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

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

10 mean sidereal revolution takes roughly 11.8 years, five revolutions take 59 mean solar years. The sixtieth year thereafter of the Jovian Cycle is considered as an Adhi- Samvatsara thereof so that the luni-Solar reckoning as well as the jovian reckoning get wedded together. Thus to get the jovian year we have simply to add 27 to the luni, solar year. The wedding of the two reckonings has its analogy in the process of intercalation which weds the solar reckon- ing with the luni-Solar only with the difference that months of the latter reckoning are set apart as Adhika or extra months. Just as the process of intercalation brings in its train what is called a Kshayamāsa as per the convention that, that month would be set apart as an Adhikamāsa, which does not carry a Samkrānti ie entrance of the Sun into the next Rasi, in which process there appears a month in which there may occur two Samkrāntis which is hence considered as a Kshayamāsa, just in a similar way that luni-Solar year in which the Jupiter enters the next Rasi, is supposed to be normal whereas that year in which such an entrance does not take place is deemed an Ādhika year and set apart while that year which carries two entrances is deemed as a Kshya year. The pro- cess of intercalation which weds together the solar and the luni-solar reckonings will be elucidated further in its appro- priate context. The word Sāmhitikas means the authors of the works called Samhitas like the Varāha-Brihatsamhita etc. Verse 31. The manushya-māna or that which men follow, adopts the year, the Ayana the Ṛtu or the season (six in number during an year) and the yuga according to the movement of the Sun ie according to Sauramāna the months and the thithis according to the luni-solar reckon- ing ie according to Chāndramāna the Vratas, upavāsas, treatment of diseases, deliveries of ladies, the names of the weeks all these according to the reckoning of civil days ie according to Sāvanamāna and finally the ghatis Vighatis etc. etc. according to the Nākshatramāna.

17 Comm. A lunation is divided into thirty tithis, which go by the names pratipat etc. of which there are fifteen in the brighter half of the lunation and fifteen in the dark fortnight. The pratipat tithi is that duration of time be- ginning from the moment of New Moon and ending when the Moon has over taken the Sun by 12°; the second tithi named dvitīyā begins at the end of pratipat and lasts upto the point of time when the moon’s elongation is 24° and so on. Thus the tithis are seen to be of unequal length in as much as both the Sun and the Moon have unequal motion. The mean duration of a tithi is seen to be a little less than a civil day since a lunation has ronghly 29½ civil days. There is a convention that the tithi which is current at a Sun-rise will be considered to be the tithi of the whole day. As per this convention it so happens that a tithi lasts just a little after Sun-rise and the next tithi vanishes during the same day so that the next but one will be taken for the next day. This vanishing tithi is known as a Kshaya tithi which is also called a Kshayāba. Again it so happens that a tithi may be current at two consecutive Sun-rises beginning a little before the Sun-rise of the first day and extending a little after the next Sun-rise. Such a tithi is called Dina- Traya or a tithi which touches three days. In this matter if seems as though we have gained a tithi but ultinately as a tithi must be less than a civil day, it so happens that on the average there will be a Kshayāha rongbly in 64 tithis. We shall see more about this matter subsequently. Verse 32. Thus there are nine mānas Mānava, Divya (or of gods), Bārhaspatya (Jovian) paitra, Nākshatra, Saura, Chāndra, Sāvana and Brāhma. But the planetary positions are to be computed by men by their own māna. Comm. Not necessary. Here ends the Adhyāya known as Kāla-māna in the Madhyādhikāra. 2

MADHYĀDHIKĀRA — SECTION II BHAGAṆĀDHYĀYA Verses 1 to 6. The number of sidereal revolutions of the Sun during a Kalpa is 4320000000. It is also the number of those of Mercury and Venus, and those of the Śīghrocclas of the planets Mars, Jupiter and Saturn. The Moon makes 57753300000 sidereal revolutions in a Kalpa, the Mars 2296828522, the Mercury's śīghroccha 17936998984, the Jupiter 364226155, the śīghroccha of Venus 7022389492 and the Saturn 146567298. The sidereal revolutions of the apogees of the Sun and the Moon and those of the aphelia of Mars, Mercury, Jupiter, Venus and Saturn in a Kalpa are respectively 480, 488105858, 292, 352, 855, 653, 41. The retrograde sidereal revolutions of the nodes of the orbits of Moon, Mars, Mercury, Jupiter, Venus and Saturn are respectively 232311168, 267, 521, 63, 893, 584. Comm. Since Mercury and Venus will be oscillating about the Sun in their apparent motion as seen from the earth in a long period of time, the number of sidereal re- volutions made by the Sun is also equal to that made by Mercury and Venus. Since, as we see in the Course of the Spaṣṭādhikāra the Sun plays the part of what is called the Śīghroccha of the three major planets named Mars, Jupiter and Saturn, the number of the Sun's sidereal revolutions is also the same as that of their śīghrocchas. The reason why the sidereal revolutions of what are called the śīghrocchas of Mercury and Venus are the same as the heliocentric sidereal revolutions of Mercury and Venus will be clarified in the spaṣṭādhikāra. The reason also why the sidereal revolutions of the major planets are

19 the same as their heliocentric ones will also be clarified in the same adhikāra. The reason why we have termed the Mandocchas as apogees in the case of the Sun and the Moon and as aphelia with respect to the other planets is that the Sun moves round the earth relatively while the Moon directly moves round the earth whereas the remainning planets move round the Sun while the Sun moves relatively round the earth. In other words the Sun and Moon have apogees whereas the remaining planets aphelia. The nodes of the planets are the points of intersection of their orbits with the ecliptic which is the apparent orbit of the Sun. In other words the planetary orbits are inclined to the ecliptic which means that their orbital planes do not coincide with the ecliptic plane. In Hindu Astronomy the Sun, the Moon and also the two nodes of the lunar orbit which go by the names Rahu and Ketu are also termed as grahas along with the other five which are known as Tāra-grahas or planets resembling stars. The etymology of the word ‘planet’ is that it moves amongst stars; in this respect the nine Hindu grahas also moving among the stars are eligible to have the same appellation though it is not permitted in modern astronomy. But the word graha has a different connotation etymologi- cally namely गृह्णातीति वा गृह्यते अनेनेति वा ग्रह : ie that which seizes the fates of men is known as a graha. In the course of this work we use the word graha and planet synony- mously so that we deem the Sun, the Moon and the nodes of the lunar orbit also as planets. When the ancient Hindu astronomers knew by obser- vation that the nodes of the lunar orbit have a retrograde motion, and could also measure their mean motion, they extended the analogy to the nodes of the other planets also, whose motion could not be measured during anybody's life-

20 time. Hence the estimate of their mean motion by the Hindu Astronomers naturally went wrong. Noticing that the Mandoccha of the Moon has a progressive motion which the Hindu astronomers could measure correctly, they extended the analogy to the apogee of the Sun and the aphelia of the other planets whose motion also being very slow could not be measured during the life-time of a man. So here also the estimate of their mean motions of the Sun's apogee and the planetary aphelia went wrong. Bhaskarāchārya has given proofs as to how the sidereal revolutions could be got but as we shall see later in the Spaṣṭādhikāra, his proof occasionally suffers from what is called ‘इतरेतराश्रयदोष’ ie ‘begging the question’. We shall however construct our own proofs at that place deferring them for the present, for, the proofs require an elucidation which obtains in the Spaṣṭadhikāra alone. Verse 7. The number of diurnal revolutions of the stars in a Kalpa is 1582236450000. Comm. In fact this number is that of the diurnal rotations of the earth which is equal to the number of apparent diurnal rotations of the stars, if the earth is deemed as fixed. The only Hindu Astronomer who made bold to say that the earth is rotating and not the stars came under criticism by Brahmagupta and the latter Hindu astronomers. Aryabhata said अनुलोमगतिः नौस्थः पश्यत्यचलं विलोमगं यद्वत्, अचलानि भानि तद्वत् समपश्चिमगानि लङ्कायाम् ” ie Even as a man stationed on a moving boat perceives that the trees etc on the banks of the canal, river or lake to be moving in the opposite direction supposing himself statio- nary, so also men stationed on the surface of the earth (which is like a moving boat) perceive the actually statio- nary stars to be moving directly from east to west at Lanka ie the equator ”.

Why none of the latter Hindu astronomers, though many of them could intuit this simple phenomenon, boldly came out asserting this, is rather mysterious. Even today there is such an irrational orthodox type of scholars who are not in touch with modern astronomy, holding the view that the earth does not move. In Hindu Astronomy it was postulated that there is what is called the pravaha wind, which effects moving of the entire stellar universe along with the planets from east to West. It is a very simple matter to visualize earths' rotation, instead of supposing that the entire stellar Universe is being driven round the earth. The absurdity in this latter supposition might not have been clear to the orthodox type of the Hindu Astronomers, for, they could not measure the dimensions of the giant stars and super- giants, which are everyone of them mighty Suns. Though, however, the dimensions of the Sun were known to them to be far greater than those of the earth, it did not occur to them why a mighty Sun should go round a pigmy earth. Or even if it occured to astronomers like Bhaskara, they dared not to go against the puranic tradition. Since the stars do not move among themselves while partaking this diurnal motion, the entire starry skies are obliged to go round the earth as a rigid structure, if we suppose that the earth is not rotating about, herself. This kind of supposition is just like a revolving person, revol- ving about himself and claiming that the entire Universe is revolving round him and not he about himself. Bhaskara gives the proof of getting this number of diurnal rotations of the stars during an year in the verses 5-7 of Madhyagati Vāsanā, समं भसूर्यावुदितौ etc. as follows. Suppose a star and the Sun rise together today. Tomorrow the star will have arisen earlier than the Sun who will have moved towards the east of the star by his own (apparent) daily motion. So tomorrow's sun-rise will get

22 belated by the duration of time that the arc of the ecliptic covered by the Sun's today's motion takes to rise. This duration of time is variable on two counts; first by the variable motion of the Sun and second by the obliquity of the ecliptic on account of which even equal arcs of the ecliptic will not rise in equal times. In other words the duration of time between two consecutive Sun-rises will not be the same. This duration of a particular day can roughly be calculated by the rule of three as follows. Let the Sun be in a particular Rasi, the rising time T of which could be computed; let the Sun cover an arc of x° in that Rasi on that day. Then the time taken by that arc to rise is xT / 30, where a Rasi consists of 30°. This time computed in Sidereal measure added to 60 Sideral ghatis is equal to the length of the day. During the course of an year ie the time taken by the Sun to move round the ecliptic starting from the Zero-point of the zodiac and again returning to the same point, the Sun will have made one revolution less than the stars. Thus if ‘ R ’ the number of diurnal revolu- tions of the Sun (where R will be not an integer) during an year or what is the same the number of Sāvana or civil days during an year, they are equal to R + 1 sideral days. Hence the number of sideral days in a kalpa will be equal to the number of civil days in a kalpa together with the number 4320000.000 which is the number of revolutions made by the Sun relative to the stars. In this context we are to know the number of civil days in a kalpa. The proof given by Bhaskara in Gaṇita- dhyāya under verses 1-6 in Bhagaṇopapatthi is as follows. Draw a circle on a horizontal plane and place a vertical pole called gnomon at the centre of the circle. Observe the point of intersection of the gnomon's shadow with the circumference of the circle at Sun-rise on a day in the Uttarāyaṇa ie during the course of the Sun’s north-word journey, just at the time when his rising point is very near the east point and also to the south thereof. Then from

that day go on counting the number of days, which will be 365 when the Sun again rises very nearly at the same point and just to the south of the east point. It will be found that the Sun will rise the next day just a little to the north of the east point, which means that the Sun has taken 365 days and a fractional part of a day to complete his revolution round the stars. Having noted the two points of intersection of the gnomonic shadow with the circle, on those two consecutive days when the Sun happens to rise just a little to the south and then on the next day just a little north of the east point, and having measured the arcs in minutes between those points of intersection and the western point of the horizontal circle (western because the gnomonic shadow of the rising Sun is cast towards west) then the following rule of three is to be applied. If during 60 ghatis of the day, the sum of the arcs in minutes is covered, what will be the time taken by the shadow to traverse the arc between the west point and the northern point of intersection. This added to 365, gives the number of civil days in an year. Here it will be noted that this year is tropical because rising in the east signifies the Sun's position at an equinox. Verse 8. The number of solar days in a kalpa is equal to 1555200000000 and of the lunar days or tithis is 1602999000000. Comm. The solar days here cited are counted at the rate of 360 per year; and the lunar days at the rate of 30 per lunation. Tithi is defined as mentioned by us under verse 31 of the previous section. Since there are thirty tithis in a lunation, their enumeration is quite alright; but there seems to be a little oddity in saying that there are 360 solar days in an year. This kind of a solar day is a little longer than a civil day and does not correspond to any particular motion of the Sun, say for example the time taken by the Sun to move a degree along the ecliptic. The definition of solar days pertains only to a stipulation

24 that 360 solar days constitute a solar year and no defini- tion is given for a single solar day. This definition of solar days, though apparently artificial, has some signi- ficance, namely that the difference of the solar and lunar days defined above constitutes 159330000000 Adhikamāsas in a kalpa at the rate of 30 tithis per month. In other words the difference between the solar and lunar days, is the number of tithis that the luni-solar reckoning gains over the solar. This topic will be dealt with later. Verse 9. The number of civil days in a kalpa is equal to 1577916450000; the number of the diurnal revolu- tions of the stars minus the number of sidereal revolutions of any particular planet constitute the days of that parti- cular planet with respect to the earth. Comm. The civil days in a kalpa are evidently the number of Sun-rises. These are as mentioned before the difference of the number of diurnal revolutions of the stars and the number of the sidereal revolutions of the Sun. By analogy, the number of the days of a particular planet with respect to the earth or what is the same the number of risings of that planet in a kalpa as seen from the earth, is the difference of the number of the diurnal revolutions of the stars and the number of the sidereal revolutions of that planet in a kalpa. Thus we have Saura-Ku-dināni, Chāndra-Ku-dināni, Bhauma-Ku-dināni etc, where the word ‘ Ku ’ means the earth. The Saura-Ku-dināni are the civil days defined before. It will be noted that the Chāndra-Ku-dināni are not the lunar days. Verse 10. The number of Adhikamāsās or intercalary months in a kalpa is equal to 1593300000 and the number of Dina-Kshayas is 25082550000. Comm. A luni-solar year as per Cāndramāna ie the luni-solar reckoning consists of twelve lunations; as such its length falls short of that of the solar year by 11 days 3 ghatis, 52 Vighatis and 30 Sukshmaghatis. If both the

25 .

solar and luni-solar years begin simultaneously this year, by the end of the luni-solar year, it will have gained over the solar year the above-mentioned days. This difference which goes by the name Adhimasa-Sesha or Suddhi accrues to the length of a lunation in about 32½ solar months. Unless this accruing difference is set apart by some device, and the beginnings of the two years again brought together, the luni-Solar year looses its significance of an year, for, it does not accord with seasons. It being a con- vention that an year should begin with the spring, if the luni-Solar year also is to begin with the spring, it is to be wedded to the solar year by some device. The device adopted in this behalf was to leave out a month in the luni- Solar reckoning as soon as it could be seen that a month has been gained by this reckoning over the solar. This knowledge is had from the following fact. The zodiac is divided into twelve equal portions called Rasi's beginning from the first point of the Hindu Zodiac, ie. from the first point of the asterism division called Aswini. The Sun traverses each of these Rasis in one Solar month, and on account of the unequal motion of the Sun, these solar months are of unequal length. The entrance of the Sun from one Rasi into another is called a Samkrānti, and the moments of Samkrāntis are held to be holy for religious pur- poses. The solar month being a little longer than a luna- tion, normally a Samkrānti occurs in a lunation. But it so happens that in a particular lunar month this Samkrānti might not occur. The Suddhi which is the time between the moment of New Moon and the subsequent Samkrānti and which is therefore the time gained by the luni-Solar reckoning over the Solar, having accrued to a lunation, it is an indication that the luni-Solar reckoning has gained a lunation over the Solar. That lunation not carrying a Samkrānti is termed an Adhikamasa and is left out. That it is left out is connoted by the word Adhika which means extra, as well as by the convention that no auspicious celebrations like a marriage etc. should not take place 4