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

316 PAÑCASIDDHĀNTIKĀ XVIII.7 also sinning against prosody, and given only 28° for the 27½ days, after the third sixty-day period, and 1½° for the next 3 days, thus, not accounting for 16° and 30½ days. ardhāṣṭakaviṃśatyā cannot mean 27½, besides being an un-Sanskritic formation. Further, for the 27½ days in that part of the synodic circle the motion should be more than 26° and for the next 3 days, more than 2¼° as can be seen by examining the actual. This error of 16° and 30½ days is doubled for the whole cycle, and the weight of this error of 32° and 61 days has been carried by them to the 60-day period of invisibility and drawn the remark on page 121, Part II: "a rather implausible conclusion. At any event, the description of the motion of Venus as given in our text seems incomplete." The footnote here is uncalled for. Jupiter and Saturn The computation of Jupiter and Saturn follows next to Venus. This is because their treatment is next simple, on account of their small mean motion and equation of conjunction, owing to their great distance. Both TS and NP have expressed inability to understand the part of the computation where the equation of the centre is obtained and applied, before the application of the eq. of conjunction. Still, they have attempted to interpret the concerned verses, changing the wordings, drastically, to yield their fancied ideas. In getting the eq. of conj., too, they have made several mistakes. As in the case of Venus, here too, the true motion is traced from one heliacal rising to the next. The method of getting the true anomaly of the eq. cent. is similar to that of the moon given by the Vāsiṣṭha in chap. II, and based on the same theory of the uniform increase and decrease of the rate of motion, forming a linear zigzag. Even the same technical term, pada, is used here. All these are reminiscent of the Babylonian astronomy of the Selucid period, as I have stated above. As between Jupiter and Saturn, their treatment is exactly similar, so that explaining one would suffice for both. Verses 6-13, deal with Jupiter and 14-23 with Saturn. My main aim here is to state and explain the procedure in the computation, a thing not understood by investigators. The verification of the epoch constants depend mainly on comparison with other systems and modern astronomy. So this will be done separately. [गुरुचारः] विचतुस्त्रिंश(द्युग)णं नाडीभिस्तावताभिरपि च गु(रोः) | (ह)त्वा 'नव नव दहनै' (रु)दया लब्धा (स्स्थि)ता दिवसाः || ६ || उदयनवां(शं) दत्वा दिनेषु षड्वर्गसंगुणैरुद(यैः) | एकनवाग्नि(च्छिन्नैः) (प)दमिति साष्टादशं शेषम् || ७ || Risings of Jupiter 6. The days of Jupiter from epoch minus 34 days, 34 nāḍikās divided by 399, give the number of risings. The remaining are days (after rising). 7. Add to these days a ninth of the number of risings. Multiply the number of risings by 36, add 18, and divide by 391. The remainder here are padas. We can conclude the following from these two verses: (i). At 34 d. 34 n. from epoch, the first period from rising to rising begins.

XVIII.8 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 317 (ii) The interval between the risings, i.e., the synodic period is 399 − 1/9 = 398 8/9 days. (iii) 391 padas make one full sidereal revolution of Jupiter, i.e., 360° of mean motion. In one synodic period, Jupiter moves 36 padas. One pada = 55' 15". 36 padas = 36° 9'. At epochal days 34 d, 34 n, Jupiter's longitude is 18 padas (= 16° 35'). But NP have taken the 18 given here as degrees. This is wrong. The difference of 1° 25' is too small to show itself in their verification, Table 32, Part II. But for Saturn this pada constant is 89, and for Mars, 85. Taking these as degrees have resulted in big differences which have puzzled them. See part II, page 124. (iv) In 391 syn. revolutions there are 391 + 36 = 427 solar sidereal revolutions = 36 Jupiter's sid. revolutions. ∴ one sid. rev. of Jupiter takes 4332-22-48 days, and one sid. rev. of the sun = 365- 15-32 days. The latter being very near Pauliśa's 365-15-30, we conclude that, there too, as in the Moon, it is mixed up with the Vāsiṣṭha's. TS and NP give the same interpretation, though making more than necessary emendations. In their verification, TS use the rough syn. period of 399 days instead of the correct 398 8/9, making the sid. period = 4333-35-0. क्रमशो म(ध्यः) स्फुटश्च खण्डौ [कार्यो] त(यो)-श्च वि(श्ले)षात् । स्फुटहाना द्युषु दद्या(न्मध्यात्) सौ(र्ये)ऽन्यथा हानिः ॥ ८ ॥ 8. One after another, mean and true segments are to be arranged. Taking their difference, if the true is less than the mean, the difference is to be added to the group of Jupiter's days (left over in the synodic cycle as the remaining days). Otherwise, (i.e., if the true is more), the difference is to be subtracted. 'True' here means 'true as corrected for the eq. of the cent'. How to get these true positions is given in verses 9-11, and the segments are to be got using these. So, this verse seems to have strayed here from after verse 11. The mean positions are to be got by using the remaining padas, extending the work done in verse 7. Kāryau is introduced to make up for the syllables wanting. Saure would mean 'pertaining to either Sun or Saturn'. But we are dealing with Jupiter Sūriḥ here. Saurya alone would mean, 'pertaining to Jupiter'. Some mss. have no dvi. Computing and arranging the mean and true segments against each other is to facilitate inter- polation to any required day. It will also be useful to prepare an ephemeride. The example worked will make things clear.

6a. A1. विवतु; A2. विवन् C.D. त्रिंशद्. A.B. द्विगुणं b. A.B.C.D. तावतीभिरपि. A.B. गुरुः c. B2.3. Unindicated om. of नव [नव....प) दमिति in 7d. d. A1. तुद्या; A2. नुद्या; B. om. line. A. ॰स्थिदिवसाः 7a. A. उदयस्वांशं; B. a-c. missing. C.D. उदयनवांशान् b. C.D. गुणे द्युदये. A. ॰रूदद्यः c. A.C.D. छिन्ने d. A. वदामिति; B. ॰मितिः. B1.3. साष्टदशं 8a. A.B. द्विक्रमशो; C. द्वि: क्रमशो; D. द्वि [हितः] क्रमशो A. प्यस्फुट b. A.B.C. खण्डैस्तयोश्च (A. थौश्च) विशेषात्; D. खण्डस्तयोश्च विशेषात् c. A.B.C.D. स्फुटहानौ B2. द्युयु d. A. तथ्यत्सौरिन्यथा; B. तत्सौरिन्यथा; C. मध्यात् सौरऽन्यथा; D. त [न्म] ध्यखण्डोऽन्यथा

318 PAÑCASIDDHĀNTIKĀ XVIII.11 TS have expressed doubts about their translation, since they have not understood verses 9-11. They have retained hi, not supplied the wanting mātrās, and not noticed the grammatical error in Saure. NP have made three drastic emendations, quite unrelated to the lettering of the text, nihitaḥ:, maṇḍalaḥ: and tanmadhya-khaṇḍe, though generally following TS. ‘रसविषयकृतशशाङ्काः’ क्षयखण्डे ‘(ख)धृतयः’ पदं यावत् | ‘विषय’ (रसेशा) वृद्धौ जीवः स्यात् पञ्चनवतिशतात् || ९ || ‘षड्वसुमनवो’ हानौ तृतीयखण्डे गुरुस्तु षोडशके | प(द)गुणिते ष्ट्यष्टकभाजिते कला पूर्वतोऽभ्युदिते || १० || नव सार्धाः कन्यांशाः प्रथमे खण्डे द्वितीयखण्डे (स्युः) | चक्रार्धं च(युगां)शाः दश(च) कला देवपूज्यस्य || ११ || 9. Jupiter being in the diminishing-motion-sector upto 180 padas, there is the constant 1456 (to work with, in order to get the eq. cent-corrected-Jupiter). Being in the increasing-motion-sector in the next 195 padas (i.e. 181 to 375), there is the constant 1165. 10. Jupiter being in the diminishing-motion-sector (again) in the next 16 padas, there is the constant 1486. (After subtracting or adding the padas for which we want computation from these numbers, in the respective sectors), multiplying them by the padas and dividing by 24, minutes of arc are got, (as the eq. cent. corrected total motion in the respective sector) at the rising in the east (and also thereafter if wanted). 11. The total of such motion of Jupiter in the first sector is 5ʳ 9° 30'. In the second sector, it is 6ʳ 4° 10'. Briefly expressed as formulae, the eq. cent. corrected Jupiter is given by: i. If padas are from 0 to 180, (1456 −padas) × padas' ÷ 24. ii. If padas are in the next increasing sector, i.e. from 181 to 375, (1165 + padas) × padas' ÷ 24 + 5ʳ 9° 30', where the padas used are those given in that sector. 9a. B. रसा B. ॰ते अष्टकभा; C. ॰तेऽष्टकभा; D. ॰ते त्वष्टभा b. B. Hapl. om: खण्डे [खधृतयः.... खण्डे] d. A.B.C.D. कलाः. A.B.1.2. ॰भ्युदिते; गुरुस्तु (10b) D. ॰भ्युदेति A. विधृतयः 11b. A.B2.3. खण्डे स्फुः c. A. रसोना; C.D. रसेना c-d. B1.2.3. missing. d. A. पचंवति c. A. चक्रार्धं व गुणाशा; C. चक्रार्धं च गुणाशाः; 10b. B. शोडशके D. चक्रार्धं द्विगुणांशाः c. A.B.C. पञ्चगुणिते; D. पञ्चविगुणिते d. A. दश श कला; C. परशकले; D. दश सदला

XVIII.11 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 319 iii. If the padas are in the next following sector, i.e. 376 to 391, (1486 – padas × padas' ÷ 24 + 5ʳ 9° 30′ + 6ʳ 4° 10′, where the padas used are those gone in that sector. Though the instructions are laconic, comparison with the Moon's computation makes things clear. The increasing-motion sector is obviously the 180° from apogee to perigee, where the rate of motion is supposed by this siddhānta to increase uniformly from a minimum to a maximum. The apogee is at 180 padas (= 166°) and the perigee at 376 padas(= 345°). The last 16 padas, continued by the first 180 padas form the diminishing half circle where the rate of motion diminishes uniformly from the perigee to the apogee. Differentiating the formula, (constant ∓ pada) pada'/24, the increase or decrease in the rate of motion is found to be 2′/24 = 1′/12 per pada. There may be a small hiatus at the junction, apogee and perigee, owing to the unequal division of 391 into 196 and 195, to avoid half pada. But the average of the rates at apogee and perigee, (1165′ and 1486′)/ 24 = 55′ 1/4, agrees with the mean motion forming one pada. (Incidentally, this justifies our amend- ment of viṣayarasonā into viṣayaraseṣāḥ. There are other justifications also, as we shall show later). Further, the first sector being a continuity of the third, the rate during the first pada in the first sec- tor must follow next to the rate during the 16th pada of the third sector. Since 1486′/24 is taken as the motion of the first pada, the motion of the 16th is (1486-30)/24 = 1456′/24. This must be the commencement of the third sector, and this is what is given. We can also see that the fastest rate, (at perigee), is 1486′/24 = 62′, and the slowest, 1165′/24 = 48′ 1/2, (at apogee), giving the mean value 55′ 1/4, of the pada, already found. But the rate for the 196th pada, ending which there is the apogee, is, (1486 − 195 × 2)′/24 = 1096′/24. But the minimum motion falling at apogee is given as 1165′/24. This hiatus*must also be due to the fact that the 2′/24 increase in the rate per pada is only approximate, and the actual is a little less than 2′/24. But the formulae are so given that the total of the first sector is (1456 − 180) 180′/24 = 5ʳ 9° 30′, as given. The total of the second sector is, (1165 + 195) 195′/24 = 6ʳ 4° 10′. The total of the third sector is, (1486 − 16) 16′/24 = 16° 20′. These add up to 12 rāśis, exactly, as they should. Incidentally, this justifies my emendation of guṇāṁśāḥ into yugamśāḥ, pāñcaguṇite into padaguṇite, and giving the meaning of tryaṣṭaka as 3 × 8 = 24. The justification for correcting vidhṛtayaḥ into khadhṛtayaḥ to get 180, and rasonā into raseṣā to get 1165, are also reinforced by this perfect agreement found here. TS and NP also give khadhṛtayaḥ:, seeing the reason for that. TS emend rasonāḥ into rasenā (= 1265), which will give the total 6ʳ 17° 43′, far from the correct 6ʳ 4° 10′. The text itself gives 6ʳ 3° 10′, one degree off. TS give 6 rāśis exactly, not knowing the peculiarity of this Siddhānta. Using cakrārdhe thus, they are left with guṇāśaḥ daśa ca kulāḥ. This they interpret as 13° (wrongly, for it can mean only 30 or 103). Emending daśa ca kalāḥ into paraśakale, they say that this 13° is the total motion of the third sector. They do not realise that the 16 padas of the third sector is near perigee, and the total motion must be greater than the mean motion, 14° 44′. Not knowing the nature of the method here, they think that the total of the third sector also should be given. It has no use, and Varāhamihira has not given it. About pāñcaguṇite tryaṣṭakabhājite, I have emended pāñca into pada, to delete the one mātrā in excess, and to give the agreement already seen. tryaṣṭaka is 24, as already said. TS retain the pañca, but emend tryaṣṭaka into aṣṭaka, making it 5/8, leading nowhere. As for NP, they generally follow TS's emendations. But, for the divisor 8 they suggest the alter- native 83 (tryaṣṭaka). Unlike TS, they realise that the three sectors must add upto 12 rāśis and make their own emendation of the last part of verse 11, as dviguṇāṁśā daśa sadalā, interpreting it as 20° 30′. NP have given the gist of verse correctly, but making a lot of unnecessary emendations. They have wondered in Part II, why such small units, as padas, have been taken. This is because, they

320 PAÑCASIDDHĀNTIKĀ XVIII.13 seem to think, that the three sectors are each taken wholly to get intermediate values by interpola- tion. An examination of the total of each sector would show how wrong it would be. The true eq. cent. corrected Jupiter is given for the end of any pada we want. We are expected to use these to get the true motion through any segmentation of the total padas, for correct interpolation, and the ends of the segments may fall anywhere, from pada 0 to pada 390. Therefore the small pada seg- ments are used. I shall work out an example at the end to make everything clear. I shall explain the rationale of the instruction in verse 8, of adding or subtracting the difference. The eq. cent-corrected Jupiter is subtracted from the Sun to get the anomaly of conjunction. So, a positive eq. cent. means less anomaly of conjunction. The days left over represent the anomaly of conj. with the 399 days of the synodic period, corresponding to 360° of anomaly. So the day is taken as roughly equal to the degree of anomaly, and the difference in degree subtracted. Vice versa for the eq. cent. corrected Jupiter, it being less then the mean. Varāhamihira is too astute to confuse day and degree, as NP think. (In verses 64-81 too, there is no confusion in the author's mind, as NP seem to think. There he has deliberately chosen the time taken by the Sun to move one degree as the unit of time, and call it 'day', for convenience. This is patent on the face of the synodic periods given, though TS have not even seen it, and are perplexed. We have reason to think that verses 64- 81 are by somebody else). दिन (षष्ट्यांशान्) द्वादश 'खकृतैर्वे (दान्)' 'कृताश्विभिर्द्वौ च । सप्ताष्टकेन वक्री षड्[भागान्] षष्टितः षट् च ॥ १२ ॥ अनुव(क्रो) ऽशीत्यार्का (न्' द्व्यू) नार्ध (श) तेन नव ततोऽस्तमितः । स्थित्वा सैकं मासं स्फुटोद (योऽष्टोत्तरैरङ्गैः) ॥ १३ ॥ ॥ बृहस्पतिः ॥ 12. By 60 days, (Jupiter moves) 12°, by 40 days 4° and by 24 days 2°. Becom- ing retrograde, by 56 days he moves 6° (i.e. -6°) and by 60 days, 6° (i.e. -6°). 13. Following after retrograde, he moves 12° in 80 days, and 9° in 48 days. Then setting, staying so for a month plus one day, he clearly rises moving 6° 8'. Ends Jupiter. The Scheme given

XVIII.13 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 321 These values agree well with actualities, considering that whole days and whole degrees are given, excepting the last 6° 8′, given to complete the value for the synodic cycle. 6° 12′ would be better at that region and for the whole number, 399 days. 56 days for – 6°, and 60 days for the same – 6° must be explained by the intention to give whole degrees and segmentation. Vargāḥ is an obvious mistake for bhāgān, and so corrected. TS have interpreted saptāṣṭakena to mean 15, which such an expression never means. It can mean either 56 or 87. They understand another 60 days by the word ca used. All this, to make up the wrong scheme used by them, based on the mistaken idea that the statement of motions here begins with conjunction and ends with the rising in the east after the next conjunction. The following is their scheme: Days Conjunction 60 40 24 15 60 60 80 45 Setting west 30 Rising east = 414 Degrees 12° 4° 2° 0° – 6° – 6° 12° 9° (15°) = 42° ddīnārdhamatena is emended by TS into dhyūnārdhaśatena but how can this word mean their 45? As for the last part, sthitvā saikam māsam, they have taken it to mean 30 days instead of the correct 31 days. Let that be. They have not given any motion for it in their interpretation. It cannot be left to be guessed and completed by an ordinary computer. They, who can be expected to know, have guessed, quite wrongly, 15° motion for 30 days, not realising that it can be only 6° and a few minutes more. For the 414 days from conj. to the rising after the conjunction, the total can only be about, 33° 9′ + 3° = 36° 9′, and not the 42° given by them. As for NP, they have emended ddīnārdhamatena into dinārdhaśatena to mean 50 days. Since they take 30 days for the setting i.e. one day less, they make the total of days, 400. They give 7° motion for the 30 days (which they make even 29 days in the last part). They have changed the wording to some ununderstandable form here, dyavantye māsasya. Further, the 7° is far too much for 30 days. But there is no 7° in the text. They have corrected the text saikam into śvam, thinking that aśvam in bhūtasaṅkhyā means 7°. Incidentally, one other matter may be considered here, viz., the degrees of heliacal rising, for Jupiter. During the set-period of 31 days, the sun moves about 30½ degrees, and Jupiter, about 6° 8′, and the relative motion is 30½° – 6° 8′ = about 24°, from setting to rising. This gives about 12°, for the heliacal rising of Jupiter, which is fairly accurate, especially for very high latitudes. (Classical Hindu astronomy gives 11°). Verse XVIII. 58 gives the Vāsiṣṭha-Pauliśa’s degrees of heliacal rising as 12°, 14°, 12°, 15°, 8°, 15° from Moon onwards, by candrādinām dvādaśamanuravitithyaṣṭatithisaṁkhyaiḥ:. 15° for Jupiter given here is too much, and 14° for Mars is too low. (Classical Hindu astronomy gives 17° for Mars). So, the scribe seems to have made a small change in the order, and the correct order is “candrādīnām dvādaśatithimanuravyaṣṭatithisaṅkhyaiḥ:”, 12°, 15°, 14°, 12°, 8°, 15°, with only one change of place. Example: Find the True Jupiter at 2415 days from epoch. (i) The beginning of the first cycle after rising next to the epoch is 34-34 days later. The days after this, required to find the number of cycles gone = 2415 – 34-34 = 2380-26. Dividing by 399, cycles gone = 2380 – 26/399 = 5, with 385-26 remainder. Adding 5 × 1/9 days, (= 0-33), we have 385-59 days left over after 5 cycles gone. (ii) The padas at 5 cycles gone = 18 + 5 × 36 = 198. Mean Jupiter = 198 padas = 198 × 360°/391 = 6ʳ 2° 18′. True Jupiter:– For the 198 padas, 180 padas forming the first sector has gone and 18 padas are left over in the second sector.

322 PAÑCASIDDHĀNTIKĀ XVIII.15 ∴ True Jupiter = 5ʳ 9° 30′ + (1165 + 18) 18′/24 = 5ʳ 9° 30′ + 14° 47′ = 5ʳ 24° 17′. Eq. cent. = True − Mean = 5ʳ 4° 17′ − 6ʳ 2° 18′ = −8° 1′. (iii) The padas at 399 days in the cycle, i.e., the beginning of 6 cycles gone = 198 + 36 = 234 = 180 + 54. Mean Jupiter = 234 × 360 ÷ 391 = 7ʳ 5° 27′. True Jupiter = 5ʳ 9° 30′ + (1165 + 54) 54′/24 = 6ʳ 25° 13′ True − mean = Eq. cent. = 10° 14′. Eq. cent. at 0 day of 6th cycle = −8° 1′ Eq. cent. at 399 days of 6th cycle = − 8° 1′ Eq. cent. at remaining days (385-59) = = (385-59) × −2° 13′ ÷ 399 + −8° 1′ = −10° 10′. (iv) True Jup. is less than Mean Jup. by 10° 10′. ∴ days of Anomaly of Conj. = 385-59 + 10-10 = 396-9. (v) True an. of conj. = for 60 days + 12° for 40 days + 4° for 24 days + 2° for 56 days − 6° for 60 days − 6° for 80 days + 12° for 48 days + 9° Total 368 days + 27° 28-9/31 × 6° 8′ = 5° 32′ for 28-9 days 5° 32′ 396-9 32° 32′ (vi) True Jup. = Mean Jup. at 0 day of An. of conj. + eq. cent. + true ano. of conj. = 6ʳ 2° 18′ − 10° 10′ + 32° 32′ = 6ʳ 24° 40′. Note 1: The need for interpolating the eq. cent. to the remaining days in the cycle can be seen by working for 399 days of the 6th cycle and 0 day of the 7th cycle and comparing. They must be the same. Note 2: The eq. cent. is computed for 0 day of each cycle, i.e., for intervals of 36 padas = 33° 9′. Interpolation using these, as we have done, can be only rough. To get better interpolations, we can divide the 36 padas into desired segments, find the eq. cent. of each, and use. We can form an ephemeride, giving the values at the ends of the day segments given, 60, 40, 24, etc. and use for interpolation. All these logically follow from the instructions, though not specifically stated. [शनिचार:] अध्यर्धशतं (स)त्र्यंशमपनयेत् सूर्यजस्य दिवसेभ्यः । 'वसुमुनिगुणो' (द्भू)तेभ्यः स्थि(ता) दिनाद्या(स्स) मभ्युदयात् ॥ १४ ॥ जह्या(दु)दयदशांशं क्षुभ्यो नवसंगुणा(न् भ)जेदुदया(न्) । 'षड्विषययमैः' शेषं पदैर्युतं त(न्र)वाशीत्या ॥ १५ ॥

XVIII.18 | XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING | 323 Motion of Saturn As already has been said, the treatment of Saturn is similar to that of Jupiter. So there will be little need for fresh explanations. 14. Regarding Saturn, 150-20 days are to be subtracted from the days from epoch. These being divided by 378, the remainder are the days from the rising gone, the quotient being the number of risings gone. 15. One tenth of the risings, (i.e., the quotient), in days, is to be subtracted from the remainder. The number of risings got is to be multiplied by 9, and divided out by 256. The remainder plus 89 padas form (the padas required for using in the computation). (The idea is that 89 is to be added to (quotient × 9), and then divided by 256, to find the padas for use). In (15), I have emended saṅguṇād and rudayāt into saṅguṇān and rudayān to agree with bhajet requiring accusatives; so also NP. But TS have kept them. In NP's emendation dinādyāptaṃ, āptaṃ does not agree with the word sthitā and, the meaning also is redundant. Both TS and NP have emended padaiḥ into pade, thinking that navāśītiḥ is degrees. Even this they doubt, as seen in the trans- lation, because as mentioned by them in Part II, page 124, it had led to disagreement. Padaiḥ, as it is, clearly, says that the 89 is padas. So is the 18 of Jupiter and the 85 of Mars. We understand from the instructions that the synodic revolution of Saturn takes 378'/10 days, that in one synodic revolution Saturn moves 9 padas, that 256 padas make nine sidereal revolutions of Saturn, that there are 256 + 9 = 265 sidereal revolutions of the sun in 256 synodic periods of Saturn, and that at 150-20 days from epoch, Saturn's mean longitude is 89 padas. (NP give in their translation, “89°?”, as mentioned already. Therefore, one sidereal revolution of Saturn takes 378'/ 10 × 256 ÷ 9 = 10754.84 days. One sid. revolution of the sun = 378'/10 × 256 ÷ 265 = 365-15-32. Again, the Sun's sid. period got is Pauliśa's. One pada = 360°/256 = 84' 22".5. The motion in one synodic revolution = 9 × 84' 22".5 = 12° 39' 22".5. Mean Saturn at 150-20 days after epoch = 89 × 84' 22".5 = 125° 9'.4. 'षड्रूपवेदपक्षाद्' वृद्धिस्रिंशत्पदानि सौरस्य | 'नवरूपविषययमला(द्)' ह्रासः 'स्वरभास्कर'पदा(न्तः) || १६ || प्रचयः 'स्वराग्निखयमान्' नवनव(ति)स्त्रिघनभागलिप्तानाम् | क्षयवृ(द्धी) द्विगुणपदैरकगुणन्नः श(नेरु)दयः || १७ || षोडश वृषभस्यांशा नवलिप्तावर्जिताः प्रथमख(ण्डे) | 'विषया' स्त्रिघ(ना) स्त्रिंश(त्) चतुर्युता मध्यमे खण्डे || १८ || 14a. A. शत्र्ये; om. the two letters. 15a. A.B. जह्याद्युदय. B. दशांश b. B. णमपानये सूर्य b. B. नवसं०. A.B.C. भजेदुदयात् c. A.B. गुणोष्टृतेभ्यः d. C.D. पदं युतं. A. तन्तवाशीत्या; B. मत्तवाशीत्या d. A.B.D. स्थितं. A.B. दिनाद्यास्स; D. दिनाद्याप्त

324 PAÑCASIDDHĀNTIKĀ XVIII.18 16. Regarding Saturn, there is an increase (of the rate of motion) for thirty padas, from 2416. Then, there is a decrease for 127 padas from 2519. 17. Next there is an increase for 99 padas from 2037. The amount of decrease and increase are by the padas multiplied by 2. The divisor of the total minutes is 27, its multiplier being one. 18. The total of the first sector is 1ʳ 15° 51' and the total of the middle sector is 5ʳ 27° 34'. Note: The multiplication by one is unnecessary, but given to clear the doubt that may arise by the instruction to multiply the padas by two for subtraction and additions coming before. The meaning is clear, and no material change has been needed. I shall give what is given in the form of formulae: The total motion upto any pada in the first sector, viz., (1-30) = (2416 + 2 × padas) padas ÷ 27, in minutes. That in the second sector, viz., (31-157) = (2519 − 2 × padas) padas ÷ 27, in minutes. That in the third sector, viz., (158-256) = (2037 + 2 × padas) padas ÷ 27, in minutes. The total of the whole of first sector given. 1ʳ 15° 51' can be verified thus: (2416 + 2 × 30) 30 ÷ 27 = 2751' = 1ʳ 15° 51' given. The total of the whole second sector = (2519 − 2 × 127) 127 ÷ 27 = 10654' = 5ʳ 27° 34', given. Being unnecessary, the total of the third sector is not given. But we can calculate it and use it to see if all those add up to 12 rāśis, as necessary, and this will verify every instruction given. The total of the third sector = (2037 + 2 × 99) × 99' ÷ 27 = 8195' = 4ʳ 16° 35'. Now, 1ʳ 15° 51'

• 5ʳ 27° 34' + 4ʳ 16° 35' = 12ʳ. Examining the constants, we find that the maximum motion per pada is 2519' ÷ 27 = 93'.3. The minimum rate is 2037' ÷ 27 = 75'.4. The mean rate is = 84' .35 as already found, as the mean motion equal to the pada. Differentiating as before, the increase or decrease in the rate is 4'/27. Actually it is slightly less than this, the multiplier being slightly less than 2, given. (2037 + 4 × 98) = 2416 shows this. The perigee falls at end of 30 padas, i.e., 1ʳ 14°, and the apogee, 127 padas later, at 7ʳ 13°. The instruction how to use the result of these verses has not been given, because it is the same as that given in verse 8 for Jupiter. Indeed, the un-emended reading saure there means, "with reference to Saturn". 16a. B. षड्भूप. A.D. पक्षा c. A. वृद्धि; C.D. वृद्धिः c. A.C.D. यमला; B यमलो c-d. C.D. द्विगुणहताश्चैकगुणघ्नः d. A.B.C.D. पदाख्यः d. A.B. शनैरुदयः 17a. A.B.C.D. यमा 18a. B. वृषभांशा b. A. नवनवतस्तिघन; B. नवनवतस्त्रिघन b. A. खण्डाः; B. खण्डा c. A.B.C.D. त्रिघनः. A.B. त्रिंशः

XVIII.20 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 325 As in the case of Jupiter, here too TS and NP have not understood what exactly is given in these verses, how it is got by applying the three formulae, how the eq. cent. is got, and why the instruction to apply this to the days remaining is given, in the manner said. So, their emendations of the readings, done without knowing the subject matter, need not be taken seriously. TS have emended the correct dviguṇapadaiḥ into dviguṇahṛtaḥ, meaning "divided by 32", applied to the risings and not to the number got in the formulae. NP have kept the reading, but given the translation as, "There is a subtraction or addition of 12 degrees and minutes, (i.e., 12° 12'). Multiply by 31 and divide (the product) by 32 (or by 32 padas). (The result is) Saturn's rising." Where is 12° 12' mentioned? They take the 32, not as a number, but as a segment of longitude equal to 32 padas, i.e., 45°. Again, how can this give the risings? And the risings have already been given in verse 14. All these show that they do not understand what is said. षड्(कृत्या त्रीनं)शान् (मुनि)भिर्लिप्ता '(श्रेषु)गुणास्सप्त' । षोडशभिश्चांशी(र्ति)कृतोनषष्ट्या '(वेदयम)'पक्षान् ॥ १९ ॥ वक्री वि'भूतषष्ट्या' (त्री)नंशान् षष्टितः 'कृतान्' सौरः । अनुगोऽर्कश(ते'ना)ष्टौ षट्कृत्या चास्त(गो) 'दहनम्' ॥ २० ॥ || शनैश्चरः || 19. Saturn (moves) 3° in 36 days, 35' in 7 days, 80' in 16 days, and 224' in 56 days. 20. Then becoming retrograde, he moves 3° in 55 days, and 4° in 60 days. Then, following up direct, he moves 8° in 112 days, and setting, he moves 3° in 36 days in the set period, (i.e., rises in the east after that). Ends Saturn. This is the scheme given Days Rising east 36 7 16 56 Retrograde 55 60 Direct 112 Setting west 36 Rising east = 378 Degrees 3° 35' 1°20' 3°44' - 3° - 4° 8° 3° = 12° 39' I shall now discuss the values given, justifying the three emendations. I have made. The corrupt ṣaḍhratāstrīṇamśān has to be emended as 3° for 36 days, considering the position, and the fact that it must practically be equal to the rate between setting and rising, 3° for 36 days. The days must add upto 378 days from rising to rising, also as from conjunction to conjunction. All the numbers for 19a. A.B. षट्कृताः; C. खण्डान्ये. D. परिहीनाः 20a. B. विभुत b. A.B. स्त्रीणांशान्; C. सिंहांशा. D. स्त्रीखांशा a-b. C. षष्टाष्टरसैस्त्रीन् षष्टितः c. A.B.D. लिप्ताश्चतुर्गुणाः A. त्रिंशान्; B. त्रिंशान्: A1. कृतात् सौरः; c. A.B. श्वाशीति; C. श्वांशामीन् B1.3. कृत्यसौरः d. A.B.C.D. षष्ट्या द्विगुणपक्षान् (A. पक्षात्, c. A. शतैर्नाष्टौ; B1. शतैर्माष्टौ (B2. र्माष्टौ); B1.2. पक्षा. B. श written after पक्षा) D. शतेनाष्टौ d. A1.2. गे दहनं || B1.2. षड्भक्त्या वास्तये दहनं ||

326 PAÑCASIDDHĀNTIKĀ XVIII.20 days are clear. Therefore, the days for the second segment must be 7. So I have emended manu into muni. The motion given there, 28', gives the rate 2', too absurd for that position, if the original 14 days are accepted, and it cannot be that the Siddhānta does not know the absurdity. Even for the emended 7 days, it is too low, being only 4' rate, while the rate on both sides is 5', and also consis- tent with facts. Therefore, ścaturguṇā is emended into śceṣuguṇā. Now, these three segments can be combined into 4° 55' for 59 days, without affecting the result. I do not know why the Siddhānta has broken it into such bits. Next, the total for the 378 days must be the mean motion for the period, i.e., 9 padas, equal to 12° 39'.4, roughly taken by the Siddhānta as 12° 39'. Therefore the motion for the fourth segment, 56 [

XVIII.20 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 327 I shall now give an example, to make the method clear. Example: Find true Saturn at 5000 days gone from epoch. i. Days 5000 To be subtracted 150-20

Divided by 378 4849-40 (12 = full cycles gone) 313-40 = (Remaining days) Days to be deducted 10/12 : 1-12

312-28 (corrected remainder) ii. Padas at 0 day of the 13th cycle: (89 + 12 × 9) / 256 = 197 remainder Mean longitude = 197 padas = 9ʳ 7° 2′ 197 = 30 + 127 + 40 (in the third sector) Eq. cent. corrected mean longitude:- = 1ʳ 15° 51′ + 5ʳ 27° 34′ + (2037 + 2 × 40) 40′ ÷ 27 = 1ʳ 15° 51′ + 5ʳ 27° 34′ + 1ʳ 22° 16′ = 9ʳ 5° 41′ Eq. cent. = 9ʳ 5° 41′ − 9ʳ 7° 2′ = −1° 21′ iii. Padas at 378 days gone in the cycle = 197 + 9 = 206 Mean longitude = 206 padas = 9ʳ 19° 41′ 206 padas = 30 + 127 + 49 (in the third sector) Eq. cent. corrected mean longitude = 1ʳ 15° 51′ + 5ʳ 27° 34′ + (2037 + 2 × 47) 47′/27 = 9ʳ 18° 0′ Eq. cent. = 9ʳ 18° 0′ − 9ʳ 19° 41′ = −1° 41′ Interpolated for days 312-28, the eq. cent = −1° 21′ − 0° 17′ = −1° 38′. iv. Correcting the remaining days 312-28 by this, 312-28 + 1-38 = 314-6 days, to be used to find anomaly of conjunction. v. 36 days +3° 7 ... + 0°.35′ Mean Sat. at 0 day 9ʳ 7° 2′ 16 ... + 1° 20′ Eq. cent − 1° 38′ 56 ... + 3° 42′ An. of conj. + 7° 40′ 55 ... − 3° --------------------------- 60 ... − 4° True Saturn = 9ʳ 13° 4′ Remaining 84-6 + 6° 1′ ---------------------------

314-6 + 7° 40′ As per Ephemeris: 9ʳ 11°.7 (sāyana)

Mars As indicated earlier, Mars, like Mercury, needed elaborate treatment owing to certain peculiarities about it, and so had been reserved by VM to the end of PS. The synodic period of

328 PAÑCASIDDHĀNTIKĀ XVIII.24 Mars, on which the equation of conjunction depends, is 780 days, during which there are more than two revolutions of the Sun, and one revolution of Mars, so that one full anomalistic period of Mars is contained within this period. This, with the large equation of the centre, and the large equation of conjunction causes large variations in its motion from sign to sign, and even in the same sign, according to the different types of motion governed by the anomaly of conjunction, like, fast, slow, retrograde etc. Hence is the need for detailed treatment. Further we have reason to think that the various motions given are all empirical, based on long observation, synodic period after synodic period. The separation into the equation of the centre, and the equation of conjunction is yet to come, it seems, unlike the cases of Jupiter and Saturn, where it is easy, and done. This would explain discrepancies found in the values given. Regarding the constants given, some can be verified by mutual comparison, and corrected where necessary, when there is a doubt about the reading itself. But some, like the epoch constants, which are peculiar to the Siddhānta itself, cannot be so verified and corrected when there is a doubt. Only in such cases, where we can argue that no Siddhānta is likely to give such wrong values, and when these values are so far from the real, that we can make some plausible corrections. TS and NP have not understood the nature of the motion of Mars, just as they have not under- stood Jupiter and Saturn. While TS have not even attempted translating some verses, wrongly interpreting those attempted, NP have attempted translating all, but many wrongly. I shall point out these after my own translation and discussion of the verses, step by step. [कुजचारः] द्युगणे ‘षट्(पञ्च)यमान्’ विहाय पञ्चाष्टकं च नाडीनाम् । ‘गगनाष्टमुनिभि’रुदया लभ्यन्ते प्राङ् महीजस्य ॥ २१ ॥ उदयगुणिता विनाड्यः ‘स्वरतिथयोऽ(ब्ध्य)’न्विता दिनक्षेपः । ‘धृतिगुणि(तांस्त्र्यग्रीन्दु)’भिरुदया(न्ह)त्वा स्थितो (तस्मिन्) ॥ २२ ॥ पञ्चाशीतिं कृत्वा प्रतिराश्य मध्यमः क्रमशः । राशिप्रमाणतोऽ(स्य) स्फुट(तश्च)रक्रमं कुर्यात् ॥ २३ ॥ स्फुटमध्यम(विश्लेषां)शान् क्षिपेत् मध्यमे[ऽधिके] द्युभ्यः । मध्यमहानौ जह्यात् गतितोऽथ चारा(न)भिधास्ये ॥ २४ ॥ Motion of Mars 21. Subtracting 256-40-0 days from the days from Epoch, and dividing by 780, the synodic risings of Mars in the East are got. 22-23. (157 plus 4) vināḍis, multiplied by the risings got, are to be added to the remaining days. Multiply the risings got by 18, and adding 85, divide by 133. The remainder, converted into rāśis is Mars at rising. According to the whole or portions of rāśis, the true motions are to be taken one after another, and pieced together.

XVIII.24 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 329 24. The difference between the mean and true degrees should be added (to the remaining days got in 21), if the mean is greater. If the mean is less, the difference should be subtracted from the remaining days. This done, I shall give the true motions, according to each type of motion: We learn from the verses the following: . 256-40-0 days from Epoch, Mars rises in the East, after which the counting of risings begin.

1. One synodic revolution takes 780 days — 161 viṇāḍis, (i.e. 779-57-19 days). The addition of viṇāḍis multiplied by revolutions, is for taking the synodic period as 780 days approximately. '. For this period of 779-57-19 days, we get 1 + 18/133 sidereal revolution of Mars, and 2 + 18/133 sidereal revolutions of the Sun. So, in one synodic period, Mars moves 408° 43′.3 ∴ In 133 syn. periods = 1,03,734-3-7 days, there are 151 sid. rev. of Mars, and 284 sid. rev. of the Sur. From this, the Sun's sid. period got is 365-15-38 days and Mars's 686-58-50 days. The Sun's period is 38 viṇāḍis more than that given for it by the Vāsiṣṭha, and near the 365-15-30 of the Pauliśa. Therefore, like the Moon, Venus, Jupiter and Saturn, Mars also is common to Pauliśa.
2. At the first rising when calculation commences, mean Mars = 85/133 rev. = 7ʳ 20° 4′.5
3. The addition or subtraction of the difference from the remaining days has been already explained with reference to Jupiter and Saturn. But, here, no method is given to find the equa- tion of the centre. Now the true motion is affected by both the eq. of the centre and eq. of con- junction. The segments of motion given in verses 25-26, below, are as affected by the eq. of con- junction alone. By making the days given for true motion in verses 27-35 conform to the seg- ments, we can get the degrees, and through that the days affected by the eq. centre alone. This can be of use only fōr the remainder of days. But the eq. of centre at the beginning of each cycle is necessary. It has not been given by any rule. Since its period is about 687 days and it has its own rise and fall of about 11° from perigee to apogee and back, it cannot be associated with the synodic period of 780 days. So this is an omission. I have corrected the corrupt tāstrayāmṛīdubhiḥ into tāms tryagnīndubhiḥ to mean 133. This is necessary for agreement with the actuals, and the effect of my emendation is seen in my discussion (3) above. TS have made it bāṇendubhiḥ. How can bāṇa, with such different lettering, come in here? Further, this will give 18/15 rev. = 432° as the mean motion of Mars in one syn. period, about 23° wrong per period. They have made pañcāśīti kṛtvā into pañcāmśonam kṛtvā and thus shut out the position constant of Mars on the first day where reckoning begins, viz. the point of time 256-40-0 days from epoch. (It will be remembered that in the cases of Moon, Venus, Jupiter and Saturn also, they have made this mistake). By this emendation they reduce the motion by 86° 24′, and make the 21a. C.D. द्युगणात् b. C.D. प्रतिराशिं; A.B. षट्कं व यमान् (B3. षट्कं). D. षट्कैकयमान् B. मध्यतः, A.B. क्रमश b. B. नाडित्वं c. A.B. प्रमाणतो स d. B. प्राक् d. A.B.C. स्फुटता चार०; D. स्फुटिता चार० 22b. A. योब्धान्विता; B. सोब्दान्विताः B. क्रकुर्यात्; D. क्रम[त्] कुर्यात् c. A.B. गुणितास्त्रयग्नी (B. त्री) दुभि; C.D. ०गुणितान् 24a. A.B. विक्षेपां बाणेन्दुभि b. A. ०शन् क्षिपेत्; B.C.D. शकान् क्षिपेत् d. A2. हत्वा; A.B. स्थितो तो समाः; c. A.B.C.D. मध्यमे द्युभ्यः C.D. स्थितोऽतोऽस्मात् d. A. गतितोथाचारमभि०; B. गतितोष्याचारमभि०; 23a. A.B. पञ्चशीति; C.D. [पञ्चांशोनं] C. गतितोऽथो चारमभि; D. गतितोऽप्याचारमभि०

330 PAÑCASIDDHĀNTIKĀ XVIII.26 mean motion of Mars 345° 36' per syn. period of 780 days! They have also wrongly emended pratirāśya to pratirāśyam. As for NP, they have made the correct emendation tryagnīndubhiḥ, giving correctly 151 revolu- tions of mean Mars in 133 syn. periods, and identified it with that given by the Babylonian astronomy of the Selucid period. But NP have not seen that pañcāśītim meaning 85, is correct as it is, and give the constant 7ʳ 20° 4'.5 at 256-40-0 days from epoch (see item 5 of discussion). They think it is the constant in degrees, though no word meaning degrees is found here. (This kind of mistake they have made in the case of Jupiter and Saturn also, as we have shown). But 85° would not do, so they have substituted satrirāśim for pratirāśya and made it 175°. But even this would not do, and therefore they have changed the days from Epoch itself into 216-40-0, by emending ṣaṭkam va yamān into ṣaṭkaikayamān. But this has led to other troubles, leading to their remark, “For Mars this would mean a longitude of 175° (instead of 194° derived on the basis of a₀ in table 32). This longitude would correspond to September 27, and a solar position at 186°, hence to an elongation of 11°’ (p.124, Part II). 11° for the first visibility of Mars is given by nobody. It is in the range of 14° to 17°. प्रागुदये षट्(चत्वार्येक)मष्टादश(ग)स्ततो वक्रम् | अध्यर्धं च शतं शीघ्रां [स्ततोऽस्तमितो द्यूनां षष्ट्या] || २५ || समतीत्य दशत्रियु(तं) निरंशगतो(ऽतस्त्रिंशतं) व्यतीत्य कुजः | उदयमुपयाति वक्ष्ये गतिचारदि(न)क्रमे चातः || २६ || 25-26. After rising in the East, Mars moves 146° (in quick motion) and then 18° each of (slow motion), retrograde and “follow up after” retrograde (anuvakra), and after that 150° of quick (śīghra) motion. Then setting, it reaches conjunction (niraṁśagataḥ) in 60 days moving 13 plus 30 (= 43) degrees. Then it rises, (moving the same degree in the same number of days). Beginning from here, I shall mention the series of motions with their days. The scheme is Rises East Moves 146°            I                      type of motion            (śīghragati)               18°           II                                                     (Mandagati)           −18°      III & IV                                                (Vakaragati)                                                                                    (Ativakragati)               18°          V                                                   (Anuvakragati)             150°          VI                                                  (Śīghragati) Sets in the West 43° VII in 60 days                                                     (Atiśīghragati) Conjunction                                                              (          ”          ) 43° VIII in 60 days Rises in the East Total 400°

XVIII.26 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 331 The numbers I have given in the scheme are practically what are found in the text, without emendation excepting three. In verse 25, I have emended captasteka into catvāryeka to get 146° the most plausible value. NP have made it ṣaṣṭāṣṭaika to get 186° which is too large. See discussion below, 150° is given by adhyardhaṃ ca śataṃ where tataḥ is emended into śatam. This is necessary to make up the total 410° motion in 780 days. Secondly, 43° motion for 60 days given from setting to conjunction is required to agree with the 17° usually given for heliacal rising. This is made up by emending viṃśatam into triṃśatam with the 13° given by daśatriyutam added. I shall now show that the motion of Mars is near 43° in 60 days, in the region of the conjunction. For its distance, nearly 1.53 that of the Sun, given by modern astronomy and also as computed from Hindu astronomy, the equation of conjunction at this region is 11' per day, (as can be verified) which, plus the daily mean motion of 31'.4 gives 42'.4 per day, making 42°.4 in 60 days, roughly 43°. This also agrees with the angle for heliacal rising of Mars, nearly 17°, given by Hindu astronomy. (In 60 days the Sun moves 59.1°. So, the elongation is 59.1 − 42.4 = 16.7°, nearly 17°). If viṃśatim is taken as it is, we get 20° + 13° = 33°, which is 9.4° short of the actual 42.4 and which also gives the angle for heliacal rising as great as 26°, so far from the 14°-17° given by all. In the mean, the motion from setting to conjunction must be equal to the motion from conj. to rising. That is why it is not given by the text separately. That the motion segments given in the two verses is mean is also clear, since no position of Mars from its apogee is taken into account. So the total motion must be equal to 409°. But the total got by adding the segments is 400°. This must be due to the defective method of the original or the empirical nature of the motions, and rounding off to whole degrees, as seen from 43° being given for 42.4°. The opposition must fall at the mid- point of the retrograde motion, − 18°, and divide it into −9, −9. The total motion from conj. to opposition must be equal to that from opposition to conj. But what we actually get is 43° + 146° + 18° − 9° = 198°, and −9° + 18° + 150° + 43° = 202°. It may be that the angle segments given are empirical and also there are errors in the apparently correct numbers giving the segments, needing emendation. It is only in the case of Mars, does VM give these eight types of motion. In II. 12-13 of the Later Sūrya Siddhānta, a set of 8 types of motion is given. But they cannot be equated to these each to each. So we have only to guess when in doubt. The days on the synodic cycles to pass each type of motion must be nearly equal to the average of the days given in verses 27-35 for that type of motion. This has been used to check the degrees of each type. But the synodic period, as also the mean motion of 1 + 18/133 revolution in that period are very nearly correct and they must have been got by analysis of the observed motions. So the Siddhānta must have known that the motions and times are half and half on both sides of the opposition. Beginning from rising type I is śīghra (quick) motion. II is manda (slow) motion. The distinction seems to be faster or slower than the mean motion. So, the dividing point must be where the tan- gent from the earth touches the synodic circle. Since the mean distance of Mars is 1.53 times that of earth from Sun, this point falls about 189.5° from conjunction. Subtracting 43.5° motion from                                                                                  d. A.B.C. द्यु (B. द्वा) नाषष्टितस्तोस्तमितः; 25a. A. षट्चप्तस्तेक॰; B. षट्वपप्तस्तेक॰;                            D. त्रिभ्रषष्टिं ततोऽस्तमितः       D. षट्कास्तैकं; C. पदसप्तकं                                 26a. A.B.C. त्रियुता; D. त्रिहतात्   b. A.B. ॰दशमस्तगस्ततो; C. ॰मष्टादशमासगस्ततो;             b. A.B.C. निरंशतो विशंति (B .विंशति)       D. ॰दश वक्रगस्ततो                                                D. [निरंशगस्त्रिंशतिं]       B. चक्रं; D. वक्रः                                                   c. B. उदयेमुपयाति   c. A.B.C. अत्यर्धं च ततः शीघ्रात्. D. गत्यर्थं च ततः शीघ्रात्      d. A.B.C. द्यु (B .द्वा) नाषष्टितस्तोस्तमितः;

332 PAÑCASIDDHĀNTIKĀ XVIII.27 conjunction to rising (given as type VIII), 146° is left for type I. This segment extends upto the point where retrogression begins. As the planet is stationary, here a small error of observation can make this lesser or greater than the actuals. The text seems to give it as 18°. Types III and IV form the retrograde motion. III is called vakra (retrograde) and IV ativakra (faster retrograde). The text is defective here, and we cannot fix the exact extent of the two segments separately. But III and IV seem to be divided in the ratio 5:7 of the total. V is anuvakra (follow up after vakra). In the detailed motions given this is the sum of III and IV but direct motion. This anuvakra must be the counter- part of II. Type VI is śīghra and so the counterpart of I. Its extent is given as 150°. Type VII is the very quick motion from setting to conjunction and given as 43° in 60 days. Type VIII is the counter- part of VII from conjunction to rising. These divisions are mostly based on convention. But as these are given only in the case of Mars and classical astronomy does not give them, we have only to guess regarding them. To add to the difficulty the text is corrupt in the places giving the numbers. TS have expressed inability to understand verse 25. Still, they have made some emendations which do not give any cogent meaning. No translation is given. There is only a question mark. In verse 26, they give 20° motion from conjunction to rising. This can give only 26 days, as against the 60 days given by the text. By this the elongation for heliacal rising would be 7 ½°, so absurdly low. As for NP, in both verses, they have needlessly emended correct forms, wrongly emended the corrupt ones, some in faulty Sanskrit and given an untenable scheme. The following is their scheme: Rising east / 186° motion / 18° retrograde motion / 180° motion / setting / 30° motion / Conj. / 30° motion / rising east. They have made the emendations and substitutions with their eye on the total motion of 409° in the synodic period. They make the total 408°, nearly correct. But they do not identify the vestiges of the different types of motion found in these verses. Further, 30° motion from setting to conj. and then from conj. to rising, is short by 12 ½° from the actual 42 ½°. The time required to move 30° is 42.2 days and the Sun would move 41.5° during this time, giving an elongation of 11.5° for heliacal rising, far short of the actual, especially for such high latitudes as the Vāsiṣṭha-Pauliśa envisages. चत्वारिश (शच्छ)शि-न(ग)-[मु] (न्य)-ष्ट-यमान्विता वि‘पक्षा’ च । प्रथमगतौ (क्रमदि)वसा मीनाद्राशिद्वयसमानाः ॥ २७ ॥ 27. In the I type motion, there are 40 + 1 (= 41), 40 + 7 (= 47), 40 + 7 (= 47), 40 + 8 (= 48), 40 + 2 (= 42), 40 – 2 (= 38), days per motion of 30° each respectively in each month of the diad of rāśis beginning from Mīna, (i.e. Pisces). The above means, that for 30° of motion, the time taken is 41 days in the rāśis Mīna (Pisces) and Meṣa (Aries), 47 days in Ṛṣabha (Taurus) and Mithuna (Gemini), 47 days in Kaṭaka (Cancer) and Siṃha (Leo), 48 days in Kanyā (Virgo) and Tulā (Libra), 42 days in Vṛścika (Scorpio) and Dhanus (Saggittarius) and 38 days in Makara (Capricorn) and Kumbha (Aquarius). 27a-b. A.B.C.चत्वारिशशिन (B.नु) मध्य (B. om. ध्य) ष्ट० c. A.B. प्रथमगतौ कुर्याद्दिवसा D. चत्वारि श[शत द्वयमष्ट] ० D. यमान्वितं विपक्षांशं C.D. प्रथमगतौ कुर्याद् दिवसा; (D. दिवसान्) d. D. समान्

XVIII.28 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 333 An examination of the rate shows that the perigee is situated at the end of the Makara (Capricorn) and the apogee at the end of Kaṭaka (Cancer), which both fairly agree with the actual. TS say that they do not understand this verse, and no translation is given; its place being taken by a question mark. NP translate thus: “In the first gati 240 plus 28 minus 1/2 (= 267 1/2) (days). One should calculate days for every two signs from Pisces.” It is clear that they do not see that this verse gives the detailed rate of motion of the Type I gati in the diads of rāśis from Pisces, as affected by the equation of the centre. They think that the first motion given in verse 25, which 186° according to them, takes 267 1/2 days, as given by them here. If so, what use is the instruction to calculate for “every two signs from Pisces”? 'विषय-[रस-] खर-(रस-)र्तु-पञ्चका[न्] 'दशगुणान् द्वि[ती]यगतौ | सहितान् 'स्वरैकपक्षर्तु-चन्द्र-शीतांशुभिः' क्रमशः || २८ || 28. The II type motion, in the same order, (i.e. for each month of the diads, Pisces–Aries, etc.) for the 18° take 5 × 10 + 7, 6 × 10 + 1, 7 × 10 + 2, 6 × 10

• 6, 6 × 10 + 1, and 5 × 10 + 1 days. This gives 57 days each to move in the signs Pisces–Aries, 61 days for each of Taurus-Gemini, 75 days for each of Cancer-Leo, 66 days for each of Virgo-Libra, 61 days for each of Scorpio-Sagit- tarius, and 51 days for each of Capricorn-Aquarius. From the days given, it can be seen that there is a slight tilt in the apogee towards Leo, and in the perigee towards Aquarius. This small difference from the findings in verse 27 shows that the values are empirical. As for the readings, rasa has been inserted because we want six numbers for the six diads, and one is wanting. Symmetry requires that it must be rasa (= 6) there. Also two mātrās are wanting. sapta is emended into rasa because, 76 for Virgo-Libra, with 72 on one side, and 61 on the other side, will take the apogee to the end of Virgo, 60° off from its place. The average for II type motion is 18° for 81 days which is about the average of the mean rate and 0 (stationary). Thus I type motion is faster than the mean, and the II type slower, as we have sur- mised. TS have expressed inability to interpret this verse also, and not translated it. Yet they have made an emendation which need not be taken seriously, since it has been done without understanding. NP have interpreted the verse as giving 57, 71, 72, 66, 61 and 51 by inserting ṛtu as the fourth. But symmetry shows that the second number 71 is wrong, and it must be 61, to avoid the jump from 57 to 72. At any rate, read with their interpretation of verse 27, we can see that they do not under- stand the use of this series of numbers. They do not even say these are days. 28a. A.B.C.D.विषयस्वरसप्तर्तु (D.र्त्तु) b. A.B.पंचकदशगुणाद्विग्गतौ (B. ०द्विद्यगतौ) C. पञ्चैकदशगुणान् द्विगगतौ; D. पञ्चकम् दशगुणान् द्वि[वी] यगतौ c. A.B. सहिता; C.D. सहिताः;

334 PAÑCASIDDHĀNTIKĀ XVIII.32 झषवृश्चिकाज(चा)पेषु (वक्रं) षट्सप्तकेन (नव) भा(गान्) । ‘(द्वि)कृतेन’ (तैर्नवा)ऽतिवक्री दिनषष्ट्या षोडशानुगतिः ॥ २९ ॥ गोमिथुनतौलिकन्या(ख'त्रिसागरैः स्व)रा'नंशान् । ‘(त्रि)कृतैर्दश त्रिष(ष्ट्या)' सप्तदश यथाक्रमं वक्राशा[त्] ॥ ३० ॥ कर्कटसिंहयो'र्वेदसागरैः 'सप्त ‘(रसार्णवैः ‘शिवा'नंशान्' । षट्(ष)ष्ट्याष्टादश क्रमात्कुजो वक्रपूर्वासु ॥ ३१ ॥ घटमृगयो' (र्नग) दहनैः' षड्भागा' (नवहु)ताश(नं)'(च) । ‘मुनिविषयैः' पञ्चदशांशकांश्च तद्व(त् त्र)येऽप्यारः ॥ ३२ ॥ 29. In the signs, Pisces, Scorpio, Aries and Sagittarius, Mars moves 7° in 42 [L11

XVIII.33 XVIII. VĀSIṢṬHA-PAULIŚA — RISING AND SETTING 335 Types III, IV and V, called, respectively, retrograde (vakra), extra-retrograde (ativakra) and follow-up after retrograde (anuvakra) are given in these verses. The first two are actual retrograde motion, and the third is the slow direct motion following. They are shown hereunder in a tabular form.