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

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 − (

41 and P₂ + (P₁ − P₂) = P₁ where P₁ and P₂ are the numbers of sidereal revolutions of a quick and slow moving planets respectively. Following the same principle as above we could obtain their positions by multiplying the equations through out by A/M and calling planets on the left-hand- side as (1) Dwiparyayāntarodbhava-graha subtracted from the quick-moving one and (2) the slow-moving planet increased by the Dwiparyayāntara-graha respectively. Verse 14. The difference of the Śīghra and Śīghra- Kendra as well as the Sum of the Mandoccha and the Manda Kendra give the planet to be computed. Or again a com- puted planet multiplied by the number of sidereal revolu- tions of a planet to be computed and divided by the number of sidereal revolutions of the computed gives the planet to be computed. Comm. U₁ − P = K₁ and P − U₂ = K₂ where U₁, P, U₂, K₁ and K₂ are respectively the number of Sidereal revolutions of the Śīghroccha, planet, the Mandoccha, the Śīghra-Kendra and the Manda Kendra ; hence, we have P = U₁ − K₁ = U₂ + K₂ from these equations also by multiplying througout by A/M , we have the planetary position as the difference of the Śīghroccha-graha and Śīghra Kendra-graha or the sum of the Mandoccha-graha and Manda-Kendra-graha. Again, if P₁, P₂ be the numbers of sidereal revolutions of a computed planet and one to be computed respectively and if p₁, p₂ be the computed planet and the one to be computed, then P₁ × A/M = p₁, P₂ × A/M = p₂ so that P₁/P₂ = p₁/p₂ ∴ (P₁ × p₂)/p₁ = P₂ which means that the position of the 6

42 planet to be computed is got by multiplying the planetary position of a planet computed by the number of sidereal revolutions of the planet to be computed and dividing by the number of sidereal revolutions of the planet computed. Verse 15. We get the Ahargaṇa by multiplying the planetary position given in number of revolutions and fraction of a revolution by the number of days in a Kalpa and dividing by the number of sidereal revolutions in a Kalpa. How by indeterminate analysis we get the same Ahargaṇa, given the number of past sidereal revolutions alone, or by the fractional part of a revolution alone, or by the sum of the fractional parts in the case of more items involved, I shall tell later. Comm. While computing the planet we have the formula (A × P) / M = p where A = Ahargaṇa, P = the num- ber of sidereal revolutions of the planet, M = number of mean solar days in a Kalpa and p = the planetary position consisting of the number of past revolutions and also the fraction of a revolution. From the above equation, the Ahargaṇa A = (M × p) / P as stated. In tne case of only the integral number of revolutions or the fraction of a revolution alone being given, or the sum of remainders if more items than one are involved, the method of fiinding the Ahargaṇa is illustrated in golādhyāya under praśna- adhyāya under verses 12—21. Verses 16, 17. Method of getting the time in solar years that has elapsed from the beginning of the Kalpa, given the Ahargaṇa. The given Ahargaṇa multiplied by the number of Kshaya-tithis in a Kalpa and divided by the number of civil days in a Kalpa gives the number of the elapsed Kshaya tithis. Adding these to the Ahargaṇa we have the lunar

43 days L. These again multiplied by the number of Adhika- masas in a Kalpa and divided by the tithis in a Kalpa gives the elapsed number of Adhikamāsas. Multiplying this number by thirty and subtracting from the above lunar days L, we have the elapsed solar days. Dividing these by thirty, we have the number of elapsed solar months, the remainder being solar days. Dividing the solar months by 12, we have the elapsed solar years and the remainder here are the solar months. Thus we have the solar years, solar months and solar days corresponding to the given Ahargaṇa. Comm. The inverse process detailed here is quite clear. Verse 18. Computation of the Ahargaṇa and the planetary positions from the beginning of the Kaliyuga. Find the Ahargaṇa from the beginning of the Kaliyuga either (according to the method described formerly with respect to a Kalpa) and this Ahargaṇa begins from Friday, Computing the mean planetary positions from this Āhar- gaṇa and adding to their mean positions at the beginning of the Kali which are known as Dhruvakas, we have their planetary positions for the day concerned. Verses 19, 20. The Dhruvakas of the planetary posi- tions at the beginning of Kali, given in a tabular form.

44 Comm. The mean Sun and the mean Moon are taken to be in conjunction at the zero-point of the Zodiac. The planetary positions given above are accepted by Bhaskara on the authority of Brahmagupta. The fact that these positions differ from those given by Aryabhata signifies that Brahmagupta observed the True positions in his own time and to obtain those positions by calculation, he must have changed the fundamental constants such as the number of civil days, and sidereal revolutions of planets etc in a Kalpa. Here ends the section known as grahā- nayana.

MADHYĀDHIKĀRA - KAKSHĀDHYĀYA Verses 1, 2. The circumference of Akāsa---Kakshā. Astronomers say that the circumference of Akāsa- Kaksha is 18712069200000000 yojanas. Some say it is the circumference of the universe whereas some say that it is the circumference of the mountain which goes by the name Lokāloka. Those who perceive the celestial sphere as a fruit of the emblic myrobalan, (Known as Āmalaka in Sans- krit) placed in the palm, say that it is the circumference of the sphere of solar radiation ie the imaginary sphere whose volume is filled by solar light. Comm. Bhāskara, in the course of the Commentary makes it clear that he does not subscribe to this idea which is only mythological. Look at his words which are signifi- cant and testify to his rational outlook “नाऽस्माकं मतमित्यर्थः, प्रमाण शून्यत्वात्” ie “This is not our view; because it is baseless”. A yojana will be seen to be equal to 5 miles approximately. Verse 3. He gives his personal view as follows. The universe may be bounded or unbounded; our view is that this dimension of the circumference is no other than the distance covered by each planet in the Kalpa. Comm. This was an assumption made by the ancient Hindu Astronomers, as well as another assumption that the distance covered by every planet during a day is the same. This we shall see later. Verse 4. The circumference of the universe given above divided by the number of the sidereal revolutions in a Kalpa of any planet gives the circumference of the plane- tary orbit, so that in a Kalpa, the total distance covered is the circumference of the universe.

46 Comm. Clear. Verse 5. The circumferences of the orbits of the Sun, Moon and the Stars. The circumference of the Sun's orbit is 4331497½ yoja- nas, that of the Moon 324000 yojanas, of the stellar sphere 259889850 yojanas. Comm. Later, we are told by Bhaskara that the cir- cumference of the earth is 4967 yojanas and its diameter is 1581. As the method given by him in the Commentary in that context, to measure the circumference of the earth is correct, we may take it that the ancient Hindu Astronomers could estimate the same correctly. If that be so, when Bhaskara gives the circumference to be 4967 yojanas, it means 4967 yoj = (3960 × 44) / 7 miles or 1 yojana = (3960 × 44) / (7 × 4967) = 5.01 miles or what is the same 1581 yoja- nas = 7920 miles ie one yojana = 5.01 miles approximately. With this measure of a yojana, the Moon's mean distance from the earth's centre should be (as given above) (324000 × 7) / 44 × 5 miles = 257725 miles approximately. This seems to be a fair estimate and we have to find out how this estimate could be made—Indeed, there are many elementary trigonometrical methods of [finding the distance of the Moon. Which of them was used by the Hindu Astronomers, we have to discuss. Bhaskara, however takes it implicitly from Brahmagupta's version. The latter does not mention from what source he derived it but simply mentions that he has resuscitated the Brahma-Siddhānta, which grew obsolete. Either he or the author of Brahma- Siddhānta must have computed this distance using trigono- metry. The following seems to be the simplest method, by which the Moon's distance was originally estimated-Refer fig. 1. Let z be the zenith-distance of the Moon as obser-

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).