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 ।

Chapter Six (VĀSIṢṬHA-) PAULIŚA-SIDDHĀNTA: LUNAR ECLIPSE ६. षष्ठोऽध्यायः वासिष्ठ-पौलिश-सिद्धान्तौ — चन्द्रग्रहणम् Introductory This chapter deals with the lunar eclipse. Nothing is given in the colophon at the end of the chapter about the Siddhānta to which this belongs. This cannot belong to the Saura for the lunar eclipse of the Saura is dealt with in Chapter X. The Sun, Moon and Rāhu of the Romaka are given in Chapter VIII, and in the same chapter the solar eclipse according to that Siddhānta occurs, and its lunar eclipse cannot be given here, earlier. Also, the method here does not have the refinement of the Romaka solar eclipse. So this chapter cannot belong to the Romaka. That it may belong to the Paitāmaha is out of question, since only the mean Sun and the Moon, and that very crudely, being given by the Paitāmaha, and Rāhu is not given. Also, the Siddhānta occupies a later chapter, the twelfth. This leaves the Vāsiṣṭha and the Pauliśa for consideration. Perhaps it belongs to both com- bined, as we have observed in the case of their Moon and its daily motion. It cannot belong to the Vāsiṣṭha separately for the Vāsiṣṭha does not give Rāhu, which we have to get from the Pauliśa. Also, it cannot belong to Pauliśa separately, for then at least part of the computation, like the duration of the eclipse will become redundant, because the duration of the lunar eclipse with its limits occurs in chapter VII also, together with the computation of the solar eclipse, which from the colophon and from the nature of the method given, must belong to the Pauliśa. Also, details usually given in connection with eclipses, like the direction of contacts, colour etc. are found only here in the VI chapter. Therefore we can conclude that chapters VI and VII belong both to the Vāsiṣṭha and the Pauliśa, and that the solar eclipse in the VII chapter belongs to the Pauliśa. [समकलौ चन्द्रसूर्यौ] नै(श्या) स्तिथिनाड़्योऽर्के देया (श्चान्द्रे) समेन्दुरवि (वि) वरात् । (दिवसोद्भवश्च) शोध्याः स भवति तत्कालशशिलिप्तः ॥ १ ॥ Sun and Moon of equal longitude

1. Minutes of arc equal to the nāḍīs of the full moon-tithi to go, after sunset, are to be added to the Sun, (which has been computed for sunset). Minutes of arc equal to the nāḍīs to go from the end of the full moon or new moon-tithi in the day-time upto sunset are to be so added to the Sun. Thus corrected, the Sun becomes equal to the Moon in (degrees and) minutes at the end of the full or new moon-tithi, (i.e. at full or new moon). The idea is that by thus finding the Sun, we can, without any trouble, get the Moon, for, if new moon, the Sun thus got is the Moon and, if full moon, the Sun plus 6 rāśis is the Moon.

VI.1 VI. VĀS.-PAUL. SIDDHĀNTA – LUNAR ECLIPSE 153 The following is the rationale of the work: The lunar eclipse occurs at the end of full moon-tithi. At that time, the Sun and the Moon are separated from each other exactly by 6 rāśis. Therefore the degrees and minutes or, which is the same, the total minutes left over after finding the rāśis at 1800 minutes a rāśi, are the same for both. Therefore they are called sama-liptas, i.e. ‘having equal minutes’. The solar eclipse is at the end of the new moon-tithi, at which time they are the same even in rāśi, not to speak of the degrees and minutes, and therefore samaliptas. So, if we know the Sun at these times, we know the Moon, for, if new moon, they are the same, and if full moon, different by 6 rāśis. Now, in the Pauliśa the ‘days from epoch’ are found for sunset, and from them the Sun and the Moon are found for sunset first. (We have already drawn the attention of the reader to this, while commenting on III.15.) Then, the ending moment of the tithi is calculated by using the difference of their motions. The Sun’s motion is roughly one minute of arc per nāḍī. Therefore if one minute per nāḍī of the time from sunset to full moon (we take only the full moon because with new moon at night there will be no solar eclipse) is added to the sunset Sun, the Sun at full moon is got, and the Moon is got from it by adding 6 rāśis. Thus the Moon is easily got, for otherwise we must calculate the Moon’s motion during the interval by proportion from its daily motion, add this to the sunset Moon, and get the Moon. In the case when new or full moon-tithi ends in the day-time, it is obvious that the minutes of arc accrued during the interval up to sunset should be deducted from the sunset Sun, to get the Moon, as a preliminary to computing either the solar eclipse or the lunar eclipse. The Siddhānta is content with thus getting the Moon roughly, for that will be sufficient considering the crudeness of its method of computation. If greater accuracy is desired, we must multiply the difference of the Sun’s daily motion from 60 minutes of arc by the time to go or time gone, and, taking the product as seconds of arc, subtract or add them, respectively, to the Moon if the daily motion is less, and add or subtract respectively if greater. The daily motion required for this is given in III.17, which we have already seen. TS have understood that the Moon at new or full moon is found here, but not the manner in which it is done so simply, for they interpret the instruction to mean that the Moon is to be got from its daily motion by proportion. To obtain this meaning, they make wild emendations of the words. But we have kept the words mostly as they are, and we can see that they are sufficient to give the correct idea. For example, in all the three readings, naiṣyāḥ, naiṣṇāḥ and vaiṣṇāḥ (ṣṇaiṣṇāḥ) there is ‘nai’ which therefore must have been in the original word. Therefore, by changing ṣ to ś, we get naiśyāḥ, meaning ‘belonging to the night’, which so well agrees with the idea. We have changed can- dram into cāndra for the sake of syntax and agreement with the idea. Between vi and va, we have introduced vi, thus reading ravi-vivarāt, which is a likely haplographical omission, and get a word that fits so well with the idea. Taking the meaningless reading, nṛvaśudbhavācca, and keeping as far as possible to the letters there, we have reconstructed the form as divasodbhavācca, fitting in with the 1a. A.B2.D. नैष्याः; B1.3. नैष्णाः; B3. वैष्णाः; C. यातैष्या A.B.C.D. नाड्योर्को c. A. नृवशूद्रवाच्च शोध्या; B. स्यु-द्धवाब्धः शोध्याः; b. A. दयाश्चन्द्रं; B. देयाश्चन्द्रं; (B2. ॰श्चन्द्र) C. See above; D. पाण्डवम्नाश्र शोध्याः C. दयतस्तत्कला विधोः शोध्याः । D. दयाच्चक्रा- d. A. तत्कालशशिदिनसार्द्धे शशिप्तितः । धोनेन्दुरविविवरात् c-d. C. स भवति तत्कालशशी दिवसैष्ये लिप्तिकायुक्तः; A. समेन्दुरविवरा; B. यमेन्दुरविवरा D. शशिलिप्तः

154 PAÑCASIDDHĀNTIKĀ VI.2 idea. NP insert vi to get the reading ravi-vivarāt but leave the other errors untouched or making unwarranted emendations. [चन्द्रग्रहणसम्भवः] राहोः स‘षट्कृति’कलां हित्वांशं तच्छाशाङ्कविवरांशैः । ग्रहणं त्रयोदशान्तः पञ्चदशान्तस्तमस्तस्य ॥ २ ॥ Probability of an eclipse 2. Deduct one degree and thirty-six minutes from Rāhu's Head or Tail (whichever is near the Moon) and find the interval in degrees between that and the Moon (at full moon found above). If it is less than thirteen, a lunar eclipse will occur then. If it is less than fifteen, (and above thirteen), there will only be a slight darkening. The following is the explanation: At the moment when the distance between the centres of the Moon and the Shadow circle is equal to the sum of their semi-diameters there is the first contact or the last contact of the eclipse, because the rims just touch each other then. see Fig. 1a, below. [Diagram: Moon's orbit, R, Ecliptic, S, M] Fig. VI. 1-a. The centre of the Shadow is always six rāśis distant from the Sun. At full moon (i.e. the end of the tithi) the Moon projected on the ecliptic (i.e. the longitude of the Moon) is 6 rāśis distant from the Sun, as we have already said. Therefore the centre of the Shadow also is there. But the actual Moon 2a. B. सषट्कृतिकलां (B2. वांशं, B3. चाशं०) b. A2. हिचांशं (A1. हित्वांशं) ; B. हिषारांतष्टांशक d. A.B. ०दशान्तःस्तमस्तस्य (B. ०न्तःस०)

VI.2 VI. VĀS.-PAUL. SIDDHĀNTA – LUNAR ECLIPSE 155 is on its orbit, at a distance equal to its latitude. Therefore only when the latitude is equal to the sum of the semi-diameters, is there at least a grazing of the rims¹. (See Fig. 1b) Fig. VI. 1-b. As according to the Pauliśa the sum of the semi-diameters is always 55', (this will be shown later), and as this much latitude can be got only if Moon ~ Rāhu is 13°, and not more² there can be no eclipse, if Moon ~ Rāhu is greater than 13°. As for the little darkening from 13° to 15°, it is due to the Moon entering the penumbra alone and getting out, instead of entering the umbra. It is well-known that if the source of light is not a point, there is a region not so dark round the shadow, which is darker and .darker as the shadow is approached, and becomes sufficient to be seen. This Siddhānta has taken this region to be about 8', round the shadow. Therefore, the Moon's orb will touch this region at full moon if the latitude is 63', and for this its distance from Rāhu must be 15', as given here. As for deducting 1° 36' from Rāhu, the author has found this is necessary by observation, and we have to accept it, as agreement with observation is necessary, otherwise people will lose faith in the Śāstra. Or the Pauliśa Siddhānta itself gives this correction for agreement with observation, for, in the phenomena intended to be seen, such correction is the practice of the writers of this Śāstra. But it may be asked how this need for correction arises at all. This implies that either the longitude of Rāhu or that of the Moon is incorrect. We showed in ch.III that at epoch Rāhu-head was 235° 59'

1. What we have said here is a little inexact and taken as such by most of the ancient authors. Actually, since the Moon's orbit is inclined to the ecliptic by about 5°, the minimum distance between the Moon and the Shadow, given by SM', the perpen- dicular on the orbit from S, is a little less than SM, and it is only when SM' is equal to the semi-diameters that the grazing occurs. Therefore even if the latitude at full moon is a little greater, an eclipse can occur, but this has been neglected as being very small, actually less than a quarter of a minute of arc.
2. As explained by us under III.31, taking 380' as the maximum latitude (i.e. for 90° distance), and taking the latitude as proportionate to the Bhuja of Moon-Rāhu, as given by the siddhānta there, we get 380' × 13° ÷ 90° = 55', for Moon ~ Rāhu equal to 13°. This agreement here is the proof of the correctness of what we said above in the explanation that the latitude is proportionate to the degrees of Bhuja. If we take the maximum latitude to be 280', as given by the reading of the text there, or to be 270' as TS have taken there without assigning any reason, neither taking the latitude as proportionate to the degrees of Moon ~ Rāhu, nor correctly as proportionate to sin (Moon ~ Rāhu) will give 55'. This is the reason why TS them- selves have, in this section, in verse 5, abandoned both 280' and 270', and taken 240' as the maximum latitude. (vide their Sanskrit and English explanations under VI.5.). 23

156 PAÑCASIDDHĀNTIKĀ VI.4 according to the Paulisa and this agrees beautifully with its position than according to modern astronomy, 236°, and tolerably well with those of other Siddhāntas. Therefore the incorrectness must be in the Moon, and as much error in the tithi is unlikely, nor in the Sun as well. On examina- tion we find it is indeed so; we find that at the period of the author the Sun and the Moon of Pauliśa were less by about a degree and a half, than those of other Siddhañtas. By this error in the Moon, the value of Moon minus Rāhu will be less by about a degree and a half, (1° 36′) and instead of correcting the error by adding it to the Moon, we subtract 1° 36′ from Rāhu, which is the same. We do not add it to the Moon, because if we do, we must add the same quantity to the Sun to keep the tithi intact, and this will affect the Saṃkramaṇas, and thus cause a lot of disturbance. If added to Rāhu nobody will even notice it. It may be asked whether it is not wrong to use the latitude calcu- lated in III.31 from uncorrected Rāhu in our work here, as we are going to do. Indeed it will be wrong, and that is why the author gives a correction below, in stanze 4, to set it right.¹ TS do not understand the nature of this correction, not even its amount and its connection with stanza 4. Their ignorance in the matter of the computation of Rāhu, which we exposed in III. 28- 29, they exhibit here also, (see their Sanskrit Comm. page 40). [ग्रहणस्थितिकालः ] विक्षेपकलाकृतिवर्जितस्य पञ्चोनषष्टिवर्गस्य । मू(लं) द्विगुणं तिथिवृद्धिभज्य काल(: स्थि)तेर्भवति ॥ ३ ॥ शशितिमिरविवरभा(गाः) त्रयोदशोनाः शराऽऽहताः क्षेप्याः । स्थि(त्यां) विनाडिकास्ता राहावधिकेऽन्यथा हानिः ॥ ४ ॥ Duration of the eclipse 3. Square the Moon's latitude, subtract it from the square of 55, (i.e. from 3025), and find its square root. Double this, and multiplying by 60, divide by the difference of the daily motions of the Sun and Moon, in minutes. The approximate time of the duration of the eclipse is got in nāḍis. 4. If Moon ~ Sun is less than 13°, multiply the degrees by 5. The result are vināḍis. Add these vināḍis to the duration if the longitude of Rāhu is greater than that of the Moon, and subtract if the Moon is greater than Rāhu. Thus the time of duration becomes correct. The following are the steps in the work: i. Using III.31, find the Moon's latitude, (using uncorrected Rāhu). ii. Uncorrected time of duration in nāḍis = √(3025 − (latitude in minutes)²) × 120 ÷ difference of daily motions of Sun and Moon in minutes. 3a. A. क्रति c. A.B. मूलो 4a. A.B.C.D. भागैः d. A.B. कालस्थि c. A.B.C.D. स्थित्या