रांतिजीवां । हरतु समलम्बज्या कलापेन भूयः ॥ स्फुट दिवस कराग्रा सा यदाऽक्षांशहीना । रविरपि षडगोले चोत्तरे लम्बकृघ्नाम् ॥ अक्षज्यया हरेद् भूयः शंकुः स्यात् सममण्डले । तद् वर्गं व्यास कृत्योर्यद् विश्लेषं तत्पदं प्रभा ॥ — MBh. III. 37-38
3. उत्तरगोलेऽग्रायां विषुवज्ज्यातो यदुक्तमूनायाम् । सममण्डलमस्तदसत् क्रान्तिज्यायां यतो भवति ॥ — BrSpSi. XI. 22

BRAHMAGUPTA A GREAT CRITIC 325 Usually by agra, we mean the arc of the celestial horizon lying between the east point and the point where a heavenly body rises ; or between the west point and the point where a heavenly body sets. Declination is krānti. 16. Brahmagupta has criticised Āryabhaṭa and his group for their expressions for determining lambana (i. e. the difference of the parallaxes, in longitude, of the Sun and the Moon), and the rule for determining the avanati or nati (i.e. the difference in parallaxes, in latitude of the Sun and Moon).¹ Lambana is obtained with the help of the five Rsines : (i) madhya-jyā, (ii) udaya-jyā, (iii) dṛk-kṣepa-jyā, (iv) dṛg-jyā, and (v) dṛg-gati-jyā. (i) The madhyajyā is the Rsine of the zenith distance of meridian ecliptic point : madhyajyā=Rsin (ϕ± declination of the meridian ecliptic point).² In this expression ϕ is the latitude of the place, and by R sin is meant R × sine, R being the radius of the celestial sphere. (ii) The udayajyā is the Rsine of the arc of the horizon intervening between the equator and the ecliptic, and is given by : udayajyā = (R sin L × R sin ε) / (R cos ϕ) where L is the longitude of the horizon ecliptic point in the east. and ε the obliquity of the ecliptic. (iii) The dṛkksepajyā is the R sine of the zenith distance of the central ecliptic point³, and is given by :

1. व्यासार्धेन विभक्ता दृग्गति जीवा चतुर्गुणा लब्धम् । लम्बननाड्यः पञ्चदशगुणिताया त्रिज्यया भक्ता ॥ दृक्क्षेपज्या भुक्त्यन्तराहता लब्धमवनतिर्भवति । स्फुटयोजनकर्णाभ्यां भूव्यासेन च विना स्पष्टे ॥ आर्यभटेनास्मिन् सति लघुनि किमर्थं महत् कृतं कर्म । गणित ज्ञानाज्जाड्यं विजानता यदि ततः सुतराम् ॥ —BrSpSi. XI. 23-25
2. The meridian ecliptic point is the point of the ecliptic on the meridian.
3. The central ecliptic point is the central point of the portion of the eclip- tic lying above the horizon.

326 BRAHMAGUPTA A GREAT CRITIC dṛkkṣepajyā = [ (madhyajyā)² - { udayajyā × madhyajyā / R }² ]^(1/2) where R is the radius of the celestial sphere. (iv) The dṛgjyā is the Rsine of the zenith distance (of the Sun) and is given by : dṛgjyā = [ R² - { dṛggatijyā × R sin (L - θ) / R }² ]^(1/2) where L is the longitude of the horizon ecliptic in the east and θ the longitude of the Sun. (v) The dṛggatijyā is the Rsine of the altitude of the central ecliptic point, and is given by : dṛggatijyā = [R² - (dṛkkṣepajyā)²]^(1/2) where R is the radius of the celestial sphere. In the Mahābhāskarīya¹, the expression for the Sun's dṛggatijyā is : (Sun's dṛggatijyā)² = (Sun's dṛgjyā)² - (Sun's dṛkkṣepajyā)² and similar is the expression for the the Moon's dṛggatijyā. Now lambana, which is the difference of the parallaxes, in longitude, of the Sun and the Moon, is given by the expres- sion : Lambana = Moon's lambana - Sun's lambana. Sun's lambana = Sun's dṛggatijyā × Earth's semidiameter / Sun's true distance in yojanas Moon's lambana = Moon's dṛggatijyā × Earth's semi-diameter / Moon's true distance in yojanas These lambanas are in terms of minutes of arc etc².

1. स्वदृग्दृक्क्षेप गुणयोर्वर्गे विश्लेषजे पदे । दृग्गतिज्ये भवेतां ते भास्करामृत तेजसः ॥ -MBh. V, 23
2. स्वदृग्गतिज्यया व्यास भेद संवर्ग संभवम् । पृथग्योजन कर्णाप्तं लिप्ताद्यं लम्बनं विदुः ॥ (Cont. on Page 225}

BRAHMAGUPTA A GREAT CRITIC 327 Thus lambana is given by subtracting the Sun's lambana from Moon's lambana. This lambana is also expressed in the following way . Lambana [ {(dṛgjyā)²–(dṛkkṣepajyā)²}¹/² × 18 = [ ────────────────────────────────── — [ Moon's true distance {(dṛgjyā)²–(dṛkkṣepajyā)²}¹/² × 18 ] — ────────────────────────────────── ] Sun's true distance ] in minutes. 60 = ── × ( lambana calculated in minutes ) is the d lambana in ghaṭīs, where d denotes the difference between the daily motions of the Sun and the Moon. Āryabhaṭa I has given his description of the determination of lambana and avanati in the Golapāda 33, 34 of the Āryabhaṭīya¹, and Bhāskara has followed his rules in the Mahābhāskarīya². Brahmagupta criticises them in his Brāhma- sphuṭasiddhānta³. (vii) We shall now take up nati or avanati (both the terms mean the same). Nati is the difference of the parallaxes in latitude, of the Sun and the Moon and is given by : [ dṛkkṣepajyā × 18 dṛkkṣepajyā × 18 ] nati = [ ───────────────── — ───────────────── ] [ Moon's true dist. Sun's true dist. ] minutes. ───────────────────────────────────────────────────────────────── (Cont. from Page 324) तद्विशेषो हतः षष्ट्या स्फुटभुक्त्यन्तरोद्धृतः । घटिकादिस्तिथेः प्राह्णे शुद्धिः छेदोऽपरे मतः ॥ दिनार्धकालनिष्पन्नं लम्बनं शोध्यते तिथेः । उदगिन्दूदयज्यायां दीयते तत्र दक्षिणे ॥ एवं पुनः पुनः कर्म यावत्तदविशिष्यते । तिथिवच्चन्द्रतीक्ष्णांशू सञ्चार्यावेव पण्डितैः ॥ — MBh. V. 24-27

1. मध्यज्योदयजीवासंवर्गे व्यासदलहृते यत् स्यात् । तन्मध्यज्याकृत्योर्विशेषमूलं स्वदृक् चे पः ॥ दृग्दृक्क्षेप कृति विशेषितस्यमूलं स्वदृग्गतिः कुवशभून् । क्षितिजे स्व दृक् छाया भूव्यासार्धं नभोमध्यात् ॥ —Arya. IV. 33-34
2. loc. cit. MBh. V. 24-27
3. loc. cit. BrSpSi. XI. 23-25