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

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

26 during that month. The next lunar month is termed as the Nija-māsa or the month which is the true. In this convention however, there may occur two Saṁkrāntis in one particular lunar month. For this to happen two cri- teria are to be satisfied namely that, that particular lunar month must be greater than the concurrent solar month and secondly one of the two Saṁkrāntis must occur imme- diately after a New-moon so as to permit the second Saṁk- rānti to occur just before the lapse of the lunar month. The Sun coming to his perigee roughly about the lunar month Mārgasira, and as such having the quickest motion at that point, he covers the length of the Rasi of 30° within the shortest span of time making that Solar month shorter than the corresponding lunar month. Since the months Kārtica and pausha, one before and the other after Mārga- sira, also being thus longer than the corresponding solar months, two Sankrāntis could occur, if at all, in these three lunar months. If they occur, that lunar month is termed a Kshayamāsa. Since two lunar months are to correspond to two Saṁkrāntis, convention has it that two lunar months lapse simultaneously during that lunation, the previous lunation running during the forenoon and the subsequent lunation during the afternoon. Thus a Kshaya māsa is also called a yugalībhūta māsa or a twin māsa, which therefore makes one lunation virtually vanish. On account of this vanishing’, it is called a Kshayamāsa. After what an inter- val of time this Kshaya māsa occurs how a month precedes it having no Saṁkrānti and how again a lunation follows it without a Saṁkrānti will be seen in a subsequent context. This convention pertaining to the institution of an Adhikamāsa goes by the name ‘Inter-calation’. It was mentioned before that on an average an Adhikamāsa occurs once in 32½ solar months roughly–Also, Adhikamāsas are the excess of lunations over the solar months in a given period. There being 51840000000 solar months in a Kalpa and 53433300000 lunations the number of Adhikamāsas is therefore 1593300000 as mentioned,

27 Kshayāhas were explained before and their number in a Kalpa is the difference of the civil days and the tithis. Verses 11, 12. The numbr of solar months in a Kalpa is 51840000000; the number of lunar months is 53433300000. The number of solar months being subtracted from the number of lunations, we have the number of Adhikamāsas—The number of solar days together with the days of Adhika months are equal to the lunar days or the tithis; or again the lunar days minus the Kshayāhas are equal to the number of civil days or the reverse will be had by a reverse process. Comm. Already explained. Verse 13. The excess of the sidereal revolutions of the moon over the number of the Sun’s sidereal revolutions is equal to the number of chāndramāsas or lunations. Or again the excess of the sum of the sidereal revolu- tions of the moon and the tithis over the sum of the luna- tions and the diurnal revolutions of the stars is equal to the number of Kshayāhas. Comm. The first part is clear. Regarding the second, let the number of lunations be x and the number of the sidereal revolutions of the moon be y. Then y —x =z, the number of the sidereal revolutions of the Sun because y — z = x from the first part above. If now, the number of the diurnal revolutions of the stars be t, then t — z = t — (y — x) = t + x — y = civil days. Subtracting these civil days from ‘U’ the number of Tithis, we have the number of Kshayāhas namely U — (t + x — y) = (U + y) — (t + x) which accords with the statement. Verse 14. The number of Adhikamāsas is equal to the excess of the number of sidereal revolutions of the Moon over thirteen times the number of sidereal revolutions of the Sun.

28 Comm. Let x and y be the numbers of sidereal revolu- tions of the Moon and the Sun respectively. The x—y is evidently the number of lunations as mentioned before. Also 12 y is the number of solar months which if subtracted from the number of lunations will give the number of Adhimāsas ie x — y — 12 y = x — 13 y = Adhikamāsas as mentioned. This completes the Bhaganādhyāya of Madhyādhi- kāra.

GRAHĀNAYANĀDHYĀYA ग्रहानयनाध्यायः Verse 1. To Compute the Ahargaṇa the collection of days from the beginning of Kalpa ie from the beginning of creation. Multiply the number of sidereal solar years from the beginning of Kalpa by 12; add the number of elapsed lunar months ; multiply by thirty ; add the number of elapsed tithis. Let this number be x. Then [ (x × A) / s ] the integral number obtained by dividing the product of x and A the number of Adhikamāsas in a Kalpa by s the number of solar days thereof gives the number of elapsed Adhikamāsas. Multiply this number of Adhikamāsas by 30 and add to x. The result gives the number of elapsed tithis, Let this be y. Then [ (y × K) / T ] the integral num- ber obtained by dividing the product of y and K the num- ber of Kshayāhas in a Kalpa by T the number of Tithis in a Kalpa, gives the number of Kshayāhas. Subtracting this number from y, we have the Ahargaṇa ie the number of the elapsed civil days from the beginning of Kalpa. This Ahargaṇa has its beginning on Sunday and is itself consti- tuted of mean solar days. While computing the Adhika- māsas or Kshayāhas, the integral numbers of the quosients alone should be taken rejecting the remainders. Comm. The elapsed number of solar years is directed to be multiplied by 12 in the beginning. This number does not constitute purely solar years. Solarity was secured upto the point of the last intercalation of an Adhi- kamāsa and thereafter one or two luni -solar years would have been added. But Construing these one or two luni- solar years as mean solar years does not make a difference while computing the Adhikamāsas, for the following reason. Adhikamāsas normally occur once in three years. Even

30 supposing that the number of the elapsed years contain three luni-solar years after the year carrying the last inter- calary month, the error in construing them to be solar will be roughly minus one solar month. In other words the difference between three mean solar years and three luni- solar years will be roughly one solar month. Since one Adhikamāsa occurs roughly in 32½ solar months, so the number of Adhikamasas obtained by construing three luni- solar years as mean Solar years will be in default by roughly 1/32½ or 2/63. As we are counting only the integral number of Adhikamāsas obtained as a quotient rejecting the remainder, the above default of 2/63 should not normally effect the quotient. Most rarely, however, it might effect the quotient by one, for which provision is made by Bhas- kara in verse 3 under Adhimāsādinirṇaya section, Madhya- mādhikara namely स्पष्टोऽधिमासः पतितोपलब्धः etc. which means that if the quotient is in default by one, where it is definitely known that one more Adhikamāsa did occur, we are directed to add one and if we certainly know that the quotient contains one more Adhikamāsa, when the Adhikamāsa is shortly to occur and has not occrued we are directed to substract one from the quotient. This principle of सैकत्वनिरेकत्व is to be observed even in the context of Kshayāhas ie we are directed to add one or subtract one from the number of civil days by adjusting the Ahargaṇa to the week-day on which the Ahargaṇa is sought to be found. Thus construing the elapsed number of years to be solar, multiplying them by twelve we have the elapsed solar months. Adding to this the number of the elapsed luni-Solar months construing these to be solar, which fact also does not affect normally the number of Adhikamāsas to be obtained (for the same reason above) and multiplying by 30 and adding the elapsed tithis we have the elapsed num- ber of solar days (This number may not be exactly the

31 elapsed number of solar days, for, we have construed the one, two or three luni-Solar years as mean solar as well as the elapsed lunations of the present year as solar months; but this difference does not affect the computation of Adhika- māsas as mentioned above). Then, as we are given that 15933,00000 Adhikamāsas occur in 1555200000000 solar days of the Kalpa, if x be the number of solar days found above, (x ✕ 1593300000) / 1555200000000 will give the number of the elapsed Adhikamāsas. Adding these Adhikamāsas multiplied by 30, to x the solar days obtained above, we have the Tithis elap- sed upto the day in question. If now, we subtract the Kshayāhās from these Tithis, we shall have the number of Sāvanāhas or the Ahargaṇa required. Since in 1602999000000 Tithis of the Kalpa there will be 25082550000 Kshayāhas, if y be the Tithis above obtained, (y ✕ 25082550000) / 1602999000000 will be the Kshayāhās from the beginning of the Kalpa upto the day in question ; subtracting these from y, we have the required Ahargaṇa. Verse 4. Computation of the planetary positions. The Ahargaṇa multiplied by the number of sidereal revolutions of a planet and divided by the number of civil days in a Kalpa gives the planet ie its number of revolu- tions upto the day concerned both integral and fractional. Comm. Applying ‘Rule of three’, if in C the number of civil days in the Kalpa, the planet makes P sidereal revolutions how many revolutions would have been made in A the Ahargaṇa ? The answer is (A ✕ P) / C. In this, the integral quotient gives the number of complete revolu- tions made; the remainder, multiplied by 12 and divided by C again, gives the number of Rasis covered by the planet from the Zero-point of the Zodiac, and again the remainder multiplied by 30 and divided by C gives the number of deg-

32 rees covered in the next Rasi; proceeding thus, the planetary position could be had next in minutes and then in seconds of arc also. This position is the mean position of the planet, at the time when the mean Sun is very nearly at the Eastern horizon at Lanka. Why it is said "very nearly at the horizon" will be clear in the context of uda- yāntara Samskāra to be dealt with later in Spaṣṭādhikāra. To obtain the planetary position at the moment when the mean Sun is exactly on the horizon, we have to apply what is called the Udayāntara correction and again to obtain the position at the moment of True Sun-rise we have to apply what is called the Bhujāntara correction, both of which will be dealt with in Spaṣṭādhikara. Having thus got the mean planetary position at True Sun-rise, we have to apply one or two as the case may be, corrections to obtain the True planetary position, besides effecting two more correc- tions known as Desāntara and chara to obtain the True planetary position at the time of True Sun-rise not at Lanka but at the place concerned. Verse 5. To obtain the position of the mean Moon, when the mean Sun is known from what is called Avama- Seṣa. The Avama Seṣa divided by 131490000000 in degrees is to be added to twelve times the elapsed tithis and the result added to the Sun's position gives the position of the Moon. Conversely the position of the Sun can be had from the position of the Moon. Comm. The moon's longitude minus the Sun's longi- tude known as elongation is called Vyarkēndu, which divided by twelve gives the number of elapsed tithis; a lunation which is the time in which the Moon overtakes the Sun by 360°, contains thirty tithis and a tithi is defined as the time, in which the Moon overtakes the Sun by 12°, beginning from the moment of conjunction ie Amāvāsyā. Hence if the Sun's longitude be x°, aud y be the number of

33 elapsed tithis, integral or fractional (x + 12y)° will be the longitude of the Moon. If, however, we consider y only as the integral number of the elapsed tithis, we will have obtained the Moon’s position from the formula (x + 12 y)° at the ending moment of the tithi on the previous day. The time in between this ending moment of the tithi and the Sun-rise of the day concerned is known as Avama- Sesha, since the Avama-days or Kshayāhas are the differ- ence of days between the number of tithis in a given period and the number of civil days during the same period. In other words, the excess of a civil day over a tithi which falls short of it, is the part of a civil day that contributes towards the number of Kshayāhās in a given period. Thus we have to compute the increase in the Moon's longitude during the aforesaid Avama-Sesha to obtain his longitude at the Sun-rise of the day concern- ed. But this Avama-Sesha is of the form F { (t × K) / T } where ‘ t ’ is the number of elapsed tithis upto the Sun-rise of the day concerned K the number of Kshayāhas in a Kalpa, T is the number of tithis in a Kalpa and F { (t × K) / T } signifies the fractional part called Avama-Sesha, the integral quotient having given the number of elapsed Kshayāhas. Hence writing F { (t × K) / T } as R / T where R is the remainder obtained by dividing (t × K) by T, the Avama-Sesha is of the form R / T. This Avama-Sesha being mean solar. it has to be rendered luni-Solar, which process is known as चान्द्रीकरणम्. The rule of three used in this behalf is ‘ If for C civil days we have T tithis of the Kalpa, what shall we have for R" / T ? The answer is (R / T) × (T / C) = R / C. This then is the balance fractional part of a tithi that is there in between the end of the elapsed tithi and the Sun-rise of the day concerned. This part of a tithi is to be multiplied by 5

34 12 to give in degrees the increase of Moon's longitude from the ending moment of the tithi on the previous day. Thus this increase is R/C × 12 ; but C = 1577916450000 ∴ R/C × 12 = R / (1577916450000 / 12) = R / 131490000000 approximately. Thus this increase is to be added to (x + 12 y)° where x° is the longitude of the Sun and y the integral part of the elapsed tithis so that the Moon's longitude is (x + 12 y)° + R° / 131490000000˙ Verses 6, 7. Computation of the positions of the Sun and the Moon from the Adhimāsa-Sesha and Avama-Sesha. The Avama-Sesha divided by 27110000000 is termed an additive constant in minutes of arc to the Sun’s position ; the same Avama-Sesha multiplied by 13 and divided by 35 is termed such an additive constant to the position of the Moon ; Construe that the Sun's position is given by as many degrees as there are elapsed tithis after the beginning of Chaitra and that the Moon's position is given by Thir- teen times the same. Let these positions of the Sun and the Moon be diminished by a number of degrees equal to what is obtained by dividing the Adhimāsa-Sesha by the number of lunations in a Kalpa. Then add the respective additive constants to the positions of the Sun and the Moon so obtained. The results will be the positions of the Mean Sun and the Mean Moon. Comm. Here the data are the Adhimāsa-Sesha and the Avama-Sesha and nothing else and the problem set is to find the Mean positions of the Sun and Moon. The Adhimāsa-Sesha is of the form (R' × 30) / S where R' is the remainder obtained while finding the elapsed Adhimāsas

35 by dividing the product of the number of lunations in a Kalpa and (12 x + y) 30 + t where x is the number of elapsed years, y the number of elapsed lunations t the number of elapsed tithis in the current lunar month at the time when the Ahargaṇa is being computed and S the number of Solar days in a Kalpa. The form of the Avama-Sesha was formerly stated as R / T. The procedure given is as follows. In the first place, we are asked to assume that the number of elapsed solar days is equal to the number of elapsed tithis from the beginning of Chaitra ie the beginning of the luni-Solar year, since we do not know when the Solar year began; so, at the rate of 1° per a Solar day (A Solar day is not a mean solar day, Vide definition given before under verse 8. Bhagaṇādh- yāya) the Sun’s position is given by as many degrees as there are elapsed tithis; and the Moon’s position must be 13 times the same since each tithi means an increase of 12° in the elougation and if x° be the Sun’s position, the Moon’s position must be 12 x° ahead of the Sun ie 13 x°. Having got thus approximate positions of the Sun and the Moon, we have to make amends for the roughness of the assumption made. In assuming that the longitude of the Sun is equal to the number of elapsed tithis, we have over- estimated the longitude since it has to begin from the beginning of the solar year. The time in between the beginnings of the luni-Solar year and the solar year is known as the Adhimāsa-Sesha at the time of the beginning of the solar year, measured in tithis; also, we have com- mitted an error in assuming the Sun’s rate to be 1° per tithi. This also is due to Adhimāsa-Sesha subsequent to the beginning of the Solar year upto the day concerned- Thus the entire error committed by assuming the Sun’s longitude to be as many degrees as there are elapsed tithis from the beginning of the luni-Solar year. is no other than the Adhi- māsa-Sesha at the day concerned, But this Adhimāsa- Sesha is of the form (R′ × 30) / S as mentioned and is in tithis.

36 This has to be converted into solar days to give us the number of degrees of the error. The conversion is effected by the rule of three as “If T tithis of a Kalpa constitute S solar days of the Kalpa, what number of solar days corres- ponds to (R' × 30) / S ?” The answer is ((R' × 30) / S) × (S / T) = (R' × 30) / T = R' / (T/30) = R' / L where L is the number of lunations in a Kalpa. This R' / L being the number of solar days corresponding to the Adhimāsa-Sesha upto the day concerned, the corresponding longitude of the Sun namely R'° / L must be subtracted from the Sun’s longitude. Now the question arises whether we have to multiply this R''° / L by 13 to be subtracted from the longitude of the Moon; not necessary, enough to subtract R'° / L only, for, the Adhimāsa- sesha extends from the preceeding Amāvāsya only when the Moon’s longitude was equal to the Sun’s longitude. We have thus got the mean positions of the Sun and the Moon at the ending moment of the tithi on the previous day. To get their mean positions at the Sun-rise of the concerned day, we have to make amends for the time in between, which is no other than Avama-Sesha as a fraction of a mean solar day. Here Bhaskara makes an ingenious approximation. The maximum Avama-Sesha could be a tithi only and the Sun moves roughly by his daily mean motion during a tithi. So, the rule of three adopted is ‘If for one tithi, the Sun’s daily motion is to be reckoned, what for the Avama-Sesha? The answer is Avama-Sesha multiplied by the Sun’s daily motion. The Avama-Sesha being of the form R / T, (R / T) × m' = (R / T) × 59 (8' / 60) =

37 R' R' R' ——— = ——————————————————————— = ——————————— very T 1602999000000 × 15 27110000000 ———— —————————————————— 59 2/15 887 approximately. This must be added to the position of the Sun to get his position at the Sun-rise concerned. In the case of the Moon the above additive constant of the Sun, is to be multiplied by 13 13/33 because the Moon's daily motion is so many times that of the Sun. Hence the additive constant in the case of the Moon is x × 13 13/33 = 13 × (1 + 1/33) where x is the additive constant in the case of the Sun. This agrees with what Bhaskara has stated. Verses 8, 9. Another way of computing the planetary positions. The mean position of the Sun in R

38 we have f' − (A × G) / M = I' + F where I' is some integer other than I. But (A × D) / M which gives us the number of diurnal rotations of the stars upto the day concerned is equal to the Ahargaṇa plus the number of revolutions of the Sun upto the day concerned, because Ahargaṇa + Revolutions of the Sun = diurnal rotations of the stars upto the day concerned. Omitting the integral Ahargaṇa and the integral number of revolutions of the Sun we have that the fractional part of (A × D) / M ie f' is no other than the Sun's position. Hence. (A × P) / M = I + F = f' − (A × G) / M and (A × G) / M = (A × G) / 1577916450000 revolutions = (12 R) / 1577916450000 Rasis = R / 131493037500 Rasis. Thus the planetary position is equal to the Sun's position minus (A × G) / 131493037500 where the integral parts on either side could be ignored. Here there is a peculiarity in this method. The planet could right away be obtained from the formula (A × P) / M where A is the Ahargaṇa, P the number of sidereal revolu- tions of the planet and M the number of mean Solar days in a Kalpa. Though the given method is more cumbrous than finding through the above formula, Bhaskara delibe- rately gives it to show the equivalence of various proce- dures at the same time giving us a beautiful technique as mentioned in his commentary. Thus in the above equation (A × P) / M = (A × D) / M − (A × G) / M the first term on the right hand side could be termed the Bha-bhrama-graha and the second term (A × G) / M the graha-Sāvana-Dina-graha. We

39 have seen above that the first term is no other than the Mean Sun ignoring the integral number. Verses 10, 11. Proof of other methods of computing planetary position. Even as the sums or differences of two or more of the numbers of Adhimāsas, Kshayāhas, lunations etc give the number of sidereal revolutions of the planets the sums or differences of two or more of the posi- tions of the imaginary planets which go by the names Adhimāsa-graha, Kshayāhagraha etc computed out of the numbers of those Adhimāsas Kshayāhas etc. give the respective planetary positions. Comm. This interesting concept is based upon the following principle. Suppose P to be the number of side- real revolutions of a planet; then (A × P) / M gives the planetary position, where A is the Ahargaṇa, and M the number of mean solar days in a Kalpa. Now suppose P = x ± y ± z where x, y, z are the numbers of Adhimāsās etc in a Kalpa, then (A × P) / M = (A × x) / M ± (A × y) / M ± (A × z) / M • The terms on the right-hand-side may be construed to be the Adhimāsa-graha etc, which are the positions of imaginary planets and their sums or differences give there- fore the position of the planet, as could be seen from the above equation. Thus for example, we have the equation P₁ — 13P₂ = a where P₁ is the number of the sidereal revolutions of the Moon and P₂ the number of the sidereal revolutions of the Sun, a stands for the Adhimāsas because Chāndramāsas — Sauramāsas = Adhimāsas; but Chāndra- māsas = P₁ — P₂ and Sauramāsas = 12 P₂ so that P₁ — P₂ — 12P₂ = a ie P₁ — 13P₂ = a. From this equation P₁ = a + 13P₂ ∴ (A × P₁) / M = (a × A) / M + (13P₂ × A) / M •

40 The first term on the right hand side is termed as the Adhimāsa-graha, and the second term is evidently 13 times the position of the Sun whereas the term on the left-hand- side is the Moon’s position. Hence, ignoring the integral number of revolutions, we have Moon’s position = Adhi- māsa — graha + 13 times Sun’s position (Ignoring the number of integral revolutions means subtracting integral revolutions or adding them if necessary). Verses 12, 13. A few more examples on the afore- said principle. The planetary position obtained by the sum of the sidereal revolutions of two planets, added to or Subtracted from another planetary position obtained by the difference of the sidereal revolutions of two planets and divided by two gives the positions of the two planets res- pectively, the quicker of the two in the first case and the slower in the second. Similarly the planetary position com- puted from the difference of the sidereal revolutions of two planets subtracted from the planetary position of the quicker of the two gives the position of the slower whereas the former planetary position added to the position of the slower gives the quicker. Comm. We have [(P₁ + P₂) + (P₁ − P₂)] / 2 = P₁ (1) and [(P₁ + P₂) − (P₁ − P₂)] / 2 = P₂ (2) where P₁ is the number of sidereal revolutions of a quick-moving planet and P₂ that of a slow-moving one. Multiplying the above equa- tions by A/M with the former notation, [A/M (P₁ + P₂) + A/M (P₁ − P₂)] / 2 = (P₁ × A) / M (3) and [A/M (P₁ + P₂) − A/M (P₁ − P₂)] / 2 = A/M × P₂ (4) Equations (3) and (4) mean what has been stated in verse (12). Again we have the equations P₁ − (