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

XIV.41 XIV. ASTRONOMICAL INSTRUMENTS 279 40-41. Assuming these vināḍīs as the time elapsed since sunrise and taking the Sun at the first point of Cancer, calculate the longitude of the rising point of the ecliptic. When (the longitude of) the Sun happens to be equal to that, then, by virtue of the graphical methods and instruments available to the science of mathematical astronomy, the sage Agastya (i.e. the star Canopus) that looks like the special red tilaka-mark on the forehead of the lady-like southern direction shines forth and delights the minds of men. Such is the divine knowledge based on time. Fig. XIV. 4 Consider Fig. 4. It represents the celestial sphere for a place in latitude Ø. SEN is the horizon and Z the zenith; ♈RET is the equator and P and Q are its north and south poles; ♈GSD is the ecliptic. 40a. A1.B. काद्यायगग्रं (B3. काझाप) D. याम्यतातो वनिता 41a-b. B. ०पलब्धः छेध्यकं (B2. क) b. B1. तदशे; B2. तदंशो; B3. तदृशो B. सहस्तांशो d. A. सुख; B. स्तुख B. विषतिल्को b. B. प्रकाशना पातम् | B. यातं c. A.B. याम्यास्ता (B. त्ता) वनिता; A. मुनिणस्त्यः c. B. खसुखपति

280 PAÑCASIDDHĀNTIKĀ XIV.41 A is Canopus at the time of its heliacal rising and S the Sun at that time. PGAQ is the hour circle of Canopus, and G the point where it intersects the ecliptic. Assuming that the celestial longitude of Canopus is 90° and the celestial latitude 75° 20' S¹, we have ♈G = 90°, AG = 75° 20' and AA' = 75° 20' − 24° = 51° 20'. Therefore Rsin A' E = Rsin (asc. diff. of Canopus A) = Rtan δ tan θ, vide formula (2), where δ is the declination AA' of Canopus and θ the latitude of the place, = (sin δ . sin θ / cos θ) × (R / cos δ) = (sin 51°20' . palabhā / 12) × (120 / cos 51°20') because, according to Varāhamihira, R = 120' = 25 × (palabhā / 2), approx. ∴ A'E = arc (in terms of degrees) corresponding to Rsine equal to 25 × (palabhā / 2), approx. EG" = asc. diff. of G, approx. = 21 × palabhā, vināḍīs, approx. because for unit palabhā, the ascensional difference for G (the first point of Cancer) is 21 vināḍīs. Also, assuming 15 to be the time-degrees for the visibility of Canopus, G'S = G"S' approx. = 15 degrees, approx. ∴ A'S' = A'E + EG" + G"S' degrees = [10 (A'E + 15) + 21 palabhā] vināḍīs, where A'E + 15 is in degrees and palabhā in digits. Degrees multiplied by 10 are vināḍīs. Now GS is the arc of the ecliptic which rises above the horizon (of Laṅkā) in the time given by the arc A'S' of the equator. Hence it is obvious that Canopus A will rise heliacally when the Sun is at S, i.e., when Sun’s longitude = longitude of G + arc GS = longitude of G (i.e., 90°) + arc of the ecliptic which rises (at Laṅkā) in the time given by the arc A'S' of the equator.

1. Actually, the longitude of Canopus is 85°4' and the latitude of Canopus is 75°50'S.

XIV.41 XIV. ASTRONOMICAL INSTRUMENTS 281 Hence the rule. Obviously, the rule is very crude. It was discarded by the later astronomers who replaced it by better rules. [इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां छेद्यकयन्त्राणि नाम चतुर्दशोऽध्यायः ||]¹ Thus ends Chapter Fourteen entitled ‘Graphical Methods and Astronomical Instruments’ in the Pañcasiddhāntikā composed by Varāhamihira

1. A. छेदकयंत्राणि चतुर्दशोऽध्यायः | B.D. इति छेदके यंत्राणि चतुर्दशोऽध्यायः | C. इति छेद्यकयंत्राणि नाम चतुर्दशोऽध्यायः |

Chapter Fifteen

SECRETS OF ASTRONOMY

पञ्चदशोऽध्यायः

ज्यौतिषोपनिषत्

[ग्रहणम्] सूर्येन्दुभगणधात्रीसंस्थानविदोऽधिकृत्य कथयामि । ग्रह(णं) सदैव भानोः स्थानविशेषात् क्वचिद्दृश्यम् ॥ १ ॥ अविदितसंस्थानानां बोधोऽपि हि जायते यथाऽऽधान्याम् । क्षीरं शंखोपहितं द(श)नविनाशक्षमं भवति ॥ २ ॥

Eclipses

1. I declare the following to those who possess pre-knowledge of the relative positions of the Sun, the Moon, the zodiac with the principal stars, and the earth. There is always an eclipse of the Sun visible somewhere in space, according to the position of the place. This is what is meant: If a solar eclipse is defined as being caused by the Moon hiding the Sun, it is obvious that at any moment some place in space will have the Sun hidden behind the Moon.
2. For those who do not know the relative positions of the above mentioned, even knowledge will become similar to the milk in the container with conch in it becoming capable of destroying the teeth. This is the meaning:- Milk and Conch are by themselves beneficial things, especially conch which can strengthen the teeth on account of its calcium content. But milk with conch soaked in it becomes positively harmful to the teeth on account of their incompatibility. So also, even know- ledge can become harmful unless the knowledge as to when, where and how to use it is also known. TS have amended the correct dhānyām into dhānyam, not realising that the word intended in the context is ādhānī, meaning the vessel for holding the milk, and also not understanding what is meant by the verse. For similar reasons, NP have emended the word as dhyānāt, and gives a meaning opposite to what is intended by VM. 1a. A. भगणा B. धात्रीं c. A. ग्रहाणां d. B. विशेषात A. क्वचिदृश्यं; B. क्वचिदृश्यम् (B3. कवि) 2a. B1.3. आअविदित A. सस्थानानां; B. सस्तानानां b. B. वेद्योपि B3. तायते B. ध्यान्याम्; C. यथा ध्यानम्; D. यथा ध्यानात् c. A. दशान; B1.3. दशानन; B2. दशा विना

XV.4 XV. SECRETS OF ASTRONOMY 283 संक्षेपसू(त्रवशतः) शशिना(ऽऽश्रि)यते दिवाकरो येषाम् । तेषां सूर्यग्रहणं स(च) देशः प्रतिदिनं क्वापि ॥ ३ ॥ सकृदेव रविं ग्रस्तं पक्षं पश्यन्ति शशिगताः पितरः । अग्रस्तमपि च पक्षं ग्रहमध्यं पौर्णमास्यां तु ॥ ४ ॥ 3. For these to whom the Sun is hidden by the Moon, according to the straight line from them through the Moon touching the Sun, for them there is a solar eclipse. And every moment (lit. every day) there is such a place somewhere. 4. The pitṛs (Manes) on the Moon see the Sun eclipsed for one whole fortnight, and not eclipsed during the other fortnight. The mid-eclipse is at full moon. [चित्र विवरण: Fig. XV. 1] S Light from the Sun Fig. XV.1. (diagrammatic, not to scale) P: The Pitṛs on the moon M₁: Position of moon at middle of dark fortnight. M₂: New moon position M₃: Position at middle light fortnight M₄: Position at full moon On the side facing the sun, the Moon has sunlight. On the other side it is darkness. E: Earth Fig. XV. 1 3a. A2. संक्षेपा A.B. सूत्रावशशिना (B. सूत्रावं); 4a. B. रविग्रस्तं D. सूत्राविशेषेण b. B3. पितराः b. A.B. त्रियते; C. ध्रियते; D. तीर्यते c. A. अग्रस्य B. ०मवि च c. B. Hapl. om of तेषां d. B. ग्रस्तं मध्या A. मध्य d. A. स व देशः

284 PAÑCASIDDHĀNTIKĀ XV.7 At $M_1$ position P (Pitṛs) just begin to get sunlight. It is sunrise to them. At $M_2$ it is midday, with Sun abovehead. At $M_3$ daylight ends and darkness begins. At $M_4$, it is midnight for the Pitṛs. On the earth, $M_2$ is new moon, $M_3$ mid-light fortnight, $M_4$ is full moon, and $M_1$ mid-dark fortnight. Thus, full moon on earth is midnight for the Pitṛs, the whole night being the eclipsed time. Note that the Pitṛs are supposed to dwell in the region opposite, along the line of sight of the Moon. Note also that the synodic month forms the whole day for the Pitṛs, full moon being mid- night and new moon being mid-day. Note again that the explanation above can hold good only if the Moon shows the same face to the Sun, as it really does. The Hindu astronomers held the same view, without any conception of it, for they held this view in the case of every planet. TS are [L1