पञ्चसिद्धान्तिका (आचार्य वराहमिहिर - सूर्य, रोमक, पौलिश, वासिष्ठ एवं पितामह सिद्धान्त)

VI.4 VI. VĀS.-PAUL. SIDDHĀNTA -,LUNAR ECLIPSE 157 iii. Corrected time of duration = uncorrected time ± 5 × degrees of Moon ~ Rāhu in vināḍis (If Rāhu is greater use the upper sign, if less use the lower sign. Moon ~ Rāhu is what we get in verse 2, in this section). Example 1. On a day, at sunset, the longitude of the Sun is rā. 10-10-12, longitude of the Moon rā. 4-8-57, Sun's daily motion 60' and Moon's daily motion 810'. (From these the end of the full moon tithi falls at 6 nāḍis after sunset). The Tail of Rāhu at full moon is rā. 4-7-42. Examine whether a lunar eclipse will occur, and if so, find the duration. As a preliminary, the Moon at full moon should be found by verse 1, thus: The time to elapse after sunset, for full moon, is 6 nāḍis. Adding 6' to the Sun at sunset, the Sun at full moon is rā. 10- 10-12 + 6' = rā. 10-10-18. Therefore the Moon is rā. 4-10-18. Now, examine whether there will be an eclipse. Rāhu at full moon = rā. 4-7-42 (given). Corrected Rāhu = rā. 4-7-42 − 1° 36′ = rā. 4-6-6. Moon ~ corrected Rāhu = 4° 12′. As this is less than 13°, there is eclipse. Now for the work of getting the duration: i. The Moon's latitude (supposed known already) = 380′ × (rā. 4-10-18 −rā. 4-7-42 in degrees) ÷ 90° = 11′. ii. Uncorrected duration in nāḍis = √(3025 − 11²) × 120 ÷ (810 − 60) = √2904 × 120 ÷ 750 = nā. 8-37. iii. Corrected duration = nā. 8-37 − 5 × 4.2 vināḍis = nā. 8-16. (subtraction because Rāhu is less than the Moon). The following is the rationale of the procedure: We have said that, according to the Siddhānta, when the distance between the centres of the Shadow and the Moon is 55′, the eclipse begins or ends, because at these times the distances are equal to the sum of the semi-diameters. [चित्रम्: Moon's orbit, Ecliptic, M1, M, M2, A, S, B] Fig. VI. 2 See SM₁ and SM₂ in Fig 2, where S is the Shadow and M₁, M₂ are the Moon. At full moon, the latitude is the distance between the centres, at that time. (SM), because at full moon the centre of the shadow is at S, because it is always 6 rāśis distant from the Sun. Since the inclination of the orbit to the ecliptic is small, the siddhānta takes it that the orbit and the ecliptic are practically parallel, and that the latitudes of the Moon at first contact (AM₁) at full moon (SM), and at last contact (BM₂) are

158 PAÑCASIDDHĀNTIKĀ VI.4 equal. As the three lines of latitude are perpendicular to the ecliptic, △ M₁ AS, and △ M₂ BS, are equal and right angled at A and B. Therefore AS = SB = √(sum of semi-diameters)² − latitude² = √55² − lat², where AS or SB are the difference of the Moon's longitude at first or last contacts from its longitude at new moon, and measured in minutes of arc. As the motion of S is the same as that of the Sun, the time taken by the longitude to move from A to S or from S to B = the minutes of difference ÷ the difference of the motions of the Sun and the Moon in minutes = √55² − lat² × 60 ÷ difference of daily motion in minutes, in nāḍīs. Therefore the total time in nāḍīs from A to B (this is the uncorrected whole duration) = 2 × 60 × √55² − lat² ÷ the difference of daily motions in minutes. But actually the latitudes at the first and last contacts differ enough from that of the full moon to justify the use of their exact value. So each should be used separately, and the half-duration before full moon, and that after full moon should be found. Using the latitudes of these times again, if necessary, they should be found again. Certain Karaṇas (manuals) like the Vākyakaraṇa apply a certain correction in the place of this repetition of work. This correction depends upon the Moon ~ Rāhu at full moon, like the correction in stanza 4, given by the author for correcting the duration, and therefore it is possible that the correction for the difference in latitude has been included in that correction. But as the correction given is rough, we cannot analyse it and find out whether the author has done so or not. Let us now consider the rationale of the correction in stanza 4. We have already hinted that this is to compensate for using the latitude got in III.31 from uncorrected Rāhu, instead of that from corrected Rāhu, which is to be used in our work here. [Diagram: Fig. VI. 3] M1 R M is the orbit without correction for Rāhu. M1′ R′ M′ is the orbit with correction for Rāhu. Fig. VI. 3 In Fig. 3, R is the uncorrected position of Rāhu, and R′ is its corrected position. The distance between them is 1° 36′, given in stanza 2. When the Moon is greater than Rāhu, (M, M′) then its uncorrected latitude is MB, and the corrected latitude is M′B, greater by M′M. This is case I. When the Moon is less than Rāhu (M₁, M₁′), then the uncorrected latitude is M₁A, and the corrected latitude is M₁′A, less by M₁′ M₁. This is case II. In case I, if the uncorrected latitude, which is less, is used in the work, √55² − lat² will be greater than what it should be, and the duration should be lessened by a correction. Therefore it is said that the correction is subtractive when the Moon is greater than Rāhu. In case II, since the uncorrected latitude is greater, √55² − lat² will be less than what it should be, and so it is said that the correction is additive if the Moon is less than Rāhu, i.e. if Rāhu is greater. The Fig. is for Rāhu-Head. It can be seen that at Rāhu-Tail too the same holds.

VI.4 VI. VĀS.-PAUL. SIDDHĀNTA – LUNAR ECLIPSE 159 We can show this by theoretical considerations as well, thus: Rāhu lessened by 1° 36′, is equiva- lent to Moon increased by 1° 36′, in its effect on Moon – Rāhu. The latitude is proportionate to the distance of the Moon from Rāhu in each quadrant. In the first and third quadrants, the Moon is greater than Rāhu, and the increase in the Moon by the correction increases the latitude. In the second and fourth quadrants, the Moon is less than Rāhu, and the increase in the Moon by the correction lessens its distance from Rāhu, with the result that the corrected latitude is less. Thus for Moon greater than Rāhu the correct latitude is greater, and for Moon less, it is less. The rest is as we have shown already. Now we shall find the quantity of correction: As the angles at R and R′ are equal, the corrected and the original orbits are parallel. (see the Fig.) Therefore the differences in the latitudes at any position like M′₁, M₁, M′M, and C R are equal. But C R, being the latitude caused by 1° 36′ of longitude, is equal to 380′ × 1° 36′ ÷ 90° = 6¾ minutes of arc. Therefore the difference in any position is 6³/4′. We shall first see how much difference this will produce in the half duration measured in minutes of arc. Clearly it is = √(55² – (lat ± 6³/4)²) – √(55² – lat²) = √(55² – lat² ∓ 13½ lat) – √(55² – lat²), (if 6³/4² is neglected, being small in comparison with 55².) = √(55² – lat²) √{1 ∓ 13½ lat/(55² – lat²)} – √(55² – lat²) = √(55² – lat²) {1 ∓ 13½ lat/2(55² – lat²)} – √(55² – lat²) (if higher powers of 13½ lat/(55² – lat²) are neglected. = ∓ 13½ lat/2√(55² – lat²), and the author has neglected lat² and taken this as ∓ 13½ lat/2 × 55. Using the difference of the mean daily motions of the Sun and the Moon, because this will not matter in the already rough result, and doubling for the whole duration, the vināḍis of correction are, 13½ lat × 2 × 60 × 60 ÷ (2 × 55 × 720) = 13½ × lat × 5/55 = 13½ × {(Moon ∼ R)° × 55/13°} × 5/55 (since lat = Moon ∼ Rāhu)° × 55/13°,) = 5 × (Moon ∼ Rāhu), roughly, as given here. The greater the latitude, the greater the roughness, but this will be submerged in the roughness caused by several other things like the incorrect semi-diameters etc., but the method has the advan- tage of being easy to apply. We shall consider the readings now. In verse 3, the need for correcting mūlaḥ into mūlam and kālasthiteḥ into kālaḥ sthiteḥ will be clear, as also for sthityā into sthityām in verse 4. As for correcting bhāgaiḥ into bhāgāḥ, this is justified by what we have shown in the explanation, viz. that it is Moon ∼ Rāhu that is to be multiplied by 5 to give the vināḍis of correction. If the word bhāgaiḥ, is taken as it is, the instruction should be taken to mean “five multiplied by the difference of Moon ∼ Rāhu and 13°. By this, the correction, instead of being zero, as it should be for zero latitude, is the maximum of 65 vināḍis. Instead of being the maximum for maximum latitude, the correction becomes zero. Further, on both sides of zero latitude, where there is a transition from the Moon being greater, to Rāhu being greater, there is a jump from – 65 vināḍis to + 65 vināḍis, which itself is an indication that the formula is incorrect. But it may be objected that if our correction into bhāgāḥ is accepted the word trayodaśonāḥ serves no purpose, for the computation will be begun only if the difference is less than 13°, and therefore this need not be mentioned. The answer is this: From our explanation of the formula for correction it may be seen that it is applicable if the difference is

160 PAÑCASIDDHĀNTIKĀ VI.5 13°, and even a little more. The author instructs that the correction should be applied only if the difference is less than 13°. But TS (also NP), have taken the word bhāgaiḥ as it is and given the interpretation, because they do not know the nature or the rationale of the correction. Here, Thibaut alone says (vide page 43 of Eng. Translation): “To the duration so found stanza 4 directs us to apply a correction whose rationale we are however unable to assign”, and thus accepts ignorance. But S. Dvivedi in his Sanskrit Commentary says that the Moon’s motion varies from time to time, and this correction is to rectify the error due to the variation. He is unable to see that the variation of the Moon’s motion has nothing to do with the correction here, and cannot be related to it. [विमर्दकालः] किन्वन्तरांशहीनैः पञ्चभिरूना हता दश ‘कृत’घ्नाः । तत्पदमेकाश्विघ्नं प(ञ्चां)शोऽस्माद् विमर्दकलाः ॥ ५ ॥ Total obscuration 5. Deduct the difference of the longitudes between the Moon and Rāhu from five degrees. Deduct this from ten degrees, and multiply the remainder by this itself and by four. Find the square root of the result and multiply it by 21. The minutes of arc of total obscuration is got. This dividend by the daily motion gives the time. The rule given is as follows: Minutes of obscuration = 21 × √{5 − (Moon ~ Rāhu)} [10 − {5 − (Moon ~ Rāhu)}] × 4/5. This can be simplified as: Minutes of obscuration = 2 × 21 √(5² − (Moon ~ Rāhu)²)/5. = 2 × 21 × √(25 − (Moon ~ Rāhu)²)/5. This multiplied by 60 and divided by the daily motion gives the duration of obscuration in nāḍikās. Here, 21 × (Moon ~ Rāhu)/5 is the Moon’s latitude at full moon. The latitude according to the Paulīśa has been shown to be (Moon ~ Rāhu) × 380′/90, where Moon ~ Rāhu is in degrees. As 380/ 90 is very nearly equal to 21/5, we can say: Latitude in minutes = 21 × (Moon ~ Rāhu)/5. It must be noted that we use here the corrected Rāhu to get Moon ~ Rāhu. So, the latitude obtained is the correct latitude. Therefore, no correction is necessary here corresponding to that of verse 4, above. Now, the rule is explained thus: In this Siddhānta, the difference between the semi-diameters of 5a. B.किं चंतराशहीनैः (B3.किं चतय रा०) b. B1.पञ्चाभी. B.हता द om श. A.क्रतघ्नाः c. B. om घ्नं d. A.B.पञ्चाशो

VI.5 VI. VĀS.-PAUL. SIDDHĀNTA – LUNAR ECLIPSE 161 the Moon and the Shadow is 21 minutes of arc. Therefore, when the difference between their centres is 21′, the total obscuration begins or ends, as at M₁ or M₂ in Fig. 4, below. Fig. VI. 4 Here M is the Moon at new-moon, and S is the centre of the Shadow. M₁S = M₂S = 21′ = the difference of the semi-diameters, constant according to this Siddhānta. SM is the latitude at full moon. Therefore, Minutes of obscuration = M₁M₂ = 2(M₁M or MM₂) = 2 √(SM₁² – SM²) = 2 √(21² – latitude²) = 2 √(21² – {21 × (moon ∼ Rahu)/5}²) = 2 × 21 √(5² – (moon ∼ Rāhu)²)/5, = 2 × 21 × √(25 – (moon ∼ Rāhu)²), the simplified rule, from which by inverse operation, we get the original rule, 21 × √[{5 – (moon ∼ Rāhu)} × [10 – {5 – (moon ∼ Rāhu)}]] 4/5. The conversion of the minutes of arc of obscuration into time is, as already given, by the propor- tion, Minutes of daily motion of (Moon – Sun): Minutes of obscuration :: 60 nāḍikās:nāḍikās of obscuration. From the simplified rule it will be readily seen that when Moon ∼ Rāhu is 5°, the minutes of obscuration, and thence the time, is zero. Therefore only when the difference is less than 5°, there is obscuration, not when greater, i.e. if the correct latitude at new moon is greater than 21′, there is no total eclipse. Example 2. The Moon at new moon is rā. 8-13-24. Rāhu (Tail) is rā. 8-12-0. The daily motion of (Moon – Sun) = 750′. Find the minutes of obscuration and the time. The corrected Rāhu = rā. 8-12-0 – 1° 36′ = rā. 8-10-24 Moon ∼ Rāhu = rā. 8-13-24 – rā. 8-10-24 = 3°. By the simplified rule, the minutes of obscuration = 2 × √(25 – 3²) × 21/5 = 2 × 4 × 21/5, minutes. The duration of obscuration = 2 × 4 × 21 × 60/(5 × 750) = nā. 2-41.

162 PAÑCASIDDHĀNTIKĀ VI.8 From verse 3, giving the general duration, we see that the sum of the semi-diameters of Shadow and Moon is 55'. Here we see that their difference is 21'. Hence, (55' + 21') = 76', is the diameter of the Shadow according to this Siddhānta, and 55' − 21' = 34', is the diameter of the Moon, giving the semi-diameters as 38' and 17' respectively. By a strange confusion of ideas TS and NP have concluded here that when Moon ~ Rāhu is less than 10°, there must be a total eclipse. (vide the Sanskrit com. p. 33, and English translation, p. 44, NP, Pt. II, p. 53). We have shown that for a total eclipse to occur the difference must be less than 5°. It is easy to see which is correct. If the difference is greater than 5°, the number under the radix becomes negative, and no real root can be obtained. For e.g. if we take 7°; as the difference, according to TS there must be total obscuration. But using it in the formula, we get, 21 × √{(5 − 7){10 − (5 − 7)} 4/5 = 21 √− 96/5 = 21 × 4 √− 6/5, which does not give a real value. Their error is due to their confusing in their work, the 21 minutes, sine of 10°, as 21 minutes, latitude to be got for 10° difference. Further the postulation by TS of a maximum latitude of 240' in this context is unwarranted. True, from the above a latitude of 55' for 13° difference, and 21' for 5° difference will follow, if the correct formula, with the sine of (Moon ~ Rāhu) is used, instead of the degree of difference. But nowhere in Hindu astronomy is 240' given, and the Vāsiṣṭha does not give the latitude at all. So when the Vāsiṣṭha wants the latitude to be used in V. 3, we have got to use only the Paulīśa formula, and the reading gives 280', which TS have translated tacitly as 270'. 280', if taken, will give 63' and 24' for differences 13° and 5°, and 270' will give 61' and 23', instead of 55' and 21', both of which are unsupported by the context. This is the reason why we changed the reading to mean 380', instead of 280', and, following the instructions strictly, gave the rule, latitude = 380' × difference in degrees/90, getting 55' and 21' for differences of 13° and 5°. This is also in keeping with the practice of the Paulīśa, which usually uses for proportion the degrees in the place of sines. [स्पर्शमोक्षदिशौ] स्थितिदलविमर्ददलयोर्विशेष(के तमः) सकलमत्तीन्दुम् । प्रग्रहणमोक्षशशिराहुविवरभागैश्च दिग् वाच्या ॥ ६ ॥ विक्षेपविपर्ययात्सान्तरीयभागे (कृ)ते त्रयोदशधा । परिधौ प्राक्प्रभृतीन्दोर्ग्रहणा(शां)शे वदेत् पर्व ॥ ७ ॥ शशिपरिधिदला(र्ध)ग्रे खेन्द्वन्तरभागसंगुणे (चा)क्षे । ‘खखरूपाष्ट’ हृते प्राग्वल(नं) वामं च्युते सव्यम् ॥ ८ ॥ Direction of the eclipse 6. During the interval from the time of first contact to the beginning of totality, Rāhu (i.e. darkness), swallows the Moon completely. The directions of the points of first and last contacts are to be calculated from the Moon ~ Rāhu of those times. 7. Divide the semi-orb of the Moon situated opposite to the direction of latitude into 13 parts, by straight lines parallel to the east-west diameter, at

VI.8 VI. VAS.-PAUL. SIDDHĀNTA – LUNAR ECLIPSE 163 equal distances from one another. At the part of the rim equal to the degrees of Moon ~ Rāhu, on the eastern or western part of the orb, are the points of first and last contacts, from which the directions can be read. 8. Multiply a fourth of the Moon's rim, (in whatever unit taken, as for e.g. minutes or digits) by the latitude, and again by the degrees of the Moon east or west of the meridian. Divide this by 8100. By so many units is the east or west point of contact bent northward or away from the north respectively if the Moon is east of the meridian, and bent away from the north and north- ward respectively, if the Moon is west of the meridian. The instructions to obtain the directions of the points of contact have been explained by figures 5 a, b, c. Fig. VI. 5-a. Fig. VI. 5-b. 6a. A. स्थितदल A. क्रते च यो; B. तते च यो; D. हते. A2. ॰दशघ्न b. A. विशेषको मे; B. विशेषका मे; C. विशेषके तमः; c. A1. परिध्ये; A2. परिधौ D. विशेषकाले d. A. ग्रहणास्तांशे; B. ग्रहणा स्वांशे; D. ग्रहणाशा (तद्) A. सकलीमतीतॊन्दुं; B. सकलमतीन्दुः (B3. ॰न्दुं); वदेत् पर्व D. ऽसकलं तमोग्तीन्दुम् 8a. A. B. ॰दलाद्द्विघ्ने c. A. प्रग्रहमोक्ष; B 1.3. प्रग्रहमा क्ष; C. D. प्रग्रहमोक्षे b. A. B. वाक्षे 7a. C. विपर्यस्ता; D. विपर्यासः c. B. रूपाष्टद्भवते b. C-D. तुरीयभागे d. A. B. वलनावामं युते (B. च्युते); D. वामं परे सव्यम् ।

164 PAÑCASIDDHĀNTIKĀ VI.8 In all the three figures, M, M₁, M₂, M' represent the centres of the Moon, and S, S₁, S₂, the centres of the Shadow. S' is the point of contact. R is Rāhu. In fig. 5a, R M₂ is part of the Moon's orbit, and RS₂ is part of the ecliptic. In position S₂ which is the limit for the occurrence of an eclipse, MS₂ = 13°, and M₂S₂ is the latitude, equal to 55', = the sum of the semi-diameters, i.e., M₂S' + S'S₂. S', the point of contact, is seen 90° from the east point, directed towards the north from the ecliptic, i.e. at the north point of the Shadow, but at the south point with reference to the Moon. In position S, the Moon is at the node, Rāhu, and (Moon ~ Rāhu) is 0°. Clearly, S', the first point of contact, is at the east point. In position S, between the above two, it is seen that S', the first point of contact, makes an angle P M₁ S₁ with the east point, on the south, with reference to the Moon. It may be seen that the sine of the angle is proportionate to PS₁, the latitude, which itself is propor- tionate to (Moon ~ Rāhu) as we have shown. Thus, at any intermediate position, the point of con- tact makes an angle with the east, whose sine is proportionate to Moon ~ Rāhu. Hence the rule to divide the Moon's half opposite to the direction of the latitude into 13 parts by parallel lines at equal intervals, and take the point of contact of that line which corresponds to the degrees, Moon ~ Rāhu. This is shown clearly in fig. 5c. Here P' S', the sine of the angle P'MS', which is the direction, is seen proportionate to PS, the latitude, which is proportionate to Moon ~ Rāhu. The figure is for Moon ~ Rāhu equal to 7°. Fig. 5b is intended to show both the first and last points of contact, and because of the increase (or decrease) of the latitude during the interval, there is an increase (or decrease) in the angle. In the figure, the angle of first contact, P'MS', corresponds to the latitude MA and is smaller; the angle of last contact, P'M'S' corresponds to the greater latitude M'B, and is greater. It is also to be noted that the last contact is at the western part of the Moon, the Moon now being east of the Shadow. The directions mentioned above are with reference to the ecliptic, taking it as east-west, (neglecting the angle of inclination of the Moon's orbit). But the directions have to be given as seen by the observer. For this, two corrections have to be applied, one to convert it with reference to the east- west of the equator, called Āyana-valana, and the other to correct it for the east-west of the place. depending on the latitude of the place, called the Ākṣavalana. Both these have been mentioned and explained in connection with the observation of the first appearance of the moon given in Chap V. The author here gives the Ākṣavalana alone following the original Siddhānta, neglecting the other one, though that is not negligible. Even in this, he takes into consideration only the northern hemis- phere. There the celestial equator is inclined south. An observer facing east looking at a body on the celestial equator sees the east point bent northward, and the west point bent southward. Simi- larly, an observer facing a body west, sees the east point bent south, and the west point bent north.

VI.10 VI. VĀS.-PAUL. SIDDHĀNTA – LUNAR ECLIPSE 165 The directions are changed accordingly. (For illustration see fig. 7, below given under example 3). All this has been explained in chap. V. The Siddhānta takes it that the bending is proportionate to the latitude, being zero at the equator and 90° at the pole. So, in terms of the length of the circumference, the bending = (latitude/90°) × circumference/4. But this amount of bending is only at the horizon. On the meridian there is no bending. In between, the Siddhānta takes it as proportionate to the angle of the Moon from the meridian. Thus, Bending = (latitude/90°) × (circumference/4) × (degrees of the Moon from the meridian/90°) = quarter circumference × latitude × degrees of the Moon from the meridian/8100, as given. As for the Moon being "devoured", it is a figurative expression, the Shadow being identified with the demon, Rāhu, in the Purāṇas. In saying that Moon ~ Rāhu should be done once for finding the point of first contact, and again for the last contact, the author recognises that the difference may be considerable, and thereby indicates that it will be good if the times also are computed separately, using the different latitudes. But TS and NP by their emendation, pragrahamokṣe, have shut out all this suggestion. TS do not seem to understand why the division into thirteen parts is instructed to be made, for Dr. Thibaut says, "We do not know the reason for the direction, given in stanza 7, to divide each quarter of the circumference into thirteen parts." (p.45). It is not each quarter, and it is not the quarter-circumference that is to be equally divided. We have emended grahaṇāsvāṃśe into grahaṇāśāṃśe, whereby we understand that the point of contact is at the point where the parallel line corresponding to Moon ~ Rāhu meets the circumference. The emendation grahaṇāśā 'tad' vadet by NP is not warranted. In the matter of the directions of Ākṣavalana, Thibaut says the opposite of what Sudhakara Dvivedi says, and neither gives the cor- rect direction. (vide. Com. and Translation) [ग्रहणकालः वर्णं च] [तिथ्यन्ते ग्रहमध्यं प्राक् परतः स्थितिदलेन चाऽऽद्यन्तौ । रक्तकपिलौ च वर्णावुच्चाऽधस्स्थे परे नितराम् || ८A ||] सर्वग्रासिन्येवं वर्णविशेषं वदेन्निशानाथे । उदयास्तमये धूम्रं खण्डग्रहणे (सलिलदाभम्) || ९ || राहुमुखोनं चक्रं [धीद्वियम] गुणं शशाङ्कसंयुक्तम् । (जूकेत्थगेऽयमुच्चः) क्रियादिकन्यान्त (गे) नीचः || १० || Moment of the eclipse and its colour 8A-9. The middle of the eclipse is at the moment of new moon. The times of first and last contacts are earlier and later than the middle, by half the time of duration. When the eclipse is total, the colour of the Moon is red or brown as it is farthest or nearest to the earth, respectively, and mixed, more or less, in between. When the eclipse is near sunset or sunrise, the Moon is smoky in colour. When the eclipse is partial, the Moon has the colour of raincloud.

166 PAÑCASIDDHĀNTIKĀ VI.10 10. Subtract the Head of Rāhu from 12 rāśis, multiply it by 228, and add the Moon's longitude. If this is between 6 and 12 rāśis, the Moon is farther, and if between 0 and 6 rāśis, it is nearer. (The idea is, that the nearer this sum is to 9 rāśis, the farther is the Moon and its colour at total eclipse is nearer to red. The nearer this sum is to 3 rāśis the nearer is the Moon, and its colour is nearer to brown). Since this Siddhānta uses the latitude, (or Moon ~ Rāhu) at full moon to find the duration, the part of the duration before full moon is equal to that after full moon, and the middle is at full moon. But other siddhāntas repeat the work, using the latitudes at first and last contacts separately, so that the two parts are not equal, and the middle does not occur at full moon. Still all siddhāntas techni- cally call the moment of full moon as the middle, since at that time the eclipse is practically the maximum. As for the colour of the Moon at eclipse, it is based on observation, and given slightly differently by different Siddhāntas. Some take the fraction of the Moon eclipsed as the criterion for the colour, others the time of the eclipse and its nearness to sunset or sunrise, etc. Here, this Siddhānta uses, in addition, a new criterion, not given by any other Siddhānta, viz. the distance of the Moon from the observer, and there is truth in what the Siddhānta says. Here, it may be asked how at all is it possible for the Moon to have any colour at eclipse. It is an opaque body, and what illumination it has comes from the Sun's rays falling on it. When it is immersed in the Sun's umbra, i.e. full shadow, (we consider the Moon in umbra alone as eclipsed, and not in pen-umbra), the Sun's rays cannot fall on it. It cannot be the earth-shine falling on the Moon and dimly illuminating it, as in the crescent Moon, giving rise to the popular belief of "the old Moon in the arms of the new". At times of new moon, when the lunar eclipse occurs, there is no earth-shine opposite the Moon to illuminate it. This is the answer: Though the Moon is in the earth's shadow geometrically speaking, the Sun's rays, refracted by the earth's atmosphere, fall on the Moon and illuminate it with a red or brown glow, red light alone being able to reach the Moon after passing through the long section of the earth's atmosphere undispersed, on account of its greater wave-length. (See fig. 6). 8A.9. Quoted by Utpala on BS 5.18 8A. Om both in A and B, but included in this edition on account of its essentiality in verses 10-14 as 11-15. this context and its being quoted as a verse of VM by Utpala in continuity c. A.B. उद्यास्तांगांसधूम्रं (b. धूम्र); C. उद्यास्तग्रासधूम्रं with verse 8. d. A.B.C. ०णे च सलिलाभम् (B. om च) While C om its this verse since it is not 10a. D. मुखोनचक्रं available in the text mss. D adds it. b. A.B.C.D. त्रियमद्भिगुणं; D. शशि [हीन] संयुक्तम् (as no. 9, and the further verses (A. संयुक्तन्) numbered as 10 etc.) since Utpala has it. c. A.B. एपिक्लेशोयमुक्ष (B. क्लेशो), (A2 यमुध); C. एभिः क्लेशोऽयमुच्चं; D. अभिक्लॆशोऽयमुच्चः 9a. A. पर्वग्रांसिन्येवं; C. ग्रासे पीनं d. D. क्रियादिः. B. गो नीयः; D. गो नीचः D. numbers the verse as 10 and the

VI.10 VI. VĀS.-PAUL. SIDDHĀNTA – LUNAR ECLIPSE 167 [चित्र: Fig. VI. 6 - Sun, Earth, Moon, Moon's orbit] Fig. VI. 6 In its nature this phenomenon is similar to the Sun apparently rising earlier and setting later, and appearing red at both times. As for the distinction between red and brown, it can be seen from the figure that there is great illumination at a greater distance, and, so, when nearer, there is less red, which gives a brown colour as mentioned by the author. The redness may vary by other causes, like the dust or water vapour in the atmosphere, and it goes to the credit of the ancients that distance was distinguished as one cause, affecting the redness. When the Moon is partially eclipsed, the glare of the illuminated part dims more or less the red- ness of the eclipsed part, so that it looks almost dark like a rain cloud. Near sunset or sunrise, the red glow from the Moon has again to pass a long distance through the atmosphere, and get filtered out, so that the colour becomes smoky. Verse 8-A is given only by Bhaṭṭotpala, the text manuscripts omitting it. The original of VM must have contained this verse, for it supplies several lacunae. The word evam in verse 9 requires a pre- vious verse mentioning colour. Omission of this verse has compelled TS to emend sarvagrāsiny evam into sarvagrāse pītam. The rule in verse 10, giving when the Moon is at ucca and nīca has a purpose only with 8-A which requires the information. If TS had verse 8-A before them, they would not have misunderstood 10, and declared that it is something pertaining to astrology. It also instructs us how to find the times of the first and last contacts, and when the middle occurs, which instruction we assumed before, in working the examples. NP rightly include this verse in brackets and give it the number 9, the further verses being numbered from 10. We have said that verse 10 gives a rule to find when the Moon is far and when it is near. We shall explain how: The rule says that when the result got by the rule is nine rāśis, the Moon is at the greatest distance, i.e. at ucca, and when the result obtained is three rāśis, it is nearest, i.e. at nīca, as we have explained. Therefore if the rule gives about nine rāśis for the Moon at ucca, i.e. when the Moon is replaced by ucca, then it must be correct. We shall show that it is so. The rule is: Moon − 228 R = ucca − 228 R (where R is the Head of Rāhu at any full-moon). At the full moon just preceding the Epoch, by II.3, taking the reading vasumuninava etc. it can be cal- culated that according to Vāsiṣṭha-Pauliśa, the ucca is rā. 8-14-21.4. (The Siddhānta does not give the ucca direct, but we can find it by the relation, Moon − Moon's kendra = ucca). Every synodic month the ucca increases by 3° 17′.3 according to this Siddhānta. Therefore after m synodic months, the ucca is rā. 8-14-21.4 + m × 3° 17′.3. The corrected Rāhu, R, at the full moon before Epoch is rā. 7-26-45.2. It decreases by 1° 33′.88 every synodic month. Therefore, after m synodic months, R = rā. 7-26-45.2 − m × 1° 33′.88 ∴ ucca − 228 R = rā. 8-14-21.4 + m × 3° 17′.3 − 228 × (rā. 7-26-45.2 − m × 1° 33′.88) = rā. 8-14-21.4 − 228 × rā. 7-26-45.2 + m (3° 17′.2 + 228 × 1° 33′.88) = rā. 8-14-21.4 − rā. 11-9-45.6 + m (3° 17′.3 + 356° 43′.9)

168 PAÑCASIDDHĀNTIKĀ VI.13 = rā. 9-4-35.8 + m × 1'.2 = practically nine rāśis, for a long time after or before Epoch. Nineteen years before Epoch, this would have been exactly nine rāśis. A small difference in Rāhu, (if it is 1'.2 more) would make it nine rāśis even at the taken time. It must be noted that one or even two rāśis either way will not matter in our context, of redness at one end and brownness at the other, for the difference between redness and brownness itself is slight. Only after two or three thousand synodic months will there be perceptible difference, and the rule cease to hold good. From the proof of the rule it will be seen that our emendation of triyamadviguṇam into dhīdviyama- guṇam is necessary. triyamadviguṇam had perhaps been wrongly written by some scribe who had the 'saros', (consisting of 223 lunations) in his mind. As antithetical to kriyādikanyāntago nīcaḥ, we have emended the meaningless group of letters, epikleśoyam ucca into jūketthageyam uccaḥ. Neither TS nor NP seem to have understood the significance of this verse. TS observes on it: "A stanza of doubtful import, see the Sanskrit commentary" (Tr., p.45) and the Sanskrit commentary suspects it to be of astrological import (com., p.34). NP gives an incorrect translation and says "The synodic months in an 18 year eclipse cycle is 223, but the role of this number in the present context remains obscure to me" (Pt. II, p.55) [ग्रहणपरिलेखा:] सप्तदशाष्टत्रिंशत् तद्द्वयलिप्ता (युतोने) सूत्रेण । शशि (राहु) स्थितिवृत्तान्येक (स्थानानि चाऽऽलिख्य ॥ ११ ॥ प्रोक्ता (शशाङ्क) लङ्का-पूर्वाऽप (रायाश्च) पार्श्वयोश्चाऽपि । आयामिन्यो रेखास्त्रयोदश समान्तराः कार्याः ॥ १२ ॥ चन्द्रच्छेदकमेतद् व्याख्यागम्यं समासतोऽभिहितम् । ग्रासविमर्दस्थितयः संस्थानेनाऽत्र दृश्यन्ते ॥ १३ ॥ Diagrammatic representation 11. Draw three concentric circles with radii 17, 38 + 17 (= 55), and 38 - 17 (= 21), minutes of arc. These circles relate to the Moon, the duration and the obscuration, respectively. (Drawing the part of the Moon's orbit forming the path of the Moon), mark the points (of first and last contacts) and also those of inversion and emergence if any). 12. Draw the diameter (making an angle equal to the Valana given in verses 7-8), with the ecliptic which, (according to this Siddhānta), is east-west with reference to the equator. (This diameter shows the east-west of the place). (As shown in fig. 5c) draw thirteen equally spaced lines parallel to this east-west diameter. (The directions of the points of contact etc. are given by this figure).

VI.13 VI. VĀS.-PAUL. SIDDHĀNTA – LUNAR ECLIPSE 169 13. The graphical representation of the lunar eclipse has here been described briefly, and can be understood properly only by explanation (followed by demonstration). From this, the total duration, the total obscuration, the magnitude, etc. can be found by inspection. The representation given by the author can do duty for all the figures used by us to explain verses 3, 5, 6 and 7, the centre of the concentric circles being the centre of the Shadow in each. An impor- tant difference is that in the previous illustrations, the orbs, of the Moon and the Shadow were shown separately, while here they are replaced by one circle for each of duration (radius = 38' + 17') and totality, (radius = 38' - 17'), the Moon being reduced to a point coinciding with its centre. As the points of contact etc. showing the direction cannot be marked on the point-Moon, another circle is drawn to represent the Moon, with the same centre. As the directions of the points on the two circles are diametrically opposite to the directions of the same points with reference to the Moon, the points can be marked on the Moon-circle by the intersection of the diameter on its oppo- site half. An examination of fig. 7 will show this, and an example will make everything clear. Example 3. (Note: This example is intended only as an illustration). The latitude of a place is 20°. The full moon occurs 4 nāḍīs after sunset. The Moon at that time is rā. 5-15-0, and the Sun, rā. 11-15-0. The uncorrected Rāhu (Head) at full moon is rā. 5-6-36, from which the latitude is 35'N. The full moon is 11 nāḍīs before mid-night. The daily motion of (Moon - Sun) = 780'. Find the times of first contact etc., and verify by a graphical representation. (i) The corrected Rāhu = rā. 5-6-36 - 1° 36' = rā. 5-5-0. Moon ~ Rāhu = rā. 5-15-0 - rā. 5-5-0 = 10°. This is less than 13°. ∴ there is a lunar eclipse. (ii) Minutes of duration = √(55² - lat²) = √(55² - 35²) = 42, minutes of arc. Now, 2 × 42 × 60 ÷ 780 = nā. 6-28 = Uncorrected duration. Difference between Moon and uncorrected Rāhu = rā. 5-15-0 ~ rā. 5-6-36 = 8° 24. Correction = 8° 24' × 5 = 42 vināḍīs. As Rāhu is less, deducting from uncorrected duration, the correct duration = nā. 5-46. Half this is nā. 2-53. Subtracting from the time of new moon, 4 nāḍīs, the first contact is at nā. 1-7 after sunset. Adding, the last contact is at nā. 6-53. (iii) Moon ~ cor. Rāhu = 10°. As this is greater than 5°, there is no total phase. (iv) The first contact is nā. 15-0 - nā. 1-7 = nā. 13-53 before midnight, i.e. the Moon is 83° east of the meridian, 11a. A.B. दशाष्टा 12a. A. प्रोक्तायां सकलङ्का; B. प्रोक्ता यो सवूलांका (B2. बू) B1.2. त्रिशतद्वय; B3. त्रिंशद्वय D. प्रोक्ताया [मं] शकलका [त्] b. A. मतेन सूत्रेण; B.C.D. मितेन सूत्रेण b. A.B.D. पूर्वापरयोश्च (A om र); C. पूर्वापरायाश्च c. A.B. शशिना बहुस्थिति (A2. नां) d. A. समात्ताराः; B. समां ताराः d. A. नि एक छेना निष्पालेख्य; B. नि एकछत्रो निवासलेख्य; 13a. A. छेदक; B. षेदक; C.D. छेद्यक D. न्येकस्थानि वा संलेख्य d. B1. मंस्थ्यानेनात्र (B2.3. सं०)

170 PAÑCASIDDHĀNTIKĀ VI.13 The Moon's circumference = 17' × 44/7 = 107'. The bending of the direction due to latitude = (107'/4) × 20 × 83 ÷ 8100 = 5½ minutes of arc. As Moon is east of the meridian, the equatorial east point is bent 5½' north of the east point of the place, i.e. the east point of the place is situated south by 5½ minutes-length. At the time of last contact, the Moon is 49° east of the meridian. The bending of the point of last contact at the western limb of the Moon =(107/4) × 20 × 49/8100 = 3.2', southward, as the Moon is east of the meridian, i.e. the west point of the place is 3½ minutes-length north of the equatorial west point. (v) We shall show all these graphically in Fig. 7. The latitude to be used in the figure is the corrected latitude = 10° × 380' ÷ 90 = 42'N. [चित्र: Fig. VI. 7] Moon's orbit, Eclipse circle M', M Moon's counterpart to mark directions West point as seen from place at last contact W, Ecliptic, E Total eclipse circle Position of last contact, Position of first contact East point as seen from place at first contact Fig. VI. 7 M, M', are the points of first and last contacts. MM' = duration in minutes of arc. MM' = 3".6 by measurement, = 3.6 × 20' = 72'. From this, the time of duration = 72 × 60 ÷ 780 = nā. 5-32.

VI. 14 VI. VĀS.-PAUL. SIDDHĀNTA – LUNAR ECLIPSE 171 See how close this is to the time calculated, viz. nā. 5-46. The Moon’s orbit does not touch the circle of totality. Therefore there is no total obscuration, as already found by calculation. The arc in the figure from the east point of place to the first point of contact gives the direction of the point at the beginning, and the arc from the west-point of place to the point of last contact gives the direction at the end. We have amended matena into yutona, nabahu into Rāhu, ekachanoni into ekasthānāni, yāmasabū into śaśāṅka and pūrvāparayośca into pūrvāparāyāśca as necessitated by the context and proximity of the lettering. [रविचन्द्रग्रहणयोर्भेदः] स्वे भूच्छायामिन्दुः स्पृशत्यतः स्पृश्यते न पश्चार्द्धे । भानुग्रहेऽर्कमिन्दुः प्राक् प्रग्रहणं रवेर्नाऽतः ॥ १४ ॥ Lunar and solar eclipses – Differences 14. In the lunar eclipse, the Moon, (moving eastward), contacts the earth’s shadow. Therefore the ‘first contact’ (occurs at the eastern limb of the Moon, and so) does not occur at the Moon’s western limb. In the solar eclipse, the Moon meets the Sun, and therefore, (the Sun being contacted as its western limb), the first contact does not occur at the eastern limb of the Sun. The Moon’s motion being more than thirteen times that of the Sun or the Shadow, (whose motion is the same as that of the Sun), it moves eastwards relative to the Sun or Shadow and con- tacts them at their western limb, and its own eastern part. As the lunar and solar eclipses are with reference to the Moon and the Sun, respectively, the first contacts are at the eastern and western limbs, respectively. It need not be mentioned that the last contacts are, respectively, on the western and eastern limbs. 14. Quoted Utpala on BS. 5.12 C. स्पृशति तथा स्पृशति दृश्यते पश्चात् । 14a. U. स्वं. B 1.2. भूयच्छायां; B3. भूच्छायां. A.B. ॰मिन्दु A. पश्चार्द्धं; B.D. पश्चार्धः b. AB.3. स्पृशतः स्पृ०; B 1. 2̂. स्पृश्यतः स्पृ० c. A.B1. भागनुग्रहे. A.B.C.D.U.मिन्दोः d. A. रवे [इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां चन्द्रग्रहणं नाम षष्ठोऽध्यायः ।]¹ Thus ends Chapter Six entitled ‘(Vāsiṣṭha-) Paulīśa Siddhānta: Lunar Eclipse’ in the Pañcasiddhāntikā composed by Varāhamihira

1. A.B.D. चन्द्रग्रहणं षष्ठोऽध्यायः C. इति चन्द्रग्रहणं नाम षष्ठोऽध्यायः 14

Chapter Seven (PAULIŚA-SIDDHĀNTA) – SOLAR ECLIPSE ७. सप्तमोऽध्यायः पौलिशसिद्धान्तः — रविग्रहणम् Introductory This chapter deals mainly with the solar eclipse according to the Pauliśa. But the last two verses giving the computation of the solar eclipse gives the lunar eclipse also. It is from this that we have to conclude that the lunar eclipse of Chap VI is that of the Vāsiṣṭha. The method of the Pauliśa for correcting the Moon’s latitude for parallax is peculiar. The correction is done on Rāhu, and thence carried to the latitude. Also, this is the earliest siddhānta to deal with the solar eclipse, and thus, with its peculiar method, historically important. [लम्बनम्] दिनमध्यमसंप्रा(प्या) यावत्यो नाडिका व्यतीता वा | ताभ्यः षड्गुणिताभ्यो ज्यात्रिंशांशस्तिथे(नामः) || १ || Parallax of longitude

1. Find the interval between mid-day and the time of new moon, in nāḍīs. Multiply this by 6. Degrees are got. Find its sine. Divide it by 30. The result is the parallax in nāḍīs to be deducted from the time of new moon if new moon is before mid-day, and to be added to the time of new moon, if after mid-day. The new moon corrected for parallax in longitude is obtained. Thus: i. nāḍīs of parallax = sine (interval in nāḍīs between mid-day and new moon × 6) ÷ 30. ii. Parallax corrected new moon = new moon ∓(i), minus for forenoon, and plus for afternoon. The rationale of parallax correction is as follows: A lunar or solar eclipse occurs when the Moon gets so close to the earth’s shadow or the Sun, that it enters the Shadow so as to be darkened by it, or hides the Sun from the observer’s view. Now, the Moon being darkened by the Shadow is practi- cally independent of the position of the observer on the earth. But the Moon hiding the Sun depends upon the observer’s position, owing to parallax. So parallax correction has to be done in the solar eclipse. The critical angular distance is the sum of the semi-diameters. The angular dis- tance between the Sun and the Moon is calculated from the longitudes of both and the latitude of 1a. A.B.C.D. संप्राप्ता c. B. गुणितान्यो b. B1.2. यावन्त्यो A. व्यतीता वत; B. व्यतीता वत्त | d. A.B. ज्यास्त्रिंशांशः A.B. तीथिनाम; C. तिथेर्नाम

VII.1 VII. PAULIŚA-SIDDHĀNTA — SOLAR ECLIPSE 173 the Moon. These being given with reference to the centre of the earth, the angular distance calcu- lated is as seen by an observer at the centre of the earth. But we want the distance as seen by an observer on the surface of the earth, and a correction has got to be made for this. This is correction for parallax or simply parallax. See Fig. 1-a. Z D | / | / | / | / | / S | / / P | / / .---. / / / | \ / M / | X / | | / \ / | | / \ / | C / \ / C Earth's centre \ / / P Position of observer \ / / Z Zenith '---' M Moon S Sun Fig. VII. 1-a. (Note: The figure is only diagrammatic and does not represent actual distances). PD = a line drawn parallel to CM. The observer at C sees the Moon along CM, and for him the Moon’s zenith distance (z.d.) is ZCM (= ZPD). But the observer at P sees the Moon along PM, and its z.d. for him is ZPM. This is equal to ZPD + DPM = ZCM + PMC, and PMC is the parallax correction to ZCM. It can be seen that the parallax correction for the Sun is PSC, and less than that for the Moon. It is actually about 2/27 of the parallax of the Moon, according to the Hindu Siddhāntas, its distance being about 27/2 times that of the Moon, according to them. (It must be noted that actually the Sun’s distance is about 390 times the Moon’s, and accordingly the Sun’s parallax is about 9", and practically negligible). The amount of parallax, PMC can be calculated trigonometrically thus: Sin PMC/PC = sin CPM/CM = sin (PCM + PMC)/CM = sin PCM/CM (∵ PMC is small). Sin PMC = sin CPM × PC/CM. Arc PMC × 120/57.3 = sin CPM × PC/CM (∵ PMC is small). Parallax of Moon in minutes = sin z.d. × (PC/CM) × 60 × 57.3

120 = sin z.d. × 28.65 × Earth’s radius/Moon’s distance. Similarly, the Sun’s parallax in minutes = sin z.d. × 28.65 × Earth’s distance ÷ Sun’s distance.

174 PAÑCASIDDHĀNTIKĀ VII.1 At a solar eclipse, z.d. is practically equal for the Moon and the Sun, and in measuring the angular distance between the Sun and the Moon relative parallax can be applied to the Moon, the Sun being supposed unaffected. We can write: Relative parallax = Moon's parallax −Sun's parallax = sin z.d. × 28.65 × earth's radius × (1/Moon's dis. − 1/Sun's dis.) When the Sun or Moon is at the horizon, sin z.d. is 120, and the relative parallax, (hereafter we shall call it merely parallax), called horizontal parallax, is a maximum, and this Siddhānta takes it as equal to 49'. ∴ Parallax = 49' sin z.d./120. Also, we can see from the fig. 1a that the Moon is depressed, away from the zenith by parallax, along the vertical circle, ZM, increasing the z.d. and that is why it is called lambanam, i.e. 'depression', in Sanskrit. This general parallax has to be resolved into two parts, correction to longitude, (p. long.) and cor- rection to latitude (p. lat.). This is shown in fig. 1b. Z Zenith N Nonagesimal (Lagna − 90°) O Orient ecliptic point (Lagna) A First point of Aries MAQ = ω Fig. VII. 1-b.

VII.4 VII. PAULIŚA-SIDDHĀNTA — SOLAR ECLIPSE 175 ZM = z.d., and MM′ is the general parallax. PM′ is the parallax in latitude, and MP, that in longitude ZQ. = latitude of the place, and ZN is the zenith distance of the nonagesimal (z.d.N.). We shall find an expression for PM′, the p.lat. p.lat = PM′ = MM′ × sin M′ MP/120 = 49′ sin ZM × sin ZM N/120² = 49′ sin ZN/120 = 49′ sin zdN/120, (from rt △ MNZ). Similarly for p.long, i.e. MP, p.long = MP = MM′ cos M′ MP/120 = 49′ sin ZM. cos ZMN/120² = 49′ cos ZN × sin MN/120² (from rt △ MNZ) = 49′ cos z.d.N × cos OM/120², OM being (lagnam − Moon). Clearly, this is positive when the Moon is east of the nonagesimal, and negative when west. It is p.long. that we are concerned with in this verse, and it must be given in terms of the hour- angle (h, or natāṃśa) cos z.d.N × cos OM = (i) cos w. cos Ø. sin h. sec δ [δ being the Moon’s declination] + (ii) [sin w × cos Moon’s longitude × {cos Ø × tan δ × sin (nāḍīs after sunrise or before sunset × 6°)/120²} − sin Ø/120]. The Siddhānta omits (ii) which is small in comparison with (i), sin Ø/120 being small in India. For the same reason, and as w and δ cannot exceed 24°, it takes cos w. cos Ø. sec δ as 120. ∴ p.long = 49′ sin h/120. As said before, this is positive, i.e. it increases the Moon’s longitude when h is east, and negative, i.e. it decreases, when h is west. Therefore the corrected time of new moon is earlier and subtractive in the forenoon, and later and additive in the afternoon. Now, 49′ of p-long, converted into time, using the mean relative daily motion of the Sun and the Moon, (731′.5), = 49′ × 60/731′.5 = 4 nāḍīs nearly. ∴ nāḍīs of parallax = 4 × sin h/120 = sin h/30, as given by the text. If (ii) is not neglected, the Moon being east or west of the nonagesimal will be the criterion for the subtraction and addition of the nāḍīkās. [नतिः] पञ्चघ्नात् ‘(त्रि)घना’ऽऽप्तादक्षान्मुखपु(च्छ)योर्धनर्णे तत् | (सशशि)चरणा(पमगुणा) धनर्णनाड्यो ‘[धृति]’विभक्ता ॥ २ ॥ उदगयने पूर्वार्धे धनमृणं दक्षिणे प्राच्याम् | पश्चाद्धनं तु याम्ये (दि)गुदग्गुणं वामतः पुच्छे ॥ ३ ॥ दिनयातशेषनाड्यश्चन्द्रा(पम)संगुणास्त्वशीतिहताः | (मे)षतुलादि ऋणधनं विपरीतं वामतः पुच्छे ॥ ४ ॥

176 PAÑCASIDDHĀNTIKĀ VII.4 Parallax in latitude 2. Multiply the degrees of latitude by 5 and divide by 27. Add or subtract the resulting degrees, respectively, to Rāhu's head or from Rāhu's tail, where the Moon is situated. (i) 3. Add three rāśis to the Moon, and find its declination in degrees. This multiplied by the nāḍīs of parallax (given by verse 1) and divided by 18, are to be added to the Head if it is forenoon and Uttarāyaṇa (i.e. the Sun is in its northward course), or afternoon and Dakṣiṇāyana. The degrees are to be sub- tracted from the Head if it is forenoon and Dakṣiṇāyana or afternoon and Uttarāyaṇa. For the Tail, the addition and subtraction should be inter- changed. (ii) 4. Take nāḍīs from sunrise to new moon if forenoon, the nāḍīs from new moon to sunset if afternoon. Multiply these by the degrees of the Moon's declination and divide by 80. The resulting degrees are to be added to the Head if the Moon's longitude is between 6 and 12 rāśis, and subtracted if bet- ween 0 and 6 rāśis. For the Tail, interchange the addition and subtraction. (iii) The corrections are: i. Ø° × 5 ÷ 27 ii. 'Degrees' × the nāḍīs of verse 1 ÷ 18, where 'Degrees' are to be got from sin 'degrees' = sin w sin (Moon + 90°)/120 iii. Degrees of Moon's declination × the time in nāḍīs from sunrise to (parallax corrected) new moon, or to sunset from parallax corrected new moon ÷ 80. The addition or subtraction is as instructed in the translation above. These rules follow from the formula derived already for parallax correction in latitude: p-lat = 49' sin z.d. N/120. 49' sin z.d. N/120 = sin w. cos Ø sin (AQ)/120³ − cos w. sin Ø/120², (from the two rt. △ s. Z NA and Z QA) = (i) − 49' cos w, sin Ø/120² + (ii) − 49' sin w cos Ø sin (Moon + 90°) × sin h. sec δ/120³ + (iii)

• 49' cos w. cos Ø tan δ. cos h/120³, (h being the hour angle of the Moon at parallax-corrected new moon). 2a. A.B1. पञ्चघ्नलि०. D. यमाप्ता b. B1. दक्षन्; D. [क्षेपे] क्षे मुख . A. पुच्छयोः; B1.2. पृष्ठयोः c. A. यक्षाधनं A.B. धनर्णं तन्न |; D. धनर्णः d. A. om दिग्; B1. दृग्गुणं; B3. दृगुड्गुणं; C.D. om दिग् c. A.B.C. राशिचरणा०; D. तद्राशिचरणा०; A. पुछे; B. पृष्ठे A.B.C.D. ०यनगुणं 4a. B. शेषं d. A.B.C. धनमृणनाड्यो (B1.3. ०ताड्यो). b. A.C.D. चन्द्रायन; B. चन्द्रानयन A. त्रशीति; A.B.C.D. दिक्० B. श्काशीति B. विभक्ता B1. ऋताः; B2.3. क्षताः 3a. B. धनं मृणं c. A.B. शेष. D. तुलाघ्णं घनं b. A. दिणे; B. दक्षिणं d. B. विपरीतं. A.B. पुछे