history of science in south asia

History of Science in South AsiaA journal for the history of all forms of scientific thought and action ancient and modern in all regions of South Asia

Karaṇakesarī of Bhāskara a 17th-centurytable text for computing eclipses

Clemency Montelle and Kim Plofker

MLA style citation form Clemency Montelle and Kim Plofker ldquoKaraṇakesarī of Bhāskara a 17th-centurytable text for computing eclipsesrdquo History of Science in South Asia 2 (2014) 1ndash62Online version available at httphssasayahnaorg

HISTORY OF SCIENCE IN SOUTH ASIAA journal for the history of all forms of scientific thought and action ancient and modern in allregions of South Asia published online at httphssasayahnaorg

Editorial Board

1 Dominik Wujastyk University of Vienna Vienna Austria2 Kim Plofker Union College Schenectady United States3 Dhruv Raina Jawaharlal Nehru University New Delhi India4 Sreeramula Rajeswara Sarma formerly Aligarh Muslim University Duumlsseldorf Germany5 Fabrizio Speziale Universiteacute Sorbonne Nouvelle ndash CNRS Paris France6 Michio Yano Kyoto Sangyo University Kyoto Japan

Principal ContactDominik Wujastyk Editor University of ViennaEmail ⟨hssasayahnaorg⟩

Mailing AddressKrishna GS Editorial Support History of Science in South AsiaSayahna jwra34 Jagathy Trivandrum 695014 Kerala India

This journal provides immediate open access to its content on the principle that making researchfreely available to the public supports a greater global exchange of knowledge

Copyrights of all the articles rest with the respective authorsThis work is licensed under a Creative Commons Attribution-ShareAlike 40 International License

The electronic versions were generated from sources marked up in LATEX in a computer runninggnulinux operating system pdf was typeset using XƎTEX from TEXLive The base font used forLatin script and oldstyle numerals was TEX Gyre Pagella developed by gust the Polish TEX UsersGroup

SAYAHNA FOUNDATION

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1 INTRODUCTION

The Karaṇakesarī of Bhāskara (fl 1681) is a set of astronomical tables along witha short accompanying versified text for computing the circumstances and

details of lunar and solar eclipses This text is divided into two chapters (adhi-kāras) the candraparvan or lunar eclipse which contains thirteen verses and thesūryaparvan or solar eclipse which contains seven verses (although the manu-scripts we have seen are not unanimous on this point) The text is both a supple-ment to the tables and a guide for using them Its last verse tells us that this workwas composed in Saudāmikā (a locality probably in Gujarāt) and the secondverse indicates that its epoch is year 1603 of the Śaka era (1681 CE) The tablesare computed for the terrestrial latitude ϕ = 223539 as discussed in the ana-lysis of verse 15 in section 4 below

Figure 1 Excerpt from Karaṇakesarī tables Poleman 4946 (Smith Indic MB) XIV f 4v (MS P1)

history of science in south asia 2 (2014) 1ndash62

2 karaṇakesarī of bhāskara

We know of at least a dozen extant manuscripts of the Karaṇakesarī1 The oneswe have consulted for our present study are the following

bull For the tables

ndash Poleman 4946 (Smith Indic MB) XIV ff 3ndash11 (denoted MS P1)ndash Poleman 4946 (Smith Indic MB) XXVII ff 2-3 (MS P2)ndash Poleman 4946 (Smith Indic MB) XXVIII 1f (MS P3)ndash Rajasthan Oriental Research Institute (RORI) Jodhpur 12792 (MS R2)

bull For the text

ndash Baroda 11268 (MS B) whose colophon asserts that it was copied in the14th day of the bright half of Aśvina in year 1819 (of the Saṃvat era)which the scribe calls a Saturday indeed it corresponds to Saturday 2October 1762 CE2

ndash RORI Jodhpur 12814 ff 1vndash2v (MS R1) allegedly copied for his ownuse by Bhāī Īcchārāma on the 15th day of the bright half of PhalgunaSaṃvat 1925 a Friday this converts to 28 March 1869 CE which how-ever was a Sunday

An additional manuscript (RORI Jaipur 9757 (MS J) copied on day 8 of the darkhalf of Jyeṣṭha in Saṃvat 1818 a Thursday ie Thursday 25 June 1761 CE) con-tains worked examples of the Karaṇakesarī procedures for two eclipses before1750 CE (see section 3) Unless otherwise noted all manuscript readings in thetables part refer to MS P1 (of which figure 1 shows a sample leaf)

While most tables pertaining to eclipses form part of comprehensive tabletexts on astronomical prediction in general the Karaṇakesarī is a less commoninstance of a work devoted exclusively to this topic (Pingree 1981 46) Its tablesdescribe characteristics of the celestial bodies implicated in eclipses elongationsbetween them their apparent diameters lsquodeflectionrsquo and parallax as well asother relevant phenomena such as eclipse duration and magnitude Appendedto the eclipse computation tables but not discussed in the text are a set of addi-tional tables concerning astrological phenomena

Our critical edition of the verse text portion attested in MSS B and R1 is in-cluded as Appendix B at the end of which we supply transcriptions and identi-fications of the titles of all the tables Preceding it is Appendix A a brief gloss-ary of the Sanskrit technical terms used in our exposition Section 4 contains a

1 See Pingree (1970ndash1994 328) and Pin-gree (1968 70) in which Pingree notes fourother works entitled Karaṇakesarī it is unclearwhether these works are related to the tablesunder consideration here

2 For the relation of Indian calendars toCommon Era dates see Knudsen and Plofker(2011 57ndash65) and the discussion of verse 12 insection 4 below


clemency montelle and kim plofker 3

complete transliteration and translation of the Karaṇakesarī rsquos edited text accom-panied by a lsquoVerse Analysisrsquo and lsquoTechnical Analysisrsquo for each verse or group ofverses The former specifies the metre(s) used as well as any notable featuresof the composition or the manuscript readings while the latter explains to thebest of our understanding the technical procedures described therein While webriefly summarise each of the associated tables and draw on their content to ex-plain the verses we have not attempted to critically edit or thoroughly analyzethe tables or their titles and notes we hope to undertake this task in a future pub-lication In the present discussion wherever we have quoted these prose titlesor notes we have transcribed them as they appear in the manuscript withoutcorrecting their orthography

In the edited text as well as in the transliteration translation and commentarywe employ the following editorial conventions

bull Square brackets [ ] indicate an editorial addition or proposed reconstruc-tion of missing text

bull Angle brackets ⟨ ⟩ indicate manuscript readings that we discard as incor-rect

bull We use the word kesarī lsquomane-bearer lionrsquo with a dental sibilant in ourstandard form of the workrsquos title Karaṇakesarī but we do not emend thelexically accepted variant keśarī with a palatal sibilant where it appears inthe manuscripts

bull Scribal variants of nāgarī orthography which are emended silently and notnoted in the critical apparatus (except where the meaning of the originalreading may be ambiguous) include the following anusvāra for a conjoinednasal omitted visarga virāma or avagraha misplaced daṇḍas reversed con-junct consonants (eg adba for abda) conjunct consonants that we cannotreproduce in our nāgarī typesetting doubled consonants after r or acrossa pāda break routinely confused consonant pairs (eg ba for va ṣa for kha)and all forms of koṣṭa for koṣṭha lsquotable entryrsquo

bull Fragments of Sanskrit words or compounds in nāgarī are indicated with asmall circle at the breakpoint

bull Folio breaks are indicated by a single vertical stroke∣∣

bull In the critical apparatus text followed by a single square close-bracket ]indicates the edited version of the manuscript reading that follows it

bull The symbol x within nāgarī text indicates an akṣara (syllable) that is illegiblein the manuscript



bull Numerals in sexagesimal or base-60 notation are shown with a semicolonseparating their integer and fractional parts and commas separating theirsuccessive sexagesimal digits The superscripts s and and prime indicate zo-diacal signs (ie 30-degree arcs of longitude) degrees and minutes of arcrespectively

2 BASIC ECLIPSE RECKONING IN THE INDIAN TRADITION

GEOCENTRIC GEOMETRY OF ECLIPSES

Eclipse reckoning is one of the most important functions of Indian astronomyIndian calendars are fundamentally keyed to the ritually significant instants

of luni-solar opposition and conjunction or full and new moon the so-calledparvans (syzygies) at which lunar and solar eclipses respectively occur (the termmay also be used for the eclipse phenomenon itself) The Indian lsquolunar dayrsquo ortithi of which there are exactly thirty in a synodic lunar month between twosuccessive conjunctions or oppositions is defined so that the parvans occur atthe ends of the fifteenth and the thirtieth tithi The regular solar day on the otherhand is the time between two successive sunrises (or midnights) conventionallydivided into 60 equal time-units called ghaṭikās or nāḍīs

Sun

Earth

Moon

Sun

Earth

Moon

Figure 2 Lunar eclipse configurations showing qualitative effects of variations in the distances ofthe moon and sun from the earth



A lunar eclipse takes place at a full moon or opposition when the earth in-terposing between the moon and the sun darkens the moon with its shadow Itis visible everywhere on earth where the moon is above the horizon As shown(greatly exaggerated) in figure 2 the apparent sizes of the disks of the moon andthe shadow are affected by small periodic changes in the geocentric distances ofthe moon and the sun When the moon is farther from the earth its disk lookssmaller and so does the cross-section of the shadow cone where it intersects themoonrsquos orbit When the sun is closer to the earth the earthrsquos shadow cone isshorter which also decreases the size of the shadow disk that the moon passesthrough

A solar eclipse occurs at new moon (conjunction) when the moon interposesbetween the earth and the sun preventing the sunrsquos rays from reaching somepoints on the earth It is visible only from a narrow strip of terrestrial localitiesbecause the much smaller moon cannot block sunlight from an entire hemisphereof the earth Observers outside that strip whose placement varies according tothe relative positions of the earth moon and sun will not notice any obscurationof the sunrsquos disk Both lunar and solar eclipses may be either total implying thatthe disk of the eclipsed body is at some point completely obscured or partial ifsome fraction of it always remains clear

Observers who can see a solar eclipse will perceive its position in the skydifferently depending on the amount of lunar parallax at their locality ie theapparent displacement of the geocentric position of the moon against its back-ground of stars due to the fact that the observer is not on the straight line betweenthe moon and the earthrsquos center This phenomenon is illustrated on the left sideof figure 3 which shows the lsquodepressingrsquo effect of the parallax angle p appearingto shift the moon downwards ie closer to the local horizon of the observer at PThe parallactic shift is zero for the moon at the observerrsquos zenith and increasesto its maximum as the moon gets closer to the observerrsquos horizon (Parallax shiftfor the sun is much smaller because of the sunrsquos greater distance and is generallyignored in eclipse computation)

COMPUTING ECLIPSE DATA IN INDIAN ASTRONOMY

Indian astronomy texts typically place the bulk of their eclipse computation al-gorithms in their discussion of lunar eclipses which precedes a shorter treat-ment of solar ones They enumerate around a dozen separate eclipse elementsto be determined by computation the seventh-century astronomer Brahmaguptacites fourteen in the lunar eclipse chapter of his Brāhmasphuṭasiddhānta (41ndash3)(Dvivedī 1901ndash1902 72) These include time of apparent syzygy (velā) dura-tion of eclipse (sthiti) instant of first contact (sparśa) instant of release (mokṣa)duration of totality (vimarda) beginning of totality (sanmīlana nimīlana) end oftotality (unmīlana) magnitude of obscuration (grāsa) obscuration at a given time



O

P p

Z

ecliptic

horizon

moon

V

Figure 3 Effect of parallax angle p on perceived position of the moon viewed from earthrsquos surface

(iṣṭagrāsa) lsquoshadowrsquo or apparent diameter of earthrsquos shadow (chāyā) directionof impact (diś) colour of eclipsed body (varṇa) inclination to cardinal directions(valana) and graphical projection or diagram (parilekha) Of course this list alsoimplies knowledge of other necessary quantities such as apparent diameters andvelocities of the luminaries at a desired time

Occurrence The astronomer is chiefly concerned to determine whetherwhen and how an eclipse at a particular syzygy will appear in the sky at hisown locality Eclipses are more rare than syzygies because the moonrsquos orbitand the apparent orbital path of the sun (the so-called ecliptic) are slightlyinclined to each other as long as the moon is sufficiently displaced in celestiallatitude north or south of the plane of the ecliptic it passes through oppositionor conjunction without ever intercepting the earthrsquos shadow or sunrsquos diskrespectively The elongation of the moonrsquos position from an orbital node ie



one of the two diametrically opposed points at which its orbit intersects theecliptic determines its latitude The latitude in turn determines the so-calledeclipse limits within which an eclipse is possible (see verse 12 in section 4below) and whether the eclipse will be partial or total

WE

S

NDirection

of lunar

motion

on orbit

shadow

Figure 4 The moon at first contact mid-eclipseand end of totality in a total lunar eclipse as seenin the sky by a south-facing observer

Phases There are five phases of atotal eclipse considered significant byIndian astronomers the moments ofsparśa sanmīlana mid-eclipse or ma-dhya unmīlana and mokṣa The firstthird and fourth of these phases areillustrated in the diagram of a totallunar eclipse in figure 4 In a par-tial eclipse by definition there is noperiod of totality so sanmīlana andunmīlana do not occur

Magnitude The amount of ob-scuration of the eclipsed body or ec-lipse magnitude is a linear quantitymeasured along the diameter of theeclipsing disk It depends on the ap-parent diameters of the eclipsed andeclipsing disks (which are inverselyproportional to their angular velocities) as well as on the lunar latitude (see verses13ndash4)

Deflection In figure 4 the moonrsquos passage through the shadow is tilted withrespect to the cardinal directions this is the so-called valana lsquoinclination deflec-tionrsquo which is deemed especially significant for astrological purposes This fea-ture has antecedents in ancient West Asian astronomy and its Greek successorswhere it is known as prosneusis lsquoinclination pointingrsquo In the Indian traditionthe valana is divided into two components (see verses 15ndash7)

bull akṣavalana accounting for the southward sag of the bodyrsquos projection ontothe celestial equator from the east-west prime vertical circle proportionalto the local terrestrial latitude and the so-called nata or depression of theluminaries from the meridian or the zenith

bull ayanavalana representing the skew between the celestial equator and theecliptic at that position computed based on the ecliptic declination of apoint determined by the sunrsquos position with respect to the equinoxes andsolstices (Consequently its celestial longitude has to be adjusted for pre-cession or the slow motion of the equinoxes and solstices over time)

The total valana is the algebraic sum of these two components



Parallax In most early astronomy the moonrsquos parallax was computed as asingle function of its zenith distance (Montelle 2011 123ndash125) In the Indian tra-dition however two components of lunar parallax are computed independentlyindeed there is no general term in Sanskrit for lsquoparallaxrsquo per se or the angle heretermed p The two orthogonal parallax components are based on the coordin-ates of celestial longitude and latitude with respect to the ecliptic or apparentorbital path of the sun (see verses 23ndash6) The right side of figure 3 illustratesthem for the moon positioned approximately on the ecliptic Longitudinal par-allax (lambana) along the ecliptic is determined chiefly by the distance of the bodyfrom the nonagesimal V a point on the ecliptic 90 west of the ascendant or inter-section point of the ecliptic with the eastern horizon Latitudinal parallax (nati)perpendicular to the ecliptic is based on the zenith distance of the nonagesimalhence dependent on the situation of the ecliptic with respect to the local zenithZ Each of these components is considered to have a theoretical maximum moreor less equivalent to the angular distance the moon moves in 4 ghaṭikās of timeie about 53 arcminutes (a little under 18 digits) This arc is in fact approxim-ately the maximum lunar parallax attested in other ancient astronomy traditions(Montelle 2011 125 241ndash244)

Diagram The valana and magnitude for any eclipse are to be depicted in agraphical representation namely a diagram or set of diagrams portraying itscharacteristics and key phases (see verses 110ndash13) Careful instructions are usu-ally given for geometrically constructing the disks of the obscuring and obscuredbodies the directions of the ecliptic and lunar orbit and the trajectory of the ec-lipsed body in the diagram Most works also enumerate the varying colours ofthe eclipsed body Various shades of reds blacks and browns are mentioned(eg dhūmra kṛṣṇa kapila rakta) and are often correlated to the phases of the ec-lipse or sometimes to its position in the sky These are presumably intended tohelp determine the astrological significance of the event

3 RELATIONSHIPS BETWEEN TABLES AND TEXTS

The verses of the Karaṇakesarī text have a different emphasis than those foundin the Sanskrit astronomical genres of treatises and handbooks (siddhāntas

and karaṇas) whose versified algorithms and parameters constitute a standalonesystem of computation for astronomical prediction The primary role of the Kara-ṇakesarī rsquos verses on the other hand is to support a set of tables to this endmuch of their content concerns the selection and manipulation of table entries toproduce a desired result These verses along with the titles and marginal notesfound in the tables themselves offer insight into the complex interplay betweennumerical data and text

However the Karaṇakesarī rsquos approach remains closely connected to those of



earlier works in the siddhānta and karaṇa genres in both direct and subtler waysWe consider some of these connections below

DEPENDENCE OF THE KARAṆAKESARĪ ON EARLIER TEXTS

Bhāskararsquos parameters for mean lunar and nodal motion are typical of thelsquoschoolrsquo in Indian astronomy known as the Brāhmapakṣa dating from at leastthe 7th century CE Moreover his assumption that the precession correction waszero in year 444 of the Śaka era (see verse 19 below) was shared by such earlierauthors as Parāśara Pṛthūdakasvāmin and Āmarāja3

In addition to this common scientific heritage the Karaṇakesarī relies directlyon previous compositions for much of its textual content Two of its sources thatcan be confidently identified due to the authorrsquos own acknowledgement or toobvious borrowing are both karaṇa texts consisting solely of verses with no ac-companying sets of tables namely the Karaṇakutūhala of Bhāskara II (Mishra1991 Balacandra Rao and Uma 2008 epoch 1183 CE) and the Grahalāghava of Ga-ṇeśa (Jośī 2006 Balacandra Rao and Uma 2006 epoch 1520 CE) The Karaṇakesa-rī rsquos indebtedness to these works manifests itself in a number of different waysBhāskara may excerpt phrases or entire pādas and integrate them into his textparaphrase an existing algorithm or technique or use data extracted from an-other text as a basis from which to compute one of his tables

For instance one of his table titles explicitly states that its numerical data isbased on a work called the Siddhāntarahasya a well-known alternative name forthe Grahalāghava (Pingree 1970ndash1994 2 94)

atha karaṇakesarigraṃthokte siddhāṃtarahasye sūryeṃdvoḥ parvana-yanārthe caṃdrasya koṣṭakā Now tables of the moon for the sake of computation of eclipses of thesun and the moon in the Siddhāntarahasya as expounded in the bookKaraṇakesarī (MS P1 f 4r)

Similarities in parts of the accompanying text as well as the procedures describedconfirm this dependence For instance Bhāskararsquos description of parallax and hisapplication of parallax to produce true half-durations (see verses 25ndash6) parallelthe procedures set out in the Grahalāghava

The inspiration that our author derived from the Karaṇakutūhala is partic-ularly evident in his description of the graphical projection of an eclipse aswell as his determination of parallax Karaṇakesarī 110ndash13 contains several pā-das identical to ones in Bhāskara IIrsquos section on graphical projection The Kara-ṇakesarī rsquos table for determining longitudinal parallax (see verse 23) is based ona much briefer versified table in Karaṇakutūhala 54ndash5 the Karaṇakesarī linearly

3 See Pingree (1981 13ndash16) for parametervalues and Pingree (1972) for a discussion of

precession in Indian astronomy



interpolates between the Karaṇakutūhalarsquos nine specified values to produce a cor-responding value for each of its 90 degrees of argument

In addition Bhāskara has drawn from non-astronomical sources in construct-ing his verses for example verse 21 imparts aphoristic wisdom lightly para-phrased from literary classics or traditional Subhāṣita works (proverbs and epi-grams see footnote 27)

INTERDEPENDENCE OF THE KARAṆAKESARĪ rsquoS TABLES AND TEXT

Bhāskararsquos verses are intended both to supplement and to explain the tablesmost of the text makes no sense without the information contained in the layoutand headings of the tables and the tables would be useless without crucial para-meters and computational steps supplied by the text (or in many cases lackingin the text and presumably supplied by the userrsquos own knowledge) The focusof the prescribed procedures shifts back and forth between computational al-gorithms in the verses and the numerical entries in the tables while many versesdirectly refer to operations with tabular values others merely clarify the conceptsbeing discussed or specify procedures that are unconnected with the table dataConciseness of expression is also balanced against practical convenience in theuserrsquos computations4

Thus the Karaṇakesarī is in some sense not so much a self-contained set oftables as an abbreviated handbook or karaṇa with some computational proced-ures and parameters replaced by table entries

MS JrsquoS WORKED EXAMPLES FOR USING THE KARAṆAKESARĪ

RORI Jaipur 9757 (MS J) a 1761 manuscript of 6 folia is written in a mixtureof regular technical Sanskrit and an early modern Indo-Aryan vernacular5 Itconsists of two detailed udāharaṇas or worked examples of calculations with theKaraṇakesarī tables according to the procedures described in its text The first ofthese is for a lunar eclipse whose date is given (f 1r) as Wednesday the fifteenthtithi of the bright half of Āśvina in Saṃvat 1801 Śaka 1666 If the Śaka year istaken as expired rather than current this appears to correspond to Wednesday21 October 1744 CE to which a (partial) lunar eclipse is also assigned by modern

4 For example the construction of thegraphical projection of an eclipse in 110ndash13 isdistinct from the table manipulations while thealgorithm for correcting longitudinal parallaxin verse 23 requires the user to divide by 40the values in the corresponding table on f 10rIt would seem more convenient from the userrsquospoint of view to have constructed the table withits entries pre-divided by 40 in the first placeLikewise limiting an angular table argument to

the first quadrant reduces the tablersquos size butimposes on the user the additional task of redu-cing arbitrary arcs to first-quadrant equivalents(which Bhāskara explains how to do in verse17)

5 A characteristic blend of the two forms isseen in these sentences from f 1r line 10

atha sūryasādhanaṃ ||prathamaje avadhīmāhegrahegrahaṇathā-

yetenosūryamadhyama



calculations6 The second eclipse a solar one is said (f 3v) to have taken placeon Sunday the new-moon tithi of the dark half of Māgha in Saṃvat 1748 or Śaka1613 ie (again taking the calendar year as expired) Sunday 17 February 1692when an annular solar eclipse would have been visible as a partial eclipse in mostof the South Asian subcontinent

We have occasionally used these examples to elucidate some point about theKaraṇakesarī text or tables in our discussion of them in section 4 but we are de-ferring a detailed study of MS J to a more thorough investigation of the tablesthemselves as proposed in section 1

4 KARAṆAKESARĪ TRANSLATION AND COMMENTARY

śrīgaṇeśāya namaḥ || śrīgurubhyo namaḥ || śrīsāradāyai namaḥ || athakaraṇakeśarī likhyate ||Homage to Gaṇeśa Homage to the Gurus Homage to Śāradā7 Now the Kara-ṇakesarī is written

11 SALUTATION

śrīkṛṣṇacandracaraṇaṃ praṇipatya bhaktyājyotirvidāṃ bahuvidām abhivandanaṃ ca ||kṛtvā kavīndrakulabhūṣaṇabhāskarākhyorāmātmajaḥ karaṇakeśarim ātanoti || 1 ||

Having bowed to the feet of Śrī Kṛṣṇa with devotion and making respectful sa-lutation to the very learned knowers of jyotiṣa the one called Bhāskara the or-nament of the Kavīndra family (kula) the son of Rāma illuminates the Karaṇa-kesarī

VERSE ANALYSIS

Metre vasantatilakāThis invocatory verse introduces the author of the Karaṇakesarī Bhāskara and

makes reference to his lineage Bhāskara is the son of Rāma a descendant of afamily (kula) called Kavīndra8 As well as being a proper name bhāskara lsquolight-

6 Both historical eclipse events mentionedhere were established via the USNO EclipsePortal astroukhogovukeclbinquery_usnocgi

7 Śāradā is another name for Sarasvatī Themanuscript gives the variant form Sāradā

8 We do not know who is meant by the term

lsquoKavīndra familyrsquo The name is reminiscent ofthe famous Kavīndrācārya (see Sastry (1921))but he is said to hail from one Puṇyabhūmiidentified with Paithan near Aurangabad in theDeccan while our author appears to identifyhimself as a member of the Moḍha Brahmanasassociated with the Gujarat area (see verse 27)



makerrsquo is a noun that means lsquothe sunrsquo Thus Bhāskara has invoked a pun on hisown name he will naturally be the one to lsquoilluminatersquo (ātanoti from the Sanskritverb ā-tan to illuminate to spread light) astronomical techniques and principlesin the Karaṇakesarī

In the last pāda or quarter-verse the accusative form of the name Karaṇakesa-rī (lsquothe mane-bearerlion among karaṇasrsquo ie pre-eminent) is rendered -kesarimrather than -kesariṇam to conform to the requirements of the metre

12 LUNAR NODE ELONGATION

śako rāmaviṣṇupadāṅgendu 1603 hīnoviyadrāmacandrair hṛto 130 labdhaśeṣau ||yutau copakarṇaṃ hy avadhyanvitaṃ tat-sapātendutātkāliko bāhubhāgaiḥ || 2 ||

The [current] Śaka year is decreased by 1603 [and] divided by 130 and [the tabu-lar entries corresponding to the amounts given by] the quotient and remainderare added [together] The lsquoresultrsquo (upakarṇa) is increasedby [the tabular amountcorresponding to the number of elapsed] avadhis That [resulting sum] is the[lunar nodersquos elongation from the sun corresponding to the] lsquolunar elongationrsquo(sapātendu) at that time

VERSE ANALYSIS

Metre bhujāṅgaprayāta9 the first pāda is hypometricThe term upakarṇa is puzzling In a mathematical context karṇa (literally

lsquoearrsquo) means lsquodiameterrsquo or lsquohypotenusersquo Context suggests it refers to a differentquantity discussed below

The last word in the fourth pāda namely bāhubhāgaiḥ belongs to the nextverse

TECHNICAL ANALYSIS

This verse gives a rule for determining the longitudinal elongation between thesun and the lunar node on the ecliptic (in zodiacal signs degrees minutes andseconds) for a specified date This elongation determines whether or not an ec-lipse will be possible

In principle the eclipse limit is determined rather by the elongation of themoon from its node as suggested by the term sapātendu literally lsquo(the longitude

9 This metre a form of jagatī (12 syllables ina pāda) is comprised of the four-gaṇa sequence

ya ya ya ya For details see for instance Apte(1970 vol 3 Appendix A p 5)



of) the moon increased by (the longitude of) its nodersquo (see section 2)10 Howeveras noted above and confirmed by the headings of the associated tables discussedbelow the quantity tabulated is instead the elongation of the sun (sūryasya) fromthe moonrsquos node We reconcile this seeming contradiction by noting that the sunand moon at an instant of true syzygy are by definition either 0 or 180 apartso the absolute value of their nodal elongations at that instant will be the same

The verse explains how to extract and combine the relevant tabular datawhose behavior we reconstruct as follows The node-sun elongation for the totaltime since epoch is dependent on the fundamental period relation between timeand the revolutions of the node The parameters appear to be derived from theBrāhmapakṣa tradition in which the moonrsquos node makes minus232311168 revolu-tions (the negative sign indicates retrograde motion) in 4320000000 years Thisproduces the following mean yearly motion for the node

minus232 311 168 times 360

4 320 000 000

= minus19 21 33 21 year

Since the sun returns to the same zero-point of longitude at the start of every(sidereal) year the mean yearly increment in the nodersquos elongation from the sunis identical to its mean yearly (negative) increment in longitude

If Y is taken to be the current Śaka year then the prescribed algorithm dividesthe difference between Y and 1603 (which we infer to be the epoch year of thetables in the Śaka era) by the constant term 130 The quotient which we will callE consists of an integer result (the number of completed 130-year periods) plusa remainder (the number of single years subsequently completed)

E =Y minus 1603

130

The total time since epoch is broken into three partsbull the number of completed 130-year intervals elapsed since the epoch (the

integer part of E called labdha or lsquoquotientrsquo in the table heading)

bull the number of completed single years elapsed in the current 130-yearperiod (the fractional part of E called śeṣa or lsquoremainderrsquo)

bull and the number of completed 14-day intervals elapsed in the current year(avadhis described in the table heading as candrasya koṣṭhakā or lsquotable entriesof the moonrsquo despite the fact that we are computing nodal-solar rather thannodal-lunar elongation)

10 The reason that this lsquoincreasersquo or sumactually signifies longitudinal elongation of themoon from its node is that the node revolvesin the lsquobackwardsrsquo or westward direction mak-

ing the nodal longitude negative Adding thepositive lunar longitude to the negative nodallongitude gives the elongation between them



These three quantities are the arguments in the three associated tables theelongation (modulo integer revolutions) computed for successive numbers of130-year intervals from 1 to 130 (f 3r) the elongation computed for numbers of1-year intervals from 0 to 130 (f 3v) and the elongation computed for successivenumbers of 14-day avadhis from 1 to 27 (f 4r) (So in principle the tables wouldbe valid for more than 130 times 130 = 16 900 years after the epoch date)

The sum of the tabular entries in the first two tables corresponding to thequotient and remainder parts of E gives the total amount of elongation corres-ponding to the beginning of the chosen year called in the verse upakarṇa Thenconsidering the number of 14-day periods that have elapsed since the start of theyear the appropriate tabular entry from the third table can be added to producethe elongation for the chosen date within that year

The first two tables are based on the assumption that the elongation changeslinearly with a constant difference from one entry to the next In the table of1-year increments this constant difference is minus19

21 derived from the periodrelation shown above (apparently truncated rather than rounded) In the tableof 130-year increments the constant difference is minus3

16 20 evidently derived asfollows

In one year the absolute value of the elongation = 19 21 34

In 130 years the elongation = 19 21 34 times 130

= 2516 43 40

Elongation modulo integer revolutions = 7 times 360 minus 2516 43 40

= 2520 minus 2516 43 40

= 3 16 20

This table of one-year increments also incorporates an epoch correction in itsfirst entrymdashthat is it begins with the value 6

s 29 24 36 which is presumably

the amount of elongation at the beginning of 1681 CE or Śaka 1603 AlthoughBhāskara does not specify exactly what calendar moment he associates with thisvalue we assume that he equates the year-beginning with caitraśuklapratipad (thefirst new moon after the vernal equinox) which according to modern retrodictioncorresponds to noon 20 March 168111

11 This converted date was computed us-ing Michio Yanorsquos online Pantildecāṅga program forthe date śuklapakṣa 1 of (amānta) month CaitraŚaka 1603 See httpwwwcckyoto-suacjp~yanompancanga In fact a solar eclipsetook place at this conjunction but it was notvisible anywhere on land Reconstructing the

epoch correction from the Brāhmapakṣa kalpaparameters with the total number of elapsedyears equal to 1972948782 (ie the accumu-lated years up to the start of the Śaka era plus1603 within the Śaka era) gives the proportionalamount of nodal longitude in degrees accumu-lated at the epoch



The third table tabulates the change in elongation for each of the 26 14-dayperiods (avadhis) during the solar year plus a 27th entry to account for theleftover day after the first 26 times 14 = 364 The division of the year into these14-day intervals is particularly useful for eclipses as conjunction and oppositionoccur roughly two weeks apart

Figure 5 An excerpt from the table of the change in elongation for each avadhi f 4r

Table 1 The changes in elongation for the 14-day avadhis during the year

Argument Row 1 Row 2 Row 31 0 2 19 55 61 46 1 ∥ ṛ2 0 16 41 1 61 21 23 1 0 56 10 60 53 24 1 15 5 26 60 26 15 1 29 10 35 60 16 0 ∥6 2 13 12 50 60 7 0 ∥ ṛ7 2 27 14 8 60 5 1 dha8 3 11 16 1 60 13 19 3 25 20 26 60 26 1

10 4 9 28 57 60 53 211 4 23 42 48 61 11 212 5 8 3 0 61 42 2

Table contdhellip

divmod((minus232311168) lowast (1972948782)4320000000) = (minus106096768 1809402624)

N(1809402624 lowast 3604320000000) =150783552000000

Recall that this nodal longitudinal increment ofapproximately 150 degrees or 5 signs is neg-ative when added to 0 of solar longitude it

comes out to a little less than 7 signs of nodal-solar elongation which more or less corres-ponds to Bhāskararsquos epoch value 6s292436The sun is not usually exactly at 0 longitudeat the caitraśuklapratipad conjunction but in thiscase it appears to have been very close to it



Table 1 contdhellip

Argument Row 1 Row 2 Row 313 5 2[2] 30 39 62 14 214 6 7 5 27 62 51 215 6 21 48 7 63 18 216 7 6 37 34 63 44 217 7 21 33 11 64 7 218 8 6 33 34 64 31 0 | dha19 8 21 35 47 64 33 0 | dha20 9 6 39 40 64 41 0 | ṛ21 9 21 43 21 64 30 122 10 6 43 21 64 16 123 10 21 46 16 6⟨4⟩[3] 59 124 11 6 39 35 63 21 1 |25 11 21 16 25 62 56 226 0 5 53 20 62 26 227 0 20 22 9 61 56 1 ṛ

Unlike the previous two tables the avadhi table contains a non-linear se-quence of values that is the differences between successive entries are not con-stant but vary sinusoidally It also differs from them markedly in structure withthree rows of data below each of its 27 argument values (See figure 5 for an ex-ample of its format in the manuscript and table 1 for a complete reproductionof its content In this reproduction which displays the original rows verticallyrather than horizontally for convenience in layout an unmistakable scribal erroris marked off by angle brackets and missing correct values are supplied withinsquare brackets)

The first row contains the increment in elongation while the second labeledgatayaḥ lsquo(daily) motionsrsquo gives the true velocity of the elongation between sunand lunar node presumably for the beginning of each avadhi The minimumvalue of this velocity is 60rsquo5 at avadhi 7 and the maximum 64

prime 41 at avadhi 20These extrema evidently fall respectively at the solar apogee where the sunrsquosmotion is slowest and the perigee where it moves fastest see verse 13 belowThe third row which is unlabeled contains entries of 0 1 and 2 in a distributionwhose rationale is unclear occasionally accompanied by symbols for the nāga-rī characters dha and ṛ standing for dhana and ṛna or lsquopositiversquo and lsquonegativersquorespectively The positive and negative symbols appear to be applied consistentlywith the transitions from increasing to decreasing values of the velocity in thesecond row The entries in the third row may thus form part of some second-difference interpolation rule that Bhāskararsquos text does not explain



It may be that the reason for the non-constant differences between the entriesin the first row is that the variation in solar velocity has been taken into accountThat is the elongation per avadhi could have been computed via the followingrelation

Elongation per avadhi = 14v(day)sun + 14v

(day)

nodewhere vsun is the true velocity of the sun and vnode is the mean velocity of thenode The average difference between entries is about 14

prime 30 which more or lessagrees with the fact that the mean sun is moving approximately 0

59 per dayand the lunar node roughly 0

44 20 in the opposite direction over 14 days

lsquoAveragersquo value of increments = 0 59 times 14 + 0 44 20

= 13 46 + 0 44 20

= 14 30 20

This reconstruction approximately describes but does not exactly reproduce theentries in the table

13 LUNAR LATITUDE AND SIZES OF DISKS

mitaiḥ koṣṭhakair aṅgulādiḥ śaraḥ syātsapātasya bhuktyupari helibimbam ||tither mānaghaṭyoparī candrabhūbhetulājādiṣaḍbhonayuk spaṣṭakubhā || 3 ||

The latitude in digits and so on should be [determined] by the table entries com-mensurate with the degrees of the bāhu [arc of elongation see the end of theprevious verse] [The diameter of] the disk of the sun [has as its] superscribed[argument] the daily velocity of the [moon increased by its] node [The diameterof] the moon and the shadow of the earth [have as their] argument the meas-ure of ghaṭikās in a tithi [That ie the diameter of the shadow of the earth] isdecreased or increased by [the appropriate tabular entry according as the sunis in] the six signs beginning with Libra or with Aries [respectively] [This is] thecorrected shadow of the earth

VERSE ANALYSIS

Metre bhujāṅgaprayātaThe metre andor the grammar in this verse are defective in several instances

and we have tried to compromise between respecting their requirements and ac-cepting the manuscript readings In the second pāda the last syllable in uparishould be heavy for the sake of the metre but we leave it light as the correct



ending for a neuter nominative singular i-stem bahuvrīhi compound agreeingwith the neuter noun bimba that it modifies Likewise the third pāda containsa dvandva compound ending in bhā (f lsquoshadowrsquo) in the manuscript we have cor-rected it to the feminine nominative dual ending -bhē and made the final vowelof upari long in the preceding bahuvrīhi compound to agree with it In the fourthpāda the syllable ku (f lsquoearthrsquo) should be long for metrical purposes

The above compound forms with upari an indeclinable adverb meaninglsquoabove upon onrsquo are peculiar to this type of technical terminology Like otheradverbs upari can be used as a modifying element in a compound as in uparikuṭīmeaning lsquoupper roomrsquo But in this verse and in many of the table titles it is usedinstead as the primary member of a bahuvrīhi compound eg bhuktyupari lsquohavingthe velocity [as its] superscribed [argument]rsquo12 The numbers indicating the val-ues of that argument are written above or at the tops of the respective columns

Interestingly the first part of the compound word heli-bimba meaning lsquodisk ofthe sunrsquo is a loan-word derived from the Greek helios

TECHNICAL ANALYSIS

The apparent sizes of the disks of the eclipsed and eclipsing bodies and the dis-tance in latitude between their centers are necessary for determining the amountof obscuration Qualitatively the latitude of the moon on its orbit is proportionalto its elongation or distance from its node with the maximum latitude (conven-tionally 4

30 or 270prime (Pingree 1981 16) equivalent to 90 aṅgulas or digits of linear

measure) occurring when the moon is 90 from the node And the apparent size

of the disk of sun or moon depends on its distance from the earth which is pro-portional to its apparent angular velocity The size of the earthrsquos shadow alsodepends on the distances of the moon and sun from the earth (as the moon getscloser it passes through a wider circular cross-section of the shadow cone whilethe sun getting closer makes the shadow cone shorter see figure 2 in section 2)The mean apparent size of the disk of the moon or the sun is realistically as-sumed in Indian astronomy to be about 32 arcminutes or 10ndash11 digits while thatof the shadow disk is about 1 21

or 27 digits (Montelle 2011 223) This verseexplains how to adjust these values to find their true sizes at a given time whichwill determine the appearance of the eclipse

The first part of the verse refers to the table of lunar latitude on f 4r whichhas three rows The first is labeled bhujāṃśa (lsquodegrees of the bhujarsquo or bāhu bothmeaning lsquoarcrsquo or lsquoangular argumentrsquo) ranging from 0 to 16 A note in the marginasserts that this argument refers to the moonrsquos elongation from its node (athasapātacandrabhujāṃśopari ) Evidently any nodal elongation greater than 16

is

12 The use of upari as a technical termmeaning the argument of a table is attested inother table texts as well including Nāgadattarsquos

Brahmatulyasāraṇī eg in MS Poleman 4735(Smith Indic 45) f 7v



considered to negate the possibility of an eclipse and thus is not tabulatedThe second row labeled śarāṃgulāḥ lsquodigits of latitudersquo runs from 0 10 (evid-

ently a scribal error for 0 0 since the accompanying value of the differencebetween the first and second entries is equal to the second entry itself) to 24 45 di-gits These latitude values appear to have been computed more or less in accord-ance with a rule from the Karaṇakutūhala of Bhāskara II (verse 45 Balacandra Raoand Uma 2008 S67) equivalent to setting the moonrsquos latitude in digits equal to 90times the (modern) sine of its nodal elongation in degrees For instance by thisformula the maximum elongation of 16

corresponds to a maximum latitude of24 48 digits nearly identical to the Karaṇakesarī rsquos 24 45

The third row gives the differences between successive entries in row 2 andit is noted in the row header that this difference is always positive (rsquoṃtara sadādhanaṃ) The differences for entries 0 to 8 are 1 34 and from 9 to 16 1 30

The second pāda of the verse refers to the final table on f 4r which tabulatesthe diameter of the disk of the sun (ravibimba) using as argument the daily elong-ation of (presumably) the sun from the lunar node (sapātasya gatiḥ) which rangesfrom 0

59 56 to 0 64 42 over twenty entries This diameter of the sun (also in

digits evidently) ranges from 10 19 to 11 11 Qualitatively this relation makessense because the variation in the nodal-solar elongation depends on the changein the sunrsquos apparent angular velocity (that of the node which has no anomaly ofits own is constant) That is when the sun is moving faster the daily nodal-solarelongation is bigger and also the sun is closer to the earth meaning that the solardisk appears larger (However this adjustment may not have been indispensablein a lunar eclipse computation the compiler of the lunar-eclipse example in MS Jfor instance neglects it)

The third pāda refers to a table on f 4v which has three rows The first quan-tifies the length of the Indian time-unit called the tithi running from 52 to 67ghaṭikās (As remarked above in section 2 a tithi on average is one-thirtieth of asynodic month or a little less than one civil day of 60 ghaṭikās The so-called lsquotruersquotithi on the other hand ie the time required for a 12

increment in the elonga-tion between the true longitudes of sun and moon is of variable length becauseit depends on their true velocities the faster they move apart in longitude theshorter the current tithi will be)

The tablersquos second row enumerates values of the diameter of the moonrsquos diskfrom 11 57 to 9 30 and the third row the diameter of the earthrsquos shadow diskranging from 30 45 to 23 11 Both these diameters like that of the sun are spe-cified in digits Their values seem to be based on the dependence of the apparentdiameters of moon and shadow upon the geocentric distance of the moon whichis inversely proportional to its velocity As previously noted fewer ghaṭikās in atithi implies higher lunar velocity and consequently a smaller lunar distance anda larger apparent disk for both the moon and the shadow



Me[ṣa] Vṛ[ṣabha] Mi[thuna] Ka[rkaṭa] Si[ṃha] K⟨ā⟩[anyā] Sign-entries011 016 020 016 01⟨6⟩[1] 00 Positive earth-shadowTu[lā] Vṛ[ścika] Dha[nus] Ma[kara] Kuṃ[bha] M⟨i⟩[īna] Sign-entries011 016 020 016 011 00 Negative earth-shadow

Table 2 Transcription of the table in figure 6

The last pāda of this verse proposes a positive or negative incremental correc-tion to the size of the disk of the shadow depending on what zodiacal sign thesun is in (positive when the sun is in the six signs beginning with Aries negat-ive otherwise see figure 6 and table 2) The correction amount ranges from 0 0

digits in signs ending at an equinox to 0 20 digits in signs ending at a solstice

Figure 6 The table of correction to thesize of the disk of the shadow f 4v

How these values were determined isnot entirely clear But we can reconstructat least a qualitative justification for themby appealing to Bhāskara IIrsquos treatment inthe Karaṇakutūhala of the position and effectof the solar apogee and the sunrsquos anomaly(that is its mean position minus the solarapogee position which in Karaṇakutūhala 21he puts at about 78

of sidereal longitude(Balacandra Rao and Uma 2008 S15)) Werecall that when the sun is closer to the earth

and consequently more distant in anomaly from the apogee it is perceived asmoving faster and shows a larger apparent disk as well as shortening the earthrsquosshadow cone The formula for correcting the diameter of the shadow in Kara-ṇakutūhala 47ndash8 (Balacandra Rao and Uma 2008 S69ndash70) takes this effect intoaccount by equating the correction to 367 of the true lunar daily motion minus17 of the solar daily motion implying that a larger true solar daily velocitymeans a smaller shadow diameter

The Karaṇakesarī likewise appears to locate the solar apogee somewhereshortly before the beginning of Cancer as the avadhi elongation table discussedin verse 12 suggests that is it takes the sun 6 avadhis or about 84 days from thestart of the year at or near the beginning of Aries to attain its slowest velocityConsequently a solar anomaly between 90

and 270 would mean more or less

that the sun is somewhere between the beginning of Libra and the beginning ofAries ie it seems faster and larger

We are told in the present verse of the Karaṇakesarī to decrease the diameterof the shadow by the tabulated correction when the sun is between Libra andAries and to increase it otherwise which qualitatively agrees with the rationaleabove Moreover the table entries are also qualitatively consistent with it since



the largest corrections (both positive and negative) occur near a solstice (whenthe sun is roughly at perigee or apogee respectively) and the zero correctionsnear the equinoxes where the true and mean solar velocity are about equal How-ever it remains unclear whether or to what extent this rationale accounts quant-itatively for the Karaṇakesarī rsquos actual correction values

14 ECLIPSE MAGNITUDE

chādyachādakamaṇḍalānvitadalaṃ kāṇḍonachannaṃ punaḥchannaṃ grāhyavivarjitaṃ tu nikhile grāse rsquopi khachannakam ||channāṅkair mitakoṣṭhake sthitir bhavenmardasya khachannataḥśrībhānor udayāt tathāstasamayāt parvāntamadhyagrahaḥ || 4 ||Half the sum of [the diameters of] the disks of the obscured and the obscurer[is] the obscured amount [when it is] diminished by the [lunar] latitude Againthe obscuration is decreased by [the disk of] the eclipsed and in a total eclipsethere is obscuration of the sky In the table commensurate with the numbers [ofthe digits] of obscuration is the half-duration [There is also a table] of the half-duration of totality [resulting] from sky-obscuration [ie in the case of a totaleclipse] [A solar eclipse is visible] from the rising of the Lord Sun just as [a lunareclipse] from [its] setting Mid-eclipse [is] the end of the parvan

VERSE ANALYSIS

Metre śārdūlavikrīḍita The second syllable ought to be heavy as should the six-teenth syllable in each of pādas 2 and 3 suggesting that the author thought of thefollowing consonant cha in each case as doubled

This verse illustrates Bhāskararsquos tendency to prescribe the steps of an al-gorithm before identifying the quantity it produces It also exemplifies hispenchant for technically ungrammatical ways of forming and modifying com-pounds For instance he expresses dalaṃ kāṇḍonaṃ channaṃ lsquothe half-sumdiminished by the latitude [is] the obscured amountrsquo as dalaṃ kāṇḍonachannaṃliterally lsquothe half-sum [is] the obscured amount diminished by the latitudersquo13

TECHNICAL ANALYSIS

The amount of the channa or obscuration (C) is stated to be

C =1

2

(R + r)minus β

where R and r are the radii of the obscurer and the obscured respectively and βis the lunar latitude all of which the user will have found from the procedures

13 This theoretically incorrect mixed con-struction is by no means unique to Bhāskara

however see eg Tubb and Boose (2007 189)



mentioned in the previous verse As illustrated in figure 4 in section 2 C rep-resents not the extent of the eclipsed bodyrsquos own diameter which is obscuredbut rather the depth to which the leading edge of the eclipsed body penetrateswithin the disc of the eclipser The second pāda defines lsquoobscuration of the skyrsquoie excess of the obscuration C over the diameter 2r of the obscured disk whichof course implies a period of totality in the eclipse (The author of the examplein MS J for reasons unknown instead (f 1v line 8) multiplies C by 20 and di-vides it by the lunar diskrsquos diameter) In principle the greater the obscurationthe longer the eclipse will last and the greater the amount of lsquosky-obscurationrsquothe longer the period of totality will be

It follows that the second half of the verse invokes two tables on f 4v thattabulate the half-duration of the eclipse from the beginning of obscuration tomid-eclipse and the half-duration of totality (if any) from the beginning of totalimmersion to mid-eclipse It also adds the fairly trivial observation that solareclipses are visible when the sun is above the horizon and lunar ones when it isnot The lsquoend of the parvanrsquo as noted in section 2 refers to the syzygy moment atthe end of a tithi when the luni-solar elongation is either exactly 180

or exactly0

The argument of the half-duration table is the digits of lsquoobscuration of themoonrsquo (caṃdrachinnāṃgula for caṃdrachannāṃgula) Its maximum value of 21digits evidently means that the maximum possible lsquoobscurationrsquo with the centerof the moon coinciding with the center of the shadow is taken to be equal tothe maximum value of the moonrsquos radius plus that of the shadowrsquos radius orapproximately 6 + 15 = 21 digits Its corresponding table entries for the meanhalf-duration (caṃdrasya madhyasthitighaṭi) are in ghaṭikās and run from 57 30 to55 17 In fact as the worked lunar eclipse example in MS J (f 1v lines 9ndash10) makesclear these tabulated values must be subtracted from 60 to give the actual extentof the mean half-duration anywhere from 2 30 to 4 43 ghaṭikās A separate tableat the bottom of this folio gives the differences of the successive tabulated entries

The table of the half-duration of totality has three rows the last of whichseems to be only an extension of the third row of a different table immediatelyto the left of it and contains only a (meaningless) zero in each cell The first rowcontains the table argument the digits of lsquosky-obscurationrsquo (khachinnāṃgulopari)running from 1 to 9 Again the maximum value suggests that if the moonrsquos andshadowrsquos centers coincide there will be 15 minus 6 = 9 digits of shadow beyond theedge of the moon The second row contains the corresponding ghaṭikās runningfrom 59 0 to 57 54 apparently indicating a range of 1 0 to 2 6 ghaṭikās for theactual half-duration of totality (mardaghaṭikā)

The last table on f 4v which is not alluded to in the text lists the meansolar longitude (with corresponding solar velocities and their interpolation dif-ferences) for each avadhi from 1 to 27 The compiler of the MS J example uses this



table to get the mean solar position for his lunar eclipse date (f 1v line 10) whichhe then adjusts to the true position (f 2r line 2)

15 BEGINNING AND ENDING OF ECLIPSES ZENITH DISTANCE ANDAKṢAVALANA

sthityā mardena hīnaḥ sparśasanmīlane syuḥmardasthityānvite rsquosminn unmilaṃmokṣasaṃjntildeam ||khāṅkair nighnaṃ 90 nanāptaṃ ghasramāne natāṃśāstānaiḥ 45 śodhyāṃśakoṣṭhe prākpare saumyayāmye || 5 ||

[The moment of mid-eclipse is] decreased by the half-duration [or] by the half-duration of totality These are the instants of first contact [or] beginning of totalimmersion [respectively] When that [ie mid-eclipse] is increased by the half-duration of totality or the half-duration it is called the beginning of emersion[or] release Multiplying [the lsquotime-arcrsquo time fromnoonorhour-angle] by90 [and]dividing by its own in [the sense of] the day-measure [the result is] the degreesof [half] the zenith distance (nata) In the tabular entry for diminished degrees by45 [one finds the akṣavalana] [When the body is] East or West [of the meridianrespectively the (akṣa)-deflection is laid off] in theNorth or South [respectively]

VERSE ANALYSIS

Metre maratatagaga14

Lexically speaking the Sanskrit word unmīla in pāda 2 (more usually unmīla-na) should have a long lsquoirsquo However it occurs in the verse with a short lsquoirsquo whichis required for the metre

The word tāna in the last pāda is associated in the Sanskrit bhūtasaṃkhyā orlsquoword-numeralrsquo system with the number 4915 But in this context the numbermust mean 45 (and indeed in MS R1 the numeral lsquo45rsquo is written out) Bhāskararsquosnonstandard approach to compounding is again apparent in the use of the instru-mental tānaiḥ lsquoby 45rsquo to modify only the individual element śodhya lsquodiminishedrsquoin the following compound

TECHNICAL ANALYSIS

This verse deals with the determination of the five significant moments in an ec-lipse (see section 2) The first part of the verse shows how to find the beginningof the eclipse or first contact and the beginning of totality if any namely sub-tract the length of the half-duration or the half-duration of totality respectively

14 This is a form of sakvarī a 14-syllablemetre Each pāda in maratatagaga is split into twogroups of seven See Apte (1970 vol 3 App A

p 19)15 See eg Sarma (2003 64)



from the time of mid-eclipse or end of the parvan (which presumably the useris expected to know from a calendar or some other source) Conversely whenthese same intervals are added to the mid-eclipse moment the results specify theinstants of release (end of the eclipse) and beginning of emersion respectively

The second half of the verse concerns the akṣavalana or lsquodeflection due to lat-itudersquo The lsquodeflectionrsquo or valana as discussed in section 2 is the angle betweenthe directional axes determined by the more-or-less easterly movement of the ec-lipsed body and the true local cardinal directions aligned with the prime verticalcircle and the prime meridian shown in figure 7 it is traditionally considered tohave ominous significance16 The valana is conventionally given in the form oftwo components one due to the tilting of the celestial equator off the prime ver-tical by the amount of terrestrial latitude (akṣavalana) and the other to the skewbetween equatorial east and ecliptic east at the bodyrsquos current tropical longitude(ayanavalana)

Instead of identifying or defining the quantity akṣavalana this half-verse givesa rule to find the zenith distance (nata lsquodepressionrsquo ζ)mdashor rather half of itmdashwhich is manipulated to become the argument in the table of akṣavalana Thezenith distance depends on the hour-angle or amount of time in ghaṭikās beforeor after noon (which somewhat confusingly is also conventionally termed nata)and the length of the bodyrsquos lsquoown dayrsquo ie the amount of time it is above thehorizon

(half) ζ =hour-angle times 90

own day measureThis rule is similar to a standard approximation appearing in eg Karaṇakutū-hala 414 (Balacandra Rao and Uma 2008 S80) which more or less converts hour-angle (time) into zenith distance (arc) by the simple proportion ζ 90

hour-angle half-day17 By using the whole lsquoown dayrsquo in place of the half-day in the di-visor of this proportion the Karaṇakesarī actually computes half the arc of zenithdistance (Note also that this procedure does not include the information neededto determine the length of the lsquoown dayrsquo which appears eventually in verse 19)

Theoretically the relation of the zenith distance ζ to the akṣavalana is based onthe following geometric rationale In figure 7 showing the celestial hemisphereabove the local horizon with zenith at Z M is the deflected body and P and Pprime arethe poles of the equator and ecliptic respectively (the latter is lifted off the rearsurface of the hemisphere for ease of viewing) Qualitatively as noted above thelocal latitude or elevation of the pole P above the northern horizon is responsiblefor the akṣavalana component of deflection while the splay of the ecliptic away

16 For further details see Montelle (2011229ndash230) The diagram in figure 7 is based on asimilar diagram in Chatterjee (1981 125)

17 The relevant part of the Karaṇakutūhala

verse (Mishra 1991 56) has clearly influencedthe phrasing of the Karaṇakesarī rsquos rulekhāṅkāhataṃ svadyudalena bhaktaṃ sparśe vimuk-tau ca nataṃ lavāḥ syuḥ



Zecliptic

S

P

Q

E

M

N

equator

prime verticalhorizon

meridian

P

Figure 7 The akṣavalana angNMP and ayanavalana angPprimeMP of a body at M

from the equator produces the ayanavalana component The total valana is thusequal to the angle angNMPprime To compute its akṣavalana part angNMP we note thatangle angMNP or arc ZQ is the projection of the bodyrsquos zenith distance ZM ontothe prime vertical while PM is the complement of its declination δ and NP theelevation of the north celestial pole above the horizon is the local latitude ϕ Bythe law of sines for spherical triangles angNMP is trigonometrically determinedby

sinangNMP =sinangMNP middot sin NP

sin PMAssuming that the hour-angle or nata converted to degrees of zenith distance ζ asdiscussed above is approximately the same asangMNP this equation is equivalentto a typical form of the rule for akṣavalana in Indian astronomy18

sin akṣavalana =sin ζ middot sin ϕ

cos δ

Note that if the body is at the zenith andor the latitude ϕ is zero there is noakṣavalana angle angNMP the akṣavalana increases with both zenith distance andlatitude The maximum akṣavalana for a given ϕ when δ = 0 and ζ = 90

is justϕ itself

The Karaṇakesarī rsquos akṣavalana table on the other hand requires a rather pecu-liar manipulation of the half-ζ computed in this verse the user is to subtract it

18 Variant forms of this rule also existedusing slightly different functions or rougher ap-proximations The differing formulas for valana

and the disagreements about it between rivalauthors are discussed in eg Montelle (2011246ndash247)



from 45 and use the result as the argument in the table which appears on f 5r

and continues onto f 5v Its first row the argument labeled natāṃśa or lsquodegreesof natarsquo runs from 0 to 45 The second row labeled valana is given in degreesminutes and seconds and runs from 22

35 39 to 0 The third column labeledantara lsquodifferencersquo gives the differences between successive entries in row 2

10 20 30 40

5

10

15

20

Figure 8 Plot of tabulated values of akṣavalana(dots) and the graph of arcsin(sin(90

minus ζ) middot sin ϕ)versus ζ2 = 0 to 45

for latitude ϕ = 22 35 39

(line)

These akṣavalana values progressbackwards with respect to the or-der one would expect from the akṣa-valana equation shown above for agiven ϕ and δ the akṣavalana oughtto increase as ζ increases As wecan see in figure 8 the values in thetable actually conform to the graphof a lsquoreversedrsquo akṣavalana functionarcsin(sin(90

minus ζ) middot sin(22 35 39)

where it is assumed that ϕ =22

35 39 and δ = 0 This andthe instruction tanaiḥ śodhya lsquodi-minished by 45rsquo inform us that theactual table argument running from

0 to 45 is 45 minus ζ2 in other words the table entries are given for every two de-

grees of zenith distance starting at 90 (when the akṣavalana attains its maximumvalue ϕ if δ is 0) and ending at 0

Bhāskararsquos final remark about directions can be justified as follows Whenthe eclipsed body is east of the meridian the equator-east direction skews northof the easterly direction parallel to the prime vertical so the akṣavalana is northWhen the body is west of the meridian on the other hand the akṣavalana is south

The awkwardness of the Karaṇakesarī rsquos presentation as a guide to actuallyusing the tables is indicated by the way the author of the worked example in MS Jhandles these calculations First he corrects for precession (f 2r line 4) the truesolar longitude that he found at the end of the procedure in the previous verseThis precession-corrected longitude is entered into a table described in verse 19to get the length of the half-day (f 2r line 5) Then he follows the directions inverse 15 to find the moment of first contact (f 2r lines 10ndash11 there is no periodof totality since the example concerns a partial eclipse) and determines the nataand its direction in accordance with the rule in verse 18 (f 2v lines 3ndash4) He thenproceeds to compute the valana as specified (f 2v line 11ndashf 3v line 4)



16 AYANAVALANA

yāmyottare diśi grahasya yutāyanāṃśāḥkoṭyāṃśakeṣu valanāntaraguṇyayuktam ||yogāntarāṃśapramite valanaṃ sphuṭaṃ syātsūryasya grāsavidhumokṣavilomadeyam || 6 ||In the south [or] north direction the precession-increased degrees of [the lon-gitude of] the planet [determine the ayanavalana in the same direction re-spectively] [The difference between the precession-increased degrees and thenearest table argument is] multiplied by the [tabulated] valana-difference andapplied to the degrees of the complement [of the precession-increased longit-ude] The valana [determined] in the amount of the degrees of the sum or thedifference [of the two components] should [now] be accurate [In the case of aneclipse] of the sun [the valana] is given in the reverse order of release [and firstcontact] from [the case of] the eclipsed moon

VERSE ANALYSIS

Metre vasantatilakāIn Bhāskararsquos prosody a consonant conjoined with a following lsquorrsquo appears

to be counted as a single consonant that does not make the preceding syllableheavy This can be seen in the present verse in eg graha in the first pāda andpramite in the third pāda each of which is preceded by a syllable that must stillbe considered as lsquolightrsquo to fit the vasantatilakā metre

TECHNICAL ANALYSIS

Like the last verse this verse does not explicitly specify the quantity it is com-puting in this case the ayanavalana or lsquodeflection due to tropical [longitude]rsquo Theayanavalana (angle angPprimeMP in figure 7) is the amount of deflection of the easterlydirection along the ecliptic from the easterly direction along the equator due tothe position of the bodyrsquos tropical longitude λT in its quadrant of the eclipticQualitatively when the body is at a solstice there is no deflection from this ef-fect since east is in the same direction along both great circles At an equinoxon the other hand the deflection is equal to the obliquity of the ecliptic (conven-tionally 24

) since the local directions of the two circles diverge by that amountQuantitatively the value of the ayanavalana angPprimeMP between those two ex-

tremes again by the law of sines for spherical triangles is given by

sinangPprimeMP =sinangPPprimeM middot sin PPprime

sin PM

The angle angPPprimeM is the complement of the (tropical) longitude of M while PMagain is the complement of its declination δ and the arc PPprime between the two



poles is the ecliptic obliquity The versersquos algorithm for using the table to findthe ayanavalana angPprimeMP simply prescribes adding the degrees of precession tothe degrees of the known (sidereal) longitude of the body to convert it to tropicallongitude and using the koṭi or complement of this result to interpolate linearlyin the table When the tropical longitude is between the winter and summersolstices (called the northern ayana) the ayanavalana is north because ecliptic-eastpoints northwards of equator-east conversely in the southern ayana it is south

The last part of this verse refers to the inverse symmetry between a lunar andsolar eclipse In a lunar eclipse the eastern part of the moonrsquos disk is eclipsedfirst so at first contact it is the western part of the shadow (the lsquoeclipserrsquo) that isbeing affected and the eastern part for release This is reversed in the case of asolar eclipse where the moon is the lsquoeclipserrsquo and the sun the lsquoeclipsedrsquo

There are three tables related to this verse on ff 6r and 6v The first is a tablegiving the ayanavalana at lsquothe time of contact [or] the time of releasersquo (which arethe chief moments for which the deflection is calculated) Of its three rows thefirst contains the argument running from 0 to 90 degrees which is evidentlythe complement of the precession-corrected longitude The second row containsthe corresponding values of the ayana-deflection from 0 0 3 when the argumentis 0 (ie when the body is at a solstice) to 24 0 1 (90

of longitude away at anequinox) The third row gives the differences between successive entries in thesecond row they change only at every tenth entry indicating that ayanavalanavalues were computed first for every tenth degree of argument and the remainingentries found by linear interpolation19

The other two tables on f 6v give the lsquocorrectedrsquo valana (spaṣṭavalana) whichis not mentioned in the verse and which despite its name does not actually intro-duce any new elements to give a more accurate value of the deflection Ratherit simply converts the combined deflection into a linear measure scaled appro-priately for the disks of the eclipsed and eclipsing bodies Consequently thereis one table for solar and one for lunar eclipses The arguments for both tablesrun from 0 to 47 which evidently represents the degrees of total valana as thealgebraic sum of akṣavalana (whose maximum for the Karaṇakesarī rsquos latitude isbetween 22 and 23 degrees) and ayanavalana (which as noted above can be asmuch as 24

) giving a possible combined maximum of 4720 The table entries

run from 0 7 to 8 2 digits for the sun and 0 4 to 13 51 digits for the moonMS Jrsquos procedure confirms this interpretation he adds his akṣavalana and

19 In fact the non-interpolated table entriesseem to conform very closely to Bhāskara IIrsquossimple expression for ayanavalana in Karaṇa-kutūhala 414ndash15 (Balacandra Rao and Uma2008 S81ndash82) which is equivalent to sin(90

minus

λT) middot 2420 See the rule for lsquocorrectedrsquo valana in eg

Karaṇakutūhala 416 (Balacandra Rao and Uma2008 S81ndash82)



ayanavalana components to get the argument he uses for the lunar corrected-va-lana table in the case of both contact (f 3r lines 1ndash2) and release (f 3v lines1ndash3)

17 REDUCTION OF ARCS TO THE FIRST QUADRANT

tryūnaṃ bhujaḥ syāt tryadhikaṃ ca ṣaḍbhātviśodhya bhārdhād adhikaṃ vibhārdham ||navādhikaṃ tad ravipātitaṃ cakoṭir bhavet syāt trigrahaṃ bhujonam || 7 ||

The bhuja should be [an arc] less than three [signs] moreover subtracting fromsix signs one [ie an arc] that is greater than three signs [or an arc] greater thana half-circle is diminished by a half-circle [or an arc] greater than nine [signs is]that taken away from 12 [signs] The koṭi should be three signs diminished by thebhuja

VERSE ANALYSIS

Metre upajātiThe grammar in this verse is particularly inconsistent in its parallel construc-

tions as frequently happens when Bhāskara tries to revamp a borrowed Karaṇa-kutūhala verse with some original phrasing (see below) However the intendedmeaning seems clear

TECHNICAL ANALYSIS

This verse gives a rule for producing a koṭi or complement from a bhujabāhu orgiven arc21 Most commonly in geometry the bhuja and koṭi are understood asperpendicular sides of a right triangle but generally in astronomical texts thebhuja is an arc eg of longitude This arc is conventionally reduced to a trigo-nometrically equivalent amount less than 90 or three zodiacal signs and thecomplement arc of this standard bhuja is the koṭi

This verse is not directed specifically toward the use of any particular tableRather it is a standard general procedure for reducing arcs to their canonicalforms in the first quadrant for which alone function values are tabulated Notethat Bhāskara in the immediately preceding verses has prescribed proceduresassuming the userrsquos ability to reduce arcs and now belatedly gives explicit dir-ections for it

21 Compare Karaṇakutūhala 24 (Mishra1991 20)tryūnaṃ bhujaḥ syāt tryadhikena hīnaṃ bhārdhaṃ

ca bhārdhād adhikaṃ vibhārdham |navādhikenonitam arkabhaṃ ca bhavec ca koṭistrigrahaṃ bhujonam ∥



18 POSITION OF NATA WITH RESPECT TO THE MERIDIAN

sūryagrahe divasam eva dinaṃ svakīyaṃrātris tathaiva śaśiparvaṇi vaipari syāt ||ahno rsquordhato niśidalendunataṃ pratīcyāṃrātridalād dyudalato rsquorkanatasya pūrve || 8 ||

In an eclipse of the sun the day is just its own day The night is likewise In alunar eclipse it should be reversed The [post-]midnight nata of the moon afterthe middle of [its own] day [ie with lsquonightrsquo and lsquodayrsquo reversed in meaning asspecified] is in the west From midnight to midday [the place] of the sunrsquos natais in the east

VERSE ANALYSIS

Metre vasantatilakāThe last pāda is defective the second syllable ought to be heavy for the metrersquos

sake

TECHNICAL ANALYSIS

This verse conveys rather simple astronomical facts about the nata or hour-anglezenith distance of the moon and the sun with respect to the meridian(here indicated by midday or midnight) lsquoIts own dayrsquo means the period of timewhen the body in question is above the horizon which is of course daytime inits usual sense for the sun but nighttime for the full moon And indeed for bothluminaries from lsquomidnightrsquo to lsquomiddayrsquo of its own day the body is to the eastof the meridian and from lsquomiddayrsquo to lsquomidnightrsquo it is west of the meridian be-cause of the westward daily rotation Again it might have been more intuitivefrom the userrsquos point of view to supply this information earlier namely beforeprescribing the calculation of the akṣavalana whose direction depends preciselyon this issue of whether the bodyrsquos nata is in the east or the west

19 PRECESSION LENGTH OF THE HALF-DAY

prabhābdhi 445 śakonāptakhāṅgo 60 rsquoyanāṃśāḥyuto rsquorkasya rāśyaṃśamāneṣu koṣṭhe ||dinārdhaviśodhyaṃ kharāmair niśardhaṃdinārdheṣṭanāḍyantaraṃ taṃ nataṃ syāt || 9 ||

445 subtracted from the [current] Śaka year [and] divided by 60 are the degreesof precession [That amount is] added to the amounts in signs and degrees ofthe sun[rsquos longitude] The subtracted [complement] of the half-day in the table



[entry] with [respect to] 30 [ghaṭikās] is the half-night [Whatever is] the differ-ence between the desired nāḍī [ie ghaṭikā] and the half-day that should bethe lsquohour-anglersquo (nata)

VERSE ANALYSIS

Metre bhujāṅgaprayāta (see verse 12) with minor deviations the third syllableof the first and third pādas is light instead of heavy

This unfamiliar bhūtasaṃkhyā numeration takes prabhā lsquolightrsquo or lsquoshadowrsquo tomean 45 The related compound prabhāvaka is elsewhere attested as meaning 8(Sarma 2003 66)

A comprehensible translation of this verse requires taking substantial liber-ties with its phrasing its literal sense is more like lsquo[The amount] which has 60 asdividing the difference between the [current] Śaka year and 445 are the degreesof precessionrsquo The mathematical meaning however is unmistakable

TECHNICAL ANALYSIS

This verse explains how to correct the solar longitude for precession or the differ-ence between sidereal and tropical longitude (which it would have been helpfulto know before calculating the ayanavalana requiring that correction) The yearŚaka 444 or 522 CE is frequently attested as the assumed date of zero precessionand the canonical rate of precession is taken as 1 degree in 60 years22 The dif-ference between 445 and the current Śaka year gives the integer number of yearselapsed between the end of 444 and the start of the current year each of whichproduces one additional arc-minute of precession

The resulting precession-corrected solar longitude in signs and degrees is theargument of the table on f 7r (see 22 below) that tabulates the sunrsquos lsquohalf-dayrsquo(dyu-dala) This means the length in ghaṭikās of half the seasonally varying periodof daylight (the solar form of the lsquoown dayrsquo previously invoked in 15)

110ndash11 GRAPHICAL PROJECTION OF ECLIPSE APPEARANCE

grāhyārdhasūtreṇa vidhāya vṛttaṃmānaikyakhaṇḍena ca sādhitāśam ||bāhye rsquotra vṛtte valanaṃ yathāśaṃprāk sparśikaṃ paścimataś ca mokṣam || 10 ||deyaṃ raveḥ paścimapūrvataś cajyāvac ca bāṇau valanāgrakābhyām ||utpādya matsyaṃ valanāgrakābhyāṃmadhyaḥ śaras tanmukhapucchasūtre || 11 ||

22 See for instance Pingree (1972 30ndash31)



When one has produced a circle by means of a [radius] string half [the size] ofthe eclipsed body and [another circle by a radius string equal in length to] the[half] part of the sum of their measures [ie diameters of eclipsed and eclipserthis is the circle of first contact (sparśa) and release (mokṣa)] with cardinal direc-tions established Now on the outer circle the valana is to be set according to[its] direction contact in the east and in the west the release [valana for a lunareclipse][and] in the case of the sun west and east [respectively] And having exten-ded two latitude [values] like a chord at the two extremities of the valana a fish[-figure is drawn with arcs centered] at the two extremities of the valana Themid-eclipse latitude is [laid off from the center] on its head-tail-line [ie the axisof the fish-figure]

VERSE ANALYSIS

Metre indravajrāThe first of these two verses is identical to Bhāskara IIrsquos Karaṇakutūhala 418

and the second nearly the same as Karaṇakutūhala 419 except for the syllableca instead of te at the end of its first pāda23 The word āśā can mean lsquoquartersof the compassrsquo this adverbial form āśam evidently means lsquorelating to cardinaldirectionsrsquo

TECHNICAL ANALYSIS

These verses and the two following give directions for producing the graphicalprojection of the eclipse The procedure does not refer directly to any of the datagiven in the tables as it is a practical description of how to represent the eclipseconfiguration using the quantities previously computed The following inter-pretation of these two verses is illustrated in figure 924 The author of the lunar-eclipse example in MS J briefly echoes the textrsquos instructions for this diagram(f 3v lines 4ndash8) but alas the manuscript preserves no drawing in illustration ofthem

The central circle represents the disk of the eclipsed body At the momentof sparśa or mokṣa the center of the eclipsing body is on the circumference ofa second larger concentric circle with radius equal to the sum of the radii of

23 Karaṇakutūhala 418ndash19 (Mishra 1991 60)grāhyārdhasūtreṇa vidhāya vṛttaṃmānaikyakhaṇḍena ca sādhitāśam |bāhye rsquotra vṛtte valanaṃ yathāśaṃprāk sparśikaṃ paścimataś ca mokṣam || 18 ||deyaṃ raveḥ paścimapūrvatas tejyāvac ca bāṇau valanāgrakābhyām |utpādya matsyaṃ valanāgrakābhyāṃ

madhyaḥ śaras tanmukhapucchasūtre || 19 ||24 As noted in the Verse Analyses for this

and the following group of verses this proced-ure is directly copied from that of the Karaṇa-kutūhala which in turn is a simplified version ofits counterpart in the same authorrsquos Siddhānta-śiromaṇi 526ndash32 (Śāstrī 1989 121ndash124)



N

S

W E

Figure 9 Initial circles and lines for the graphical projection of an eclipse

eclipsed and eclipser so the two bodies are tangent The trajectory between sparśaand mokṣa is determined by the values of the valana and lunar latitude

The local cardinal directions are laid out on this outer circle as shown in fig-ures 9 and 10 Note that this direction grid represents a left-right-reversed imageof the eclipsed body as it would appear in the southern part of the sky wherethe southward-facing observer would see east on the left hand and west on theright25

The amount of the appropriately scaled linear combined valana (see 16 above)for sparśa or mokṣa is marked off north or south of the appropriate extremity onthe east-west line The radii between those two points will not be exactly collin-ear unless the sparśa and mokṣa values of valana are exactly equal but they willindicate approximately the direction of the ecliptic with respect to local east-west

Then the corresponding values of the latitude are marked off from the valanapoints lsquolike a chordrsquo so that they fall on the circumference of the outer circle to in-dicate the positions of the center of the eclipser at sparśa and mokṣa (Presumablysince the possible values of latitude are small the fact that the lsquochord-likersquo line

25 See eg Sūryasiddhānta 612 viparyayodiśāṃ kāryaḥ pūrvāparakapālayoḥ lsquoa reversal ofthe directions of the east and west hemispheresis to be madersquo (Pāṇḍeya 1991 69) The graphical

projection shows how the eclipse would look ina mirror image eg in a tray of water or otherreflector set on the ground next to the diagram



N

S

W E

Figure 10 Depicting the three moments of contact mid-eclipse and release in the projection

segments representing them are not quite perpendicular to the approximate lineof ecliptic direction determined by the valana points is considered negligible)Then a lsquofish-figurersquo or perpendicular bisector to the ecliptic direction is construc-ted through the center of the circle and a line segment representing mid-eclipselatitude is laid off along it

112ndash13 GRAPHICAL PROJECTION CONTINUED

kendrād yathāśo rsquotha śarāgrakebhyovṛttaiḥ kṛtair grāhakakhaṇḍakenasyuḥ sparśamadhyagrahamokṣasaṃsthāathāṅkayenmadhyaśarāgracihnāt || 12 ||

mānāntarārdhena vidhāya vṛttaṃkendre rsquotha tanmārgayutidvaye rsquopi ||tamo rsquordhasūtreṇa vilikhya vṛttaṃsanmīlanonmīlanake ca vedye || 13 ||Bymeansof circles constructed fromacentre in the [appropriate] directionat thetips of the latitude [line segments] now [at that time] with [radius equal to] the[half] part [of the diameter] of the eclipser the forms [of the eclipser] at contactmid-eclipse and release should be [shown]

Now producing a circle with [radius equal to] half the difference of the amounts



N

S

W E

Figure 11 Depicting the beginning and end of totality

[of the diameters of the twobodies] one should draw [it] from the point of the tipof the mid-eclipse latitude Now [there are] two centres at the two intersectionsof that [circle] and the path [of the eclipser] [Upon] drawing a circle [at eachcentre] with radius [equal to that of] the eclipser the beginning of totality andend of totality too are to be known

VERSE ANALYSIS

Metre upajātiThese verses like the preceding ones closely track their counterparts in the

Karaṇakutūhala26 The Karaṇakesarī version has introduced a couple of unusualor awkward grammatical constructions such as the nominativeaccusative formof tamas lsquodarknessrsquo that we take to refer to the eclipser in a genitive sense andwhat appears to be an instance of the particle atha lsquonowrsquo more or less prepended

26 Karaṇakutūhala 420ndash23(Mishra 1991 61)kendrād yathāśaṃ svaśarāgrakebhyovṛttaiḥ kṛtair grāhakakhaṇḍakena |syuḥ sparśamadhyagrahamokṣasaṃsthāathāṅkayen madhyaśarāgracihnāt || 20 ||ādyantyabāṇāgragate ca rekhejntildeeyāv imau pragrahamuktimārgau |mānāntarārdhena vilikhya vṛttaṃ

kendre rsquotha tanmārgayutadvaye rsquopi ∥ 21 ∥bhūbhārdhasūtreṇa vidhāya vṛttesammīlanonmīlanake ca vedye |mārgapramāṇe vigaṇayya pūrvaṃmārgāṅgulaghnaṃ sthitibhaktam iṣṭam ∥ 22 ∥iṣṭāṅgulāni syur atha svamārgodadyādamūniṣṭavaśāt tadagre |vṛtte kṛte grāhakakhaṇḍakenasyād iṣṭakāle grahaṇasya saṃsthā ∥ 23 ∥



to śara lsquolatitudersquo instead of sva lsquoits ownrsquo to imply lsquothe now-latitudersquo or lsquolatitudeat that timersquo

TECHNICAL ANALYSIS

Figure 10 shows an arc drawn through the aforementioned three latitude pointsto mark the lsquopathrsquo of the eclipse and the disk of the eclipsing body centred ateach of them To find where the eclipsing body sits at the remaining two prin-cipal moments of the eclipse ie the beginning and end of totality the user isinstructed to draw a circle (shown shaded in figure 11) whose radius equals thedifference between the radii of eclipsed and eclipser The two points where thiscircle intersects the lsquopathrsquo mark the positions of the centre of the eclipser at thosetwo moments

iti candraparvādhikāraḥ ||

Thus the chapter on lunar eclipses

21 DIDACTIC APHORISM

śāstrāṇy anekāni mahārthasūtrā-ṇy anantavidyālpamatir jano rsquoyam ||kalau na dīrghāyur ato hi [grā]hyaṃtattvaṃ yathā kṣīravidhau ca haṃsaḥ || 1 ||

[There are] many treatises (śāstras) rules (sūtras) with great purpose [and] end-less knowledge this person [ie the speaker is] of small intelligence In the Kali-yuga life is not long therefore truth is to be grasped just as the swan [does] inthe act of [separating] milk [from water]

VERSE ANALYSIS

Metre upajātiWe have taken considerable liberties with the arrangement of the pādas and

especially with the reconstruction and translation of the third pāda in which thepenultimate syllable seems to be missing The final syllable looks in the manu-script more like hvaṃ than hyaṃ so the final word in the original might havebeen something quite different But the reference is clearly to the well-knownanalogy between the soul discerning valid knowledge from a mixture of truthand untruth and the proverbial ability of a goose or swan to separate out the



components of a mixture of milk and water27 The Kaliyuga or Indian lsquoIron Agersquois traditionally considered a time of degeneration and decay in which for ex-ample the standard human lifespan is much shorter than it is said to have beenin earlier ages

22 HALF-DAY AND OBLIQUE ASCENSION

sāyanārkabhalavādikoṣṭhakāḥyukte ceṣṭhaghaṭiyuktatatsamān ||koṣṭhakāṅkamitiṃ rāśipūrvakaṃlabdhalagnam ayanāṃśakonnatam || 2 ||

The table entries [for half-day and oblique ascension have as argument] theprecession-increased [longitude of the] sun in zodiacal signs and degrees andso on And when [the longitude in signs and degrees] is increased [by minutesuse the digits] corresponding to those [minutes] increased by the ghaṭikās [cor-responding to] the desired [argument these digits are] commensurate with thenumber in the table cell headed by [its] zodiacal sign The obtained oblique as-cension [has] the precession-increased [solar longitude] beginningwith zodiacalsigns [as its] argument (unnata)

VERSE ANALYSIS

Metre rathoddhatāAs usual with the verses that appear to be original to the Karaṇakesarī there

are a few metrical irregularities namely in pāda 2 the second syllable should belight as should the sixth syllable in pāda 3

TECHNICAL ANALYSIS

This verse apparently applies to two tables although it explicitly mentions onlythe second of them The first on f 7r tabulates half the seasonally varying lengthof daylight in ghaṭikās (see 19 above) and is titled atha sāyanaravirāśyoparidyu-dalaṃ lsquoNow [the length of] the half-day with argument the signs [and degrees]of the precession-corrected [longitude of the] sunrsquo Its values for each degree of

27 This simile is attested in Sanskrit liter-ature as far back as eg Mahābhārata 16910Smith (1999)prājntildeas tu jalpatāṃ puṃsāṃ śrutvā

vācaḥ śubhāśubhāḥ |guṇavad vākyam ādatte haṃsaḥ kṣīram

ivāmbhasaḥ ∥

But the direct inspiration for the Karaṇakesarī rsquosverse was apparently closer to the form foundin eg Garuḍapurāṇa 1684 (Wood and Subrah-manyam 1911 164)anekāni ca śāstrāṇi svalpāyur vighrakoṭayaḥ |tasmāt sāraṃ vijntildeānīyāt kṣīraṃ haṃsa ivāmbhasi ∥



the sunrsquos (tropical) longitude are arranged in double entry format zodiacal signsagainst degrees along the vertical axis are the signs numbered 1ndash11 followed by0 and along the horizontal degrees numbered 0ndash29 The maximum entry 16 48

ghaṭikās occurs at Cancer 0 and the minimum is 13 13 ghaṭikās in Capricorn 0

the equinoctial half-day is of course 15 ghaṭikās one-quarter of a full nychthem-eron or 60 ghaṭikās28

As noted above this table seems more relevant to computations involving thenata in the lunar eclipse chapter Its placement here may have been intended toallow the user to check whether the sun will be visible above the horizon at thetime of the predicted solar eclipse

The remaining instructions in the verse deal with computing the lagna usu-ally meaning the lsquohoroscopic pointrsquo or ascendant where the ecliptic intersectsthe eastern horizon but in this case evidently intended to signify the obliqueascension in ghaṭikās of the ecliptic arc between the sun and the vernal equinoxThe computation employs a two-part set of tables the first part on ff 7vndash8r isanother double entry table for each degree of (tropical) solar longitude with zo-diacal signs 0ndash11 on the vertical axis and degrees 0ndash29 on the horizontal The tab-ulated values begin with 0 at 0

of argument and increase to 60 ghaṭikās at 360

meaning that these oblique ascensions are cumulative The differences betweentabular entries for successive degrees within a given zodiacal sign are constantand their values are symmetrical about the equinoxes as shown in table 3

Zodiacal sign Constant difference (ghaṭikās)

AriesPisces 0736TaurusAquarius 0838GeminiCapricorn 01012CancerSagittarius 01120LeoScorpio 01118VirgoLibra 01056

Table 3 The constant differences between oblique ascension values for successive degrees

The other table in this set on ff 8vndash9v allows the user to make the oblique as-cension calculation precise for arcminutes of longitude it is somewhat mislead-

28 We assume that zodiacal sign 1 in thefirst row is Taurus and zodiacal sign 0 (thelast row) is Aries The half-day extreme val-ues imply that the so-called lsquomaximum half-equation of daylightrsquo ωmax ie the differencebetween half the equinoctial day and half thesolstitial day is 1 47 ghaṭikās We used this

quantity to confirm the value of the terrestriallatitude ϕ = 22 35 39

that we inferred fromthe akṣavalana table used in verse 15 pluggingit into the formula for the terrestrial latitudeϕ = arctan(sin ωmax tan ε) where ε is the ca-nonical ecliptic obliquity of 24

yields a valuefor ϕ of approximately 22 38



ingly titled atha lagnasya kalākoṣṭakā lsquoNow the tabular entries for the arcminutesof the oblique ascensionrsquo Actually in each of its six sub-tables (one for eachpair of zodiacal signs from AriesPisces to VirgoLibra) it is the argument thatis measured from 1 to 60 kalās or arcminutes within a degree not the tabulatedoblique ascension amounts

The table entries in sixtieths of a ghaṭikā are the corresponding incrementsadded by those extra arcminutes to the oblique ascension already computed forthe degree of tropical solar longitude specified by the previous tablersquos argumentAgain the differences between entries in each sub-table are constant the entriesfor successive arcminutes differ by just one-sixtieth of the previous tablersquos con-stant difference between entries for successive degrees of the appropriate zodi-acal sign (Thus for example every additional arcminute in a solar longitudefalling in the sign Aries adds 0 0 7 36 ghaṭikās to the corresponding oblique as-cension) By the combination of the table for degrees and the one for arcminutesthe accumulated oblique ascension can be determined for each of the possible21600 arcminutes of tropical solar longitude

The version of this procedure in the solar eclipse example in MS J starts byfinding the usual lagna or longitude of the ascendant and computing the longit-ude difference between that and the tropical solar longitude to give the argumentfor the algorithm in the following verse

23 LONGITUDINAL PARALLAX

darśāntakāle rsquorkatanoviśeṣaṃkāryaṃ tadaṃśamitispaṣṭhakoṣṭhe ||yallambanaṃ sve guṇakena guṇyaṃkhaveda 40 bhaktaṃ sphuṭalambanaṃ syāt || 3 ||

At the time of the end of the conjunction [tithi] the difference [of the longitudes]of the sun and the ascendant ( tanas) is to be computed Whatever longitudinalparallax (lambana) [corresponds to that difference] multiplied by the multiplierin its own true-table entry commensurate with the degree of that [ascendant]and divided by 40 should be the corrected longitudinal parallax

VERSE ANALYSIS

Metre upajātiIn pāda 2 the second syllable of aṃśa ought to be heavy and the second syl-

lable of mitispaṣṭa ought to be light for the sake of the metre We know of noother use of the word tanas as a synonym for lagna or ascendant but that is themeaning required by the procedure



TECHNICAL ANALYSIS

Figure 12 Plots of lsquomeanrsquo lambana in ghaṭikās against elongation λA minusλS in degrees in three formsthe trigonometric expression 4 sin(λA minus λS) (line) the short list of values in the Karaṇakutūhala(large dots) and the tabulated values in the Karaṇakesarī (small dots)

The first of the two tables referenced in this verse lists the lsquomeanrsquo longitudinalparallax or lambana in ghaṭikās for values of elongation between the sun and theascendant ranging from 0 to 91

The table entries start with 3 40 ghaṭikās at 0

maximising to 4 0 at 24 and decreasing to 0 1 at 91

The next table which givesthe laṃbanaspaṣṭaguṇakāḥ lsquomultipliers for correcting the longitudinal parallaxrsquohas as its argument the longitude of the ascendant (lagna) adjusted for precessionThis is a double-entry table with 0ndash11 zodiacal signs along the vertical and 0ndash29degrees along the horizontal The multiplierrsquos maximum is 35 59 (for Virgo 9ndash12

) and its minimum is 27 21 (for Pisces 29) According to the instructions in

the verse the multiplier is divided by 40 when applied to the lsquomeanrsquo longitudinalparallax to correct it

This application of table data roughly parallels the algorithmic approach toparallax calculation in Karaṇakutūhala 52ndash3 (Balacandra Rao and Uma 2008 S92ndashS94) which also begins with finding a so-called lsquomeanrsquo lambana This lsquomeanrsquo orapproximate lambana as in the corresponding Karaṇakesarī table depends onlyon the elongation of the sunrsquos position λS (which in a solar eclipse more or lesscoincides with that of the moon) from a given point on the ecliptic In the Karaṇa-kutūhalarsquos formula the reference point for elongation is the so-called nonagesimalwith longitude exactly 90

less than the longitude of the ascendant λA a bodyat the nonagesimal has no longitudinal parallax The lsquomeanrsquo lambana is just thesine of that sun-nonagesimal elongation linearly scaled from its minimum of 0when the sun and nonagesimal coincide to a maximum absolute value of 4 ghaṭi-kās when the sun is on the horizon However the Karaṇakutūhala also provides a



Z

ecliptic

S

P

V

E

N

equator

autumnal equinoxhorizon

meridian

sun andmoon

Q

Figure 13 Celestial hemisphere with the autumnal equinox at the ascendant and the nonagesimalon the meridian near the zenith

brief versified table of nine lsquomeanrsquo lambana values that do not exactly correspondto this relation but which apparently inspired the Karaṇakesarī rsquos tabulated valueson f 9v Figure 12 shows the two sets of values along with the curve representingthe scaled sine of the elongation

The lsquomeanrsquo lambana is subsequently corrected to its lsquotruersquo counterpart by tak-ing into account the contribution from the depression of the nonagesimal fromthe zenith Figure 13 illustrates a case where the nonagesimal V is at the summersolstice and consequently high in the sky near the zenith Z the small nonages-imal zenith distance ZV does not greatly affect the parallax due to the elongationbetween V and the sun Bhāskara IIrsquos correction represents this by multiplyingthe lsquomeanrsquo lambana by the cosine of ZV which will make the lambana zero if thenonagesimal V is on the horizon but leave the lsquomeanrsquo lambana unchanged if itis at the zenith The zenith distance arc ZV is taken as approximately equal tothe local latitude ϕ or ZQ diminished by the nonagesimalrsquos ecliptic declinationor VQ (Note that this combination usually only approximates the exact zenithdistance because the declination and latitude are measured along different arcsunless the nonagesimal the zenith and the celestial pole all fall on the same greatcircle as shown in figure 13) So this true lambana or longitudinal parallax cor-rection to the time of mid-eclipse is given by the equation

(true) lambana = 4 middot sin |λS minus (λA minus 90)| middot cos(ϕminus arcsin(sin 24

sin(λA minus 90)))

The individual lsquomultipliersrsquo in the second Karaṇakesarī table on f 10r whichmust be divided by 40 and then multiplied by the lsquomeanrsquo lambana have been com-puted to imitate this correction The graphs in figure 14 reproduce the lsquomeanrsquolambana values from figure 12 and also show their corrected versions for two



Figure 14 Plots of lsquomeanrsquo and corrected lambana in ghaṭikās against elongation λA minus λS in degreeswith ϕ = 22 35 39

for λA = 0 (left) and λA = 90

(right) the corrected quantity in bothtrigonometric (dashed line) and tabulated (dots) form is smaller than its uncorrected counterpart

sample values of λA at the Karaṇakesarī rsquos terrestrial latitude ϕ = 22 35 39 The

above trigonometric expression for true lambana is graphed as a dashed linewhile the Karaṇakesarī rsquos tabulated lambana values corrected by their appropri-ate tabulated multipliers are shown as small dots As λA increases from 0

andV gets closer to the zenith each corrected version of lambana approaches its ori-ginal lsquomeanrsquo value so that by about λA = 150

the lsquomeanrsquo and true versionsbecome indistinguishable on the graph



24 LONGITUDINAL AND LATITUDINAL PARALLAX

viśleṣas tribhyo rsquobhyadhikonakaś cettithyāntadā svarṇam idaṃ kramāt syāt ||lagnasya rāśyaṃśamiteṣu koṣṭhenatis tathā lambanako guṇo rsquosti ||4||

If the difference [between the longitudes of the sun and the ascendant] is greateror less than three [signs] at the end of the tithi this [longitudinal parallax] shouldbe [applied] positively or negatively respectively The latitudinal parallax (nati) isin the table [entry] among those [entries] determined by the signs and degreesof the ascendant Now the lambana is [to be] multiplied [see next verse]

VERSE ANALYSIS

Metre upajātiThe third syllable in pāda 1 ought to be light The rather odd construction

tithyāntadā in the second pāda appears to be intended as an adverb of time alongthe lines of tadā lsquothenrsquo

TECHNICAL ANALYSIS

Bhāskararsquos discussion of longitudinal parallax continues with a rule for applyingit to the time of conjunction If the luminaries are westward of the nonagesimal(roughly post meridian or more precisely farther than one quadrant away fromthe ascendant) it makes the conjunction later than initially calculated since theparallax depresses the moon toward the (western) horizon Likewise parallaxapplied to a position east of the nonagesimal makes the conjunction earlier Sothe longitudinal parallax in ghaṭikās calculated in the previous verse is appliedto the mid-eclipse time positively or negatively respectively

The other parallax component latitudinal parallax or nati accounts for thefact that the nonagesimal or midpoint of the visible semicircle of the ecliptic usu-ally does not coincide with the local zenith The parallactic depression of thenonagesimal toward the horizon is treated as a correction to the moonrsquos latitudenorth or south of the ecliptic

From the graph in figure 15 it seems almost certain that the Karaṇakesarī rsquosnati values in the table on f 10v whose argument is the sign and degree of thelongitude λA of the ascendant were derived from a formula in Karaṇakutūhala53 (Balacandra Rao and Uma 2008 S92ndash93) In this rule the ecliptic declinationof the nonagesimal is again arithmetically combined with the local latitude ϕto produce the nonagesimalrsquos approximate zenith distance whose sine is then



Figure 15 Plot of nati in digits (northward positive southward negative) against λA in degreesshowing the Karaṇakesarī rsquos tabulated values of nati at the initial degree of each sign (dots) and theKaraṇakutūhalarsquos nati function with ϕ = 22 35 39

(line)

linearly scaled to give the nati in digits

nati = 1

8

middot 13

12

middot 120 sin(arcsin(sin 24 sin(λA + 90

))minus ϕ)

Qualitatively we can understand the behavior of this function by recollectingthat the nonagesimal is the point where longitudinal parallax is zero so the lat-itudinal component of parallax or nati depends only on how much (and in whichdirection) the nonagesimal is depressed from the zenith

Note that in the above expression subtracting ϕ from the declination meansthat the zenith distance ZV will be negative if the nonagesimal V falls south ofthe zenith and positive if it is north (see figure 13) Consequently the nonages-imal will approximately occupy the zenith point and the latitudinal parallax willvanish when the northern declination VQ is equal to the local terrestrial latitudeϕ = ZQ here about 22 35 39

This value is not far from the maximum declin-ation of 24

that the nonagesimal reaches in the southern or northern directionwhen it falls at the winter or summer solstice respectively while the ascendantoccupies the following equinox In other words when the ascendant falls a littlebefore or a little after the autumnal equinox at tropical Libra 0

we should expectthe nati for this latitude to be minimised

Indeed the values in the Karaṇakesarī rsquos nati table reach their minimum of0 1 digits when the argument λA is Virgo 12

or Libra 18 between these two

positions the nati is to be applied in the northern direction (because then thenonagesimal is slightly north of the zenith) although it is a southward correc-tion everywhere else Conversely the natirsquos maximum (absolute) value of 11 46



digits occurs when the ascendant is around the vernal equinox and the nonages-imal around the winter solstice producing the largest possible depression of thenonagesimal south of the zenith29 Most of the other nati table entries appear tohave been linearly interpolated between trigonometrically computed values at15-degree intervals

The final reference to the longitudinal parallax goes with the subsequentverse

25 ADJUSTING ECLIPSE DATA FOR PARALLAX

viśva 13 ghnalambanakalās tithivad yutona-pātācca kāṇḍamataṣaḍguṇalambanāṃśaiḥ ||yugyaṃ vilagnatanatiś ca śarau vidadyātspaṣṭo bhavec ca viśayāt sthitichannasādhyam || 5 ||

[There are] arcminutes [produced by] the lambana multiplied by 13 and [bythem the elongation of the sun] from the node [is] increased or decreased [ac-cording] as [the time to the end of] the tithi [is increased or decreased by thelambana to give the moment of apparent conjunction] By the degrees [pro-duced by] the lambana considered as a [separate] quantity [and] multiplied by6 [the longitude of the nonagesimal] is to be increased [or decreased respect-ively] One should apply the nonagesimal-[derived] nati to the [lunar] latitudeand [it] should be correct The determination of the half-duration and the ob-scuration is [derived] from [the corrected time of] the middle [of the eclipse]

VERSE ANALYSIS

Metre vasantatilakāOur translation understands nati in the accusative in the third pāda which is

grammatically incorrect but represents our best guess at the sense The unusualterm vilagnata if we have not misread or misinterpreted it seems to be inten-ded as a sort of portmanteau word implying the correction of the nonagesimal(vilagna) by the lambana as well as the recalculation of its nata or zenith distanceand its resulting new nati value

TECHNICAL ANALYSIS

This and the following verse rather sketchily describe a procedure evidentlymodeled on that of the Grahalāghava for correcting the eclipse based on the com-

29 If ϕ were zero the nati would fluctuatesymmetrically between zero and equal northand south extremes of about 66 digits corres-

ponding to the maximum nonagesimal depres-sion of 24

north or south (see section 2 for therelation between arcs and digits of parallax)



puted values of the parallax components30 The flowchart in figure 16 illustratesthe main sequence of steps in such a procedure starting with the time of trueconjunction and the positions of the bodies at that time These quantities andthe corresponding longitudinal parallax are computed in turn until they becomefixed at the time of apparent conjunction The resulting latitudinal parallax cor-rects the mid-eclipse latitude and magnitude and consequently the half-durationof the eclipse A similar cycle of correction is then applied to each half to find thepositions and times of the apparent contact and release

The present verse starts the Karaṇakesarī rsquos version of this process by comput-ing a quantity more or less equivalent to the change in arcminutes in the moonrsquoslongitude between true and apparent conjunction Specifically the lambana inghaṭikās multiplied by 13 is taken as arcminutes applied to the elongation of thesun from the node positively if the lambana in ghaṭikās is positive (ie increas-ing the time to conjunction) and negatively if it is negative This scale factor isqualitatively justified by recollecting that the sunrsquos position at the time of a solareclipse is nearly coincident with the moonrsquos and if the moon travels approxim-ately 13 degrees per day then it moves approximately 13 arcminutes per ghaṭikāOf course strictly speaking the lambana corresponds to an arc of time-degreesor right ascension along the equator rather than to elongation along the eclipticbut this distinction is disregarded for the sake of convenience

Meanwhile the original lambana in ghaṭikās is also converted to degrees bymutiplying it by 6 Again this quantity is applied as though it were an arc oflongitude to correct an ecliptic position in this case the longitude of the non-agesimal From the new position of the nonagesimal a new value of nati is tobe computed which then requires recomputing the times of contact and releaseand the extent of the obscuration

30 The key verses are Grahalāghava 63ndash5(Balacandra Rao and Uma 2006 S394)trikunighnavilambanaṃ kalās tatsahitonas

tithivad vyaguḥ śaro rsquotaḥ |atha ṣaḍguṇalambanaṃ lavās tair

yugayugvitribhataḥ punar natāṃśāḥ || 3 ||daśahṛtanatabhāgonāhatāṣṭendavastad

rahitasadhṛtiliptaiḥ ṣaḍbhir āptāsta eva |svadigiti natir etatsaṃskṛtaḥ so rsquoṅgulādiḥ sphuṭa

iṣuramuto rsquotra syāt sthiticchannapūrvam || 4 ||sthitirasahatiramśā vitribhaṃ taiḥ pṛthaksthaṃ

rahitasahitamābhyāṃ lambane ye tu tābhyām |sthitivirahitayuktaḥ samskṛto madhyadarśaḥkramaśa iti bhavetāṃ sparśamuktyos tu kālau ||5||Compare the Karaṇakutūhala counterpartquoted in the discussion of the followingverse and the worked example in BalacandraRao and Uma (2008 S99ndashS108) As noted inBalacandra Rao and Venugopal (2008 68) thesolar eclipse methods in the Karaṇakutūhalaand Grahalāghava give quite similar results



Time of

conjunction

Longitudinal

parallaxLongitudes at

conjunction

Latitudinal

parallax

Latitude

Half-

duration

Time of

release

Time of

contactLong

parallax

Contact

half-

duration

Release

half-

durationLat

parallax

Longitudes

Latitude

Longitudes

Lat

parallax

Latitude

Long

parallax

Figure 16 The cycles of parallax correction to get from the time of true conjunction to the momentsand positions of apparent conjunction contact and release

26 ADJUSTING ECLIPSE DATA FOR PARALLAX CONTINUED

sthitiśūlasūta 6 ghnalavonayuktāḥpṛthaksthatanvoḥ kṛtalambanaṃ svam ||ṛṇaṃ sthitihīnayuter vidarśetataḥ sphuṭasaṃspṛśi[mo]kṣikālau || 6 ||

[The degrees of the nonagesimalrsquos longitude] are [separately] diminished andincreased by the degrees [produced by] multiplying the half-duration by 6 For[each of] the two separate results [ tanu] the lambana [is] made [and applied]additively [or] subtractively [to thehalf-duration to get the interval betweenmid-eclipse and release and the interval between contact and mid-eclipse respect-ively] When themid-eclipse [time] is decreasedor increasedby the [correspond-ing] half-duration then the two times of true contact and release [result]



VERSE ANALYSIS

Metre upajātiThe metre and sense of this verse as transcribed here are both far from sat-

isfactory but it seems to borrow some of its phrasing not only from the above-mentioned Grahalāghava verses but also from the Karaṇakutūhala31

As reconstructed in upajāti metre the first syllable makes the first pāda hy-permetric and the fourth syllable in each of the last two pādas ought to be heavyinstead of light but neither the manuscript nor the sense gives any pretext forenforcing corrections We are not sure how to interpret śūlasūta to supply the re-quired content of a word-numeral meaning six Perhaps this form is a corruptionof the reading śūlisuta 6 lsquoson of the spear-bearerrsquo quoted twice (f 5r line 11 andf 5v line 4) in the worked example of MS J we assume this alludes to the six-headed deity Skanda but we know of no other instance of his name invoked as aword-numeral The word tanu in presumably the genitive dual must mean theadjusted nonagesimal longitude although we are not familiar with this meaning(compare tanas for lsquoascendantrsquo in verse 23 above)

TECHNICAL ANALYSIS

Although the versersquos grammar is very hard to follow there seems no doubt thatit is attempting to continue describing the above Grahalāghava-inspired proced-ure Namely the corrected half-duration in ghaṭikās is converted to degrees bymultiplying it by 6 and then as before that time-degree arc is treated as an arc ofthe ecliptic applied negatively or positively to the longitude of the nonagesimalto give approximate positions of the nonagesimal at the moments of contact andrelease respectively A new lambana is computed for each of those nonagesimalpositions and the time of half-duration is adjusted separately by each of thoselambana values to give the actual duration of each lsquohalfrsquo of the eclipse These twodistinct lsquohalfrsquo durations no longer equal halves of the total eclipse duration givethe moments of contact and release when applied with the appropriate sign tothe moment of mid-eclipse

This completes the Karaṇakesarī rsquos determination of the times of the beginningmiddle and end of a solar eclipse Indefinite cycles of iterative recalculation arediscarded in favour of a few specified corrections and re-corrections Nor are

31 Karaṇakutūhala 56ndash8(Mishra 1991 68ndash69)spaṣṭo rsquotra bāṇo natisaṃskṛtaḥ syāc channaṃ

tataḥ prāgvad ataḥ sthitiś ca |sthityonayuktād gaṇitāgatāc ca tithyantato

lambanakaṃ pṛthakstham ∥ 6 ∥svarṇaṃ ca tasmin pravidhāya sādhyas tātkālikaḥ

spaṣṭaśaraḥ sthitiś ca |tayonayukte gaṇitāgate tat svarṇaṃ pṛthaksthaṃ

muhur evam etau ∥ 7 ∥syātāṃ sphuṭau pragrahamuktikālau sakṛt kṛte

lambanake sakṛtsnaḥ |tanmadhyakālāntarage sthitī sphuṭe śeṣaṃ

śaśāṅkagrahaṇoktam atra hi ∥ 8 ∥



there specific directions stated for computing the times of beginning and end oftotality if any

The example in MS J appears to follow these directions faithfully finishingwith the sum of the disksrsquo radii (presumably for the purpose of an eclipse dia-gram) and identifying the eclipse lsquolordrsquo (parveśa) as Brahmā

27 CONCLUSION

asti vaiṣṇavadhāmni sajjanavati saudāmikāhve pureśrautasmārttavicārasāracaturo moḍho hi rāmāhvayaḥ ||jyotirvittilakopamanyava iti khyātaḥ kṣitau svair guṇaiḥtatsūnuḥ karaṇākhyakeśarim imaṃ cakre kavir bhāskaraḥ || 7 ||

In the city called Saudāmikā filled with good people in the Vaiṣṇava clan thereis one called Rāma aMoḍha learned in the essence of investigations of śrūti andsmṛti a [member of the] Aupamanyava [gotra who is] an ornament of the jyoṭiḥ-knowers renowned on the earth for his own good qualities His son Bhāskarathe poet wrote this [work] the Karaṇakesarī

VERSE ANALYSIS

Metre śārdūlavikrīḍita with minor discrepancies the second and twelfth syl-lables of the first pāda should be heavy rather than light

Bhāskararsquos idiosyncratic method of compounding appears to be at work inthe last compound where the name of the text is unpacked as lsquoKeśari calledKaraṇarsquo instead of the other way around The term moḍha seems to refer to aBrahmana group primarily associated with the Gujarat region (which is con-sistent with the terrestrial latitude used in the Karaṇakesarī) but we have noother information on Bhāskararsquos location or any identification of the placenameSaudāmikā with a modern locality

iti śrīdaivajntildearāmātmajabhāskaraviracite karaṇakeśariye sūrya-parvādhikāraḥ || saṃpūrṇo rsquoyaṃ granthaḥ || śubhaṃ lekhakapāṭhakayoḥ ||śubhaṃbhavet kalyāṇam astu || saṃ 1819 varṣe śāke 1684mitī aśvinaśudi 14śanau dine lipīkṛtaṃ ||

Thus the chapter on solar eclipses in the Karaṇakesarī composed by Bhāskarathe son of RāmaDaivajntildea This book is complete Good fortune to thewriter andthe readermay there be good fortunemay there beprosperity [Themanuscriptwas] written in the year Saṃvat 1819 Śaka 1684 on the 14th [tithi] of the bright[fortnight] of Āśvina on Saturday [ie Saturday 2 October 1762 CE]



ACKNOWLEDGEMENTS

We are indebted to the authors and maintainers of Davide P Cervonersquos Ob-jectTEX and CritTEX macro packages the Ledmac macro package and

Frans Velthuisrsquos devnag preprocessor at sarovarorg in conjunction with theTEXLive distribution for enabling us to produce the critical edition and accom-panying commentary Our thanks are also due to Michio Yano for his Pancangacalendar conversion program and to the developers of the Inkscape and Sageopen source software tools supporting graphics and computation as well as tothe typesetters for their elegant and meticulous work We are especially grate-ful to Anuj Misra our doctoral student at the University of Canterbury for hisefforts in procuring copies of manuscript materials

The libraries of Brown University and Columbia University kindly suppliedimages of manuscript copies (from the John Hay Libraryrsquos David E Pingree Col-lection and the Smith Indic Collection respectively) as did the Rajasthan Ori-ental Research Society (Jodhpur and Jaipur) We wish to thank the Royal Societyof New Zealand for their generous support Agathe Keller for the initial inspira-tion and the anonymous reviewers of the paper for their helpful corrections andcomments

REFERENCES

Apte Vaman Shivaram 1970 The Studentrsquos Sanskrit-English Dictionary 2nd edDelhi Motilal Banarsidass

Athavale R B (ed) 1981 Kavīndrakalpadruma of Kavīndrācārya Sarasvatī AsiaticSociety of Bombay Monograph No 6

Balacandra Rao S and Uma S K 2006 Grahalāghavam of Gaṇeśa Daivajntildea AnEnglish Exposition with Mathematical Explnantion Derivations Examples Tablesand Diagrams New Delhi Indian National Science Academy Reprinted fromIndian Journal of History of Science 41(1ndash4) (2006)

Balacandra Rao S and Uma S K 2008 Karaṇakutūhalam of Bhāskarācārya II AnEnglish Translation with Notes and Appendices New Delhi Indian National Sci-ence Academy

Balacandra Rao S and Venugopal Padmaja 2008 Eclipses in Indian AstronomyBangalore Bharatiya Vidya Bhavan

Balacandra Rao S and Venugopal Padmaja 2009 lsquoEffects of Moonrsquos Parallaxlambana and nati in Indian AstronomymdashA Studyrsquo Gaṇita Bhāratī 31 57ndash81

Chatterjee Bina 1981 Śiṣyadhīvṛddhida Tantra of Lalla with the commentary ofMallikārjuna Sūri 2 vols New Delhi Indian National Science Academy

Dvivedī Sudhākara 1901ndash1902 Brāhmasphuṭasiddhānta (The Pandit NS 23ndash24)Benares Government Sanskrit College



Jośī Kedāradatta 2006 Gaṇeśadaivajntildearsquos Grahalāghavam 5th edition DelhiMotilal Banarsidass

Knudsen Toke and Plofker Kim 2011 lsquoCalendars in Indiarsquo in Calendars andYears II Astronomy and Time in the Ancient and Medieval World ed John M SteeleOxfordOakville Oxbow Books pp 53ndash68

Mehra Anjani Kumar 1982 lsquoSolar Eclipse Theory and Observations in the 18thCentury Indiarsquo Indian Journal of History of Science 17(2) 252ndash259

Mishra Satyendra ed 1991 Karaṇakutūhala of Bhāskarācārya Varanasi Krish-nadas Academy

Montelle Clemency 2011 Chasing Shadows Baltimore Johns Hopkins Univer-sity Press

Pāṇḍeya Śrīcandra ed 1991 Sūryasiddhānta Varanasi Sampurnanand SanskritUniversity

Pingree David 1968 lsquoSanskrit Astronomical Tables in the United Statesrsquo Trans-actions of the American Philosophical Society New Series 58(3) 1ndash77

Pingree David 1970ndash1994 Census of the Exact Sciences in Sanskrit series A vols1ndash5 Philadelphia American Philosophical Society

Pingree David 1970 lsquoOn the Classification of Indian Planetary Tablesrsquo Journalfor the History of Astronomy 1(2) 95ndash108

Pingree David 1972 lsquoPrecession and Trepidation in Indian Astronomy beforeAD 1200rsquo Journal for the History of Astronomy 3 27ndash35

Pingree David 1973 Sanskrit Astronomical Tables in England Madras Kup-puswami Sastri Research Institute

Pingree David 1981 Jyotiḥśāstra Wiesbaden Otto HarrassowitzPlofker Kim 2009 Mathematics in India Princeton Princeton University PressSarma K V 2003 lsquoWord and Alphabetic Numerical Systems in Indiarsquo in The

Concept of Sunya ed A K Bag and S R Sarma New Delhi ignca insa ampAryan book International pp 37ndash71

Sastri Bapu Deva 1861 Translation of the Sūryasiddhānta and the SiddhāntaśiromaṇiCalcutta Baptist Mission Press

Śāstrī Bāpū Deva ed 1989 The Siddhāntaśiromaṇi a Treatise on Astronomy 2nd edrev Gaṇapati Deva Śāstrī (Kashi Sanskrit Series 72) Varanasi ChaukhambhaSanskrit Sansthan

Sastry R Ananta Krishna ed 1921 Kavīndrāchāryarsquos List (Gaekwadrsquos OrientalSeries XVII) Baroda Central Library

Smith John C ed 1999 The Electronic Text of the Mahābhārata httpbombayindologyinfomahabharatastatementhtml

Sternbach Ludwik 1974 Mahāsubhāṣitasamgraha Being an extensive collection ofwise sayings in Sanskrit critically edited with Introduction English translation Crit-ical Notes etc Volume I Hoshiapur Vishveshvaranand Vedic Research Insti-tute



Tubb Gary A and Boose Emery R 2007 Scholastic Sanskrit A Manual for Stu-dents New York Columbia University

Wood Ernest and Subrahmanyam S V 1911 Garuḍapurāṇa AllahabadBhuvaneswarī Āśrama

APPENDIX A GLOSSARY OF TECHNICAL TERMS

akṣavalana The component of valana or directional deflectioninclination thatdepends on the akṣa or local terrestrial latitude

aṅgula Literally lsquodigitrsquo (in the sense of finger-breadth) a linear measure appliedto quantities like disk diameter and lunar latitude taken as equivalent tothree arc-minutes

avadhi A period of fourteen mean solar days

ayanavalana The component of valana that depends on the tropical longitudeof the bodies

bāhu bhuja Arc or angle usually referring to some table argument measured inunits of arc

ghaṭikā One-sixtieth of a day equal to twenty-four minutes

karaṇa Usually a brief astronomical handbook in verse (may also mean one-halfof a tithi)

kalā One-sixtieth of a degree or arc-minute

koṭi The complement of a given arc or angle (bāhu or bhuja)

koṣṭha koṣṭhaka A numerical table or an entry in such a table

lagna Ascendant the point of intersection of the ecliptic and the local horizonin the east

lambana Longitudinal component of parallax usually measured in time-units

madhya The moment of mid-eclipse

mokṣa The moment of release or end of eclipse when the eclipsed body is nolonger obscured

nāḍī A synonym for ghaṭikā



nāgarī The most common North Indic script in which these manuscripts of theKaraṇakesarī are written

nata Literally lsquodepressionrsquo or angular distance from an apex used both forzenith distance and for hour-angle separation from the meridian

nati Latitudinal component of parallax usually measured in digits

nimīlana The moment of immersion or beginning of totality

parvan Either an eclipse itself or the instant of syzygy when mid-eclipse occurs

pāda One line of a (usually) four-line Sanskrit verse

sāyana Precession-corrected referring to tropical longitude

sanmīlana A synonym for nimīlana

sparśa The moment of first contact between eclipsed and eclipsing bodies

tana Used idiosyncratically in the Karaṇakesarī to mean ascendant

tithi One-thirtieth of a synodic month

unmīlana The moment of emersion or end of totality

upari Literally lsquoaboversquo but in this case referring to the argument at the top of atable

valana lsquoDeflectionrsquo or lsquoinclinationrsquo ie angle between the local east-west direc-tion and the eastward movement of the moon in an eclipse

vilagna Nonagesimal the point of the ecliptic 90 west of the ascendant



B Critical Edition of the text

ौीगणशाय नमः ौीगो नमः ौीसारदाय नमः अथ करण- f [1]v Bf 1v R1

कशरी िलत

ौीकचचरण िणप भाोितिव दा बिवदामिभवन च

5 का कवीकलभषणभाराोरामाजः करणकशिरमातनोित १

शको रामिवपदा १६०३ हीनोिवयिामचतो १३० लशषौयतौ चोपकण वित त-

10 पातताािलको बाभागः २

िमतः कोकरलािदः शरः ात सपात भपिर हिलिबमितथमा नघोपरी चभभ तलाजािदषोनयक कभा ३

15 छाछादकमडलाितदल काडोनछ पनःछ मािवविज त त िनिखल मासऽिप खछकम छािम तकोक िितभ वद खछतःौीभानोदयाथासमयात पवा ममहः ४

1ndash2 ौीगmdashिलत ] ौीगो नमः ौीसारदाय नमः B अथ कणकशिर िल-त ोक R1 3 ौीब R1 नीप R1 4 ोतीिव दा बिवदामिवभ जन चः R1

5 कविकलभषण R1 7 िवप दाग १६०३ हीनो B िवपदागिहनो १६०३ R1 8 चि१३० तो R1 9 यतो चोपकण R1 त om R1 10 सपात R1 11 गलािदः शरः ]गलािदशरः B गलािदसर R1 12 सपातभपिर R1 13 घोपिरिभभा B R1

14 कभा R1 15 छाः छाद B R1 का ढोनछ B काडोनछन R1 15ndash16 पनछ R1

16 िनिखलमा B कछनक R1 17 िितभ वद ] िितभव B ितभवमडR1 खछतः ] खछतो B R1 18 दायाथासमयात B दयाथासमयािा R1 पवाम B पचम R1



िा मदन हीनः शसीलन ःमदिाितऽििल मोसमखािन ९० नना घॐमान नताशा-ानः ४५ शोाशको ार सौया ५

5 याोर िदिश मह यतायनाशाःकोाशकष वलनारगययम योगाराशिमत वलन ट ात सय मासिवधमोिवलोमद |यम ६ f [2]r B

न भजः ािध | क च षात f 2r R1

10 िवशो भाधा दिधक िवभाध मनवािधक तििवपाितत चकोिटभ वािमह भजोनम ७

सय मह िदवसमव िदन कीयरािऽथव शिशपविण वपिर ात

15 अोऽध तो िनिशदलनत तीारािऽदलादलतोऽकनत पव ८

भाि ४४५ शकोनाखाो ६०ऽयनाशाःयतोऽक रायशमानष को

1ndash2 हीनशसीलन ः मिाित B यः शसिलनम िन R1

2 ऽििल ] िनिल B तिमीलन R1 3 खाकिनR1 ९० om R1 नताR1

4 ४५ om B सोसको R1 सौ ] सो R1 5 यामोर िदशी R1 6 कोाशकष ]कोशकष R1 गय ] गय R1 7 िमत] मीत R1 8 मासिवध ] मासिवध Bमासिवध R1 9 न] न B R1 भजः ] भजः R1 ािधक च ] ात िधकन R1

षात ] षा B 10 भाधा दिधक ] भाा दिक B 11 नवाधीक R1 12 कोिटभ वाि ]कोिटभ वाि७ B कोटीभवाी R1 भजोनम ] भजोन R1 13 िदवसमव ] िविदवशएव R1 िकय R1 14 राऽीथव शसीपव णी R1 15 िनिश ] िनशी R1 तीा ] चाा R1 16 रािऽदलादलतो B राऽीदलादलतो R1 17 भाि ] भाि R1 खाो ]खाग R1 ६० om B 18 यतोऽक ] यताक R1 ऽक ] का B रायश ]राश R1 को ] कोटः R1



िदनाध िवशो खरामिन शाधिदनाधनार त नत ात ९

मााध सऽण िवधाय वमानखडन च सािधताशम

5 बाऽऽ व वलन यथाशाक िशक पिमत मोम १०

दय रवः पिमपव ताव बाणौ वलनामकाामउा म वलनामकाा

10 मः शरखपसऽ ११

काथाशोऽथ शरामकोवः कतमा हकखडकनः शममहमोसाअथाकयशरामिचात १२

15 मानाराधन िवधाय वकऽथ ताग यितयऽिप तमोऽध सऽण िविल वसीलनोीलनक च व १३

इित चपवा िधकारः

1 िदनाध िवशो ] िदना वीसो R1 खरामिनशा R1 2 िदनाध ] दीना R1 नतात R1 3 मााध सऽण ] मामाणसदजन R1 4 मानखडन च सािधताशम ] मा-नाितच दलन च सािधताशा R1 5 व वलनयथाश ] ववजनिदय यथाशाR1 वलन]ववलन B 6 िशक पिमत मोम ] शमोमपर रवीदय R1 verses 11ndash12व-लनामका शराबतदयपव वलनामकाच जखलमखासरासा बाणा क िकतमाहकखडकन स मह मोशरामिचात ११ R1 7 पव त ] पव ता B 10 मः ] म B प ] पछ B 12 वः ] व B 15 िवधाय ] िविवधाय B व ] व R1 16 ऽिप ]पीः R1 17 सऽण िवल व R1 18 च व १३ ] च - - - वध १२ R1 19 इितचपवा िधकारः ] इती चिपव समा R1



शाायनकािन महाथ सऽा-यनिवामितज नोऽयमकलौ न दीघा यरतो िह [मा]त यथा ीरिवधौ च हसः १

5 |सायनाकभलवािदकोकाः f [2]v B

य चघिटयतमान कोकािमित रािशपव कललमयनाशकोतम २

दशा कालऽकतनोिवशष10 काय तदशिमितको

य |न गणकन गय f 2v R1

खवद ४० भ टलनात ३

िवषिोऽिधकोनकत ितादाणिमद बमाात

15 ल रायशिमतष कोनितथा लनको गणोऽि ४

िव १३ लनकलाििथवतोन-पाता काडमतषणलनाशःयय िवलतनित शरौ िवदात

verse 1 om R1 1 महाथ ] माहाथ B 3 िह मा ] िह B 4 हसः ] हस B5 सायनाक ] x x नाकB 6 यचघटीयतान R1 घिट ] घटीB 7 िमित ]त R1 रासीपव क R1 8 ललम R1 कोनतम B किनत R1 २ ] १ R1

9 दशा काल ] x x तकाल B दशा तकाल R1 ऽकतनोिवष B कतनोव शष R1 fol-lowed by two akṣaras crossed out B 10 काय च तदशिमतकोः R1 11 यबन B य |बन R1 गय ] गय R1 12 ४० om R1 ३ ] २ R1 13 िवषिो ]िवषिऽो B R1 नकत ] नक R1 14 िततदाणमीद बमात R1 15 लराशश R1 16 गणो ] ग णो R1 ४ ] ३ R1 17 १३ om R1 वतोन ] वतोन R1

18 काडकतखण R1 लनश B 19 यय ] यगयम with overline on म Note inmargin ग B ययग R1 िवलतनित ] िवलतनित R1 शरौ ] शरो B शर R1



ो भव िवशयात िितछसाम ५

िितशलसत ६ लवोनयाःपथतोः कतलन-मण िितहीनयतिव दश

5 ततः टसिश[मो]िकालौ ६

अि ववधाि सनवित सौदािमका परौौता िवचारसारचतरो मोढो िह रामायःोितिव िलकोपमव इित ातः ितौ ग णःतनः करणाकशिरिमम चब किवभा रः ७

10 इित ौीदवरामाजभारिवरिचतकरणकशिरय सय पवा िधकारःस-पणऽय मः

MS B concludes

शभ लखकपाठकयोः शभ भवत काणम स १८१९ वष शाक१६८४ िमती अिनशिद १४ शनौ िदन िलपीकत

15 MS R1 concludes

सयोम हणष गाजलवासोपमा वाडवाःशः ानमथारा हवनक ामानाय ण [ ]राऽाविशत िपतयजन चोपराग तान दानमथायो िनगिदतो सावथयौ नः सित १

1 भव िवशयात ] भवत िविशषात R1 ५ ] ४ R1 2 ितीशलीसत R1 ६ om R1

3 तोः ] तनो R1 लबन R1 3ndash4 मण ] मण B 4 हीनयतिव ] हीचयत-िव B िहनयतिव R1 5 ततः ] भवतः R1 िशमोि ] िशि B शमि R1

६ ] ५ R1 verse 7 om R1 6 सनवित ] सनवित B सौदािमका ] सौदिमकाB 8 ितिव िलको ] ितिव लको B ातः ] ातो B ग णः ] गणः B10ndash11 ईती कण कशिरमथ सय पव समा R1 11 मः ] मथः १ B 13 भवत ]भवत B काणम] काणमः B 14 अिनशिद ] अनसिद B 16 सयौ R1

17 ानमथातराहबनक R1 ामानाणः R1 18 तः R1 19 न सित R1



सवनािप कत ौा वरायदशन अकव न नािोक गौिरव सीदत २

दािसदाससतो वा यिद भवित कलािऽपऽोऽिप भः कया िडदान महणमपगत भार िसतग वा विताीणलोकान सऽा िपतरः [] िक

5 पनः पऽपौऽ तावत िपडाडसिहतितलः िक कतौ कोिटिपडौध

इित सपण स १९२५ फानशिद १५ शब िलिखत भाइः इारामःयमवाथ इद पक ौी

1 सवनिप R1 वरादशनः R1 2 अकवा णR1 नािोक R1 िसदत R1 3 कलान िमऽपऽोपीभः R1 कया डदान R1 4 सीतग वाः R1 वािछतान िणलोकान R1 सऽ R1

5 पनः ] प नः R1 पऽपौऽः R1 तात किव त R1 िपडात गडशिहतितलः R1 कोटीिपडौR1 7 ईती R1 नाफाग णशिद R1 शबः R1 लीिषत R1 ईछाराम R1 8 ईद R1



अथ सय लपिचब Table I on f 3r elongation between the sun and lunar node argument from

1 to 130 at intervals of 130-year periods

अथ ौीकरणकशिरमथ सय शषपिचब १२ ३०Table II on f 3v elongation between the sun and lunar node argument from

0 to 130 at intervals of single years

अथ करनकसिरमथो िसारह सयोः पव नयनाथ चि कोका अवोपिरTable III on f 4r elongation between the sun and lunar node argument from

1 to 27 at intervals of 14-day avadhis

शरागलाःTable IV on f 4r digits of lunar latitude argument from 0 to 16 degrees of

nodal-solar elongation

अथ सपातचिगपिररिविबबागलािद Table V on f 4r digits of size of solar disk argument from 59 56 to 64 42

minutes of daily nodal-solar elongation

अथ ितिथमा नघोपिर चििबब तथाभतागलािद Table VI on f 4v digits of apparent lunar diameter argument from 52 to 67

ghaṭikās in a day

रवगलािदफलसतTable VII on f 4v digits of increments to apparent shadow diameter argu-

ment from Aries to Virgo and from Libra to Pisces

खिछागलोपिर मद घिटकाचबTable VIII on f 4v ghaṭikās of half-duration of totality argument from 1 to 9

digits of sky-obscuration

अथ चििछागलोपिर चि मिितघिटचब ६० Table IX on f 4v ghaṭikās of half-duration argument from 1 to 21 digits of

lunar obscuration

िततका महाः वलनाथ शकाल तथा मोकाल साधन चिमहणTable X on f 4v degrees of solar true longitude and true daily motion argu-

ment from 1 to 27 avadhis

चििछागलोपिर िितनो अतरTable XI on f 4v differences in entries of table IX argument from 1 to 21

digits of lunar obscuration

अथ चििछागलोपिर ििभागःTable XII on f 5r agnibhāga ( ) of the half-duration argument from 1 to 21




अथ मबाणागलोपिर बाणफलमिितTable XIII on f 5r latitude-result ( ) argument from 1 to 21 digits of lunar

obscuration

अथ खाकहत घौमानन भ लनताशा त तान ४५ शोत शोाशा उपिर अ-वलन

Table XIV on f 5rndash5v degrees of valana argument from 0 to 45 degrees ofzenith distance

अथ सपातचिसय रायपिर पवशान Table XV on f 5v lords of parvans argument 9 eclipse-possibility month

numbers between 0 and 24

अथ शकाल तथा मोकाल सायनमह कोशोपिर अशा अयनज वलन सायनसह-कका दौ दिण मकरादौ उर वलन दय

Table XVI on f 6r degrees of valana argument from 0 to 90 degrees of thecomplement of tropical solar longitude

अथ सय वलन आा तथा आयनज वलनयोयगातराशोपिर अगला वलन सय सय माशवलन पिम दय मोवलन पव दय सय शवलन िवपरीतदय उर जात दिण दय दिण जात उर दय

Table XVII on f 6v scale-factor for conversion to digits of solar valana argu-ment from 0 to 47 degrees of combined valana

अथ चि वलन अावलन तथा आयनज वलनयोयगातराशोपिर अगला वलनचिमहण शवलन पव दय मोवलन पिम दय चि मोवलन िवपरीत दय उर जातदिण दय दिण जात उर दय

Table XVIII on f 6v scale-factor for conversion to digits of lunar valana ar-gument from 0 to 47 degrees of combined valana

अथ सायनरिवरयोपिर दल Table XIX on f 7r ghaṭikās of half-lengths of daylight argument from 0 to 29

degrees of 1 to 12 signs of tropical solar longitude

सायनरिवरयशोपिर ल[को ]काः Table XX on ff 7vndash8r ghaṭikās of oblique ascension argument from 0 to 29


अथ ल कलाकोका Table XXI on ff 8vndash9v ghaṭikās of oblique ascension argument from 1 to 60

arcminutes of any degree of signs 1 to 6 and equivalently 12 to 7 (decreasing)

अथ दशात लाकयोिव वरबाभागिमत कोक ममलबन घिटकािद Table XXII on f 9v ghaṭikās of longitudinal parallax argument from 0 to 90

degrees of elongation between ascendant and sun



अथ शायनलरायशोपिर लबनगणकाः Table XXIII on f 10r scale-factor for longitudinal parallax argument from 0

to 29 degrees of 0 to 11 signs of tropical longitude of ascendant

अथ सायन लरायशोपिर नित अगलािदTable XXIV on f 10v digits of latitudinal parallax argument from 0 to 29

degrees of 0 to 11 signs of tropical longitude of ascendant

अथ रिवछगलात मिितघिटकािदTable XXV on f 11r ghaṭikās of half-duration of solar eclipse argument from

1 to 12 digits of solar obscuration

अथ जातक महाणा िवकलाना कोकािद नािदःTable XXVI on f 11r the arc-seconds per ghaṭikā of the planets in horoscopy

argument from 1 to 60

अथ नऽाणा योिनिवचारःTable XXVII on f 11v type of birth category for each of the nakṣatras argu-

ment the 28 nakṣatras


Please write to ⟨infosayahnaorg⟩ to file bugsproblem reports feature requests and to get involvedSayahna Foundation bull jwra 34 Jagathy bull Trivandrum 695014 bull india

Introduction

Basic Eclipse Reckoning in the Indian Tradition

Geocentric geometry of eclipses

Computing eclipse data in Indian astronomy

Relationships between Tables and Texts

Dependence of the Karaṇakesarī on earlier texts

Interdependence of the Karaṇakesarīs tables and text

MS Js worked examples for using the Karaṇakesarī

Karaṇakesarī Translation and Commentary

Glossary of technical terms

HISTORY OF SCIENCE IN SOUTH ASIAA journal for the history of all forms of scientific thought and action ancient and modern in allregions of South Asia published online at httphssasayahnaorg

Editorial Board

1 Dominik Wujastyk University of Vienna Vienna Austria2 Kim Plofker Union College Schenectady United States3 Dhruv Raina Jawaharlal Nehru University New Delhi India4 Sreeramula Rajeswara Sarma formerly Aligarh Muslim University Duumlsseldorf Germany5 Fabrizio Speziale Universiteacute Sorbonne Nouvelle ndash CNRS Paris France6 Michio Yano Kyoto Sangyo University Kyoto Japan

Principal ContactDominik Wujastyk Editor University of ViennaEmail ⟨hssasayahnaorg⟩

Mailing AddressKrishna GS Editorial Support History of Science in South AsiaSayahna jwra34 Jagathy Trivandrum 695014 Kerala India

This journal provides immediate open access to its content on the principle that making researchfreely available to the public supports a greater global exchange of knowledge

Copyrights of all the articles rest with the respective authorsThis work is licensed under a Creative Commons Attribution-ShareAlike 40 International License

The electronic versions were generated from sources marked up in LATEX in a computer runninggnulinux operating system pdf was typeset using XƎTEX from TEXLive The base font used forLatin script and oldstyle numerals was TEX Gyre Pagella developed by gust the Polish TEX UsersGroup

SAYAHNA FOUNDATION

jwra 34 bull Jagathy bull Trivandrum bull 695014 bull india bull ⟨infosayahnaorg⟩ bull wwwsayahnaorg



1 INTRODUCTION







bull For the tables


bull For the text



























Sun

Earth

Moon

Sun

Earth

Moon











O

P p

Z

ecliptic

horizon

moon

V







WE

S

NDirection

of lunar

motion

on orbit

shadow
















































11 SALUTATION



VERSE ANALYSIS














VERSE ANALYSIS




TECHNICAL ANALYSIS









minus232 311 168 times 360

4 320 000 000

= minus19 21 33 21 year



E =Y minus 1603

130
















= 2516 43 40


= 2520 minus 2516 43 40

= 3 16 20












10 4 9 28 57 60 53 211 4 23 42 48 61 11 212 5 8 3 0 61 42 2

Table contdhellip


N(1809402624 lowast 3604320000000) =150783552000000





Table 1 contdhellip









(day)






= 13 46 + 0 44 20

= 14 30 20





VERSE ANALYSIS








TECHNICAL ANALYSIS







is


































VERSE ANALYSIS



TECHNICAL ANALYSIS


C =1

2

(R + r)minus β








or exactly0










VERSE ANALYSIS




TECHNICAL ANALYSIS



p 19)15 See eg Sarma (2003 64)















Zecliptic

S

P

Q

E

M

N

equator


meridian

P






cos δ


is justϕ itself









10 20 30 40

5

10

15

20




(line)











16 AYANAVALANA


VERSE ANALYSIS



TECHNICAL ANALYSIS





sin PM












minus









VERSE ANALYSIS



TECHNICAL ANALYSIS










VERSE ANALYSIS


sake

TECHNICAL ANALYSIS








VERSE ANALYSIS




TECHNICAL ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS








N

S

W E










N

S

W E









N

S

W E



VERSE ANALYSIS








TECHNICAL ANALYSIS







VERSE ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS




ivāmbhasaḥ ∥



























VERSE ANALYSIS





TECHNICAL ANALYSIS












Z

ecliptic

S

P

V

E

N

equator


meridian

sun andmoon

Q





sin(λA minus 90)))
















VERSE ANALYSIS



TECHNICAL ANALYSIS







(line)


nati = 1

8

middot 13

12


))minus ϕ)
















VERSE ANALYSIS



TECHNICAL ANALYSIS


















Time of

conjunction

Longitudinal


conjunction

Latitudinal

parallax

Latitude

Half-

duration

Time of

release

Time of

contactLong

parallax

Contact

half-

duration

Release

half-

durationLat

parallax

Longitudes

Latitude

Longitudes

Lat

parallax

Latitude

Long

parallax







VERSE ANALYSIS




TECHNICAL ANALYSIS














27 CONCLUSION



VERSE ANALYSIS







ACKNOWLEDGEMENTS




REFERENCES






























































































































MS B concludes


15 MS R1 concludes

































lunar obscuration










obscuration

































Introduction












1 INTRODUCTION







bull For the tables


bull For the text



























Sun

Earth

Moon

Sun

Earth

Moon











O

P p

Z

ecliptic

horizon

moon

V







WE

S

NDirection

of lunar

motion

on orbit

shadow
















































11 SALUTATION



VERSE ANALYSIS














VERSE ANALYSIS




TECHNICAL ANALYSIS









minus232 311 168 times 360

4 320 000 000

= minus19 21 33 21 year



E =Y minus 1603

130
















= 2516 43 40


= 2520 minus 2516 43 40

= 3 16 20












10 4 9 28 57 60 53 211 4 23 42 48 61 11 212 5 8 3 0 61 42 2

Table contdhellip


N(1809402624 lowast 3604320000000) =150783552000000





Table 1 contdhellip









(day)






= 13 46 + 0 44 20

= 14 30 20





VERSE ANALYSIS








TECHNICAL ANALYSIS







is


































VERSE ANALYSIS



TECHNICAL ANALYSIS


C =1

2

(R + r)minus β








or exactly0










VERSE ANALYSIS




TECHNICAL ANALYSIS



p 19)15 See eg Sarma (2003 64)















Zecliptic

S

P

Q

E

M

N

equator


meridian

P






cos δ


is justϕ itself









10 20 30 40

5

10

15

20




(line)











16 AYANAVALANA


VERSE ANALYSIS



TECHNICAL ANALYSIS





sin PM












minus









VERSE ANALYSIS



TECHNICAL ANALYSIS










VERSE ANALYSIS


sake

TECHNICAL ANALYSIS








VERSE ANALYSIS




TECHNICAL ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS








N

S

W E










N

S

W E









N

S

W E



VERSE ANALYSIS








TECHNICAL ANALYSIS







VERSE ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS




ivāmbhasaḥ ∥



























VERSE ANALYSIS





TECHNICAL ANALYSIS












Z

ecliptic

S

P

V

E

N

equator


meridian

sun andmoon

Q





sin(λA minus 90)))
















VERSE ANALYSIS



TECHNICAL ANALYSIS







(line)


nati = 1

8

middot 13

12


))minus ϕ)
















VERSE ANALYSIS



TECHNICAL ANALYSIS


















Time of

conjunction

Longitudinal


conjunction

Latitudinal

parallax

Latitude

Half-

duration

Time of

release

Time of

contactLong

parallax

Contact

half-

duration

Release

half-

durationLat

parallax

Longitudes

Latitude

Longitudes

Lat

parallax

Latitude

Long

parallax







VERSE ANALYSIS




TECHNICAL ANALYSIS














27 CONCLUSION



VERSE ANALYSIS







ACKNOWLEDGEMENTS




REFERENCES






























































































































MS B concludes


15 MS R1 concludes

































lunar obscuration










obscuration

































Introduction












bull For the tables


bull For the text



























Sun

Earth

Moon

Sun

Earth

Moon











O

P p

Z

ecliptic

horizon

moon

V







WE

S

NDirection

of lunar

motion

on orbit

shadow
















































11 SALUTATION



VERSE ANALYSIS














VERSE ANALYSIS




TECHNICAL ANALYSIS









minus232 311 168 times 360

4 320 000 000

= minus19 21 33 21 year



E =Y minus 1603

130
















= 2516 43 40


= 2520 minus 2516 43 40

= 3 16 20












10 4 9 28 57 60 53 211 4 23 42 48 61 11 212 5 8 3 0 61 42 2

Table contdhellip


N(1809402624 lowast 3604320000000) =150783552000000





Table 1 contdhellip









(day)






= 13 46 + 0 44 20

= 14 30 20





VERSE ANALYSIS








TECHNICAL ANALYSIS







is


































VERSE ANALYSIS



TECHNICAL ANALYSIS


C =1

2

(R + r)minus β








or exactly0










VERSE ANALYSIS




TECHNICAL ANALYSIS



p 19)15 See eg Sarma (2003 64)















Zecliptic

S

P

Q

E

M

N

equator


meridian

P






cos δ


is justϕ itself









10 20 30 40

5

10

15

20




(line)











16 AYANAVALANA


VERSE ANALYSIS



TECHNICAL ANALYSIS





sin PM












minus









VERSE ANALYSIS



TECHNICAL ANALYSIS










VERSE ANALYSIS


sake

TECHNICAL ANALYSIS








VERSE ANALYSIS




TECHNICAL ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS








N

S

W E










N

S

W E









N

S

W E



VERSE ANALYSIS








TECHNICAL ANALYSIS







VERSE ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS




ivāmbhasaḥ ∥



























VERSE ANALYSIS





TECHNICAL ANALYSIS












Z

ecliptic

S

P

V

E

N

equator


meridian

sun andmoon

Q





sin(λA minus 90)))
















VERSE ANALYSIS



TECHNICAL ANALYSIS







(line)


nati = 1

8

middot 13

12


))minus ϕ)
















VERSE ANALYSIS



TECHNICAL ANALYSIS


















Time of

conjunction

Longitudinal


conjunction

Latitudinal

parallax

Latitude

Half-

duration

Time of

release

Time of

contactLong

parallax

Contact

half-

duration

Release

half-

durationLat

parallax

Longitudes

Latitude

Longitudes

Lat

parallax

Latitude

Long

parallax







VERSE ANALYSIS




TECHNICAL ANALYSIS














27 CONCLUSION



VERSE ANALYSIS







ACKNOWLEDGEMENTS




REFERENCES






























































































































MS B concludes


15 MS R1 concludes

































lunar obscuration










obscuration

































Introduction




























Sun

Earth

Moon

Sun

Earth

Moon











O

P p

Z

ecliptic

horizon

moon

V







WE

S

NDirection

of lunar

motion

on orbit

shadow
















































11 SALUTATION



VERSE ANALYSIS














VERSE ANALYSIS




TECHNICAL ANALYSIS









minus232 311 168 times 360

4 320 000 000

= minus19 21 33 21 year



E =Y minus 1603

130
















= 2516 43 40


= 2520 minus 2516 43 40

= 3 16 20












10 4 9 28 57 60 53 211 4 23 42 48 61 11 212 5 8 3 0 61 42 2

Table contdhellip


N(1809402624 lowast 3604320000000) =150783552000000





Table 1 contdhellip









(day)






= 13 46 + 0 44 20

= 14 30 20





VERSE ANALYSIS








TECHNICAL ANALYSIS







is


































VERSE ANALYSIS



TECHNICAL ANALYSIS


C =1

2

(R + r)minus β








or exactly0










VERSE ANALYSIS




TECHNICAL ANALYSIS



p 19)15 See eg Sarma (2003 64)















Zecliptic

S

P

Q

E

M

N

equator


meridian

P






cos δ


is justϕ itself









10 20 30 40

5

10

15

20




(line)











16 AYANAVALANA


VERSE ANALYSIS



TECHNICAL ANALYSIS





sin PM












minus









VERSE ANALYSIS



TECHNICAL ANALYSIS










VERSE ANALYSIS


sake

TECHNICAL ANALYSIS








VERSE ANALYSIS




TECHNICAL ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS








N

S

W E










N

S

W E









N

S

W E



VERSE ANALYSIS








TECHNICAL ANALYSIS







VERSE ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS




ivāmbhasaḥ ∥



























VERSE ANALYSIS





TECHNICAL ANALYSIS












Z

ecliptic

S

P

V

E

N

equator


meridian

sun andmoon

Q





sin(λA minus 90)))
















VERSE ANALYSIS



TECHNICAL ANALYSIS







(line)


nati = 1

8

middot 13

12


))minus ϕ)
















VERSE ANALYSIS



TECHNICAL ANALYSIS


















Time of

conjunction

Longitudinal


conjunction

Latitudinal

parallax

Latitude

Half-

duration

Time of

release

Time of

contactLong

parallax

Contact

half-

duration

Release

half-

durationLat

parallax

Longitudes

Latitude

Longitudes

Lat

parallax

Latitude

Long

parallax







VERSE ANALYSIS




TECHNICAL ANALYSIS














27 CONCLUSION



VERSE ANALYSIS







ACKNOWLEDGEMENTS




REFERENCES






























































































































MS B concludes


15 MS R1 concludes

































lunar obscuration










obscuration

































Introduction


















O

P p

Z

ecliptic

horizon

moon

V







WE

S

NDirection

of lunar

motion

on orbit

shadow
















































11 SALUTATION



VERSE ANALYSIS














VERSE ANALYSIS




TECHNICAL ANALYSIS









minus232 311 168 times 360

4 320 000 000

= minus19 21 33 21 year



E =Y minus 1603

130
















= 2516 43 40


= 2520 minus 2516 43 40

= 3 16 20












10 4 9 28 57 60 53 211 4 23 42 48 61 11 212 5 8 3 0 61 42 2

Table contdhellip


N(1809402624 lowast 3604320000000) =150783552000000





Table 1 contdhellip









(day)






= 13 46 + 0 44 20

= 14 30 20





VERSE ANALYSIS








TECHNICAL ANALYSIS







is


































VERSE ANALYSIS



TECHNICAL ANALYSIS


C =1

2

(R + r)minus β








or exactly0










VERSE ANALYSIS




TECHNICAL ANALYSIS



p 19)15 See eg Sarma (2003 64)















Zecliptic

S

P

Q

E

M

N

equator


meridian

P






cos δ


is justϕ itself









10 20 30 40

5

10

15

20




(line)











16 AYANAVALANA


VERSE ANALYSIS



TECHNICAL ANALYSIS





sin PM












minus









VERSE ANALYSIS



TECHNICAL ANALYSIS










VERSE ANALYSIS


sake

TECHNICAL ANALYSIS








VERSE ANALYSIS




TECHNICAL ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS








N

S

W E










N

S

W E









N

S

W E



VERSE ANALYSIS








TECHNICAL ANALYSIS







VERSE ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS




ivāmbhasaḥ ∥



























VERSE ANALYSIS





TECHNICAL ANALYSIS












Z

ecliptic

S

P

V

E

N

equator


meridian

sun andmoon

Q





sin(λA minus 90)))
















VERSE ANALYSIS



TECHNICAL ANALYSIS







(line)


nati = 1

8

middot 13

12


))minus ϕ)
















VERSE ANALYSIS



TECHNICAL ANALYSIS


















Time of

conjunction

Longitudinal


conjunction

Latitudinal

parallax

Latitude

Half-

duration

Time of

release

Time of

contactLong

parallax

Contact

half-

duration

Release

half-

durationLat

parallax

Longitudes

Latitude

Longitudes

Lat

parallax

Latitude

Long

parallax







VERSE ANALYSIS




TECHNICAL ANALYSIS














27 CONCLUSION



VERSE ANALYSIS







ACKNOWLEDGEMENTS




REFERENCES






























































































































MS B concludes


15 MS R1 concludes

































lunar obscuration










obscuration

































Introduction












WE

S

NDirection

of lunar

motion

on orbit

shadow
















































11 SALUTATION



VERSE ANALYSIS














VERSE ANALYSIS




TECHNICAL ANALYSIS









minus232 311 168 times 360

4 320 000 000

= minus19 21 33 21 year



E =Y minus 1603

130
















= 2516 43 40


= 2520 minus 2516 43 40

= 3 16 20












10 4 9 28 57 60 53 211 4 23 42 48 61 11 212 5 8 3 0 61 42 2

Table contdhellip


N(1809402624 lowast 3604320000000) =150783552000000





Table 1 contdhellip









(day)






= 13 46 + 0 44 20

= 14 30 20





VERSE ANALYSIS








TECHNICAL ANALYSIS







is


































VERSE ANALYSIS



TECHNICAL ANALYSIS


C =1

2

(R + r)minus β








or exactly0










VERSE ANALYSIS




TECHNICAL ANALYSIS



p 19)15 See eg Sarma (2003 64)















Zecliptic

S

P

Q

E

M

N

equator


meridian

P






cos δ


is justϕ itself









10 20 30 40

5

10

15

20




(line)











16 AYANAVALANA


VERSE ANALYSIS



TECHNICAL ANALYSIS





sin PM












minus









VERSE ANALYSIS



TECHNICAL ANALYSIS










VERSE ANALYSIS


sake

TECHNICAL ANALYSIS








VERSE ANALYSIS




TECHNICAL ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS








N

S

W E










N

S

W E









N

S

W E



VERSE ANALYSIS








TECHNICAL ANALYSIS







VERSE ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS




ivāmbhasaḥ ∥



























VERSE ANALYSIS





TECHNICAL ANALYSIS












Z

ecliptic

S

P

V

E

N

equator


meridian

sun andmoon

Q





sin(λA minus 90)))
















VERSE ANALYSIS



TECHNICAL ANALYSIS







(line)


nati = 1

8

middot 13

12


))minus ϕ)
















VERSE ANALYSIS



TECHNICAL ANALYSIS


















Time of

conjunction

Longitudinal


conjunction

Latitudinal

parallax

Latitude

Half-

duration

Time of

release

Time of

contactLong

parallax

Contact

half-

duration

Release

half-

durationLat

parallax

Longitudes

Latitude

Longitudes

Lat

parallax

Latitude

Long

parallax







VERSE ANALYSIS




TECHNICAL ANALYSIS














27 CONCLUSION



VERSE ANALYSIS







ACKNOWLEDGEMENTS




REFERENCES






























































































































MS B concludes


15 MS R1 concludes

































lunar obscuration










obscuration

































Introduction

















































11 SALUTATION



VERSE ANALYSIS














VERSE ANALYSIS




TECHNICAL ANALYSIS









minus232 311 168 times 360

4 320 000 000

= minus19 21 33 21 year



E =Y minus 1603

130
















= 2516 43 40


= 2520 minus 2516 43 40

= 3 16 20












10 4 9 28 57 60 53 211 4 23 42 48 61 11 212 5 8 3 0 61 42 2

Table contdhellip


N(1809402624 lowast 3604320000000) =150783552000000





Table 1 contdhellip









(day)






= 13 46 + 0 44 20

= 14 30 20





VERSE ANALYSIS








TECHNICAL ANALYSIS







is


































VERSE ANALYSIS



TECHNICAL ANALYSIS


C =1

2

(R + r)minus β








or exactly0










VERSE ANALYSIS




TECHNICAL ANALYSIS



p 19)15 See eg Sarma (2003 64)















Zecliptic

S

P

Q

E

M

N

equator


meridian

P






cos δ


is justϕ itself









10 20 30 40

5

10

15

20




(line)











16 AYANAVALANA


VERSE ANALYSIS



TECHNICAL ANALYSIS





sin PM












minus









VERSE ANALYSIS



TECHNICAL ANALYSIS










VERSE ANALYSIS


sake

TECHNICAL ANALYSIS








VERSE ANALYSIS




TECHNICAL ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS








N

S

W E










N

S

W E









N

S

W E



VERSE ANALYSIS








TECHNICAL ANALYSIS







VERSE ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS




ivāmbhasaḥ ∥



























VERSE ANALYSIS





TECHNICAL ANALYSIS












Z

ecliptic

S

P

V

E

N

equator


meridian

sun andmoon

Q





sin(λA minus 90)))
















VERSE ANALYSIS



TECHNICAL ANALYSIS







(line)


nati = 1

8

middot 13

12


))minus ϕ)
















VERSE ANALYSIS



TECHNICAL ANALYSIS


















Time of

conjunction

Longitudinal


conjunction

Latitudinal

parallax

Latitude

Half-

duration

Time of

release

Time of

contactLong

parallax

Contact

half-

duration

Release

half-

durationLat

parallax

Longitudes

Latitude

Longitudes

Lat

parallax

Latitude

Long

parallax







VERSE ANALYSIS




TECHNICAL ANALYSIS














27 CONCLUSION



VERSE ANALYSIS







ACKNOWLEDGEMENTS




REFERENCES






























































































































MS B concludes


15 MS R1 concludes

































lunar obscuration










obscuration

































Introduction
















VERSE ANALYSIS




TECHNICAL ANALYSIS









minus232 311 168 times 360

4 320 000 000

= minus19 21 33 21 year



E =Y minus 1603

130
















= 2516 43 40


= 2520 minus 2516 43 40

= 3 16 20












10 4 9 28 57 60 53 211 4 23 42 48 61 11 212 5 8 3 0 61 42 2

Table contdhellip


N(1809402624 lowast 3604320000000) =150783552000000





Table 1 contdhellip









(day)






= 13 46 + 0 44 20

= 14 30 20





VERSE ANALYSIS








TECHNICAL ANALYSIS







is


































VERSE ANALYSIS



TECHNICAL ANALYSIS


C =1

2

(R + r)minus β








or exactly0










VERSE ANALYSIS




TECHNICAL ANALYSIS



p 19)15 See eg Sarma (2003 64)















Zecliptic

S

P

Q

E

M

N

equator


meridian

P






cos δ


is justϕ itself









10 20 30 40

5

10

15

20




(line)











16 AYANAVALANA


VERSE ANALYSIS



TECHNICAL ANALYSIS





sin PM












minus









VERSE ANALYSIS



TECHNICAL ANALYSIS










VERSE ANALYSIS


sake

TECHNICAL ANALYSIS








VERSE ANALYSIS




TECHNICAL ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS








N

S

W E










N

S

W E









N

S

W E



VERSE ANALYSIS








TECHNICAL ANALYSIS







VERSE ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS




ivāmbhasaḥ ∥



























VERSE ANALYSIS





TECHNICAL ANALYSIS












Z

ecliptic

S

P

V

E

N

equator


meridian

sun andmoon

Q





sin(λA minus 90)))
















VERSE ANALYSIS



TECHNICAL ANALYSIS







(line)


nati = 1

8

middot 13

12


))minus ϕ)
















VERSE ANALYSIS



TECHNICAL ANALYSIS


















Time of

conjunction

Longitudinal


conjunction

Latitudinal

parallax

Latitude

Half-

duration

Time of

release

Time of

contactLong

parallax

Contact

half-

duration

Release

half-

durationLat

parallax

Longitudes

Latitude

Longitudes

Lat

parallax

Latitude

Long

parallax







VERSE ANALYSIS




TECHNICAL ANALYSIS














27 CONCLUSION



VERSE ANALYSIS







ACKNOWLEDGEMENTS




REFERENCES






























































































































MS B concludes


15 MS R1 concludes

































lunar obscuration










obscuration

































Introduction













minus232 311 168 times 360

4 320 000 000

= minus19 21 33 21 year



E =Y minus 1603

130
















= 2516 43 40


= 2520 minus 2516 43 40

= 3 16 20












10 4 9 28 57 60 53 211 4 23 42 48 61 11 212 5 8 3 0 61 42 2

Table contdhellip


N(1809402624 lowast 3604320000000) =150783552000000





Table 1 contdhellip









(day)






= 13 46 + 0 44 20

= 14 30 20





VERSE ANALYSIS








TECHNICAL ANALYSIS







is


































VERSE ANALYSIS



TECHNICAL ANALYSIS


C =1

2

(R + r)minus β








or exactly0










VERSE ANALYSIS




TECHNICAL ANALYSIS



p 19)15 See eg Sarma (2003 64)















Zecliptic

S

P

Q

E

M

N

equator


meridian

P






cos δ


is justϕ itself









10 20 30 40

5

10

15

20




(line)











16 AYANAVALANA


VERSE ANALYSIS



TECHNICAL ANALYSIS





sin PM












minus









VERSE ANALYSIS



TECHNICAL ANALYSIS










VERSE ANALYSIS


sake

TECHNICAL ANALYSIS








VERSE ANALYSIS




TECHNICAL ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS








N

S

W E










N

S

W E









N

S

W E



VERSE ANALYSIS








TECHNICAL ANALYSIS







VERSE ANALYSIS









VERSE ANALYSIS



TECHNICAL ANALYSIS




ivāmbhasaḥ ∥



























VERSE ANALYSIS





TECHNICAL ANALYSIS












Z

ecliptic

S

P

V

E

N

equator


meridian

sun andmoon

Q





sin(λA minus 90)))
















VERSE ANALYSIS



TECHNICAL ANALYSIS







(line)


nati = 1

8

middot 13

12


))minus ϕ)
















VERSE ANALYSIS



TECHNICAL ANALYSIS


















Time of

conjunction

Longitudinal


conjunction

Latitudinal

parallax

Latitude

Half-

duration

Time of

release

Time of

contactLong

parallax

Contact

half-

duration

Release

half-

durationLat

parallax

Longitudes

Latitude

Longitudes

Lat

parallax

Latitude

Long

parallax







VERSE ANALYSIS




TECHNICAL ANALYSIS














27 CONCLUSION



VERSE ANALYSIS







ACKNOWLEDGEMENTS




REFERENCES






























































































































MS B concludes


15 MS R1 concludes

































lunar obscuration










obscuration

































Introduction










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