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

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

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

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

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

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

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

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

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

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