सिद्धान्तशिरोमणि: गणिताध्याय (भास्कराचार्य - ग्रहगणित, मध्यमाधिकार व स्पष्टाधिकार सटीक)

xx In the course of this book, I shall have occasion to quote from most of the Astronomers of ancient India like Aryabhaṭa I, Varāhamihira, Lalla, Aryabhaṭa II, Brahmagupta, Bhāskara I, Muñjāla, Vateśvara and Śrīpati not to speak of some others, whose books are available in print or manuscript. One thought that lurks in my mind, I make bold to present in this preface. I say ‘I make bold’ because, there are some historians of Hindu Astronomy, who are too ready to attack me if I claim that there must have been not a primitive astronomical activity in ancient India prior to Aryabhaṭa I. The reason for their attack is that prior to Aryabhaṭa’s work, only one crude Vedāṅga Jyotiṣa has come to light. Also some of these historians have a strong impression that the galaxy of Hindu Astronomers ranging from Aryabhaṭa derived an incentive from a foreign source especially the Greek. If such historians agree to keep an open mind, I make bold to present my thought as follows. In my humble opinion there must have been considerable astronomical activity in ancient India even prior to Aryabhaṭa. This is borne out by the following expressions of Aryabhaṭa and others. Aryabhaṭa says¹ that he dived into the then extant astro- nomical lore, which got mixed up with mathematical and non-mathematical (mythological or otherwise) knowledge, and by his intellect and the grace of his Goddess, brought- out the truly mathematical. Varāhamihira² says that he was codifying the then extant five Siddhāntas, out of

1. सदसत्ज्ञानसमुद्रात् समुद्धृतं देवताप्रसादेन सद्ज्ञानोत्तमरत्नं मया निमग्नं स्वमतिनावा Verse 49—Golapāda
2. पूर्वाचार्यमतेभ्यो यद्यत् श्रेष्ठं लघुस्फुटं बीजम् तत्तदिहाऽविकलमहं रहस्यमभ्युद्यतो वक्तुम् Verse 2—Ch. I. पोलिशकृतः स्फुतोऽसौ तस्यासन्नस्तु रोमकप्रोक्तः स्पष्टतरः सावित्रः परिशेषौ दूरविभ्रष्टौ Verse 4—Ch. I.

which the ‘Sāvitra’ was more accurate. Brahmagupta³ says, again, that he was giving a clear presentation of the ancient Brahma Siddhānta which got obsolete by a long lapse of time. In the wake of these statements and a number of others, is it not right to construe that there did exist some ancient astronomical texts which are lost to us. If, as asserted by a host of modern interpreters, we think that there was only the crude Vedāṅgajyotiṣa, before Aryabhaṭa, does it not tantamount to saying that these three great astronomers (leave others) were impostors, who, having derived their knowledge from a foreign source, simply claimed that there were existent before them, Saura, Brāhma and some other Siddhāntas which dealt with Graha-gaṇita. It is uncharitable to say that three rational astronomers were sueh impostors. So, we must conclude that there did exist some astronomical activity, which we have no right to call primitive like the Vedāṅgajyotiṣa. Some might think that there might be existing some other texts in between the times of Vedāṅgajyotiṣa and Aryabha but that at the time of the Vedāṅgajyotiṣa⁴ (roughly 1180 B.C.) astronomy in India was that crude. Even this conclusion need not be

“I am going to tell clearly what has been, the best, what has been kept secret and briefly-worded in a nut shell from out of the works of ancient Acāryas”. “The Siddhānta of pauliśa is allright, the Romaka is very nearly the same, but the Sūryasiddhānta is more accurate, whereas the two others Vāsiṣṭha and paitāmaha are crude”. 3. ब्रह्मोक्तं ग्रहगणितं महता कालेन यत् खिलीभूतम् अभिधीयते स्फुटं तत् जिष्णुसुतब्रह्मगुप्तेन । Verse 2—Ch. I. 4. As per statement आश्लेषाधर्धात् दक्षिणमुत्तरमयनं रवेर्धनिष्ठाद्यम् of Varāhamihira quoting the meaning of the verse. प्रपद्येते श्रविष्ठादौ सूर्याचन्द्रमसावुदक् सार्पार्धे दक्षिणा......Verse 7—Vedāṅgajyotiṣa, the vernal equinoctial point was at the beginning of Dhaniṣṭha, which means that the time was 1180 B.C. approximately.

xxii correct, for the simple reason that, even today, crude works exist side by side with advanced works. The simple fact that a Vedāṅgajyautiṣa belonging to 1180 B.C. has been unearthed and nothing else, is not a complete proof that there did not exist more advanced texts some where else in such a big country like India , especially when locomotion or transport was difficult. Just one more thought, I place before the learned historians, which is pertinent to this context. According to geology, biology and some other similar modern sciences, the first man came into the cosmic picture some millions of years ago. If that be so, how is it that we say that man remained stupid all these years and happened suddenly to blossom into a genius one fine morning round about Eighteen fifties (1850 A.D.) whereafter came all Scientific discoveries in a series, as if by the waving of a magic wand ? It could not be so, as if, we are the chosen few of God. Civilizations must have been there, which got buried in the bosom of the earth, as has been revealed by the Mohenzadaro excavations. Hence we are not right in saying that the ancient Hindu civilization was so primitive at 1150 B.C. as is reflected in the Vedāṅga Jyotiṣa. This is another and more important reason that this author makes bold to place before the historians as to why he (the author) does have faith in the statements of Aryabhaṭa, Varāhamihira and Brahmagupta who said that they were peacing out the knowledge contained in the then extant works signifying at the same time that many more books were lost even to them. In this commentary of mine, I have chosen to keep the commentary away from the Sanskrit text and Bhās- kara’s own Vāsanā Bhāṣya, for, otherwise the book grows bulkily unwieldy. I have chosen to keep silent over passages which do not call for a mathematical elucidation. Here and there I have chosen to present what modern astronomy presents in some particular contexts, so that,

xxiii mathematics students who do not happen to study modern astronomy may have a better perspective of the ancient Hindu astronomy, presented in Juxtaposition to the corresponding modern treatment. However, I do not propose to enter into the intricacies of modern astronomy which are not called for to elucidate the text. Before concluding, it is my sacred duty to thank the Rāṣṭrīya Sanskrit Samsthan, under the Ministry of Education, Social Welfare and Culture, Government of India, for having accepted my work under the publication series of K. S. Vidyapeetha, Tirupati and the late- lamented Dr. M. Ananthasayanam Ayyangar, former Chairman of KSV and Ex-Governor of Bihar, who encouraged me in my studies on Hindu Astronomy. Dr. M. D. Balasubrahmanyam deserves my thanks for the encouraging interest he has shown in the publication of this annotation, D. ARKASOMAYAJI

C O N T E N T S Page Foreword ... xv Preface ... xix

1. सिद्धान्तशिरोमणौ गणिताध्याये मध्याधिकारे कालमानाध्यायः ... 1
2. Madhyādhikāra – Section II Bhagaṇādhyāya ... 18
3. Grahānayanādhyāya ... 29
4. Madhyādhikāra - Kakṣādhyāya ... 45
5. The Adhyāya known as Pratyabda Śuddhi in Madhyādhikāra ... 53
6. The Section known as Adhimāsādi- nirṇaya in Ch. I ... 70
7. The Section Bhū-Paridhi-Mānādikara — Circumference of the Earth ... 83
8. Spaṣṭādhikāra - Rectification of Planets ... 102
9. The Tripraśnādhikāra ... 223
10. Parvasambhavādhikāra ... 331
11. Lunar Eclipses ... 346
12. Grahacchāyādhikāra ... 449
13. Grahodayāstādhikāra ... 471
14. Sṛṅgonnatyadhikāra ... 482
15. Grahayutyadhikāra ... 501
16. Pātādhyāya ... 521 Appendix ... 545

ओं श्रीमहागणपतये नमः, श्रीगुरुभ्यो नमः सिद्धान्तशिरोमणौ गणिताध्याये मध्याधिकारे कालमानाध्यायः Verse 1. May the Sun at once give expression to our tongue in meaning ful words-that Sun, who rises to protect this world, whose duty it is to expel darkness, who is reported to be the Spouse of the lotus-creeper, who purges out the sins of all those that supplicate him, and on whose rising take place Vedic sacrifices and thereby the gods in heaven headed by Indra get feasted. Commentary (Comm.). Needs no Commentary. Verse 2. Excels that blessed Brahmagupta, the son of Jishnu, who is hailed as the crest-jewel of Mathemati- cians ; excel also those like Varāhamihira who were the authors of well known works, who were adepts in reason- ing, and u age of beautiful expression and on a study of whose works, one like me even of lesser intellect, will be able to produce monumental works. Comm. Needless to comment. However, we may state one thing. Bhaskara’s quoting the name of Brahma- gupta at the very outset, and that too with reverence, informs us that he is going to accept the Āgama of Brahmagupta in preference to that of others like Arya- bhata. This means that he is going to adopt the number of revolutions of planets in a Yuga and such other astro- nomical constants as were adopted by Brahmagupta. This point will be clarified later. Verse 3 This blessed author Bhaskara is now writing the work named Siddhānta-Siromani, the crest-jewel of all astronomical works, for the pleasure of good minded astronomers, after having bowed to the lotus-feet of his

2 father, from whom he has derived his knowledge; this work, containing good metres, will be easy to understand, besides being flawless and clear, and will enable intelligent readers to develop their ability to understand things. Comm. Not neeessary. Verse 4. Ancient astronomers did write, of course, works abounding in intelligent expression; nonethe- less, this work is started to give expression to some lacunae in their works. I am going to make amends for the deficiencies of the older works and these improvements will be found here and there in their respective places; So, I beseech the good-minded mathematicians to go through this entire work of mine also (for, otherwise, they may not locate my own contribution). Comm. Not necessary. Verse 5. May the good people be pleased with the particulars of my contribution! May the ill-minded people also derive pleasure out of ridiculing me with ignorance unable to understand my contribution! Comm. Not necessary. Verse 6. A Siddhānta work ie an astronomical trea- tise is such a one which deals with the various measures of time ranging from a Tṛti (to be explained shortly) upto the duration of a Kalpa which culminates in a deluge; planetary theory, arithmatical computations as well as algebraical processes, Questions with respect to intricate ideas and their answers, location of the earth, the star and the planets, and description and usage of instruments Comm. Not necessary. Verse 7. Though one knows astrology and that par of the science of Jyotisha which is known as Samhita (an which deals with various subjects like Muhurtas i.e. auspi

8 cious moments to be prescribed for various functions, Desarishtas ie Calamitous occurences to the countries etc.) which form a part of the Science, he cannot answer so many intricate problems pertaining to Astronomy. Such a person, who does not know the astronomical part of the science, which abounds in innumerable reasonings, is one like a king depicted in a drawing, or a lion fast tied to a pole. Comm. Not necessary. Verse 8. The Science of Jyautisha without Astronomy, is like a king's army without roaring elephants though ex- celling in horses etc; is like a garden without mango trees, or like a lake without water, or again like a lady parted with her newly married lover. Comm. Not necessary. Verse 9. The Vedic lore prescribes Sacrifices to be performed; these sacrifices are based upon a knowledge of appropriate time to perform them. This science of astronomy gives a knowledge of time; hence it has been reckoned as one of the six Vedāngas or limbs of the Veda. Comm. Not necessary. Verse 10. (Out of the six Vedangas) The science of grammar is like the face of the person of the Veda, the science of Jyautisha takes the place of the eyes, the Nirukta that of the ears; the Kalpa that of the hands; the Siksha that of the nose and the Chandas the place of the feet. Comm. Not necessary. Verse 11. This science of Jyautisha being depicted as the very eyes of the Person of the Veda, so it has been acclaimed as the most important of the six Angas or limbs of the Veda, in as much as, even if a person be endowed

24 with limbs like the ears, nose etc, if he be devoid of vision, he could not do anything. Comm. Not necessary. Verse 12. (Since this Science of Astronomy has been declared as the most important of the Vedangas) hence this has to be studied by the Dwijas (ie Brahmins Kshatri- yas and the Vaishyas who form the three higher castes), also because it is sacred, secret and the best discipline. By so doing they would acquire Dharma, Artha, Kāma as well as fame (Life is depicted as having a four-fold purpose out of which, Dharma, Artha and Kāma form the first trio, Moksha being the ultimate goal of life). Comm. Not neceasary. Verses 13 and 14. The creator having created the stellar circle along with the planets, placed the latter at the beginning of the circle, put them in constant revolution, at the same time putting the extreme two stars (on either side) in a fixed position. Comm. Bhaskara's own commentary Vāsanā Bhasha under this verse mentions the following points, which are to be noted. The twenty seven stars known as Aswini, Bharani etc. occupy positions roughly at equal distances along the Zodiac, arranged from west to east. The planets were all placed in the beginning of the stellar circle in such a way that they were in a straight line the moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn occupying consecutive positions from the earth in increasing dist- ances, not of uniform measure. The circle of stars known as the Zodiac lies far behind all the planets. There is a wind known as pravaha which keeps the entire Zodiac, as well as the planets below going round and round in the Westerly direction. At the same time, the planets while participating in this westerly motion, have themselves individual motion towards the east. Two stars, are placed

5 one at the north-pole and the other at the south, and they are fixed¹. The diurnal motion due to pravaha is far greater than the individual motion of the planets among the stars in a direction from west to east ie the direction opposite to that of diurnal motion. In this context, it is worth-noting that Aryabhata I mentioned the diurnal rotation of the earth in the verse “अनुलोमगतिः...” Bhaskara must have been aware of this ; yet, in spite of his rational outlook in all matters, mis- guided himself in this respect (vide verse 3 Madhyagati- vāsanā-Goladhyaya), apparently for fear of tradition. Verse 15. The first Mahayuga, the first year, the first day of the bright half of the first month named Madhu,² all of them began simultaneously at the Sun-rise³ at Lankā on Sun-day, at the beginning of the first Kalpa which marked the beginning of creation. Comm. Though, in the Commentary before, Bhaskara gave expression to the fact that Time is eternal with no beginning and no end, herein he mentions the point of Time when creation commenced. So, at the back of his mind, the concept of Time arose only at the beginning of creation, whereas before creation, as well as after deluge the आत्यन्तिकप्रलय, there was and would be neither the concept of Time nor space. In other words, both space and Time are manifest only after creation, and get extin- guished after deluge. Verses 16, 17 and 18. The unit of Time named Tat- para is 1/30th of what is known as Nimeṣa or the time taken during the fall of an eyelid; One-hundredth of a Tatpara is known as a Tṛti ie the time taken to pierce a lotus-leaf with the finest needle. Eighteen Nimeṣas are equal to a Kāṣṭhā. Thirty Kāṣṭhas are equal to a Kalā, and thirty Kalās are equal to one sidereal ghatī. Two

6 ghatis make a Kshana and thirty Kshanas make a sidereal day. Thirty sidereal days are equal to a sidereal month and twelve sidereal months make a sidereal year (not the sidereal solar year). The Zodiac, divided into twelve rasis, and 360 degrees, a degree divided into 60 minutes of arc and a minute divided into 60 seconds of arc all correspond to the year and its successive divisions. Comm. In the Commentary under these verses, Bhas- kara gives further details of division of time as follows. The time taken to pronounce a guru, i.e. double the time of pronouncing a short vowel, is one-tenth of a Prāna, which is the time required by a healthy person to inhale and exhale once. Six Prāṇas make one Vighaṭī and sixty ghatis make one sidereal day. It may be clearly noted here that this sidereal month consisting of thirty sidereal days is not the sidereal month that is the time taken by the Moon to go round the Zodiac, nor the sidereal year defined above is the time that the Sun takes to go round the Zodiac once in his aoparent annual motion. To distin- guish these latter divisions of time, we shall use the nomenclature sidereal lunar month and sidereal solar year. There are further other divisions of time which will be later elucidated. Verses 19, 20. The time taken by the Sun to com- plete one revolution with respect to the stars goes by the name ‘The sidereal solar year’. This will be a day for the gods and demons. The time that elapses between two consecutive new moons or conjunctions of the Moon with the Sun is called a Chāndra-māsa or a lunar month or simply a lunation. This again is the day of the Pitṛs or the Manes. The time that elapses between two consecutive Sun- rises at a place is termed the Sāvana day or civil day. This is called Saura-Sāvana day and it is also the day of the earth.

7 The sidereal day is the time taken by the stars to go round the earth once. It is called Nākshatra-dina. Comm. In Hindu mythology gods are supposed to reside at the north-pole, where one civil day is the same as one sidereal solar year for the other places; also the demons are supposed to reside at the south pole so that their civil day also is equal to a sidereal solar year. But, what is day to the gods is night to the demons and vice- versa. If we call the earth, the moon of the moon so to say, which it is so, when it is the moment of new moon for the earth, it is the moment of Full Moon to the Moon. Thus what is a Chāndra-māsa to the earth may very well be called with respect to the Moon a Bhauma-māsa. In Hindu mythology the manes are supposed to take residence on the surface of the Moon. We know from modern astronomy that the moon revolves about her axis once roughly in a lunation. Thus for 'the man on the moon', a day is roughly equal to our lunation. So if the manes were to reside on the moon, their day is equal roughly to a lunation of ours. We say 'roughly' because as we see under the chapter of the lunar eclipse, the moon almost shows the same face to the earth on account of what are called 'librations in longitude'. On this count the time of rotation of the moon does not exactly coincide with the time of a lunation. A civil day for a place is also the civil day for every place of the earth so that it is called the earth's day. By 'day' here we do not mean the time when the Sun is above the horizon, for that time differs from place to place on the earth. A civil day is the sum-total of the duration of day and the duration of night for any place and it will be seen that this is the same for the entire earth except at the places having perpetual day. The word Saura Sāvana is used to signify that the time that elapses between two consecutive rises of any other planet is termed the Sāvana

8 day pertaining to that planet. Thus the Chāndra-Sāvana day is roughly equal to 24 hrs–52' and this is the longest of the Sāvana days pertaining to the planets. The shortest Sāvana days is that of the Saturn, since Saturn moves a very little along the ecliptic and his Sāvana day is there- fore just a little longer than the sidereal day. This nomen- clature brings in the phenomenon that the Sāvana day of a retrograde planet happens to be less than a sidereal day. But in a given time, which is sufficiently long like a yuga, the Sāvana days of a planet are equal to the number of sidereal revolutions in that period minus the number of the planetary revolutions in the same period. This will be constant for a given planet in such a long period, though individual days happen to be some shorter and some longer than a sidereal day. The modern sidereal day is the time between two consecutive rises of the equinoctial points and as the equi- noctial points have a slow retrograde motion, the modern sidereal day is just a little shorter than the Hindu sidereal day, which does not take cognizance of the revolution of the equinoctial points round the earth in reckoning diurnal motion. The Hindu astronomers speak of the revolution of the stars only around the earth in the context of diurnal motion. It will be noted that the Saura-Savana day or the civil day will not be of the same duration, since the Sun has unequal motion amongst the stars from day to day. When the Sun is in perigee and has the max. daily velocity, his Sāvana day at that moment is the longest and the Saura- Sāvana day when the Sun is in apogee will be the shortest. Though in Indian Chronology a day is divided into sixty ghatis for convenience, these ghatis are evidently longer than the Nākshatra-ghatis or the sidereal ghatis, which are of a fixed duration. Thus a Sāvana ghati is a little longer than a Nākshatra-ghati and what is more, a Saura-Sāvana ghati is of a variable duration from day to day, the variation being of course very small.

9 The concept of a month originally arose out of the phenomenon of new-moons, for, this phenomenon alone appeals to every lay man, when he could not sight the Moon. The concept of an year arose originally out of what is called a tropical year, which is the time between two consecutive conjunctions of the Sun with an equi- noctial point and which it is that makes the seasons recur. Thus the primitive man must have had the concept of an year when he once saw the mango trees blossoming and again when he saw them blossom. This is why the Hindus celebrate the new year's day with eating the neem flower with unripe mangos, which goes by the phrase Nimba- Kusuma bhakshanam or eating the neem flower. In the Vedic times, however, the year began with the luni-Solar month called Mārga-Śīrṣa, which brings in the new year crops. In the Vedic sacrifices, there is thus what is called the Āgrahāyaṇeṣṭi, where the word Āgra-hāyaṇa means the Mārga-Śīrṣa month. The etymology of the word is that अग्रे हायनं यस्य तत् आग्रहायणम् ie the year is ahead of this month, which therefore is the beginning month of the year. This is also why the ancient lexicon named Amarakośa enumerates the months from Mārga-Śīrṣa. This is the month when the full moon occurs when the Moon is in the star Mṛgaśira. We have also an inkling from this that the vernal equinoctial point was probably situated in the star Mṛgaśira. The Veda however enumerates the stars from the star Kritticā, and we have a statement in the Śata- patha-Brāhmaṇa that "एता ह वै कृत्तिकाः प्राच्यै दिशो न श्च्यवन्ते ie Behold ! these stars which go by the name the Krit- ticas do not deflect from the east-point. As this group of Krittikas is situated on the ecliptic, the statement that they were rising in the east signifies that the Vernal equi- noctial point was situated in the Kritticas". Arguing about the situation of the vernal equinoctial point in the so-called Vedic times, the late Lokamānya Bāla Gangā- dhara Tilak concluded that Vedic literature must have had its beginning about eight thousand years ago. 2

10 From the original concepts of the month and the year, further concepts of the different kinds of month and the year arose with the advance of astronomical knowledge. We shall deal with these different kinds of the month and the year in their respective contexts. In these two verses, we have the definitions of Saura- māna, Daiva-māna, chāndra-māna, Paitra-māna, Sāvana- māna and Nākṣatra-māna, six of the nine mānas, ie mea- sures of time. Verses 21, 22, 23, 24, 25. The four yuga-pādas named Kṛita, Tretā, Dwāpara and Kali consist of 4 × 432000, 3 × 432000, 2 × 432000 and 432000 mean solar years respectively, the sum total of which consisting of (4+3+2+1) × 43200 = 43,20000 mean solar years, is calleda yuga. Each of the yuga-pādas above are inclusive of what are called their respective Sandhyās and Sandh- yāmsās which constitute one-twelfth of their own durations. A Manu's duration consists of 71 yugas and 14 Manus duration is reported to be the day-time of Brahma, whose night is also of an equal duration. The duration of a Manu, known as a Manvantara has a Sandhyā-Kāla on either side, ie before and after, equal to one Kṛita. If these are taken into account, the day- time of Brahma amounts to one thousand yugas and it goes by the name a Kalpa so that a complete day of Brahma equals two Kalpas. The life-duration of Brahma consists of one hundred years on this scale (where one year = 360 days). This life-duration of Brahma goes by the name Mahā-Kalpa, as reported by elders. In as much as Time was without a beginning and will have no end either, I do not know how many Brahmas have gone before. Comm. In these verses we are given what is known as Brāhma-māna, the seventh of the nine mānas. Inciden- tally we are also given the measures yuga-pādas, yugas,

11 Manvantaras and a Kaḷpa. Since in many ancient astro- nomical works, the revolutions of the planets and the planetary points like nodes, apogees or aphelia are given as integers during the course of a yuga, the concept of a yuga must have arisen as follows. The durations of the sidereal revolutions of the planets and the apogee of the Moon and its node having been ascertained by observation, a period was calculated in which are contained integral multiples of those durations. In other words a yuga of 4320000 mean solar years is construed as the period in which the planets the node and apogee of the Moon make an integral number of revolutions with respect to the stars. We have excluded here the aphelia and the nodes of the planets, as we shall see later that their sidereal revolu- tions were not based on observation but by an assumption that those points also must be having an integral number of revolutions during the course of a Kaḷpa, having started at the beginning of the Hindu Zodiac ie the beginning point of Aswini at the beginning of the Kaḷpa. On this assumption cited, and using indeterminate analysis the number of their sidereal revolutions were got as reported by Bhāskarāchārya in his Commentary in the chapter Bhaga- nādhyāya. He has given us a clue that the numbers of sidereal revolutions of the planets including the node and apogee of the Moon were originally determined by observa- tions though he appeals to Āgama that those numbers were given by Āgana, as accepted and transmitted by Brahma- guptāchārya. Even in the Upapatthis or proofs that Bhās- karāchārya gives regarding the numbers of sidereal revolu- tions known as Bhagaṇas, we perceive that he consciously commits the logical flaw known as इतरेतराश्रयदोष as we are going to show in that context. The proofs he adduces are indeed based upon Aryabhatāchārya's verse 48, Golapāda namely क्षितिरवियोगाद्दिनकृत्, रवीन्दुयोगात्प्रसाधितश्चेन्दुः शशिताराग्रहयोगात् तथैव ताराग्रहाः सर्वे

12 and on Brahmaguptācharya's verse 12, ch. 20 namely ज्ञातं कृत्वा मध्यं भूयोऽन्यदिने तदन्तरं भुक्तिः त्रैराशिकेन भुक्त्या कल्पग्रहमण्डलानयनम् Aryabhatācharya and his immediate followers Lallā- charya and Vateswarācharya make the yuga-pādas of equal duration. Brahmaguptācharya criticises Aryabhata for having said so against the Canons of the Smṛtis as well as Romaka for having ignored the concept of yugas, manvan- taras and Kalpa, as this he deems as a heresy (Vide verses 9 & 13 ch. I). Indeed there seem to be two schools among the ancient Hindu Astronomers one of Brahmagupta who was followed by Sripati, Bhaskara and a number of others and the other of Aryabhata who was followed by Lalla, Vateswara, Bhas- kara I, and a host of others mostly hailing from Kerala. There is an Aryabhata who has been termed Aryabhata II and who was the author of a book named Brihad-Arya- bhatiyam or Mahā-Siddhanta as it is also called. M. M. Sudhakara Dwivedi mentions in his Ganaka-Tarangani, that this Aryabhata should have existed after the author of Modern Surya Siddhanta. Aryabhata I, Aryabhata II, many of the Kerala astronomers used a different nomenclature to signify numbers, denoting them by letters. Thus one of the distinguishing features of the Kerala school of astrono- mers (not all of them) seems to be to use letters for numbers. One Kalpa = 14 Manvantaras = 14 × 71 yugas + 15 Sandhis in between the Manvantaras each equal to a Krita ie 4 Kalis = 994 yugas + 60 Kalis = 994 + 6 yugas = 1000 yugas = 4320000000 mean Solar years. Bhaskara speaks of Sandhyas and Sandhyamsas each of them equal to 1/12 of the yugapādas. Thus one Kali = 432000 years = 1200 Divyābdas (gods' years each year being equal to 360 Solar years) = 1000 +

18 100 + 100 Divyābdas since ¹/₁₂ × 1200 = 100 = measure of the Sandhyā and Sandhyamsa times. At this rate Dwa- para = 2000 + 200 + 200 Divyābdas and so on. However, the measure of the Sandhyas with respect to a Manvantara does not follow this one-twelfth rule, because a Krita is not ¹/₁₂th of the Manvantara. Thus 1000 Divyābdas = one Kali excluding Sandhyā and Sandhyāṁsa, whereas 1000 yugas = one Kalpa including the Sandhyas and Sandh- yamsas. When a yuga was conceived as a period wherein the planets make an integral number of revolutions, it goes without saying that they make integral numbers of revolu- tions in a Manvantara or a Kalpa. When a Kalpa was con- ceived as the period in which the slow-moving planetary points aphelia and nodes also make an integral number of revolutions, one wonders how a manvantara was conceived. It is further peculiar why such an odd number 71 was chosen, when it was said that 71 yugas make a Manvantara. One also wonders why the modern Suryasiddhānta says that after the Kalpa began, creation started only after 47400 Divyābdas whereas neither Brahmagupta nor Bhāskara speaks of this. As a matter of fact Bhāskara mentions later the number of years that had elapsed upto the begin- ning of the Saka era, as equal to 1972947179, but does not speak when the creation of planets and stars began actually. Verse 26. Half the life-period of the present Brahma has elapsed; some said that only eight and half years of his life has elapsed — Let the Āgama or tradition be whatsoever; we don't have any need of knowing it be- cause the planetary positions have to be computed only from the beginning of this Kalpa. Comm. Vateswara it was that mentioned that only eight and half years of the present Brahma had elapsed (Vide verse 10 Madhyādhikāra ch. I Vateswara Siddhānta) Vateswara prescribes that Ahargaṇa or the collection of

14 days has to be calculated from the birth-time of this Brahma, but Bhāskara rightly points out that it is a waste of labour, for, all the planets must have returned to the Zero-point of the zodiac ie the beginning of the Star Aswini at the beginning of this Kalpa and hence it is sufficient to calculate only from the beginning of this Kalpa. Further Bhāskara states that when the very planets did not exist during the last elapsed night of Brahma, what is the fun of calculating their positions. There is also a tradition that there are nine Brahmas and that the present one is the very first. This tradition Bhāskara does not mention, because he exclaims that he does not know how many Brahmas have gone by, Time being without a beginning. Verse 27. In as much as the creation started only from the beginning of this Kalpa which is the present day- time of Brahma, and because deluge takes place at the end of the day-time, the question of Computing the planetary positions arises only when the planets exist. If some (the allusion is to Vateswara) propose to compute the planetary positions even when the very planets did not exist, may we salute those great people ! Comm. Not necessary. Verse 28. Six Manus have elapsed in this Kalpa, thereafter twentyseven yugas, as well as three yugapādas namely Kṛita, Tretā and Dwāpara. Further 3179 years of this fourth yugapāda namely Kali have elapsed by the end of the Saka king (which moment was the beginning of the Saka era). Hence in the present Kalpa ie the day-time of this Brahma, 19729 47179 years had elapsed upto the beginning of the Saka era. Comm. The computation is as follows : 6 Manvantaras = 6 × 71 × 10 Kaliyugas since each Manvantara Consists of 71 yugas and a yuga Consists of

18 ten Kaliyugas (one yuga = Krita + Tretā + Dwāpara + Kali = 4 + 3 + 2 + 1 = 10 Kaliyugas). The Sandhis that were there in between the Manus and in the beginning of the first Manu are seven and each Sandhi being equal to one Krita or four Kalis, the seven Sandhis = 7 × 4 = 28 Kaliyugas. Further it is stated that 27 yugas had elapsed in the present seventh Manvantara known as Vaivasvata which are equal to 27 × 10 = 270 Kaliyugas. Further in the present yuga, Krita, Tretā and Dwāpara had elapsed equal to 4 + 3 + 2 = 9 Kaliyugas. Thereafter in the present Kaliyuga 3179 years elapsed upto the beginning of the Saka era. Thus totalling we have 4260 + 28 + 270 + 9 kalis + 3179 years. = 4567 kalis + 3179 years. = 4567 × 432000 + 3179 years. = 179 294 7179 year as mentioned. Verse 29. The six Manus that went before the present Vaivasvata were Swāyambhuva, Swārociṣa, Auttama Tāmasa, Raivata and Cākshusha. Comm. Clear—Upto this point we have seen the Brāhmamāna, the seventh of the nine mānas. Verse 30. The Sāmhitikas declare that a Samvatsara is equal to the time of a mean sidereal revolution of Guru the Jupiter (This is the Bārhaspatyamāna). The ninth māna named the Manushyamāna is composite of the four mānas (as detailed in verse 31). Comm. The years of the Bārhaspatya māna are enu- merated as Vijaya, Jaya etc which are sixty in number. The same names are also adopted in the cāndramāna ie the luni-Solar reckoning (In vogue in the Andhra Pradesh and some other provinces too) only with the difference that prabhava is taken as the starting year in which sequence Vijaya happens to be the twenty-seventh year. Since the