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

of days in a lunar month) multiplied by 36 × ²/₆₅ × ¹/₃₀ of an adhikamāsa, assuming that an adhikamāsa would occur at the latest in 36 solar months. (In fact, an adhikamāsa would occur on the average in 32½ solar months, but we have taken 36 roughly as the maximum figure in as much as the occurence of the Adhikamāsa might be belated on account of the convention stipulated). Thus the error would be 36 × 2 × ²/₆₅ × ¹/₃₀ = ¹/₁₃th of an adhikamāsa at the maximum. Hence, we are directed not only to construe that all the years elapsed to be solar but also the subsequent lunations of the current luni-solar year also to be solar months. Thus getting the number of elapsed months from the beginning of the Kaliyuga, the computation of the Adhikamāsas is formulated as follows. If in the course of 51840000 solar months of the Yuga there be 1593300 Adhikamāsas then during the elapsed solar months x, what is the number of elapsed Adhika- māsas? The result is x × 1593300 x × 1593300 ----------- = --------------- 51840000 796650

51840000

796650 = 2 × x --------- . Since Bhāskara knows that there will be two 65-4-21 Adhikamāsas roughly in 65 solar months, he performed the above operation. This shows that for every 65 solar months roughly there occur two Adhikamāsas or more accurately a little less than two Adhikamās. So, taking, in the first instance 2/65 as the ratio of Adhikamāsas to the number of solar months, Bhāskara tries to find as to what quantity is to be subtracted from 2. That is found as follows. If there be A adhikamāsas in s solar months what will be the number of Adhikamāsas in x solar months? The result is A x --- . Again if there be two Adhikamāsas roughly in 65 s solar months, how many will be there in x solar months? 2 x The answer is --- . But we have seen about that the 65

accurate number should be (2 x / 65) - λ ie. a little less than (2 x / 65) . The question is now to find the value of λ. So, equating (2 x / 65) - λ to (A x / s), λ = (2 x / 65) - (A x / s) = x ((2 / 65) - (A / s)) = x ((2 s - 65 A) / (65 s)). Substituting for 2 s - 65 A namely 2 × 51840000 - 65 × 1593300 = 115500 λ = (x × 115500) / (65 × 51840000) = (x × 2 × 57750) / (65 × 51840000) = (2 x / 65) × 1 / (51840000 / 57750) = (2 x) / (65 × 898) ∴ (A x) / s = (2 x / 65) - λ = (2 x / 65) - (2 x) / (65 × 898) = (2 x / 65) (1 - 1 / 898) as given. The procedure, adopted as above, is in a way a short cut in Hindu Astronomy to obtaining a convenient con- vergent to a continued fraction. Let us use the method of continued fractions; the number of Adhikamāsas in x solar months is (A x / s) ie. x × A/s = (x × 1593300) / 51840000 = (x × 5311) / 172800 . Converting 172800 / 5311 into a continued fraction we have 32 + 1/(1+) 1/(1+) 1/(6+) 1/2 + 1/(1+) 1/(1+) 1/(18+) 1/4 to which 65/2 is a convergent but a good convergent is 245 / 69 . As this good convergent is unwieldy, Bhāskara used 2/65 and made amends for the roughness introduced by adopting it. Wherever a con- venient convergent is not available, an easy and rough convergent is used and amends will be made for the rough-

334 ness resulting as follows. Let M/N be a fraction to which m/n is a convergent having small numbers as numerator and denominator, so that M/N is taken to be equal to m/n (1 + 1/λ). Thus M/N = m/n (1 + 1/λ) or Mn = Nm (1 + 1/λ) ∴ Mn − Nm = Nm/λ or λ = Nm / (Mn − Nm). In the present case M is the number of Adhikamāsas, and N the number of solar months. m/n if taken to be 2/65 − 2 × Solar months ∴ In this case λ = -------------------------------------------------- 65 × Adhikamāsas − 2 × Solar months which is indicated in the commentary by Bhāskara. In this context, it may be mentioned that a Karaṇa- grantha named Nārasiṁha based upon Sūryasiddhānta (A Karaṇagrantha is a manual according which the Hindu calendar is computed with easy numbers without under- going the laborious process indicated in the treatises called Siddhāntas like the present Siddhānta Śiromaṇi. In these Karaṇagranthas, instead of taking the beginning of the Kalpa or Mahāyuga or the Yuga, as the epoch, a recent date ie. the date of the author of the Karaṇagrantha is taken as the epoch, and processes using approximations are adopted for the sake of ease. Naturally therefore these Karaṇas (as they are also called) get easily obsolete within the course of a few hnudreds of years so that a fresh Karaṇa is called for preparation, if the calculations were to accord with the Siddhāntas which those Karaṇas pro- fess to follow. In fact, the present Karaṇa of Nārasiṁha written in 1333 Śaka year ie. in 1411 A.D. declares that a previous Karaṇa named Tithicakra reported to have been written by one Mallikārjuna Suri grew obsolete and

"? Let's zoom in on that glyph: It has:

• vertical line
• a loop to the left Wait, does it have a loop? Yes! A closed oval loop on the left of the vertical stem! Wait! An oval loop on the left of a vertical stem with a flat top is... 'व'! Wait, does 'च' have an oval loop? No, 'च' has an open curve! Wait, but look at the letter 'व' in "एतदेव": In "एतदेव", the 'व' has the EXACT same shape! Wait, could it be a misprint in the original book? Yes, old Indian presses often confused च and व. Wait, let's look at the letter again: Does it have an open top or closed loop? Looking at: त ित िथ [letter] क ं Wait, between 'थि' and 'क', look at the glyph: Wait, is the loop closed or open? It is completely closed, like 'व'! Wait, but wait! What if it's 'च'? In some fonts, ink fill makes 'च' look closed, or the typesetter mistakenly picked 'व' instead of 'च'. Wait, what about the 'क'? Does it have a repha or r-stroke? Wait! Look at "तिथिवक्रं" vs "तिथिचक्रं": Wait, is it "तिथिव्रतं"? No, the letter after

336 and s and m the angular semidiameters of the Sun and the Moon. (This formnla we shall see later). This higher limit comes to 88.5'. The limit of 56' for the occurence of a lunar eclipse is the value of pm + ps — s + m as we shall see later. The latitude of 56' of the Moon arises out of a longi- tude of 12° of the Moon with respect to a node, whereas the latitude of 32' arises out of a longitude of 7° with respect to the node. Since at an eclipse solar or lunar, the longitude of the Moon with respect to a node, is the same as the longitude of the Sun with respect to the same or opposite node, the latter must be 12° for the occurence of a lunar eclipse. But as the difference between the mean and true Suns is about 2°, the longitude is stipulated as 14°. In other words, for the occurence of a lunar eclipse, the longitude of the Sun on the full-Moon day with respect to the nearer node shall be less than 14°. To compute this longitude of the Sun with respect to the nearer node on a full-Moon day, we are given the subse- quent procedure indicated in the verse. In 53433500000 lunations of the Kalpa, the sum of the sidereal revolutions of the Sun and the Node (Rāhu) (Sum because Rāhu has a retrograde motion) is equal to 455231168 which is equal to 455231168 × 12 = 54627734016 Rasis. Then in one lunation what will be the increase of the longitude with respect to the Node? The result is 54627734016 / 53433300000 = 1 Rasi + 3583302048° / 5343330000 (= 74652126 / 111319375) dividing by 48 both the numerator and denominator. Taking the first two digits in the numerator and denomi- nator of the fraction the fraction is approximately equal to 74/111 or 2/3. Taking this as a convergent we make amends for the roughness as follows. 74652126 / 111319375 = 2/3 (1 + 1/λ) ∴ λ = (2 × 111319375) / (3 × 74652126 — 2 × 111319375)

387 = 222638750 / 1317628 = 169 approximately. Hence the increase of the Sun's longitude with respect to a node is 1 Rāśi + 2/3 (1 + 1/169)° I. In the beginning of the Kali- yuga, the longitude of the node was 5 Rāśis—3°—13′ and the arc moved by the Sun with respect to the node during the course of half a lunation is 0—15—20, so that their sum is 5 —18 —33. Here we have added for half a lunation because the context is a lunar eclipse and the beginning of the Kaliyuga was a New Moon day. Also, at the begin- ning of the Kali, the Mean Sun being at the zero-point of the zodiac, the negative longitude of the node only is the longitude of the Sun with respect to the node. Hence we have to add the above longitude of 5 —18 —33 to the longitude obtained through the above formulation I, which means 168°—33′ is to be added to 2x/3 (1 + 1/169) where x is the elapsed number of lunations. Taking 168°—33′ as nearly equal to 168°—40′, 2x/3 (1 + 1/169) + 168 2/3° = 2x/3 (1 + 1/169) + 506/3 = 2x/3 (1 + 1/169) + 503/3 (1 + 1/169) approximately = (2x + 503)/3 (1 + 1/169) as formulated. Thus for x lunations, the longitude of the Sun with respect to the node is x Rāśis + (2x + 503)/3 (1 + 1/169)°. If this longitude falls short of 14°, we could except a lunar eclipse. Latter half of verse 3 and verses 4, 5. Particularity with respect to a solar eclipse. Add half a Rāśi to the longitude previously obtained; find out on which side the Sun lies, north or south; com- pute the longitude of the Sun from the number of days 43

338 elapsed after the Saṃkramaṇa day (ie. the day on which the Sun has left one Rāśi and entered another); obtain the hour-angle in nāḍīs of the Sun at the ending moment of the Amāvāsyā ie. at the moment of New Moon; add or subtract one-fourth thereof in Rāśis from the position of the Sun according as the Sun is in the Western or Eastern hemisphere; then finding the declination of that point and from the sum or difference of the declination and latitude of the place, obtain the zenith-distance of the culminating point of the ecliptic; taking that point to be roughly the Vitribha ie. the point of the ecliptic which is 90° behind the Sun on the ecliptic, find one-sixth of the zenith-distance; taking the sum or difference of the result and the longitude of the Sun with respect to the node (got in the beginning by adding half a Rāśi to his position at full-Moon) if the result happens to fall short of 7°, then we could expect a solar eclipse. If there be no eclipse at the current New Moon, then go on adding 1 Rāśi - 0° - 40′ - 15″ to the longitude of the Sun with respect to the Node (which will be his longi- tude for the moment of the next New Moon) and repeating the procedure indicated, the occurence of an eclipse or otherwise could be known. If occurence be indicated then compute the actual positions of the Sun, Moon and Rāhu and following the procedure to be indicated in the chapter on solar eclipses, the moment of the occurence of the eclipse and other relevant details could be computed. Comm. In the case of a lunar eclipse, the Manaik- yārdha ie. half the sum of the diameters of the eclipsing and eclipsed bodies (namely the cross-section of the Earth's shadow at the lunar orbit and the Moon) is 56′. This is the maximum limit to the celestial latitude of the Moon if an eclipse were to occur, and this latitude will be there if the longitude of the Moon with respect to the nearer node is 12°. Since a lunar eclipse occurs at the moment of a full Moon, the distance of the Sun then from the opposite node

339

should be also 12° for the occurence of an eclipse. Since it is customary to check the occurence of an eclipse through Sapātasūrya ie. longitude of the Sun with respect to the node (the prefix Sa is to signify that the sum of the Sun's longitude and that of the node should be taken, as the longitude of a node is measured in the opposite dire- ction from the zero-point of the ecliptic) it is stipulated that the Sapātasūrya should be 12° for the occurence of a lunar eclipse. But, as the True Sun might differ from the Mean by about 2°, and as we are concerned with the True Sun only, the limit is increased by 2°, so that a lunar eclipse may occur if the Sapātasūrya happens to be less than 14°. Thus there is no more complication with res- pect to the occurence of a lunar eclipse than requiring the longitude of the Sapātasūrya to be less than 14° for the occurence of a lunar eclipse. If this condition be satisfied, there will be an eclipse and that will be visible at all places, where there is night, since a body in shadow will not be seen from any place whatsoever. But, there is a complication with respect to the occur- ence of a solar eclipse namely that it is not a question of the Sun entering a shadow. The Sun could never be shadowed. A solar eclipse occurs when the disc. of the Moon comes in between the Sun and an observer and obstructs a vision of the Sun's disc. The Moon being very near us compared with the Sun, its coming in between the Sun and an observer may well be compared with a cloud obstructing the vision of the Sun. Just as, when a cloud obstructs the vision of the Sun for an observer, it could not do so with respect to another who is situated at a distance, so also, if the Moon effects a solar eclipse for a particular place, it could not do so for all places. This is said to be due to parallax or Lambana as it is called (Refer fig. 64). It is so called because, when there is an eclipse of the Sun for an imaginary observer at the centre C of the Earth, the Moon intersecting in the line of sight to

340 Fig. 64 the Sun, for an observer 'O' on the surface of the Earth, the Moon is not in the line of sight namely OS but hangs down that line (लम्बते अनेनेति लम्बनम् = That phenomenon by which the Moon hangs down the line of sight). Hence it is not sufficient to say that the Sapātasūrya has a longitude of 7° to conclude the occurence of a solar eclipse for a place. If the Sapātasūrya be less than 7°, certainly there will be a solar eclipse for some place on the earth but not for all places. So, to decide whether there will be a solar eclipse for a given place, we are to take into account the phenomenon of parallax. At every New Moon, the longitudes of the Sun and the Moon will be no doubt equal; yet the Moon may not obstruct a vision of the Sun, not being situated in the ecliptic plane. He may be above the ecliptic plane or below it and if he be within 32' from the plane, a part of the Moon's globe may hide a part of the Sun's from the vision of certain observers who are situated about the point o' of fig. 64. But suppose an observer is at O. For him. there is no eclipse at all as could be seen from the figure, As the observer moves away and away from O' towards O, the effect of parallax will be greater and greater in longi- tude, whereas, as the observer moves away and away from O' towards O₁ (where O₁ is the geocentric pole of the circle

241 (c) shown in the figure) along the circle of intersection of the Earth with a plane through CO' perpendicular to the plane of the paper, where O₁ is a point on the earth such that SĈO=90° the effect of parallax will be greater and greater in latitude. In other words the parallax has both an effect in longitude as well as in latitude. When it has an effect in longitude only it is called Lambana, whereas, when it has an effect in latitude, it is called Nati. (Thus the translation of 'parallax' as Lambana alone is not fully correct, though at times the parallax may have its com- plete effect in longitude only or in latitude only). For the observer who moves in the ecliptic plane only as the one moving from O' towards O, the parallax will have its entire effect in longitude only and for the observer moving in the perpendicular plane from O' to O₁ mentioned before, the parallax will have its entire effect in latitude only. For observes other than the two above, it will have effect both in longitude and latitude also. When parallax effects longitude the time of conjunction is preponed or post- poned, whereas when it effects latitude, the latitude of the Moon appears to have increased or decreased. When it increases, no eclipse occurs and when it decreases an eclipse does occur. At O', the latitude will be exactly what has been computed; at the point of intersection O'' of the join of the centres of the Sun and Moon with the surface of the Earth, parallax nullifies the latitude and on the great circle O' O'' there will be parallax in latitude only effecting the magnitude of the latitude. It will be seen that for the point O'', the Sun and the Moon are in the zenith, so that neither will suffer from parallax. For the point O the Sun will be on the horizon and the Moon being depressed below the horizon though he is in geocentric conjunction the parallax in longitude or lambana is maximum and the occurence of the New Moon had already elapsed 4 nādis ago. Further

342

it will be seen that at the points O₁ and O₂ where O₁ is the other geocentric pole of the circle (c) drawn there cannot be an eclipse, the latitude being increased (as per Hindu astronomy) by 48' – 46''. In fact, there will be eclipse for the places on either side of O'' (the point of intersection of the join of the centres of the Sun and Moon with the Earth's surface) to such a distance as will increase the latitude to 32' only. For the other places on O' O'' beyond these points, the latitude of the Moon exceeds this limit and so there will be no eclipse. Further clarification of Lambana and Nati will be given later. The above analysis underlies our investigation for the occurence of a solar eclipse. Sapātasūrya might be less than 7°, but it does not mean that every place will enjoy an eclipse. So, for the place concerned, we have to see that, even after taking into account the parallax in latitude ie. in Nati, still the latitude will be less than 32'. To obtain this parallax in latitude, the method adopted is to find it at the point called Vithribha, (nonagesimal) ie. the point which is behind the Lagna the rising point of the Ecliptic by 90°; for, as we shall see in the Chapter on Solar eclipses, the parallax in latitude at the Vitribha will be equal to the parallax in latitude at any point of the ecliptic. In other words, wherever be the Sun and Moon on the ecliptic (Moon also being very near the node may be taken roughly to lie on the ecliptic) to compute the amount by which the latitude is increased, we compute it for the Vitribha, and this will hold good for the arbitrary position of the Moon, for, there also the latitude will be increased by the same amount. The procedure given in verse 4 is to locate the Vitribha from the position of the Sun and to find its zenith-distance to compute the Nati; or rather, it is to locate the culminating point and taking it roughly to be Vitribha, to compute the influence of Nati on the latitude or Śara. If it were only to find the Vitribha, it could be computed from the

lagna of the moment of New Moon. Computation of the zenith-distance of the Vitribha is a little cumbrous, so that, for brevity, it is sought to compute the culminating point, obtain its declination and thereby its zenith-distance which could be taken to be the zenith-distance of the Vitribha also, from which the Nati is calculated. We are directed to obtain first the Nata or the hour angle of the Sun for the moment of New Moon. We know on that particular New Moon day how long Amāvāsya will last after Sun-rise ie. we know when the actual moment of New Moon occurs on that day. We also know the duration of day time on that day so that subtracting the time of occurence of the New Moon after Sun-rise from half the duration of day, we obtain the hour angle of the Sun (Nata) in nādis. Now the Moon's longitude is effected by parallax, the effect being depression of the Moon. It is roughly estimated that the hour angle ex- pressed in nādis is increased by ¼ of its value on account of this. Strictly speaking the effect of parallax is far more on the position of the Moon than on the Sun. But the Hindu procedure apparently treats the Sun alone for parallax. The reason is that at the moment of geocentric conjunction of the Sun and the Moon, when we consider the combined effect of parallax on the Moon and the Sun at once, for a given place, we may as well compute the relative position of the Sun effected by parallax. Let the hour-angle of the Sun in nādis be x. Then effected hour- angle will be x (1 + ¼) = 5x/4 nādis. But each Rāsi being taken roughly to rise in 5 nādis, the hour-angle in Rāsis will be 5x/4 ÷ 5 Rāsis x/4. Hence we are directed to divide the hour-angle of the Sun in nādis to divide by 4. This x/4 being substracted from the longitude of the Sun, we get the longitude of the culminating point. Then we are directed to obtain the declination of the culminating point from the formula H sin δ = H sin ω × H sin λ ÷ R,

344 This declination of the culminating point being known, and the latitude being known, its meridian zenith-distance could be got. Take that to be roughly the zenith-distance of the Vitribha. Then an approximate estimate of the effect in latitude is obtained as follows. Let the zenith- distance of the Vitribha be x. Then if we have for H sin z = R, the maximum effect of 48′ — 46″ in the latitude, what shall we have for H sin 45° ? The result is (H sin 45 × 48′ — 46″) / 3438 = 2431 / 3438 × 48′ — 46″ = 34′ — 30″. Then again another approximate estimate is made as follows. Let the Sapātasūrya (Sapātasūrya = longitude of the Sun or what is the same of the Moon with respect to the Node) be λ. Then for λ = 15°, we have a latitude of 70′. That being so, for a variation of 34′ — 30″ in the latitude, what will be the corresponding variation in the longitude λ ? The result is = (69 / 2) × (15 / 70) = 207 / 28 = 7° 11/28 which is roughly one-sixth of 45°. Hence if the zenith-distance of the culminating point taken to be the Vitribha be 45° the variation in the Sapāta-sūrya (= Sapāta chandra) will be one-sixth thereof. Hence we are asked to increase or decrease as the may be, the Sapāta-sūrya by 1/6th. If the resulting Sapātasūrya be less than 7°. there may be an eclipse. When the lunar orbit lies north of the ecliptic, the culminating point of ecliptic being south of the zenith, the latitude of the Moon is decreased by parallax, so that, we have to decrease the Sapātasūrya; whereas when the lunar orbit is then south of the ecliptic, the effect of parallax is to increase the latitude and consequently, we have to increase the Sapātasūrya. Or again when the culminating point of the ecliptic is north of the zenith and the lunar orbit is north of the ecliptic, parallax appears to increase so that we have to increase the Sapātasūrya ; and when at that moment the lunar orbit is south of

345 the ecliptic, parallax appears to decrease the latitude so that we have to decrease the Sapātasūrya. The proportion that 70' of latitude correspond to 15° of Sapāta-chandra is due to the formula H sin 15° ✕ H sin 4½ —————————— = H sin β. Using logarithms, R log sin β = 9·4130 + 8·8946 = 8·3076 so that β = 1° - 10' = 70'. This proportion could be used because the Moon is within 15° of the Node. Hence when we are investigating the occurence of a solar eclipse, application of rule of three is not unjustified or crude. Herein, Bhāskara assumed the zenith-distance of the nonagesimal to be round about 45° and drew a conclusion that the effect in the longitude on account of parallax in latitude is one-sixth of the zenith-distance. Instead if the zenith-distance be assumed to be z, the result would be (48' - 43" ✕ sin z ✕ 15) / 70 = 21/2 sin z which may be taken as a better approximation. If, on the other hand the modern value of 57' of the lunar parallax be taken, the result would be (57 sin z ✕ 15) / 70 = 12 sin z approximately which is a better value. In this calculation, it is better to compute the zenith- distance of the nonagesinal using modern methods instead of assuming the nonagesimal to be on the meridian. 44

LUNAR ECLIPSES Verse 1. The ritualistic purpose served at the time of an eclipse. Scholars of Smṛtis and pūrāṇas declare that prayer, charity or offerings to gods made in fire at the moment of an eclipse conduce to much spirituality. Hence I give hereunder the methods of computing the moment of an eclipse (lunar or solar) in as much as such a knowledge apart from its religious importance, is also wrought with a beautiful mathematical treatment. Verse 2. The initial procedure to be adopted to compute an eclipse. To know the occurence of a solar eclipse, find the exact moment of the New Moon, which is indicated by the equality of longitudes of the Sun and the Moon, and to know the occurence of a lunar eclipse compute the exact moment of the full Moon which is indicated by the fact that M = S + 180° where M and S are the longitudes of the Moon and the Sun, agreeing in degrees, minutes and seconds, though differing in Rasis by six. Also compute the longitude of the Node (Rāhu) for the moment as directed. Comm. Having ascertained the possibility for the occurence of a solar eclipse, we are directed to compute the positions of the Sun, the Moon and Rāhu for the day of the New Moon. The Sun and the Moon are to be rectified for corrections like Desāntara, Bhujāntara, Udayāntara etc. From the elongation of the Moon the tithi is to be computed and the method of successive approximations called Chālana Karma is to be used to obtain the exact moment of conjunction. This process of

347 Chālana is as follows. At first knowing the elongation of the Moon at Sunrise, by rule of three, using the then daily motions of the Sun and Moon, the moment of the conjunction is to be computed. Again the positions of the Sun and Moon are to be computed for that moment and also their daily motions are to be rectified for the moment. With these rectified daily motions and with the then positions of the Sun and the Moon, again the moment of conjunction is to be computed. Repeating the process till an invariable answer is reached, we have the exact moment of conjunction. For that moment, the position of the Rāhu is also to be calculated. Similar is the procedure for a lunar eclipse also. It is to be noted that the correction called Natakarma, which we formerly identified to be the correction for ' Astronomical Refraction' is also prescribed here as particularly mentioned by Bhāskara, in the commentary. (Vide verses 68, 69 Spaṣṭādhikāra). Verse 3. The magnitndes of the orbits and the . orbital radii of the Sun and Moon. The distances of the centres of the globes of the Sun and the Moon from the centre of the Earth in yojanas are respectively 689377 and 51566. Comm. We saw in the Kakshādhyāya of the first chapter as to how these distances were estimated. Some scholars pronounced these distances are parameters; but as per the modern estimate of the Earth's radius as compared with that of Bhāskara, (The method indicated by him in the Bhuparidhimānādhyāya of chapter I, is quite mathematical) a yojana equals five modern miles and with this correspondence, the distance of the Moon is very near the truth. So to say that the distances given above are mere parameters is wrong; also if one of the parameters gives a correct value of the quan-

848 tity in question, the other also should; but because the latter does not, to call them parameters is merely meaningless. It is interssting to note that in the commentary under this verse, Bhāskara says “ If for a circumference of 3927, the diameter will be 1250, ...” This means that Bhāskara takes π = 3927 / 1250 = 3·1416 which compares very well with the modern value 3·14159. The value of π adopted by Bhāskara in this, seems to have been taken from Lallā- chārya’s Śiṣyadhīvṛddhida, Chandragrahaṇādhikāra verse no. 3. “ शरयमाङ्गहता 625 भनवाग्निहृत् ग्रहवृतिश्रवणः फलमुच्यते ” Verse 4. Computation of what is called the ‘ Kalā- karṇa ’. The radius vector is to be computed even in the case of the Equation of centre as we did in the case of Śīghra- phala. If it be ‘ K ’, R² / (2 R - K) will be what is Kalākarṇa both in the case of the Sun, as well as the Moon. Comm. While obtaining the Equation of centre, the formula used was (r sin m) / R whereas, strictly speaking, it should have been as in the case of the Śīghraphala, after effecting the so called ‘ Karṇānupāta ’ (r / K) sinm. While trying to answer why this Karṇānupāta was not done there also, Brahmagupta gave such an answer as made Bhāskara exclaim ‘ यतो विचित्रा फलवासनाऽत्र ’ i.e. ‘ It is really curious in this respect.’ Bhāskara was really a most rational type of astronomer, and one will not fail to appreciate his sense of rationality when he declares that (1) “ अस्मिन् गणितस्कन्धे उपपत्तिमानेव आगमः प्रमाणम् ”: and when he was

349 unable to adduce a proof he declares (2) 'उपलब्धिरेव वासना' meaning thereby (1) 'In this branch of science, we reckon only such an authority which has a proof behind it ie. which could be substantiated' and (2) 'There is no proof in this but accordance with observations alone has to be taken as a proof'. Though in the case of formulating the equation of centre Karṇānupāta was not stipulated to simplify matters, as there was not much difference, the Equation of centre being generally small. It is to answer such contexts as this that Bhāskara said in the Golādhyāya 'स्वल्पान्तरत्वात्, अबहूपयोगात्, प्रसिद्धभावाच्च बहुप्रयासात्, ग्रन्थस्य तद्वैर्गुरुताभयेन यत्त्यज्यतेऽर्थो न स दूषणाय" ie. 'If we in some particular context do not mention certain things it should not be condemned because in such contexts, (1) there is not much difference or (2) no useful purpose is served to a good extent or (3) it is too clear as does not require to be mentioned (4) the procedure implies a lot of cumbrous calculations and the result is after all negli- gible and (5) Mention will make the work on hand too unwieldy and brevity which is the soul of wit is to be sacrificed. In the present context this procedure of 'rectification of the Karṇa' is sought to improve matters. Bhāskara's words 'यदा ग्रहस्य कर्ण उत्पन्नः तदा कर्णो व्यासार्धं ग्रहकक्षायाः' seem really to imply that in the formula r/R H sin m for the equation of centre in the place of R, we are called upon to substitute really K. Though we had been in default for not doing so in the context of the Equation of centre, there is no reason why we should not make up the deficiency in this context. So, therefore, this rectification of Karṇa is stipulated here.

350 The procedure originally called for a rectification is that taking K to be R, we have to compute r and again taking the resulting K to be R, we have to compute r and so on repeating the process till an invariable value for K is obtained. This means that we should go on substituting for r, r K / R . Instead of following this laborious process of ' Asakṛt-Karma ' ie. method of successive approxi- mations, Bhāskara gives an alternative in the verse, whieh is as follows. Let K be the value of the Karṇa, for a value r of r. Since we are directed to make this K as R, ie. we have to add R—K to K thus making it R, we add also R—K to R, to keep the relative position of R and K to be almost the same. In other words considering the fraction K / R , adding R—K to both the numerator and denominator we have R / (2 R—K) whieh means that for a radius 2 R—K, the Karṇa will be R; that being so for a Radius R what will be the Karṇa ? The result is R² / (2 R—K) as given. It will be noted that the above interpolative procedure is adopted as a short cut technique to the otherwise laborious process. The mathematical correctness of this procedure will be seen from the following analysis. The problem is to change the Mandaparidhi to a radius K of the deferent by the formula (as indicated by Bhāskara in the course of the commentary) r' = r K / R so that δr = r¹ — r = r K / R — r = r (K—R) / R (1) Now we have K² = R² + r³ + 2 R r cos m construing R and m as constants we have to find δ K for δ r

351 Differentiating 2 K δ K = 2 r δ r + 2 R cos m δ r ie. δ K = [δ r (r + R cos m)] / K . But from fig. 65 (which is a portion of the epicyclic figure) M̂₁ = 180—m Ô₁ = θ = Mandaphala M̂₂ = m—θ Fig. 65 r = K cos m̅—̅θ̅ — R cos m so that r + R cos m = K cos (m — θ). Substituting in the above, δ K = [δ r × K cos m̅—̅θ̅] / K = δ r cos m̅—̅θ̅ But δ r from (1) is [r (K—R)] / R ∴ δ K = [r (K—R)] / R cos m̅—̅θ̅. But from the tri- angle of fig. 65. K = R cos θ + r cos m̅—̅θ̅ so that r cos m̅—̅θ̅ = K—R cos θ. But θ being small cos θ may be taken to be unity so that r cos m̅—̅θ̅ = K—R. Again substituting in the above δ K = [(K—R)²] / R (2) Now as per the formulation of Bhāskara K¹ = R² / (2 R—K) = R² / (R + R — K) = R / [1 + (R—K)/R] = R / [1 — (K—R)/R] = R (1 — (K—R)/R)⁻¹ Since |(K—R)/R| < 1, expanding binomially,