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

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

LAGHU-BHĀSKARĪYA 73

Mahā-Bhāskarīya. Shukla further says that the arrangement of the contents of the Laghu-Bhāskarīya is more systematic and logical than that of the Mahā-Bhāskarīya, and is, at the same time, in keeping with the general practice followed by the other Hindu astronomers. Numerous quotations of this work occur in the annotative works of Sūryadeva (b. 1191 A.D.), Yallaya (1480 A.D.), Nīlakaṇṭha (1500 A.D.), Raghunātha Rāja (1597 A.D.), Govinda Somayājī and Viṣṇu Śarmā and in the Prayoga- racanā, an anonymous commentary on the Mahā-Bhāskarīyam. We find the commentaries of this abridged work in Malayalam and Tamil also. All this speaks of the great popularity of this work. There are circumstantial evidences to show that Bhāskara I had associations with the countries of Aśmaka and Surāṣṭra. His commentary on the Āryabhaṭīya was probably written in the city of Valabhī in Surāṣṭra. Perhaps Bhāskara I was born and educated in Aśmaka and later on he migrated to Valabhī, where he wrote his commentary on the Āryabhaṭīya or that he was a native of Valabhī and got his education in the Aśmaka country. Perhaps there was a strong school of Astronomy in the Aśmaka country, which was founded by the followers of Āryabhaṭa, so much so that at places, Bhāskara I has also called Āryabhaṭa as Aśmaka, his Āryabhaṭīyam by the name Aśmaka- Tantra or the Āśmakīya and the followers of Āryabhaṭa as Āśmakīyāḥ. This Aśmaka country or Aśmaka Janapada is men- tioned in the Buddhist literature also. It was somewhere either in the north-west of India, or was situated between the rivers Narmadā and Godāvarī. Bhāskara I was evidently a resident of the latter Aśmaka (which was between the Narmadā and Godāvarī). Brahamagupta and Bhāskara I were contemporaries. Both of them developed their systems in the earlier part of the seventh century A.D. (3700 years of Kaliyuga). Brahāmasphuṭasiddhānta was written in 628 A.D. and the commentary on the Ārya- bhaṭīyam by Bhāskara I was composed in 629 A. D. Bhāskara closely followed Āryabhaṭa, but Brahmagupta had the guts to oppose the views expressed by this great master and he not only contradicted him at places, but also propounded many new ideas, methods of calculation and constants of greater accuracy.

74 BRĀHMASPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDYAKA The classification of the contents of Astronomy in adhikāras appears to be the original concept of Brahmagupta; this system was to some extent adopted in the modern Sūrya-Siddhānta in the case of a few chapters. The Vaṭeśvara-Siddhānta by Vaṭeś- varācārya (born 802 Śaka or 880 A.D. in Ānandapura city, Punjab, son of Mahādatta) also adopts the terminology : Madhya- madhikāra, Spaṣṭādhikāra and Tripraśnādhikāra. We owe this type of caption-nomenclature to Brahmagupta. Contents of the Brahmasphuṭasiddhānta Now we shall summarise the contents of the Chapters of this great treatise and also enumerate the number of verses in each chapter. The author has himself given the total number of verses in the Chapter in the ending verse of each chapter. Some- times, the verse specifying this number itself is not taken into account while giving the total number of verses in that chapter, and therefore, there occurs a minor discrepancy in the actual number and the number specified by the author himself. The following table records both these numbers separately : ${cccc} & & Number of & Actual num- Chapter & Title & verses indi- & ber of verses & & cated by the & & & author & $ PŪRVA-DAŚĀDHYĀYĪ (First Ten Chapters) $I$ \quad Madhyamādhikāraḥ (Madh- \qquad yamagati-radhyāyaḥ) \qquad\qquad\qquad\quad 63 \qquad\qquad\quad 63 $II$ \quad Spaṣṭādhikāraḥ (Sphuṭagati- \qquad radhyāyaḥ) \qquad\qquad\qquad\qquad\qquad 67 \qquad\qquad\quad 68 $III$ \quad Tripraśnādhikāraḥ (Tripraś- \qquad nādhyāyaḥ) \qquad\qquad\qquad\qquad\quad 66 \qquad\qquad\quad 66 $IV$ \quad Candragrahaṇādhikāraḥ \qquad (Candragrahaṇādhyāyaḥ) \qquad\qquad\quad 20 \qquad\qquad\quad 20 $V$ \quad Sūryagrahaṇādhikāraḥ \qquad (Ārka-grahaṇam or Ravigra- \qquad\quad 26 \qquad\qquad\quad 27 \qquad haṇādhyāyaḥ) $VI$ \quad Udayāstādhikāraḥ \qquad (Udayāstamayādhyāyaḥ) \qquad\qquad\quad 12 \qquad\qquad\quad 13

CONTENTS OF THE BRĀHMASPHUṬASIDDHĀNTA 75 VII Candraśṛṅgonnatyadhikāraḥ 18 18 (Candraśṛṅgonnati-adhyāyaḥ) VIII Candracchāyādhikāraḥ 9 9 (Candracchāyā-adhyāyaḥ) IX Grahayutyadhikāraḥ 26 26 (Grahamelanādhyāyaḥ) X Bhagrahayutyadhikāraḥ 70 70 (Bhagrahayutiḥ-adhyāyaḥ) ——— ——— Total (of Daśādhyāyī) 377* 380* XI Tantra-Parīkṣādhyāyaḥ 63 63 (Dūṣaṇādhyāyaḥ) XII Gaṇitādhyāyaḥ Miśraka-vyavahāraḥ 1—16 Śreḍhī-vyavahāraḥ 17—20 Kṣetra-vyavahāraḥ 21—39 Vṛttakṣetra-gaṇitam 40—43 Khāta-vyavahāraḥ 44—46 Citi-vyavahāraḥ 47— Kākacika-vyavahāre karaṇasūtre 48—49 Rāśi-vyavahāraḥ 50—51 Chāyā-vyavahāraḥ 52—66 66 66 XIII Praśnādhyāyaḥ 49 48 (Madhyagatyuttarādhyāyaḥ) XIV Sphuṭagatyuttarādhyāyaḥ 54 55 XV Tripraśnottarādhyāyaḥ 60 60 XVI Grahaṇottarādhyāyaḥ 46 47 XVII Śṛṅgonnatyuttarādhyāyaḥ 10 10 XVIII Kuṭṭādhyāyaḥ 103 102 (Kuṭṭakādhyāyaḥ) Kuṭṭākāraḥ 1—29 Dhanarṇa-śūnyānām saṁkalanam 30—42 Ekavarṇa.samīkaraṇa- bījam 42—50 Anekavarṇa-samīkaraṇa-

76 BRAHMASPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDYAKA bījam 51—59 Bhāvita-bījam 60—63 Varga-prakṛtiḥ 64—74 Udāharaṇāni 75—102 XIX Śaṅkucchāyādijñanādhyāyaḥ 20 20 XX Chandaścityuttarādhyāyaḥ 20 19 XXI Golādhyāyaḥ 70 70 Sāmānya-golaprakara- ṇam 1—16 Jyā-prakaraṇam 17—23 Sphuṭagati-vāsanā 24—30 Bimba-sādhanam 31—35 Grahaṇa-vāsanā 36—48 Golabandhādhikā- raḥ 49—70 XXII Yantrādhyāyaḥ 53 57 XXIII Mānādhyāyaḥ 12 12 XXIV Sañjñādhyāyaḥ 13 13 Total 1016 1022 In one of the verses, Brahmagupta states that he has composed the treatise containing 1008 verses. Sudhākara Dvivedī has given the total as 1021, whereas he says, this number according to Brahmagupta's own statement should be 1020. If one de- ducts the concluding 12 verses of the Sañjñānadhyāya, the num- ber should be 1008. Sudhākara Dvivedī, in his addition of the Brahma-sphuṭa- siddhānta (published in the Paṇḍita, 1901 and 1902) gives as a supplement a small treatise of Brahmagupta known as Dhyāna- grahaṇopadeśādhyāya or Dhyānagrahaṇādhikāraḥ which has 72 verses. It would be worthwhile to give here the details of the Khaṇḍakhādyaka also, a book of Brahmagupta about which we have spoken so much. The titles to the chapters have not been indicated in the Text ; most likely they have been assigned by the commentator, Pṛthūdaka Svāmī known as the Khaṇḍakhā- dyaka-vivaraṇam.

CONTENTS OF THE BRĀHMASPHUṬASIDDHĀNTA 77 Chapter Title Number of verses I Tithi-nakṣatrādhikārādhyāyaḥ 32 (On tithis, nakṣatras etc.) II Grahagatyadhyāyaḥ 19 (On the mean and true places of 'star' planets) III Tripraśnādhyāyaḥ 16 (On the three problems relat- ing to diurnal motion) IV Candragrahaṇādhyāyaḥ 7 (On lunar eclipses) V Suryagrahaṇādhyāyaḥ 6 (On solar eclipses) VI Udayāstādhikāraḥ 7 (On the rising and setting of planets) VII Candraśṛṅgonnatyadhyāyaḥ 4 (On the position of the Moon's cusps) VIII Samāgamādhyāyaḥ 6 (On conjunction of planets) UTTARA KHAṆḌAKHĀDYAKA—APPENDIX IX Corrections and new methods 14 X On conjunction of stars and planets 16 Total 127 Bhaṭṭotpala, in his commentary on the Khaṇḍakhādyaka has given several additional verses in the main or proper treatise and also in its Uttara portion or the Appendix. P.C. Sengupta's edition (Sanskrit Text 1941) has given at the end of this pub- lication the account of these additional verses. The English

78 BRAHMASPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDYAKA edition (1934) classifies the Uttara-Khaṇḍakhādyaka into two chapters (which the author calls as Chapters IX and X), the Sanskrit edition gives 3 verses in Chapter IX, 21 verses in Chapter X and 24½ verses in Chapter XI. Of these three chapters, the Chapter X has been given the title "Pātādhikāra" and Chapter XI the title "Parilekhādhyāya" by Bhaṭṭotpala. T A B L E Arrangement of contents in different treatises

ĀRYABHAṬA AND BRAHMAGUPTA CONTROVERSY 79

1. प्रणिपत्य महादेवं जगदुत्पत्तिस्थितिप्रलयहेतुम् । वक्ष्यामि खण्डखाद्यकमाचार्यार्य्यभटतुल्यफलम् ॥

60 BRAHMASPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDYAKA treatise (i.e. Khaṇḍakhādyaka, literally meaning food prepared from sugar-candy) is made so as to yield the same results as that.¹ 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 Āryabhaṭa's midnight system.² In the Brāhmasphuṭasiddhānta, Brahmagupta accepts the astronomical day to begin with the sunrise at Laṅkā, and the calculations of days, months, years, Yugas, and Kalpas all begin from Caitra Śukla Pratipadā (the first tithi of the month Caitra in the bright-half of the Moon) and the first day is regarded as Sunday.³ Varāhamihira in his epicyclic cast to the Sūryasiddhānta in his Pañcasiddhāntikā adopts the ardharātrika system or the system of reckoning days from midnight. The question why Brahmagupta who was so bitter an opponent of Āryabhaṭa I in his younger days (628 A.D.) claimbed down to describe and teach one of the systems of Āryabhaṭa's astronomy in his sixty- seventh year (665 A.D.) is difficult to explain. In fact so great was Āryabhaṭa's reputation and fame that in spite of Brahma- gupta's severe ricticisms of the former in Chapter XI of the Brāhmasphuṭasiddhānta, it perhaps was undiminished and it was Āryabhaṭa who continued to be universally followed. Some authorities have thus expressed the view that to meet the popular demand Brahmagupta in the Khaṇḍakhādyaka took upon himself the task of simplifying Āryabhaṭa's ardharātrika system and in this task he became eminently successful. But it has been supposed that in this task he could not be a mere simplifier or expounder. प्रायेणार्यभटेन व्यवहारः प्रतिदिनं यतोऽशक्यः । उद्वाहजातकादिषु तत्समफल लघूक्तरोक्ति रतः ॥ तिसृभिः शनिज्ञैशीघ्रं द्वाविंशत्या कुजोऽधिको द्वाभ्याम् । चतसृभिरधिको जीवोऽर्द्धरात्रिकार्य्यभट मध्य समाः ॥ —KK. I. 1,2,7 ☛ चैत्रसितादेरुदयाद्भानोर्दिनमासवर्षयुगकल्पाः । सृष्ट्यादौ लंकायां समं प्रवृत्तः दिनेऽर्कस्य ॥ —BrSpSi I.4

ARYABHAṬA & BRAHMAGUPTA CONTROVERSY 81 The minor work of Brahmagupta known as Khaṇḍakhādyaka has two distinct parts : Khaṇḍakhādyaka proper and the Uttara Khaṇḍakhādyaka. In the first part the astronomical constants are the same as those of Āryabhaṭa's ardharātrika system, but the methods of spherical astronomy, calculation of eclipses and other topics are almost the same as in the Brāhmasphuṭasidd- hānta. The corrections for parallax in calculating a solar eclipse is here an important illustration.¹ In the Uttarakhaṇḍakhādyaka, Brahmagupta gives correc- tions to the Khaṇḍakhādyaka proper. In it are to be found the neat and original methods of interpolation and correction to the longitudes of the aphelia, as also to the dimensions to the epicycles of apsis of the Sun and the Moon.² while a few addi- tional chapters supply what else is necessary to the first seven chapters of the first part, to make the whole a complete treatise on Indian scientific astronomy. It was perhaps through the in- fluence of this supplementary part of the Khaṇḍakhādyaka that Brahmagupta's great work, the Brāhmasphuṭasiddhānta came to be valued among a distinct school of Indian astronomers. For long in this country India, this Siddhānta of Brahmagupta has been forming the basis for the calculation of almanacs by astro- nomers of the orthodox school of Rājasthān, Bombay and others. We might at this stage take up the question : Was Āryabhaṭa the author of two distinct systems of astronomy ? Undoubtedly he was. Several authors have written on this subject. I may specially mention the name of Prabodha Chandra Sengupta (Journal of the Department of Letters, Calcutta University, vol. XVIII; Bulletin, Calcutta Mathemati- cal Society, vol. XXII. Nos. 2 and 3). The reasons advanced by him may be restated in slight details thus. In his Brāhma-

1. व्यासार्धेन विभक्ता दृग्गतिजीवा चतुर्गुणा लब्धम् । लम्बननाड्यः पञ्चदश गुणिताया त्रिज्यया भक्ता ॥ दृक्क्षेपज्या मुहुत्यन्तरा हता लब्धमवनतिर्भवति । स्फुटयोजनकर्णाभ्यां भू-यासेन च विना स्फ०टे ॥ आर्यभटेनास्मिन् सति लघुनि किमर्थं महत् कृतं कर्म । गणिताज्ञानाज् जाड्यं विजानता यदि ततः सुतराम् ॥ —BrSpSi. XI. 23-25, also KK. V.
2. See UKK. 9.

82 BRAHMASPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDYAKA sphuṭasiddhānta, Brahmagupta thus speaks of the two works of Āryabhaṭa: As in both the works the number of the Sun's revolutions is spoken of as 432,000 years, their planetary cycle is clear, i. e., of 4,320,000 years. Why then is there difference of 300 civil days in the same cycle of the two books ?¹ Again, he says : In 14,400 years elapsed of the Mahāyuga, there is produced a difference of one day in the audayika and ardharāt- rika systems.² Varāhamihira in the Pañcasiddhāntikā writes - Āryabhaṭa maintains that the beginning of the day is to be reckoned from midnight at Laṅkā; and the same teacher again says that the day begins from sunrise at Laṅkā. ³ Thus from the writings of Brahmagupta and Varāhamihira, it is clear that Āryabhaṭa I was the author of both the audayika and ardharātrika systems of astronomy. In Varāhamihira's verse the phrase sa eva (स एव), meaning "he undoubtedly" is of special significance. It removes the least doubt as to Ārya- bhaṭa's authorship of both these systems. The audayika and ardharātrika astronomical constants are respectively to be found from the Āryabhaṭīya and may be deduced from the Khaṇḍa- khādyaka as well. The following is the comparative view of the constants of Varāhamihira and of the present day Sūrya- siddhānta. TABLE I Planetary revolutions in a mahāyuga of 4,320,000 years, according to various authorities.

1. युगरविभगणाः रव्यूधृति यत् प्रोक्तं तत् तयोर्यु गं स्पष्टम् । त्रिशती रव्युदयानां तदन्तरं हेतुना केन ॥ —BrSpSi XI. 5
2. अधिकैः शतैश्चतुर्भिर्वर्ष सहस्रैश्चतुर्दशभिरेकः । युगायातैर्दिनवारान्तर मौदयिकार्ध रात्रिकयोः ॥ —BrSpSi XI. 13
3. लङ्कार्द्धरात्रसमये दिनप्रवृत्तिं जगाद चार्य्यभटः । भूयः स एव सूर्योदयात् प्रभृत्याह लङ्कायाम् ॥ —PSi. XV. 20

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 83 Planet BrSpSi Āryabhaṭīya Khaṇḍa- Varāha- Later or khādyaka Sūrya- modern siddhānta Sūrya- (PSi.) siddhānta Moon 57,753,300 57,753,336 57,753,336 57,753,336 57,753,336 Sun 4,320,000 4,320,000 4,320,000 4,320,000 3,320,000 Mars 2,296,828.522 2,296,824 2,296,824 2,296,824 2,296,832 Jupiter 364,226.455 364,224 364,220 364,220 364,220 Saturn 146,567.298 146,564 146,564 146,564 146,568 Moon's apogee — 488,219 488,219 488,219 488,203 Venus 7,022,389.492 7,022,338 7,022,388 7,022,388 7,022,376 Mercury 17,936,998.984 17,937,020 17937,000 17,937,000 17,937.060 Moon's nodes 232,311.168 232,226 232,226 232,226 232,238 TABLE II Longitudes of the apogees of the orbits of Planets. Āryabhaṭīya Khaṇḍa- Varāha Modern Planets khādyaka Sūrya-sid- Sūrya-siddh- dhānta. ānta Sun 78° 80° 80° 77° 07′ Mercury 210° 220° 220° 220° 26′ Venus 90° 80° 80° 79° 49′ Mars 118° 110° 110° 130° 00′ Jupiter 180° 160° 160° 171° 16′ Saturn 236° 240° 240° 236° 37′ TABLE III Dimensions of the epicycles of Apsis Planets Ārybhatīya Khaṇḍa- Varāha. Modern khādyaka Sūrya- Sūrya- siddhānta siddhānta Sun 13°-30′ 14° 14° 13½°-14° Moon 31°-30′ 31° 31° 31½°-32° Mercury 22½°-31½° 28° 28° 28°-30° Venus 9°-18° 14° 14° 11°-12° Mars 63°-81° 70° 70° 72°-75° Jupiter 31½-36½° 32° 32° 32°-33° Saturn 40½-58½ 60° 60° 48°-49°

84 BRĀHMASPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDYAKA Table IV Dimensions of the Śīghra epicycles (i.e. conjunctions) Planet Āryabhaṭīya Khaṇḍa- Varāha Modern Khādyaka Sūrya- Sūrya- siddhānta siddhānta Saturn 36½°- 40° 40° 40° 39°- 40° Jupiter 67½°- 72° 72° 72° 70°- 72° Mars 229½°-239½° 234° 234° 232°-235° Venus 256½°-265½° 260° 260° 262°-262° Mercury 130½°-139½° 132° 132° 132°-133° Table V Longitudes of the nodes of the orbits of planets Planets Āryabhaṭīya Khaṇḍa- Varāha Modern khādyaka Sūrya- Sūrya- siddhānta siddhānta siddhānta Saturn 40° 40° Not stated Have to be Jupiter 20° 20° in the calculated Mars 80° 80° Text from the Venus 60° 60° data of the Mercury 100° 100° text Table VI Orbital inclinations (geocentric) to the ecliptic Planets Āryabhaṭīya Khaṇḍa- Varāha Modern khādyaka Sūrya- Sūrya- siddhānta siddhānta Mars 90' 90' 10' 90' Mercury 120' 120' 135' 120' Jupiter 60' 60' 101' 60' Venus 120' 120' 101' 120' Saturn 120' 120' 135' 100' The Mahābhāskarīya of Bhāskara I (522 A. D.) contains a passage which corroborates the fact that Āryabhaṭa I was the author of both the audayika and the ardharātrika systems of Indian Astronomy. According to Pṛthūdakasvāmin, whose

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 85 commentary on the Brāhmasphuṭasiddhānta we have the privilege of presenting to the public, it is clear that in certain respects Bhāskara and others may be wrong but the Āryabhaṭa's authenticity cannot be questioned. Pṛthūdakasvāmin while commenting on the Brāhmasphuṭasiddhānta, XI. 26 writes: Such a mistake may have been made by Bhāskara and others; they have not understood his (Āryabhaṭa's) intention. The passage in the Mahābhāskarīya giving constants of the ardharātrika system runs as follows (we are giving the translation from Kripa Shankar Shukla's edition on the Mahābhāskarīya) : The astronomical processes which have been set forth above come under the sunrise day - reckoning (audayika system). In the midnight day-reckoning (ardharātrika system) too, all this is found to occur: the difference that exists is being stated (below). ¹ The next fourteen stanzas relate to the midnight day- reckoning of Āryabhaṭa I. (i) Civil days and omitted lunar days in a yuga and revolution numbers of Mercury and Jupiter are thus given : (To get the corresponding elements of the midnight day-reckoning) add 300 to the number of civil days (in a yuga) and subtract the same (number) from the number of omitted lunar days (in a yuga); and from the revolution numbers of (the śīghrocca of) Mercury and Jupiter subtract 20 and 4 respectively. ² Thus according to the midnight day-reckoning, we get civil days in a yuga = 1,577,917,800 omitted lunar days in a yuga = 25,082,280 revolution number of the śīghrocca of Mercury = 17,937,000 revolution number of Jupiter = 364,220

1. निबन्धः कर्मणां प्रोक्तो योऽसावौदयिको विधिः । अर्धरात्रे त्वयं सर्वो यो विशेषः स कथ्यते ॥ —MBh. VII. 21
2. त्रिशती भूर्दिने क्षेप्या ह्यवमेभ्यो विशोध्यते । ज्ञ गुर्वोर्भगणेभ्योऽपि विंशतिश्च ततोऽब्धयः ॥ —MBh. VII. 22

86 BRAHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA (ii) Diameters of the Earth, the Sun and the Moon are thus given : (In the midnight day-reckoning) the diameter of the Earth is ( stated to be ) 1,600 yojanas; of the Sun 6,480 (yojanas) and of the Moon, 480 (yojanas)¹. (iii) Mean distances of the Sun and the Moon are as follows : The (mean) distance of the Sun is stated to be 689,358 (yojanas), and of the Moon 51,566 (yojanas).² (iv) Longitudes of the apogees of the planets are as follows: 160, 80, 240, 110, and 220 are in degrees the longitudes of the apogees of Jupiter, Venus, Saturn, Mars and Mercury respectively.³ (v) Manda and Śīghra epicycles of the planets are as follows : The Manda epicycles ( of the same planets ) are 32, 14, 60, 70, and 28 (degrees) respectively; and the Śīghra epicycles are 72, 260, 40, 234, and 132 ( degrees ) respectively. The Sun's apogee and epicycle are the same as those of Venus (i. e. 80° and 14° respectively). The Moon's epicycle in the midnight day-reckoning is stated to be 31 (degrees). ⁴ (vi) The positions of the so called manda and śīghra pātas of the planets are given below: ( The following directions for) the degrees of the (manda and śīghra) pātas of the planets as devised

1. अष्टिशतगुणा व्यासो योजनानां भुवो रवेः । खाष्टाब्ध्यक्ष्यङ्गानि शीतांशोः शून्यवस्वब्धयस्तथा ॥ —23
2. वस्विन्द्रिय गुणच्छिद्रवस्वङ्गानि विभावसोः । अङ्गाङ्गेष्वेक भूतानि चन्द्रकर्णः प्रकीर्तितः ॥ —24
3. अष्टिरष्टौ जिनारुद्रा विंशतिर्द्वयधिकाः क्रमात् । दशघ्ना गुरुशुकार्कि भौमज्ञांशाः स्वमन्दजाः ॥ —25
4. मन्दवृत्तानि द्वात्रिंशन्मनवः षष्टिरेव च । खाद्रयो वसुदस्त्राः स्युः शीघ्रवृत्तान्यथ क्रमात् ॥ —26 द्वयद्वयः खाङ्कनेत्राणि खाब्धयोऽब्ध्यग्निदस्रकाः । द्वयग्नीन्दवो रवेर्मन्दं शुक्रवद् वृत्तमेव च ॥ —27 एकत्रिंशत्क्षपाभर्तु र्धरात्रे विधीयते । —MBh. VII 26-28 (i)

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 87 (under the midnight day-reckoning) should be noted carefully by learned scholars : Add 180° to the longitudes of the mandoccas (apogees) apd śīghroccas of Mercury and Venus, and subtract 3 signs from the mandoccas (apogees) and śīghroccas of the remaining planets. Then are obtained the longitudes of the manda and śīghra pātas of the planets. (Also) add 2 degrees to the longitudes of the manda pātas and śīghroccas of Venus, Saturn and Jupiter, and 1½ degrees to those of Mars and Mercury. (It should be noted that) the śīghra pātas have been stated for all the planets excepting Mercury. (Mercury does not have a śīghra pāta).¹ That is to say, the longitudes of the manda pātas of Mars, Mercury, Jupiter, Venus, and Saturn are 21.5, 41.5, 72, 262 and 152 degrees respectively; and the longitudes of the śīghra pātas of Mars, Jupiter, Venus and Saturn are (śīghrocca--88.5°), (śīghrocca —88°), (śīghrocca+182°) and (śīghrocca--88°) respectively. (vii) A rule for finding the celestial latitude of a planet is as follows : (From the longitude of a planet severally) subtract the longitudes of its (manda and śīghra) pātas and there- from calculate (as usual) the corresponding celestial latitude of that planet. Add them or take their differ- ence according as they are of like or unlike directions. Then is obtained the true celestial latitude of that particular planet. The true celestial latitude of any other planet is also obtained in the same way. The remaining (astronomical) determinations are the same as stated before. This all is in brief the difference of the other tantra (embodying the midnight day-reckon-

1. पातभागाश्च विज्ञेयाः पण्डितैः परिकल्पिताः ॥ —28 मन्द शीघ्रोच्चयोः क्षेप्यं चक्रार्धं बुधशुक्रयोः । राशित्रयं तु शेषाणां पात्यते पातसिद्धये ॥ —29 शुक्राऽर्किदेव पूज्यानां भागौ द्वावेव संयुतौ । मन्दपाताच्च शीघ्रोच्चात् सार्धांशस्तु कुजज्ञयोः ॥ —30 विबुधानां च सर्वेषां शीघ्रपाताः प्रकीर्तिताः । MBh. VII. 28 (ii) 31 (i)