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

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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

328 BRAHMAGUPTA A GREAT CRITIC (viii) Moon’s true latitude = Moon’s latitude × nati. The present Sūryasiddhānta and Brahmagupta both utilise the following expressions for lambana and nati which give more accurate values : lambana = [Rsin (M - ☉) × dṛggatijyā] / {Rsin (30°)}² ghaṭis where M and ☉ denote the longitudes of the meridian ecliptic and the Sun respectively. nati = (dṛkkṣepajyā × d) / (15 × R) where R is the radius of the celestial sphere and d denotes the difference between the daily motions of the Sun and the Moon¹. Brahmagupta has raised objections to the Āryabhaṭa system regarding lambana (XI. 26-28), dṛkkṣepa (XI. 30-31) ayanadṛk- karma (XI. 35), elevation of Moon’s horns (śṛṅgonnati) (XI. 39) and similar several other points. He is so vehemently opposed to Āryabhaṭa that finally he declares : “It is beyond my capacity to enumerate all the defects of Āryabhaṭa. Only a few have been given here as illustration. Intelligent people can easily find out others².” He also says : Āryabhaṭa is neither conversant with the Gaṇita (mathematics), nor Kāla (time calculations) nor Gola (celestial or spherical calculations). It is difficult to enumerate separately the fallacies committed by him in the respective chapters of the Gaṇitapāda, Kālakri- yapāda and Golapāda³.

1. loc. cit. BrSpSi. XI. 23.
2. आर्यभटदूषणानां संख्या वक्तुं न शक्यते यस्मात् । तस्मादयमुद्देशो बुद्धिमताऽन्यानि योज्यानि ॥ —BrSpSi. XI. 44
3. जानात्येकमपि यतो नार्यभटो गणितकालगोलानाम् । न मया प्रोक्तानि ततः पृथक् पृथग् दूषणान्येषाम् ॥ —BrSpSi. XI. 43.

BRAHMAGUPTA AND ŚRĪṢEṆA 329 Brahmagupta and Śrīṣeṇa In Varāhamihira's Pañcasiddhāntikā we have a critical review of the five Siddhāntas or five systems of astronomical study : Puliśa Siddhānta, Romaka Siddhānta, Vasiṣṭha Siddhānta, Sūrya Siddhānta and Brāhma Siddhānta. Colebrooke in his Paper “On the notion of the Hindu Astronomers concerning the precession of the equinoxes and motions of the planets”, publi- shed in the Asiatic Researches, vol. xii. p. 209-250, Calcutta. 1816,4 to, reproduced in the Miscellaneous Essays. Vol. II. 1872, says the following in regards to the authorship of these schools of astronomy : All these books are frequently cited in the astrono- mical compilations and are occasionally referred to their real or supposed authors. The first is everywhere assigned to Puliśa, whose name it bears. The Romaka Siddhānta is ascribed by the scholiast of Brahmagupta and by a commentator of the Sūrya Siddhānta to Śrīṣeṇa. The Vāsiṣṭha Siddhānta is by the same authority given to Viṣṇucandra. Both these authors are repeatedly mentioned with censure by Brahma- gupta ; and it is acknowledged that they are entitled to no particular deference. The Brāhma Siddhānta, which is the basis of Brahma- gupta’s work, is not anywhere attributed to a known author ; but referred to in all quotations of it which have fallen under observation, either to the Viṣṇudhar- mottara Purāṇa, of which it is considered as forming a part, or to Brahmā (also called Pitāmaha) who is introduced into it as the speaker in a dialogue with Bhṛgu, or it is acknowledged to be the work of some unknown person. The true author it may be now impracticable to discover, and would be vain to con- jecture. The Sūrya Siddhānta (if the same which we now pos- sess) is in the like manner ascribed to no certain author unless in the passage cited by my colleague Mr. Bently (Asiatic Researches, vol. vi. p. 572) who says that “in the commentary of the Bhāsvatī, it is declared, that

330 BRAHMAGUPTA A GREAT CRITIC Varāha was the author of the Sūrya Siddhānata¹, and who adds, that “Satānanda, the author of the Bhāsvatī was a pupil of Varāha under whose directions, he himself acknowledges, he wrote that work”. This concluding remark alludes to the following verse of the Bhāsvatī Karaṇa : “Next I will propound succinctly., from Mihira’s instruction, (the system) equal to the Sūrya Siddhānta¹, (Miscellaneous Essays, p. 388-90) (The word ‘Mihira’ has double meaning : it might be an abbreviation of Varāhamihira, or it may mean sun or Sūrya). Thus on the authority of Colebrooke, Śrīṣeṇa may by re- garded as the initiator of the Romaka system. Brahmagupta him- self mentions in one of his passages the name of Śrīṣeṇa in connec- tion with the Romaka system, and further the conceptions of the Romaka system came down as Vāsiṣṭha system through Viṣṇucan- dra,² Lāṭadeva also derived from Śrīṣeṇa the concepts of the mean motions of the Sun, the Moon, the Moon’s apogee and her node and the mean motions of Mars, Mercury’s. Śīghra, Jupiter, Venus’ śīghra and Saturn. I have indicated elsewhere, which is also the view of Sankara Bālakṛṣṇa Dīkṣita, that the original Romaka and Paulīśa Siddhāntas were introduced to Indians by Lāṭadeva, and the latter Romaka Siddhānta by Śrīṣeṇa (Original Romaka Siddhānta was prevalent before Śaka 427 and this is the one which is mentioned by Varāhamihira who makes no reference to Śrīṣeṇa and Viṣṇucandra in the Pañcasidhāntikā, and the latter Romaka Siddhānta was introduced by Śrīṣeṇa as is indicated by Brahmagupta. Thus we have two Vāsiṣṭha Siddhāntas and two Romaka Siddhāntas). My personal view is that Lāṭadeva, Śrīṣeṇa and possibly Viṣṇucandra also, were naturalised Greeks, settled in India and they had adopted themselve to Indian life. They were conversant in Greek and Indian Astronomy both and had contributed substantially to Indian astronomy. Brahmagupta was opposed to any of these f oreign influences dominating Indian

1. अथ प्रवक्ष्ये मिहिरोपदेशात् तत्सूर्यसिद्धान्त समं समासात् ॥ —Bhāsvatī Karaṇa
2. श्रीषेणेन गृहीत्वा रचोन्त्यरोमकः कृतः कन्था । एतानेव गृहीत्वा वासिष्ठो विष्णुचन्द्रेण ॥ —BrSpSi. XI. 50

BRAHMAGUPTA AND ŚRĪṢEṆA 331 systems, and he very much resented such interferences in pure academic life of this country. He was opposed to Āryabhaṭa for a different reason. Āryabhaṭa was universally regarded as an authority in this country, and the conservatism was so deep that even where it could be shown by direct observation or on valid theoretical grounds, that a particular concept was erroneous or less accurate, people still chose to adhere to it, since they had the backing of Āryabhaṭa's authority. Brahma- gupta was against this nonscientific attitude. Needless to say, Brahmagupta was not always fair to Āryabhaṭa in his criticism ; he overdid in enumerating the shortcomings of Āryabhaṭa's system, as if he was personally jealous of his wide popularity. Brahmagupta's feelings against Lāṭadeva, Śrīṣeṇa, Viṣṇu- candra and others would be seen from the following passage in the Brāhmasphuṭasiddhānta : From the fact that Śrīṣeṇa, Viṣṇucandra, Pradyumna, Āryabhaṭa, Lāṭa, and Siṁha contradict one another regarding eclipses and similar topics, their ignorance is proved daily. The criticisms which I have passed on Āryabhaṭa are, with the requisite modifications, to be applied to the doctrines of each of these teachers as well. I will, however, make some further critical re- marks on Śrīṣeṇa and others. Śrīṣeṇa took from Lāṭa the rules concerning the mean motions of the Sun, and the Moon, the Moon's apogee and her node, and the mean motions of Mars, Mercu- ry's Śīghra, Jupiter, Venus's Śīghra, and Saturn ; he took elapsed years and the revolutions of yuga (yuga- yāta-varṣa-bhagaṇa) from Vasiṣṭha and the Padakaraṇa of Vijayānandi; further took from Āryabhaṭa the rules concerning the apogee, epicycles and nodes, and those referring to the true motions of the planets and thus the Romaka Siddhānta which was (or is) a heap of jewels (as it were) has, by Śrīṣeṇa, been made into a patched rag (as it were)¹.

1. श्रीषेणविष्णुचन्द्र प्रद्युम्नार्यभटलाटसिंहानाम् । ग्रहणादिविसंवादात् प्रतिदिवसं द्विगुणमज्ञत्वम् ॥ [Cont. on Page 330]

332 BRAHMAGUPTA A GREAT CRITIC Brahmagupta very emphatically says about his system that so long as people would be finding concordance between the observed and theoretical results (dṛggaṇitaikyam) in respect of solar and lunar eclipses, his Brāhma Siddhānta would be held in esteem¹. In other systems, whatever concordance appears to be bet- ween the observation and calculation, of eclipses etc., it is, Brahmagupta says, merely accidental or by chance, as the maxim of letters bored by an insect in wood or paper². युक्त्याऽऽर्यभटोक्तानि प्रत्येकं दूषणानि योज्यानि । [Cont. from Page 329] श्रीषेणप्रभृतीनां कानि चिदन्यानि वक्ष्यामि ॥ लाटात् सूर्यशशांकौ मध्याबिन्दूच्च चन्द्रपातौ च । कुजबुधशीघ्रबृहस्पति सितशीघ्र शनैश्चरान् मध्यान् ॥ युगयातवर्षभगणान् वासिष्ठाद्विजयनन्दि कृतपादात् । मन्दोच्च परिधिपातस्पष्टीकरणाद्यमार्यभटात् ॥ श्रीषेणेन गृहीत्वा रत्नोच्चयरोमकः कृतः कन्था । एतानेव गृहीत्वा वासिष्ठो विष्णुचन्द्रेण ॥ —BrSpSi. XII. 46-50 2. चन्द्ररवि ग्रहणेन्दुच्छायादिषु सर्वदा यतो ब्राह्मे । दृग्गणितैक्यं भवति स्फुटसिद्धान्तस्ततो ब्राह्मः ॥ —BrSpSi. XI. 61 3. अन्योर्न कदाचिदपि ग्रहणादिषु भवति दृष्टिगणितैक्यम् । यद्भवति तद् घुणाक्षरमतोऽस्फुटभ्यां किमेताभ्याम् ॥ —BrSpSi. XI, 51. —: 0 :— Reference Brahmagupta : Tantraparīkṣādhyāya in BrSpSi. K.S. Shukla : The Mahābhāskarīya and the Laghubhāskarīya. H.T. Colebrooke : Miscellaneous Essays, Vol. II., 1872. G. Thibaut and Sudhākara Dvivedi : The Pañcasiddhāntikā, Preface, 1889.

CHAPTER XIII Brahmagupta and Astronomical Instruments The Twenty-second Chapter of the Brāhmasphuṭasiddhānta is known as the Yantrādhyāya or a chapter on instruments. There is a description of seventeen types of time-reckoning instru- ments (Kāla-yantra)¹ :

1. Dhanuryantra—Bow instrument.
2. Turyagolaka yantra—Quadrant (one-fourth sphere)
3. Cakra yantra=wheel or circle.
4. Yaṣṭi yantra—a pole or staff instrument.
5. Śaṅku yantra—Gnomon.
6. Ghaṭikā yantra—a clock or pot instrument.
7. Kapāla yantra—Bowl or potsherd instrument.
8. Karttarī yantra—Scissor or knife ; cutter.
9. Pīṭha yantra=Pedastal or seat instrument.
10. Salila yantra—Water-leveller.
11. Brahma or Śāṇa yantra—For describing circles.
12. Avalamba Sūtra—Threads with plumbs (Plumb lines).
13. Karṇa or chāyā-karṇa—A set of squares for diagonals.
14. Chāyā or śaṅku-chāyā—Sundial. ──────────────────────────────────────────────────
15. सप्तदश कालयन्त्राण्यतो धनुस्तुर्यगोलकं चक्रम् । यष्टिः शंकुर्घटिका कपालकं कर्त्तरी पीठम् ॥ सलिलं भ्रमीऽवलम्बः कर्णश्छाया दिनार्थमर्कोच्चः । नतकालज्ञानार्थं तेषां संसाधनान्यष्टौ ॥ —BrSpSi. XXIII, 5-6

334 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS 15. Dinārdha yantra—Midday measure instrument. 16. Arka yantra—Sun-instrument. 17. Akṣa or Palāñśa yantra—Small degree measure arc instrument. Salila yantra is used for levelling; since a liquid such as water seeks its own level, it can be utilised to know whether a surface has been levelled or not.¹ Bhrama or Śāṇa is used for drawing circles. Avalambaka or plambline is used for adjusting vertical line. Karṇa is used in connection with angles and diagonals. From Salila (no. 10) to the last (no. 17); these eight are used for adjustments and are basically important. The dhanuryantra is used for nata and unnata kāla ghaṭikās. On the paridhi or the circumference of the cakra-yantra are indicated the twelve rāśis, ending up to Mīna (XXII. 18). Brahmagupta has described the yaṣṭi yantra and shown how it could be used to give time at different parts of the day, and from its shadow dṛgjyā and other characteristics can be calculat- ed. This instrument can also be used for ascertaining the solar- lunar differences, and for fixing up the directions. It can be used for determining various heights and altitudes. The karttarī yantra is of the shape of a pair of scissors with two semi-circular blades, fastended to a string at the centre; at the centre is fixed a pin or a pole which casts shadows. Setting up of the Gnomon Here it would be interesting to describe the setting of a gnomon, which K.S. Shukla has given in details while commenting on the Mahābhāskarīya (IV. 1) : After having tested the level of the ground by means of water, draw a neat circle with a pair of compasses (karkaṭa) (At the centre of that circle, set up a vertical gnomon). The gnomon should be large, cylindrical, massive, and tested for its perpendicularity by means of four threads with plumbs (avalmbaka) tied to them.

1. सलिलेन समं साध्यं भ्रमेण वृत्तमवलम्बकेनोर्ध्वम् । तिर्यक् कर्णेनान्यैः कथितैश्च नव प्रवक्ष्यामि ॥ —BrSpSi XXII. 7.

SETTING UP OF THE GNOMON 335 Bhāskara I in his commentary on the Āryabhaṭīya tells us that there was a difference of opinion amongst astronomers in his time regarding the shape and size of gnomon (also called style). Some astronomers prescribed a gnomon with its one-third in the bottom of the shape of a prism on a square base (caturasra), one- third in the middle of the shape of a cow's tail (go-pucchākāra). and one-third at the top of the shape of a spear-head (Śūlākāra) and some others prescribed a square prismoidal (samacaturasra). gnomon. The followers of Āryabhaṭa I, he informs us, prescrib- ed the use of a broad (pṛthu), massive (guru), and large (dīrgha cylindrical gnomon, made of excellent timber and free from any hole, a scar or knot on its body. In the above stanza, Bhāskara I prescribes this last kind of gnomon : the other two kinds he proves in the commentary to be defective and so he rejects them. For getting the shadow'end easily and correctly the cylindri- cal gnomon was surmounted by a fine cylindrical iron or wooden nail fixed vertically at the centre of the upper end. The nail was taken to be longer than the radius of the gnomon, so that its shadow was always seen on the ground. Certain writers, Bhāskara I tells us in the commentary, prescribed a gnomon of half a cubit (=12 aṅgulas) in length and having twelve divisions. But according to Bhāskara I (although it was the usual custom) there was no such hard and fast rule. The gnomon could be of any length and any number of divisions. The gnomon should, however, be large enough, so that the rings of graduation on the gnomon may be clearly seen on the shadow. A broad and massive gnomon was preferred because it was unaffected by the wind. Brahmagupta describes gnomon which at the bottom is two aṅgulas wide, pointed as a needle., 12 aṅgulas in length, and full of holes from the basic circular part to the pointed extremity. (BrSpSi. XXII. 39). As regards testing the level of the ground, Bhāskara I observes : When there is no wind, place a jar (full) of water upon a tripod on the ground which has been made plane by means of eye or thread, and bore a (fine) hole (at the bottom of the jar) so that the water may

336 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS have continuous flow. Where the water falling on the ground spreads in a circle, there the ground is in perfect level; where the water accumulates after departing from the circle of water, it is low; and where the water does not reach, there it is high. (Bhāskara's Commentary on the Āryabhaṭīya, II. 13). After the ground was levelled, a prominently distinct circle was drawn on the ground as stated in the text (MBh. III. 1). In the time of Śaṅkaranārāyaṇa (869 A.D.), there it seems that all lines were drawn on the ground with sandal paste (candana- kṣodārdra). The above circle having been thus drawn and coated with sandal paste, another small concentric circle was drawn with the radius of the gnomon. The gnomon was then placed vertically with the periphery of its base in coincidence with that circle. The gnomon was thus set up exactly in the middle of the bigger circle. The verticality of the gnomon was tested by means of four plumb lines hung on the four sides of the gnomon Gnomon Used for Finding the Directions The rule in this connection has been described by Brahmagupta in BrSpSi. III. 1. The same rule in other words has been described by Bhāskara I in MBh. III. 2. In the Vāsanā Bhāṣya, Pṛthūdaka Svāmī describes the details of determining the directions. The level of the ground is ascertained by means of water and a gnomon of 12 aṅgulas is set up. Find out two points where the shadow of the gnomon enters into and passes out of the circle. Bhāskara prescribes drawing out a fish figure with these points. The thread line which goes through the mouth and tail of the fish figure indicates the north and south directions with respect to the gnomon. Brahmagupta says that if the Sun is on the eastern side, then where the shadow-point enters circle (in the forenoon) that point would be the west, and the point where it emerges out (in the afternoon) is the east. As the Sun moves along the ecliptic, its declination changes. By the the time the shadow moves between the forenoon and afternoon points as given above, the Sun , traverses some distance of the ecliptic and, so, theoretically speaking, its

GNOMON USED FOR FINDING THE DIRECTIONS 357 declination gets changed. It follows. therefore, that the East- West line in the above determination is not the true position of the actual East-West line. Brahmagupta (628 A. D.) was the first Hindu astronomer who prescribed the determination of the East-West line with proper allowance for the change in the Sun's declination. (Shukla) The details of the method intended by him have been supplied by his commentator Pṛthūdaka Svāmī (860 A. D.). Bhāskara and Brahmagupta both give another method of determining directions : (BrSpSi. III. 2 ; MBh. III. 3) : With the three points (at the ends of the three shadows of the gnomon) corresponding to (any three) different times (in the day), draw two fish-figures (each with two of the three points) in accordance with the usual method. From the point of intersection of the lines passing through the mouth and tail (of the two fish-figures), determine the north and south directions. (MBh. III. 3). Brahmagupta in his rule is more precise : The point where the lines passing through the two fish-figures, which are drawn by means of three shadow ends (of the gnomon), intersect each other is for places in the northern hemisphere, the south direction, (if the midday shadow falls to the north of the foot of the gnomon). If the midday shadow falls towards the south of the foot of the gnomon, it is the north direction. (BrSpSi. III. 2). This rule is obviously based on the assumption that the the locus of the end of the shadow of the gnomon is a circle. In tact the locus for places whose latitude is less than (90°—the obliquity of the ecliptic), this locus is a hyperbola. Brahmagupta has made numerous uses of gnomon. He and Bhāskara, for example, both give the rules for finding the latitude and colatiude and the zenith distance and altitude of the Sun by finding out the length of the shadow and the length of the gnomon (BrSpSi. III. 10 ; MBh. III. 5) ; also rule for the determination of the latitude with the help of the Sun's meridian zenith distance and declination (BrSpSi. III. 13 ; MBh.

338 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS III. 17) ; also rule for finding the Sun's altitude (BrSpSi. III. 27 ; MBh. III. 24) (The Sun's altitude for the night has been called by Brahmagupta as pātāla-śaṅku, BrSpSi. XV. 9). Golayantra or Armillary Sphere The first mention of the Golayantra or the armillary sphere is in the Āryabhaṭīya (Golapāda. 22)¹ which was a uniformly round circle made of wood or of bamboo and which was of uniform weight or density alround. It was levelled with mercury, oil or water. A śalākā or pin (or rod) was fixed in it in the south-north direction. Its description from the com- mentary Bhaṭadīpikā of Paramādiśvara is given here : A sphere of wood. uniformly round on all sides and with uniform density, and also light is made to revolve round an iron axis fixed north-south without friction (oil may be introduced to avoid friction). To the backside of the sphere, is fixed a nālaka full of water which has the length equal to the circumference of the sphere: and which has a hole at the bottom. Now a thread, connected to the hook of the wooden ball (on the top side) passing over another small ball (in the same axis of the wooden ball) is attached to the mercury lobe by its other end. The mercury lobe is placed on the level of water and water is allo- wed to flow through the bottom hole and with water mercury lobe also goes down. The time in which the above hook of ball comes to bottom (180°) is noted. The experiment is repeated with oil. The use of this mechanism is to revolve the ball by water or oil¹

1. काष्ठमयं समवृत्तं समन्ततस्सम गुरु लघु गोलम् । पारतं तैलजलैस्तं भ्रमयेत्स्वधिया च कालसमम् ॥ Arya. IV. 22 काष्ठमयं वंशादि काष्ठेन निर्मितं समवृत्तं सर्वतोवृत्तं समन्ततस्सम गुरुं सर्वावयवेषु समं गुरुत्वं यथा भवति तथा कृतं । लघुमगुरुं एवं भूतं गोलं कृत्वा पारतादिभिस्तं स्वधिया च कालसमं भ्रमयेत् । अयमर्थः । भूमिष्ठ दक्षिणोत्तरस्तम्भयोरुपरि गोलप्रोतायश्शलाकाया अग्रे स्थापयेत् । गोलदक्षिणोत्तरा- च्छिद्रे च तैलेन सिञ्चेत् यथा निस्सङ्गो गोलो भ्रमति । गोलस्यापरतो गोलपरिधिसंमित दैर्ध्यं साधश्छिद्रं जलपूर्णं नलकं निदध्यात् ततो गोलस्यापरस्वस्तिक कीलकं विधाय तस्मिन्सूत्रस्यैक मग्रं बद्ध्वादौ विषुवन्मण्डलपृष्ठेन प्राङ्मुखं नीत्वा तदग्रबद्धं पारतपूर्णमलावु जलपूर्णे नलके निदध्यात् ततो नलकस्याध [Cont. on Page 337]