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

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)

88 BRĀHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA ing of Āryabhaṭa I).¹ (viii) A rule for finding the longitude of the true mean planet according to midnight-day reckoning is as follows : Apply half the Śīghraphala and (then) half the mandaphala to the longitude of the planet's own mandocca (reversely). From the resulting longitude of the planet's mandocca cal- culate the (mandaphala and apply it to mean longitude of the planet : the resulting longitude of planet is stated to be) the true-mean longitude of the planet. This is stated to be another difference (of the midnight day-reckoning) ² (ix) Length of the circle of the sky and derivation of the lengths of the orbits of the planets are given as follows : Multiply the revolutions of the Moon (in a yuga) by 3, 240,000 and then discard the zero in the unit's place : (this is the length of the circle of the sky in terms of yojanas). (Severally) divide that by the revolutions of the planets (in a yuga): thus are obtained the lengths of the orbits of the respective planets in terms of yojanas.³ From these stanzas (from 20-35), it is evident that one yojana of the sunrise day-reckoning is one and a half times that of the midnight day-reckoning. Now from stanza 22, it appears that 300 is to be added to the number of civil days in a Mahāyuga. According to the Āryabhaṭīya. the number of civil days in this cycle is 1,577,917, 500, which increased by 300, becomes 1,577,917,800, the number of

1. शोधयित्वा क्रमात् पातान् विक्षेपार्शान् प्रसाधयेत् । —31 योगविश्लेषणोत्पत्तिरेकानेकस्वदिग्वशात् । विक्षेपः स स्फुटो ज्ञेयो ग्रहस्यैकस्य कीर्तितः ॥ —32 अन्यस्याप्येवमेव स्याच्छेषाः प्रागुक्त कल्पनाः । एतत्सर्वं समासेन तन्त्रान्तरमुदाहृतम् ॥ —MBh. VII. 31 (ii)–33
2. शीघ्रमन्दोच्चचापाधैसंस्कृता त् स्वीयमन्दतः । स्फुटमध्यग्रहः स धै विशेषः परिकीर्तितः ॥ —MBh. VII. 34
3. वेदाश्विरामगुणितान्य युताहतानि । चन्द्रस्य शून्यरहितान्यथ मण्डलानि ॥ र : स्वैर्हृतानि भगणैः क्रमश्रो ग्रहाणां । कक्ष्या भवन्ति खलु योजनमण्ड्रस्था ॥ —MBh. VII. 35

civil days in a Mahāyuga according to Brahmagupta as referred to in the Khaṇḍakhādyaka. Again, the same stanza tells us to subtract 20 and 4 respec- tively from the revolutions of Mercury and Jupiter, and we arrive at the figures 17,937,000 and 364, 220, which are the revolutions of Mercury and Jupiter in a Mahāyuga according to Brahma- gupta as given in the Khaṇḍakhādyaka. Again we can compare the figures for the diameters of the Earth, the Sun and the Moon given in the Mahābhāskarīya, in the present day Sūrya-siddhānta and the Āryabhaṭīya. Diameter Mahābhāskarīya Modern. Āryabhaṭīya of Sūrya-siddhānta Earth 1,600 yojanas 1,600 yojanas 1,050, yojanas Sun 6,480 6,500 4,410 Moon 480 480 315 Then in stanza 24, we are given the distances of the Sun and the Moon as 689, 358 and 51, 566 yojanas respectively. The same figures are worked out by Lalla according to the Āryabhaṭīya and quoted in the Śiṣyadhwṛddhida (IV.3.4) and they come to be 459,585 and 34,377. The stanza 25 states the longitudes of the aphelia of planets these figures tally with the corresponding figures given by Brah- magupta in the Khaṇḍakhādyaka : Longitude of aphelion of Jupiter 160°, of Venus 80°, of Saturn 240°, of Mars 110° and of Mercury 220°. Similarly the stanza 26 gives the peripheries of planets’ epicycle of apsis, which also is in concordance with the values given by Brahmagupta : Periphery of epicycle of apsis of Jupiter 32°, of Venus 14°, of Saturn 60°, of Mars 70° and of Merury 28°. In the stanza 27 of the Mahābhāskarīya, we have the dimensions of the epicycles of conjunction for planets; these figures are also the same as given by Brahmagupta : Epicycles of conjuction for Jupiter 72°, for Venus 260°, for Saturn 40°, for Mars 234° and for Mercury 132°. In the stanza 28, we have the Sun’s epicycle having a periphery of 14° and the Moon’s epicycle 31° ; the longitudes of

90 BRĀHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA the nodes of the planets to be the same as in the Āryabhaṭīya. All these are the same as given by Brahmagupta in the Khaṇḍa- khādyaka. In the stanza 33 we have rules for finding the geocentric longitudes of planets which may be taken to be the same as in the Khaṇḍakhādyaka¹; compare these values with those in the Sūryasiddhānta of Varāhamihira in the Pañcasiddhāntikā, XVII. 6, but slight different from the Āryabhaṭīya.² The last stanza of the Mahābhāskarīya (35) gives the dimen- sions in yojanas of the orbits of planets ; these are the same as in the modern Sūryasiddhānta. Thus we find a great semblance in the constants as given by the Mahābhāskarīya of Bhāskara I, of the Sūryasiddhānta as given by the Pañcasiddhāntikā and understood by Varāhamihira, and also the constants as given by Brahmagupta in his treatises, specially the Khaṇḍakhādyaka. It must not be forgotten that the same Āryabhaṭa I who is the celeberated author of the Āryabhaṭīya is also the author of another treatise very often referred to as the Tantra. I shall quote Prabodha Chandra Sengupta in connection with these similarities, and the great influence of Āryabhaṭa on Indian Astronomy. He writes in his Introduction to the Khaṇḍakhādyaka as follows : We have shown that there is much resemblance in the constants between the Sūryasiddhānta of Varāha and the Khaṇḍakhādyaka and for the matter of that with the Tantrāntara of Āryabhaṭa I. In my papers “Ārya- bhaṭa and Āryabhaṭa's Lost Work”, I have establish- ed the fact that the Sūryasiddhānta as it existed before the time of Varāha, was made more accurate by him by borrowing the constants from Āryabhaṭa's ārdharātrika system. That there was a Sūryasiddhānta १. शीघ्रफलाद्धं मध्ये मन्दफलाद्धं च मन्दशीघ्रफले । सकले मध्ये स्पष्टः शीघ्रं मध्योनकं केन्द्रम् ॥ —KK. II. 18 २. मन्दोच्चाच्छीघ्रोच्चादर्धमृणधनं ग्रहेषु मन्द्येषु । मन्दोच्चात्स्फुट मध्याच्छीघ्रोच्चाच्च स्फुटा ज्ञेयाः ॥ शीघ्रोच्चादर्धोनं कर्तव्यमृणं धनं स्वमन्दोच्चे । स्फुटमध्यौ तु भृगुबुधौ सिद्धान्मन्दात्स्फुटौ भवतः ॥ —Arya. III. 23-24

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 91 before the time of Varāha, is seen from Section 6 of the Table on page xii given before. This point is made clear from another consideration, viz., the star table in the modern Sūryasiddhānta, which unmistakably points to the conclusion that the longitudes of some stars, e.g., Spica etc., correspond to a time much ante- rior to that of Āryabhaṭa I. The great fame of Ārya- bhaṭa I induced Varāha, the first maker of a neo-Sūrya- siddhānta to use the elements of Āryabhaṭa's ardha- rātrika system to supplant the older materials in it. No wonder, therefore, that there is an opinion in favour of the hypothesis that Āryabhaṭa I was the author of the Sūryasiddhānta. If there were a shadow of truth in it, Varāha would have admitted it. Albe- runi indeed says that the Sūryasiddhānta was compos- ed by Lāṭa (Alberuni's India, translated by Sachau, Vol. I.p. 153). We now know that this Lāṭa or Lāṭa- deva was one of the first pupils of Āryabhaṭa I. He was the expounder of the Romaka and Paulīśa Siddhāntas as we learn from Varāhamihira's Pañcasiddhāntikā, (I.3). As Alberuni's statement is not corroborated by Varāha, we are not inclined to take it as correct. None of the earlier writers suggest that the Sūryasiddhānta was in any way modified or changed by Ārya- bhaṭa I. It has now been established beyond doubt that the same Āryabhaṭa was the author of the Aryabhaṭīya and another Tantra which is now lost. There is reason in support of hypothesis that this Tantra itself was the first work of Āryabhaṭa I and that the Aryabhaṭīya was the second work from the order in which Varāha mentions them in the Stanza quoted earlier. If this hypothesis be true, the stanza in the Aryabhaṭīya¹, which was translated by me as : “Now when sixty times sixty years and three quarter yugas also have elapsed, twenty increased by three years have elapsed since my birth.”

1. षष्ट्यब्दानां षष्टिर्यदा व्यतीतास्त्रयश्च युगपादाः । व्यधिका विंशतिरब्दास्तदेह मम जन्मनोऽतीताः ॥ —Ārya. III. 10.