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

47 Fig. 1 ved from a place A on the primary meridian going through Lanka, Ujjain, Kerukshetra etc by an instrumeut (A pro- tractor) described by Bhaskara under verse 5 Chandragraha- ṇādhikāra, grahagaṇita at the time of transitting. Let B be a sublunar point on the earth at the moment of that obser- vation, where B is also on the same primary meridian. Since both the places happen to be on the primary meridian, and such places were primarily known, both the observa- tions could be made simultaneously. Knowing the distance A B between the two places, the angle θ subtended by A B could be easily got from the triangle A O M where O is the earth's centre and M the Moon. As a first approximation, d / Sin θ = (d + a) / Sin Z = a / (Sin Z − Sin θ) taking O M roughly to be equal to (a + d). In the above equation, a being known

48 d = a Sin θ / (Sin Z - (Sin θ)). Strictly speaking d / Sin θ = a / Sin (Z - θ) so that d = a Sin θ / Sin (Z - θ). Since (Sin Z - Sin θ) < Sin (Z - θ) ∴ the estimate of d obtained is a little greater than its true value. We shall discuss other possible methods of finding the Moon's distance in the chapter on lunar eclipses. Having got the distance of the Moon as afore-said, it was easy to obtain the angle ∝ marked in fig. 1 for, Sin ∝ = M₁ M¹ / O M¹ = a / d = 2πa / 2πd = 4967 / 324000 Since ∝ is small Sin ∝ = ∝ radians. ∴ ∝ = (4967 × 3438) / 324000 = (4967 × 191) / 18000 = 52.7' Since the Moon's daily average motion 790' — 35" is had in sixty Nadis, 52.7' of motion is covered in (52.7 × 60) / 790.5 approximately = 60 / 15 = 4 Nadis. Thus the fact that we are given the horizontal parallax as 4 Nadis in the context of lunar eclipses is based on this. Having obtained thus the distance of the Moon from the centre of the earth E, and having measured the angular diameter of the Moon's disc with the help of the protractor mentioned above, from the triangle EAM (Ref. fig. 2) where [चित्र: Fig. 2 - त्रिभुज EAM जिसमें E पर कोण β, भुजा EM = d, बिन्दु M चन्द्रमा का केन्द्र, AM = r' त्रिज्या, और स्पर्शरेखा EA है।] Fig. 2

49 εA is a tangent to the Moon's disc and M the centre of the Moon, Sin B = r' / d where B is the angular semi-dia- meter of the Moon's disc, r' = the spherical radius of the Moon's disc measured in yojanas, so that since Sin B = B' in Hindu trigonometry when the angle is small, 2πr' / 2πd = 2πr' / 324000 = 16 / 3438 ∴ 2πr' = (324000 × 16) / 3438 ; hence r' = (18000 × 16) / 191 × 7 / 44 = 504000 / 2101 = 240 yojanas. Bhaskara gives the semi-diameter of the Moon's disc to be 16'-0"-9"' so that the above r' is very approximately 240 yojanas as given by Bhaskara. Now with these constants pertaining to the Moon's distance, and his spherical radius, it was sought to find the spatial distance traversed by the Moon during a day. The rule of three used was " If 16'-0"-19"' of the Moon's angu- lar semi-diameter pertains to a spatial distance of 240 yoja- nas at his orbit, what does the mean daily motion of 790'-35" pertain to ? " The answer is 683064000 / 57609 = 11858¾ yojanas as given by Bhaskara elsewhere or taking the Moon's mean semi-angular diameter to be 16' only, and using the rule of three " If 16' at the lunar orbit correspond to 240 yojanas, what do 790'-35" correspond to " we have (790'-35" × 240) / 16' = 15 × 790 7/12 = 5/4 × 9487 = 9487 + 2371¾ = 11858¾ yojanas as given by Bhaskara. Thus having obtained the daily spatial motion, the Hindu Astronomers assumed all the other planets (includ- ing the Sun) to have the same daily mean spatial motion. On this assumption since the, Moon's orbit will be 27.3217 × 11858¾ = 324000 yojanas, and the circumference of the universe will be 324000 × 57753300000 = 18712069200000000 yojanas the Sun's orbit will be 7

50 circumference of the universe —————————————————————————————— = 4331497½ yojanas Number of sidereal revolutions (because (324000 × 57753300000) / 4320000000 = (3240 / 432) × 577533 = (15 / 2) × 577533 = 8662995 / 2 = 4331497½). Also, the circumference of the stellar universe was presumed to be sixty times that of the Sun’s orbit. With the constants obtained for the Moon’s diameter and distance and with assumption that all the planets have the same daily motion, the constants pertaining to the Sun and the other planets were obtained. We shall resume this topic in another context. Verse 6. The mean daily motion of the planets. The circumference of the universe divided by the num- ber of days in the Kalpa, gives the daily spatial motion of a planet. The planets move thus a distance of 11858¾ yojanas in a day. Comm. Already explained. Verse 7, and half the verse 8. The Ahargaṇa multiplied by 11859 decreased by the quotient obtained by dividing the product of the Ahargaṇa and 9921 by 35419 gives the distance covered by a planet in yojanas. These yojanas divided by the circumference of the planet’s orbit gives the fraction of a revolution and the integral number of revolu- tions made. Comm. Let the Ahargaṇa be A; every planet should have described a space equal to A × D where D is the dis- tance traversed per day and is just a little less than 11859, or more correctly should have described a space equal to A × (C / c) where C is the afore-said circumference of the

51 universe, and c the number of civil days or mean solar days in a Kalpa. ' For the sake of an easy computation Bhaskara gives here an interpolation. In the first place, we are asked to multiply A by 11859 by which, an excess is there in the result; then this excess is sought to be removed. The excess is A × 11859 - (A × C) / c because A × C / c is the correct distance described. In other words, the distance described by any planet is (A × C) / c = A × 11859 - (A × 11859 - (A × C) / c) I But C / c = 18712069200000000 / 1577916450000 = 420024000 / 35419 dividing by the common factor 44550000 ∴ The space described by any planet is from I A × 11859 - A { (c × 11859 - C) / c } = A × 11859 - A { (35419 × 11859 - 420024000) / 35419 } But the numerator within the brackets is 9921 ∴ Space described = A × 11859 - (A × 9921) / 35419 as stated. This distance divided by the individual orbital lengths, we have the integral number of revolutions made by each planet, rejecting which we have the fractional part of a revolution which gives the position of the planet. Verses 8, 9. The orbit of the planet itself is no doubt the orbit of the Mandoccha (apogee with respect to the Sun and the Moon and aphelion with respect to the other Star- planets ie Mars, Mercury, Jupiter, Venus and Saturn) and of the node (point of intersection of the orbits of the Star- planets with the ecliptic); but while Computing the posi- tions of these Mandocchas and the Nodes, as per the method indicated above ie according to the method of

52 Kakshādhyāya, their orbits are taken to differ from those of the planets. (because slow-moving, points will have longer orbits as per the assumption made namely that the circumference of the universe divided by the number of the sidereal revolutions gives the length of the orbit). Simi- larly the orbit of the Sun itself will be the orbit of Mercury and Venus, and the orbits of their S'īghrocchas are their real orbits wherein Mercury and Venus are taken to move with the velocity of the Sun. Comm. The prescription of the computation of a planetary position as per the method of Kakshādhyāya, has brought in an awkward situation. Let us consider the Computation of the positions of Mercury and Venus. These two planets oscillate about the mean position of the Sun, because their orbits happen to lie within the earth's orbit. Hence their mean sidereal periods coincide with that of the Sun, which means that their numbers of sidereal revolu- tions coincide with the number of sidereal revolutions of the Sun. Hence the Kakshādhyāya method of Computing a planetary position brings in the idea that the orbits of Mercury and Venus coincide with the orbit of the Sun. Bhāskara perceived the awkwardness of this situation and says therefore, the above coincidence of the orbits must not be taken to be a reality but is intended only for the sake of computation. The actual planets Mercury and Venus in fact revolve says Bhaskara in the orbits of their Sīghroc- chas, with the velocity of the Sun-Even this supposition that the actual planets move with the velocity of the Sun sounds odd, but this will be clarified later in the spashta- adhikāra, wherein we propose to explain the peculiar con- cept of S'īghroccha at length. Here ends the Grahānayanādhyāya according to the Kakshādhyāya method.

THE ADHYĀYA KNOWN AS PRATYABDA S'UDDHI IN MADHYĀDHIKĀRA Verse 1. The number of years which have elapsed from the beginning of the Kalpa, respectively multiplied by 2, 4 and 3 and divided by 8, gives what is called Dinādya in days, ghatis and Vighatis respectively. If this be added to the number of years and divided by seven, the remainder gives the Abdapa or the lord of the year, (under whose name the week-day of the commencement of the year stands). Comm. One mean solar year consists of 365 days, 15 ghatis, 30 palas, and 22½ Vipalas where the units are all mean solar and one mean solar day is equal to 60 mean solar ghatis, one mean solar ghati is equal to 60 mean solar palas, one mean solar pala is equal to 60 mean solar Vipalas and so on in sexagesimal sub-division. The fraction of the day over and above 365 days, namely 0-15-30-22-30 multiplied by 8 gives 2 days, 4 ghatis and 3 palas so that by the rule of three i.e. ‘If in 8 mean solar years, the fraction accrues to 2 days, 4 ghatis and 3 palas, what will it accrue to in x elapsed mean solar years from the beginning of the Kalpa ? we have the answer (x × 2)/8 days, (x × 4)/8 ghatis and (x × 3)/8 palas. If this is added to the number of the elapsed years, and the result divided by seven, the remainder gives the week-day of the commencement of the concerned year, be- cause the remainder got by dividing 365 by 7 is one, and the week day advances at the rate of one per year. Also the Kalpa began on Sun-day. Verse 2. Alternate method. Half the number of elapsed years added to 1/60 of itself, then divided by 60 and added to 1/4 of the elapsed years, gives the Dinādya.

54 Comm. Since the Dinādya per year is 0-15-30-22-30, in x elapsed years it accrues to x × 0-15-30-22-30 = (x × 15)/60 days + (x × 30)/60 ghatis + x × 22½ palas = x/4 days + x/2 g + (x × 45)/(2 × 60) palas = x/4 d + x/2 g + 3x/8 × 1/60 g = x/4 d + x/2 (1 + 1/80) g = x/4 d + (x/2 (1 + 1/80))/60 d which is the given formula. Verse 2. Alternative method. The number of elapsed years divided by respectively 4, 120, and 9600 and the Sum taken gives the Dinādya. Comm. Let x be the number of elapsed years. The Dinādya as before is x × 0-15-30-22-30 = x/4 d + (x × 30)/(60 × 60) d

• (x × 45)/(2 × 60) × d/(60 × 60) = (x/4 + x/120 + x/9600) d as given. Verse 3. To obtain what is known as Kshayāhādya. The number of elapsed Kshayāhās from the beginning of Kalpa upto the commencement of the year, is obtained as follows. Let x be the number of elapsed years; then x − (x (1 + 1/80) + 30x)/160 = Kshayāhās. Comm. The number of Kshayāhās in a Kalpa of 4320000000 solar years is 25082550000 so that per year their number is 5-48-22-7-30. In this 0-48-22-7-30 is said to be Kshayāhādya per year = 1 − (0-11-37-52-30) putting the quantity within the brackets into a fraction, 52½ vipalas = 105/2 × 1/60 = 7/8 palas; (37 + 7/8) palas = 101/8 × 1/60 ghatis = 101/160 ghatis ; 11 101/160 ghatis = 1861/(160 × 60) days

55 = 1/160 of 1861/60 days = 1/160 of 31 days, one ghati. Hence per year the Kshayāhādya is 1 − 1/160 of (31ᵈ - 1ᵍ) so that for x years it would be {x − x/160 (31ᵈ - 1ᵍ)} d = x − x/160 (31 1/60) = x − 1/160 { 30x + x + x/60 } = x − 1/160 { 30x + x (1 + 1/60) } which is the formula given. Verse 4. Alternative method. The Dinādya obtained before multiplied by three, is to be diminished by 1/400th of the number of years; the result increased by 1/30th of the number of years gives the number of Kshayāhās. Comm. The Dinādya pertaining to one year is 0-15-30-22-30 and the Kshayāhādya is 0-48-22-7-30. Multiply the former by 3 and subtract from the latter; we have 0-1-51 = 0-1-51/60 = 0-1-17/20 = 0-37/20 = 37/20 × 1/60 = 37/1200 day Hence K−3D = 37/1200 day where K = Kshayāhādya and D = Dinādya ∴ K = 3D + 37/1200 hence per x years it will be 3D × x + 37x/1200 = 3D × x + (40 − 3) x / 1200 = 3D × x + x/30 − x/400 which is the formula given. Verse 4 contd. Alternative method. Or else K = K/160 (1 − 1/60) + x(1 − 1/5) Comm. The Kshayāhādya for an year is 0-48-22-7-30 48 ghatis = (1 − 1/5) day; hence for x years x (1 − 1/5).

56 The remaining fraction = 0-0-22-7½ = 0-0-(22 + 15/2 × 1/60) = 0-0-22⅛ = 0 — 177/8 × 1/60 = 0 — 59/160 = (60 — 1) / (160 × 60) day = 1/160 — 1/(60 × 160) = 1/160 (1 — 1/60) ; hence for x years x/160 (1 — 1/60) Adding the two we have the required formula. Verse 5. To obtain the elapsed number of Adhikamā- sās and what is called Suddhi. The sum of the Dinādya, Kshayāhādya and ten times the number oi elapsed years divided by 30, gives the num- ber of the elapsed Adhikamāsas; the remainder is known as Suddhi if diminished by the fraction of the Kshayāhas. Comm. The number of mean solar days in an year is 365-15-30-22-30; the Kshayāhās in an year are 5-48-22-7-30; adding the two we have the number of tithis in an year equal to 371-3-52-30. The number of solar days being 360, the number of tithis which constitute the Adhimāsas is equal to 11-3-52-30. The sum of the Dinādya and Ksha- yāhādya in an year = 0-15-30-22-30 + 0-48-22-7-30 = 1-3-52-30. Hence the above number of tithis which cons- titute the Adhimāsas namely 11-3-52-30 = 10 + Sum of Dinādya and Kshayāhādya pertaining to an year. Hence for x years, the Adhimāsa days are equal to x × 10 + Sum of Dinādya and Kshāhādya for x years. These Adhimāsa days divided by 30 give the Adhimāsas, and the remainder is called Suddhi, so called because while finding the Abargaṇa from the beginning of a solar year, this remainder has to be subtracted. Why the fraction of Kshayahas, which is there in this Suddhi is prescribed to be subtracted, will be explai- ned in another context. Verse 6. An alternative method to find the Adhimā- sas. The number of years divided separately by 32 and 30,

57 the sum increased by the number of years multiplied by eleven and the result divided by 30 gives the number of elapsed Adhimāsas. The remainder diminished by the fraction of Kshayāhās as mentioned before, is the S'uddhi. Comm. Every year, the number of Tithis that consti- tute the Adhimāsās accruing in the course of years, is 11-3-52-30. The fractional part of this namely 0-3-52-30 = 0-3-52½ = 0 — 3 + 105/(2×60) = 0 — 3⅞ = 0 — 31/8 = 31/8 × 1/60 = 31/480 = (16+15)/480 = 1/30 + 1/32 converted into days. Hence for x years x/30 + x/32 . Adding the number of days 11x, we have 11x + x/30 + x/32 as stated. Dividing these days by 30, we have the number of elapsed Adhimāsas and the remainder is S'uddhi, if the fraction of the Kshayāhās is subtracted therefrom as indicated. Verse 7. Method of finding the lord of the year with- out a knowledge of Dinādya. The week-day at the Commencement of the Solar year, which pertains to the lord of the year, is the remain- der got by dividing by seven the S'uddhi which is itself diminished by the remainders got by dividing by seven separately firstly double the excess of the elapsed years over the elapsed Adhikamāsās and secondly the elapsed Ksha- yāhās. This may be put in symbols as R₇ { S—R₇ (2y—A) — R₇. K. } = R₇ { S' — R₇ (2y — 2A + K) } where R₇ signi- fies the remainder got by dividing by seven the quantity which follows, S signifies S'uddhi, y means the elapsed Solar years A the elapsed Adhikamāsās and K the elapsed Kshayāhās. 8

58 Comm. The lord of the year is the lord of the week- day at the Commencement of the solar year. To get this week-day we have if the number of Tithis elapsed upto the beginning of the luni-solar year be T, the Suddhi S', the Kshayāhās upto the Commencement of the solar year be K, then since the number of civil days is equal to T + K, R₇ (civil days) = R₇ (T + S — K). But T = 360y + 30A where y is the number of elapsed solar years and A the number of elapsed Adhikamāsās. ∴ R₇ (T) = R₇ (360y + 30A) = R₇ (3y + 2A) since the remainder got by dividing 360 and 30 by seven are respectively 3 and 2. Hence R₇ (civil days = Ahargaṇa) = R₇ (3y + 2A) + R₇S' — R₇K. The number of Kshayāhās K = y (5-48-22-7-30) = 5y + y (0-48-22-7-30). But y (0-48-22-7-30) = K where K is the previously calcu- lated Kshayāhādya. Hence R₇K = R₇5y + R₇K. Substi- tuting this in the above R₇ (civil days) = R₇ (3y + 2A) + R₇S' — R₇ (5y + K) = R₇S' — R₇ (5y — 3y) + R₇ (2A) — R₇K = R₇ [S' — R₇ (2y — 2A) — R₇K ] which is the given formula = R₇ [S — R₇ (2y — 2A + K) ]. Verse 8. To obtain the fractional part of the elapsed Kshayāhās even without a knowledge of the number of Kshayāhās. The excess of the ghatis of the Adhimāsa Sesha ie the remainder got by dividing the sum of the Dinādya, Ksha- yāhādya and ten times the number of elapsed years by 30 as indicated in verse 5, over the ghatis of the Dinādya, obtai- ned under verse 1 gives the fractional part of the Kshayāhās. Comm. The ghatis or the fractional part of the Adhi- māsa Sesha is the-sum of the fractional parts of Dinādha and the Kshayāhādya so that the [excess of the ghatis of the Adhimāsa Sesha over the ghatis or the fractional part of the Dinādya gives the ghatis or [the fractional part of the Kshayāhās. Verse 9. The planetary positions at the end of the elapsed solar year.

59 The number of the elapsed solar years multiplied by the number of sidereal revolutions of the respective planets in a Kalpa and divided by the number of Solar years in a Kalpa gives the planetary positions at the end of the last Solar year. Comm. This is a simple rule of three. The plane- tary positions so got, leaving out the integral numbers of revolutions made which are not required, are called the Dhruvakas of the respective planets for the ensuing solar year. With respect to the apogee of the Sun and the aphelia and nodes of the star—planets these Dhruvakas themselves give their positions for the whole ensuing year, as their motion is very very slow. Verse 10. An alternative method of obtaining the Dhruvaka of the Moon. The Adhimāsa Sesha multiplied by 12 gives the posi- tion of the Moon at the Commencement of the Solar year. Comm. Since the position of the Sun is at the Zero- point of the Zodiac at the Commencement of the solar year, and since the Adhimāsa Sesha is the difference of the solar and luni-Solar systems of reckoning, or what is the same, the arc gained by the Moon over the Sun, which is no other than the elongation of the Moon measured in Tithis each tithi being of 12° gain of elongation, so the Adhimāsa Sesha at the Commencement of the solar year in Tithis multiplied by 12 gives the longitude of the Moon at that Commencement. (Note - Bhaskara waxes into poetic eloquence in the second half of the verse, having given the procedure in the first half). Verse 11. Procedure prescribed in the event of com- puting planetary positions from the beginning of the Kaliyuga for the sake of convenience.

60 The Dinādya may be also obtained from the beginning of the Kali, which begins with Friday. The Dhruvas calculated for the commencement of the solar year are to be added to the planetary positions at the beginning of the Kali, in the event of computing the Ahargaṇa and thereby the planetary positions from the beginning of the Kaliyuga for the sake of convenience. Comm. The Dinādya at the commencement of Kali is Zero, since the number of civil days during the length of time equal to a Kaliyuga is integral and equal to 157791645 according to Brahmagupta and Bhaskara who follows him. The Suryasiddhānta, it may be noted here, gives the number of days in a Kaliyuga as not integral but equal to 157791782.8. Hence the Dinādya computed from the beginning of the Kali is to be increased by .8 to obtain its value according to Suryasiddhānta. The plane- tary positions at the commencement of Kali were given by Bhaskara already. Verse 12. The number of what are called Kshepadinās to obtain the Ahargaṇa. Hereafter Bhaskara is going to obtain the planetary positions for any day during the current solar year having obtained the Dhruvakas or the planetary positions for the beginning of the Solar year. In that behalf the Ahargaṇa or the collection of days which have elapsed from the commencement of the Solar year is to be found. This Ahargaṇa is obtained by subtracting the number of Ksha- yāhās from the number of tithis that have elapsed. In finding these Kshayāhās, we have to take note that there is a little remnant of Kshayāhās at the beginning of the Solar year which is also to be taken into account while computing the number of Kshayāhās during the course of the year. In other words, the number of Kshayāhās that are going to be computed during the course of the year, for the elapsed part of the year will be in default of the

61 actual number if we ignore the accrued fraction of Ksha- yāhās at the commencement of the Solar year. To make amends for that default we have to add some number to the numerator of the improper fraction which is going to give us the number of the elapsed Kshayāhās during the course of the year. The formula that is going to be used to obtain the number of Kshayāhās during the elapsed tithis is x/64 (1 + 1/702). This formula arises out of the fact that there are 55739 Kshayāhas during 3562220 tithis, so that for 64 tithis the number of Kshayāhās is equal to (64 × 55739) / 3562220 = 3567296 / 3562220 = 1 + 5076 / 3562220 = 1 + 1 / (3562220 / 5076) = 1 + 1/702. Let the fraction of Kshayāha at the commencement of the solar year, to be taken into account be x ghatis i.e. x/60 of a tithis (for Kshayāhās are are computed out of tithis) i.e. (x tithis) / 60. Let y be the number of tithis elapsed after the commencement of the solar year. Then to compute the Kshayāhās that ensue after the commencement of the solar year upto the day concerned during the course of the year, the formula to be used is y/64 (1 + 1/702). To this we have to add (x tithis) / 60 as the balance of Kshāyāha at the commencement of the solar year to be taken into acccount (x tithis) / 60 = (x × 64) / (64 × 60) = (x × 64/60) / 60 Now, in computing the number of tithis which have elapsed after the commencement of the Solar year, we subtract Śuddhi from the number of tithis that have elapsed after the beginning of the luni-Solar year. But

62 in this Śuddhi we have subtracted the fractional part of the Kshayāhas for a different purpose so that in sub- tracting the Suddhi, we have increased the tithis by the fractional part of the Ksbayāhās. This increase there- fore should be nullified, which means that the quantity to be added is (x × 64/60) / 64 − x/60 = (x × 63/60) / 64 = (21/20) (x/64) = x (1 + 1/20) / 64 Hence the Kshepadinas or the tithis to be added to the number of tithis which have elapsed from the commence- ment of the Solar year, are x (1 + 1/20). This means that the ghaṭis x which form the fractional part of the Kshayāhas are to be increased by one twentieth part of themselves and are to be viewed as tithis and not ghaṭis as mentioned. Verse 12 (contd

next Sun-rise, T₃ the ending moment of the tithi preceding the Sun-rise R₄ upto which point the Ahargaṇa has to be found from the commencement of the Solar year. Thus the Ahargaṇa to be found is SR₄ = ST₃ + T₃ R₄. Here T₃ R₄ is the Kshayāghatis, at the Sun-rise concerned. SR₃ is a fraction of a day that is to be there in the Ahar- gaṇa SR₄ we are seeking. While subtracting the Śuddhi T₁ S diminished by the Kshayāghatis T₂ R₃ from the number of elapsed integral number of tithis from the commencement of the luni-Solar year namely T₁ T₃ we have T₁ T₃ — (T₁ S — T₂ R₃) = T₁ T₃ — (T₁ T₂ — SR₃) = T₂ T₃ + SR₃. Thus instead of SR₃ + R₃ R₄ we have by the above procedure SR₃ + T₂ T₃. Though both T₂ T₃ and R₃ R₄ are integral numbers and should be the same if the interval is small, they may differ by an integral number, if the interval happens to be long. Bhaskara says that this difference of an integral number will be rectified by the Kshepadinas found under verse No. 12, for, from these Kshepadinas, the Kshayāhas are found and subtracted from the tithis. It will be noted that the difference of an integral number between T₂ T₃ and R₃ R₄ is no other than Kshayāhas. Or, this may be seen in an other way. We are to find SR₄ = T₁ R₄ — T₁ S = T₁ T₃ + T₃ R₄ — T₁ S = T₁ T₃ — (T₁ S — T₃ R₄) = No. of elapsed tithis — (Suddhi — Kshayaghatis at the day concerned). But instead of the Kshayaghatis at the day concerned Bhaskara prescribes the subtraction of the Kshayāghatis at the end of the Solar year ie instead of subtracting T₃ R₄ it is prescribed to sub- tract T₂ R₃. This difference, Bhaskara says is made up by taking into account the Kshepadinas pertaining to the Kshayāhas. Verse 14. In case the Ahargaṇa is required for a day preceeding the commencement of the Solar year, then the elapsed tithis are less than the Tithis of the Suddhi; so subtraction is not feasible. In this case, it is prescribed

64 to take the elapsed tithis from the previous Chaitra and the Suddhi of the previous year. Also in this case the Dhruvakas pertain to the commencement of the previous Solar year. Comm. Easy. Verse 15. To obtain the position of the Sun. The number of the days in the Ahargaṇa is to be diminished by 1/60 part of itself to obtain the number of degrees, and fractions thereof; then the Ahargaṇa multi- plied by three and divided by 22 gives the minutes and fractions thereof. Adding the two results we get the position of the Sun. Comm. The mean daily motion of the Sun is 0-59-8-10-21. Here 59' = 1° - 1' = (1 - 1/60)°; for x days x (1 - 1/60)° = x° - x°/60 as mentioned. The remain- ing part namely 0-0-8-10-21 = 9807'/72000; Converting this into a continued fraction it is equal to 1/(7 +) 1/(2 +) 1/(1 +) 1/(12 +) ... of which a good convergent is 3'/22. Hence for x days 3x/22 as stated in the verse. Verse 16. To obtain the position of the Moon. The number of elapsed integral tithis multiplied by 12 and added to the Sun's position in degrees, gives the Moon's position in degrees at the ending moment of the tithi preceeding the day at the Sun-rise of which the planetary positions are sought. To find the position at the Sun-rise required, ten times the Kshaya-dina-Sesha increased by 1/7 th of itself gives the number of minutes to be added to the position got above.

65 Comm. The first part is clear, because for every tithi, there will be an increase of elongation of 12°. To get the fractional part of the tithi in between the ending moment of the tithi and the subsequent Sun-rise, the interval which is Sāvana ie expressed in the unit of civil day has to be converted into luni-Solar units. The rule of three is ‘ If for 63 Sāvana days there are 64 tithis, what will it be for the above interval ?’ Here the approximate ratio of 64/63 is used because the Ahargaṇa which is Laghu is small ie is less than 365. The Kshaya-dina-Sesha which was got under verses 12, 13, has a divisor of 64, and is of the form y/64 . Hence the quantity to be added is (y × 64)/(64 × 63) = y/63 of a tithi = y/63 × 12 × 60 minutes of arc = 8/7 × 10 y = 10 y (1 + 1/7) minutes as given in the verse. This has to be added to 12t°, got before where t is the number of elapsed tithis. Verse 17. Computation of Mars. The mean daily motion of Mars is 0-31-26-28-7=0-30

• 0-0-90 minus 0-0-3-31-53. If x be the number of days x × 0-30' = x°/2 ; x × 0-0-90" = x/2 × 3' ; thus the rule prescribed is “ Half of the number of days gives the degrees ; half the number of days multiplied by 3 gives the number of minutes ” from this we have to subtract x × (0-0-3-31-53). The quantity within the brackets is approximately 1/17 of a minute for 1/17' = 0-0-3"-31"'-46"'. Since the Ahargaṇa is small the precision required is there. Thus the rule is x°/2 + x/2 × 3' + x'/17 + the Dhruvaka at the beginning of the Solar year. Verse 18. Computation of the S'īghroccha of Mercury. The mean daily motion of the Budha-S'īghra is 4°-5'-32"-18"'-28"". 9

66 Rule. If the Ahargaṇa be x days then the Śīghra will be 4x° + (4x × 3') / 130 + Dhruvaka. Proof. For x days, the mean motion is x × 4° + x × (0-5′-32″-18‴-28⁗). The second part is taken to be 4x × 3° / 130 ie x × 12° / 130 = x × 6/65° for 6/65 of a degree = 0-5′-32″-18‴-"28"". The fraction 6/65 can be seen to be a convergent of the remainder for 5′-32″- 18‴-28⁗ = 299077° / 3240000 = 1/(10 +) 1/(1 +) 1/(4 +) 1/(1 +) 1/9968 of which the penultimate convergent is 6/65. Verse 19. The Ahargaṇa divided by 12 and by 71 gives respectively the positive degrees and negative minutes to be added to the Dhruvaka of Jupiter. Comm. If x be the Ahargaṇa x° / 12 − x' / 71 + Dhruvaka = Guru. The mean daily motion of Jupiter is 0-4-59-9.9 = 0-5′ minus 0-0-0-50-51. Evidently 0-5′ × x = x° / 12. The remainder 0-0-0-50-51 = 61′ / 4320 of which the continued fraction is 1/(70 +) 1/(1 +) 1/4. A very approximate convergent is 1/71 as given. Verse 19 contd. To obtain the position of the Śīghra of venus. The Ahargaṇa multiplied by 10 and divided by 6 and 155 respectively gives degrees positive and negative to be added to the Dhruvaka of Venus to give his position. Comm. The formula is 10 x / 6 − 10 x / 155. The mean daily motion of the Śīghra of Venus is 1°-36′-7″-44‴-35⁗ =