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

XIII.41 XIII. SITUATION OF THE EARTH 259 असितात् सिताच्च पक्षाद् असितं पक्षार्धमर्कमीक्षन्ते । राशित्रयादुभयतो नभो यतः शीतकरसंस्थाः ॥ ३८ ॥ 38. Anywhere on the Moon, its denizens, (the Pitṛs, in this case) see the Sun for half the time during each fortnight, (on the whole, not seeing the Sun for a fortnight's time, and seeing it for a fortnight's time), because the visible part of the sky extends only upto 90° from the zenith. [ग्रहाणां स्थानम्] चन्द्रादूर्ध्वं बुधसितरविकुजजीवाऽर्कजास्ततो भानि । प्राग्गतयस्तुल्यजवा ग्रहास्तु सर्वे स्वमण्डलगाः ॥ ३९ ॥ तैलिकचक्रस्य यथा विवरमराणां घनं भवति नाभ्याम् । नेम्यां स्यान्महदेवं स्थितानि राश्यन्तराण्यूर्ध्वम् ॥ ४० ॥ पर्यति शशी शीघ्रं स्वल्पं नक्षत्रमण्डलमधःस्थः । ऊर्ध्वस्थस्तुल्यजवो विचरति तथा न महदर्कसुतः ॥ ४१ ॥ The Planets and their situation 39. Beyond the moon are orbiting higher and higher, Mercury, Venus, the Sun, Mars, Jupiter and Saturn, and beyond that there are fixed stars. All the planets (from Mercury to Saturn) move in their own individual orbits at a con- stant speed. Note. All this is Hindu theory. 40. Just as the spokes of the oil-press wheel are thick, (close to one another), near the navel, and the space between one another increases as the rim is approached, so the linear extension of the rāśi increases as the orbits are situated higher and higher. 35a. Quoted by Pṛthūdaka on BrSS d. B. इमं वयं हरिस्यान्तः 21.8 end, 36 quoted by Sūryadeva on 37a. A.C.D. ॰मर्कात् Abh. Gola. 5, and Makkibhaṭṭa on b. B. विदोषेण शौक्य A. परिवृधिः Sid. Śekhara 1.1. d. B. घटस्येवा 35a. A2. भित्यमथ A. स्पेंदोः; D. ॰स्येद्रोः A.B. भवति भानोः 38a. B. अप्रसिता सिताश्च c. A. स्वछायान्द्वसितं b. B. ॰मीक्तन्ते 36a. B. सलिलमपे. A. ये च शशिनि. A. दिंधयो (A2. दी); c. A. ॰त्रयादूभ; D. ॰क्षयादुभ B. - धत्तयो d. A. मभोयतः; B.C. नभो यत (C. ॰तः); c. A.B. क्षपयन्ति D. न भान्य [था] तु. B. गेतीतकर

260 PAÑCASIDDHĀNTIKĀ XIII.42 41. Situated near-most, the Moon goes round in the shortest time, its orbit being the shortest. But Saturn situated farther-most, in its longest orbit, cannot move so fast, i.e. moves slowest. [मास-दिन-वर्षाधिपाः] मासाऽधिपा यथो (र्ध्वं) चन्द्रात् सौरादधश्च होरेशाः । ऊर्ध्वक्रमेण दिनपाश्च पञ्चमा वर्षपाः (षष्ठा) ॥ ४२ ॥ Lords of the Months, Days and Year 42. The successive Lords of the Month are the successive farther planets, beginning from the Moon. The Lords of the Horās are the successive nearer and nearer planets, beginning from Saturn. The successive fifth in the ascending order of its distance is the successive Lord of the Day. The sixth in its ascending distance order is successively the Lord of the year. Note. The month meant here is the sāvana month of 30 days, and the year, the sāvana year of 360 days. We get the lords of the horās; Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon, Saturn etc. the lords of the day, Sunday, Monday, etc., the lords of the months, Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn, Moon etc., and the lords of the year, Moon, Jupiter, Sun, Mercury, Saturn, Mars, Venus, Moon etc. [इति पञ्चसिद्धान्तिकायाम् वराहमिहिरविरचितायां त्रैलोक्यसंस्थानं नाम त्रयोदशोऽध्यायः ॥]¹ Thus ends Chapter Thirteen entitled ‘Situation of the Earth: Cosmogony’ in the Pañcasiddhāntikā composed by Varāhamihira 39-41. Quoted by Utpala on BS, 2, 41a. B. राशिशीघ्रनक्षत्रं (B2.3.शीघ्रं न) pp. 42-43 b. B.मध्यस्थः 39a. A. चन्द्रादूर्ध्ववुधसितं (A2. बुध) ; c. A.ऊर्ध्वस्तस्तुलजवो B1.2. चन्द्राधुधखित; B3. चन्द्राद्बुधस्वीत d. B.विवरति c. B. प्रागातरा (B3. प्राणां) A.B. om तथा; C.D. om न; A. स्तुल्यजवा; B.-ल्पजवा U.°जवोऽपि संस्थितस्तथा० d. A. ग्रहाः स्तु. B1.3. मण्डलर्गाः 40a. B1.3. तैलक 42a. A.B. मासाधिपो यथोर्ध्वां (B. द्धर्वाः) b. B. विपर. B. ना धुभ्या b. चन्द्रा c. B. मेम्यं स्या c. A.ऊर्ध्व; C.D. ऊर्ध्वं. A.दिनपा च; B. ष्पिपा च c-d. U.नेम्यां महदेवं संस्थितानि d. A.B. पंचमास्याः A.B.C.D. स्पष्टाः d. B.देवस्तिष्ठानि. A.°यूर्ध्व; B.°यूर्द्धम् 1.col. A. त्रैलोक्यसंस्थानं त्रयोदशोध्यायः B.C.D. इति (B.om इति) त्रै (B1.2. सै) लोक्यसंस्थानं नाम त्रयोदशोऽध्यायः

Chapter Fourteen GRAPHICAL METHODS AND ASTRONOMICAL INSTRUMENTS* १४. चतुर्दशोऽध्यायः छेद्यक-यन्त्राणि Introductory In chapter I, vss. 5-7, Varāhamihira enumerated yantra and chedyaka among the topics to be dealt with in the present work. The present chapter deals with these two topics. The compound word chedyakayantrāṇi is equivalent to the compound word yantracchedyāni of I. 7. The word chedyaka means graphics or graphical methods and the word yantra, in the present con- text, means astronomical instruments. Dvivedi interpreted the word chedyakayantrāṇi as follows: “That which cuts or removes doubts is chedyaka; the instruments which serve as chedyaka are chedyakayantrāṇi.” If it were so, there would be no difference between yantra and chedyakayantra. Chedyaka and Yantra, in fact, are two distinct topics of Indian astronomical works called Gola or Spherics. Lalla, Vaṭeśvara and Bhāskara II, for example, have earmarked two separate chapters for their treatment in their works on spherics. Pingree, on the other hand, translates chedyakayantrāṇi as “the Magical Diagrams of the (Graphical) Construc- tions.” But the diagrams or astronomical instruments discussed in the present chapter bear no magical significance. [चरः] साशीतिकाङ्गुलशतं विस्तीर्णवृत्तमविषमं धरित्र्याम् । समराश्यंशकचिह्नं परिधौ सापक्रमं कुर्यात् ॥ १ ॥ याम्योदक्समसूत्रादपक्रमांशावगाहिभिः सूत्रैः । प्रथमवदङ्कक्षिप्तं वृत्तत्रयमालिखेन्मध्यात् ॥ २ ॥ अक्षे क्षिप्तां लेखां [प्र]कुर्याच्च भगणचिह्नपर्यन्ताम् । अक्षोत्तरलेखान्तरमपक्रमांशोत्थमादाय ॥ ३ ॥ द्विगुणं प्रसार्य वृत्ते स्वे [दिक्] तच्चापांशदलाभ्यस्ताः । प्रथमर्शचरविनाड्यो ज्ञेयाः परिशेषयोर्मिश्राः ॥ ४ ॥ GRAPHICAL METHODS Ascensional differences of the zodiacal signs. In III, 10-12, Varāhamihira stated an approximate practical method for finding the ascensional differences of the signs for places living between the Indian Ocean and the Himalayas and prom-

• This chapter was left untranslated by T.S. Kuppanna Sastri. The translation given here was supplied by K.S. Shukla.

262 PAÑCASIDDHĀNTIKĀ XIV.4 ised to give a method for other places in a subsequent chapter devoted to chedyaka. The following rule is in fulfilment of that promise.

1. Construct on the ground a level circle with diameter equal to 180 digits. On its circumference put down, at equal distances, marks showing signs and degrees (etc.). Also put down marks showing the declinations (of the end- points of the signs Aries, Taurus and Gemini). (Through the centre of the circle draw the north-south line and at right angles to it draw three chords through the marks showing the declinations for the ends-points of the signs Aries, Taurus and Gemini).
2. From the centre, draw three circles with diameters equal to the three chords which have been drawn through the declination marks at right angles to the north-south line, and graduate them with marks (of signs and degrees) like the first circle.
3. Then (from the same centre) draw a line towards the latitude (i.e. towards that point of the first circle which marks the latitude of the place) and extend it upto the mark (indicating the latitude of the place) on the circumference of the first circle. (This is the latitude-line). On the chord corresponding to the desired declination (i.e. declination of the end-point of the desired sign), mea- sure the portion lying between the latitude-line and the north line (i.e. the line drawn from the centre to the north point).
4. Lay off the double of that (like a chord) on the corresponding circle. Ten multiplied by one-half of the degrees in the arc subtended by that chord are to be known as the vināḍīs of ascensional difference in the case of the first sign. In the case of the other two signs, they are the vināḍīs of the mixed ascen- sional difference (i.e., the mixed ascensional difference of Aries and Taurus and the mixed ascensional difference of Aries, Taurus and Gemini). Consider Fig. 1. ENWS is the level circle of diameter 180 digits drawn on the ground, E, W, N and S being the east, west, north and south cardinal points, respectively. EW is the east-west line and NS the north-south line. The arcs Ed₁, Ed₂ and Ed₃ are equal to the declinations δ₁, δ₂ and δ₃ of the end-points, of the signs Aries, Taurus and Gemini respectively. Wd' = Ed₁. Wd" = Ed² and Wd"' = Ed₃. d₁d', d₂d" and d₃d"' are the three chords corresponding to the declinations δ₁, δ₂ and δ₃, respectively. These are at right angles to the north-south line. 1c. A. राश्यंकचिह्नं C. राश्यंकू चिह्नं c. B. अक्शेतरे d. A. मापक्रमं; B. मायक्रमः d. A.B. ॰शोच्छमादाय (B2. शोछ) 2b. B. पद for दप B2.3. क्रमाशा c. AB. ॰वदंकाक्षिप्रं; (B. gap for वं) 4a. B. द्विगुण B1.3. प्रसार्य; B2. प्रस्तार्य B. वृशौ d. A2. वृत्तं; B. वृत्तनुत्रय b. A.B. om दिक् B. नवापांशकादलाभ्य (B1. भ) स्ताः 3b. A. om प्र A. कुर्याक्व भगणपर्यन्तां; c. A. विनाड्ये; B. विनाप्रे B. कुर्यार्कलंगणपर्यन्तात् (B1.2. ॰कर्यार्क॰) d. B2. त्तेथाः B3. मिश्रा

XIV.4 XIV. ASTRONOMICAL INSTRUMENTS 263 Fig. XIV.1 The arc NL is equal to the latitude Ø of the place, L being the point marking the latitude of the place. OL is the latitude-line drawn from 0 to meet the point L. ON is the north line. enws is the circle drawn with centre 0 and diameter d₁ d’. (The other two circles drawn with diameters d₂ d” and d₃ d’’’ are not shown in the figure). dl is the portion of the chord d₁ d’ lying bet- ween OL and ON. In the triangle Odl, Od = Rsin δ₁ dl = Od. tan Ø = Rsin δ₁.tan Ø, (1) R denoting the radius of the circle ENWS.

264 PAÑCASIDDHĀNTIKĀ XIV.6 The chord DMD' of the circle enws, with its middle point at M, is equal to 2dl. Let 2c₁ be the angle subtended by DD' at 0 (or, wdhat is the same thing, the arc DWD' subtended by DD'). Then ½ DD' = DM = OD sin c₁ = Rcos δ₁ . sin c₁. ½ DD' = dl = Rsin δ₁ . tan Ø, from (1). ∴ Rcos δ₁ sin c₁ = Rsin ε₁ . tan Ø. ∴ sin c₁ = tan δ₁ . tan Ø (2). This c₁ is the ascensional difference of the end point of the sign Aries, or the ascensional differ- ence for the sign Aries. See supra, IV. 26. See also IV. 34. Since c₁ is equal to the number of degrees lying in half the arc DwD', therefore the degrees in half the arc DwD' give the ascensional difference of the sign Aries. These degrees multiplied by 10 give the corresponding vinâḍîs. Similarly, in the case of Taurus and Gemini. But in these cases if c₂ and c₃ be the ascensional differences for the end-points of Taurus and Gemini respectively, then ascensional difference for Taurus = c₂ - c₁ and ascensional difference for Gemini = c₃ - c₂. [नाडीतः छाया छायातः नाडी च] नाड्यः (षड्घ्न्यो) भागास्तज्ज्या व्यासार्धशोधिता छाया । माध्यन्दिनीसमेता नाड्यर्थे सा तया हीना ॥५॥ छायाहरिजाभ्यन्तरजीवाचापांशषष्ठभागो यः । ता नाड्यः (प्राग् याताः) पश्चाच्छेषास्तथा (प्राप्ताः) ॥६॥ Rsine of the Sun's zenith distance for the given time, (and vice versa). 5. The nâḍîs (elapsed since sunrise in the forenoon or to elapse before sunset in the afternoon) multiplied by 6 are degrees; the (versed) Rsine of that sub- tracted from the radius (R) and then increased by the Rsine of the Sun's zenith distance for midday gives the Rsine of the Sun's zenith distance (for that time). In order to find the nâḍîs (from the given Rsine of the Sun's zenith dis- tance) the (given) Rsine of the Sun's zenith distance should be diminished by the Rsine of the Sun's zenith distance for midday. 6. Whatever is the sixth part of the degrees of the arc corresponding to the (versed) Rsine equal to the difference between the given Rsine of the Sun's zenith distance (as diminished by the Rsine of the Sun's zenith distance for midday) and the radius (chāyāharijābhyantara), gives the nāḍīs elapsed since sunrise in the forenoon or to elapse before sunset in the afternoon. Let n be the nāḍīs elapsed since sunrise in the forenoon or to elapse before sunset in the after- noon, and zₒ the Sun's zenith distance at midday. Then, according to the rule stated in verse 5 above, the Rsine of the Sun's zenith distance for that time (which we shall denote by Rsin z) is given by Rsinz = R - Rvers (6n) + Rsin zₒ, (3) where R stands for the radius.

XIV.6 XIV. ASTRONOMICAL INSTRUMENTS 265 This relation is approximately correct for a place on the equator, and is accurate at an equinox. For any other place it is incorrect, for Rsin z really depends upon Ø, the latitude of the place, and the Sun's declination on the day in question, besides the time of the day elapsed since sunrise in the forenoon or to elapse before sunset in the afternoon. Any such formula must therefore involve these elements. Formula (3) stated above does not occur in any other work on Indian astronomy. But analogous formulae do occur. Of these mention may be made of the following. (i) Vasiṣṭha's formula. Let s₀ denote the gnomonic shadow at midday and s the gnomonic shadow at any other time. Then, in the forenoon: s = 64800 / [(lagna − Sun) in mins.] − 12 + s₀ and in the afternoon s = 64800 / [10800 − (lagna − Sun) in mins.] − 12 + s₀ (see above II.12-13) (ii) Paulīśa's formula s = (6 × day-length) / (day elapsed) − 12 + s₀ (gnomon = 12. See IV. 49. This formula was later restated by Mahāvīra in the form: s = gnomon / (2 day elapsed / day-length) − 12 + s₀ (gnomon = 12. See Gaṇita-sāra-saṅgraha, IX. 18). (iii) Śrīdhara's formula s = (½ gnomon) / d − gnomon, where d = (day elapsed or to elapse) / (day-length) See Triśatikā (ed. Sudhākara Dvivedi), Rule 65. This may be derived from Paulīśa formula by assuming s₀ = 0. Śrīdhara's formula was restated by Nārāyaṇa Paṇḍita in the form: s = { (½ day-length) / (day elapsed or to elpase) − 1 } × gnomon See Gaṇita-kaumudī, Rule 13, p. 207. 5a. A. षड्या भागा; B. ष (B2. श्र) मध्याभागा 6a. B2. छया d. A. B. पश्चाल्लेषा० A. स्तथा C. ॰स्तुया (B2. °सागा) ab. B. ॰भ्यन्तरजा जीवा A. प्राप्नो; B. प्राप; C. [प्राची] b. A. तज्या b. A. षष्ट; B1.2. षष्टं; B3. षष्ठं c. A माध्यंदिनी B समेत्ता c. A. प्राग्यतः; B. प्राग्युता

266 PAÑCASIDDHĀNTIKĀ Formula (3) may also be stated in the form: Rvers (6n) = R − (Rsin z − Rsin z₀), so that n = one-sixth of the degrees in the arc corresponding to the versed Rsine equal to R − (Rsin z − Rsin z₀). Hence the rule stated in vs. 6. [राश्युदयः] तिर्यग्रेखा समदक्षिणोत्तरापक्रमांशरेखायाम् | तच्चापांशा दिग्घ्नाः राश्युदयविनाडिकाः क्रमशः || ७ || Right ascensions of the signs (defined by means of the armillary sphere) 7. The degrees in the arcs of the equator which lies orthogonally (tiryak) bet- ween the north-south declination arcs for the ends of the signs, multiplied by 10, are the vināḍīs of the right ascensions of the signs (Aries, Taurus and Gemini) in their respective order. Fig. XIV.2 In Fig. 2, ♈'R is the equator and ♈'C the ecliptic. AP, BQ and CR are the declinations for the end- points of the signs Aries, Taurus and Gemini, respectively. Then ♈P is the right ascension of Aries, PQ is the right ascension of Taurus, and QR is the right ascension of Gemini. 7a. C. रेखासम० c. A. दिघ्नाः; B. दि—राश्युं b. A. ०रावक्रमांश; B. ०राचक्रमांशा A.B. रेखायाः d. A. विनाडिकां क्र० A. क्रमशं; B. क्रमश i. Chāyāharijābhyantara literally means the distance between the chāyā (i.e. Rsine of the Sun's zenith distance) and the hori- zon. This is equal to the difference between the Rsine of the Sun's zenith distance and the radius, for the former being diminished by the Rsine of the Sun's zenith distance for midday as instructed in verse 5.

XIV.10 XIV. ASTRONOMICAL INSTRUMENTS 267 The degrees of the equator multiplied by 10 are obviously vinâḍîs. The vināḍīs of the right ascensions of the signs could also have been defined with the help of Fig.

1. Let the arc WP (in Fig. 1) be equal to 30° (i.e., the tropical longitude of the end-point of the final sign Aries), OQ, the perpendicular dropped from P on OW intersecting the circle drawn with radious Rcos δ₁, at R, and ORT, the line drawn from O through R. Then the number of degrees in the arc WT, multiplied by 10, are the vināḍīs of the right ascension of the first sign Aries. The vināḍī-s of the right ascensions of the second and third signs may also be defined similarly. यन्त्राणि शङ्कुः [रविक्रान्तिः] माध्यन्दिनीसमेता नाड्यर्थे सा तया हीना ॥ ५ ॥ (शङ्कग्रयातसूत्राद् विषुवान्तरे या क्रान्तिरुक्ता सा) ॥ ८ ॥ ASTRONOMICAL INSTRUMENTS Gnomon The Sun’s declination defined by means of the moving gnomon and the armillary sphere.
2. Whatever be the position of the (moving) gnomon in the path described by it (lit. in the shadow), whether at midday, or when towards the east or elsewhere, the angular distance between the equator (viṣuvat) and the thread that proceeds from the centre and passes through the vertex of the gnomon is called the (Sun’s) declination. [मध्याह्नच्छायातः अक्षांशः] विन्यस्योदक् छायां छायाग्राच्छङ्कुरपरतः (पात्यः) | तत्कर्णसमं मध्यात् प्रसारयेत् सूत्रमापरिधेः ॥९ || तद्विषुवान्तरमक्षोऽतोऽक्षाच्चैवं प्रकल्पयेच्छायाम् | 8a. A. मध्यानां प्रांतथा; B. मध्यान्त्यां धातपा; c. A. यातंत्सूत्रा; B. यातत्सूत्राः C. [खग्रान्तं सूत्रं] C. [मध्ये विन्यस्य] तथा d. A.B. विषुवान्तरयाश्च्व; C. विषुवान्तरभांशका उदिताः b. A.B. छायायामन्य (B. ख) तो; C. छायाऽभावे A. कांद्गदिता:; B. कान्दिः दिताः (B2. ॰कान्द्रिदिनाः) स्वतो गते D. विषुवान्तरं यश्च कांक्षितम् A.B. गते ततः शं B. शकौ; D. शङ्कौ

268 PAÑCASIDDHĀNTIKĀ XIV.11 Local latitude (from the equinoctial midday shadow and vice versa) 9-10 (a-b). Lay off the (equinoctial midday) shadow towards the north, and let a gnomon be caused to fall to the west (or east) from the tip of the shadow. Then stretch a thread from the centre along the hypotenuse (of the gnomon triangle) up to the circumference (of the circle drawn on the ground). The (angular) distance between the point thus reached and the equinoctial point (i.e. the west or east point, whichever is nearer) is the latitude of the place. Similarly, from the latitude one may find the (equinoctial midday) shadow. In Fig. 3, let ENWS be the circle drawn on level ground, E, N, W, and S being the east, north, west and south cardinal points. OA is the equinoctial midday shadow of the gnomon, AB the gnomon which is caused to fall to the west from the tip of A of the equinoctial midday shadow. OBP is the thread stretched along the hypotenuse OB up the point P on the circumference of the circle. Then, the angle PON or the arc PN is equal to the altitude of the equinoctial midday Sun, i.e., the colatitude of the place, and the complementary angle POW or arc PW is equal to the zenith dis- tance of the equinoctial midday Sun, i.e., the latitude of the place. Fig. XIV.3 [रेखांशः] इष्टेऽहनि बुद्ध्वाऽपममक्षादधिकं यदूनं वा ॥ १० ॥ तज्ज्या तिर्यग्रेखा विषुवद्रेखा स्थिता स्पृशति यस्मिन् । तच्चापांशसमानो ज्ञेयोऽर्को गोलभागेन ॥ ११ ॥ Sun’s longitude 10 (c-d)-11. On the desired day find the Sun’s declination, no matter whether it is greater or less than the latitude. Find the Rsine of that and insert it bet- ween the ecliptic and the equator (at right angles to the latter, in its own quadrant). The arc of the ecliptic measured eastwards from the first point of Aries up to the point where the Rsine touches the ecliptic, should be known as (the longitude of) the Sun. 9a. A.विन्यस्यो; B.विष्टभ्यस्यो (B2.विन्यं) B.दृक् for दृक् B. छाया and hapl. om of one छाया b. A.पात्पः; B.पाताः d. A.सूत्रं मा B2.परिधो; B1.3.एरिधो 10a. A.तद्विष्टंतरं; B.तद्विष्टंतरं b. A.कल्पये. B. छाया 10c. A.बुध्यायन्; B.बुद्ध्ययनक्षा०; D.बुध्धायन d. A1.०महाधिकं 11a. A1.तज्या; A2.तज्पा a-b. B. Hapl om. ग्रे-खादिषु c. B1.2.चापाश d. B.०मान ज्ञेयोको B.भागेना

XIV.14 XIV. ASTRONOMICAL INSTRUMENTS 269 यष्टिः [स्फुटतिथिः] छेद्यार्धयष्टिवेधादर्कैनूद्धोरन्तरांशकार्काशः । स्फुटनष्टतिथिज्ज्ञेया तस्मात्कार्या तथा चान्या ॥ १२ ॥ V-Shaped Yaṣṭi True tithi 12. One-twelfth of the degrees intervening between the Sun and Moon observed by means of a (V-shaped) Yaṣṭi of length equal to the semi-diameter of the level circle (with one arm pointed towards the Sun and the other towards the Moon) is to be known as the true tithi which being destroyed is to be known. From that (true tithi) one may derive another. Similar rules are found to occur in Lalla's Gola (VIII 42-43) and Śrīpati's Siddhānta-śekhara (XIX 26). [चन्द्रांशः] दत्त्वांशकेषु तेष्वेव भास्करं छेद्यकेन विज्ञातम् । स भवति तस्मिन् काले निशाकरश्छेद्यकेनैव ॥ १३ ॥ Moons longitude 13. When the Sun's longitude, obtained graphically, (vide above. vv, 10 c-d-11) is added to those degrees (intervening between the Moon and the Sun), the result is the Moon's longitude for that time. This is how the Moon's longitude is obtained graphically. [शङ्कुच्छायातः दिगानयनम्] नाभ्याः शङ्कुच्छायाग्रमङ्कयेत् त्रिस्ततो लिखेन्मत्स्यौ । तन्मत्स्यवदननिःसृतसूत्रद्वयपातमध्येन ॥ १४ ॥ 12a. A. छेद्यर्ध; B. छेद्यद्य; C. केन्द्रार्ध 13a. A. ॰केषु; B1.2. read: द चंशकेषु (B2. अंशकेषु) b B. दकेद्वोरंत्वशशवतकाशिः । B3. दचंशकेषु c. A. तिथिशेया b. B. भास्करछेद्यकेन d. A2. चान्पा c. A.B1. भवति हि तस्मिन् d. A. ॰करात् छिद्यं; B. ॰कर—य (B2.3. ध) केनैव

270 PAÑCASIDDHĀNTIKĀ XIV.18 (सूत्रेण) बिन्दुत्रयसंस्पर्शसमेन मण्डलं यत् स्यात् । तेन तदह्निच्छाया शङ्कोर्गच्छत्यमुञ्चन्ती ॥ १५ ॥ तन्मण्डलमध्यं यच्छङ्कुतश्च दक्षिणोत्तरं भवति । तच्छङ्कुविवरमुदगास्थितं च माध्यन्दिनी छाया ॥ १६ ॥ Cardinal directions (by means of three shadows of a gnomon) 14-16. Mark three times (at intervals) the tip of the shadow of a gnomon set up at the centre of the circle (drawn on level ground). With the help of those (three points) construct two fishes. Taking the point of intersection of the two strings passing through the head and tail of those fishes as centre and a thread so long as to touch the three points as radius, construct a circle (passing through the three points). The tip of the shadow of the gnomon that day moves on that circle without swerving from it. 16. The centre of that circle lies to the north or south of the gnomon. Bet- ween that and the gnomon is the midday shadow lying to the north (or south) of the gnomon. The midday shadow falls towards the north or south according as the Sun’s northern declination δ is less or greater than the latitude ϕ of the place. When the Sun’s declination is south, the shadow always falls towards the north. [खगोलः] हरिजमिति गगनमवनौ प्रसक्तमिव यत् प्रदृश्यतेऽन्तेषु । सममिति पूर्वापरतो ह्येवमेव दक्षिणोत्तरतः ॥ १७ ॥ ध्रुवहरिजविवरमक्षोऽक्षनवतिविवरं च लम्बकोऽभिहितः । (लम्बो) नमति ख (मध्याद्) द्युव्यासोऽस्तोदयाख्यस्य ॥ १८ ॥ The Celestial Sphere 17. (The circle) where the sky appears to meet the earth at their skirts is called the horizon (harija). The (vertical) circle which runs from east to west is called 14a. A.B. नाभ्यासंन छा (B2. च्छा) याग्रं (A. ०छाद्या) ; D. नाभ्यासन्नच्छायाग्र० D. तदाह्नि छाया; B. तदद्धि छाया b. B1. ०ग्रंनकये (B2. न कपे-त्रिसुतो) 16a. A. मध्या; B. मध्यात्; D. मध्याधश्छङ्कु; c. B. मत्स्यव - निसृतसूत्रं द्वय b. A. छंकुः श्व B3. छड़क. B. दक्षिणोत्तरे d. A.B.C.D. तुल्येन c. A. तछंकु; B. तछजषविवर 15a. A. सूर्येण; B. स्तार्येण d. B. ०गास्थितश्च c. B1. तिन for तेन

XIV.20 XIV. ASTRONOMICAL INSTRUMENTS 271 the prime vertical (samamaṇḍala). Similarly, the (vertical) circle which runs north to south is called the meridian (dakṣiṇottara) 18. The (arcual) distance between the north pole and the horizon is called the latitude (of the place). The difference between 90° and the latitude is called the colatitude. The colatitude is the depression (of the north pole) from the zenith. The day-diameter is the diameter of the so called diurnal circle (astodayākhya). [अर्धकपालयंत्रम्] छेद्यवदर्धकपालं सचिन्हमक्षोन्नतं सदिक्कचक्रम् । सुसमावटविन्यस्तं कुर्याच्छङ्कं सनाभ्यङ्कम् ॥ १९ ॥ सूत्रद्वयसम्पातच्छाया [भुजांशका] रवौ देयाः । स भवत्युदयो राशिर्दिनस्य नाड्यश्च [षड्भक्ताः] ॥ २० ॥ Hemispherical Bowl and its use 19. Construct a hemispherical bowl of the radius chosen for our construc- tions with a gnomon fitted at its centre. Graduate its circular rim with marks of cardinal directions and circular divisions (signs and degrees etc.). Place it in a smooth (hemispherical) cavity in the ground with the gnomon inclined to the horizon at an angle equal to the latitude of the place (and pointing to the north pole). 20. Read the degrees traversed since sunrise by the shadow (of the gnomon) which passes through the intersection of the two lines (viz., east-west and north-south), and add them to the Sun's longitude. The sum thus obtained is the longitude of the rising point of the ecliptic. The degrees crossed over by the shadow divided by six are the nāḍīs of the day (elapsed since sunrise).

17a. A.B. हरियमिति B. गमतेमवनौ b. A1. प्रदृश्यंते तेषु; B1. प्रदिश्यन्तेषु (B3. ॰न्तेवु) c.d. A. ह्योवमृगं दक्षि; B. परतो येव मृणं दक्षि; C. ह्योवमतो दक्षि D. परेरेखैवं च दक्षि d. B1.3. ॰त्तरगत; B2. ॰त्तरत; D. ॰त्तर गता 18a. B. हिरेज A-B. विवरमक्ष: b. A. क्षितिरवदिविवरं; B. क्षिभिरवविकिवर च B. ॰भिगृहित: c. A.B.D. लग्नो (B. मणो) नमिति A.B. खमध्य; D. खमध्यं d. A. स्तोदय ||; B. स्तो—||; D. स्तोदय [चक्रस्य] 19a. B. ॰वदर्द्धा D. ॰वदर्ध्यकपालं b. B. स (B3. ख) चिड्डु [B2. द्रु] मक्षौत्र च स दिक्॰ c. B. सु (B3. स) यवाढ (B3. रु) विन्यस्तं (B3. स्त) C. सुसमावनिविन्यस्तं d. A.B. कुर्याद्दिक्कः (B1.2. क B3. ख) A.B. सनाथ्यंकं (B. कुं) 20a. B2. खम्पात; D. सम्पाता b. A.B. मुक्ताशका; C.D. भुक्तांशका c. B. राशिर्दिनाड्यस्य d. A.B.C.D. नाड्यश्च ता याताः (B. यात:)

272 PAÑCASIDDHĀNTIKĀ XIV.23 [चक्रयंत्रम्] समभगणांशकचक्रमर्धाङ्गुलबहलमायतं हस्तम् । विस्तारमध्यभागे छिद्रं तद्गामि तिर्यक् च ॥ २१ ॥ मध्याह्नार्कमयूखं प्रवेश्य सूक्ष्मेण परिधिविवरेण । मध्यावलम्बिसूत्रात् तदान्तरांशा (स्तदा खाक्षः) ॥ २२ ॥ Hoop and it use 21. Construct a circular hoop with diameter equal to one cubit (= 24 digits) and rim half a digit broad. Graduate its (inner) rim evenly with the marks of signs and degrees (etc). At one place in the middle of its broad rim, pierce a fine hole at right angles to the rim. 22. When it is noon, let a ray of the Sun pass through the fine hole in the rim (and fall on the diametrically opposite point of the rim). The degrees that lie between the thread hanging vertically through the centre and that (ray of light) indicate the Sun's meridian zenith distance. The word khākṣa is a technical term used in Indian astronomy in the sense of "meridian zenith distance". Also see supra, IV. 21, where the same term has been used in the same sense. [गोलबन्धः] समवृत्तपृष्ठमानं सूक्ष्मं गोलं प्रसाध्य धातुमयम् । स्थगितार्कसमाङ्कितकालभोगरेखाद्वये परिधौ ॥ २३ ॥ Armillary Sphere 23. Construct a perfectly round light (Armillary) Sphere by means of wooden strips (or metallic spokes). On the surface construct two circles, one representing the equator (kāla-rekhā) and the other the ecliptic (bhoga-rekhā) being marks where the Sun stops (i.e. at the two solstices). 21a. A. ॰भगणांकक॰; B. भगणांफक॰ 22a. B. मध्याद्वा (B2. ह्ना) र्का 23a. A. पृष्ठमानं; B. षष्ट (B2. षष्ठ) मानं B. वक्र for चक्रे A. मयूषं; B. मसूष b. B. शूक्ष्मं (B3. सूक्ष्मं) प्रसाध b. B1.3. मर्द्धगुलवदज्म त हं b. B. विचरेण; D. [विचरणेन] B.C.D. दारुमयँ; A2. धानुमयं A. बहल B. हस्त c-d. A. सूत्रांतलांतरांश॰; B. सूत्रान्तल्पान्तरांश॰ c. A. स्थगितार्कमङ्कित; B. स्थगिचावर्मिकित; c-d. B. छिद्रं --- C. सूत्रात् तलान्तरांश॰; D. सूत्रात्तपान्तरांश॰ D. स्थगितार्कसमङ्कित d. A.B. तिर्यक्ता d. A. तदन्यक्षः; B. तदन्यक्षम्; C.D. तदन्याक्षः

XIV.26 XIV. ASTRONOMICAL INSTRUMENTS 273 याम्योदग्रेखायां झषाजसंध्युभयतो न्यसेद्धेधात् | अपमांशकाङ्कतुल्यांस्तिर्यग्वेधप्रकाशकरान् ॥ २४ ॥ अक्षोत्क्षिप्तस्योदक् तिर्यग्वेधप्रकाशहरिजस्थाः | या नाड्यस्ता याता षडंशकसमन्विता मध्ये ॥ २५ ॥ 24. On either side of the junction of Pisces and Aries, in the north-south (hour) circles at points denoting the degrees of the (Sun's) declination, fasten, by means of observation, light points which may illumine the (Sun's) oblique diurnal circles. 25. Mount the (Armillary) Sphere in such a way that it is elevated towards the north by an amount equal to the local latitude. Then the nāḍīs that lie (on the Sun's diurnal circle) between the light point on the (Sun's) oblique diurnal circle and the horizon denote the nāḍīs that have elapsed (since sunrise in the forenoon), each nāḍī being made up of six degrees. [रवेः उत्तर-दक्षिणगती] यदुपेति कालचक्रे मृगादिकमुद[गयनं] द्युवृद्धिः स्यात् | व्यत्यासे तद्धानिर्व्याख्याताच्छेषमिति गम्यम् ॥ २६ ॥ Sun's northward and southward journeys 26. In the cycle of time, as long as the Sun is in the six signs beginning with Capricorn, it is Uttrāyaṇa (i.e. the period of the Sun's northward journey) and there is increase of day; in the contrary case (i.e., when the Sun is in the six signs beginning with Cancer), there is decrease of day. What remains to be said (here) is to be understood from what has already been stated. 24a-b. B. रेखायाश्चरवाजसन्ध्याभयसो unknowingly three lines: या [नाड्य b. A2. न्यमेत् B1.3. वेधम्; C. वेधान् to गुण] सलिल, 27a. c. A.B.D. अयनांश०. d. A.B. समिन्वताः B. ०काक तुल्या (A2. तुल्या); 26a. A.C.D. यदुदद्यति d. B. तिर्यग्वैधप्र०; b. A. भागादिक०; B. भादिक०; C. प्रागादिक० A2. करात्; B. कारान् A.B.C. मुदयते; D. मुद [गयने] 25a-b. A. अक्षोक्षिप्त०; B. अक्षोक्षिपूषोदक् B. तेषु वृद्धिः सा | (B3. पूथिनोशकाकन्येस्योदक्) c. A. तद्धति व्या; B. तद्धानिव्या० b. B. जास्थाः d. A1. ०मितिम्य०; A2. ०मिति-म्य०; B. om गम्यम् c. A. यान्याड्यः; B3. Scribe omits

274 PAÑCASIDDHĀNTIKĀ XIV.30 [कालमानयंत्राणि] गुणसलिलपांशुभिर्योजितानि बीजानि सर्वयंत्राणाम् । तैः फलके कूर्ममानवयथेष्टरूपाणि कार्याणि ॥ २७ ॥ गुरुरचपलाय दद्याच्छिष्यायैतान्यवाप्य शिष्योऽपि । पुत्रेणाऽप्यज्ञातं बीजं संयोजयेद् यंत्रे ॥२८ ॥ Instruments for measuring time 27. The bījas (i.e. seeds or basic necessities) of all instruments (for measuring time) are furnished by string, water and sand¹. By means of them one may construct instruments resembling a tortoise, a man, or any other desired shape, and mount them on a wooden board. 28. The teacher should impart (this knowledge) only to a devoted (lit. steadfast) pupil, and the pupil too, after having learnt it, should apply the bījas (string, water and sand) to his instruments, keeping the secret unknown to his son even. Varāhamihira has not described these instruments here only to keep their mystery a secret. But these instruments have been described in some of the later works. For example, see the chapter on astronomical instruments in the works of Lalla and Śrīpati. But there too the description is not very explicit. [देशान्तरम्] अभिमतदेशाक्षवशात् कृतवेधेनेन्दुपूर्णिमाकर्म । दृष्टिघटिकोदयांशानधिकल्पत्वे वियुतयुतम् ॥ २९ ॥ तिथिवद्द्विभज्य लब्धं चरकालेनान्वितं क्रियाद्येषु । जूकादिषु च विहीनं विषुवति देशान्तरं स्पष्टम् ॥ ३० ॥ Local longitude in terms of time 29. By means of observation at the desired place in a given latitude, perform the Moon’s pūrṇimāntakarma (i.e. find out how much time after or before local sunset the Moon rises at the local place). The degrees of the ecliptic which rise during that time should be subtracted from or added to the Moon’s longitude according as the Moon rises later or earlier than sunset. 28a-b. A. चपालाय; B. गुरुञ्चपल (B2. ०पदेले)—दद्या० 27a. A.B.गुणं A. दद्याच्छि० b. A.बीजानि B. योजातानि c. A1.०प्यज्ञानं c. B1.3.कूर्म; B2.कूर्य d. A.वीजं d. A2.०ष्टरू । पाणि B.संयोज्ये-नो ||

1. When a heavier substance than sand was required, Mercury was used.

XIV.32 XIV. ASTRONOMICAL INSTRUMENTS 275 30. Dividing as in the case of tithi, find the local time in terms of ghaṭīs for the end of full moon tithi (as reckoned from local sunset). Increase or diminish the ghaṭīs obtained by the time corresponding to the Sun’s ascensional difference, according as the Sun is the six signs beginning with Aries or in the six signs beginning with Libra. The result is the local time for the end of the full moon tithi at the local equatorial place. (The difference of this and the local time for the end of the full moon tithi at Laṅkā) is the true longitude for the local place. The rule is obvious, because: (a) Local time for the end of the full moon tithi at the local place (as measured since sunset at the local place) ± time corresponding to the Sun’s ascensional differ- ence = local time for the end of the full moon tithi at the local place (as measured since sunset at the local equatorial place),

• − sign are to be taken according as the Sun is in the six signs beginning with Aries or in the six signs beginning with Libra. (b) Also, local time for the end of the full moon tithi at the local equatorial place ~ local time for the end of the full moon tithi at Laṅkā = local longitude in terms of time. The local time for the end of the full moon tithi at the local place has to be obtained by the process of successive approximations, but the whole process has not been described in the text. [नाडी (घटी) मानम्] द्युनिशिविनिःसृततोयादिष्टच्छिद्रेण षष्टिभागो यः । सा नाडी (खम्थथो) वा श्वासाशीतिः शतं पुंसः ॥ ३१ ॥ कुम्भार्धाकारं ताम्रं पात्रं कार्यं मूले छिद्रं स्वच्छे तोये कुण्डे न्यस्तं तस्मिन् पूर्णे नाडी स्यात् । मूलाल्पत्वाद्वेधो वा षष्टियोर्ज्या चाह्ना रात्र्या वर्णाः षष्टिर्वक्राः श्लोको यत्तत् षष्ट्या वा सा स्यात् ॥ ३२ ॥ 8. The Nāḍī or Ghaṭī

31. One-sixtieth of the time taken by water to flow out through a desired hole during a nychthemeron is defined as the duration of a nāḍī. Or, it is the time of 180 breaths of a man. 29a-b. B. देशाक्षतवशाक्षतपधेनोडुपतप्तरामिकर्मा; d. A.C.D. वियुतयुक्तं; B. विड्युत (B2. विधुत) युक्तं A. कृतवधेनोडुपूूर्णमाकर्म; D. कृतवेधेनोडुपपूर्णमाकर्म 30a. B. तिथिविभाज्यं D. विकृत्य c. B. देष्टि b. A. चकालादिनान्वितः; B. चकालदनान्वितं c-d. A.B. घटिकोदयांसंदुल्पान्यत्वे; (B. ०स तुल्यान्यत्वे); c. A.B. जूकादिषु पति हीनं C. घटिकान्तरांशानधिकल्पत्वे; d. B. विषुवती D. घटिकोदयांशतुल्यान्यत्वे

276 PAÑCASIDDHĀNTIKĀ XIV.35 32. Construct a copper vessel resembling one-half of a spherical pot and pierce a hòle at its bottom. Put it in pure water in a basin. The time in which the vessel is filled up is the duration of a nāḍī. The hole at the bottom of the vessel should be so small that on account of its small size, the vessel may sink into water exactly sixty times during nychthemeron. Or, it is the time in which one may recite 60 times a verse composed of 60 long syllables (as vs. 32 is). [शशितारायोगः] बुध्वा शशिविक्षेपं दृष्ट्वा ताराशशाङ्कविवरं च । संसाध्यैवं वाच्यः पश्चात्तारासमायोगः ॥ ३३ ॥ Conjunction of the Moon with a Star 33. Having ascertained the Moon's celestial latitude and observed the distance of the Moon from a star and having made the requisite calculations, one should predict the time of conjunction of (the Moon with) the star which is to take place in the future. [नक्षत्रस्थानानि] बहुला षष्ठांशान्ते सार्द्धे हस्तत्रये च भगणोदक् । रोहिण्यष्टदलान्ते दक्षिणतश्चार्द्धषष्ठेषु ॥ ३४ ॥ हस्तेऽष्टमेऽष्टमंशे पुनर्वसोर्दक्षिणोत्तरे तारे । अर्द्धचतुर्थे हस्ते पुष्यस्योदक् चतुर्थेऽशे ॥ ३५ ॥ 31a. B. षुनिशि. A. विनिसृतं; B. विनिसृत 33. Quoted by Utpala on BS 24.5-6. b. A.B. तोय दिष्ट छिद्रेण a. U. कृत्वा c. B1.3. खतचो; B2. खमन्त्रो; D. खमता B. After बुध्वा occurs the passage d. A.B. स्वासाशीतसमं. B. घुंसः ति हेतुः (of XV. 8) to सः || अ (of XV. 23), 32a. B1.3. कुम्भाद्धिकारं; B2. कुम्भार्द्धकारं B. मुले being transferred here, apparently due B1.3. छिद्रं to the misplacement of a folio in the b. A. खछे; B. स्वच्छे—तोये common archetype of the B mss., B. कुन्डे न्यस्ते B1,2 and 3. B. पुर्सों नाडी c. D. मूला पातात् B. राशिविक्षेपं A.B. षष्ठिजोज्याम्न्हा (B. द्वा) राज्या b. B. नाराशशाङ्क d. A.B. षष्टिवक्त्राः c. A. संसाध्यैव वाच्यः; B. संसाध्यैव—वाच्यः; A1. यत्ततष्टृया वा; A2. यत्रतष्टृया U. संसाध्य च वक्तव्यः A. साम्यात्

XIV.38 XIV. ASTRONOMICAL INSTRUMENTS 277 दक्षिणतारा हस्ते सार्पस्यांशे तथोत्तरा तारा । पित्र्यस्य स्वक्षेत्रे षष्ठे चांशे समायॊगः ॥ ३६ ॥ चित्रार्धाष्टमभागे दक्षिणतः संस्थिते त्रिभिर्हस्तैः । विक्षेपकलान्तादङ्गुलानि मध्याच्छशाङ्कस्य ॥ ३७ ॥ Positions of Certain Junction-Stars 34. (The junction-star of) Kṛttikā is at the end of the sixth degree (from the initial point of that nakṣatra) and 3 1/2 cubits to the north of the ecliptic. (The junction-star of) Rohiṇī is at the end of the eighth degree (of that nakṣatra) and 5 1/2 (cubits) to the south. 35. The southern and northern (junction-) stars of Punarvasu are at the end of the eighth degree (of the nakṣatra) and 8 (cubits) south and north (respectively). The (junction-)star of Puṣya is at the end of the fourth degree (of that nakṣatra) and 3 1/2 cubits to the north. 36. The southern (junction-) star of Āśleṣā is at the end of the first degree (of that nakṣatra) and one cubit (to the south); so also is the northern (junction- star). The conjunction (of the Moon) (with the junction-star of) Maghā occurs in its own field at the end of the sixth degree (of that nakṣatra). 37. (The junction-star of) Citrā is at the end of 7 1/2 degrees (of that nakṣatra) and 3 cubits to the south. The digits are counted from the centre of the Moon where the minutes of the latitude end. [शशितारयोरङ्गुलमानम् योगकालश्च] विक्षेपात्सप्तदशापनीय तिथिसङ्गुणात् कृताग्न्यंशः । विद्यादङ्गुलमानं कालं दिनभोगविवरेण ॥ ३८ ॥ Digits between the Moon and a Star in Conjunction And Time of Conjunction 38. Having subtracted 17 from the latitude (of the star with respect to the Moon) multiply by 15 and take a thirty-fourth of that; this is to be known as the number of digits (between the Moon and the star). 34a. B. चन्द्रा; D. बहुलाः 35a. B. ॰मेऽष्टमेशे d. A.B. षष्ठे A. वांशे; B. वाशे A. षष्ठांशात्ते; B. षष्ठाशाति; B2. हातं b. A.B.D. पुनर्वसौ दक्षि 37a. A. चित्रार्थाश्रमभागे b. A. भगणोदक; B. भगणोदकः c. A2. तरे for तारे B. हस्तेषु पुष्य b. B. त्रिहस्तैः c. A. दलात्ते d. A. ॰स्पोदक् A. चतुर्थेशे c. A. कलातादंगु; B. कलातादृगु d. A.B. दक्षिणस्तश्च A. स्वार्ध; B. स्वार्द्धे; 36a. A. सार्पस्यांसेत्तथोत्तरात्तारा d. A. मध्याछशां; B. मध्याछः शकस्य A. षष्टेषु c. A. पित्र्यस्य स्वछेत्रे; B. स्वष्टे च त्रे

278 PAÑCASIDDHĀNTIKĀ XIV.41 The time (of conjunction) is to be known from the distance between the star and the Moon and the daily motion of the Moon. According to Varāhamihira, the measure of the Moon's diameter is 34 in terms of minutes and 15 in terms of digits. Hence the above rule.

Polar longitudes and latitudes of the Junction-stars

39. Multiply the square of 5 (i.e. 25) by half the equinoctial midday shadow; (treating it as the Rsine of an arc) find the corresponding arc (in terms of degrees) and add 15 (degrees) to that. Multiply that by 10 and add 21 times the equinoctial midday shadow. These are vināḍīs. 39-40 Quoted by Utpala on BS 12.21 38a. B. ॰दशापनीय 39a. A. विषुवच्छाया B. द्विगुणा b. A. संगुणाकृता b. A.B. पंचकृतेस्तत्कलास्ततः; D. पञ्चकृतिस्तत्कला ततः c. A. ॰माणं c. A. छायानृसप्तक॰; B. छायात्तृप्तक A1. युत A.B. भोगोविरेण (B. ॰चिरेण); d. A.B. गुणिता