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

ARABS ON MOTIONS OF FIXED STARS 45 Here we finish our review of ancient astronomy. We have omitted as not coming within our province several valuable contributions to science which did not deal with cosmology or planetary theory. But even with this limitation enough has been said to show that when Europeans again began to occupy themselves with science they found astronomy practically in the same state in which Ptolemy had left it in the second century. But the Arabs had put a powerful tool into their hands by alter- ing the calculus of chords of Ptolemy into the calculus of sines or trigonometry, and hereby they influenced the advancement of astronomy in a most important manner. References

1. Peter Doig : A Concise History of Astronomy London
3. J.L.E. Dreyer : A History of Astronomy from Thales to Kepler Dover publications, 1953 (chapter XI reproduced).
4. Satya Prakash : Founders of Sciences in Ancient India, Delhi, 1965.

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CHAPTER II Personal References of Brahmagupta Sudhākara Dvivedī in his Gaṇaka Taraṅgiṇī, a small book on biographical skethes of astronomers and astrologers of this country gives a brief account of Brahmagupta thus : Brahmagupta was born in 520 Śaka (655 Vikramī or 589 A.D.) in the reign of King Vyāghramukha, belonging to the Cāpa family ; his father was Jiṣṇugupta ; and at the age of 30, he wrote in 550 Śaka (628 A. D.) his well known treatise on Astronomy known as the Brāhmasphuṭasiddhānta, which is corroborated by the statement in the Viṣṇudharmottara Purāṇa, (Chapter on the Brahma-siddhānta), His other treatise, entitled the Khaṇḍa- Khādyaka, which is a karaṇa book, was completed in 587 Śaka (665 A. D.). According to some authorities, Brahmagupta was the grandson of Viṣṇugupta, and the family suffix (Gupta) indicated that he belonged to the Vaiśya family, and he was in the service of the King of Rewah, known as Vyāghrabhaṭa. Brahmagupta was a great critic ; he did not spare any of his predecessors like Āryabhaṭa, Varāhamihira, Śrīṣeṇa, Viṣṇucandra and others. Later on, his influence on the writing of the succeeding generations has been immense. Bhāskarācārya II in his Bījagaṇita has acknowledged him as a great authority on algebra and has given him as the first place amongst the galaxy consisting of Brahmagupta, Śrīdhara, Padmanābha etc. The Eighteenth Chap- ter of the Brāhmasphuṭasiddhānta, known as Kuṭṭaka Chapter (on Pulveriser) has been translated by H. T. Colebrooke in English in 1817. The English translation of the Twelfth Chapter on Gaṇita or Calculations from the Brāhmasphuṭasiddhānta is also available in English. (See Colebrook's Algebra with Arithmetic and Mensuration from the Sanskrit of Brahmagupta and Bhās- kara, London, 1817). The Vāsanā Commentary on the Brāhmasphuṭasiddhānta by Pṛthudakasvāmī (860 A. D.) is also available though with diffi- culty (as indicated by Sudhākara Dvivedi) ; its incorrect manus-

48 PERSONAL REFERENCES OF BRAHMAGUPTA cript is available in the Library of the King of Banaras (Kāśīrāja) which has the colophony at the end as : श्री चापवंशतिलके श्री व्याघ्रमुखे नृपे शकनृपात् , पञ्चाशत्संयुक्तैर्वर्षशतैः पञ्चभिरतीतैः । ब्राह्मस्फुटसिद्धान्तः सज्जनगणितज्ञगोलवित्प्रीत्यै, त्रिंशद्वर्षेण कृतो जिष्णुसुतब्रह्मगुप्तेन ॥ Bhāskara II has written the famous treatise Siddhāntaśiromaṇi (1150), which is almost based on the Brāhmasphuṭasiddhānta. It has been edited by the author's own gloss (Vāsanābhāṣya) by Bāpu Deva Śāstrī (Vārāṇasī); by Murlidhar Jha with the com- mentaries, Vāsanāvārttika of Nṛsiṁha (1621) and Marīci of Munīśvara (1635), vol. I (containing chapter 1 of the Gaṇitā- dhyāya) (Vārāṇasī, 1917); by Girija Prasad Dvivedi with original commentaries in Sanskrit and Hindi, vols. I and II (Lucknow, 1911, 1926); English translation of the text only by Bāpu Deva Śāstrī and Wilkinson (Calcutta, 1861). In the very first Chapter (verse 2), Brahmagupta writes : The old calculations dealing with planets (i.e. the old astronomy), based on the system of Brahmā have become erroneous in course of past ages and therefore, I, the son of Jiṣṇugupta would like to clarify them. Brahmagupta was not a mere theorist, he based his calcula- tions on direct observations with the help of instruments or devices (nalikādi yantra); he was in favour of making correc- tions on the basis of these observations. He was himself an expert observer. In his Khaṇḍakhādyaka also he has emphasised the need of direct observation. At many places, Brahmagupta has severely criticised the Romaka and Pauliśa systems of astronomy which were introduced in this country by Lāṭadeva and Śrīṣeṇa. There are many passages where this criticism would be available with vehe- mance. Brhamagupta was opposed to the system of Āryabhaṭa I. He never spares the school of Āryabhaṭa which was regarded as the most authoritative then. Sudhākara Dvivedi says that as Brahmagupta was opposed to the system of Āryabhaṭa, so the Vaṭeśvara Siddhānta was opposed to that of Brahmagupta. The Institute has already published the Vaṭeśvara Siddhānta and now it has the privilege of publishing the Brahmasphuṭasiddhānta.

A NOTE ON BHILLAMĀLA 49 A Note on Bhillamāla It is said that Brahmagupta completed his Brāhmasphuṭa- siddhānta in Śaka 550, and he has come to be known as Bhilla- mālakācārya or a teacher residing in "Bhillamālaka." In this connection, therefore, it would be interesting to reproduce a note on Bhillamāla from G. Bhūler's article on Gurjara Inscriptions, No. III, published in the Indian Antiquary, July 1888, vol. 17, p. 192 : With a single exception all the complete inscriptions call the princes enumerated above, scions of the Gurjara race; and Khe I. and II. highly extol the greatness and wide extent of this family. Na. alone names the Mahārāja Karṇa as their ancestor. With respect to this personage it is for the present impossible to say whether the famous hero of the Mahābhārata may be meant, or some real historical king. But the name Gurjara makes it evident that this dynasty belonged to the great tribe which is still found in Northern and Western India and after which two provinces, one in the Bombay Presidency and one in the Pañjāba, have been named. The Gurjaras or Gūjars are at present pretty numerous in the western Himālaya, in the Pañjāba and in Eastern Rājputānā. In Kachh and Gujarāt their number is much smal- ler. It would, therefore, seem that they came into Western India from the north. Their immigration must have taken place in early times, about the beginning of our era or shortly after- wards. In Western India they founded, besides the kingdom of Broach, another larger state which lay some hundred miles further north. Hiuen Tsiang mentions in his travels¹ the kingdom of Kiu-che-lo and its capital Pi-lo-mi-lo. It has been long known that the former word corresponds to Gurjara. But the name of the town has been incorrectly connected by the French scholars with Bālmer in the Jēsalmīr territory, and this ident ification has been accepted in Mr. Beal's new transla- tion of Siyuki. As I have stated already formerly² following Colonel. J. Watson, Pilomilo corresponds exactly to Bhillamāla

1. Beal, Siyuki, Vol. II, p. 269f. Hiuen Tsiang assigns to the nor- thern Gurjara State an extent about double of that given for the kingdom of Broach.
2. Ante, Vol. VI. P. 63.

50 PERSONAL REFERENCES OF BRAHMAGUPTA one of the old names of the modern Bhīnmāl or Śrīmāl¹ in southern Mārwāḍ close to the northern frontier of Gujarāt. Another work, which was composed a few years before Hiuen Tsiang's visit to Gujarāt, contains likewise a notice of this northern kingdom of the Gurjaras. The astronomer, Brahma- gupta, who completed his Siddhānta in Śaka-Samvat 550 or 628 A.D. calls himself Bhillamālakakācārya², "the teacher residing in Bhillamālaka and is called so by his commentator Pṛthūda- kasvāmin. He further states that he wrote under king Vyāghra- mukha who was 'an ornament of the Cāpa race.' This family, whose name recurs in the Haḍḍāla grant of Dharaṇīvarāha³ prince of Vadhvān, thus seems to have been the reigning house of Bhillamāla. It is most probably identical with the Cāuḍas, Cāvōṭakas⁴ or Chāpōtkaṭas, who from 756 to 941 A.D. held Aṇhilvāḍ and still possess various small districts in northern Gujarāt. The Gurjara kingdom of Broach was without a doubt an offshoot of the larger State in the north, and it may be that its rulers, too, belonged to the Cāpa family.

1. Bhillamāla means etymologically 'the field of the Bhil' and Śrī- māla 'the field of Śrī'. The latter name must also be ancient, as the Śrī- mālī Brāhmaṇ as are called after it. The Jainas narrate various, of course incredible, legends, which explain how Śrīmāla came to be called Bhillamāla. Merutuṅga says that king Bhoja invented the latter name, because the people of Śrīmāla let the poet Māgha die of starvation. According to another authority, the town had a different name in each Yuga. It is in India very common for ancient towns to have two or even more names. Thus Kanauj was called, Kānyakubja, Gādhipura, and Mahodaya.
2. See Professor A. Weber, Die Sanskrit und Prakrit Handschriften der Berliner Bibliothek Vol. II. pp. 297-298. In the first passage the MSS. offers incorrectly Bhilamācārya; in the second which occurs in the commen- tary on the Khaṇḍakhādyaka, we have Bhillamālavakācārya, a slightly cor- rupt reading. This latter varia lectio occurs also in other MSS., see Weber, Indische Streifen. Vol. III, p. 90, and has given rise to erroneous suppositions regarding Brahmagupta's home. The Gujarātī Joshīs still pre- serve the tradition that Brahmagupta was a native of Bhinmāla.
3. Ante, Vol. XII. p. 190ff. The remark which I have made there that the Cāpas are not named elsewhere, of course requires correction.
4. The form Cāvoṭaka, which occurs in Dr. Bhagavanlal's grant of the Gujarāt Cālukya king Pulakeśin of Samvat 490, is the immediate prede- cessor of the word Cāuḍa. Its Sanskrit original is certainly not cāpotkaṭa which probably has been coined in comparatively speaking modern times, in order to explain the difficult Prakrit word, just as the bards of Rajputana have invented Rāstrauḍha as etymon for Rāṭhoḍ.

BRAHMAGUPTA'S OWN REFERENCES 51 Brahmagupta's own References In the Twenty-fourth Chapter (Sañjñādhyāya), of the Brāhmasphuṭasiddhānta, Brahmagupta has made a reference to his own biography : In the reign of Vyāghramukha belonging to the family of Cāpa, in the year 550 Śaka the treatise Brāhmasphuṭasiddhānta was composed for the benefit of benevo- lent astronomers by Brahmagupta, son of Jiṣṇugupta at the age of 30. (BrSpSi. XXIV. 7. 8) Then again he says : The Brāhmasphuṭasiddhānta has been written by Brahmagupta, son of Jiṣṇu, in 1008 verses of Āryāch- anda. (ibid 10 ) In the beginning of this Sañjñādhyāya, he refers to the differences in fundamental notions created by the various existing systems of astronomy as the Sūrya-siddhānta, Pulisa-siddhānta. Romaka-siddhānta, Vasiṣṭha-siddhānta and other Yavana-siddhā- ntas, which have caused anomalies in the calculations of eclipses. He also refers to the anomalies due to the calculations based on midnight day-reckoning and sunrise day-reckoning. From the point of view of own references, the following would be of interest : Brahmagupta, son of Jiṣṇugupta (Jiṣṇusuta-Brahmagupta) : BrSpSi. I. 2; XVI. 35. 37; XXIV. 8. 10; XXV. 73. It is strange that in the Khaṇḍakhādyaka, Brahmagupta has not given his name nor his father's name anywhere. At least the reading of the Khaṇḍakhādyaka as given by Pṛthūdakasvāmī does not contain this name. In the edition of Bhaṭṭotpala, there are three more chapters in the Khaṇḍakhādyaka (Chapters IX, X and XI). In the Chapter XI (known as Pātādhikāra), we have 21 verses and in the last 21st verse we find the name of Brahmagupta,¹ son of Jiṣṇu mentioned : Those who are eager to have the knowledge of the motion of stars and planets, for them and for the benefit of disciples in this field, Brahmagupta son of Jiṣṇu has composed this Khaṇḍakhādyaka.

1. खण्डखाद्यकमिदं तृप्त्यर्थं ग्रहगतिश्रुतात्तांनाम् । शिष्याणां हितार्थं प्रोक्तं जिष्णुसुतब्रह्मगुप्तेन ॥ — KK. XI. 21

52 PERSONAL REFERENCES OF BRAHMAGUPTA Reference to Āryabhaṭa We have said that it appears that Brahmagupta was a bitter opponent of Āryabhaṭa in his younger days (628 A. D.), but later on (in 665 A. D.), he climbed down to describe and teach one of the Āryabhaṭa's system of astronomy. Āryabhaṭa was universally revered. and it was difficult for Brahmagupta to have ignored him and thus he has to refer to this great authority some times to oppose some of his views and some times to expound his views. The following are the pasages in the Brahmasphuṭasiddha- nta, where the author has referred to the name of Āryabhaṭa. There are many other passages where the name "Aryabhaṭa" does not occur but where Brahmagupta indirectly means to quote the views of this great master. BrSpSi. I. 9. 12. 28. 32. 60. 61 ; II. 33. 46; V. 21.25; VI. 13; IX.10; XI. 4. 9. 10. 12. 25. 29. 33. 41. 42. 43. 44, 46, 47. 49, 62; XIII, 27; XIV. 45; XVI. 37. 46; XXI. 39. The largest references are in Chapter XI, where Brahmagupta has made an attempt to show the discrepancies of the Āryabhaṭī- aya system of reckoning astronomical observations and constants. In the Khaṇḍakhādyaka also there is a reference to the name of Āryabhaṭa but on very few occasions. In Chapter I, we have this reference at three places- Having made obeisance to God Mahādeva, who is the great cause of this world's rise (i.e. creation), existence and destruc- tion, I shall declare the Khaṇḍakhādyaka (i.e. a short treatise on astronomy, which is as pleasant as food prepared with sugarcandy), which will yield the same results as the great astronomical treatise of Āryabhaṭa. As in most cases calculation by the great work of Āryabhaṭa, for (the know- ledge of time and longitude of planets etc. at ) marriage, nativity and the like, is impracticable for common use every day, this smaller treatise is made so as to yield the same results as that. ¹

1. प्रणिपत्य महादेवं जगदुत्पत्तिस्थितिप्रलयहेतुम् । वक्ष्यामि खण्डखाद्यकमाचार्य्यार्य्यभट तुल्यफलम् ॥ प्रायेणार्य्यभटेन व्यवहारः प्रतिदिनं यतोऽशक्यः । उद्वाहजातकादिषु तत्समफललघुतरोक्तिरतः ॥ —KK. I, 1-2

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;