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

XVII.9 XVII. SAURA: TRUE PLANETS 303 centre). Find its arc-sine. Add half this arc to the half rectified longitude of apogee if the anomaly of apsis is from 0° to 180° and subtract if 180° to 360°. Thus the apogee is rectified completely. मध्यात् पु(न)र्विशोध्य (त)स्माद्बा[हुर्न] तस्य यच्चापम् । तन्मध्यमे क्षयधनं कर्तव्यं मन्दकेन्द्रवशात् ॥ ८ ॥ Third step 8. Substract this rectified apogee from the mean and thus get the anomaly of apsis. Find its bhuja and multiply it by the epicycle of the apsis and divide by 360°. The bhuja-result, (this is the equation of the centre), is got. Find the arc- sine of this, and subtract the whole of this arc from the mean if the anomaly of apsis is from 0° to 180°, and add it from 180° to 360°. The result is rectified mean. एवं स्फुटमध्याख्यं शीघ्रात् संशोध्य पूर्वविधिनैव । आदिवदा(प्तं) चापं स्फुटमध्या(ख्ये) चयापच(यम्) ॥ ९ ॥ Fourth step 9. Deduct the rectified mean from the śīghra. The anomaly of conjunction is got. Find the bhuja and koṭi of this in the same manner as we did in the first step. Multiply the bhuja by the epicycle of conjunction and divide by 360°. Sine anomaly of conj. is got. Multiply the koṭi, i.e., cos. anomaly of conjunction, by the epicycle of conj. and divide by 360°. The related cosine is got. Add this to 120 if the anomaly is from 270° to 90° and subtract from 120 if from 90° to 270°. Square this, add the square of the bhuja (i.e. equation of conjunction) and find the square root. Divide the equation of conj. × 120 by this square root. The arc sine of this is the result. Add this result to the rectified mean if the anomaly of conj. is from 0° to 180°. Subtract otherwise. The geocentric true planet is got. 6a. A1. तद्भुज; B. तद्भुजयौग             d. B. धनहानिः; C.D. धनहानि b. A. भाजयेन्नभू (A2. भु) जखं;             B 1.2.3. repeat the verse twice and   B. भाजयेन्नभुजखं; C. विभजेद् भुजं फलं खं;       give them two consecutive verse   D. भाजयेत्ततो भुजं ख               numbers.   B. सूर्यघ्न्यः (B2. घ्नः)              8a. B. मध्यासुरो A. पुरो विशोध्य; C.D. पुनर्विशोध्यः c. A.B. तज्झापा (B. या) र्धं             b. A.B.C.D. तस्माद् बाहुं न d. B. शीघ्रं B. वशातात्               9a. a. मध्याख्यां; C. मध्याख्यान् 7a. B. स्फुटयि (B1. पि, B3. यी) त्वैव मन्द          b. D. संशोध्यं b. C. विशोधितस्य; D. विशोध्यं तस्य           c. A.B.D. आदिवदाप्ते B. चाल्पं c. B2. परिणाम B3. कामुकार्धं             d. A.B. मध्याख्योप (B. च) चयापचयः (B. पापचयः)

304 PAÑCASIDDHĀNTIKĀ XVII.9 Note 1. In verse 4, I follow TS's reading, except that I have emended ṣaḍbhyaḥ into ṣaḍbhiḥ, instead TS's ṣaḍbhyaḥ, because my reading allows us subtraction or addition, as is wanted. In verse 5, I follow TS's except in the second foot, where I give the Bṛhatsaṃhitā reading. Either reading gives the same sense. In verse 6, I follow TS's except in the second foot, where I have given bhājyam for vibhajet, as being more likely. But the meaning is the same. In verse 7, the text required no emending, and TS's dhanabāni is unnecessary. In verse 8, like TS I have corrected puro into punaḥ but I have also corrected bāhum into bāhur which is required by grammar. In verse 9, I have corrected madhyākhyām into madhyākhyam, which is the reading of some of the manuscripts. Otherwise I follow TS. Note 2. VM, here, as elsewhere in the PS, uses his tabular sine where R is 120', as given in chap IV. So we must use his tabular values to get the R sines and R cosines. Of course, we may use the modern table, or the Siddhāntic table with R = 3438. But then the R, 1 for the modern tables, and 3438 for the Siddhāntic tables is to be used instead of 120' which is instructed here. (VM uses bhuja to mean sine, and koṭi to mean cosine, instead using the word jyā). Note 3. The method is the same as what is found in the Later Sūrya Siddhānta, with some changes for convenience. But in the matter of the number or order of the steps, the Āryabhaṭīya and the Siddhānta-Śiromaṇi differ. This is because, correctly speaking, the first two steps are useless, and the last two steps alone are necessary. In essence, the third serves to get the true heliocentric position, and the fourth to convert the heliocentric position into geocentric. The earlier steps are in the fond hope of getting correct positions agreeing with observation, while the real trouble is in the inexact parameters followed by the Siddhāntas. Note 4. The second and third steps are merely akin to finding the equation of the centre and applying to the mean. The first and fourth steps are conversion of heliocentric to geocentric posi- tions, neglecting the latitude, which is small and does not affect the result much. The work can be illustrated thus: Epicycle of Conjunction Planet Bhuja Anomaly of Conjunction Q Koti Sun East <— 120' Chapter XVII. Fig. 1

XVII.9 XVII. SAURA: TRUE PLANETS 305 The fig, is for Mercury and Venus, having the Sun as the mean planet. For Mars, Jupiter and Saturn, interchange the planet and the Sun. The geocentric position in both cases is mean planet plus Q. If Q is formed below the horizontal line, it is so much negative and must be subtracted from the mean. Example 1. Find the true, i.e. geocentric, Mars at 1,20,553 days from epoch. Given: Mean Mars, already found with bīja corr. 8ʳ 9° 2′ Śīghra Mars = mean Sun of date = 0ʳ 17° 18′ Aphelion (apogee) of Mars assumed, 3ʳ 20° (for 120° given) Epicycle of apsis = 70°, Apsis of conj. = 234°. First step. Anomaly of conj. = Śīghra − mean = 17° 18′ − 8ʳ 9° 2′ = 128° 16′. This is more than 90° and less than 180°. So, subtracting from 180°, bhujāṃśa is 51° 44′, Koṭiaṃsa = 38° 16′ Bhuja = 94′ 2″. Koṭi = 74′ 17″ Bhuja-result = 94′ 2″ × 234° ÷ 360° = 61′ 14″ Koṭi-result = 74′ 17″ × 234° ÷ 360° = 48′ 17″ As anomaly of conj. is between 90° and 270°, this is subtractive from 120′. 120′ − 48′ 17″ = 71′ 43″. 120′ × bhuja-result ÷ √(71′ 43″² + _bhuja-result_²) = 120′ × 61′ 14″ ────────────────────── √(71′ 43″² + 61′ 14″²) = 77′ 55″. Arc for 77′ 55″ = 40° 30′ ½ arc = 20° 15′ Subtracting from aphelion (since anomaly of conj. is from 0° to 180°), 110° − 20° 15′ = 89° 5′, which is the half-corrected aphelion. Second step Anomaly of apsis = 249° 2′ − 89° 45′ = 150° 17′ This is more than 90° and less than 180°. So, Bhuja degrees = 20° 43′. Bhujā = 42′ 24″. Bhuja-result (i.e. eq. of the centre) = 42′ 24″ × 70°/360° = 8′ 15″ Arc for this = 3° 56′; ½ arc = 1° 58′ This is additive because An. of apsis is between 0° and 180°. Half rectified aphelion + 1° 58′ = 91° 43′ = full rectified aphelion. Third step The corrected anomaly of apsis = 249° 2′ − 91° 43′ = 157° 19′. The bhuja degrees = 22° 41′. Bhuja = 46′ 17″. The Bhuja-result = 46′ 17″ × 70°/360° = 9′ 0″ . Arc of 9′ 0″ = 4° 18′, deductive because an. of apsis is between 0° and 180°. Mean − arc = 249° 2′ − 4° 18′ = 244° 44′ = mean corrected (for eq. centre).

306 PAÑCASIDDHĀNTIKĀ XVII.9 Fourth step An. conj = 17° 18′ − 244° 44′ = 132° 34′. Bhujāṁśa = 180° − 132° 44′ = 47° 26′, Koṭyaṁśa = 42° 34′. Bhuja = 88′ 20″. Koṭi = 81′ 8″. Bhuja-result = 88′ 20″ × 234° ÷ 360° = 57′ 25″. Koṭi-result = 81 8″ × 234° ÷ 360° = 52′ 44″. As an. of conj. is in 270° to 90°, this is deductive from 120′. So, 120′ − 52′ 44″ = 67′ 16″. 120′ × bhuja-result ÷ √(bhuja-result² + 67′ 16″²) = 77′ 32″. Arc of this taken as sine = 46° 16′, additive because an. conj. is from 0° to 180°. So, corrected mean arc = 244° 44′ + 40° 16′ = 285° 0′ = geocentric true Mars. Example 2. Find the geocentric Venus at 1,20,553 days from epoch. Given: The śīghra of Venus = 2ʳ 27° 16′ 20″ = 87° 16′ Mean Venus = Mean Sun = 17° 18′ Aphelion of Venus = 2ʳ 20° = 80°. Epicycle of Conj. of Venus = 260°. Epicycles of the apsis = 14°. First step An. Conj = 87° 16′ − 17° 18′ = 69° 58′. The Bhuja degrees = 69° 58′. Koṭi degrees = 20° 2′. Bhuja = 112′ 42″. Koṭi = 41′ 5″. Bhuja-result = 112′ 42″ × 260° ÷ 360° = 81′ 24″. Koṭi-result = 41′ 5″ × 260° ÷ 360° = 29′ 40″. An. Conj. is between 270° and 90°. So the Koṭi-result is additive to 120′. So, 120′ + 29′ 40″ = 149′ 40″. Bhuja-result × 120′ ÷ √(81′ 24″² + 149′ 40″²) = 57′ 20″. Arc 57′ 20″ = 28° 33′. Half arc = 14° 17′, subtractive to aphelion (as An. conj. is from 0° to 180°) 80° − 14° 17′ = 65° 43′ = Half corrected aphelion. Second step An. of apsis = mean − half cor. aphelion = 17° 18′ − 65° 43′ = 311° 35′. The Bhuja degrees are 48° 25′.. Bhuja = 89′ 44″. Bhuja-result = 89′ 44″ × 14° ÷ 360° = 3′ 29″. Arc. 3′ 29″ = 1° 40′. Half arc = 50′, subtractive, as an. conj. is from 180° to 360°. Corrected aphelion = Half-corrected aphelion − 50′ = 65° 43′ − 50′ = 64° 53′. Third step An. of apsis = mean-corrected aphelion = 17° 18′ − 64° 53′ = 312° 25′ Bhuja degrees = 47° 35′. Bhuja = 88′ 33″. Bhuja-result = 88′ 33″ × 14° ÷ 360° = 3′ 27″. Arc sine 3′ 27″ = 1° 39′, additive as an. of apsis is 180° to 360°. So, mean Venus + arc = 17° 18′ + 1° 39′ = 18° 57′, is the eq. cent. corrected mean.

XVII.10 XVII. SAURA: TRUE PLANETS 307 Fourth step An. conj. = Śīghra-corrected mean = 87° 16′ − 18° 57′ = 68° 19′. Bhuja degrees = 68° 19′. Koṭi degrees = 21° 41′ Bhuja = 111′ 29″. Koṭi = 44′ 19″. Bhuja-result = 111′ 29″ × 260° ÷ 360° = 80° 31′ Koṭi-result = 44′ 19″ × 260° ÷ 360° = 32′ 0″. As an. conj is from 270° to 90°, additive to 120°. So, 32′ 0″ + 120′ = 152′ 0″. Bhuja-result × 120 ÷ √(Bhuja-result² + 152²) = 62′ 12″. Arc sine = 62′ 12″. Half arc = 31° 14′, additive as an. conj is 0° to 180°. Geocentric true Venus = 18° 57′ + 31° 14′ = 50° 11′. In verse 11 below, VM requires us to subtract 67′ or 1° 7′ constant, as bīja-correction, after all work is over. So, geocentric True Venus = 50° 11′ − 1° 7′ = 49° 41′. [बुधशुक्रयोः विशेषक्रिया] सर्वे स्फुटाः स्युरेवं ज्ञस्य तु शीघ्राद्विहाय रविमन्दम् । रविपरिधिनतं बाहुं (बुधेऽर्कवत्) क्षयधने कुर्यात् ॥ १० ॥ शुक्रस्य सप्त(षष्टि)र्लिप्ताः शोध्याः स्फुटीकृ(तस्यै)व । Special work for Mercury and Venus 10. All star-planets are (geocentrically) made true in the above manner. But in the case of Mercury, this additional work is to be done: Subtract its apogee from the śīghra and, using the Sun’s epicycle, find the bhuja-result and apply it to the mean Mercury (which, of course, is the same as the Sun’s), with the addition or subtraction done, as the Sun’s bhuja-result is additive or subtractive. 11 a-b. From Venus, subtract 67′, constant, after all the earlier sphuṭa work instructed has been done. Note 1. The reading kṣayadhane is better as it is, and TS need not have corrected it into kṣayadhānam. The reading budhavat is deficient by two syllables, Lalla’s reading budherkavat supplies these and makes the meaning more clear. So I have adopted it. TS’s emendation budhaphalavat, with phala added for the two mātrās wanting, is not different in meaning from budhavat. Being an arbitrary rule, we cannot decide which gives the original meaning, budherkavat or budhavat. But since Lalla’s reading is not defective, at least as far as the mātrās are concerned, I have adopted it. 10a. B. omस्फूटः B. कोरेवं           11a. A. व्यष्टिः; B. व्यष्टि b. A2. ज्ञस्य पुर्शी; B. ज्ञेक्यं धुशीघ्रा; D. ज्ञेइयेषु शी    b. A.B. शोध्या A. स्फुटिकृतस्यैव; B. स्फुटितस्यैव c. A2. बाहु; B. वादं              (B2.3. ॰त्त तस्यैव) d. A.B. बुधवक्ष; C. बुध्फलवत्; D. बुधे (क) व(ौ) क्षयधनं

308 PAÑCASIDDHĀNTIKĀ XVII.11 Also, it is clear that it is not a substitute for any of the four steps because, if so the separate epicycle for Mercury will be useless. Note 2. It is clear that the rules given here are VM's own, to secure, in his opinion, better agree- ment with observation, because they are not given in the Ārdharātrika-pakṣa etc. and the original four steps are all in line with them, as also the modern Sūrya-Siddhānta and the Siddhānta Śiromaṇi. Note 3. The whole work of finding the true positions, especially of the star-planets is defective in Hindu astronomy, in that the equation of the centre of Hindu astronomy neglects the second, third, etc. terms, which is considerable in the case of the Moon, Mars, Saturn and Mercury, in which last case the second term is as large as 3°. In the case of Mercury and Venus it is applicable to the Sun, instead of their Śīghra which is really their mean. In the equation of conjunction, the Sun's true distance from the earth and true longitude should be used, instead of the mean distance and mean longitude, as is done in Hindu astronomy. On account of these defects, computation does not agree with observation, and all sorts of hotch-potch rules are given in different astronomical works. The disagreement among themselves would itself show that they are beside the mark. When these defects are remedied, the third and fourth steps alone would be necessary, the third step giving the heliocentric true planet; and the fourth step converting the heliocentric position to the geocentric. Note 4. In the case of Venus, there is another kind of defect. Its maximum eq. of cent. being small, it is confused with the Sun's, and the Sun's epicycle and apogee are given to Venus also. While its aphelion position is 290°, according to modern astronomy, its apogee is given as 80°, the same as the Sun's. Table of Heliocentric Star-planets at epoch. (For mutual comparison)

• This needs explanation: Perhaps the reading is śūnyāśvi in Sūrya Siddhānta I.31, which will reduce the degrees by 12. But the commentator Raṅganātha takes it as śūnyartuḥ.

XVII.12 XVII. SAURA: TRUE PLANETS 309 Table of Synodic periods of the Star-planets

310 PAÑCASIDDHĀNTIKĀ XVII.13 Heliacal rising of the planets 12. The heliacal rising and setting of the Moon, Mars, Mercury, Jupiter, Venus and Saturn are when their elongation (from the true Sun) are 12°, 17°, 13°, 11°, 9°, and 15°. Note 1. I generally adopt TS's readings. But Śaśi is extra, and evidently a mistake which has crept into the reading. To make up for this they have removed rudra which is necessary, and this emen- dation has spoiled the correct agreement with other siddhāntas. Note 2. These are time-degrees, i.e. time expressed in degrees (kālabhāga) and are arbitrary in essence, and depend on the keennees of the observer's eyesight, as also the atmospheric conditions. The later Sūrya-Siddhānta gives 10° and 8° for Venus at superior and inferior conjunctions, and 14° and 12° for Mercury, respectively, while the Sūrya-Siddhānta here and some others give the mean of each. (The Mahābhāskarīya gives even 4° or 4 1/2° for Venus at inferior conj. and 8° at superior conj.) [ग्रहविक्षेपाः] मन्दग्रहान्तरज्या स्वाष्टांशयुतार्कजीवशुक्राणाम् । सौम्यान्ययोः पदो(ना) विक्षेपोऽन्यश्च शीघ्रविधौ ॥ १३ ॥ गुरुभूतनयाऽऽस्फुजितां पादोना (ज्ञयमयोस्तु सा)ष्टांशाः । त्रिज्याघ्नी कर्णाप्ता (वि)यो(गाश) विक्षेपः ॥ १४ ॥ Latitudes of planets 13. Add one eighth of itself to the R (120') sine of (mean planet − apogee), in the case of Saturn, Jupiter and Venus. For, the two others, (i.e., Mercury and Mars), subtract one fourth of itself. (This is one part of latitude). There is another part of latitude using the Anomaly of conjunction. 14. From the R sine anomaly of conjunction of Jupiter, Mars and Venus sub- tract one fourth of itself. From that of the rest, (viz., Mercury and Saturn) add an eighth. Add both algebraically and note the direction, north or south. Multiply this by R (i.e. 120') and divide by the hypotenuse got in the last step. The latitude is got, its direction being that of the noted direction. 12a. A. स्फुट दिनकरांतरांतरांशाः; c. B. सौम्यपन्योः; B. स्फुरदिनकरांतरांतराशाः D. सौम्यारयोः A. पदोनां; B 1.2. पदनां b. A. च दर्शनी ज्ञेयाः; B. व दर्शज्ञेयाः d. A. विक्षेपोप्यश्च c. A.B. विशातिरु (B. रू) ना 14a. B. गुरुभूततया d. A.B.C. वसु शशिशिखि A. रुद्रेदियैः; B. रुद्रेद्रिणैः; b. B. पदोना A. ज्ञयममयोमुशांष्टांशाः; B. ज्ञयमयोमुष्ट्यंशां C. नवकेन्द्रियैः C. ज्ञमयोश्च C.D. साष्टांशा 13a. B. ग्रहातरज्या (B2. नरज्या, B3. तरुज्या) c. B. कर्णप्ता A. नियोगयोशम विक्षेपः; b. B. स्वाष्टांशसुर्किजीव०; D. युता कुजेज्यशुक्राणाम् B. नियोगशास विक्षेपः; C. वियोगजाशः स विक्षेपः; D. वियोगयोगः स विक्षेपः

XVII. 14 XVII. SAURA: TRUE PLANETS 311 Note 1. This is a peculiar primitive way of finding the latitude of the star-planets. It is not found in the allied Khaṇḍakhādyaka and the quoted part of the Bhaṭṭotpala-quoted Pauliśa. It is found in Āryabhaṭa’s Ārdharātrika-pakṣa given in the Mahābhāskarīya (VII. 28-33). But there are some differ- ences between the two, and we cannot decide which follows the original Saura here, and which has slightly modified the original. They both mention two kinds of latitudes for each star-planet which are to be added algebraically. But there is a difference in the maximum latitudes, and in the ascending nodes to be subtracted from the mean longitudes or śīghras. VM’s Saura implies the max. latitude 90′, 90′, 135′, 135′, and 135′, for Mars, Mercury, Jupiter, Venus and Saturn, respectively, to be multiplied by sine anomaly of conjunction, and 90′, 135′, 90′, 90′, and 135′ to be multiplied by sine anomaly of conjunction, no separate node being given, which means that the apogee itself is the node for one kind of latitude, and the mean planet itself for the other. But the Ārdharātrika gives only one set of maximum latitudes for both, viz., 90′, 120′, 60′, 120′, 120°. It gives the nodes, 20°, 40°, 70°, 260°, and 150° for the former and 20°, nil, 70±, 260° and 150° for the latter. Govindasvāmi's Bhāṣya on the Mahābhāskarīya, being meagre, does not help us. Note 2. By implication, we had better take the arguments of the eq. cent. used in the third step for the former, and the anomaly of conj. used and hypotenuse obtained in the fourth step for the latter. Example: Find the latitude of Mars at 1,20,553 days from epoch. In the third step of the earlier example, the sine of the argument of eq. conj. is 42′ 24″. As it is Mars, deducting quarter of itself, the latitude is 31′ 48″, north, as this argument is between 0° and 180°. In the fourth step, the sine of the argument of conj. is 88′ 20″. For Mars one fourth is to be subtracted. So, the latitude due to this is 66′ 15″, again north, since the argument is from 0° to 180°. Adding, 31′ 48″ + 66′ 15″ = 98′ 3″, north. The hypotenuse obtained there, in the fourth step, is 88′ 52″. 98′ 3″ × 120 ÷ 88′ 52″ = 132″ north, is the true latitude of Mars for the day. (As it is, this is far from the latitude obtained from using the later Siddhāntas.) Note 3. The treatment of Saura is taken up by VM in XVIII. 57-60, where the latitude found here is used to correct the mean elongation given in verse 12, for the heliacal setting and rising. The exposition of these verses is given in XVIII. [इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां ताराग्रहस्फुटीकरणं नाम सप्तदशोऽध्यायः ||] Thus ends Chapter Seventeen entitled ‘Saura-Siddhānta – True Planets’ in the Pañcasiddhāntikā composed by Varāhamihira Col.: A.B.D. ताराग्रहस्फुटीकरणं षोडशोऽध्यायः; C. इति ताराग्रहस्फुटीकरणं नाम सप्तदशोऽध्यायः

Chapter Eighteen (VĀSIṢṬHA-) PAULIŚA-SIDDHĀNTA — RISING AND SETTING OF PLANETS अष्टादशोऽध्यायः वासिष्ठ-पौलिशसिद्धान्तौ — ग्रहोदयास्ताधिकारः Introductory Chapter XVIII of PS (NP’s ch. XVII) deals primarily with the star-planets according to the Vāsiṣṭha and Pauliśa siddhāntas. The true motion of the planets is traced from one heliacal rising to the next. The method of getting the true anomaly of the equation of the centre is similar to that of the Moon given by the Vāsiṣṭha in ch. II and based on the same theory 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. The text as available too is not very pure and this too has made the interpretation of this chapter difficult for TS. In my paper on ‘Some misinterpretations and omissions of Thibaut and Sudhakara Dvivedi in the PS of VM’ (VIJ., 11 (1973) 107-18) I have indicated the errors occurring in the said publication. NP have improved upon TS’s interpretation in some places, but have committed worse mistakes in other places. TS and NP have also failed to understand how the equation of the centre has been computed and applied to the equation of con- junction of Jupiter and Saturn. Since they had to interpret, in some way or other, the related verses, they have altered the verses, in all sorts of ways, to yield what they thought the meaning might be. Even in the case of Venus, where computation has been made simpler by neglecting the small equation of the centre, they have committed several errors which have been pointed out in the exposition of these verses given below. [शुक्रचारः] हित्वा ‘(जलमुनि) चन्द्रा (न्)’ द्युगणाद् ‘वेदाष्ट-भूत’हृतलब्धाः | शुक्रोदया गुणाप्तैः सार्धाः पञ्चालिनो भोगाः || १ || कन्यांशा [न्] षड्विंशतिमित्वा शुक्रोऽपरेण या [त्यु] दयम् | उदयैकादशभागान् दिनेषु दत्वा तत [श्चा] राः || २ || Motion of Venus

1. Subtracting 174 from the days from epoch, the quotient got by dividing the remainder by 584 are the heliacal risings of Venus. Its motions during the periods is 7ʳ 5° 30' 20" each.
2. Having gone to 26° of Virgo, i.e. at, 5ʳ 26°, Venus rises in the west (for the first time after epoch). Adding an eleventh of the quotient (given in verse 1) to the remaining days, the motions (are to be taken from the Table given in verses 3-5).

XVIII.3 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 313 174 days after epoch, Venus rises heliacally in the west. We have emended munijala as jalamuni because 147 will not agree with the longitude of Venus at rising given as 5ʳ 26°. If Venus is 5ʳ 26°, the sun must be 5ʳ 18°, to satisfy the 8° given for Venus's heliacal risng in verse 58. But since at epoch the sun is in the neighbourhood of 358°, it can be only near 4ʳ 25° after 147 days, i.e., 38° from Venus. After 174 days from epoch, the Sun would be near 169° according to the Vāsiṣṭha- Pauliśa. (Vide my paper ‘The epoch of the Romaka etc.’, Indian Journal of History of Science, 13 (1978) ii. 155-58). This gives the elongation as 7° instead of the required 8°. But this small discrepancy can be put to an accumulated error in computing from the original. TS have not felt the need for this emendation because they have emended kanyāṃśān into kālāṃśān and omitted the necessary kṣepa viz. the constant equal to 5ʳ 26°, for beginning the motion. They have not understood the meaning of the word kālāṃśān which they have brought in. It is the same as the 8° mentioned above, for Venus. NP have interpreted kanyāṃśān correctly. But since they have kept 147 days to be deducted intact, they find a serious discrepancy expressed by them on page 124 of Part II. However, they derive satisfaction from the fact that the September 10th position of the sun would agree with the time of Venus's rising and their longitudes. This at least must have shown them that the rising takes place only more than 20 days later. They have made all sorts of unnecessary emendations, but they have failed to do this necessary one. We can infer from the instruction to add an eleventh of the quotient, that one synodic revolution takes 583 10/11 days. In this period the sun has moved 1 revolution 7ʳ 5° 30′ 20″ and Venus, 2 revolutions 7ʳ 5° 30′ 20″. From this we can infer that the sun takes 365-15-25 days for a sidereal revolution. From the methods given for the other planets also we can see that the sidereal period of the sun used is c. 365-15-30, which is an evidence for the system given here being connected with the Pauliśa also, as in the case of the Moon. TS have unnecessarily emended bhogāḥ into bhāgāḥ and taking sārdhāḥ to mean ‘together with’, instead of ‘with half’, have given a motion of 7ʳ 5° 20′ per synodic period. They are unaware that this would make the sidereal year c. 365-22 days, so wrong. The error of 10′ 20″ in Venus would accumulate by 1° every nine years. NP interpret sārdhāḥ correctly, but take guṇāptaiḥ to mean ‘with 1/3 degree’ and given 7ʳ 5° 50′, which would make the sun's sidereal year c. 365-3-0, so very wrong. By this the error in Venus would accumulate by 1° in every 5 years. Next the days for segments of motion in the synodic cycle are given by verses 3-5. षष्टि[र]येण ῾वेदाग्नि-यम-᾿युतामंशसप्ततिं भुङ्क्ते | (अर्था)ष्ट[कैर्द्वा] (त्रिंशतं) विंश(त्या) [वि]त्रिभिः सपा(दार्थान्)᾽ || ३ ||

1. A.B.C.D.मुनिजल (B. जाल). A.B. चंद्रा 2a. A.D. कन्यांशाः; B. कन्याशाः;. C. कालांशाः b. B. ॰ष्टमूत. A2. दूत corrected to हत; B. हत B. षड्वति c. A. शुक्रेन्द्या. C. गुणांशौ:; D. गुणांशाः b. A1. यात्पदयं; A2. यात्पदयं d. A. सार्द्धा; B. सार्धा c. B. उदैका. C.D. भागं B1.3. पंचलिनो; B2. पञ्चालिनो. C. भागाः; d. A.C. सतस्ताराः;. D. ततश्चवारः D. भोगः

314 PAÑCASIDDHĀNTIKĀ XVIII.5 वक्रमतस्तिथिभिर्द्वौ पञ्चभिरेवं ततोऽपरास्तमितः । दशभिः प्रागुदितः स्यान्नखैश्च जलधीन् मि(तान्) गत्वा ॥ ४ ॥ अनुवक्री परिगत्वा विपरीतमस्तमैत्यै(न्द्रयाम्) । षष्ट्यांशपञ्चसप्ततिमित्वाऽपरतो भृगुर्दृश्यः ॥ ५ ॥ ॥ वसिष्ठसिद्धान्ते शुक्रः ॥ Days for segments of motion in the synodic cycle 3. In three periods of 60 days Venus moves 74°, 73°, 72°, respectively. In 40 days it moves 32° and in 17 days, 5 1/4° 4. From here retrograde motion (begins). In 15 days these are 2°; in 5 days the same, i.e., 2°. Then, setting in the west, it rises in the east after ten days. Venus is in follow-up retrograde for 20 days, moving 4°. 5. Then continuing the (direct) motion round, in the order of days and motions reversed, Venus sets in the east. Then moving 75° in 60 days, it becomes visible in the west. The argument to be used in the above table of motions are the days left over, together with the eleventh of the quotient, as mentioned. It can be seen that in my emendations of some of these I have done very little violence to the text. I have been guided in these by the actual motion that must have been observed, putting it to observational or other error, where the numbers are clear, but deviate from the actual. The ratio of Venus's distance from the sun to the earth is c. 0.72 given by all Hindu and modern astronomy, and this I have used to compute the segments of actual motion for comparison. (See Table on next page) Another guide is that the days and motions from the superior to the inferior conjunction must add up to half of the whole, i.e., 292 days, and 287 3/4°, since the equation of the centre has been dispensed with. 72° motion for the third 60 days is about 4° in excess. For the next 40 days the motion has to be 36°, and I could have filled up the lacuna by (khaissat) instead of (khairdvā) to get this. But the Siddhānta seems to have compensated the earlier 4° excess by the 4° defect here, which of course is an error. From this we can see that the emendation of 'dviṁśat' into 'triṁśat' is necessary lest the motion be reduced to 26°, which is too small. 3a. A.B.C.D. षष्टित्रयेण B. सेप्ताग्नि B. मंशसप्त | ति; B. साप्तति भुक्ते c. A1. अर्धाष्टकेन विंशति (A1. om केन); B. अर्धाष्टकेन विंशातं (B2. विंशति) C. अर्धाष्टकेन सप्त; D. अर्धाष्टकविंशत्या d. A.B. विंशत्यैस्त्रिभिः सपादांशं; C. सप्तत्यंशांस्त्रिभिः सपादांशम्; D. विंशत्यं [शक्र] स्त्रिभिः सपादांशम् 4a. B. वक्रमतं क्षितिभिः d. B. स्रान्तरवैश्च A. मिता गत्वा; B. मिता गता; D. सितो for मितान् 5. B. verse missing 5a. C. दन्तक्वैः for परिगत्वा b. A1. मस्तमत्यैड्य्यां; A2. मस्तमत्यैड्यं; C. रवशरयमानस्तमेत्यैन्द्रयाम्; D. विपरीतं चास्तमेत्यैन्द्रयाम्

XVIII.5 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 315 TABLE I Motion on the synodic circle (computed)

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 ॥