चन्द्रच्छायागणितम्

चन्द्रच्छायागणितम् [[लेखकः :|]] १९५२

चन्द्रच्छायागणितम् OF Edited K. V. SARVA ८८ - by ऽि H O S H I A R P U R OF SANSKRIT AND INDOLOGICA STUDIES प्रधान-सम्पादकः–के. वी. शम Printed by DEVA DATTA Shastri at the V. V. R. I. Press. and published by 1. S. cf I. S. Panjab Universit:y. Hoshiarpur Panjab University चन्द्रच्छायागणितम् Indological Series-6 O _ F Critically edited with Introduction, Translation and Appendices K. V. SARMA Acting Director, V.V.B.I.S. & I.S. Panjab University, H०ऽhlarpur 0F SANSKRIT AND IND0L00GICAL STUDIDS HOSHIARPUR 1 9 7 6 सर्वेऽधिकाराः सुरक्षिताः प्रथमं संस्करणम्, २०३३ (वि.) प्रकाशकृत् विश्वेश्वरानन्द-विश्वबन्धु -संस्कृत भारती-शोध-संस्थानम् पञ्जाब-विश्वविद्यालयः साधुआश्रमः, होशियारपुरम् (पं, भारतम्)

ष्ट Publishers PANJAB UNIVERSIT P.(). Sadhu Ashram, H०shiarpur (Pb., India) INTRODUCTION C 0 N T E N T S C 7dr८८clayag८१ita1- Nilakantha's authorship of the work- Manuscript material- Assessment of the manu scripts- Nilakaptha, the author- Personal details Birth-place and Family- Satikara, the brother, and Netranārāyapa, the patron- Ravi and Damodara, the teachers- Works of Nilakantha-Chronology of his works- Versatility of Nilakantha- Acknowledgements I. कालात् छायागणितम् (Shadow from Time) वस्तुनिर्देशः (Introduction) शशिनः स्फुटविक्षेपः (Moon's true latitude) चन्द्रस्य द्वितीयस्फुटः (Moon's second imequality correction) ... गतिसंस्कारः (Correction to daily motion) काललग्नं प्राग्लग्नं च (Kalalag70 and Orient ecliptic point) दृक्क्षेपः (Zenith distance of the Nomagesimal) चन्द्रच्छाया (Moon's Shadow) चन्द्रच्छायानयने प्रकारान्तरम् (Alternative method for Moon's Shadow) I. छायायाः कालानयनम् (Time from Shadow) विपरीतच्छायानयने साधनानि (Bases for Wip010claya) चतुज्यसाधनम् (The four main R sines) दृक्कर्म (Reduction to the Polar longitude) उदयादि-स्थूललग्नं तत्स्फुटीकरणं च . (Approximate Orient ecliptic point etc. and their correction) छायातः कालानयनम् (Time from shadow) प्रन्थसमाप्तिः (Conclusion) APPENDICES 1. Index of Half-verses Page७ 10 12 14 16 18 18 22 24 26-28 29-30 3] [ N T R 0 0 U C T [ (0 N The Carldranchayagapitar edited here critically for the first time, from original manuscripts, with Translation and auto-commentary, is an interesting astronomical manual by Nilakantha Somayaji, a reputed mediaeval astronomer of Kerala (c. 143-1545 A.D.) Chayagarita or 'Computations concerning the Shadow' constitutes the Hindu astronomer's method for ascertaining the exact time of occurrence of any event on the basis of the shadow cast by the Sun or the Moon. The expediency of this device during ancient and mediaeval times would be apparent when one considers the limitations of such chronometrical devices as the hour-glass, water-disc, etc. The gाmom101, being one of the simplest of astronomical instruments that could be set up at any time and at any place, and readings made independent of circumscribing factors like 2ero-point etc., is the most handy instrument to record such abrupt 0ccurrences as the birth of a child, death of a person and setting in of the initial menstruum . A knowledge of the exact m10ument of these happenings, as computed by astronomical Imethods from the measure of the shadow, facilitates the prediction of the future based on them in accordance with the dictums of astrology Obviously, the shadow cast by the Sun is measured during day and that by the Moon during might The utility Clayagapita in the daily life of the mediaeval Hindu whose prescribed way of life required the performance of numerous religious rites, is not far to seek. (Ordinary astronomical computation enabled him to calculate auspicious times (null॥rta) for sacred rites and ms of time-units like 1alkds and wina lkds, say, after sumrise or sunset. But, in the absence of accurate chrom10umeters. it was not easy to ascertain when that auspicious moment, as calculated had arrived. Chayagapita came to his rescue in such a situation. For, it was possible to calculate , In advance, the length that the Sun's or the Moon's shadow would attain at the appointed time. One could, then, wii 111 set up the grommon and watch for the shadow to reach the stipulated length and perform the rite at the right moment. This process of computing the shadow for any specific Time is called Kal72-chaya, Direct (process of) Shadow (computation) (process of) Shadow (computation)', according to which Time is calculated from the length of the Shadow, was, equally, if not more, important in everyday life, in that it enabled the accurate ascertainment of the time of abrupt, nonplammed occurrences like m10ments of birth and death, unexpected arrivals, untoward happenings and the like Popularity of Chayagapita in Kerala advantage of this 17atural phenomenon and devised highly intricate calculations to get the exact times corresponding to the measures of the shadow, taking into consideration also the factors that affected the shadow, such as the latitude and longitude of the place, time of the year, precession of the equinoxes, etc. Practically every one of the numerous astronomical manuals (Kara10-3ra11tlas) produced in Kerala contain sections devoted to Claygarita, b0th of the Sun and of different texts have been identified and documented, C7aya is one of the five subjects dealt with, the others being yoatipata, Gral474, Sigondati and Mau4lya. Several texts which are devoted solely to computations based on shadow have also come to be composed. What is more interesting, c011putatio1s concerning the Moon's shadow being more important, several manuals exclusively devoted to Carldrarchy4 have also come to be comip0sed, though the calculations 1. See K.V. Sarm1a, history of the Kerala Scl001 0f Hiradlt 4ऽtror107], (Hoshiarpur, 1972), SIn. on 'Bibliography of Kerala Jyotisa', pp. 134-37 2. See , for instance, Chayaga10ाrdstaka, Clayag4 tit, I-III, Chayast८ka of Acyuta Pisarati (op.cit., p. 118), Sryac८rdra०chayagapita (p. 176), Suryachayadiguita, I-II (p. 17 iMTRODUCTtoM tx here are more involved and intricate. These include five different works, all entitled Candracchayaganita, and CandracchHyUnayanopayah. 1 The Candracchayaganita of Nilakant,ha Somayaji, belongs to this genre of texts and sets out the processes for the computation both of Kramacchaya, 'Shadow from Time' (verses 1-17), and ViparitacchSya, Time from Shadow' (verses 18-32), of the Moon. The commentary by Nllakantha himself gives a lucid exposition of the textual verses. The Translation in English, with footnotes wherever needed, as presented on pages facing the text, is intended to set out the sense of the textual verses in terms of modern mathematics, reserving a full-fledged demonstration thereof for being presented elsewhere on another occasion. Nllakantha' s authorship of the work The Candracchayaganita carries no benedictory verse but commences abruptly with the subject-matter. No indication, therefore, is available in the beginning of the work either about the author or the title of the work. The authorship is, however, ascertained from the concluding verse of the work which reads as follows (p. 25) :

• forc»i5t*attt 3iTfr?cTf^T3$^?tferrfr: i

'Victorious shines, illuminating everything, the 'Moon of Shadow- computation', with its brilliant rays of rules, having been extracted by Nllakantha (Somayaji) from the ocean of astronomical lore.' The commentary, however, carries an introductory verse which specifies the title of the work and also indicates that the author himself composed the commentary (p. 2, below) : 3T?*forf?T§?m: fgwncft uwts srnrwr 'Having paid obeisance to that Brahman, whence (take place) the origin, subsistence and extinction of the worlds, (the work) Candracchayaganita is being commented upon by its author himself, viz,, Gargya (Kerala-Nilakan$ha Somayaji).' 1. Op. cit., pp. 115-16. Candr*. 2 4 CANDRAdCHAYAdAjsiITAM Manuscript material The present edition of Candracchayaganita and its auto-commentary is based on the only three manuscripts of the text and one manuscript of the commentary thus far identified, all of them being preserved in the Oriental Research Institute and Manuscripts Library of the Kerala University. Trivandrum. A. Ms. No. 5862-B, a well-preserved palmleaf manuscript in Malayalam script procured from Shri Tuppan Nampntiri of the Ponnorkkoftu Mana, an old family of traditional scholars of Central Kerala. It contains both the text and the commentary. The manu- script is inked and the writing is very readable. However, it is not dated nor is any mention made of the scribe. The codex 1 contains the following three works, all on astronomy : LaghukalBrpanam, an anonymous karana work, Candracchayaganita, with auto-commentary, being the work edited here, and Uparagakriyakrama, an anonymous work on the computation of eclipses according to procedures enunciated by Nllakantha (Somayajl), as stated in its opening verse : B. Uncatalogued folios 173-77 'of Ms. No. 5877, being a codex of astronomical works including Nllakantha SomayajI's Golasara and Siddhantadarpana. The manuscript is well preserved, but it is not dated nor any mention of the scribe made. The text alone of Candracchayaganita is contained herein. C. Ms. No. 475-1, occurring in a codex of astronomical works procured from Nareri Mana (Kutalhlr Mana), a reputed scholarly family of Nampntiri brahmans in Central Kerala. The manuscript is old, brittle and frayed at the edges. It is inked and the writing is very legible, though not attractive. It contains only the textual verses which, how- ever, are accurately-inscribed. The works contained in the codex are : A. Aryabhatiyam, B. Mahabhaskariyam, C. Laghubhaskariyam, D. Siddh- Qntadarpanam, E. Tantrasahgraha, F. Lilavati, G. Pancabodha, H. Laghu- mdnasa, I. Candracchayaganita, J. Goladipika, and K. Grahana$taka. 1. A transcript of this codex is available also in the Govt. Or. Mss. Library, Madras, No. 5185 a,b,c. INTRODUCTION ii The codex can be dated by the chronogram sevyo dugdhabdhi- talpah, viz., the 1699847th day of the Kali era, given as the date of transcription of one of the works contained herein. This date works out to A.D. 1551, being almost contemporaneous with the author Nilakantha Somayaji who passed away in about 1545 A.D. Assessment of the manuscripts The tendency found in technical manuscripts to go corrupt and the ease with which errors find their way into them have not seriously affected the manuscripts used in the present edition. In fact, the text and the commentary as preserved in our manuscripts are mostly accurate and free from errors. The variants are few and just one or two alternate readings are indicated. Possibly, these emendations go back to the author himself. The lone manuscript of the commentary does not carry the concluding verse and its exposition. Since the manuscript does not indicate any break here, it is likely that this verse, which does not contain any technical matter, has not been commented upon by the author. Appendices Ms. No. 5862-B contains, in continuation of the commentary, an expanded version of verses 28 and 29, in three verses with an additional correction. After this is found directions, in the Malayalam language, for a geometrical demonstration of verse 19, paramapa- kramakotya etc. Since these two fragments are related to Candracchaya- ganita edited here, they have been added as Appendices I and II, towards the end of this publication. Nilakantha, the author Our author is generally referred to with the title Somayaji, Somasut, Somamtvan or Comatiri, the last being the Malayalam derivative of the Sanskrit word. A detailed colophon occurring at the end of his Bhasya on the Ganitapada of the Aryabhatiya contains a good deal of information about him : ffa «ft-5mnnfcT triiitzu 3n**?n**T

• tt<£;t %TOrontMg^T ^-^m^5rm-q^?:^«iiTrg^^T«[cTr^^ *nffta
• !i CANDRACCHAYAGASITAM

f?T5l ? cT5l7T^T% JT^THT^ etc. 1 Personal details The above-quoted passage informs that Nllakantha belonged to the Gargya gotra } 2 was a follower of the Ahalsyana-sutra of the Rgveda and was a Bhatta. He was the son of Jatavedas and had a younger brother named Saiikara. He had an uncle Jatavedas by name and a close friend Subrahmanya. He was a performer of the Soma sacrifice. He had composed several works on astronomy, in which subject he had made deep and extensive investigations, a fact which is well borne out by his available works. Some more personal details about Nllakantha seem to be forthcoming from a Malayalam work Laghuramayana? This work des- cribes itself as a work of Rama, son of Nllakantha of the Gargya-gotra and resident of Kundagrama. Cf. the colophon at its end : fffT f^n^ im*g*firafci tft-ifenTORiT** arnrfflri-ini^JT TOff^mfa-tr^- nfwv-irraT-m-qRhmT tfwi^l^taSft ^ fcfcr Mhnromr » This Nllakantha is identified by the editor of the work with our author. 4 If this identification is correct, Nllakantha's wife was named Arya, and he had two sons Rama and Daksinamurti, the latter of whom was well versed in the Dharmasastras and learned in the three languages Sanskrit, Tamil and Malayalam. The great Malayalam poet Tuncattu' 1Qln , L 1D Ed " T, ' iya ^rum Sanskrit Series (TSS), No. 101, (Trivandrum, 1930), p. 180. ' 2. Generally the term Gargya is affixed to his name in references It may also be noted that in the commentary edited here, he refers to himself merely as 'Gargya', (cf Candracchayaganitam vyakhyayate'sya Gargyena p. 2), obviously, on account of his full name not being amenable for insertion in the verse. v , 3 ' E< L P ' R< Men ° n ' Tunchattu Grant/avail, No. 3, Tunchattu Karyalayam, Chittoor, 2nd edn., 1931. 4. Vide P.R. Menon in his article 'Tunchattu Ezhuttacchan' in the Malayalam monthly Tunchattu Ezhuttacchan, 3 (1952-53) 127-35, INTRODUCTION xiil Ezhuttacchan is said to have been a student of Nllakaitfha. Nllakantha is also said to have composed, at the request of a friend, a panegyric in Malayalam on the Goddess Parvati, the presiding deity of the temple of Urakam in Cochin, in order to ward off the predicted premature death of that friend's daughter. 1 The authenticity of the above work and the source of the information are, however, not quite certain, and corroborative evidences have to be found before accepting the above statements. Birth-place and Family Nllakantha hailed from Tr-k-kantiy-ur (Sanskritised into Sri- Kunda-pura or Sri-Kunda-grama), near Tirur, S. Rly., Ponnani taluk, South Malabar, a famous seat of learning in Kerala during the middle ages. The name of his Warn, as the house of a NampTitiri brahman is called, was Kelallur (sometimes spelt also as Kerallur and Kelannur), Sanskritised into Kerala-sad-grama corresponding to the Malayalam word Kerala-nall-nr.* Nilakant.ha's house is identified as the present Etamana Illam, situated a little to the south of the local temple. 3 It is stated that Nllakant.ha's family became extinct and the family property was inherited by the nearest dayadi relations, viz., the Etamana family. 4 Nilakantha's favourite deity was Lord Siva installed at the famous temple at Trpparannod (Sanskrit Svetaranya) near his village ; cf, ^T^rnq-qT^^^W^frfefl!"T, in the colophon to the ABh.Bhasya quoted above (p. xi). Sankara, the brother, and Netranarayana, the patron Nilakantha refers to his younger brother Sankara in several places in the ABh.Bhasya. Sankara too seems to have been well versed 1. Ibid.. This stotra is published in a collection of stotras in Malayalam script, Stavaratnamala, Pt. I. 2. It may be noted that in the expression Gar gy a- Kerala prefixed to the author's name, the word Kerala refers to the name of his house and not to his state, as is sometimes taken. 3. Cf, Vatakkumkur Rajaraja Varma, History of Skt, Lit. in Kerala, vol. I, p. 384. 4. I am thankful for this information to the late Sri Rama Varma Maru Thampuran, Chalakkudi (Cochin), CANDRACCHAYAGATSITAM in astronomy and followed his elder brother's studies. Thus, after describing some methods on the Rule of three {Trairatika) in his ABh Bhasya, Ganita., 26, Nilakantha says how his brother who was teachmg at the house of his patron explained to the latter some of those theories ; cf fcrffc, ^ ^ ^ ^ WWW ^m$* w tfwtetu , ( TSS 101 , p. 156) Nilakantha observes at the close of the Bhusya on the Qolapada that he was entrusting the Bhasya to Sankara for its proper propagation Thus just before the final colophon, Nilakantha says : (TSS 185, p. 164) That N 5 lakantha was intimately connected to and was patronised byKausitakiAdhya NetranarJyana, known locally a s AzhvSnceri Tamp- rttkal. the religious head of the Nampntiri brahmans of Kerala, is known from several references in his writings. It is als0 dear that the patron hadgreat esteem for Nllakantha's erudition in astronomy, in which JSLt 100 ;r interest : d and med to aiscuss ^ ^ **> Nilakantha. Thus, m » he discussion on the calculation of the mot.on of planets (ABh., Kala., 22-25), Nilakantha says : aTOTOfer ^tfegrirfcT TOPron*<ir SR* II WW cT^g^r WW?s-i tffrfasft STfa; I {TSS 110, p. 63) INTRODUCTION Again, in the long discussion on the calculation of the apparent position of celestial bodies ( ABh., Kala., 17-21 ), speaking on a method to derive the sakrt-karna, our author says : 3T?*refa (TSS 110, p. 47) This would indicate the intimacy that existed between Niiakantha and his patron and the common interest that bound them together. On the compilation of the ABh.Bhasya, Niiakantha observes in one place :

• j?*mre %<*ife5T ^i*ri eratfiT: srfaqra ^Wtaf^TT snOT'T ^R^THs^

3*nsJ?H *flTfT?TO, 3?cI^T^m fo*s*Tf* I (TSS 101, p. 113). Again, at another context, he remarks : r?t% ^ET^fW 3Rf*TT%*T finffcreirfa foKta ^WrTf^T ^TfTcW I cTftRiT g?r: snWTSTOTOEW I (T^S 101, p. 156). It is clear from the above that the credit of enthusing Niiakantha in his investigations, and, in fact, to have prompted him to write his ABh.Bhasya., goes to Netranarayana, 1 the members of whose family are known all through the annals of Kerala history to have been good scholars and, at the same time, patrons of scholarship. Ravi and Damodara, the Teachers Niiakantha informs us in his ABh.Bhasya that he studied Vedanta under Ravi, cf. Ravita atta-Vedanta-'sastrena, (TSS 101, p. 180). That Ravi was well versed also in jyotissastra and that Niiakantha imbibed some of his knowledge in astronomy from Ravi is clear from the introductory verse to Nllakantha's Siddhantadarpana, where Ravi, his teacher, has been mentioned by double entendre : 1. Even with regard to Nllakantha's Tantwsahgraha, its introductory verse, has a veiled reference to his patron (Netra)-Narayana at whose instance that work too seems to have been written. CANDiACCHlYAGANlf AM A work on astrology, Acaradipikn, which is a detailed commentary, in verse, on Muhurta$taka is ascribed to this Ravi. 1 The teacher of Nilakantha who actually initiated him into the science of astronomy and instructed him on the various principles underlying mathematical calculations was Damodara, son of the Kemla-Drgganita author Paramesvara, 2 of the Bhargavagotra and resident of the village of Alattur (Sanskritised into Asvatthagrama) which was situated quite near Nilakantha s own village. In his ABh.BM$ya, as also in his other works, Nilakantha reverentially refers to his teacher and his studies under him. He speaks of how even as a boy he stayed with his guru at the latter's residence prosecuting his studies; cf wn % StfcTT STTe* ft* etc. (ABh.Bhasya, TSS 110, p. 48). He also refers, often, to his teacher's views and quotes him ; cf st^rrt^i etc. (N's Grahana-grantha* in the Trivandrum Palace Collection," Ms. No 975 ; transcript with me, p. 61); ere^tanwqsreiqf: (ABh.Bhasya, TSS 101, p. 47); Pro * fT?T at* q^pfa: qs^fircqsnfafa: <3Tcp^;j m ;i 5T$*fa' etc." (ibid., p. 48); rRfa - " ^^^WfacW (Siddhuntadarpana-vyukhya, on verse 27, Ms. Trivandrum Palace Collection, No. 975 ; transcript with me, p. 30). Similar quotations and other references, which Nilakantha and later authors make, proclaim Damodara not only to be a prominent astronomer of the times but also as the author of erudite works on the subject, manuscripts of which, are yet to come to light. Nilakantha followed in the footsteps of Paramesvara, founder of the Drgganita system of astronomy in Kerala and one of. the fore- most astronomers of the land. For him Paramesvara was not only the revered father of his Guru but was also his Parama-Acarya, by which term he generally refers to him in his works; cf, m «n«fe- 1. Ulloor, Kerala Sahitya Caritram, vol. II, p. 114. For a Ms of this work, see Kerala Uni. Mss. Lib., No. 3336-B. 2. Cf. the detailed colophon quoted above, pp. xi-xii. 3. On this work, see below, p. xx. परमेश्वराचार्येण अस्मत्परमगुरुणा ‘चलांशास्स्वं' (4546) इति कल्यब्दे परीक्ष्य पञ्च दशांशपूर्तिनिर्णीता । etc. (Siddlutadorp010-)yakly, verse 18) ; प्रस्मत परमगुरुणापि सिद्धान्तदीपिकायाम. एतत् प्रतिपादितम् । (48). Bhasy4 , Golapada, verse 3) Works of Nilalkaptha Nilakantha has written several works which reflect his deep study of and ripe scholarship in astronomy, embodying the results of 1his investigations in the subject and interpreting the science lucidly A mention of his works may, advantageously, be made here : 1. Goldऽ '0' ('Quintessence of Spherical astronomy)' in pariched0s, embodying the basic astronomical elements procedures 2. Siddld॥tularp010, a short work in thirty-two (I114stubls , enunciating the astronomical constants with reference to the Kalp0 and specifying his views on the main astronomical concepts and topics on which there is difference of opinion among authorities. title it is sometimes cited, 4. A commentary on present edition. a short the cast by the Moon and vice-versa, being the work edited here. work 5. Tantrasaigral८* divided 432 verses. This is a major work in thirty-two which verses on the three and Carld)'accld y'gd१it, included into eight chapters of Nilakantha and is in the comprising an erudite 1. Cr. cdin. with Translation, by K.V. Sarma, W.W.R. Institute, Hoshiarpur, 1970 2. Critically Ed. with Translation by K.V. Sarma, Adyar, Library, Madras, 1955. Two short anonymous tracts, entitled Siddhanta darparasiddlhor-p0ary()yadaya} and] antatla724asild-paryaya-bha dinani, added as Appendices to this edition, vouch for the popularity of this text 3. Ed. with the commentary Lagluyivrti by Sarlkara , in Tऽऽ 188 (1958) ±viil CANDRACCHAYAGAISllrAM treatise on astronomy. As a work belonging to the Tantra class, it takes the commencement of the Yuga as the starting point for calculations. In the several chapters, it deals with : I. Astronomical constants and general principles and conceptions. II. Geocentric positions of the planets. III. The Sun's shadow. IV. Eclipses of the Moon and the Sun. V. Specialities in the Sun's eclipse. VI. Vyatipata. VII. The Phases of the Moon, etc. VIII. Srhgonnati of the Moon. 6. AryabhatTya-Bhasya, 1 an elaborate commentary on the cryptic and sutraAike text of Aryabha^a which comprehends in 121 aryas the fields of Mathematics and Astronomy. A perusal of the commentary will amply prove that it is no false claim that Nilakantha make* when he designates his wo,k as a 'mahabhasya' and explains the method of exposition adopted by him: ^m^fTO^TrgrTfeT?fT-5iire!Ti% Mvti ^^rs-qmsfasft^sm^m ww?m (TSS 101, p. 180). In another context, recalling how he came to write the commentary, Nilakantha remarks : surtmr mm g^cft: srferarefai (TSS 101, p. 156). The lucid manner in which the difficult conceptions about the celestial globe and astronomical calculations are made clear, the wealth of quotations, and the results of personal investigations and comparative studies presented herein amply justify the appellation 'Mahabhasya' which Nilakantha has given to his work. Nllakaatha has commented only on the Ganita, Kalakriya and Gala padas of the Aryabhatiya, leaving out the GitikSpada, which he says is covered by the commentary on the other three sections; cf. ?Nr TOTfcri (TSS 101, p. 1). 7. Siddhantadarpana-vyakhyZ, a commentary on his own Siddhantadarpana, of which an incomplete Ms. is available in the Palace Library Collection, Trivandrum, No. 975. The commentary is elaborate 1. Ed. in TSS 101, 110, 185 (1930, 1931, 1957). INTRODUCTION xix and resembles, in diction and treatment, his Aryabhatiya-bhasya. It is in this work that Nilakantha gives the actual date of his birth (see below, p. xxiii). 8. Grahananirnaya, a work on the computation of lunar and solar eclipses. Manuscripts of this work are yet to be discovered, but later authors and Nilakantha himself in his ABhBhn$ya quote from this work; cf. m I tosht*^ § fe^q^feTOSTFretfTO^sft: qw^cttsfq "refer: ^pNttr^ u (TSS 185, p. 102) These verses are quoted also by Saiikara in his commentary on Nilakanfha's Tantrasahgraha (on ch. IV, verse 27, TSS 188, p. 107) with the introductory remark : HfsfcTJT^'^r ^wf^Ft I 9. Sundararajaprasnottara. Sundararaja, son of Ananta- narayana, was an astronomer of the Tamil country contemporaneous with Nilakantha and author of a detailed commentary on Vnkyakarana or Vakyapancudhyayi which is a manual on the basis of which almanacs in the Tamil districts are computed. 1 Sundararaja had the greatest respect for Nilakantha whom he addressed for clarification of certain points in astronomy. Nllakantha's detailed answers to these questions formed a regular work, Sundararajaprasnottara. Manuscripts of this work are yet to come to light, but both authors refer to this work. Sundararaja in his commentary on the last verse of ch. V of the Vakya- karana says 2 : 1. Cr. ed. by T.S. Kuppanna Sastri and K.V. Sarma, K.SR, Inst., Madras, 1962. 2. Ibid., p. 119. I* C ANDR ACCHA Y AGAtflTA M Nilakanfha too refers to this work in his ABhMasya, Gola., 48 : cf. ^!&jsmft^rx& V$Vm***%BP*m (TSS 185, p. 149). ' 10. A Grahana-grantha, copied in continuation of Nllakantha's Siddhantadarpana-vyakhya- in the Trivandrum Palace manuscript No. 975. The work as available in this manuscript begins ( m a nd without any more introduction, continues : sr^swfq S^TT* ^ nifirtbX- wrnm: i irkw v&tm [>] STrJTsrcMr*: i It goes on to describe the necessity of correcting old astronomical constants by observation, deals in detail with the Sakabda-samskara, quotes the verses of N's Parama- guru Paramesvara from his Siddhtotadipikn (Mahabhaskanya-bhasya- vyakhya), 1 on the latter's observation of some, eclipses, and after some more discussions ends abruptly. There is no doubt that this work is from Nllakantha's pen. References herein to his own works, teacher, etc. fully confirm this point. One of his own works is referred to herein thus : aw txar mj s^ptr^ <Tc?rra?m*T ^h*^*— •sRn^urf etc. (p. 60 of my transcript). The verses quoted are from Nllakantha's Candracchuyaganita, vv. 8-10. He refers to his grand-teacher Parames- vara and his ABhMasya too, herein : cf, mm %£R^^Tf<nfa |Tfaf*rfew: fwf<tfw^ irw^n* H (pp. 57-58 of my transcript). The ABh. Bhasya aiso is referred to elsewhere in this work (cf. pp. 62, 63 of the transcript). For a characteristic reference to N's teacher, see : HttTJT, cfaf^Rfq an*TCtarcrfa? || (p. 61 of the transcript). 11. Grahapariksakrama (?). The well-known Kerala astrologer Puliynr Purushottaman NampOtiri has edited 2 an old, incomplete 3 1. Ed. Madras Govt. Or. Ser., 130 (1957). 2. Pub. by the Astrological Research Institute, Bombay-25, 1950. 3. The colophonic words at the end of the edition indicating its completion form only the editor's addition. Malayalam summary of a Sanskrit work under the title Gralhapariks6 krama. The textual verses were not available to the editor and he presumed that the author was Drganita-Parameswara. * These verses are however, found in Nilakantha's Bhasya on the Golapadld of the 2,४५0 blafrya, under verse 48 (7SS 185, pp. 132-49). It is a long tract of about 200 verses, summing up the principles and methods followed in Hindu astronomy a:nd forms a veritable handbook on the subject. [ t ends : इति संक्षेपतः प्रोक्ता परीक्षा ज्योतिषामिह । कालमानचतष्कस्य श्रुतस्य विवृतिस्त्वियम् । It is not however very definite whether this is an independent work with the title G’(alhap4ark:Sakra1714 and is quoted in extens0 in the Bhasya or is but a part of the Blsya. Nilaka11tha should have written more works than those mentioned above since there are uotations attributed to him in later works, for instance, in Sarikara's commentary Ldgl। s८igrala, which could not be traced to his known works. Again the Trivandrum Palace Ms. No. 975 containing Nilakaptha's Siddldn1a darp070ाyakly and the work on eclipses described above, has, in continuation, some incomplete tracts with no definite titles, which again, to all appearances, seem to be Nilaka11tha's writings According to some, Nlakantha has composed a work entitled Gralami1040.* It is likely, however, that this is only the Graha१10 Imir१10)va, noticed above. UIloor attributes t0 Nilakantha a work called Ga१ity!ikti. Thus, speaking about a Bhaऽyuktiblds, he says that “it is 10t the work of Kelallur Comatiri, author of Gapitayukti.”* The ascription is wrong and the fact is that while our author belonged to the Gargya-gotra, this latter work is by an anonymous author belonging to 1. Vide the cditor's Introduction, p. i ; See also Shri Namputiri's review and opinion of Gotif(१praka5ikd by K. V. A. Rama Poduval, Canammore, 1950, p. xiv 2. Vatakkumkur, Hist. (9/ Skr. Lit. ir Kerala, vol. 1. p. 389 , UIloor, Kerala Salitya Critra71, vol. II, p. 117. 3. UI100r, ibid., p. 122. 11 the Bhāradvaja-gotra as follows : as is clear from its first verse , which runs विदित्वार्यभटप्रोक्तगोलतत्वेन केनचित् । भारद्वाजेन तन्यन्ते काश्चित् गणितयुक्तयः । Chr01010gy of Nilakantha's Works It has been 10ticed that Naka11tha's 719l. Bhasya is later than his 7a71trasaigrala and G.lassara which are quoted in the former. But 10thing could be said about the chronology of his other works. The present writer's investigations have, however , shed some light 011 The first five works (enumerated above , wiट., Gol25ara, Siddld110 (darp070, C21d'acclajagapita, the commentary thereon, and 7a17d scrigrald do not refer to any other work, but are , in their turn, quoted in other works of Nilakantha. Of these, the 7a711tra56ailgroll40 is the most comprehensive of the five and gives the date of its composition as 1500 A.D., i.e., it was written when the author was fify-eight, and on these considerations it may be presumed that the other four works were written before this date . The Grallar07i"10)'0 and the Sundard॥rajopra5101(170 , of which manuscripts have yet to be discovered and which are quoted in the 481.Blasy0, lhave also to be ascribed to this period. This Bhasya, 1his mature work, Nilakatha wrote when he was very old, as he himself remarks : मयाद्य प्रवयसा *** यथाकथंचिदेव याख्यानमारब्धम् (7SS 101, p. 156) (/., com. (on verse 25 : एतत सर्व मया प्रार्यभटीयव्याख्याने प्रपञ्चितमिति विरम्यते । p. 22 of my transcript) is still later. And so also his discursive treatise on eclipses which to refers to the 431.Blass)'0 more than once ; /, तत्र कालक्रियापादे सूचितं मया विवृतम् (p. 63 of my transcript); एतत् सर्च गणितपादे विस्तरेणोपपादित: (ibil., 63) Indisputable evidences are available regarding the date of our author. Sarikara, Nilakaptha's pupil, in his commentary on his teacher's 1. Ms. : Kerala U।। the M:durals, Mal. D. 339, pp. 83-89, 10w, transferred ss. Library, Trivauldrum to Introduction ixiii Tantrasahgraha, points out that the first and last verses of the work contain chronograms specifying the dates of commencement and completion of the work. Thus, after giving the natural meaning of the first verse of the work : 'I fectjft f?rf|a ^nra I Sankara says : 3?TrJTqfa %H S^foP 3TTf%fft ^nT SPWnl* SWUTWfor-qfJW- These two Kali dates, 16,80,548, and 16,80,553, work out to Kali Year 4601, Mina 26, and 4602 , Mesa 1, both dates occurring in April 1500. The SiddhantaJar pana and NTlakan^ha's own commentary thereon give the year and actual date of his birth. Cf. : Text : ^f?m?&W*Ti% FOTcTOnsft ^ | (Sid. Dar. 18) Com. : feswwcTfacTt g?r r>t% tT^^r wmli stowst: myfsssns<j£?3*T^; i *t sr ?fkrssHT qs^anft^-siflfacT. (4500) i ?t?st TTriTST: q^^T?^?^: (45) i cTcf: ?W5TcTt??T^: 'fawf^'fa (4545) I sai wfswJftsrrrafa ar^-q^gw '^ri^Tt^' (4536) q*te*r ^N^cR 1 <RT ST^T^R^ '^T^^T TO (16,60,181) ffcT I (Trivandrum Palace Ms. 975 ; p. 14 of my transcript). Here Nlia- kant.ha himself says that he was born on the Kali day 16,60,181, which works out to A.D. 1443 Dec. (Kali 4545 Vrscika). That NTlakantha lived to a ripe old age, even to become a centenarian, is attested by a contemporary reference made of him in a Malayalam work on astrology, viz., the Pras/iasara by Madhava, a Namputiri brahman of the Incakkazhva house in Kerala, who wrote his work in A.D. 1542-43. Here, Madhava says that he could count upon reputed authorities like 'Kelanaliur' to recommend his work. Cf. ; itiV candracchAyAganitam al-ayat-adaravil adiyil Attimattam lokottaran punar-itinn-iha 'Kelanallur' j Sbhasar allarivatujlavar adarippan porum prasiddhi perikoljavar untanekam // The date of composition of this work, Prasnasara, is given as Kali 4644 (A D. 1542-43) by the following verse in the work itself : ezhunuttorupattetfSvatu Kollam ataya nal / varunna visuvad bhdvatattvam (4464) kalyabdam ayatu // Rightly does Nilakantha remark in his A.Bhnsya : W*TZ sr^r iwwacfwfc vtmvmTW* (TSS 101, p. 156). Moreover, we know of at least two more works composed by him subsequent to his writing the ABh.Bhasya, viz., the commentary on the Siddhmtadarpana and the work on eclipses, both of which quote the ABh.BhSsya. Versatility of Nilakantha For a mere Jyautisika and one who had specialised only on its astronomical aspect, Nilakantha seems to be very well read. Every other page of his writings substantiate his knowledge of the several branches of Indian philosophy and culture. Sundararaja, the Tamil astronomer, calls him sad-darsanhparahgata, 'one who had mastered the six systems of philosophy'.* Nilakantha himself informs us that he studied Vedanta under Ravi : cf, Ravita atta-Vedantasastrena. He can refer to a Mlmamsa authority to establish a mathematical point 2 and with equal felicity apply a grammatical dictum to the same purpose 3 Pingala's Chandas-smra" and the lexicons are quoted as the occasion 1. Cf. his commentary on the Vakyakarana, 5.19 (edn., p, 119). , 2, < Cf' AB h.Bhasya,TSS IM, pp. 54, 158, where P3rthasarathi Mi^ sVyaptimrnaya and Advaitavivarana, and AJita (com. on Sloka- Z Tn I ItS , commentar y V 'Jaya come in for quotation. On Gola- pada, 50, the B r hattiku of Kumarila Bhat{a is cited. 3. Cf. quotations from the Vakyapadiya, ABh.Bha$ya, TSS 110, 4. See ABh.Bhasya, TSS 101, p. 4. INTRODUCTION xxv demanded. The scriptures and the DharmaSastra texts also come in for citation. 1 And, so also the Puranas 2 like the Bhagavata z and the Visnu* As for Jyotisa works, Nllakantha exhibits a surprising familiarity with a large number of them, from the Vedahga- Jyotisa down to the treatises of his own times. He cites all types of jyotisa texts, Ganita, SafnhitH and Hors, but as became his subject of specialisation, his quotations are mainly from texts dealing with astronomy proper. Some of the more important texts of all-India prevalence that Nilakan$ha quotes are : Vedahga-Jyotisa, Aryabhatiya, Varahamihira's Pancasiddhs- ntika, Brhajjataka and Brhatsamhitd, the Saryasiddhanta, Srlpati's Siddhantasekhara and Munjala's Laghumanasa. Of texts common only in Kerala may be mentioned the Parahitaganita or Grahacdranibandhana of Haridatta, Bhasya by Bhaskara I on the Aryabhatiya, and Bhaskara's Laghu and Maha-Bhaskariyas, Govindasvamin's Bhasya on the latter and Paramesvara's super-commentary thereon ; other works of Paramesvara like his Aryabhatlya-vyakhya also come in for citation as also passages from his own teacher Damodara. Another Kerala author whom Nllakantha quotes profusely h Madhava, often styled 'Golavid', 5 who was a reputed astronomer of the times. Manuscripts of several works quoted by Nllakantha are yet to be unearthed. Indeed, a detailed study of the numerous authorities quoted by Nllakantha is bound to throw much light on the history of Hindu astronomy. Acknowledgements As indicated earlier, all the manuscripts used for the present 1. See Com. on Siddhantadarpana, verse 1 ; the Grahana work, pp, 48, 49 ; and ABh. Bhasya, GolapSda, verse 48, where the Taittinya- upanisad, Kalanirnaya of Sayana, Manusmrti, etc. are quoted. 2. See Com. on Siddhantadarpana, verse 1. 3. Cf., ABh.Bhasya, TSS 110, pp, 16, 26. 4. Cf„ ibid., p. 8. 5. On this Madhava, (c. 1340-1425), who was a teacher of Drgganita-Paramesvara, see the present writer's Introduction to his edition of Madhava's Venvaroha (Trippunithura, Cochin, 1957), and Sphuacandraptih (Hoshiarpur, 1973). Candra. 4 1 edition of Ca71d0cclayag62if( with commentary are preserved in the magniffcent manuscripts collection of the Oriental Research Institute and Mss. Library of the Kerala University. I am grateful to the authorities of the Library for making available to me the said manuscripts for use in this edition. A sad interest, however, relates to this publication, in that Shri N. Rama Sastri, Senior Pandit in the Library, wh0 copied the main manuscript for me, is no more. To Prof. T.S. Kuppamma Sastry, lately of the Presidency College, Madras, I am highly obliged for the help rendered towards the translation of the work. The W. V. R. 1. Press deserves to be complimented for the efficient and prompt printing of the work in spite of its heavy and pressing work schedule. W.W.B.I.S. & I.S., Panjab University, May 4, 1976 K.V. SARMA नीलकण्ठ-सोमयाजि-विरचितं चन्द्रच्छायागणितम् [ ssgfwr: ] 1 H#T KWaf 5RT II ? II ^^l^W^^ s^qftrftr^Jmsrf ^^it^r^M^

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Mss. used: A. No. 5862-B ; B. No. 5867 (ff. 173-77); C. No. 475 (f. 65), all from the Kerala Univ. Or. Mss. Inst, and Mss. Lib., Trivandrum. 2. B. ; C 5^ frayed out. 3. A. adds here wnw — 1. Ms. used : A. No, 5862-B. being the same as text no. A. COMPUTATIONS CONCERNING MOON'S SHADOW By NILAKANTHA SOMAYAJl I. COMPUTATION OF SHADOW FROM TIME (Introduction) 1. Let the Moon's Shadow (prabha) 1 be computed after determining the true sun and moon, the moon's apogee and ascending node, the ayanamsa 2 at the time and the latitude of the place (for which the computation is made). (The Moon's True Latitude) 2. The R 3 sine of the 'true moon-wmws-ascending node" multiplied by 270 and divided by the 'Last Hypotenuse', is the true latitude of the moon. It is north or south, (respectively, for the first two quadrants and the last two quadrants). The square root of lits square subtracted from R 2 J, is the 'perpendicular' (koti). 1. Prabha is used in the sense of 'shadow' on account of the equivalence between their measure. 2. Ayanamsa is the distance of the Vernal equinox-point from the first point of the Indian Zodiac, /. e., the First point of the sign Mesa. 3. R^fa^i of 3438 units. s sftqf *Tf^qr eft iTmfofa: 'sr^req: (^o) fc^q- ^wrniT- ^rr-cq-^'TFT fg^q^s^rnr ^ t ^ q-^ssf % *fsw fq#r:

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5Tl%q: I sq-cftq-r^- g fosq#cr ^qrss^q: I ^ feftq^^tffjT I ^ crfefhf ^'fffTczr fawnrofa fq-^ftsq* ^fqfa i at fasfr^Ftfe: i ^ f^TrTc^ TSTWtq" II ^ II [ WW ftcftqf^: ] fk??w fkwiw ^tT^t sffar f^tor at ^fcr- nwrcr^qr f rqrss<% ^:3rtfCTs% ftt^r 1 q - ^ ^TfeW ^ f^uww $qrfa r i <rw ^ ^T^^q - ^ q^m^n^ ^ sr^sroT j ^^f«r ^5n^w fa«rw . f^q-qr ^rs^r?^^ f c^rss^f ^rt^ ^5"f§£ f^^rm: t ^ ww fscrt^^f 1 चन्द्रच्छायागणितम् (The Sec011d-Ime१uatity-C0rrection) 3. Thc R cosine of the 'true sur1-7॥i॥५७-m100n's apoge' X R sine, or R cosine, respectively, of the eduation of the 'centre-corrected m00n-11irus-true Sun' x the eguation of the 'centre-corrected-1901's daily motion'–:- (20xR x the moon's daily mearn motion) is, respectively, the moon's bht]ja-phala and koti-plhald 4. Add to or subtract from R, the koti-plala according as the 'true sum-71i115-moon's apoge' and the 'cguation of the centre-corrected moon-72inux-true sum' are both in the same a y201a (i.e., from 90° to 270* or from 270* to 90"), or in different ()'a70s (i.e., (one is from 90 to 270° and the other is from 270° to 90°). Suare this and add to the square of the bluja-plhala. Find the square root. This is the "Last Hypotenuse 5. R x bht!jd-plala -- the 'Last Hypotenuse ' is to be added (or subtracted from the euation of the 'centre-corrected-me001', as the 'true sum-7irus-apoge' is from Cancer or Capricorm, (i.e., between 90° and 270', or between 270° and 90") if it is the light fortmight (i.e., if 'no n piru७-sum' is between 0" and 180°) . Reverse the addition1 and sub traction if it is the dark fortunight (.e ., if 'Im10 -11irtus-sun' is between 180° and 360-). 4. This is called Evection . First, Vatesvar॥ (904 A.D.), then Muljala (932 A. D.), and after him Sripati (999 A. D.) gave this correction in Hindu astronomy. <sit?tteh StTTTsr- ^faster Mw^— [vfffcaq] ^% i ^srtstr [ flH^ SW*T * ] g^T^l^w, nl^T^ %Tfa$jt: s II vs II ^tht^t: i ^s^ft^^T ?iT^n": i giRfqr ^rcT^r i srm- TO^T?rnctf^raT T%*rfn"T^r Traffic srfasrfcr i fafa %^ w^MTcsrwcj-

• T?R — 4. C. ^ft frayed out. 5. C. f^TffRft: frayed out.

6. C. f[|?f frayed. 7. C. aw to iTTwit (verse lOd), which occur in the bottom line of the obverse of the folio and top line of the reverse, are frayed out, leaving but traces of the letters, which however, do not indicate any variant reading. (Correction to daily motion) 6. The daily motion of the equation of the centre-corrected- raoon X R -7- the Last Hypotenuse is the daily motion to the used here, i.e., in the computation of the moon's shadow. 5 Add the avandmsa to this corrected moon, and to the true sun (and use them). (Kalalagna and the Orient ecliptic point) 7. Add to or subtract from the true sun of the time taken, the Reduction to the equator (Red. eq.) at one minute of arc per prd$a as the sun is in the even or odd quadrants. Add or subtract the ascensional difference (cam) of the (latitude of the) place in pranas (taken as minutes of arc) as the sun is (in the six signs) from Libra, (T ula), or Aries (Me fa), respectively. Also add or subtract the prSrias (taken as minutes of arc), of the time gone after sunrise, or to go before sunrise, respectively. 8. This is called Ka~la{lagna). Place this separately (in two places) and find the ascensional difference and the reduction to the equator pertaining to this (in minutes of arc). Add or subtract them in the reverse order (to that mentioned in verse 7), to the Kalalagna. Repeat this operation several times. The exact Orient ecliptic point (Oep) (Lagna or Prag-lagna) is obtained. 5. Though this rate of motion is given as per day, it pertains to the moment taken for computing the shadow. दयावधिकाः प्राणाश्चरप्राणकलान्तरसंस्कृतरवेविशोध्या : । एवं त्रिभि: संस्कृत स रविः कालात्मक लग्नं, घटिकामण्डलगतं लग्नमिति यावत् । ततोऽपक्रमलग्न मानेयम् । कथम् ? तदर्थमेतत् काललग्नं पृथग्विन्यस्य प्राग्वद् भुजायाश्चरं प्राणकलान्तरं च गृहीत्वा तस्मिन्नेव काललग्ने व्यस्तं कुर्यात् । प्राणकलान्तरम् ओजे धनम् ऋणं युग्मे, चरदलम् मेषादौ धनं तुलादावृणं च कुर्यात् । एवमानीतं क्षेत्रलग्नं स्थूलम् । अतोऽस्याऽविशेषः कार्यः । तदुक्तम्--ततश्चाप्तं मुहुर्लग्नं च तद् दृढमिति । तदुभयसंस्कृतात् काललग्नाच्चरं प्राणकलान्तरं च गृहीत्वा केवल एव काललग्ने प्राणकलान्तरम् ओजे धनं युग्मे ऋणं च चरमपि गोलयोर्धनमृणं च कुर्याद् इति चकारेण द्योत्यते । एवं मुहुः पुनः पुनः कुर्याद् यावदविशेषः । दृढमविशिष्टं तत् प्रसिद्धं लग्नं च स्यात् ।। ७-८ ।। मूलम्– नीलकण्ठ-सोमयाजि-विरचितम् अन्त्यद्युज्याहताक्षाद् यत् त्रिज्याप्तं यश्च लम्बकः । काललग्नोत्थकोटिघ्नः करार्थाब्ध्युरगैर्हतः ।। ९ ।। दृक्क्षेपस्तद्रिदैक्यं च काले ककिंमृगादिगे' । विश्लेषे लम्बजाधिक्ये सौम्यो, याम्योऽन्यदा सदा" ।। १० ।। अथ श्लोकद्वयेन दृक्क्षेपानयनमुच्यते – [ अन्त्यद्युज्याहताक्षाद् ] इति । अन्त्यस्वाहोरात्रज्या परमापक्रमकोटिः । सा ‘कविकुलम्’ (३१४१) इति। तया अन्त्यद्युज्यया स्वदेशाक्षज्यां हत्वा त्रिज्या हृत्वाऽऽप्तं यत् तत् स्वदेशजमवधार्यम् । ततो लग्नार्थमानीतस्य काललग्नस्य कोटिज्यां गृहीत्वा तया स्वदेशलम्बज्यां हत्वा करार्थाब्ध्युरगैः प्रणवादि'(८४५२) संख्यैराप्तं च यत् काललग्ने मृगादिगे तद्द्वयं संयोज्यम् । कक्र्यादिगे तु वियोज्यम् । स दृक्क्षेपः । स च दक्षिण: । यदा पुनरक्षात् सिद्धं लम्बकानीताद् विशोध्यते तदैव सौम्यः वियोगेऽप्यक्षाल्लब्धस्याऽधिक्ये याम्य एव, अक्षस्य नित्यदक्षिणत्वात् ।। ९-१० ।। 8. 9. | दृक्क्षेपः ] 10. A. B. give an alt. reading के for गे B. (C. विश्लेष for विश्लेषे B. C. यदा ; B. gives also सदा as alt. reading (The Zenith distance of the Nonagesimal, Zdn) 9. (a) Multiply the R sine of the latitude of the place by the final semi-day-diameter (i.e., R cos 24°) and divide by R. (b) Multiply the R cosine of latitude of the place by the R cosine of the Kojalagna, and divide by 8452. 10. The difference of the two, or sum, is the Zenith distance of the nonagesimal (Zdn), when the Kalalagna is, respectively, in the six signs beginning from Cancer or in the six signs beginning from Capricorn. In the case of the difference, if the part using R cosine of place (/.*., b) is greater, the Zdn is north. Otherwise, and in the case of the sum, it is south, always.

1. 7$te^T ^T, fir^qf^T *m!J 3^ II ^ It

Wi^ a?£tft«feft srcr II ?3 ii fq-?ftsq* 9ig3*mrFfrr sirrfrrsrf^q-^ i crafts [shfjfcnrr ^ f^i ?^<t^3^t ^rra; f^irzTT ^c^TssRi Ferrer wfffsrT^^^rr i <rer $r*: *reu^** m^fsnp: *rf$ : i aerify n^rar i s#?r ^tfesrs^r ^f^jq-R^cTTT^sir^ II <|3 II ijsra — 11. C. ^7 to 3ffT?tesTr (verse 12), frayed away, leaving only traces. 12. C. sin in faster omitted. (The Moon's Shadow) 11. The sum or difference of the Zdn and the moon's latitude, when they are of the same or different directions, respectively, is called NatL The R sine of the angle got by subtracting the Nati from three signs, is called Para-Saiiku. 12. R minus or plus R sine (Oep-m/nt/j-moon), respectively, when (Oep-m/mtf-moon) is in the six signs from Aries or Libra, multiplied by R cos moon's latitude and divided by R, is called Bana. 13. BVna multiplied by R cos Zdn and divided by R is to be subtracted from the Para-Sahku. This is the Great Gnomon {Mahatt prabha or Maha-Sahku) of the moon, (/.<?., Maha-$ahku=R sine altitude the moon). Its R cosine is the^Great Shadow {Maha-Chaya). T^TSS^T TO* *TCcftf itfftmr I! ?tf II ^RTT<fTfr I ^ ttq-iT W^f ^ ^T^T ^ I ^T^Tf^^Tf^ ) SR?, ^Ip^N^ ^ *TS<ft 5RT1I % II ^T^lff : STT^mT TO: I — 13. B. C. for *f££ 14. C. sift* STSf frayed out. 15. C. £ frayed out 1* 14. The Great Shadow, multiplied by a gnomon of any length, and divided by the 'Great Gnomon -wim-the fifteenth part of the rate of motion of the moon per day 8 is the Correct Shadow of the moon (measured in the same unit as the gnomon used), (AlternatiYe method for the Moon's Shadow) 15. Find the product of the Bina got above (in verse 12) and the R sine of Zdn, divided by R. The difference between this and the R sine of Nati (got in verse II) is called Baku {te., base), if the Natl is of the same direction as the Zdn. If of different directions their sum (instead of their difference) is the Baku* 16. R cos latitude of the moon should be multiplied by R cos 'moon-mmi/j-Oep' and divided by R. The square root got by adding the square of this and the square of Balm is the Great Shadow. 17. The perpendicular, (/. e. t R cosine), of this, (/. e., >/(R*— Great Shadow 2 ), is the Great Gnomon. From this, the moon's, shadow of the desired time is to be got, (as before). 7 6. The purpose of subtracting the" 15th part of the daily rate of motion is to correct the Great Gnomon for parallax of the moon, which depresses the moon by about 53' in the mean near the horizon and proportionately elsewhere along the vertical circle, and lessens its apparent altitude upon which the shadow depends Hindu astronomers take the horizontal parallax proportionate to the true daily rate of motion, though this is a bit rough. 7. The second half is not clear : tf«n«T35* "The moon will be visible provided the Great Gnomon is greater than the quantity to be subtracted, (i.e., the 15th part of the daily motion, in verse 14)." Otherwise the moon will be below the horizon. This half is not commented upon by the author. ^Kftfftfa: to: *ra?8f 3r*ro M *rcft 1 q^mwftfflfaiTT ^TTOOT*TO*rT STOW 3^TWfefaf%<T- i q^fcraw srnTffsrrerc ^ *rftoT ^to*t >^^^r 16. C. 51 frayed out. IL COMPUTING THE TIME FROM THE SHADOW (Bases for Viparitacchaya) 18 If the problem is to find the exact time, given the moon's shadow together with other requirements (like the date and rough time of day) 8 then, by using the shadow and the rough time and tagna, the sun and the moon, and the moon's node, and its apogee for the sake of computing the second inequality corrected moon, are to be computed. 8 Time from shadow is multivalued. Thus, for a given shadow, even on the same day, we can have two times, one with the moon being east of the meridian, and the other west, not to speak of differen days That is why the rough time when such length of shadow will occur, is required, which is obtained from tables constructed for different places , connecting the shadow with the time. 5Rt*faT$T5rra fetftw forc^ wto <r^rfa^rr mmfaf? rr^ismfe^fafa i <pr: f4^Twr?^^r^f faster mfV^T [ ^nfawnr ] S^to: qwr^w: i ^ Turret:, sfsr^srs^ftacr?^ i a^fe^Tq" (2715; 'Sift «W (1398) 3?*T^ 5i«r*<T f%5T I 17. Verses 19 and 20 are quotations from Msdhava of Sarigamagrarna and have been quoted by Nilakantha himself in his Aryabhafiya-Bhasya, GoldpSda {TSS 185, p. 1.08) with the introductory statement : «TcT ^ Tl^fa^I f^T^at ^PIKTTWiraSt Tf^Tcrf^^'T: srsfacT: t See also the Karanapaddhati of Putumana Somayaji, 9. 9-10. 18. B. fagfa left out. (The four main R sines) 19 Multiply the R sine of moon's latitude by the R cosine of the greatest declination (viz., 24° according to Hindu astronomy). Multi- ply the R sine of the declination by the R cosine of the moon's latitude. Divide each product by R, These two products are to be added or subtracted one from the other (according to the instruction in verse 20) 20. Add the two products, if the declination and the latitude are of the same direction ; if not, subtract one from the other. The result is the R sine of the true declination of the body in its own orbit. Its R cosine is the radius of the diurnal circle of the body. 21. The ft sine of the true decimation, multiplied by the equinoctial shadow and divided by 12, is called the Earth-sine (K$itijya). This multiplied by R, and divided by the radius of the diurnal circle is the sine of the Cara, its arc being the Cara in pranas {Carasu or Carapr&na) . [ ] EFT ^ II ^ II 3^ma^r% sr^^r _ wi

• t*t^t wxt( $1 Jnwi^ it ^v? n (Reduction to Polar longitude : Drk-karma)

22. Multiply R sine (of the latitude of the moon etc.) by R sine Zdn and again by R. Divide this by the product of their R cosines (i.e., the product of R cos lat- and R cos Zdn). Find its arc. This should be added to the moon etc. at rise if the latitude and the Zdn are of the same direction, and subtracted, if of different directions. At their setting, the addition and subtraction are to be reversed. 23. When (the moon etc. are) at mid-heaven, 9 the declination of the Kalalagna itself is the R sine Zdn. But, in this case, (i.e., in doing the drk-karma of verse 22, for the moon etc. at mid-heaven), the addition is to be done if the directions of the latitude and the Zdn are different, and subtraction, if they are of the same direction. (The approximate Orient Ecliptic Points etc. and their Correction) 24 a. The longitude of the planet (i.e., the moon etc.) corrected thus, (Le., acc. to verses 22, 23) is the respective (approximate) ecliptic point at rising (i.e., Orient), setting (i.e., Occident) and mid-heaven (i.e., meridian) (as the case may be). 9. By 'mid-heaven' is meant here the meridian. ^TO^T^Tfa%*r ^ft^5#WT^— [wirwwAs] I ^fefa - i aft- ^r^rospreto: ^ i fpr: ? ^f^^r^T ^r^r fkfk% *rc^r *FSk: ^^^^r^rsRsrJTrTT^RT ^r^tstvPT^^sft- sp^fa^r g^tcf ?ppnr i 5R*pf srR^h* ^^r^T^^TT: wr 5% 1 ^ ^> <{*] ^fwr 1 (pTfoFTCT ^W?TT: sni*5TFcTT f$^rr ?TS3T^ *f*f q% I cF*TS2T- «ih: sura i ^t^rw JTsq^p^fafa" tr^cr 1 *r ^ TOT^^ft Trfsr- mmfir^q-: I TT^ iTSqT^flTvPRTT fa $1 ^ JfR3T 35*raPT**»T5PftTf^rq'- stfjpfstt^t; ^ i srappfr § t>^9H*rerrc> TTffts^^^FTt TTf^r^^r-

• r%n snt?R ^rt i ^Rr ^TsrTW^^rTT^rt ^^cf^T^jf *w i

19. C TO frayed out, 20, C* *W ifi? for <HHlfc 11 24 b. The Meridian ecliptic point, with its Right ascensional difference applied, is its time, (when corrected). 25. This plus three signs is the Kahlagna. Using the declination moon etc. of this Kalalagna time, the Meridian ecliptic point is to be found repeatedly, till there is no difference (between the previous result and the next). 26-27. To the Orient ecliptic point got (in verse 24a), cara and also Right ascensional difference are to be applied to get its corresponding trme. The Occident ecliptic point got, plus six signs, is its correspond- ing Orient ecliptic point. From its corresponding time (got by applying its cara and Right ascensional difference), the moon, Zdn, the ecliptic point etc. are to be computed, and this Orient ecliptic point time is to be made correct by successive approximation. The difference between this time got and the time of the sun (with the cara and the Right ascensional difference applied to it also) is the time of the moon etc. (in their own rising, transiting and setting) day or night. ^ -O * NO S!*qT%S;5 *#5*?f firatft: Wlt^ II 3? II 21. C. crnssT for sftejiT 22. C. £<TT: for $*zn; 23. C Hterf *RfT^ (The Time from Shadow proper) 28 The square root of the square of the shad ow-p/ttf- 144, is called the Shadow-hypotenuse. The radius of the diurnal circle dimini- shed or increased, by the Earth-sine (got in verse 21), respectively as the declination is south or north, is called Antya. 28b-29. The Equinoctial Shadow-hypotenuse is to be multiplied by R and divided by the Shadow- hypotenuse. This should be sub- tracted from the Antya, multiplied by R and divided by the radius of the diurnal circle. The arc of this is to be found by using the R versed-sine 10 table. This is the Hour-angle of the moon in pranas. 30 a. These prams are to be subtracted from the time of the mid-day (/.*., the time of the Meridian ecliptic point) if the moon is east of the meridian, and added to, if west. 30b-31. These pranas are to be subtracted from the time taken at first, (see verse 18). Using the motion during this period to correct the sun, moon etc., the whole operation is to be repeated several times, as before, so that two consecutive times, got by using the sun and the moon, become equal. 10. If the 'sine* is the bow-string, the Versine' is like the arrow placed on it. Hence the expression banaih.

• • ST«®ItniTfoT?T1

24. A. omits this verse and the colophon following. 25, In place of this colophon A reads only i sft : I sft B. reads sm«^r %iPT I C. carries no colophon. 32. Victorious shines, illuminating everything, the 'Moon of Shadow-computation', with its brilliant rays of rules, having been extracted by Nilakantha from the ocean of astronomical lore. [ Thus ends THE COMPUTATIONS CONCERNING MOON'S SHADOW By Gargya-Kerala-Nilakantha-Somayaji ] APPENDIX I 'i 1. These verses are found in A, Le. t Ms. No. 5862-B, in continuation of the Commentary. APPENDIX I DERIVATION OF THE HOUR-ANGLE OF THE MOON IN PR AN AS 1 The square root of 144-p/ws-the square of the Shadow is called the 'Shadow-hypotenuse'. The radius of the diurnal circle, diminished or increased, respectively, by the Earth-sine (got in verse 21), as the declination is south or north, is called Antyn. 2, A third of the moon's daily motion in degrees is to be multiplied by the Shadow in ahgulas and added to R. This should be multiplied by the Equinoctical Shadow-hypotenuse and divided by the Shadow-hypotenuse. 3. The result is to be subtracted from the Antya got above and the product multiplied by R, and divided by the radius of the diurnal circle. The arc of this, using the versed-sine, is the Hour-angle in pranas} 1. The three verses here instruct the same thing as given by verses 28-29, with the exception that a correction is made to the first R used in verse 29. This is necessary and does duty for the correction to be made for the sake of the moon's parallax and corresponds to the correction applied in verse 14. See note there. APPENDIX n 'wraTOten* ^f^-^rfc^T: ^ro^forft: i ^qTmq^TT^toin' tysrfir^ ^ren vtfa *|tt tost, ^qr-^firotenr^w- qfss=3n^ aq%*3^£ 'ftfes «ro tai*g TOrfa*, sit ffcfws Tf^^H ^rtssfa^j q*fe ^^r sistfaff, an wrspwwr 3*3 i tV? asuwsr sns wra i 3T5T «pte*wwn srffT^ snrf^ra-?? trfew*r- anrowfasg rrg ^*ro ^Tiftwr i fTf«R?r « sroq*?tfcni fa^q^tfor q?g fefsra^Tfcr i art>£ TO^'fo^sr TfeN grs%* i c^tst stwt TOi?Tfas| *re*T i fe^sNf q»"n?E ?^r^tw»r ( wife f 3raRF&«s9g 'att% qwf Sifafa (? i 1. This demonstration, in the Malayalam language, of verse 19 of the Candracchayaganita is found inscribed in Kerala Uni. Ms* 5862-B, in continuation of the text and Sanskrit commentary edited here. Short V and V which occur in Malayalam are represented in the following passage by " andT, respectively. 2# गाग्र्य-केरल-नीलकण्ठसोमयाजि-विरचित चन्द्रच्छायागणितान्तर्गत श्लोकार्धानाम् अनुक्रमणिका अङ्गुलषष्टयंशो हि, व्या० अन्तरं बाहु, २५ अन्यद्युज्याहताक्षात् । ९ अन्त्या तत्फलहीना, Ap. I अर्काङ्गुलशङ्कोः, व्या० अस्तलग्नं स, २७ अहिस्तुङ्गो, २८ इन्दूच्चोनार्क, ३ इष्टक्रान्ति चोभे, १९ उदयास्तखभध्येषु, २८ एभिर्विश्लिष्यपूर्वा, ३० कक्र्येणादौ वितुङ्गक, ५ काललग्नोत्थ, ९ क्रान्तिक्षेपदिशोः, २३ क्रान्तिज्या विषुवद्, २१ क्षेपकोटया हृता, १२ क्षेपवृक्क्षेप, ११ खागाश्विघ्न्यान्त्य, २ गत्यंशत्र्यंशगुणित, Ap. I पृष्ठम् 12 12 26 16 12 20 | 14 16, 28 18 22 18 16 10 10 26 ! 29 चन्द्रच्छायागणितं, व्या० चन्द्रदृक्क्षप, २७ चण्द्रोनलग्न, १२ चन्द्रोऽयमयन, ६ चरं चोदयलग्नस्य, २६ जन्मस्थितिहृतयः, व्या० जयत्युद्योतयन्, ३२ ततो दृक्क्षेपकोटि, १३ तत्कोटिश्च महा, १७ तत्र च शिष्टं, व्या० तद्दशाक्ष च, १॥ तद्वाहुफलवगक्य, ८ तुलजादिक्रान्त्युत्थ, व्या० तुलाजाद्योश्चर, ७ त्रिराशिज्याभक्तं, व्या त्रिज्यांशौ स्फट, ३ त्रिज्याघ्नं दो:फलं, ५ त्रिज्याध्नायाः स्वकणप्ति, २९ त्रिज्यायां कोटिज, ४ वृक्क्षेपः काललग्नस्य, २३ पृष्ठम् 20 10 20 24 10 12 12 12 12 22 18 2*5* g 16 22 26 awmfaw TOST, $ 6 18 «rensrfa q^*<wi?f , Ap. I 26 22 22, 26 24 12 16, 28 12 12 10 IS 22 18 16 12 18 14 6 22 26 22 2 2 10 sftfr: *tw, o^tto 16 12 16 18 12 8 12 6 Aryabharya-Bhasya of Nilakantha Somayāji Madhava of Saigamagrama Putumana-Somayāj1 31 • • • Page 12 16 16 12 16 20-00 PUBLICATIONS of the > Vishveshvaranand Vishva Bandhu Institute of Sanskrit and Indological Studies (P. U.) HOSHIARPUR (Pb.) Rs. P. 1-2. G. A. Grier son's Linguistic Survey of India— A Summary, by Siddheshwar Varma, Pts. l-II, each ... 50-00 3. tt ft Pt. Ill Appendices and Indices (in press) 4. Upanisad-uddhara-koSa, by Vishva Bandhu — 30-00 5. New Varttikas to Panini's grammar, by Vishva Bandhu and Munishwar Deo ••• 8-00 6 A comparative and critical dictionary of Vedic interpretation : A specimen, by Vishva Bandhu and S. Bhaskaran Nair — 7 "°° 7. A History of the Kerala School of Hindu Astronomy, by K. V. Sarma 8, Bibliography of Kerala and Kerala-based works on Hindu Astronomy, by K.V. Sarma — 17-50 ' 9. 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