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

REFERENCE TO ĀRYABHAṬA 53 These two verses show that Brahmagupta was not hostile to Ārya- bhaṭa when he wrote this Khaṇḍakhādyaka ; he merely intends presenting the subject matter on simple lines and furnishing the results obtained by Āryabhaṭa in a simpler way. In the following verse of the same chapter, he refers to Āryabhaṭa’s midnight day-reckoning system : The mean Saturn diminished by 3 seconds, the Śīghrocca of Mercury diminished by 22 seconds, the mean Mars increased by 2 seconds and the mean Jupiter increased by 4 seconds are equal to the respective mean planets of Ārya- bhaṭa's “midnight-system”.¹ In the Appendix of the Khaṇḍakhādyaka known as Khaṇḍa- khādyakottara, we again find a few verses where the name of Āryabhaṭa occurs; Three of these verses have been reproduced from the Brāhmasphuṭasiddhānta (BrSpSi I. 62, 6

54 PERSONAL REFERENCES OF BRAHMAGUPTA In the Appendix of the Khaṇḍakhādyaka, there is another verse which also speaks in the same strain against Āryabhaṭa (this verse does not occur in the Brāhmasphuṭasiddhānta) : As the process of finding the apparent places of pla- nets as given by Āryabhaṭa does not make them agree with observation, I shall, therefore, speak of this process. Of the Sun the apogee is at two signs and seventeen degrees (KK. IX. 4)¹ Brahmagupta opposed to Śrīṣeṇa— Viṣṇucandra, Lāṭa, Vijayanandī and others Brahmagupta was a great critic; he did not spare Āryabhaṭa, and along with him he was vehemently opposed to the doctrine of Śrīṣeṇa, Viṣṇucandra, Lāṭadeva, Vijayanandī and Pradyumna also. He was opposed to the Romaka and Pauliśa Siddhāntas, which were the systems of foreign astronomy, derived from Greece, Babylonia and other centres of learning. He did his best to resist the foreign influences on astronomy. The following are the verses in the Brāhmasphuṭasiddhānta, where Brahmagupta expressed his note of discord against the sys- tems or notions of Āryabhaṭa, Śrīṣeṇa and Viṣṇucandra : BrSpSi. I.60; II 46, 47; X 13, 62; XI.31, 46, 47, 48-50, 55, XVI. 36, 46; XXI.39; XXII. 2. In two of the verses, he refers to Lāṭasiṃha : BrSpSi. XI. 46, 48. In the following verse, he refers to Aṅkaciti, Vijayanandī, Prad- yumna and others: BrSpSi. XI. 58. In the Khaṇḍakhādyaka we do not find the names of these adversaries of Brahmagupta ; we, however, have a reference of Śrīṣeṇa, Āryabhaṭa and Viṣṇucandra in the verses already quoted, occurring in the Appendix of the Khaṇḍakhādyaka (Khaṇḍa- khādyakottara), Chapter IX. 2 and 3. As we have said before, these verses of the Appendix have been reproduced from the Brāhmasphuṭasiddhānta. (I.62 and II.47). Reference to Romaka and Pauliśa systems Brahmagupta speaks of his system as if expounded by Brahmā himself for the first time, later on deteriorated, and then revived by Brahmagupta himself. The very second verse of the Brāhma-

1. न स्फुटमार्यभटोक्तं स्पष्टीकरणं यतस्ततो वक्ष्ये । भानुमतो मन्दोच्चं राशिद्वयमंशकांश्च सप्तदश ॥ —KK. IX. 4.

REFERENCE TO ROMAKA AND PAULIŚA SYSTEMS 55 sphuṭasiddhānta (I. 2) substantiates this view : The science of astronomy (or the calculations of heavenly bodies) in course of long duration became ineffective or erro- neous; this was revived by Brahmagupta, son of Jiṣṇu.¹ (BrSpSi. I.2). The system of astronomy which goes by the name Brahma (Brahmasiddhānta) has been handed down to us in three forms : (i) one is as treated by the Śākalya Saṁhitā, (ii) one as described in prose in the Viṣṇudharmottara Purāṇa, and (iii) the one des- cribed by Varāhamihira in the Pañcasiddhāntikā, which re- cognises the yuga of the duration of five years. Which of these three was accepted by Brahmagupta is not clear. But from the measures of the number of revolutions performed by a planet in a given period (graha-bhagaṇa) etc., it is clear that Brahmagupta acknowledges the system as propounded in the Viṣṇudharmottara Purāṇa. In his chapter "Tantra-parīkṣādhyāya or Dūṣaṇā- dhyāya, he contradicts the notions of the Vedāṅga Jyotiṣa which accepts the yuga of five years. We may further emphasize the fact that Brahmagupta has not clearly detailed out the errors to which the Brahma-siddhānta succumbed in course of time, and how these errors were eradi- cated by him. During the days of Brahmagupta, Romaka and Pauliśa systems were getting currency in this country. Reference to these two are found in the Brāhmasphuṭasiddhānta at several places as follows : ROMAK : I.13; XI.50; XXIV.3 PAULIŚA : XIV.45; XXIV.3 In fact, BrSpSi. XXIV.3, we find the line "Sūryendu-Puliśa- Romaka-Vasiṣṭha-Yavanādyaiḥ", where we have a reference to all the then existing systems : Sūrya-siddhānta, Indu-siddhānta, Puliśa-siddhānta, Vasiṣṭha-siddhānta, and other Yavana-siddhā- ntas. Just as the Sun is one, so the astronomical system is also one ; this is a different thing that calculations in different systems may vary according to different sunrises in different places. Brahmagupta refers at one place to Varāhamihira in BrSpSi. XXI.39, where he has been spoken of in connection with a list of

१. ब्रह्मोक्तं ग्रहगणितं महता कालेन यत् खिलीभूतम् । अभिधीयते स्फुटं तज्जिष्णुसुतब्रह्मगुप्तेन ॥ — BrSpSi. I. 2.

56 PERSONAL REFERENCES OF BRAHMAGUPTA anti-authoritative versions of astronomical systems : Evam Varāhamihira-Śrīṣeṇācāryabhaṭa-Viṣṇucandrādyaiḥ Lokaviruddhamabhihitam Veda-smṛtisaṃhitābāhyam. At one place, he mentions the difference between the calculations based on the system of Āryabhaṭa and the Pañca-siddhāntas (Five systems) : Pauliśa, Romaka, Vasiṣṭha, Saura, and Paitāmaha. (BrSpSi. XIV.46). Brahmagupta was also familiar with the Jaina systems of astronomy; for example, at one place he uses the term "Jinoktam" (i.e., one propounded by the Jainas) : He repudiates the concept of two Suns and two Moons "Dvāvarkaindavau" (do canda do sujja) (BrSpSi. XI.3) as enunciated by the Jainas. At several places we find a reference to the Vasiṣṭha-sid- dhānta (BrSpSi. XI.49, 50; XXIV.3) Wherever, Brahmagupta has to press for his views in pre- ference to the views of others, he uses the words Brāhma or Brahmokta : BrSpSi. Brāhma : I.32; X.62; XI, 61; XVI.37 Brahmokta : II.31, 33; X.63, 69; XV.59; XVI.33 Reference Sudhākara Dvivedi : Gaṇaka Taraṅgiṇī G. Bühler : Gurjara Inscriptions, Indian Anti- quary, 1888. P. C. Sengupta : Khaṇḍakhādyaka, Calcutta, 1934. — . o : —

CHAPTER III Manuscripts of the Brāhmasphuṭa Siddhānta Sudhākara Dvivedī has given an account of some of the manuscripts of the Brāhmasphuṭasiddhānta in his Bhūmikā or Introduction appended to the edition published in the PAṆḌITA, Vol.XXIV, 1902 (New Series) : (i) available in the Library of the Government College, Kashi (Vārāṇasī) i. e. Kāśīka-Rājakīya-Pāṭha- laya,(ii) Dr.Thibaut's Manuscript,(iii) the Manuscript in possession with Yajñadatta Śarmā, the Chief Astronomer attached to the Prince of Ayodhyā. It is further mentioned that Dr. Thibaut's Manuscript was a copy of a Manuscript available in the Deccan College, Poona. The Manuscripts (ii) and (iii) were identical. The Manuscripts (iii) was very faulty and incorrect. The Manuscript from which Colebrooke translated out in English the Twelfth and the Eighteenth Chapters on Gaṇita or Mathematics and Kuṭṭaka (the Chapter on Pulveriser) respectively, appeared to be different from the three Manuscripts described above. The readings differed considerably. The Kuṭṭaka- Chapter of that book is, writes Sudhākara Dvivedī, still available in the India Office Library. (See Catalogue of the Sanskrit¹ Manuscripts in the Library of India Office, Part V.p. 995).¹

1. एतत्कृतस्यास्य सिद्धान्तग्रन्थस्यैका प्रतिः काशिकराजकीय पाठालयतो द्वितीया डा० थिबौ साहिब महाशयतस्तृतीया चायोध्यानरेशप्रधानज्योतिर्विच्छ्री यज्ञदत्तशर्मणो मया लब्धा । डा० थिबौमहा- शयस्य पुस्तकं कस्यचिद्दक्षिणदेशीय (Deccan College, Poona ) डेकन पुस्तकस्य प्रत्यन्तरम् । इदं पुस्तकं तथा पं० श्री यज्ञदत्त पुस्तकं चैकमातृकमेव । इदं पुस्तकत्रयमतीवाशुद्धं बहु च स्खलितं चास्ति । यत्पुस्तकानुसारेण व्यक्ताव्यक्तयोर्द्वादशाष्टादशसंख्ययोरंगलभाषायामनुवादः कोलब्रूकसा- हिबेन कृतस्तत्पुस्तकमेतत्त्रयतो भिन्नमित्यसंशयं विभाति पाठविभेदात् । तत्पुस्तकस्य कुट्टकाध्यायः संप्रति इण्डिया-आफिस-सरस्वती भवने वर्तते (See Catalogue of the Sanskrit Manuscripts in the Library of India Office, Part V, Page 995) पृथूदककृताऽस्य सिद्धान्तस्य टीका या कोलब्रूकसाहिबेन भारतवर्षे ह्युपलब्धा सा च संप्रति लण्डननगरे इण्डिया-आफिस-सरस्वती भवने वर्तते । तस्या एका प्रतिर्मद्द्वारेण श्री डा० थिबौ साहिबमहाशयैरुत्पादिता सापि संप्रति मन्निकटेऽस्ति । अहो इण्डिया-आफिस-सरस्वती (Continued on next page)

58 MANUSCRIPTS OF THE BRĀHMASPHUṬASIDDHĀNTA Colebroke procured a Manuscript of the Pṛthūdaka Svāmī's Annotation on the Brāhmasphuṭasiddhānta from somewhere in India. This is also available in the India office Library, London. Undoubtedly it appears that this Manuscript is a copy from one written in the Maithila script. Dvivedī describes a few character- istics of this Manuscript. For example, on the Folio 11. 7 (७) is written instead of 9 (९). At a few other places also, the same has happened. Following a visarga, (kva क्व) is inscribed instead of 'ka' (क), for example refer to Folio 12, line 6. At some places instead of visarga (:) we find ṣa (ष) inscribed, for example on Folio 12, line 1. Sometimes we find a sandhi at the virāma or end of a sententence ; e. g. in the Golādhyāya, Folio 21, line 2: sarva- mupapannamuktamakhaṇḍena (सर्वमुपपन्नमुक्तमखण्डेन). At some places we find nū (नू) written in place of nta (न्त). In one of the Folios, Śrī Gaṇeśāya namaḥ. श्रीगणेशायनमः is written in Maithila script.¹ Sudhākara Dvivedī prepared a copy of this Manuscript for Dr. Thibaut, and this copy was available with Sudhākara Dvivedī when he published his commentary with Text in the Paṇḍita. During the course of binding, on account of carelessness, many of the Folios got misarranged, and many of them got fragmented. Sudhākara Dvivedī emphasises in his Introduction the need of careful research on the arrangement of these Folios and their readings. (See Catalogue of the Sanskrit Manuscripts in the Lib- rary of India Office, Part V,p. 993-995). Sudhākara Dvivedī with considerable efforts could rearrange the Text : Golādhyaya-Bhāṣyam 1 to 45, mutilated at the beginning Madhyamādhikāra-Bhāṣyam 45 to 59 (Continued from previous page) भवने अस्य पुस्तकस्य पुटकबन्धनकालेऽनवधानतया पत्राण्यसंगतानि जातानि, बहुत्र खण्डितानि च सन्ति । तानि कदाचिदनुपयुक्तपत्राणां मध्ये स्युरिति तेषां सम्यगन्वेषणं समुचितम् । (See Catalogue of the Sanskrit Manuscripts in the Library of India Office Part, V, Page 993-995)

1. इदं पुस्तकं कस्यचिन्मिथिलाक्षरैर्लिखितस्य पुस्तकस्य प्रत्यन्तरमिति निःसंशयं प्रतिभाति । ११ पत्रे, ९ इत्यस्य स्थाने ७ इति लेखात् । एवमन्यत्रापि । विसर्गात् परतः ‘क’ स्थाने ‘क्व’ इति लेखात् । यथा १२ पत्रे, ६ पंक्तौ । तथा विसर्गस्थाने ‘ष’ इति लेखात् , यथा १२ पत्रे, १ पंक्तौ । वाक्य- विरामेपि सन्धिकरणात् । यथा गोलाध्यायस्य २१ पत्रे, २ पंक्तौ सर्वमुपपन्नमुक्तमखण्डेन । कुत्रचित् ‘न्त’ स्थाने ‘नू’ इति लेखात् । एकस्मिन् पत्रे मैथिलाक्षरैः श्री गणेशायनमः इति लेखाच्च ।

MANUSCRIPTS OF THE BRĀHMASPHUṬASIDDHĀNTA · 59 In this, the commentary is up to verse 31. In the Spaṣṭādhikāra, the commentary begins from verse 29. Folio 60. At the top of the Folio, we have the old numbering 1 and 115. After this, 68 folios are mis-arranged, and there the old numbering is marked 9. After that, up to 87 folio, we have in the bound volume a commentary up to the verse 6 of Tripraśnādhikāra. Here the old numbering is marked 28. Then we have the commentary up to verse 27 of the Tripraśnādhikāra. Here the numbering of folios is marked 1 and 159 (old numbering) and 218 new. After this, then we have in the bound volume, commentary up to verse 33 of the Tripraśnādhikāra. On the last folio, the old numbering is marked 5 and 163, the new numbering 222. After this begins the commentary on the Candragrahaṇādhikāra from verse 4, old numbering 1 and 297 and new numbering 257. Then follows the commentary on Sūryagrahaṇa up to verse 23. The last old folio numbering is 36 and 232, the new numbering 292. This is the last folio-numbering of the volume in the India Office Library. After this, begins the commentary on Grahayutyadhikāra, verse 11, old numbering 1 and 164 and the new numbering 223. Then follows the Madhyagatyuttarādhyāya up to verse 40, old folio numbering 119, and new numbering 178. Then we have the commentary of the Madhyagatyuttarādhyāya beginning from the verse 45; the old folio numbering 120 and the new numbering 179. Then we have the commentary on Tripraśnottarādhyāya up to verse 56. Here the last old folio number is 48 and 158, and the new numbering 217. In this way, we have the commentary on : the mutilated Golādhyāya, mutilated Madhyamādhikāra, mutilated Spaṣṭā- dhikāra, mutilated Tripraśnādhikāra, mutilated Candra-graha- ṇādhikāra, mutilated Sūrya-grahaṇādhikāra, mutilated Grahayū- tyadhikāra, Bhagrahayutyadhikāra, Tantra-parīkṣādhyāya (also known as Dūṣaṇādhikāra), Gaṇitādhyāya (Arithmetic), mutilated Madhyagatyuttarādhyāya, Sphuṭagatyuttarādhyāya, mutilated Tripraśnottarādhyāya.¹

1. मया महताऽऽयासेन तत्प्रति-पाठक्रम एवं नियोजितः । गोलाध्यायभाष्यम् १-४५ आदौ खण्डितम् । मध्यमाधिकारभाष्यम् ४५-५१ । (Continued on next page)

60 MANUSCRIPTS OF THE BRAHMASPHUṬASIDDHĀNTA Sudhākara Dvivedī says that nowhere in the Brahma- Sphuṭasiddhānta, Madhyamādhikāra, is found the verse: Saṁsādhya spaṣṭataraṁ bījaṁ nalikādiyantreṇa. Tat-saṁsṛtagrahebhyaḥ kartavyau nirṇayādeśau. (This verse has been quoted by Dvivedī in the Gaṇaka- Taraṅgiṇī, page 19).¹ This verse has been quoted by Dvivedī from the Trans- lation of Grahālāghava by Mallāri. · Dvivedī further says that in the Manuscripts available, there is mentioned a Twenty-fifth Chapter under the title "Dhyā- nagrahopadeśādhyāya". Dvivedī thinks that this Chapter does not constitute the Brahmasphuṭasiddhānta proper, which ends in fact with twenty-four chapters.² In his commentary and Edition in the Paṇḍita, he has published it as a separate treatise of Brahma- gupta. Thus he has named his Edition as: अत्र मध्यमाधिकारे ३१ श्लोकपर्यन्तमेव भाष्यम् । अतः स्पष्टाधिकारस्य २६ श्लोकतष्टीकारब्धा ५० पत्रतोऽत्र प्राचीन संख्या पत्रोपरि १, तथा ११५ । अग्रे ऽच ६८ पत्रमसंगतम् , यत्र प्राचीन संख्या ६ । ततः ८७ पत्रपर्यन्त संलग्नग्रन्थस्त्रिप्रश्नाधिकारस्य ६ श्लोकपर्यन्तं टीका । अत्र प्राचीनपत्रसंख्या २८। ततस्त्रिप्रश्नाधिकारस्य २७ श्लोकतष्टीका, पत्रसंख्या प्राचीना १ तथा १५६, नवाना २१८ । अग्रे संलग्न ग्रंथस्य त्रिप्रश्नाधिकारस्य ३३ श्लोकपर्यन्तं टीकान्तिमपत्रप्राचीनसंख्या ५ तथा १६३, नवीना संख्या च २२२ । ततश्चन्द्रग्रहणाधिकारस्य ४ श्लोक टीकारब्धा प्राचीनपत्रसंख्या १ तथा २६७ । नवीना संख्या च २५७ । ततः सूर्यग्रहणास्य २३ श्लोकपर्यन्तं टीका । अन्तिम प्राचीन पत्र- संख्या ३६ तथा २३२, नवीना संख्या च २६२ । इण्डिया-आफिस पुस्तकपुटके चेयमन्तिमपत्रसंख्या । ततो ग्रहद्युत्यधिकारस्य ११ श्लोकतष्टीकारब्धा यत्र प्राचीनपत्रसंख्या १ तथा १६४, नवीना संख्या २२३ । ततो मध्यमगत्युत्तराध्यायस्य । ४० श्लोकपर्यन्तं संलग्नग्रन्थौ यत्रान्तिमप्राचीनपत्र संख्या ११६ नवीना संख्या च १७८ । ततो मध्यगत्युत्तराध्यायस्य ४५ श्लोकतष्टीकारब्धा । अत्र प्राचीनपत्रसंख्या १२० नवीनासंख्या च १७६ । ततस्त्रिप्रश्नोत्तराध्यायस्य ५६ श्लोकपर्यन्तं टीका । अत्रान्तिमप्राचीनपत्रसंख्या ४६ तथा १५८, नवीना संख्या च २१७ । खण्डितगोलाध्यायस्य । खण्डितमध्यमाधिकारस्य । खण्डितस्पष्टाधिकारस्य । खण्डित- त्रिप्रश्नाधिकारस्य । खण्डितचन्द्रग्रहणाधिकारस्य । खण्डितसूर्यग्रहणाधिकारस्य । खण्डितग्रहद्युत्यधि- कारस्य । भगृहद्युत्यधिकारस्य । तन्त्रपरीक्षाध्यायस्य (दूषणाधिकारस्य) । गणिताध्यायस्य पाटीग- णितस्य) । खण्डितमध्यगत्युत्तराध्यायस्य । स्फुटगत्युत्तराध्यायस्य । खण्डितत्रिप्रश्नोत्तराध्यायस्य च टीका वर्त्तते ।

1. सर्वेष्वपि पुस्तकेषु 'संसाध्य स्पष्टतरं बीजं नलिकादियंत्रेण' इत्यादिश्लोको मध्यमा- धिकारे नास्ति । मया गणकतरंगिण्यां मल्लारिकृतग्रहलाघवटीकातो व्यलेखि (द्रष्टव्या गणकतरंगिण्यो पृ० -१८।१९) ।

MANUSCRIPTS OF THE BRAHMASPHUṬASIDDHĀNTA 61 Brāhmasphuṭasiddhānto Dhyānagrahopadeśādhyāyaśca or Brāhmaphuṭasiddhānta and Dhyānagrahopadeśādhyāya by Brahma- gupta (1902)¹ The manuscript of this small treatise was also mutilated, and Dvivedī took special pains in editing it, and he revised the calculations also incorporated in this treatise. The small treatise, Dhyānagrahopadeśādhyāya must have been composed prior to the Brāhmasphuṭasiddhānta, since we find a verse in the last Chapter (the 24th Chapter also known as the Sañjñādhyāya of the Brāhmasphuṭasiddhānta), verse 9, a reference to this book : How could this result be obtained in a simple way has been shown by me in the Dhyānagrahopadeśādhyāya of 72 Āryā verses, and therefore, it is not repeated here. BrSpSi. XXIV. 9) In the Dhyanagrahopadeśādhyāya, we have a verse 61, which is also found in the Khaṇḍakhādyaka (KK. I. 21) : Navatithayaḥ (159) divided by aṣṭi (16), pañcarasāḥ (65) divided by vasu (8), 10 divided by 3, each multiplied by the equinoctial shadow are the (tabular differences of) ascensional difference expressed in vināḍīs. (KK. I. 21; DhGr. 61) This verse then indicates that the Dhyānagrahopadeśādhyāya has been composed after the Khaṇḍakhādyaka. It may also be possible that the Brāhmasphuṭasiddhānta and the Dhyānagraho- padeśādhyāya were simultaneously written, and the above verse (DhGr. 61) was repeated again in the Khaṇḍakhādyaka. Sudhākara Dvivedī has taken the help from the commentary of Pṛthūdaka Svāmī in the Chapters on Pāṭīgaṇita (Arithmetic), and has quoted the examples from this commentary. At many places he has corrected the readings which were mutilated in manuscripts.

1. उपलब्धमूलपुस्तकत्रये पंचविंशतितमेऽध्याये वस्तुतो ब्रह्मगुप्तकृतो ध्यानग्रहोपदेशाध्यायो वर्त्ततेऽतो मयायं पृथक्त्वेन तन्नाम्ना मुद्रितः । अत्र बहुत्र स्खलितानि पदानि तानि गणितेन संशोध्य मुद्रितान्यपि सुधीभिर्भृशं विचिन्त्यानि । अस्य सिद्धान्तस्य चतुर्विंशतितमेऽन्तिमे संज्ञाध्याये (पृ० ४०८) गणितेन फले सिद्धिर्ब्राह्मे ध्यानग्रहे यतोऽध्याये । ध्यानग्रहो द्विसप्ततिरायां न लिखितोऽत्र मया ॥ इति नवमश्लोकेन ध्यानग्रहोपदेशाध्यायस्य रचनेतरसिद्धान्तरचनातः पूर्वं विभाति परन्तु तत्राहर्गणसाधनप्रकारेण सिद्धान्तरचनाकाल एवास्य रचना सिध्यति तथा 'नवतथ्योष्टिविभक्ता' इत्यादि- पद्येन खण्डखाद्यरचनातः पश्चात् सिध्यति ।

62 MANUSCRIPTS OF THE BRĀHMASPHUṬASIDDHĀNTA Reference Sudhākara Dvivedī : His Bhūmikā on BrSpSi., Pandita, Vol. XXIV, 1902 (New Series) ─: 0 :─

Chapter IV

Subject Matter Classified in the

Brāhmasphuṭasiddhānta

& Khaṇḍakhādyaka

It needs no emphasis that Āryabhaṭa commanded a great influence as an astronomer not only prior to Brahmagupta but during his days also, and we have seen how Brahmagupta quoted this great authority in his writings, sometimes borrowing from him and sometimes contradicting him or improving upon his calculations. Āryabhaṭa's great work is known as the Aryabh- atīyam written in 499 A. D. Āryabhaṭa was born in 476 A. D. ( Kaliyuga Saṁvat 3577 ). The Aryabhaṭīyam is also known as the Aryasiddhānta. For details about Āryabhaṭa the reader is referred to the Chapter entitled "Aryabhaṭa lays Foundations of Algebra" (Founders of Sciences in Ancient India, 1965, Chapter X). This great work was written in Kusumapura, (modern Patna, Bihar).

Divisions of Āryabhaṭīyam

The Aryabhaṭīyam is divided into four chapters called Pāda: (i) the Gītikā Pāda, with ten verses; (ii) the Gaṇita Pāda with 33 verses; (iii) the Kālakriyā Pāda with 25 verses and (vi) the Gola Pāda with 50 verses. In the Gaṇita Pāda, a chapter on mathematics, we have such subjects : squares (varga), cubes (ghana) (verse 3), square-root (vargamūla) (4), cube-root (ghanamūla) (5) ; area of a triangle, volume of a prism (6), area of a circle, volume of a sphere (7), area of a quadrilateral (viṣamacaturasra) (8); circumfe- rence of a circle (10), Rsine (radius x sine) (Jīvā) (11); determina- tion of the Rsine of the zenith distance, the base (bāhu) of a right- angled triangle, and the upright (koṭi) of a right-angled triangle (16); hypotenuse (karṇa) of a right-angled triangle and ardhajyā (17), Reversed-sine (śara) (18), areas of series-figures (śreḍhīphala) (19), rule of three (trairāśika) (23), reduction of fractions (savar- ṇīkaraṇa of bhinna) (27) ; inverse-rule of three (vyasta) (28) ;

• line by line check: 64 BRAHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA evaluation of unknown values (mūlya-pradarśana of avyakta mūlya (30); and the theory of pulveriser (Kuṭṭaka) (32, 33). In this chapter, Āryabhaṭa gives the solutions of quadratic equa- tions, and thus he earns for himself the credit of founding the science of algebra. (iii) The Kālakriyā Pāda with 25 verses, which enumerates the units of time [1 year (varṣa)=12 months (māsa); 1 month= 30 days (divasa); 1 day =60 nāḍīs; 1 nāḍī=60 vināḍī; vināḍī or vināḍikā is the same as vighaṭikā equivalent to our 24 seconds; nāḍī or nāḍikā or ghaṭī is equal to 24 minutes]¹;

DIVISIONS OF ĀRYABHAṬĪYAM 65 day should begin from the month Caitra, Śukla Pratipadā (the first day of the brighter half of the month Caitra) (11); man- docca (apex of slowest motion) and śīghrocca (apex of fastest motion) (17—24). (iv) The Gola Pāda with 50 verses. In Verse 1, there is a reference to a point in the Sun's path, the commencement of meṣa (meṣādi); this must have been vasanta-equinox. The ascen- ding nodes (pāta) of planets and the shadow of the Earth move on the path of the Sun (arka-apamaṇḍala) (2-3); the angular difference in relation to the Sun at the appearance of Moon (12 degree or aṁśa), of the Venus (9 degrees or 9 vināḍika); of the Jupiter (2 more than the Venus, i.e. 11 vināḍikas); the Mercury or Budha (13 vināḍikas), of the Saturn or Śani (15 vināḍikas); and of Mars or kuja (17 vināḍikas). The half of the Earth, Moon, planets and stars is dark, since those parts happen to be under their own shadow; the other half is bright as it faces the Sun (this is not true with respect to stars—author) (5). The Earth is surrounded with an atmos- phere of air and water (6, 7). In the Brahma-Divasa (Brahma's day), the sphere of the Earth is increased by one yojana and decreased by this amount during the Brahmā's night (8). Just as a person sitting on a moving boat sees the stationary trees etc. on the bank of a river moving in the opposite direction, similarly the stationary stars are seen moving from Laṅkā (or equator) moving towards the West(9). On account of pravahavāyu (air) the nakṣatra system and planets rise and set receding towards the West (10). The dimensions of Sumeru Parvata (North pole) is given to be one yojana, and it shines like a jewel (11) and in the next verse is given the position of Sumeru and the Bāḍa- vamukha (South pole) (12). The four cities situated at a diffe- rence of 90° each on the equator are given (13). The distance of Ujjayinī from Laṅkā (thus giving the latitude of Ujjayinī) is given (14). On account of the thickness of the Earth-sphere, the Khagola (celestial sphere) is seen less than the hemisphere (15). The next verse describes how moving appears the Khagola on the North and South poles (16). Then is given the measure of day and night of devas (gods), pitar (fathers), asura (demons) and manuja (men) (17). Then are given a few technical definitions

66 BRAHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA of celestial mathematics (18—21) like dṛṣṭi-sthāna (intersection of horizontal and vertical axis—pūrvāparadiggatā rekhā and adha- ūrdhva-diggatā rekhā). Dṛṅ-maṇḍala, dṛgkṣepamaṇḍala (dṛgkṣepa is the zenith distance of that point of a planet's orbit which is at the shortest distance from the zenith). Then the Bhūbhagola instrument is described (22, 23). Then follow the formulæ for calculating lagna (the horizon ecliptic point in the East), kāla etc. comprising the Tripraśnādhikāra (24—33). In the next verses, we have lambaka or Rsine of colatitude (34); dṛk-karma (35) and ayana-dṛk-karma (36). Then follow the calculations of lunar and solar eclipses (37--47). Verse 48 describes the coordi- nates of the Sun which are determined by the conjunction of hori- zon with the Sun, of the Moon by the conjunction of the Sun and the Moon, of planets by the conjunction of the Moon and planets or stars. Verse 49 describes how this jewel treatise has been pro- cured out of an ocean of true and untrue knowledge with the help of a boat of intellect. This means that the author has taken special pains in discriminating true knowledge from falsehood with respect to the prevalent notions of astronomy. In the last verse he says that he has not given anything new; he has given that very knowledge which was imparted by the Svayambhu in the earliest times (50). The Pañchāṅgas prepared according to the rules and for- mulæ of the Āryabhaṭīyam are still regarded with reverence by the Vaiṣṇavas in the South. Brahmagupta was a great critic of Āryabhaṭa, but finally he wrote his treatise, the Khaṇḍakhād- yaka, on the basis of this very Āryabhaṭīya (this treatise is a kara- ṇagrantha i.e. one containing a principal element of the Indian Calendar). The four commentaries of the Āryabhaṭīya available in Sanskrit are of Bhāskara I, Sūryadeva Yajvā, Parameśvara and Nīlakaṇṭha. Two English translations by P.C. Senagupta (1927) and W.E. Clark (1930) are also available. Major Landmarks Indian astronomy which reached its zenith in the times of Āryabhaṭa and Brahmagupta shows its evolution in the following stages :

1. Rudiments of astronomy in the Vedas and Brāhmaṇa Books like the Taittirīya Saṁhitā and the Śatapatha Brāhmaṇa. This period is associated with the dis-

MAJOR LANDMARKS 67 covery of the Vedic era, some of the planets, the twenty- four nakṣatras, cycle of seasons, concept of leap year, the dimensions of a yuga, solar and lunar years and the like. II. Astronomy of the Vedāṅga, also known as the Vedāṅga Jyotiṣ. Lagadha is the most prominent figure of this period (1400 B.C. to 850 B.C.) ; he is the first compiler of a text on astronomy. In his work, we for the first time in history find a reference to the Jñeya Rāśi (the knowable or the unknown quantity) and the Jñāna- Rāśi (known quantity). He lays the foundations of astronomical calculations. In his treatise we find a mention of such subjects as : Solstices (northern and southern journey of the Sun), increase in days and nights in the ayanas or solstices, solstitial tithis omis- sion of tithis, parva rāśi, acceptable and non-acceptable parvas, addition of day, acceptable parvas, concept of yoga (a term applied to the joint space which would be travelled by the Sun and the Moon in a given period of time on the presumption that these two bodies have travelled in directions opposite to each other), method of finding out a nakṣatra on any parva day, distinction between parva-nakṣatra and tithi-nakṣatra, correlation of solar and lunar dates, measure of a nāḍikā (unit of time), nakṣatra of the Sun, yoga and its nakṣatra, parva- bhaśeṣa and equivalent kalās, solar year, lunar revolu- tions (risings of nakṣatras) deities of nakṣtras, lunar and sāvana day differences (āhika māsa), divisions of a sāvana day and length of day in two ayanas. This author probably belonged to Kashmir. III. The period of Siddhāntas : Varāhamihira in his well known treatise, the Pañcasiddhāntikā, refers to five Siddhāntas or systems of Astronomy : Paitāmaha, Vasiṣṭha, Romaka, Pauliśa and Sūrya (Saura). As re- gards its importance, he gives the first place to the Sūryasiddhānta, places next the Romaka and Pauliśa, and declares the remaining two to be definitely inferior to the former. We do not possess the full treatise of these Siddhāntas, except the Sūryasiddhānta. Here too,

68 BRĀHMSPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDAYAK we have difficulty. The Sūryasiddhānta, as summarised by Varāhamihira in his Pañcasiddhāntikā, in many essential features differs from the system prescribed by the Text of the Sūryasiddhānta now available. So we have two versions, the one with which Varāhmihira was familiar and the modern one. The present Sūryasiddhānta comprises of fourteen chapters called adhyāyas. We have an authentic commetary on it by Parameśvara. The first two chapters of the Sūryasiddhānta have no special name. The classification of chapters is as follows : Modern Sūrya Siddhānta Chapter Name of the Number Chapter of verses Subject I — 69 Mean longitude of planets. II — 68 True motion and true longitude of planet ; and elements of Pañ- cāṅga. III Tripraśnādhyāya 50 Directions, place and time. IV Candra-grahaṇā- 29 Lunar eclipse. dhyāya V Sūrya-grahaṇādh- 17 Solar eclipse. yāya VI Chedyakādhyāya 24 Projection of eclipse on a plane surface. VII Graha-samāgama- 24 Conjunction of one yuddhādhyāya planet with another. VIII Tārāviṣayodhyāya 21 Conjunction of a Planet with the junc- tion star of a nakṣatra. IX Udayāstamaya-vi- 18 Heliacal rising and ṣayodhyāya setting of planets. X Candrāstamayādi 16 Moon rise and eleva- viṣayāḥ tion of Moon-horn. XI Vyatīpāta-viṣayāḥ 23 Pāta (vyatīpāta) XII — 87 Cosmogony and geo- graphy

MODERN SŪRYA SIDDHĀNTA 69 Chapter Name of the Number Chapter of verses Subject XIII Vyatīpāta-viṣayāḥ 26 Armillary sphere and astronomical instru- ments XIV — 28 Modes of reckoning time Total 500 From the contents of the Modern Sūryasiddhānta, it appe- ars that its great author (or rather compiler, since this Siddhānta has been existing much before he compiled this treatise) intends giving a complete but brief account of the entire Indian astron- omy. He has been careful enough to avoid controversies, and he has omitted the methods occurring in other Siddhāntas and Mahā-tantras as alternative rules. He describes only those which are most important and of general nature. Part II of the book dealing with cosmogony and geography etc.. is very brief , a nd the entire subject has been summarised in three chap- ters (Chapters XII to XIV). The Chapter on Astronomical instruments (XIII) is sketchy and rather incomplete; the author has given the names only of the instruments, as if the purpose was to keep the details secret and guarded. The whole text of the Sūryasiddhānta is finished in 500 verses (compare this with the length of other Indian books on Astronomy: the Brāhmasph- uṭasiddhānta 1008 verses, the Siddhānta-śekhara 890 verses and Siddhānta-śiromaṇi proper 962 verses). The Modern Sūryasiddhānta is a composition or compila- tion of a period when the Āryabhaṭīyam and the Brāhmasphuṭa- siddhānta had already become popular. This Sūryasiddhānta is indebted to both these systems, though it has nowhere acknowledged them in the Text. We shall refer to this matter later at an appro- priate stage of our discussion. It may casually be mentioned here that Alberuni has ascribed the authorship of the Sūryasiddhānta to Lāṭadeva while Munīśvara (603 A. D.) has ascribed it to Āryabhaṭa I. Though there is little or no support to these views, it is not improbable, that the works of Āryabhaṭa I and Lāṭadeva which followed mid- night-day-reckoning were based on the Sūryasiddhānta prevalent at the time. P. C. Senagupta on the other hand is of the opinion that “ the old Sūryasiddhānta was made up-to-date by Varāhami- hira by replacing the old constants in it by new ones from Ārya-

70 BRAHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA bhaṭa I's midnight system. " In this connection, it must be remembered that Varāhamihira nowhere expresses his indebted- ness to Āryabhaṭa I. The Sūryasiddhānta undoubtedly is the most popular book on Astronomy in this country. It has been so for the last 1000 years as is seen from the list of commentators on its text (K. S. Sukla has given a list of 28 commentators includiug those who wrote commentaries in the South Indian languges like Telugu and Kannaḍa.) : Allanarya Suri Maheśvara Amaredya Mallikārjuna Sūri (1178 A. D.) Bhaṭṭotpala (966 A. D.) Nārāyaṇa Bhūdhara (1572 A. D.) Nrsinha Daivajña (b. 1586 A.) Bhūtiviṣṇu Nṛsiṃhadeva Caṇḍeśvara (1178) Parameśvara (1432 A. D.) Cola Rāghava Śarmā (1592 A. D.) Dādābhai (1719 A. D.) Rāmakṛṣṇa Ārādhya (1472 A. D.) Devīdāsa Raṅganātha (1603 A. D.) Kamala Kara Daneśvarah Sārvabhauma (1618 A. D.) Kāmabhaṭṭa Tamma Yajva (1599 A. D.) Kṛṣṇa Daivajña Yallaya (1472 A. D.) Madanapāla Viśvanātha (1628 A. D.) Mādhavācārya A large number of astronomical books in this country were written on the basis of the Sūryasiddhānta, as the Gaṇakānanda by Sūrya (1387-1447 A. D.), Gaṇitādarśa by Dharmapathin, Makaranda Sāriṇī by Makaranda (1478 A D.), Grahacakra by Kucanācārya (1299 A.D.), Viṣṇukaraṇa by Viṣṇu (1556 A.D.) besides many others with indefinite dates such as the Sūrya- siddhānta nayana-prakāraḥ, Sūryasiddhānta-gaṇita Sūryasid- dhānta-saṃgraha by Viśvanātha Suri, and Sūryasiddhānta Sāriṇī by Rāmadatta Daivajña. K. S. Sukla has given a list of this literature in his Introduction to the Sūryasiddhānta. (iv) Period of Bhāskara I : The author cr the compiler ot the Sūryasiddhānta is not known nor its date of composition. Bhāskara and Brahmagupta are the brilliant names of a contem- porary period. Bhāskara I lived in the seventh century of the Christian era and was a contemporary of Brahmagupta (628

DIVISION OF MAHĀBHĀSKARĪYA 71 A.D.). He wrote three works on Astronomy which were most likely composed in the following order : (i) the Mahā-Bhāskarīya, (ii) a commentary on the Āryabhāṭīya, and (iii) the Laghu- Bhāskarīya, His commentary on the Āryabhāṭīya was written in 629, i.e., only one year after the Brāhmasphuṭasiddhānta. His commentary on the Āryabhaṭīya was written in 629 A.D., i.e. one year after the completion of the Brāhmasphuṭasiddhānta. Undoubtedly Bhāskara was the follower of Āryabhaṭa I. Shukla says, that his works provide us with a detailed exposition of the astronomical methods taught by Āryabhaṭa I and throw light on the development of Astronomy in India during the sixth and early seventh centuries A. D. which was the most brilliant period in the history of Indian Astonomy-Shukla has brought a critical edition with English translation of the Mahābhāskarīya and the Laghu-Bhāskarīya and he proposes to bring out a volume on the life and works of this great astronomer, Division of Mahābhāskarīya The Mahābhāskarīya is divided into Chapters called the adhyāyas. The adhyāyas have not been named as some of the Chapters of the Sūryasiddhānta or the Brāhmaspuhtasiddhānta. The number of verses in the Mahābhāskarīya is as follows : Subject Number of verses Adhyāya I Mean Longitude of a Planet and Planetary Pulveriser 52 Adhyāya II The Longitude Correction 10 Adhyāya III Direction, Place and Time. Junction-stars of the Zodiacal Asterisms and conjunction of Planets with them. 75 Adhyāya IV The Longitude of a Planet 64 Adhyāya V Eclipses 78 Adhyāya VI Rising, Setting and Con- junction of Planets 62 Adhyāya VII Astronomical Constants 35 Adhyāya VIII Examples 27 These titles to the subject-matter given in the above table have been assigned by Dr. Kripa Shanker Shukla in his critical

72 BRAHMASPHUṬA SIDDHĀNTA & KHAṆḌKHĀDYAKA edition. The total number of verses in this work is 403. The Sūryasiddhānta has, as stated above, 500 verses in all. Laghu-bhāskarīya The Laghu-Bhāskarīya is also divided into eight chapters, each chapter is known as Adhyāya. The subject matter in the book has been dealt with as follows : Subject Number of verses Adhyāya I Mean longitudes of the Planets 37 Adhyāya II True Longitudes of the Planets 41 Adhyāya III Direction, Place and Time from Shadow 35 Adhyāya IV The Lunar Eclipse 32 Ādhyāya V The Solar Eclipse 15 Adhyāya VI Visibility, Phases and Rising and Setting of the Moon 25 Adhyāya VII Visibility and Con- junction of the Planets 10 Ādhyāya VIII Conjunction of a Pla- net and a Star 19 The total number of verses in this text is 214. The Laghu- Bhāskarīyam is thus an abridged edition of the Mahā-Bhāskarīyam. From the closing stanza of this work, it is clear that the author wrote this work for the benefit of young students with immature mind by condensing and simplifying the contents of his larger work, the Mahā-Bhāsarīyam (also known as Karma-nibandha). Thus we have a little of the parallelism: Brahmagupta, after finishing his bigger treatise, the Brahma-sphuṭasiddhānta, wrote a minor abridged work, the Khaṇḍa-khādyaka, as a karaṇa grantha. This latter work of Brahmagupta, however, incorporates some original ideas not included in the earlier work. Shukla has given an analytical table indicating the rules of the Mahā- Bhāskarīyam incorporated in the Laghu-Bhāskarīyam also in an abridged or modified form, and also a list of the rules which have been omitted in the Laghu-Bhāsakrīya. There are a few rules in the Laghu-Bhāskarīya also which have no counterpart in the