132 PAÑCASIDDHĀNTIKĀ IV. 56 Radius: 'Bhuja' :: shadow-hypotenuse : shadow-Bhuja, So we have, (Sūryāgrā ± Agrā) × shadow-hypotenuse ÷ 120 = shadow Bhuja. Our author calls this Koṭi or ‘Perpendicular’, as we have already said. But this does not matter, for in a right angled triangle, with the hypotenuse given (as here the shadow), the other two sides are perpendicular to each other, and any one may be taken as the base, and the other as the perpen- dicular. (v) Base: When the ‘Perpendicular’ is got from the well-known formula of the right angled triangle, Base² + Perpendicular ² = hypotenuse², (the shadow being the hypotenuse here,) we have, ‘Base’ = √shadow² − Perpendicular². Since the Perpendicular is north-south, the ‘Base’ is east-west, and is a part of the east-west line, as the foot of the shadow is on the east-west line. Since the east-west line is known, we can lay the ‘Base’ on it, lay the ‘Perpendicular’, at right angle, and draw the shadow. Clearly, if initially we have the shadow marked on the ground, we can get the directions by using this method. The angle between the shadow and the base gives the direction of the shadow. Obviously the direction of the Sun is given by the equal angle vertically opposite. What has been said here for the shadow may be said for the sun without reversing the direction as we have done for the sake of the shadow, and the Sun’s direction can be got. From that the direc- tion of the shadow may be got as being vertically opposite. But it is clear that the author says every- thing here for the shadow, and not for the Sun. [छायातः रव्यानयनम्] छायासमरेखान्तरगुणिता त्रिज्या स्वकर्णभक्ताऽस्याः । एकत्वे (विश्ले) ष्या सूर्याग्रा संयुताऽन्यत्वे ॥ ५५ ॥ लम्ब [क] गुणिता (भा) ज्या काष्ठामौर्व्या (ततो) ऽर्कः स्यात् । सूर्योद्गवेन विधिना ग्रहा (स्त) तोऽन्येऽपि कर्तव्याः ॥ ५६ ॥ Sun from Shadow 55. (Explanatory translation): By a process reverse to the previous one, the longitude of the Sun can be computed from the shadow, thus: Take the dis- tance of the tip of the shadow in aṅgulas, from the east-west line, multiply it by 120′, and divide by the aṅgulas of the shadow hypotenuse (mentioned in the previous work). This is ‘the sine’. (It may be seen that this is the Sūryāgrā ∓ Agrā, of the previous work). If the shadow is north of the east-west line then ‘the sine’ also is north. If the shadow is south, ‘the sine’ is south. Compute the Sūryāgrā as given already in the previous work. This is to be taken as always north (as already mentioned). If ‘the sine’ and Sūryāgrā are of different direc- tions, then ‘the sine’ plus Sūryāgrā is Agrā. (It must be remembered that they will be of different directions only when the Sun is in the northern hemis- phere, i.e. within the six signs from Aries). If they are of the same direction, then the Agrā is one deducted from the other. (If ‘the sine’ is greater, then the Sun is in the southern hemisphere, i.e. within the six signs from Libra. If

IV. 56 IV. THREE PROBLEMS 133 Sūryāgrā is greater, then the Sun is in the northern hemisphere, i.e. in the six signs from Aries). 56. The Agrā thus got multiplied by the sine of colatitude, and divided by 48′ 48″. is the sine of the Sun’s longitude and from that the sun is obtained. (From this sine, first the degrees of Bhuja, D, is got. If the Sun is in the northern hemisphere, then the Sun’s longitude is D, or (six signs − D). If the Sun is in southern hemisphere, the Sun’s longitude is six signs + D, or (twelve signs − D). What exactly it is of the diad must be determined by the Sun’s ayana). (Following the method for the Sun, the other grahas also can be got). The following are the steps in the work:– (i) As already seen, shadow-hypotenuse = √(shadow² + 144). (ii) As already seen, Sūryāgrā = 12 × 120 × sin lat ÷ (shadow-hypotenuse × sin colat). (iii) ‘The sine’ = the distance in aṅgulas from the east-west line to the tip of the shadow × 120′ ÷ shadow-hypotenuse. (If the shadow is north of the east-west line, ‘the sine’ is north, if the shadow is south of the east-west line, ‘the sine’ is south). (iv) (a) If ‘the sine’ is north, and greater than the Sūryāgrā, Agrā = ‘the sine’ − Sūryagrā, and the Sun is in the southern hemisphere. (b) If ‘the sine’ is north and less than the Sūryāgrā, Agrā = Sūryāgrā − ‘the sine’, and the Sun is in the northern hemisphere. (c) If ‘the sine’ is south, Agrā = Sūryāgrā + ‘the sine’, and the Sun is in the northern hemis- phere. (v) Sine longitude of the Sun = Agrā × sin colat ÷ 48′ 48″ = Agrā × sin colat × 5 ÷ 244. (vi) From the sine of longitude, the Bhuja degrees D, and using that the longitude of the Sun by examination, are to be obtained. As in the previous work, (iii), (iv) and (v) can be simplified thus: Sine sun’s longitude = (12 ×sin lat ± sin colat × the distance in aṅgulas between the tip of the shadow and the east-west line) × 150 ÷ (61 × shadow hypotenuse). In ± if the shadow is south of the east-west line then the upper sign is to be taken, and the Sun then is in the northern hemisphere. If the shadow is north, the lower sign is to be taken. In this case, if 12 × sin lat is greater, the Sun is in the northern hemisphere, and if sin colat × distance in aṅgulas, is greater, the sun is in the southern hemisphere. Example 25. Of a certain place, sin lat = 60′, sin colat = 103′ 55″. There, on a day during Uttarāyaṇā, when the length of the shadow is 5 aṅgulas, the distance of the shadow tip from the east-west line is measured to be aṅg. 1-36.6, north of the east-west line. Find the longitude of the Sun. (i) Shadow-hypotenuse = √(5² + 144) = 13 aṅg. 55a. A1. ॰न्वे तितेष्या; A2. ॰न्वे तिरतेष्या; A.D. सा ज्या D. ॰न्वेज्जारितैष्या b. A. काष्टा. A. मनोर्कः; D. हतार्कः d. A. सूर्याग्रा. A2. न्यवे c. A2. सूर्यो-वेन 56a. A. लम्बगुणिता d. A. ग्रहक्षतो

134 PAÑCASIDDHĀNTIKĀ IV. 56 (ii) Sūryāgrā = 12 × 120' × 60' ÷ (13 × 103' 55") = 63' 57".2 (iii) ‘The Sine’ = aṅg. 2-36.6 × 120 ÷ 13 aṅg = 24' 6". (This is north as shadow is north). (iv) As ‘the sine’ is north, the lower sign is to be used, i.e. the difference is to be found. There, as Sūryāgrā is greater, Agrā = 63' 57".2 – 24' 6" = 39' 51" (The Sun is in the northern hemisphere). (v) The sine of Sun’s longitude = 39' 51" × 103' 55" × 5 ÷ 244 = 84' 51". (vi) The Bhuja degrees D = Arc of 84' 51" = rāśi. 1-15. As the sun is in the northern hemisphere, the longitude is rāśi 1-15, or rāśi 6-0 — rāśi 1-15, i.e. rāśi 4-15. As the Sun is in Uttarāyaṇa, the longi- tude of the sun is rāśi 1-15. Using the simplified method, and taking the lower sign since the distance is north, sin Sun’s long = (12 aṅg × 60' ~ aṅg 2-36.6 × 103' 55") × 150 ÷ (61 × 13 aṅg.) = (720' – 271' 30") × 150 ÷ (61 × 13) = 84' 51". (As 12 × sin lat is greater, the sun is in the northern hemisphere). The rest of the work is the same. Example 26. Of a certain place, sin lat = 60', sin colat = 103' 55". On a Dakṣiṇāyana day, when the shadow is aṅg. 27-30, its tip is found to be aṅg 3-2.15 south of the east-west line. Find the Sun. (i) Shadow-hypotenuse = √(144 + 27½²) = aṅg. 30. (ii) Sūryāgrā = 12 × 120' × 60' ÷ (30 × 103' 55") = 27' 42".8. (iii) ‘The sine’ = aṅg. 3-2.15 × 120 ÷ aṅg. 30 = 12' 8".6 (south, as the shadow is south). (iv) As the sine is south, the upper sign is to be taken, and the Sun is in the northern hemisphere, and therefore, Agrā = 27' 42".8 + 12' 8".6 = 39' 51". (v) Sin longitude of Sun = 39' 51" × 103' 55" × 5 ÷ 244' = 84' 51". (vi) The degrees of Bhuja = Arc 84' 51" = rāśi 1-15 As the Sun is in the northern hemisphere, the longitude is rāśi 1-15 or rāśi 4-15. As it is Dakṣiṇāyana, the Sun is rāśi 4-15. Applying the simplified method for this, as the upper sign is to be taken, since the shadow tip lies south of the east-west line, sin long = (12 × 60 + 103' 55" × 3.2) × 150 ÷ (61 × 30) = 84' 51", and the Sun must be in the northern hemisphere. The rest of the work is the same as done already. Example 27. Of a certain place sin lat = 72', and sin colat = 96'. There, on a certain day in Uttarāyaṇa, when the shadow is aṅg. 27-30, the distance of its tip from the east-west line is aṅg. 16-37.5 north. Find the Sun. (i) Shadow hypotenuse = √(144 + 27½²) = 30. (ii) Sūryāgrā = 12 × 120' × 72' ÷ (30 × 96) = 36'. (iii) ‘The sine’ = aṅg. 16-37.5 × 120 ÷ aṅg. 30 = 66' 30", (north, as the distance is north).

IV. 56 IV. THREE PROBLEMS 135 (iv) As ‘the sine’ is north, the difference is to be taken. As ‘the sine’ is greater, Agrā = 66′ 30″ − 36′ = 30′ 30″, (and the Sun is in the southern hemisphere). (v) Sine latitude of Sun = 30′ 30″ × 96 × 5 ÷ 244 = 60′. (vi) The degrees of Bhuja = rāśi 1-0. As the Sun is in the southern hemisphere, the Sun is rāśi 6-0

• rāśi 1-0, i.e. rāśi 7, or rāśi 12-0 − rāśi 1-0, i.e. rāśi 11. As it is Uttarāyaṇa, the Sun’s longitude must be rā. 11. Applying the simplified method, since the lower sign is to taken as the distance is north, sin long = (12 × 72 ∼ 96 × 16-37.5) × 150 ÷ (61 × 30). Here since distance × sin colat is greater, Sin long = (96 × 16-37.5 − 12 × 72) × 150 ÷ (61 × 30) = 60′, and the Sun must be in the south- ern hemisphere. The rest of the work is the same. The proof of the above rules is as follows: In the previous work, the ‘Perpendicular’, i.e. the dis- tance of the tip of the shadow from the east-west line, was calculated, given the Sun and the shadow, and from that the ‘Base’ and the direction were calculated. Here, given the distance and the ‘Perpendicular’, the Sun is computed. Therefore this is the converse of the previous work, and can be derived from that. Steps (i) and (ii) are the same as steps (i) and (ii) of the previous work, and have been derived there. We shall therefore derive (iii), (iv) and (v) from (iii), (iv) and (v) there. In the previous work in (iv), ‘Perpendicular’ = (Sūryāgrā ∓ Agrā) × shadow hypotenuse ÷ 120. ∴ ‘The sine = (Sūryāgrā ∓ Agrā) = Perpendicular × 120 ÷ Shadow hypotenuse, as in (iii) here. Since, ‘the sine’ = (Sūryāgrā ∓ Agrā), when the Sun is in the northern and southern hemispheres, respectively, Agrā = Sūryāgrā ∼ ‘the sine’. It has been mentioned that Sūryāgrā is always north, ‘the sine’ is either south or north according to the line to the tip of the shadow from the east-west line, and Agrā is south if the Sun is in the northern hemisphere and vice versa. Therefore, when Agrā is north, (i.e. when the Sun in the southern hemisphere,) ‘sine’ is north, and greater than Sūryāgrā. Therefore, in using (‘the sine’ − Sūryāgrā), we get that the Sun is in the southern hemisphere. If Agrā is south, and therefore to be got negative by the addition of Sūryāgrā, (i.e. when the Sun is in the northern hemisphere), and ‘the sine’ is north, Sūryāgrā is greater than ‘the sine’. Here we have to use (Sūryāgrā − ‘the sine’), and we get that when the Sun is in the northern hemisphere. If Agrā is south again, (i.e. the Sun is in the northern hemisphere, again), and ‘the sine’ is also south, then we have the case, Agrā = Sūryāgrā + ‘the sine’, in which case also the Sun is in the northern hemisphere. From the Agrā, the sine of Sun’s longitude is got thus: In step (iii) of the previous work, Agrā = Maximum declination × sine longitude of the Sun ÷ sin colatitude. ∴ sin long. of the Sun = Agrā × sine colatitude ÷ max. dec. = Agrā sin colat ÷ 48′ 48″, as we get here in step (v). The explanation of getting the Sun’s longitude from its sine has already been given in connection with getting the sines for degrees (IV.1-15). Another point to be noted in this connection is this: In what the author gives, there is nothing to say about the addition of ‘the sine’ and Sūryāgrā when they are of different directions, and therefore

136 PAÑCASIDDHĀNTIKĀ IV. 56 about the Sun being in the northern hemisphere. But when they are of the same direction, and one is to be deducted from the other, the author mentions only the case where Sūryāgrā is to be deducted from the Sun (thereby assuming ‘the sine’ to be greater) the case in which the Sun is taken to be in the southern hemisphere. We have seen that when ‘the sine’ and Sūryagrā are of the same direction, the Sun is to be taken as in the northern hemisphere in the case (iv) in which Sūryagrā is greater, and ‘the sine’ is deducted from it. The author has omitted to mention this case. Has he forgotten it? We think not. He hopes that the reader himself will infer the changes to be made in this contingency, viz, that ‘the sine’ is to be deducted from Sūryagrā, and as the result is to be considered negative, and as the Agrā thus got is negative it is to be taken as south, and as south Agrā is for the Sun in the northern hemisphere, the Sun in the northern hemisphere will be inferred. As for TS and NP, here too they interpret Sūryāgrā as Agrā. They are not aware of the error that would be caused by this in the situation of the Sun, the hemispheres being reversed. For the matter of that they do not refer to the Sun’s situation at all, nor to the contingency of the reversal the sub- tractor and the subtrahend. [इति पञ्चसिद्धान्तिकायाम् वराहमिहिरविरचितायां करणाध्यायश्चतुर्थः]¹ Thus ends Chapter Four entitled ‘Three Problems: Time, Place and Direction in the Pañcasiddhāntikā composed by Varāhamihira

1. Col. A.C.D. इति करणाध्यायश्चतुर्थः

Chapter Five

PAULIŚA-SIDDHĀNTA — MOON’S CUSPS ५. पञ्चमोऽध्यायः पौलिशसिद्धान्तः — चन्द्रशृङ्गोन्नतिः Introductory In this chapter the Moon's visibility after or before heliacal setting, the appearance of its horns at the time of visibility with its geometrical representation, and the daily rising and setting of the Moon with its time of reaching the meridian are dealt with. We can surmise that this chapter is a part of the Pauliśa Siddhānta because the things required for the computations like the declination of the Sun and the Moon with the latitude of the Moon, are available to us only from the Pauliśa, the Romaka and the Saura having not been dealt with as yet, and because the methods are too rough to be attributed to the Saura. [चन्द्रदर्शनकालः] अपमान्तरसंयुक्तात् तदूनगुणिताच्छशाङ्करविविवरात् । मूलेनापमविवरे छिन्ने विक्षेपसंगुणिते ॥ १ ॥ फलमिन्द्वर्कविशेषाच्छोध्यं त्वयनानुकूलविक्षिप्ते । तद्व्यत्यासे देयं विपरीतं पूर्वसन्ध्यायाम् ॥ २ ॥ Time of Moon’s visibility

1. Find the difference in longitude of the Sun and the Moon, as also the difference of their declinations, (the mean declination of the Moon being used for this purpose). Multiply the sum of these two differences by their difference and find the square root of the product. By this ‘square root’ divide the product of the Moon’s latitude and the difference of declination already found.
2. The ‘result’ is to be subtracted from the difference in longitude, if visibility in the west is in question and the latitude and ayana (i.e. course northward or southward) of the Moon are of the same direction, or added to the difference in longitude if of opposite directions. If visibility in the east is in question, reverse the subtraction and addition. 1-3. Quoted by Utpala on BS 4.15. 1a. A.D. अयातान्तर \qquad\qquad\qquad\qquad\qquad\quad 2a. A. फलसिंध्वर्क \ \ b. A. ॰त्तद्वनयुक्ताछशांकविविरान् \qquad\qquad\qquad\quad b. A1. छोध्यचयनानु॰; A2. छोध्यं च यनानु॰ \ \ c. A.D. मूलेनायनविवरे \qquad\qquad\qquad\qquad\quad\ \ c. U. च्छेद्यमपमानुकूल

138 PAÑCASIDDHĀNTIKĀ V. 3 दिनकृत्सप्तमभवनात्तेनोदयनाडिकाद्वयं यदि वा | वियति विमले (तदे)न्दोर्लोकस्यालोक (आ)याति || ३ || 3. In the case of the visibility pertaining to the west, if a segment equal to the corrected difference in longitude takes at least two nāḍīs to rise in the east as reckoned by using the ascensional difference (for the place) of the seventh rāśi from the Sun, then the Moon will be visible, provided the sky is clear. In the case of visibility in the east, use the ascensional difference of the Sun’s rāśi itself. The following are the steps in the operation:- i. Find the difference in the longitude of the Sun and the Moon. ii. Find the difference of the Sun’s declination, and the Moon’s mean declination. iii. The square root = √[(diff. in long. + diff. in dec.) × (diff. in long. − diff. in dec.)] iv. Result = diff. in dec. × Moon’s lat ÷ the square root. v. Corrected diff. in long = diff. in long ∓ result. (Of ∓ the upper sign is to be used if visibility pertains to the west, and the latitude and the course of the Moon are of the same direction, or if the visibility pertains to the east and the latitude and course of the Moon are of different directions. The lower sign is to be used otherwise. vi. If the visibility pertains to the west, find the time of rising of an ecliptic segment equal to (v), by using the ascensional difference (for the place) of the seventh sign from the Sun and Moon. If it pertains to the east, use the ascensional difference of the Sun’s rāśi itself. If the time so found is greater than two nāḍīs, the Moon is visible; otherwise it is not visible. The time that we find in (vi) is the time of Moonset after sunset in the west, and the time of moon- rise before sunrise in the east. The sun, Moon, declinations and latitude of these times should be used and the work repeated for greater accuracy. Other siddhāntas mention this, though the author here has not done so specifically. Or, even before beginning the work, we can know the approximate times of moonset and moonrise, and do the work using the elements of these times. Near the time of new moon the Moon is invisible because the lighted up part is very small, and the sky itself is bright by the nearness of the Sun below the horizon. It has been fixed by the ancient authors by observation, that if the Moon sets within two nāḍīs after sunset, or if the sun rises within two nāḍīs after moonrise, the moon is not visible. (In places near the equator this criterion will be satisfied if the elongation of the Moon is in the neighbourhood of twelve degrees.) It is this we are finding by the computation, and it is obvious that the nearer the time of the elements used to the result, the better will be the result itself. Therefore is the need for successive approximation. If it is only for the sheer beauty of its appearance in the sky which has been described by poets like Kālidāsa, the first digit of the Moon is fit to be sought. But it is necessary for religious purposes 3b. A. ॰नोदया. c.A. तदिन्दो d. A.C.D. U. लोकमायाति ।

V. 3 V. PAULISA MOON'S CUSPS 139 also. The Baudhāyanas have to avoid Iṣṭi being performed on the day of the first appearance of the Moon, and do it on the previous day, and the offerings to the manes have to be done on the day pre- vious to the Iṣṭi. The Dharmaśāstras describe the seeing of the first digit of the Moon as meritorious. The Muslims consider their months ending with the first appearance of the Moon, and so this is important to them for calendrical purposes. The observance of the last digit of the Moon was neces- sary in ancient times, for from that they had to determine whether the same day or the next one would be the new moon day, so necessary for their religious rites. The importance can be guessed from the special names they had for the days at new moon, Sinīvālī and Kuhū in which the streak of the Moon will be visible and invisible, respectively. Example 1. At a certain place having lat. 30°N. examine the visibility of the Moon in the evening, given, the Sun at sunset = rāśi 1-0, the Moon at sunset = rāśi 1-15, and the Moon's latitude = 240' south. From the Sun and the Moon, their respective declinations are 704'N and 1004'N (mean). From the latitude 30°N, and Sun's declination the vināḍis of ascension at the place, of Scorpio, the seventh rāśi from Sun and Moon, can be calculated to be 355. From these, i. Diff. in long = rā. 1-15 − rā. 1-0 = rā. 0-15 = 15°. ii. Diff. in dec. = 1004' − 704' = 300' = 5° iii. The square root = √[(15° + 5°) × (15° − 5°)] = 14° 8'.4 iv. The Result = 5° × 4° ÷ 14° 8'.4 = 85'. v. Corrected diff. in long = 15° + 85' = 16° 25', (the lower sign, because the work pertains to the west (evening) and the Moon's latitude is south, while its ayana is north), vi. As the work pertains to the west, the seventh rāśi measure is to be used, which we have found to be 355 vināḍis. Using this, the time taken for 16° 25' to rise is 16° 25' × 355 ÷ 30° = 194 vināḍis. This is more than 2 nāḍis and so the Moon will be visible. As the time found is far above the requirement, we need not repeat the work using the elements of the time of moonset. Example 2. At a certain place (north of the equator) on a particular day in the evening the Sun rā 6-0. The Moon is rā. 6-15. The Moon's latitude is 4° 40'. The equinoctial shadow of the place is 4 digits. Examine for Moon's visibility. From the Sun, its declination is 0', and from the Moon its mean declination is 363' S. From the equinoctial shadow and the Sun's declination, the measure of the ascension of Aries, (seventh from Sun and Moon, since the computation pertains to the west) can be calculated to be 228 vināḍis. Using these, i. diff. in long. = rā 6-15 − rā 6-0 = 15° = 900' ii. diff. in dec. = 363' − 0' = 363'. iii. The square root = √[(900' + 363') × (900' − 363')] = 823'.5 iv. The result = 363' × 280' ÷ 823.5 = 123'.4 v. The corrected diff. in long. = 900' − 123'.4 = 12° 56'.6 (The upper sign because, the work pertains to the west, and the ayana and latitude of the Moon are of the same direction.) vi. As the work refers to the west, using the measure of Aries, the seventh rāśi from Sun and 12

140 PAÑCASIDDHĀNTIKĀ V. 3 Moon, the time for a segment equal to 12° 56′.6 to rise is, 228 × 12° 56′.6 ÷ 30 = 99 vināḍis. This is less than 2 nāḍis and so the Moon will not be visible that day. As the time got is far less than the requirement, repetition of the work is unnecessary. The steps are explained thus: [चित्र : Fig. V. 1 - Cel. Eq., D, M, M1 (M'), Ecliptic, Diurnal Circle, South, W, S, L, North] Fig. V. 1 Here, WD is the celestial equator. SM'C is the ecliptic and LM' is the diurnal circle of the Moon projected on the ecliptic. S is the Sun, M is the Moon and M' is the same projected on the ecliptic. MM' is the Moon’s latitude. SM' is the difference in longitude which is found in step (i). WS is the Sun’s declination, and DM' is the Moon’s mean declination. ∴ SL is the difference of the declinations, found in step (ii). Assuming the triangles as plane triangles, in the right angled triangle SLM', LM'² = SM'² − SL² = (SM' + SL) (SM' − SL) ∴ LM' = √(SM' + SL) (SM' − SL), and LM' being the square root, it is equal to √(diff. in long. + diff. in dec.) (diff. in long − diff. in dec.).... (step iii) M'C is the result and it is found thus: As MM' is perpendicular to SC, triangle MM'C is right angled at M'. ∴ angle SM'L = angle CMM' Therefore the two triangles are similar. ∴ M'C/MM' = SL/LM .

V. 3 V. PAULISA MOON'S CUSPS 141 ∴ M'C = SL × MM' ÷ LM', i.e. 'the result' = difference in declination × moon's latitude ÷ 'the square root', (which is step iv). Now for the additive or subtractive nature of 'the result': If the Moon's ayana is northward, i.e. if the ecliptic is inclined northwards (as in fig. 1), the Moon having south latitude, being at the end of a perpendicular to it, is lifted up. Therefore the Moon projected at M' is projected at C, as it were, and the difference in longitude which is the distance between S and M', is increased. So, in this case, 'the result' M'C is to be added. Now consider the case, when the ayana does not change, but the latitude also is north, like the ayana, as in Fig. 2. M¹ Fig. V. 2 C M South ← S L → North Now, clearly the Moon M at the end of M'M is bent downwards, with the result that M'C is deduc- tive in this case, as the instruction says. Let us next consider the case when the Moon's ayana is southward as in Fig. 3. C M M¹ Fig. V. 3 M C South ← L S → North Clearly in this case the Moon having north latitude is lifted up, and 'the result', CM', is additive, and the Moon having south latitude is depressed, and CM' is subtractive. Thus we have, for ayana and latitude having identical direction, 'the result' is subtractive and having different directions it is additive. This is for visibility in the west.

142 PAÑCASIDDHĀNTIKĀ V. 3 Now, for the visibility in the east: we are now looking eastward and successive points on the ecliptic are lower and lower towards the horizon. Therefore in figs. 1, 2 and 3, other things being the same, the ecliptic alone is to be represented as being directed downwards, as in Fig. 4. M¹ C M South S L North Fig. V. 4 Therefore, in each case taken up for consideration above, the direction of the ayana being changed, we see that for the ayana and latitude having different directions, 'the result' is subtrac- tive, and having the same direction it is additive. Thus step (v) is explained. Now for step (vi). We have already said that the Moon will be visible if it does not set within two nāḍīs after sunset, or if it rises before 2 nāḍīs before sunrise. (As visibility depends actually on other factors like the keenness of the eyesight of the observer, we have only to take the authority of the Śāstras in this matter). So in the evening we have to find the time by which the Moon will set after sunset, i.e. the segment constituting the corrected difference in longitude will set. As the distance between the rising and setting points in always in 6 rāśis, this time is equal to that of the rising of an equal segment in the east, which can be calculated by using the vināḍīs of the ascensional difference of the rising sign, which being six rāśis away, is the seventh from the Sun (or Moon). If this time is greater than 2 nāḍīs, the Moon would not have set, and therefore be visible. In the matter of visibility in the east, the same explanation holds, except that now the time of rising of the segment in the east is wanted, using the ascensional difference of the rising sign in which the Sun (or Moon) itself is situated, and hence the instruction to use that sign. This instruction to use the ascensional difference of the same sign as the Sun in the case of visibility in the east is implied by the use of the word vā, though not explicitly stated, and can also be inferred from the nature of the explanation. TS-NP do not seem to have noted the difference in the methods to be pursued in the operation. Another mistake they have made is that they have discarded the correct reading, ayanānukūlavikṣipte (v. 2) and chosen the incorrect reading apamānukūlavikṣipte and accordingly, have given the condition for additiveness or subtractiveness, "If the moon's latitude is

V. 5 V.-PAULISA MOON'S CUSPS 143 of the same direction as the difference in declination etc." Declination had direction, but what direction can be attributed to the difference in declination as given in the text? Or how can the word for declination mean difference in declination? Whatever the latitude, 'the result' is zero at the junction of the ayanas, which means its sign, i.e. its additiveness or subtractiveness changes there, and therefore the ayana should be a criterion for additiveness or subtractiveness. The very name of this correction, Āyanadṛkkarma (this name is not mentioned here by the author, but it is this) will suggest that the ayana of the Moon must form part of the criterion. Another thing must be mentioned: The work given here is very rough, because spherical tri- angles are taken as plane triangles, and another correction called Ākṣadṛkkarma which is to be done for the sake of the latitude of the observer has been omitted. Therefore the reader should refer to works like the Mahābhāskarīya and Siddhānta Śiromaṇi for greater accuracy. [चन्द्रशृङ्गोन्नतिः तत्परिलेखाश्च] द्विगुणेऽ(क्षे) 'तिथ्यंशः' शृङ्गमुदक् तुङ्गमुडुगणाऽधिपतेः । देयं च भुजादेतच्छौक्ल्यं कर्णाद् द्विषट्कांशम् ॥ ४ ॥ अपमान्तरविक्षेपा(वे)कान्यत्वे युतोनितौ कोटिः । कर्णो रवीन्दुविवरं तत्कृतिविवरात् पदं बाहुः ॥ ५ ॥ Diagram of the Moon's cusps 4. Multiply the latitude of the place in degrees by two and divide by fifteen. By the resulting number of aṅgulas or digits (measured along the rim), the northern tip of the horn of the Moon should be raised upwards (as caused by the latitude at the time of first visibility). This raising should be directed upwards like the 'Bhuja' which we are going to mention. The number of digits of illumination of the Moon's orb, (usually called merely digits), is the twelfth part of the difference in longitude in degrees, last found, and should be directed like 'Hypotenuse', which we are going to mention. 5. The difference in declination last found should be added to the Moon's latitude or subtracted from it, as the directions of the Moon's ayana and its latitude be the same or different. (This refers to the visibility in the west in the evening. With reference to the visibility in the east in the morning, the addi- tion and subtraction, is done vice versa). The result is called 'Koṭi'. The differ- ence in longitude is called 'the Hypotenuse'. The 'Bhuja' is the square root of the difference of the squares of the 'Hypotenuse' and the 'Koṭi'. 4-7. Quoted by Utpala on BS 4.15. 5a. A.अनान्तर; C.D.अयनान्तर. A1.विक्षेपा; 4a. A.द्विगुणेच्छे; C.U.दिनगुणेच्छा; D.द्विगुणाक्षे A2.धिक्षेपा b. A.शृंगमुदकुंमुदुगुणाधिपतिः b. A1.2.वैकानले; A2.वैकानचे U.वैकान्यत्वे d. A.कर्णाद्विष्टकांश: D.कर्णाद्विष्टकांशः A.यातोनिता; C.युतोनिता c. A.रवींदुविवरं

144 PAÑCASIDDHĀNTIKĀ V. 7 सविता यतः शशाङ्कात् कोट्या परिकल्पितस्ततः कोटिः । देयांशकाङ्गुलसमा भुजकर्णौ चाङ्गुलैरेव ॥ ६ ॥ शशिमध्यात् प्राक् कर्णः कोटिरतोऽतो भुजः शशाङ्कगतः । परिधावक्षो(न्ना)मः शौक्ल्यं मध्याद्धनुस्तत्र ॥ ७ ॥ 6. The ‘Koṭi’ is to be drawn on that side of the Moon towards the Sun, north or south, which is got in computing it, using the scale, one aṅgula = one degree of ‘Koṭi’. The Bhuja and the Hypotenuse also should be drawn to the same scale. 7. Thus, first there is the Hypotenuse from the centre of the Moon to that of the Sun. From the centre of the Sun the ‘Koṭi’ is laid in the direction com- puted for it. Then from its termination the ‘Bhuja’ is laid towards the Moon’s centre. On the rim of the Moon represented by a circle of fifteen aṅgulas, the raising of the horn in aṅgulas due to the latitude of the place is to be done. At the centre of the two ends of the horn the illumination in digits is to be represented on the diameter. There the arc (forming the upper boundary of the illumination) is to be drawn (by making the arc pass through the two ends of the horn and the point in the middle to which the illumination extends). Though it is plain that these four verses give instructions for the graphical representation of the Moon at the times of visibility, (specifically its first visibility in the evening in the west), yet on account of possible incorrect copyings, and because we are not sure of the degree of roughness of the result intended by the Siddhānta, we encounter a lot of difficulty in ordering the words and interpretting them. The author has not given the diameter of the Moon in aṅgulas, which is neces- sary to draw the orb, and represent in it the illumination and the uplifting of the horn. But we can infer the diameter to be fifteen aṅgulas thus: On Aṣṭamī, at the middle of either fortnight, when the hypotenuse is 90°, according to the rule for getting the illumination, we have 90/12 = 7½ aṅgulas of illumination. We know that half the Moon is illuminated then, and therefore the whole Moon should have a diameter of fifteen aṅgulas, as we have stated. This agrees also with the ‘elevation of the horn’ due to the latitude, which can be shown thus: The line joining the tips of the horn seen horizontal by a person on the equator, is seen vertical by a person at the pole, i.e. at 90° latitude, because the celestial equator is inclined by 90° there, so as to be coincident with the horizon. As the hypotenuse at the time for which the elevation is required is small, we can take it that the elevation of the horā is proportionate to the degrees of latitude. According to the rule for elevation given by the author, it is for 90° and 90 × 2 ÷ 15 = 12 aṅgulas, along the rim of the quadrant, from the horizontal to the vertical. Therefore the whole rim, i.e. the circumference, is 4 × 12 = aṅgulas and this shows that the diameter must be 48 × 7/22 = fifteen aṅgulas very nearly. This agreement in the diameter, as calculated by the two rules, itself is a criterion for the correctness of the rules. 7c. A.°वक्षोनामः; C.D.U.°वक्षो नाम d. D.शौक्ल्यमध्यात्॰ 6b. A.कोज्यापरिकल्पितकोटिः A.तदनु सूत्रं; D.तदनु [च] सूत्रम्’

V. 7 V. PAULISA MOON'S CUSPS 145 Now, we shall show why this elevation is always on the northern limb. As mentioned several times before, when latitude is used in the rules given, it is always north latitude that the author means. As seen from north latitudes, the circles on the stellar sphere are all bent towards the south above the horizon. Therefore the hypotenuse also is inclined south, the angle of inclination being equal to the latitude, the hypotenuse being small and taken as a straight line. By this inclination south, the line joining the tips of the horns, which is perpendicular to the Hypotenuse is elevated in the north and depressed in the south, the angle of elevation being equal to the latitude. This elevation, measured on the rim in aṅgulas is, as we have shown, twice the latitude divided by fifteen. In the matter of the addition or subtraction of the difference of declination and the Moon's latitude, we have said that the author has in view only the visibility in the west in the evening, for then alone is the statement correct. Perhaps the author thinks that this is enough, because the ele- vation of the horn at evening appearance alone is observed anxiously by people, as an omen of good or evil. Or the author thinks that the readers themselves will understand the reversal of addi- tion and subtraction for the morning appearance, by analogy with what was done before in the case of visibility. It must also be noted that the object here is only to represent the orb of the Moon as it appears, and the Hypotenuse, Bhuja and Koṭi are given to serve this end. Therefore it would not matter if these are represented on a different scale from that on which the Moon is given, as for instance an aṅgula per degree here. (On this scale the Moon will have to be represented by a dia- meter of a half-aṅgula.) There is a view that the elevation of the horns should be observed when the orb of the Moon is on the horizon. In that case, the Sun will be below the horizon, and the question of the difference in scale will not arise at all. So, these are the steps in the work:- i. The elevation of the horn due to latitude in aṅgulas = latitude in degrees × 2 ÷ 15. ii. Illumination or digit of illumination in aṅgulas = the difference in longitude in degrees ÷ 12. iii. Koṭi in aṅgulas = diff. in declination in degrees ± latitude in degrees. (For evening in the west, if the Moon's latitude and ayana are of like direction, addition, and if of different directions, sub- traction. For morning in the east, reverse the addition and subtraction). iv. Hypotenuse = aṅgulas equal in number to difference in longitude. v. Bhuja in aṅgulas = √(Hypotenuse² – Koṭ i²). vi. See fig. 5, below. On the surface on which the phenomenon is to be represented draw a hori- zontal line and mark the north and south sides on both ends. Mark the point S on it to represent the Sun. Mark a point A on the horizontal line on the side in which the Moon is situated, (this is known when finding the Koṭi) such that SA = Koṭi. From A draw a perpendicular upwards equal to the Bhuja and at the end mark M, the centre of the Moon. MS is the Hypotenuse. With M as centre draw the orb of the Moon having a diameter of 15 aṅgulas. At M draw a diameter BC perpen- dicular to the Hypotenuse. From the northern end the diameter, say C, measure the aṅgulas of ele- vation due to the latitude of the place, along the rim, and mark the point D. Draw the diameter DME. D and E are the tips of the horns. On the lower semicircle caused by DE, mark its mid-point, F. Draw the radius FM. On this mark a point G, such that FG = the aṅgulas of illumination. Draw the arc DGE by the well-known method of making a circle pass through 3 points. This is the upper limit of the illumination. The figure of the Moon is now as it will be seen in the sky. The horizon is between the Sun and the Moon, parallel to the original horizontal line. It must be remembered that

146 PAÑCASIDDHĀNTIKĀ V. 7 what the Siddhānta gives is only approximate, though easy to do, and for greater accuracy, we have to do a lot of work like calculating the Great gnomons of the Moon and the Sun etc. Example 3. Represent graphically the Moon of example 1. There, we are given, latitude of the place = 30°N, the Moon’s ayana is northward, and its latitude 4°S, and we get the difference in longitude = 15°, and the difference in declination = 5°. Horizon South North To Sun S A Fig. V. 5 From the data given above: i. The aṅgulas of elevation due to the latitude of the plane = 30 × 2 ÷ 15 = 4. ii. Aṅgulas of illumination = 15 ÷ 12 = 1¼. iii. Koṭi = 5° − 4° = 1°, and ∴ 1 aṅg., the Sun being to the south, (because it is evening observa- tion, and Moon’s ayana and lat. are of diff. direction). iv. Hypotenuse = 15 aṅgulas. v. Bhuja = √225-1 = nearly 15 aṅgulas. vi. Representation: Fig. 5: (Scale 1' unit = 6 aṅg.) It should be remembered that the fig. is intended only for the appearance of the Moon, with the illumination, and elevation of the horns represented on it, and none else. The line DGEFD is the part illuminated, D and E being the tips of the horns. Actually the Sun is down, on the line MF.

V. 9 V. PAULIŚA MOON'S CUSPS 147 Now for the readings: As the elevation due to the latitude of the place is considerable, it cannot be neglected and must be represented; therefore we have corrected dviguṇeche tithyaṃśa into dviguṇe'kṣe tithyaṃśa, by changing cha into kṣa. But TS have adopted the reading dviguṇecchātithyaṃśa and considering it a combination of dviguṇecchā and atithyaṃśa thinking that the subject matter is astrological, (which is obviously unlikely). We have corrected paridhāvakṣonāmāḥ into paridhāvakṣonnāmaḥ, for the instruction to apply the elevation due to the latitude must be given. But TS and NP take the reading as it is, and say that something on the rim of the Moon is named akṣa, which is purposeless. Their readings themselves in these two cases are from their own edition of Bhaṭṭotpala's commentary on the Bṛhatsaṁhitā, and to say that their (TS's) readings agree with those of the Bhaṭṭotpala may be improper, for probably they have themselves put the readings there. [चन्द्रस्य दैनन्दिनोदयास्तौ] याम्योदग्विक्षेपाद्विषुव (द्भा) घ्ना 'द्रवि'भिरवाप्तांशाः । उदये शशिनो वृद्धिः क्षयो विपर्यस्तमस्तमये ॥ ८ ॥ एवं व्यर्काच्चन्द्राद्यद्द्युना राशयः षडधिका वा । तदुदयकालेन दिवा निशि च शशाङ्कोदयो वाच्यः ॥ ९ ॥ Daily rising and setting of the Moon 8. Multiply the Moon's latitude in degrees by the equinoctial shadow and divide by twelve. Add the resulting degrees to the longitude of the Moon, or subtract from it, according as the Moon's latitude is south or north, if the times of daily moonrise is to be computed. If the times of daily moonset is to be found, reverse the addition and subtraction, i.e. subtract and add, respec- tively. 9. Subtract the longitude of the Sun from that of the Moon corrected thus. Find the time for this segment of the ecliptic to rise, after sunrise. By so much time after sunrise, the Moon will rise. If this segment is less than six rāśis, then the moonrise will fall in the day-time, if greater, the moon will rise at night. 8-10. Quoted by Utpala on BS 4.15. 9a-b. A. व्यर्काच्चाद्येनोना 8b. A. ०द्विषुवज्याघ्नाद्; C. द्विषुवत्याघ्नाद्; b. C. षट्कोनाः; D. द्येनोना. A. षडधिका या D. ०द्विषुवच्छा[या]घ्नाद् c. A. तदुदया A. रविरुत्तरांशाः; D. रविभक्तांशाः; U. रविभिरवाप्तांशः B1.2.3. Commence again from ०न दिवा after the big gap which commenced at IV.20. d. U. विपर्यस्तमय एवम् । d. A. निशे. B3. शशाकोदयो

148 PAÑCASIDDHĀNTIKĀ V. 10 कृत्वैवं क्षयवृद्धी व्य(र्काच्चन्द्राद् वि)शोध्य चक्रा(र्धम्) | शेषोदयकालसमे शशिदिवसा(न्ते) शशी मध्ये || १० || 10. In the manner given (in verse 8), correct the Moon for moonset, deduct the Sun from this corrected Moon, and deduct 6 rāśis from the remainder. Find the time by which the remaining segment will rise, after sunrise. This is the time from sunrise when the Moon will set. At the time exactly midway bet- ween moonrise and moonset, the Moon will reach the meridian, (i.e. will be at upper culmination). The following is the work to be done: i. The correction (for latitude) = the Moon's latitude × the equinoctial shadow ÷ 12 (This is known as Akṣadṛkkarma). ii. This correction is to be applied to the true Moon. Corrected Moon = True Moon ± Correction. (If the time of moonrise is to be found, then the correction is subtractive if the Moon's latitude is north, and additive if it is south. If moonset is wanted, if the Moon's latitude is south, the correction is subtractive, if north it is additive). iii. The time of moonrise is found thus: The corrected Moon - True Sun = elongation. The time of rise of the segment of elongation from sunrise is the time of moonrise. (In other words, the corrected Moon's position on the ecliptic being known, the time when that point rises is the time of moonrise). When the elongation is less than 6 rāśis, moonrise is in the day-time, otherwise at night. iv. The time of moonset is found thus: Corrected Moon - Sun = elongation. The time of rise of (the segment of elongation - 6 rāśis) from sunrise, is the time of moonset. (In other words, the time of rising of the point diametrically opposite to corrected Moon is the time of moonset). Here, if the elongation is less than 6 rāśis, then the moonset is in the night, and if greater, it is in the day- time. v. Moonrise to moonset is the moon-day-time. It is obvious that at the middle of its day time it is on the meridian. It is obvious that the times of rising and setting will be correct if the longitudes and Moon's latitude of those times are used. But as the computation as done here is only approximate, we can guess the approximate times of moonrise and moonset for the day from the tithi of the day, and use the elements of those times, to get tolerably accurate times. Example 4. The equinoctial shadow for a certain place (in the northern hemisphere) is 4 aṅgulas. The ascen- sional differences for the place are for Aries 236 vināḍīs, Taurus 265, Gemini 309, Cancer 337, Leo 333, 10a. A.क्रत्वैवं; B. तच्चैवं. A.B. वृद्धि b. A.B.C.D. U. व्यर्कै; (B2. व्यर्क) चन्द्रं विशोध्य d. B3. पशि and C.D. निशि for शशि चक्रार्धात् A.B.U. दिवसाद्धे;. D. दिवसेऽस्तं c. A. शेखोदय; B. मेषोदय A.B. शशिमध्ये; C. शशी याति

V. 10 V. PAULISA MOON'S CUSPS 149 Virgo 320, Libra 320, Scorpio 330, Sagittarius 337, Capricorn 309, Aquarius 265, and Pisces 236. There, on a certain day, the true longitude of the Moon at sunrise is rā. 1-18, the true Sun is rā. 10-3, the Sun's daily motion is 60', the Moon's daily motion is 840', the latitude of the Moon is 272' S, and its motion per day 8' S. Find the moonrise, moonset and upper culmination. The distance of the Moon from the Sun = rā. 1-18 − rā. 10-3 = rā. 3-15, (equal to 8¾ tithis). From this the approximate time of moonrise is, 8¾ × 2 = 17½ nāḍis. Therefore the time of moonset is approximately, 17½ + 31 = 48½ nāḍīs. The Sun at approx. moonrise is rā. 10-3-18, the moon rā. 1-22-5, and its latitude 274' S. At approximate moonset, the sun is rā. 10-3-49, the Moon rā. 1-29-19, and its latitude 278' S. Using each set, the computation is as follows: i. The correction for moonrise = 274' × 4 ÷ 12 = 91'. The correction for moonset = 278' × 4 ÷ 12 = 93'. ii. The corrected Moon for moonrise = rā. 1-22-5 + 91' = rā. 1-23-36. The corrected moon for moonset = rā. 1-29-19 − 93' = rā. 1-27-46. iii. Computing Moonrise: Elongation = Corrected Moon − Sun = rā. 1-23-36 − rā. 10-3-18 = rā. 3-20-18. As this is less than six rāśis, the moonrise is in the day-time. The time for the segment, rā. 3-20-18, to rise after sunrise is found thus: For the rest of Aquarius, which is the sign occupied by the Sun, to rise, the time taken is 265 × 1602' ÷ 1800' = 236 vināḍis. For Pisces to rise, 236, for Aries 236, for the corrected Moon to rise in Taurus, 265 × 1416' ÷ 1800' = 208 vināḍis. So the total time taken is, 236 + 236 + 236 + 208 = nāḍis 15-16. This is the time of moonrise. iv. Moonset: Elongation = Corrected Moon − Sun = rā. 1-27-46 − rā. 10-3-49 = rā. 3-23-57. Deducting six rāśis from this, we have rā. 9-23-57. The time for the rise of this much segment is found thus: For the rest of Aquarius to rise, the time taken is 265 × 1571' ÷ 1800' = 231 vināḍis. For Pisces 236, Aries 236, Taurus 265, Gemini 309, Cancer 337, Leo 333, Virgo 320 and Libra 320. For Scorpio to rise upto the point diametrically opposite to the corrected Moon, 333 × 1666' ÷ 1800' = 308. Adding up, the time of moonset is nā. 48-15. This agrees with what we can infer from elongation, for the elongation found is less than 6 rāśis, and the moonset must be in the night. v. The duration of the lunar day is nā. 48-15 − nā. 15-16 = nā. 32-59. Half this is nā. 16-30. Adding this to moonrise, mid-moon-day, the time of upper culmination of the Moon is nā. 15-16

• nā. 16-30 = nā 31-46, after sunrise. The instruction is thus explained: The problem is to find the time of rising or setting of the Moon, which is in its orbit, at a distance equal to its latitude from the ecliptic. If it can be projected

on the ecliptic in such a way that its projected position rises and sets at the same time as it itself rises and sets, then the time can be found like lagma, by the method given in Chap. IV for that purpose. In order to effect the said projection two corrections have to be applied to the Moon, one for the inclination of the ecliptic called Ayana-dṛkkarma (we did this for visibility), and the other for the latitude of the observer called Akṣa-dṛkkarma. The Siddhānta gives the correction for latitude alone here, which is got by multiplying the latitude of the Moon by the equinoctial shadow and dividing by 12. The following is its rationals: The latitude is measured on the great circle perpendicular to the ecliptic and directed towards the pole of the ecliptic. If the latitude is projected on secondaries to the pole, and we get the true declination of the Moon by adding this to the mean declination, we can find its time of rising and setting directly, as we find the rising and setting of the Sun, by com- puting its cara etc. and getting its own ascensional differences. But if the latitude is small, it can roughly be taken as the correction for the Moon's mean declination, given by its longitudes. So the correction to the vināḍis of true cara can be found by the latitude taken as part of the declination, by proportion from the cara-vināḍis for the mean declination already used in finding the ascen- sional differences. Therefore, as in getting the cara-vināḍis, here too we have to multiply by the equinoctial shadow and divide by 12. But the division by the diurnal radius is not done, because here we are not finding actually the vināḍis of cara, but an element of the ecliptic corresponding to the cara, for the sake of which we have to multiply again by the radius of the diurnal circle, and the two cancel out. Thus the correction will permit us to consider the Moon to be on the ecliptic. We shall now consider when it is additive, and when subtractive. In the northern hemisphere, the Unmaṇḍala is elevated above the horizon, the elevation increasing towards the north. Therefore if the Moon is a little to the south of the ecliptic on account of its south latitude, it rises later. As successive points on the ecliptic rise later and later, the correction got from the Moon's south latitude is equi- valent to an increase in the Moon's longitude, and so the correction to the longitude is additive. From this we can see why the correction is subtractive if the Moon's latitude is north. As for the time of the setting of the Moon, the further north a body is, the later it sets, and therefore the correction is additive if the latitude is north, from which we see it is subtractive if the latitude is south. The Siddhānta has in view only observers in the northern hemisphere, as we have already said. We have already drawn the attention of the reader to the omission of the correction due to the inclination of the ecliptic (Āyana-dṛkkarma). It may be that the author expects us to make this correction also, taking the hint from the computation of the heliacal rising of the Moon (visibility). The part of the instruction to find the time when the corrected Moon rises is explained thus: The corrected Moon minus Sun is the segment of the ecliptic between them, and the time taken for its rise after sunrise is the time of the rise of the corrected Moon itself. If this segment is less than six rāśis, the Moon must rise in the day-time, because just after the rise of six rāśis from sunrise the sun sets, but this segment is less. Clearly, if it is more than six rāśis the Sun has set, and it is night when the Moon rises. As for moonset, the point of the corrected Moon plus (or minus) six rāśis rises at that time. So, the time of its rise, or which is the same, the time taken by the corrected Moon minus Sun ± six rāśis, after sunrise is the time. Of ±, the author has chosen minus, because the effect of both is the same. By analogy with mid-day Sun, the Moon is on the meridian at the middle of its day-time, provided its motion and change of declination is tolerably uniform. We must add here that it would have been sufficient if the author had said, 'Treat the corrected moon as Lagna, and its time of rise is the time of moonrise. Treat the corrected Moon plus (or minus) six rāśis as Lagna, and its time of rise is the time of moonset'.

V. 10 · V. PAULIŚA MOON'S CUSPS 151 Now for the readings: In verse ten if the meaning is taken as it is, then we shall be getting the time of sunset after moonrise, which serves no purpose and cannot be the intention of the author to get, and it is incompatible with the time of the meridian Moon sought to be found in the fourth foot. This is the middle of the Moon's day-time and for this, moonset has to be found. (TS and NP too inter- pret this verse as giving moonset). Therefore we have corrected vyarkam candram viśodhyā cakrārdhāt into vyarkāt candrāt viśodhya cakrārdham, by interchanging the case endings, and thus got the time of moonset, required for meridian Moon. The reading meṣodayakāla for śeṣodayakāla has been dis- carded as being unconnected with the problem. TS and NP have given the impossible correction, niśi divase'stam śaśī yāti for śaśidivasārdhe śaśīmadhya found in the manuscripts. Their aim, viz. to get the time of moonset, is all right, but their interpretation of the stanza to get this is wrong, and also self-contradictory. See the Sanskrit com- mentary for the said interpretation: 'Deduct the Moon minus Sun from 6 rāśis; the time of the rising of this is the time of moonset, reckoned from sunrise. Here, if the Moon sets in the day-time then Moon-minus-Sun must be deducted from 6 rāśis. If in the night, the Moon itself is to be deducted from 6 rāśis. This order of procedure should be understood.' If their instruction in the first sen- tence is followed, the time of sunset after moonrise will be got, as we have already said, but not the time of moonset. It is to avoid this that we interchanged the case endings. As for the instruction in the second sentence, the first part of it disagrees with the second part. We shall illustrate these defects found in their interpretation, by applying them to two examples. (a) The Sun is rā. 0-15, Moon is rā. 2-0, the Moon's latitude is zero, i.e. there is no corrections. In this case, the moonset according to TS is to be found thus: Moon − Sun = rā. 2-0 − rā. 0-15 = rā. 1-15. Deducting this from 6 rāśis, the remainder is rā. 4-15. They say, the time taken for rā. 4-15 to rise, after sunrise, is the time of moonset after sunrise. The absurdity of this can be seen by find- ing the time of moonrise, which is the time taken by the Moon-Sun to rise, i.e. for rā. 1-15 to rise; i.e. the interval between moonrise and moonset is the rising time of 3 rāśis. Or, by the instruction in the second sentence, as the Moon does not set in the day, it sets in the night, and therefore deducting the Moon itself from 6 rāśis, we get rā. 4-0, and they say by the time of rise of rā. 4-0 from sunrise the Moon sets. Does it occur in the night at all? Perhaps they meant sunset. Then, let us take another case. (b) The Sun is rā. 0-15, the Moon is rā. 8-0, and Moon's latitude is 0, again. Then, Moon − Sun = rā. 7-15. The moonset is in the day-time, clearly. Therefore deducting this from 6 rāśis, we have rā. 6-0 − rā. 7-15 = rā. 10-15. According to them the Moon sets by the rise of this segment after sun- rise. Clearly according to this the moonset will fall in the night, and not in the day-time as required. Assuming sunrise is a mistake for sunset, reckoned from sunset also it will be wrong, for then the moonset will be rā. 4-15 from sunrise, which is wrong, for it is correctly the time of rising of rā. 1-15 from sunrise. This demonstration shows their interpretation to be wrong, and at the same time justifies our interchanging the case-endings, by which alone the time of moonset can be got correctly. [इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां शशिदर्शनम् नाम पञ्चमोऽध्यायः ] ¹ Thus ends Chapter Five entitled Pauliśa-Siddhānta: Moon's Cusps in the Pañcasiddhāntikā composed by Varāhamihira

1. Col.: A. शशिदर्शनं | B.C.D. इति शशिदर्शनम् ।