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

328 Agrā / s = S. S. / 12 = Taddhṛti / k = (Agrā + S. S. + Taddhṛti) / (s + 12 + k) (1) (2) (3) = 1800 / (9 + 12 + 15) = 1800 / 36 = 50 II (4) Equating (1) and (4) Agrā = 9 × 50 = 450 Equating (2) and (4) S. S. = 12 × 50 = 600 Thirdly Taddhṛti = 15 × 50 = 750 Again Equating (1) and (6) of I Agrā / Krāntijyā = 5 / 4 = 450 / Krāntijyā ∴ H sin δ = (450 × 4) / 5 = 360 from which λ could be computed. Verse 98. The chara at a place where s = 9, is equal to 3 nādis. If you could compute the longitude of the Sun, then certainly you are a leader among astronomers, Oh ! Scholar ! Verse 99. Answer to the problem above. 12 Carajyā / √((12 × Carajyā / R)² + s²) = H sin δ where from λ the longitude of the Sun could be computed. Comm. Let H sin δ = x; then from the third lati- tudinal triangle Kujyā / Krāntijyā = s / 12 = 9 / 12 = 3 / 4 ∴ Kujyā = 3 x / 4 since Krāntijyā means H sin δ ∴ Carajyā = 3 x / 4 × R / (H cos δ) = 3 R x / (4 √(R² - x²)) = H sin (3 × 6) = H sin 18° ∴ Squaring 9 R² x² = 16 (R² - x²) H sin² 18 = 16 Carajyā² (R² - x²)

324 ∴ x² (9 R² + 16 Carajyā²) = 16 R² Carajyā² ∴ x² = (16 R² Carajyā²) / (9 R² + 16 Carajyā²) ∴ x = (4 R Carajyā) / √(9 R² + 16 Carajyā²) = (12 Carajyā) / √(81 + (12² Carajyā² / R²)) = (12 Carajyā) / √(9² + (12 Carajyā / R)²) Here Carajyā being known, H sin δ could be computed. Verse 100. If you studied what is known as Madh- yamāharaṇa, then compute λ the longitude of the Sun given that H sin δ + H cos δ + H sin λ = 5000. Verse 101. Answer to the problem above. Let the given sum multiplied by 4 and divided by 15 be Ādya ; then H sin δ = Ādya − √(910678 − (2 square of the given sum / 337)) . Comm. Let H sin δ = x ; then H cos δ = √(R² − x²) and since H sin δ = (H sin ω H sin λ) / R ∴ H sin λ = (x R) / (H sin ω) = (x R) / 1397 ∴ The given sum = x + √(R² − x²) + (x R / 1397) = 5000 ∴ √(R² − x²) = 5000 − x (1 + R / 1397) = 5000 − (4835 / 1397) x ∴ R² − x² = 5000² + x² (4835 / 1397)² − (2 × 5000 × 4835 / 1397) x ∴ x² {1 + 4835² / 1397²} − (2 × 5000 × 4835 / 1397) = R² − 5000²

325 ie. x² (1397² + 4835²) - 2 × 4835 × 1397 × 5000 x = 1397² (R² - 5000²) ie. x² (25328834) - 2 × 4835 × 1397 × 5000 x = 1397² (R² - 5000²) ∴ x² - (2 × 4835 × 1397 × 5000 x) / 25328834 = (1397² (R² - 5000²)) / 25328834 ∴ x² - 2 × 5000 x × 6754495 / 25328834 = ,, Converting 675 / 2533 into a continued fraction we have 1/(3 +) 1/(1 +) 1/3 = 4 / 15 so that the equation could be written as x² - 2 × 5000 (x 4) / 15 = (1397² (R² - 5000²)) / 25328834 Here (5000 × 4) / 15 is symbolized as Ādya so that we have x² - 2 Ādya x = (1397² (3438² - 5000²)) / 2532883 ∴ (x - Ādya)² = Ādya² + (1397² (3438² - 5000²)) / 2532883 = 5000² × 16 / 225 - (5000² × 1397²) / 2532883 + (1397² × 3438²) / 2532883 = 5000² (16 / 225 - 1397² / 2532883) + (1397² × 3438²) / 2532883 Here 16 / 225 - 1397² / 2532883 is approximated to -2 / 337 and (1397² × 3438²) / 2532883 is approximated to 910678 so that we have x = Ādya ± √(910678 - (2 s²) / 337) where s is the given sum. Since the positive sign of the radical is invalid because H sin δ ≯ R, so the negative sign is taken.

326 Verse 102. In a place where s = 5″, the sum of H sin δ, S. S., Taddhṛti, Kujyā and Agrā is 6500 ; find them individually oh, mathematician, if thou art adept in understanding the sphere and dealing with the latitudinal triangles. Verse 103. Answer to the problem above. Assuming H sin δ to be equal to 12 s and computing the various quantities cited ; take their sum. Then by rule of three “ If for this sum got, the individual magni- tudes are such and such what will they be for the given sum ” each can be had. Comm. The cited magnitudes are respectively H sin δ, (R H sin δ) / (H sin ϕ), (R² H sin δ) / (H sin ϕ H sin ϕ), (H sin δ H sin ϕ) / (H sin φ) and (R H sin δ) / (H cos ϕ) which are all proportional to H sin δ, ϕ being given through ‘ s ’. With this idea of proportionality at the back of his mind, Bhāskara sets this ingenious ques- tion, and gives an easy way of solving it by assuming H sin δ to be 5 × 12 = 60, so that the others can be got rationally. With this H sin δ, S. S. (3438 × 60) / (3438 × 5/13) = 156, Taddhṛti = (3438² × 60) / (3438 × 5/13 × 3438 × 12/13) = 169 ; Kujyā = (60 × 5) / (13 × 12/13) = 25 Agrā = (3438 × 60) / (3438 × 12/13) = 65 The sum of these is 475. So, by the rule of three men- tioned above, H sin δ = 1200, S. S. = 3120, Taddhṛti = 3380, Kujyā = 500 and Agrā = 1300. Or alternatively given s = 5, k = 13 so that H sin ϕ = (3438 × 5) / 13, H cos ϕ = (3438 × 12) / 13. Hence the values of

327 the various magnitudes are H sin δ, (H sin δ × 13) / 5 , (H sin δ × 13²) / 60 , (H sin δ × 5) / 12 and (H sin δ × 13) / 12 The sum of these is H sin δ (1 + 13/5 + 169/60 + 5/12 + 13/12) = H sin δ ((60 + 156 + 169 + 25 + 65) / 60) = 475/60 H sin δ = 95/12 H sin δ = 9500 ∴ H sin δ = 1200 from which by substi- tution the remaining magnitudes could be obtained. Verse 104. If the sum of Agrā, H sin δ and Kujyā be 2000 find them individually, oh ! mathematician if thou be an adept in the geometry of the sphere and compu- tation. Comm. Here the quantities are respectively (R H sin δ) / (H cos φ), H sin δ and (H sin δ H sin φ) / (H cos φ) so that their sum is H sin δ (1 + R / (H cos φ) + (H sin φ) / (H cos φ)) = (H sin δ (H sin φ + H cos φ + R)) / (H cos φ) Here also we are to presume s = 5 so that the above sum is (H sin δ (s + 12 + k)) / 12 (by proportion of the first and second latitudinal triangles) = (H sin δ (5 + 12 + 13)) / 12 = 5/2 H sin δ = 2000 ∴ H sin δ = 800. Substituting this value in the above formula, Agrā = (RH sin δ) / (H cos φ) = (3438 × 800 × 13) / (3438 × 12) = 10400 / 12 = 866-40 ; Kujyā = (H sin δ H sin φ) / (H cos φ) = (800 × 5) / 12 = 4000 / 12 = 333-20

328 OG = Gnomon ; OP = Shadow of the gnomon ; PM = the Bhuja drawn from the extremity of the shadow P perpendicular on the East-west line ; OM = Koti of the shadow extending along the East-west line. AB is the Nalaka placed along the Chayakarṇa PG. The eye is placed at A and the planet ☉ is visible through the tube of the Nalaka AB. Fig. 62 Verses 105, 106 and 107. The method of observing through the instrument called Nalaka, the planetary position. On a horizontal plane mark a point and through it draw the East-west line and also the North-south ; if the planet is in the East mark off the computed Koti of the shadow towards on the East-west line; if the planet is in the Western hemisphere, mark this Koṭi towards the East. From the extremity of the Koti mark the computed Bhuja perpendicular to the East-west line and draw the computed shadow from the point so as to form a right-angled

329 triangle with the Bhuja and Koti. Extend a thread from the point of intersection of the bhuja and shadow to meet the gnomon's top so as to form the Chāyākarṇa or the hypotenuse of the right-angled triangle of which the other sides are the gnomon and the shadow. Along this thread place the Nalaka such that the lower extremity of the Nalaka coincides with the eye. Seeing through the Nalaka, the planet is to be seen. I shall tell how the planet could be seen in water as well. Comm. The Nalaka is a simple tube formed generally of bamboo. The purpose of this is to verify the correct- ness of the computation of the shadow and its bhuja. If the computation is wrong the planet will not be seen in that direction. It might be asked how the shadow and bhuja are pertinent with respect to a planet, whose shadow cannot be observed as that of the Sun. True, but the computation of the shadow and bhuja are done as will be done with respect to the Sun, knowing the declination etc. as in the case of the Sun. Computation does not depend on the observation of the actual shadow. Computing the magnitudes of the Bhuja and Koti, the direction of the Chāyākarṇa points to the planet in the sky. Verse 108. Observing the planet through the Nalaka in water. Fig. 63 42

330 Place the Śaṅku at the point of intersection of the Bhuja and shadow and holding the Nalaka along the join of the top of Śaṅku and the point, the planet could be seen in a basin of water placed at the point. Comm. Let P be the planet casting the shadow AC of the gnomon AB. C the extremity of the shadow is the point of intersection of the shadow and the Bhuja. Though we have shown the gnomon in the position AB, it need not have been placed there in as much as we have the computed magnitudes of the shadow, the Bhuja and the Koti. Now we are directed to place the Śaṅku act- ually at C the point of intersection of the shadow and the Bhuja. Thus CD is the Śaṅku. Since CD=AB and and both are vertical evidently Δs DBA and DCB are congruent. Hence DĈB = DÂB. But DĈB = zenith- distance of the planet and as such is equal to BÂD (also the zenith-distance of the planet) ∴ DÂB = BÂP. Hence if a tray of water is placed at A, the planet will be visible as seen through DE, the Nalaka since the angle DAB is the angle of incidence and BÂP the angle of reflection are equal. Verse 109. The planet is to be shown to the king, who has an eye of appreciation for the same, either direct- ly (as shown in fig. 62) in the sky or through water as shown in the fig. 63, having finished the preliminaries indicated. Comm. Clear. End of the Tripraśnādhyāya.

PARVASAMBHAVĀDHIKĀRA Investigation into the occurence of an eclipse Verses 1–2. Multiply the number of years that have elapsed from the beginning of the Kaliyuga by twelve and add the number of months elapsed from the beginning of the luni-solar year. Let the result be x. Then add [2 x (1 - 1/898)] / 65 to x. Let the result by y. Then the longi- tude of what is called Sapāta-Sūrya or the longitude of the Sun with respect to a node will be x Rasis + [(2 y + 503) (1 + 1/169)] / (3 × 30) Rasis. If this longitude be less than 14°, then a lunar eclipse is likely to occur. Comm. The first operation indicated above in direct- ing x to be added to [2 x (1 - 1/898)] / 65 is intended to obtain the lunations that have elapsed from the beginning of the Kaliyuga. In this behalf we are asked to multiply the elapsed years by twelve to get the number of solar months. Here there is one subtlety to be noticed. The years that have elapsed are not entirely solar. In fact the years reckoned according to the luni-solar system were all originally luni-solar; but according to the convention of intercalary months, they were rendered solar upto the point of the latest intercalation, for, solar months plus intercalary months are equal to the elapsed lunations. From the moment of the end of the latest intercalary month, the subsequent years or year or fraction thereof would be luni-solar only. Nonetheless, no difference will be there in the computed Adhikamāsas in adding a few lunar months to the solar and taking them all to be solar. The maximum error committed in so doing will be of the order of (no. of days in a solar month minus no.

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

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• 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

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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