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

CHAPTER XI Brahmagupta’s Astronomy : Its Highlights Beginning or Starting Point Very often in Indian astronomy, we come across a term ahargaṇa (literally meaning collection of days), which means the number of mean civil days elapsed at mean Sunrise at Laṅkā on a given lunar day (tithi), since the beginning of Kaliyuga. It is the beginning of Kaliyuga, which is taken as the starting point for the reckoning of ahargaṇa. This happened on Friday, Febru- ary 18, B.C. 3102, at mean sunrise at Laṅkā, when the Sun, Moon, and the planets are supposed to have been in conjunction at the first point of the nakṣatra Aśvinī (which is a fixed point situated near the star ζ-Piscium). According to Āryabhaṭa and Bhāskara I, the duration of Kaliyuga is 1,080,000 solar years. Four times this (4,320,000) is the duration in solar years of a bigger unit called Mahāyuga or even yuga. Laṅkā in Indian astronomy is a hypothetical place where the meridian of Ujjain (latitude 23° 11′ N, longitude 75° 52′ E from Greenwich) intersects the equator. It is one of the four hypothetical cities on the equator called Laṅkā, Romaka, Siddhapur and Yamakoṭi (or Yavakoṭi). The Sūrya-siddhānta describes Laṅkā as a great city (mahāpurī) situated on an island to the south of Bhāratavarṣa.¹ The present Ceylon is not the

1. समन्तान्मेरुमध्यात्तु तुल्यभागेषु तोयधेः । द्वीपेषु दिक्षु पूर्वादिन्नगर्यो देवनिर्मिताः ।। भूवृत्त पादे पूर्वस्यां यवकोटीति विश्रु ता । भद्राश्व वर्षे नगरी स्वर्णप्राकारतोरणा । याम्यायां भारतेवर्षे लङ्का तद्वन्महापुरी ।। (Cont. on page 290)

290 BRAHMAGUPTA'S ASTRONOMY ITS HIGHLIGHT astronomical Laṅkā, as it is about six degrees to the north of equator. The astronomical Laṅkā is mentioned by Brahmagupta in the beginning of his very first Chapter¹. According to Brahmagupta all the four yugas of a Catur- yuga or mahāyuga are not of the equal duration : Kaliyuga is of 432,000 years. Dvāpara of 864,000 years, Tretā of 1,296,000 years and Kṛtayuga of 1,728,000 years; total of the four is 4,320,000 years. Āryabhaṭa regards all yugas of equal duration, 1,080,000 years². The Saka era, which is usually used in Indian astronomy for the reckoning of years commenced 3179 years after the beginning of Kaliyuga. The number of lunar months in a yuga does not coincide with the number of solar months. Thus we have the conception of the Intercalary months : the number of intercalary months in a yuga denotes the excess of the number of lunar months in a yuga over the number of solar months in a yuga. Thus in a yuga we have Lunar months 53,433,336 Solar months 51,840,000 Intercalary months 1,593,336 Lunar days 1,603,000,080 Civil days 1,577,917,500 Omitted lunar days 25,082,580 The number of omitted lunar days in a yuga is equal to the number of lunar days in a yuga minus the number of civil days in a yuga. (Cont. from page 289) पश्चिमेकेतुमालाख्ये रोमकाख्या प्रकीर्तिता । उदक्सिद्धपुरी नाम कुरुवर्षे प्रतिष्ठिता ॥ --Sūrya. XII. 36-39

1. चैत्रसितादेरुदयाद्भानोर्दिनमासवर्षयुगकल्पाः । सृष्ट्यादौ लंकायां समं प्रवृत्ता दिनेऽर्कस्य ॥ —BrSpSi. I. 4
2. युगदशभागो गुणितः कृतं चतुर्भिस्त्रिभिर्गुणस्त्रेता । द्विगुणो द्वापरमेकेन संगुणः कलियुगं भवति ॥ युगपादानायैभटश्चत्वारि समानि कृतयुगादीनि । यदभिहितवान् न तेषां स्मृत्युक्तसमानमेकमपि ॥ —BrSpSi. I. 8-9

UNITS OF TIME 291 Units of time For the measurements of durations, it is necessary to have units of time. Brahmagupta gives the following units :¹ 6 prāṇas or Asus=1 Ṛkṣa-vināḍikā or nakṣatra-vighaṭikā or one pala (24 seconds) 60 palas =1 ghaṭikā (24 minutes) 60 ghaṭikās =1 divasa or dina (day) (24 hours) 30 dinas =1 māsa (month) 12 māsas =1 varṣa or year Similar to the divisions of time, we have the divisions of an arc :² Vikalā (or viliptā or viliptikā) =second of arc. 60 vikalās =1 kalā (minute of arc) 60 kalās =1 aṁśa (degree of arc) 30 aṁśas =1 rāśi 12 rāśis =1 bhagaṇa (complete circle, 360°) Unlike Āryabhaṭa and others, who take kali, dvāpara, tretā, and kṛta of equal number of years, Brahmagupta regards kali consisting of 432,000 years, dvāpara twice of it, consisting of 864,000 years, tretā thrice of Kali consisting of 1,296,000 years and kṛta four times of kali conisting of thus 1,728,000 years, all the four to be making a yuga of 4,320,000 years.³ Further in the beginning of kṛta there is a sandhyā of 1728000/12 years (=144,000 years), and at the end of Kṛta, there is a sandhyāṁśa of 144,000 years; similarly in the beginning of tretā, we have a

1. प्राणैर्विनाडिकाऽर्क्षी षड्भिर्घटिका षष्ट्या । घटिका षष्ट्या दिवसो दिवसानां त्रिंशता मासः ॥ —BrSpSi. I. 6
2. मासा द्वादशवर्षं विकलालिप्तांशराशिभगणांतः । क्षेत्र विभागस्तुल्यः कालेन विनाडिकाद्ये न ॥ —BrSpSi. I. 6 वर्षं द्वादश मासास्त्रिंशद्दिवसो भवेत्स मासस्तु । षष्टि र्नाड्यो दिवसष्षष्टिश्च विनाडिका नाडी ॥ —Ārya. III. 1 गुर्वक्षराणि षष्टिर्विनाडिकार्क्षी षडेव वा प्राणाः । एवं काल विभागः क्षेत्र विभागस्तथा भगणात् ॥ —Ārya, III. 2
3. खचतुष्टयरदवेदा रवि वर्षाणां चतुर्यु गं भवति । सन्ध्या सन्ध्यांशैः सह चत्वारि पृथक् कृतादीनि ॥ युगदशभागो गुणितः कृतं चतुर्भिस्रिभिर्गुणस्त्रे ता । द्विगुणो द्वापरमेकेन संगुणः कलियुगं भवति ॥ —BrSpSi. I. 7. 8

292 BRAHMAGUPTA'S ASTRONOMY ITS HIGHLIGHT sandhyā of 1.296.000/12; i. e. 108.000 years and at the close of tretā a sandhyāṁśa of 108,000 years. Again, in the beginning of dvāpara we have a sandhyā of 864.000/12, i. e. 72.000 years and at the close of dvāpara a sandhyāṁśa of 72000 years; and similarly at the beginning a sandhyā and at the close a sandhyāṁśa of 432.000/ 1, i. e. of 36,000 years in the case of kali. In this respect Brahmagupta appears to follow Manu. the first author or giver of law. He regards further the following divisions of time :¹ 71 yugas =1 manu 14 manus =1 kalpa Again, in the beginning, at the middle and at the close of each manu, there are sandhis, each equal to the measure of kṛta. Thus, taken as a whole 1 kalpa=71×14 yugas+15 sandhyā-sandhyāṁśa =994 yugas+15×duration of kṛta =994 yugas+15×(4×432,000) years =994 yugas + 6 yugas =1000 yugas = 1 Brahma-dina (Brahmā's day) Thus Brahmā's day is regarded as 1 kalpa or one thousand caturyugis or 1000 yugas or the same as 1000 mahāyugas). Āryabhaṭa regards a manu to consist of 72 yugas and therefore a kalpa according to him would be of 14×72 yugas, or 1008 yugas.² Since in the foreign Siddhāntas like Romaka, there is no reference to yuga, manu and kalpa. Brahmagupta regards these systems to be unauthoritative.³ We have said that our starting point was the beginning of Kaliyuga, Friday February 18, B.C. 3102, at mean rise at Laṅkā, when the Sun, Moon and the planets are supposed to have been in conjunction at the first point of the Nakṣatra Aśviṇī. This type of conjunction would again happen after a period of kalpa.

1. मनुरेकसप्ततियुगः कल्पो मनवश्चतुर्दश मनूनाम् । आद्यन्तरान्त सन्धिषु कृतकालोऽस्माद्युग सहस्रम् ॥ —BrSpSi. I. 10
2. दिव्यं वर्ष सहस्रम् ग्रहसामान्यं युगं द्विषट्क गुणम् । अष्टोत्तरसहस्रं ब्राह्मो दिवसो ग्रहयुगानाम् ॥ —Ārya. III. 8
3. युगमन्वन्तरकल्पाः कालपरिच्छेदकाः स्मृतावुक्ताः । यस्मान्नरोमके ते स्मृतिबाह्यो रोमकस्तस्मात् ॥ —BrSpSi. I- 13

UNITS OF TIME 293 We shall have the same type of conjunction of grahocca, mandocca śīghrocca and pāta after a complete cycle of kalpa as we had in the beginning of creation. This is a natural observable cycle which is recognised in Indian astronomy and in no other foreign system; and hence only the Indian system recognises the time measure of kalpa. Ucca or apex is of two kinds : mandocca (apex of the slowest motion) and śīghrocca (apex of the fastest motion). The mandocca is that point of a planet's orbit which is at the remotest distance and where the motion of the planet is slowest. In the case of the Sun and the Moon, it is the apogee; and in the case of other planets, it is the apogee or aphelion, the geocentric longi- tude of the apogee being equal to the helio-centric longitude of the aphelion. The śīghrocca of a superior planet (Mars, Jupiter and Saturn) is defined as the mean Sun; that of an inferior planet Mercury or Venus) śīghrocca is an imaginary body which is supposed to move in such a way that its direction from the earth is always approximately the same as that of the actual planet from the Sun. The bhagaṇas or the revolution numbers of a planet have been given by Āryabhaṭa and Brahmagupta¹ both; they mean the number of revolutions that a planet performs in a certain period, say a kalpa of 4,320,000,000 years. The bhagaṇas of planets are given as follows :

1. कल्पेऽर्कबुधसितानां भगणाः शून्यानि सप्त रदवेदाः । प्राग्ब्रजता कुजगुरुशनि शीघ्रोच्चानां स्वकक्षासु ॥ पञ्चाम्बराणि गुण-गुण पञ्चमुनि स्वरैर्मिताः शशिनः । भौमस्य द्वियमशराष्टपञ्चवसूरसनवद्वियमाः ॥ कृतवसुखवाष्टनन्द-नवषट्‌त्रि नवागेन्द्वो ज्ञशीघ्रस्य । जीवस्य शरेष्वद्रि षट्पञ्चद्वि कृतरसरामाः ॥ सितशीघ्रस्य यमलगोवेदनवाष्टाग्निपञ्चयमखनगाः । अष्टनवपञ्च मुनि रसशररस मनवोऽर्कपुत्रस्य ॥ खाष्टाब्धयो वसुशरवसुपञ्चखचन्द्रक्सुवसुसमुद्राः । द्विनवयमा द्वित्रिगुणाः शरेषु वसवस्त्रि पञ्चरसाः ॥ शशिवेदा मन्दानामर्कादीनां विलोमपातानाम् । वसूरसरुद्रेन्दुगुणा द्वित्रियमाः सप्तरसपञ्चाः ॥ शशियमाशरा गुणरसास्त्रिनन्दवसवः समुद्रवसु विषयाः । चन्द्रादीनां पश्चात् व्रजतोऽश्विन्यादि भगणस्य ॥ BrSpSi. I. 15-21

294 BRAHMAGUPTA'S ASTRONOMY ITS : HIGHLIGHTS | Planet or a body | Bhagaṇas | | Ravi or Sun | 4,320,000,000 | | Budha or Mercury | 4,320,000,000 | | Śukra or Venus | 4,320,000,000 | | Candra or Moon | 57,753,300,000 | | Kuja or Bhauma or Mars | 2,296,828,522 | | Budha-śīghrocca | 17,936,998,984 | | Bṛhaspati or Jupiter | 364,226,455 | | Śukra-śīghrocca | 7,022,389,492 | | Śani or Saturn | 146,567,298 | | Arka or Ravi-mandocca | 480 | | Candra mandocca | 488,105,858 | | Kuja or Bhauma mandocca | 292 | | Budha-mandocca | 332 | | Bṛhaspati or Jīva-mandocca | 855 | | Śukra-mandocca | 653 | | Śani-mandocca | 41 | | Candra-pāta | 232,311,168 | | Kuja or Bhauma-pāta | 267 | | Budha-pāta | 521 | | Bṛhaspati or Guru-pāta | 63 | | Śukra-pāta | 893 | | Śani-pāta | 584 | By pāta is meant the ascending node of a planet's orbit (on the ecliptic). In a kalpa, the number of bha-bhramas (sidereal days) or also known as bha-parivartas is 51,040,000,000. If we subtract out from this number the bhagaṇa of the Sun, we get what is known as ku-dinas or Savana days or the solar or sacrificial days. (51,040,000,000—4,320,000,000=46,720,000,000 Sāvana days or kudinas). In a kalpa, the number of Ravi-bhagaṇas also correspond to the number of solar years (Saura-varṣas), i.e., 4,320,000,000; this number multiplied by 12 gives the number (i.e. 51,840,000,000) of solar months. The difference between the candra-bhagaṇas and the Ravi- bhagaṇas in a kalpa gives the number of lunar months (Cāndra-

UNITS OF TIME 295 māsa) in a kalpa (57,753,300,000-4,320,000,000=53,433,300,000 lunar months). By subtracting the number of solar months from the number of lunar months in a kalpa, one gets the number of adhi-māsas (additional-months) : 53,433,300,000-51,840,000,000= 1,593,300,000 adhimāsas. This multiplied by 30 gives the number of lunar days (śaśi-divasa) in a kalpa ; 53,433,300,000×30= 1,602,999,000,000 lunar days. The difference between the lunar days and kudinas in a kalpa gives the number of avama-dinas in a kalpa : 1,602,999,000,000-45,720,000,000=1,556,279,000,000.¹ Brahmagupta calculates out the sṛṣṭi-saṁvatsara or the Creation Era during his year of composition of the Treatise. He says : Six manus have gone in the kalpa ; the seventh manu is now running of which have lapsed 27 caturyugis ; of the twenty eighth caturyugī, the three yugas, kṛta, dvāpara and tretā have gone by and also of the present kaliyuga 3179 years have lapsed. The total period thus lapsed on calculation comes to be 1,972,947,179 years :² Total Period=6 manus+7 manu-sandhis+27 yugas +kṛta+dvāpara+tretā+3179 years of kali. =(6×71×4,320,000 years) +(7×4×432,000 years)+ (27×4,320,000 years)+(1,728,000+1,296,000+864,000)+ 3,179=1,972,947,179 years. =1,840,320,000+12,096,000+116,640,000+3,888,000+ 3,179=1,972,947,179 years. Calculation of Ahargaṇa : The method of calculating ahar- gaṇa (number of days elapsed since the beginning of kaliyuga)

1. परिवर्त्ताः स्वचतुष्टयशराश्वि रसगुणयमद्विवसुतिथयः । रवि भगणोना भानोः सावनदिवसाः कुदिवसास्ते ॥ रवि भगणाख्यब्दा द्वादशगुणिता भवन्ति रविमासाः । भगणान्तरं रवीन्द्वोः शशिमासाः सूर्यमासोनाः ॥ अधिमासाः शशिमासास्त्रिंशद्गुणिता भवन्ति शशिदिवसाः । शशिसावनदिवसान्तरमवमानि तिथिः शशांकदिनम् ॥ —BrSpSi. I. 22-24
2. कल्पपरार्द्धं मनवः षट् कस्य गताश्चतुर्युगत्रिघनाः । त्रीणिकृतादीनिकलेर्गोऽङ्गैक गुणाः शकान्तेऽब्दाः ॥ नवनगशशि मुनिकृत नव यमनगनन्देन्दवः शकनृपान्ते । सार्धमतीतमनूनां सन्धिभिराद्यन्तरान्तर्गैः ॥ BrSpSi. I. 26-27

: is it a footnote mark, or is the print literally: - खराऽष्टेषु? Wait, look at line 39: -खराऽष्टेषुद्वयष्टशून्यशरारिभिः । Wait, look at the first character: संगुण्याख्या? No, look at line 39 again very carefully: First word: संगुण्याख्या? Wait! Could it be: संगुण्याख्या- No, look at the image: संगुण्याख्या-बराऽष्टेषुद्वयष्टशून्यशरारिभिः । Wait, why बरा? Could it be ऽबरा? Wait, look at the English text above: "25,082,580" Why would the Sanskrit have those characters? Wait, this is an Indian printed book from around 1970-1980 (looks like Indian National Science Academy, INSA, "BRAHMAGUPTA'S ASTRONOMY ITS : HIGHLIGHTS", page 296). In these INSA publications (often edited by K. S. Shukla or similar scholars), let's look at how the typesetters set Sanskrit text. Often the typesetter made typos, or used specific ligatures! Let's transcribe what is VISUALLY PRINTED on the page, because the user wants: "Transcribe all text on the ancient Indian

UNITS OF TIME 297 Addendum : The mean lunar day (madhyama tithi) may, however, differ from a true lunar day (spaṣṭa tithi) by one, so that the ahargaṇa obtained by the above process may sometimes be in excess or defect by one. To test whether the ahargaṇa (obtained by the above process) is correct, it is divided by seven and the remainder counted with Friday. If this leads to the day of calculation, the ahargaṇa is correct; if it leads to the preceding day, the ahargaṇa is in defect; and if that leads to the succeeding day, the ahargaṇa is in excess. When the ahargaṇa is found to be in defect, it is increased by one; when it is found to be in excess, it is diminished by one. (K.S. Shukla : MBh. p. 4-5) Example—Calculate the ahargaṇa on October 1,1965. From Indian Calendar we find that October 1,1965 falls on Friday. 7th lunar day (tithi) in the light half of the 7th month Āśvina in the Saka year 1887 (elapsed). Let us proceed as follows : Adding 3,179 to 1,887. we get 5,066. (1) Multiplying this by 12 and adding 6 (i. e. the number of lunar months elapsed since the beginning of Caitra) we get 60,798. ... ... (2). Multiplying this by 1,593,336 and dividing the product by 51,840,000, we get 1,868 as quotient. (The remainder is discarded as unnecessary) (3) Adding this number (i.e. 1,868) to the previous one (i.e.) 60,798) we get 62,666. (4) Multiplying this by 30 and adding 6 (i.e. the number of lunar days elapsed since the beginning of the current month) to the product, we get 1,879,986. (5) Multiplying this by 25,082,580 and dividing the product by 1,603,000,080, we get 29,416 as the quotient. (The remainder is discarded as not necessary). (6) Subtracting this number (i.e. 29,416) from the previous one (i.e. 1,879,986) we get 1,850,570. (7) This is the required ahargaṇa. Since division by 7 leaves

298 BRAHMAGUPTA'S ASTRONOMY: ITS HIGHLIGHTS 1 as the remainder. we subtract one from it, and get 1,850,569 as the correct ahargaṇa for the day. An Alternative Rule for Ahargaṇa Both Bhāskara I and Brahmagupta give an alternative rule for calculating out ahargaṇa¹ : Multiply the number of (solar months) elapsed since the beginning of kaliyuga by the number of lunar months (in a yuga) and divide by the number of solar months (in a yuga). Reduce the quotient to days (and add the number of lunar days elapsed since the begin- ning of the current lunar month); then multiply by the number of civil days (in a yuga) and divide by the number of lunar days (in a yuga); the quotient denotes the ahargaṇa. Mean Longitude of a Planet (i) The mean longitude of a planet in revolutions is given by the expression : (Brahmagupta² and also Bhāskara³).

1. शशांकमासैरभिताडितान् हरेदतीतमासानथ वार्कसम्भवैः । दिनीकृतान् भूमिदिनैर्हतान् दिनैर्विभज्य लब्धश्शशिजैर्हरर्गणः ॥ MBh. I. 7 युगगतशशिमासवधाद्रविमासाप्तं दिनीकृतं सदिनम् । भूदिनगुणितं शशिदिनहृतमाप्तमहर्गणः सैकः ॥ —BrSpSi. XIII. 18
2. इष्टग्रह भगण गुणादहर्गणात् कल्पसावन द्यु हृतात् । भगणादि फलं मध्यो लंकायां भास्करौदयिकः ॥ BrSpSi. I. 31
3. उद्दारितान् यान् भगणान् क्षमादिनैर्लभामहे कान् कलियातवासरैः । इति प्रलब्धा भगणास्ततः क्रमाद् गृहाशलिप्ता विकलाः सतत्पराः । —MBh. I. 8 पर्यायाहर्गणाभ्यासो ह्रियते भूदिनैस्ततः । लभ्यन्ते पर्यायाः शेषाद्राशि भागकलादयः ॥ भास्करै स्त्रिंशता षष्ट्या सङ् गुणय्य पृथक् पृथक् । तेनैव भागहारेण लभ्यन्तेऽर्कोदयावधेः ॥ —LBh. I. 15-17 (Divide the product of the revolution-number of a planet and the ahargaṇa by the (number of) civil days (in a yuga); thus are obtained the (number of) revolutions (performed by that planet). From the (successive remainders multiplied respectively by 12,30 and 60 and divided by the same divisor (i.e. the number of civil days in yuga) are obtained the signs, degrees and minutes etc. (of the mean longitude of that planet) for (mean) sunrise (at Laṅkā).

MEAN LONGITUDE OF PLANET 299 revolution number of planet × ahargaṇa Mean longitude = —————————————————————————————————————— civil days in a yuga Similar expression is given by more recent Indian astro- nomers also. (ii) Mean longitude of desired planets in minutes (mean longitude of the known planet in revolu- tions etc. reduced to minutes) × (revolution number of the desired planet) = ———————————————————————————————————————————————— revolution number of the known planet. This rule is common to Brahmagupta¹ and Bhāskara I². (iii) An alternative rule for deriving the mean longitude of the Moon from that of the Sun and vice versa has been given by Bhāsakara I and Brahmagupta both. Multiply the ahargaṇa by the number of intercalary months in a yuga and divide (the product) by the number of civil days (in a yuga) : the result is in the terms of revolutions etc. Add that to thirteen times the mean longitude of the Sun. (This is the process) to obtain the mean longitude of the Moon³. Mean longitude of the Moon (intercalary months in a yuga) × ahargaṇa = ————————————————————————————————————————— revolutions civil days in a yuga

1. ज्ञातभगणादिभुक्तं सविकलमिष्टयुग भगणसंगुणितम् । ज्ञात युगभगणभक्तं मध्यो भगणादि फलमिष्टः ॥ — BrSpSi. XIII. 27
2. निशाकरं वाग्रहमुच्चमेव वा कलीकृतं तत्सहयातमण्डलैः । यथेष्ट नक्षत्रगणैर्हतं हरेत् तदीयनक्षत्र गणैस्ततः कला ॥ MBh. I. 10 The (mean) longitude of the Moon, the planet, or the Ucca (whichever is known) together with the revolutions performed should be reduced to minutes. The resulting minutes should then be multiplied by the revolution-number of the desired planet and (the product obtained should be) divided by the revolution-number of that (known) planet. The result is (the mean longitude of the desired planet) in minutes.
3. द्युगणं युगाधिमासैर्गुणितं युग भूदिनैर्हृतेऽलब्धम् । भगणादि मध्यमार्के त्रयोदश गुणार्धिकं चन्द्रः ॥ —BrSpSi. XIII. 33 युगाधिमासैर्धुगणं हतं हरेत् क्ष्मादिनैर्वां भगणादि लभ्यते । त्रयोदशघ्ने सवितर्यथा क्षिपेन्निशीथिनीनां पतिचारसिद्धये ॥ —MBh. I. 11

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.