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\let\*=\d
\def\deg{^{\circ}}
\def\elp{$\ldots$}
\def\Sin{\mathop{\rm Sin}\nolimits}
\def\Cos{\mathop{\rm Cos}\nolimits}
\def\Vers{\mathop{\rm Vers}\nolimits}
%\def\Crd{\mathop{\rm Crd}\nolimits}
\def\arcSin{\mathop{\rm arcSin}\nolimits}
\def\signs{^{\rm s}}
% \def\disp{\displaystyle}

% Proper name and technical term macros
%%%%%
%
\def\KK{\textit{Kara\-\d{n}a\-kut\=u\-hala}}
\def\kks{\textit{Kara\-\d{n}a\-kesa\-r\={\i}}}
\def\KKD{\textit{Kav\={\i}ndra\-kalpa\-dr\=uma}}
\def\GP{\textit{Garu\-\*da\-pur\=a\-\*na}}
\def\GL{\textit{Graha\-l\=agha\-va}}
\def\BSS{\textit{Br\=ahma\-sphu\*ta\-siddh\=anta}}
\def\MBharat{\textit{Mah\=a\-bh\=ara\-ta}}
\def\SDV{\textit{\'Si\*syadh\={\i}v\*rddhidatantra}}
\def\SidSir{\textit{Siddh\=anta\-\'siro\-ma\*ni}}
\def\SurSid{\textit{S\=urya\-siddh\=a\-nta}}
%
\def\ROne{\xtsc{R}$_1$}
\def\RTwo{\xtsc{R}$_2$}
%
\def\Kavindracarya{Ka\-v\={\i}ndr\=a\-c\=a\-rya}
\def\Krsna{K\*r\*s\*na}
\def\Ganesa{Ga\-\*ne\-\'sa}
\def\Daivajna{Dai\-va\-j\~na}
\def\Brahmagupta{Brahma\-gupta}
\def\Bhatta{Bha\-\*t\*ta}
\def\Bhaskara{Bh\=askara}
%
%
\def\Brahmapaksa{Br\=ahma\-pak\*sa}
\def\Saka{\'Sa\-ka}

\def\aksa{\textit{ak\*sa}}
\def\aksara{\textit{ak\*sa\-ra}}
\def\aksavalana{\textit{ak\*sa\-valana}}
\def\angula{\textit{a\.ngu\-la}}
\def\angulas{\textit{a\.ngu\-las}}
\def\adhikara{\textit{adhi\-k\=a\-ra}}
\def\adhikaras{\textit{adhi\-k\=a\-ras}}
\def\anusvara{\textit{anu\-sv\=ara}}
\def\ayana{\textit{aya\-na}}
\def\ayanavalana{\textit{ayana\-valana}}
\def\avagraha{\textit{a\-va\-gra\-ha}}
\def\avadhi{\textit{a\-va\-dhi}}
\def\avadhis{\textit{a\-va\-dhis}}
%
\def\unmilana{\textit{unm\={\i}\-la\-na}}
\def\upari{\textit{upa\-ri}}
%
\def\r{\textit{\*r}}
\def\rna{\textit{\*rna}}
%
\def\karana{\textit{ka\-ra\-\*na}}
\def\karanas{\textit{ka\-ra\-\*nas}}
\def\karani{\textit{ka\-ra\-\*n\={\i}}}
\def\karanis{\textit{ka\-ra\-\*n\={\i}s}}
\def\kalpa{\textit{kalpa}}
\def\kala{\textit{ka\-l\=a}}
\def\kalas{\textit{ka\-l\=as}}
\def\krsnapaksa{\textit{k\*r\*s\*na\-pak\*sa}}
\def\koti{\textit{ko\-\*ti}}
\def\kostha{\textit{ko\-\*s\*tha}}
\def\kosthaka{\textit{ko\-\*s\*tha\-ka}}
\def\kosthakas{\textit{ko\-\*s\*tha\-kas}}
%
\def\ganita{\textit{ga\-\*ni\-ta}}
\def\gotra{\textit{go\-tra}}
\def\ghati{\textit{gha\-\*ti\-k\=a}}
\def\ghatis{\textit{gha\-\*ti\-k\=as}}
%
\def\jati{\textit{j\=a\-ti}}
\def\jyotisa{\textit{jyo\-ti\-\*sa}}
\def\jyotisis{\textit{jyo\-ti\-\*s\={\i}s}}
%
\def\tana{\textit{ta\-na}}
\def\tithi{\textit{ti\-thi}}
\def\tithis{\textit{ti\-this}}
%
\def\dandas{\textit{da\*n\*das}}
\def\dvandva{\textit{dvandva}}
\def\dha{\textit{dha}}
\def\dhana{\textit{dha\-na}}
%
\def\nata{\textit{na\-ta}}
\def\nati{\textit{na\-ti}}
\def\nagari{\textit{n\=a\-ga\-r\={\i}}}
\def\nadi{\textit{n\=a\-\*d\={\i}}}
\def\nadis{\textit{n\=a\-\*d\={\i}s}}
\def\nimilana{\textit{ni\-m\={\i}lana}}
%
\def\parilekha{\textit{pari\-lekha}}
\def\parvan{\textit{parvan}}
\def\parvans{\textit{parvans}}
\def\pada{\textit{p\=a\-da}}
\def\padas{\textit{p\=a\-das}}
%
\def\bahuvrihi{\textit{bahu\-vr\={\i}\-hi}}
\def\bahu{\textit{b\=a\-hu}}
\def\bhuja{\textit{bhu\-ja}}
\def\bhut{\textit{bh\=u\-ta\-sa\*mkhy\=a}}
%
\def\madhya{\textit{madhya}}
\def\moksa{\textit{mo\-k\*sa}}
%
\def\lagna{\textit{lagna}}
\def\labdha{\textit{labdha}}
\def\lambana{\textit{lamba\-na}}
%
\def\valana{\textit{va\-la\-na}}
\def\virama{\textit{vi\-r\=ama}}
\def\vilagna{\textit{vi\-lagna}}
\def\visarga{\textit{vi\-sarga}}
%
\def\suklapaksa{\textit{\'sukla\-pak\*sa}}
\def\sesa{\textit{\'se\-\*sa}}
\def\sruti{\textit{\'sr\=u\-ti}}
%
\def\sandhi{\textit{sa\*m\-dhi}}
\def\sanmilana{\textit{sanm\={\i}lana}}
\def\sayana{\textit{s\=a\-ya\-na}}
\def\siddhanta{\textit{siddh\=anta}}
\def\siddhantas{\textit{siddh\=antas}}
\def\sparsa{\textit{spar\'sa}}
\def\smrti{\textit{sm\*r\-ti}}

% Verse metre macros
%%%
%
\def\indravajra{\textit{indra\-vajr\=a}}
\def\upajati{\textit{upa\-j\=ati}}
\def\jagati{\textit{jaga\-t\={\i}}}
\def\bhuj{\textit{bhuj\=a\.nga\-pray\=ata}}
\def\marata{\textit{maratatagaga}}
\def\rath{\textit{ratho\-ddha\-t\=a}}
\def\vas{\textit{va\-santa\-tila\-k\=a}}
\def\sard{\textit{\'s\=ard\=ula\-vikr\={\i}\*dita}}
\def\sakvari{\textit{sakvar\={\i}}}

%%%

\def\tsc#1{\bgroup\footnotesize\uppercase{#1}\egroup\xspace}
\def\xtsc#1{\bgroup\footnotesize\uppercase{#1}\egroup}


\usepackage{natbib}
\setcitestyle{authoryear,round,comma,aysep={}}
\begin{document}
\bibliographystyle{sanmed}

\verso{kara\-\d{n}a\-kesa\-r\={\i} of bh\=askara} 
%{: a 17th-century table text for computing  eclipses}

\recto{clemency montelle and kim plofker}

\title{\kks\ of \Bhaskara: a 17th-century \\
table text for computing eclipses}

\author{Clemency Montelle and Kim Plofker}

%\date{\today}

\firstpage{1}
\lastpage{62}

\covercitation{Clemency Montelle and Kim Plofker, ``\kks\ of
  \Bhaskara: a 17th-century  \\
  table text for computing eclipses." \textit{History of 
  Science in South Asia} 2 (2014): \thefirstpage--\thelastpage.}

\coverpage

\copyrightspage

\ifcover\setcounter{page}{0}\else\setcounter{page}{1}\fi

\maketitle

\section{Introduction} \label{Intro}

\setcounter{page}{1}

\lettrine[lines=2,loversize=.1]{T}{he} \textit{Kara\*nakesar\={\i}} of
Bh\=askara (fl.\ 1681) is a set of astronomical tables along with a
short accompanying versified text for computing the circumstances and
details of lunar and solar eclipses.  This text is divided into two
chapters (\adhikaras): the \textit{candraparvan} or lunar eclipse
which contains thirteen verses, and the \textit{s\=uryaparvan} or
solar eclipse which contains seven verses (although the manuscripts we
have seen are not unanimous on this point).  The text is both a
supplement to the tables and a guide for using them.  Its last verse
tells us that this work was composed in Saud\=amik\=a (a locality
probably in Gujar\=at) and the second verse indicates that its epoch
is year 1603 of the \Saka\ era (1681 \tsc{ce}).  The tables are
computed for the terrestrial latitude $\phi =$ 22;35,39$\deg$, as
discussed in the analysis of verse 1.5 in section~\ref{TransComm}
below.

\begin{figure}[h]
  \includegraphics[width=\linewidth]{KKef4vnice.pdf}
  \caption{Excerpt from \kks\ tables, Poleman 4946 (Smith Indic
    \tsc{mb}) XIV f.~4v (\xtsc{MS.\ P}$_1$)}
\label{kkstablesf4v}
\end{figure}

We know of at least a dozen extant manuscripts of the \kks.%
\footnoteA{See \citet[328]{pingreecess} and \citet[70]{satius}, in which
  Pingree notes four other works entitled \kks; it is unclear whether
  these works are related to the tables under consideration here.}
%
The ones we have consulted for our present study are the following:
\begin{itemize}
\item For the tables:
  \begin{itemize}
  \item Poleman 4946 (Smith Indic \tsc{mb}) \tsc{XIV} ff.~3--11
    (denoted \tsc{MS.\ P$_1$})
  \item Poleman 4946 (Smith Indic \tsc{mb}) \tsc{XXVII} ff.~2-3
    (\tsc{MS.\ P$_2$})
  \item Poleman 4946 (Smith Indic \tsc{mb}) \tsc{XXVIII} 1f (\tsc{MS.\
      P$_3$})
  \item Rajasthan Oriental Research Institute (\tsc{RORI}) Jodhpur
    12792 (\tsc{MS.\ \RTwo})
  \end{itemize}
\item For the text:
  \begin{itemize}
  \item Baroda 11268 (\tsc{MS.\ B}), whose colophon asserts that it
    was copied in the 14th day of the bright half of A\'svina in year
    1819 (of the Sa\*mvat era), which the scribe calls a Saturday;
    indeed it corresponds to Saturday 2 October, 1762 \tsc{ce}.%
     %
    \footnoteA{For the relation of Indian calendars to Common Era
      dates, see \citet[57--65] {KnuPloIndCalendars} and the discussion
      of verse~1.2 in section~\ref{TransComm} below.}
    %
    % the only manuscript of the text currently available to us).
  \item \tsc{RORI} Jodhpur 12814 ff.~1v--2v (\tsc{MS.\ \ROne}),
    allegedly copied for his own use by Bh\=a\={\i} \=Icch\=ar\=ama on
    the 15th day of the bright half of Phalguna Sa\*mvat 1925, a
    Friday; this converts to 28 March 1869 \tsc{ce}, which however was
    a Sunday.
  \end{itemize}
\end{itemize}
%
An additional manuscript (\tsc{RORI} Jaipur 9757 (\tsc{MS.\ J}),
copied on day 8 of the dark half of Jye\*s\*tha in Sa\*mvat 1818, a
Thursday: i.e., Thursday 25 June 1761 \tsc{ce}) contains worked
examples of the \kks\ procedures for two eclipses before 1750 \tsc{ce}
(see section~\ref{MSJ}).  Unless otherwise noted, all manuscript
readings in the tables part refer to \tsc{MS.\ P$_1$} (of which figure
\ref{kkstablesf4v} shows a sample leaf).

While most tables pertaining to eclipses form part of comprehensive
table texts on astronomical prediction in general, the \kks\ is a less
common instance of a work devoted exclusively to this topic
\citep[46]{pingree81}.  Its tables describe characteristics of the
celestial bodies implicated in eclipses: elongations between them,
their apparent diameters, `deflection', and parallax, as well as other
relevant phenomena such as eclipse duration and magnitude.  Appended
to the eclipse computation tables but not discussed in the text are a
set of additional tables concerning astrological phenomena.

Our critical edition of the verse text portion attested in \tsc{MSS.\
  B} and \ROne\
%%%%% accompanying the tables  
is included as Appendix~B, %% [NOTE: APPENDIX REF IS HARD-CODED]
at the end of which we supply transcriptions and identifications of
the titles of all the tables. Preceding it is \ref{Glossary}, a brief
glossary of the Sanskrit technical terms used in our exposition.
Section~\ref{TransComm} contains a complete transliteration and
translation of the \kks's edited text, accompanied by a `Verse
Analysis' and `Technical Analysis' for each verse or group of verses.
The former specifies the metre(s) used as well as any notable features
of the composition or the manuscript readings, while the latter
explains to the best of our understanding the technical procedures
described therein.  While we briefly summarise each of the associated
tables and draw on their content to explain the verses, we have not
attempted to critically edit or thoroughly analyze the tables or their
titles and notes; we hope to undertake this task in a future
publication. In the present discussion, wherever we have quoted these
prose titles or notes we have transcribed them as they appear in the
manuscript, without correcting their orthography.

In the edited text as well as in the transliteration, translation and
commentary we employ the following editorial conventions:
%
\begin{itemize}
\item Square brackets [~] indicate an editorial addition or proposed
  reconstruction of missing text.
%
\item Angle brackets $\langle$~$\rangle$ indicate manuscript readings that we
  discard as incorrect.
%
\item We use the word \textit{kesar\={\i}} `mane-bearer, lion' with a
  dental sibilant in our standard form of the work's title \kks, but
  we do not emend the lexically accepted variant
  \textit{ke\'sar\={\i}} with a palatal sibilant where it appears in
  the manuscripts.
%
\item Scribal variants of \nagari\ orthography which are emended
  silently and not noted in the critical apparatus (except where the
  meaning of the original reading may be ambiguous) include the
  following: \anusvara\ for a conjoined nasal, omitted \visarga,
  \virama\ or \avagraha, misplaced \dandas, reversed conjunct
  consonants (e.g., \textit{adba} for \textit{abda}), conjunct
  consonants that we cannot reproduce in our \nagari\ typesetting,
  doubled consonants after \textit{r} or across a \pada\ break,
  routinely confused consonant pairs (e.g., \textit{ba} for
  \textit{va}, \textit{\*sa} for \textit{kha}), and all forms of
  \textit{ko\*s\*ta} for \textit{ko\*s\*tha} `table entry'.
%
\item Fragments of Sanskrit words or compounds in \nagari\ are
  indicated with a small circle $\circ$ at the breakpoint.
%
\item Folio breaks are indicated by a single vertical stroke $\bigl|$.
%
\item In the critical apparatus, text followed by a single square
  close-bracket ] indicates the edited version of the manuscript
  reading that follows it.
%
\item The symbol \textit{x} within \nagari\ text indicates an \aksara\
  (syllable) that is illegible in the manuscript.
%
\item Numerals in sexagesimal or base-60 notation are shown with a
  semicolon separating their integer and fractional parts, and commas
  separating their successive sexagesimal digits. The superscripts
  $\signs$ and $\deg$ and $'$ indicate zodiacal signs (i.e., 30-degree
  arcs of longitude), degrees and minutes of arc, respectively.
%
\end{itemize}

%%% SECTION 2
%%% SECTION 2
%%% SECTION 2
%%% 
\section{Basic Eclipse Reckoning in the Indian Tradition}

\label{EclipseTheory}

\subsection{Geocentric geometry of eclipses}

\label{EclipseTheoryGeom}

\lettrine[lines=2,loversize=.1]{E}{clipse} reckoning is one of the
most important functions of Indian astronomy.  Indian calendars are
fundamentally keyed to the ritually significant instants of luni-solar
opposition and conjunction or full and new moon, the
so-called \parvans\ (syzygies) at which lunar and solar eclipses
respectively occur (the term may also be used for the eclipse
phenomenon itself). The Indian `lunar day' or \tithi, of which there
are exactly thirty in a synodic lunar month between two successive
conjunctions or oppositions, is defined so that the \parvans\ occur at
the ends of the fifteenth and the thirtieth \tithi. The regular solar
day, on the other hand, is the time between two successive sunrises
(or midnights), conventionally divided into 60 equal time-units called
\ghatis\ or \nadis.

\begin{figure}[h]
  \begin{center}
    \includegraphics[width=3.25in] %%
    {KKeShadowCone}
    \caption{Lunar eclipse configurations showing qualitative effects
      of variations in the distances of the moon and sun from the
      earth}
    \label{ShadowConeDiagram}
  \end{center}
\end{figure}

A lunar eclipse takes place at a full moon or opposition when the
earth interposing between the moon and the sun darkens the moon with
its shadow. It is visible everywhere on earth where the moon is above
the horizon. As shown (greatly exaggerated) in
figure~\ref{ShadowConeDiagram}, the apparent sizes of the disks of the
moon and the shadow are affected by small periodic changes in the
geocentric distances of the moon and the sun.  When the moon is
farther from the earth, its disk looks smaller and so does the
cross-section of the shadow cone where it intersects the moon's orbit.
When the sun is closer to the earth, the earth's shadow cone is
shorter, which also decreases the size of the shadow disk that the
moon passes through.

A solar eclipse occurs at new moon (conjunction) when the moon
interposes between the earth and the sun, preventing the sun's rays
from reaching some points on the earth.  It is visible only from a
narrow strip of terrestrial localities because the much smaller moon
cannot block sunlight from an entire hemisphere of the earth.
Observers outside that strip, whose placement varies according to the
relative positions of the earth, moon and sun, will not notice any
obscuration of the sun's disk.  Both lunar and solar eclipses may be
either total, implying that the disk of the eclipsed body is at some
point completely obscured, or partial if some fraction of it always
remains clear.

\begin{figure}[h]
  \begin{center}
    \includegraphics[width=3.5in] %%
    {KKeParallaxMoonStars}
    \qquad\qquad
    \includegraphics[width=1.75in] %%
    {KKeParallax}
    \caption{Effect of parallax angle $p$ on perceived position of the
      moon viewed from earth's surface}
    \label{ParallaxDiagram}
  \end{center}
\end{figure}

Observers who can see a solar eclipse will perceive its position in
the sky differently depending on the amount of lunar parallax at their
locality: i.e., the apparent displacement of the geocentric position
of the moon against its background of stars due to the fact that the
observer is not on the straight line between the moon and the earth's
center.  This phenomenon is illustrated on the left side of
figure~\ref{ParallaxDiagram}, which shows the `depressing' effect of
the parallax angle $p$ appearing to shift the moon downwards, i.e.,
closer to the local horizon of the observer at $P$.  The parallactic
shift is zero for the moon at the observer's zenith and increases to
its maximum as the moon gets closer to the observer's
horizon. (Parallax shift for the sun is much smaller because of the
sun's greater distance, and is generally ignored in eclipse
computation.)


\subsection{Computing eclipse data in Indian astronomy}
\label{EclipseTheoryComp}

Indian astronomy texts typically place the bulk of their eclipse
computation algorithms in their discussion of lunar eclipses, which
precedes a shorter treatment of solar ones.  They enumerate around a
dozen separate eclipse elements to be determined by computation; the
seventh-century astronomer Brahmagupta cites fourteen in the lunar
eclipse chapter of his \BSS\ (4.1--3) \citep[72]{DvivediBSS}.  These
include time of apparent syzygy (\textit{vel\=a}), duration of eclipse
(\textit{sthiti}), instant of first contact (\textit{spar\'sa}),
instant of release (\moksa), duration of totality (\textit{vimarda}),
beginning of totality (\sanmilana, \textit{nim\={\i}lana}), end of
totality (\unmilana), magnitude of obscuration (\textit{gr\=asa}),
obscuration at a given time (\textit{i\*s\*tagr\=asa}), `shadow' or
apparent diameter of earth's shadow (\textit{ch\=ay\=a}), direction of
impact (\textit{di\'s}), colour of eclipsed body (\textit{var\*na}),
inclination to cardinal directions (\textit{valana}), and graphical
projection or diagram (\textit{parilekha}).  Of course, this list also
implies knowledge of other necessary quantities, such as apparent
diameters and velocities of the luminaries at a desired time.

%\begin{figure}[h]

\subhead{Occurrence.}  The astronomer is chiefly concerned to
determine whether, when and how an eclipse at a particular syzygy will
appear in the sky at his own locality.  Eclipses are more rare than
syzygies because the moon's orbit and the apparent orbital path of the
sun (the so-called ecliptic) are slightly inclined to each other: as
long as the moon is sufficiently displaced in celestial latitude north
or south of the plane of the ecliptic, it passes through opposition or
conjunction without ever intercepting the earth's shadow or sun's
disk, respectively.  The elongation of the moon's position from an
orbital node, i.e., one of the two diametrically opposed points at
which its orbit intersects the ecliptic, determines its latitude.  The
latitude in turn determines the so-called eclipse limits within which
an eclipse is possible (see verse 1.2 in section~\ref{TransComm}
below) and whether the eclipse will be partial or total.

\begin{wrapfigure}[18]{r}{2.7in}
  \begin{center}
    \includegraphics[width=2.65in] {KKeMagnitude}
    \caption{The moon at first contact, mid-eclipse, and end of
      totality in a total lunar eclipse, as seen in the sky by a
      south-facing observer}
    \label{ImmersionDiagram}
  \end{center}
\end{wrapfigure}
\subhead{Phases.}  There are five phases of a total eclipse considered
significant by Indian astronomers: the moments of \sparsa, \sanmilana,
mid-eclipse or \textit{ma\-dhya}, \unmilana\ and \moksa.  The first, third and
fourth of these phases are illustrated in the diagram of a total lunar
eclipse in figure~\ref{ImmersionDiagram}.  In a partial eclipse, by
definition, there is no period of totality so \sanmilana\ and
\unmilana\ do not occur.

\subhead{Magnitude.}  The amount of obscuration of the eclipsed body,
or eclipse magnitude, is a linear quantity measured along the diameter
of the eclipsing disk.  It depends on the apparent diameters of the
eclipsed and eclipsing disks (which are inversely proportional to
their angular velocities) as well as on the lunar latitude (see verses
1.3--4).

\subhead{Deflection.}  In figure~\ref{ImmersionDiagram} the moon's
passage through the shadow is tilted with respect to the cardinal
directions: this is the so-called \valana\ `inclination, deflection',
which is deemed especially significant for astrological purposes.
This feature has antecedents in ancient West Asian astronomy and its
Greek successors, where it is known as \textit{prosneusis}
`inclination, pointing'.  In the Indian tradition, the \valana\ is
divided into two components (see verses 1.5--7):
%
\begin{itemize}
\item \aksavalana\ accounting for the southward sag of the body's
  projection onto the celestial equator from the east-west prime
  vertical circle, proportional to the local terrestrial latitude and
  the so-called \nata\ or depression of the luminaries from the
  meridian or the zenith;
\item \ayanavalana\ representing the skew between the celestial
  equator and the ecliptic at that position, computed based on the
  ecliptic declination of a point determined by the sun's position
  with respect to the equinoxes and solstices. (Consequently, its
  celestial longitude has to be adjusted for precession or the slow
  motion of the equinoxes and solstices over time.)
\end{itemize}
%
The total \valana\ is the algebraic sum of these two components.

\subhead{Parallax.}  In most early astronomy, the moon's parallax was
computed as a single function of its zenith distance
\citep[123--125]{montelle11}.  In the Indian tradition, however, two
components of lunar parallax are computed independently; indeed, there
is no general term in Sanskrit for `parallax' \emph{per se}, or the
angle here termed $p$.  The two orthogonal parallax components are
based on the coordinates of celestial longitude and latitude with
respect to the ecliptic or apparent orbital path of the sun (see
verses 2.3--6).  The right side of figure~\ref{ParallaxDiagram}
illustrates them for the moon positioned approximately on the
ecliptic.  Longitudinal parallax (\lambana) along the ecliptic is
determined chiefly by the distance of the body from the nonagesimal
$V$, a point on the ecliptic 90$\deg$ west of the ascendant or
intersection point of the ecliptic with the eastern horizon.
Latitudinal parallax (\nati) perpendicular to the ecliptic is based on
the zenith distance of the nonagesimal, hence dependent on the
situation of the ecliptic with respect to the local zenith $Z$.  Each
of these components is considered to have a theoretical maximum more
or less equivalent to the angular distance the moon moves in 4
\ghatis\ of time, i.e., about 53 arcminutes (a little under 18
digits). This arc is in fact approximately the maximum lunar parallax
attested in other ancient astronomy traditions \citep[125,
241--244]{montelle11}.

\subhead{Diagram.}  The \valana\ and magnitude for any eclipse are to
be depicted in a graphical representation: namely, a diagram or set of
diagrams portraying its characteristics and key phases (see verses
1.10--13).  Careful instructions are usually given for geometrically
constructing the disks of the obscuring and obscured bodies, the
directions of the ecliptic and lunar orbit, and the trajectory of the
eclipsed body in the diagram.  Most works also enumerate the varying
colours of the eclipsed body. Various shades of reds, blacks, and
browns are mentioned (e.g., \textit{dh\=umra, k\*r\*s\*na, kapila,
  rakta}) and are often correlated to the phases of the eclipse, or
sometimes to its position in the sky.  These are presumably intended
to help determine the astrological significance of the event.

%%% SECTION 3
%%% SECTION 3
%%% SECTION 3

\section{Relationships between Tables and Texts} 
\label{IntertextualRelations}

\lettrine[lines=2,loversize=.1]{T}{he} verses of the \kks\ text have a
different emphasis than those found in the Sanskrit astronomical
genres of treatises and handbooks (\siddhantas\ and \karanas), whose
versified algorithms and parameters constitute a standalone system of
computation for astronomical prediction.  The primary role of the
\kks's verses, on the other hand, is to support a set of tables; to
this end, much of their content concerns the selection and
manipulation of table entries to produce a desired result.  These
verses, along with the titles and marginal notes found in the tables
themselves, offer insight into the complex interplay between numerical
data and text.

However, the \kks's approach remains closely connected to those of
earlier works in the \siddhanta\ and \karana\ genres, in both direct
and subtler ways.  We consider some of these connections below.


\subsection{Dependence of the \kks\ on earlier texts}

\Bhaskara's parameters for mean lunar and nodal motion are typical of
the `school' in Indian astronomy known as the \Brahmapaksa, dating
from at least the 7th century \tsc{ce}.  Moreover, his assumption that
the precession correction was zero in year 444 of the \Saka\ era (see
verse 1.9 below) was shared by such earlier authors as Par\=a\'sara,
P\*rth\=udaka\-sv\=amin, and \=Amar\=aja.%
%
\footnoteA{See \citet[13--16]{pingree81} for parameter values and
  \citet{pingree72} for a discussion of precession in Indian
  astronomy.}

In addition to this common scientific heritage, the \kks\ relies
directly on previous compositions for much of its textual content. Two
of its sources that can be confidently identified, due to the author's
own acknowledgement or to obvious borrowing, are both \karana\ texts
consisting solely of verses with no accompanying sets of tables:
namely, the \KK\ of \Bhaskara~II (\citealt{mishra91}, \citealt{raouma08},
epoch 1183 \tsc{ce}) and the \GL\ of \Ganesa\ (\citealt{joshi06},
\citealt{raouma06}, epoch 1520 \tsc{ce}).  The \kks's indebtedness to
these works manifests itself in a number of different ways: Bh\=askara
may excerpt phrases or entire \padas\ and integrate them into his
text, paraphrase an existing algorithm or technique, or use data
extracted from another 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 is based on a work called the
\textit{Siddh\=antarahasya}, a well-known alternative name for the
\GL\ \citep[2, 94]{pingreecess}:
%
\begin{quote}
  \textit{atha kara\*nakesarigra\*mthokte siddh\=a\*mtarahasye
    s\=urye\*mdvo\*h
    parvana\-yan\=arthe ca\*mdrasya ko\*s\*tak\=a} \elp  \\
  Now, tables of the moon for the sake of computation of eclipses of
  the sun and the moon in the \textit{Siddh\=antarahasya} as expounded
  in the book \kks.  (\xtsc{MS.\ P}$_1$, f.\ 4r)
\end{quote}
%
Similarities in parts of the accompanying text as well as the
procedures described confirm this dependence.  For instance,
\Bhaskara's description of parallax and his application of parallax to
produce true half-durations (see verses 2.5--6) parallel the
procedures set out in the \GL.

The inspiration that our author derived from the \KK\ is particularly
evident in his description of the graphical projection of an eclipse,
as well as his determination of parallax.  \kks\ 1.10--13 contains
several \padas\ identical to ones in \Bhaskara~II's section on
graphical projection.  The \kks's table for determining longitudinal
parallax (see verse~2.3) is based on a much briefer versified table in
\KK\ 5.4--5: the \kks\ linearly interpolates between the \KK's
nine specified values to produce a corresponding value for each of its
90 degrees of argument.

In addition, \Bhaskara\ has drawn from non-astronomical sources in
constructing his verses: for example, verse 2.1 imparts aphoristic
wisdom lightly paraphrased from literary classics or traditional
\textit{Subh\=a\*sita} works (proverbs and epigrams; see
footnote~\ref{SubhasitaRefs}).

\subsection{Interdependence of the  \kks's tables and text}

\Bhaskara's verses are intended both to supplement and to explain the
tables: most of the text makes no sense without the information
contained in the layout and headings of the tables, and the tables
would be useless without crucial parameters and computational steps
supplied by the text (or, in many cases, lacking in the text and
presumably supplied by the user's own knowledge).  The focus of the
prescribed procedures shifts back and forth between computational
algorithms in the verses and the numerical entries in the tables:
while many verses directly refer to operations with tabular values,
others merely clarify the concepts being discussed or specify
procedures that are unconnected with the table data.  Conciseness of
expression is also balanced against practical convenience in the
user's computations.%
%
\footnoteA{For example, the construction of the graphical projection
  of an eclipse in 1.10--13 is distinct from the table manipulations,
  while the algorithm for correcting longitudinal parallax in verse
  2.3 requires the user to divide by 40 the values in the
  corresponding table on f.~10r.  It would seem more convenient from
  the user's point of view to have constructed the table with its
  entries pre-divided by 40 in the first place.  Likewise, limiting an
  angular table argument to the first quadrant reduces the table's
  size, but imposes on the user the additional task of reducing
  arbitrary arcs to first-quadrant equivalents (which \Bhaskara\
  explains how to do in verse 1.7).}
%

Thus the \kks\ is in some sense not so much a self-contained set of
tables as an abbreviated handbook or \karana\ with some computational
procedures and parameters replaced by table entries.

\subsection{MS.\ J's worked examples for using the \kks}
\label{MSJ}

RORI Jaipur 9757 (\xtsc{MS.~J}), a 1761 manuscript of 6 folia, is written in
a mixture of regular technical Sanskrit and an early modern Indo-Aryan
vernacular.%
%
\footnoteA{A characteristic blend of the two forms is seen in these
  sentences from f.~1r,  line 10: \\ 
  {\mbox\null\quad \itshape atha s\=uryas\=adhana\*m $||$ \\
    \mbox\null\quad prathamaje avadh\={\i}m\=ahegrahegraha\*nath\=a-\\
    \mbox\null\qquad yetenos\=uryamadhyama} }
%
It consists of two detailed \textit{ud\=ahara\*nas} or worked examples
of calculations with the \kks\ tables according to the procedures
described in its text.  The first of these is for a lunar eclipse
whose date is given (f.~1r) as Wednesday, the fifteenth \tithi\ of the
bright half of \=A\'svina in Sa\*mvat 1801, \'Saka 1666.  If the
\'Saka year is taken as expired rather than current, this appears to
correspond to Wednesday 21 October 1744 \tsc{ce}, to which a (partial)
lunar eclipse is also assigned by modern calculations.%
%
\footnoteA{Both historical eclipse events mentioned here were
  established via the USNO Eclipse Portal, \url{astro.ukho.gov.uk/
    eclbin/query_usno.cgi}.}
%
The second eclipse, a solar one, is said (f.~3v) to have taken place
on Sunday, the new-moon \tithi\ of the dark half of M\=agha in
Sa\*mvat 1748 or \'Saka 1613: i.e. (again taking the calendar year as
expired), Sunday 17 February 1692, when an annular solar eclipse would
have been visible as a partial eclipse in most of the South Asian
subcontinent.

We have occasionally used these examples to elucidate some point about
the \kks\ text or tables in our discussion of them in
section~\ref{TransComm}, but we are deferring a detailed study of
\tsc{MS.~J} to a more thorough investigation of the tables themselves, as
proposed in section~\ref{Intro}.


%%% SECTION 4
%%% SECTION 4
%%% SECTION 4

\section{\kks: Translation and Commentary}
\label{TransComm}

\begin{tika}%
  \'sr\={\i}ga\*ne\'s\=aya nama\*h $||$ \'sr\={\i}gurubhyo nama\*h
  $||$ \'sr\={\i}s\=arad\=ayai nama\*h $||$ atha kara\*nake\'sar\={\i}
  likhyate $||$

    \sflight  \medskip\noindent Homage to Ga\*ne\'sa.  Homage to the Gurus.
    Homage to \'S\=arad\=a.{\rm \footnoteA{\'S\=arad\=a is another name for
    Sarasvat\={\i}. The manuscript gives the variant form S\=arad\=a.}}  Now the
    \kks\ is written.
%DW kludge here, putting the \footnoteA inside an \rm environment to stop
%\sflight interfering with the vertical spacing of the footnotenum.
\end{tika}



\begin{mula}{1.1 Salutation}%
  \'sr\={\i}k\*r\*s\*nacandracara\*na\*m pra\*nipatya bhakty\=a\\
  jyotirvid\=a\*m bahuvid\=am abhivandana\*m ca $||$\\
  k\*rtv\=a kav\={\i}ndrakulabh\=u\*sa\*nabh\=askar\=akhyo \\
  r\=am\=atmaja\*h kara\*nake\'sarim \=atanoti $||$ 1 $||$\\

	\sflight
  \medskip\noindent Having bowed to the feet of \'Sr\={\i} \Krsna\
  with devotion, and making respectful salutation to the very learned
  knowers of \jyotisa, the one called \Bhaskara, the ornament of the
  {Kav\={\i}ndra} family (\textit{kula}), the son of R\=ama,
  illuminates the \kks.
\end{mula}

\subsection*{Verse Analysis}

Metre: \vas.

This invocatory verse introduces the author of the \kks, \Bhaskara,
and makes reference to his lineage.  \Bhaskara\ is the son of R\=ama,
a descendant of a family (\textit{kula}) called {Kav\={\i}ndra}.%
%
\footnoteA{We do not know who is meant by the term `{Kav\={\i}ndra}
  family'.  The name is reminiscent of the famous \Kavindracarya\ (see
  \citet{sastry21}), but he is said to hail from one Pu\*nyabh\=umi
  identified with Paithan near Aurangabad in the Deccan, while our
  author appears to identify himself as a member of the Mo\*dha
  Brahmanas associated with the Gujarat area (see verse 2.7).}  As
well as being a proper name, \textit{bh\=askara} `light-maker' is a
noun that means `the sun'.  Thus \Bhaskara\ has invoked a pun on his
own name; he will naturally be the one to `illuminate'
(\textit{\=atanoti}, from the Sanskrit verb \textit{\=a-tan}, to
illuminate, to spread light) astronomical techniques and principles in
the \kks.

In the last \pada\ or quarter-verse, the accusative form of the name
\kks\ (`the mane-bearer/lion among \karanas', i.e., pre-eminent) is
rendered \textit{-kesarim} rather than \textit{-kesari\*nam} to
conform to the requirements of the metre.

\begin{mula}{1.2 Lunar node elongation}%
%
  \'sako r\=amavi\*s\*nupad\=a\.ngendu 1603 h\={\i}no\\
  viyadr\=amacandrair h\*rto 130 labdha\'se\*sau $||$\\
  yutau copakar\*na\*m hy avadhyanvita\*m tat-\\
  sap\=atendut\=atk\=aliko b\=ahubh\=agai\*h $||$ 2 $||$\\

  \sflight\medskip\noindent The [current] \Saka\ year is decreased by 1603
  [and] divided by 130, and [the tabular entries corresponding to the
  amounts given by] the quotient and remainder are added [together].
  The `result' (\textit{upakar\*na}) is increased by [the tabular
  amount corresponding to the number of elapsed] \avadhis.  That
  [resulting sum] is the [lunar node's elongation from the sun
  corresponding to the] `lunar elongation' (\textit{sap\=atendu}) at
  that time.
%
\end{mula}

\subsection*{Verse Analysis}

Metre: \bhuj;%
%
\footnoteA{This metre, a form of \jagati\ (12 syllables in a \pada),
  is comprised of the four-\textit{ga\*na} sequence \textit{ya ya ya
    ya}.  For details, see for instance, \citet[vol.~3, Appendix A,
  p.~5]{apte70}.}
%
the first \pada\ is hypometric.

The term \textit{upakar\*na} is puzzling.  In a mathematical context,
\textit{kar\*na} (literally `ear') means `diameter' or `hypotenuse'.
Context suggests it refers to a different quantity, discussed below.
%% 
%% In a mathematical context, \textit{\'se\*sa} means `remainder' and
%% \textit{kar\*na} (literally `ear') means `diameter' or
%% `hypotenuse'.  These two words when compounded together would
%% translate then as `remainder-hypotenuse', which makes little sense
%% here. Context suggests it refers to a different quantity, discussed
%% below.

The last word in the fourth \pada, namely \textit{b\=ahubh\=agai\*h},
belongs to the next verse.

\subsection*{Technical Analysis}

This verse gives a rule for determining the longitudinal elongation
between the sun and the lunar node on the ecliptic (in zodiacal signs,
degrees, minutes, and seconds) for a specified date.  This elongation
determines whether or not an eclipse will be possible.

In principle, the eclipse limit is determined rather by the elongation
of the \emph{moon} from its node, as suggested by the term
\textit{sap\=atendu}, literally `(the longitude of) the moon increased
by (the longitude of) its node' (see
section~\ref{EclipseTheoryComp}).%
%
\footnoteA{The reason that this `increase' or sum actually signifies
  longitudinal \emph{elongation} of the moon from its node is that the
  node revolves in the `backwards' or westward direction, making the
  nodal longitude negative.  Adding the positive lunar longitude to
  the negative nodal longitude gives the elongation between them.}
%
However, as noted above and confirmed by the headings of the
associated tables discussed below, the quantity tabulated is instead
the elongation of the \emph{sun} (\textit{s\=uryasya}) from the moon's
node.  We reconcile this seeming contradiction by noting that the sun
and moon at an instant of true syzygy are by definition either 0$\deg$
or 180$\deg$ apart, so 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
data, whose behavior we reconstruct as follows.  The node-sun
elongation for the total time since epoch is dependent on the
fundamental period relation between time and the revolutions of the
node.  The parameters appear to be derived from the \Brahmapaksa\
tradition in which the moon's node makes $-$232,311,168 revolutions
(the negative sign indicates retrograde motion) in 4,320,000,000
years.  This produces the following mean yearly motion for the node:
%
\begin{equation*}
{\displaystyle\frac{{-232,311,168 \times 360}}{{4,320,000,000}}} = -19;21,33,21,... ^{{\circ}/\hbox{\rm year}}
\end{equation*}
%
Since the sun returns to the same zero-point of longitude at the start
of every (sidereal) year, the mean yearly increment in the node's
elongation from the sun is identical to its mean yearly (negative)
increment in longitude.

If $Y$ is taken to be the current \Saka\ year, then the prescribed
algorithm divides the difference between $Y$ and 1603 (which we infer
to be the epoch year of the tables in the \Saka\ era) by the constant
term 130.  The quotient, which we will call $E$, consists of an
integer result (the number of completed 130-year periods) plus a
remainder (the number of single years subsequently completed):
%
\begin{equation*}
E = {\displaystyle{\frac{{Y -1603}}{{130}}}}.
\end{equation*}
%
The total time since epoch is broken into three parts:
\begin{itemize}
\item the number of completed 130-year intervals elapsed since the
  epoch (the integer part of $E$, called \labdha\ or `quotient' in the
  table heading);
\item the number of completed single years elapsed in the current
  130-year period (the fractional part of $E$, called \sesa\ or
  `remainder');
\item and the number of completed 14-day intervals elapsed in the
  current year (\avadhis, described in the table heading as
  \textit{candrasya ko\*s\*thak\=a} or `table entries of the moon'
  despite the fact that we are computing nodal-\emph{solar} rather
  than nodal-lunar elongation).
\end{itemize}

These three quantities are the arguments in the three associated
tables: the elongation (modulo integer revolutions) computed for
successive numbers of 130-year intervals from 1 to 130 (f.~3r), the
elongation computed for numbers of 1-year intervals from 0 to 130
(f.~3v), and the elongation computed for successive numbers of 14-day
\avadhis\ from 1 to 27 (f.~4r).  (So in principle, the tables would be
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 the quotient and remainder parts of $E$ gives the total amount of
elongation corresponding to the beginning of the chosen year, called
in the verse \textit{upakar\*na}.  Then, considering the number of
14-day periods that have elapsed since the start of the year, the
appropriate tabular entry from the third table can be added to produce
the elongation for the chosen date within that year.

The first two tables are based on the assumption that the elongation
changes linearly, with a constant difference from one entry to the
next.  In the table of 1-year increments this constant difference is
$-19\deg;21$, derived from the period relation shown above (apparently
truncated rather than rounded).
%
In the table of 130-year increments the constant difference is
$-3\deg;16,20$, evidently derived as follows: 

\medskip

\begin{tabular}{l c l}
  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 - 2516;43,40$\\
  &$=$&$ 2520 - 2516;43,40$\\
  & $=$&$ 3;16,20$\\
\end{tabular}

\medskip

This table of one-year increments also incorporates an epoch
correction in its first entry---that is, it begins with the value
$6\signs,29\deg;24,36$, which is presumably the amount of elongation
at the beginning of 1681 \tsc{ce} or \'Saka 1603.  Although Bh\=askara
does not specify exactly what calendar moment he associates with this
value, we assume that he equates the year-beginning with
\textit{caitra\'suklapratipad} (the first new moon after the vernal
equinox) which according to modern retrodiction corresponds to noon 20
March 1681.%
%
\footnoteA{This converted date was computed using Michio Yano's online
  Pa\~nc\=a\.nga program for the date \suklapaksa\ 1 of
  (\textit{am\=anta}) month Caitra, \Saka\ 1603.  See
  \url{http://www.cc.kyoto-su.ac.jp/~yanom/pancanga/.}  In fact, a
  solar eclipse took place at this conjunction, but it was not visible
  anywhere on land.  Reconstructing the epoch correction from the
  \Brahmapaksa\ \kalpa\ parameters with the total number of elapsed
  years equal to 1972948782 (i.e., the accumulated years up to the
  start of the \Saka\ era plus 1603 within the \Saka\ era) gives the
  proportional amount of nodal longitude in degrees accumulated at the
  epoch:
  \\
  \bgroup\ttfamily
  \mbox\null divmod$\mathtt{((-232311168)*(1972948782)}$, \\
  \mbox\null\quad  $\mathtt{4320000000)=(-106096768,1809402624)}$ \\
  \mbox\null N$\mathtt{(1809402624 * 360/4320000000) =}$\\
  \mbox\null\quad $\mathtt{150.783552000000}$
  \egroup\\
%
%
  Recall that this nodal longitudinal increment of approximately 150
  degrees or 5 signs is negative; when added to 0$\deg$ of solar
  longitude it comes out to a little less than 7 signs of nodal-solar
  elongation, which more or less corresponds to \Bhaskara's epoch
  value 6$\signs$,29$\deg$;24,36.  The sun is not usually exactly at
  0$\deg$ longitude at the \textit{caitra\'suklapratipad} conjunction,
  but in this case it appears to have been very close to it.}

The third table tabulates the change in elongation for each of the 26
14-day periods (\avadhis) during the solar year, plus a 27th entry to
account for the leftover day after the first 26 $\times$ 14 = 364.
The division of the year into these 14-day intervals is particularly
useful for eclipses as conjunction and opposition occur roughly two
weeks apart.

\begin{figure}[h]
  \includegraphics[width=\linewidth]{KKe-tables-f4r-elongation}
  % \centering
  \caption{An excerpt from the table of the change in elongation for
    each \avadhi, f.~4r}
  \label{KKStablesf4relongation}
\end{figure}

%%%%%%%%%%%%%% 	HARD PAGE BREAK FOR LONG TABLE 
% \newpage
\def\0{\phantom{0}}
\begingroup
\tabcolsep=10pt
\footnotesize
\begin{longtable}{l|l|l|l}

  \caption{The changes in elongation for the 14-day \avadhis\ during
    the year
    \label{kkselongation}}\\
  \hline
  % 14 day period  &Elongation  &Velocity   &\\
  Argument & Row 1 & Row 2 & Row 3 \\ \hline
  \endfirsthead
  \multicolumn{4}{c}{Table \thetable\space contd\dots}\\[6pt]
  \hline Argument & Row 1 & Row 2 & Row 3 \\ \hline
  \endhead
  \hline \multicolumn{4}{r}{\rule{0pt}{15pt}Table contd\dots}
  \endfoot
  \hline
  \endlastfoot
%
%\begin{table}
%  \footnotesize \tabcolsep=8pt
%  \centering
%  \begin{tabular}{c|l|l|l}\hline
%    %    14 day period &Elongation &Velocity &\\
%    Argument & Row 1 & Row 2 & Row 3 \\  \hline
  \01&    0    2    19    55&    61    46&    1 $\|$  \r \\
  \02&    0    16    41    1&    61    21&    2\\
  \03&    1    0    56    10&    60    53&    2\\
  \04&    1    15    5    26&    60    26&    1\\
  \05&    1    29    10    35&    60    16&    0 $\|$ \\
  \06&    2    13    12    50&    60    7&    0 $\|$ \r \\
  \07&    2    27    14    8&    60    5&    1 \dha \\
  \08&    3    11    16    1&    60    13&    1\\
  \09&    3    25    20    26&    60    26&    1\\
  10&    4    9    28    57&    60    53&    2\\
  11    &4    23    42    48&    61    11&    2\\
  12& 5 8 3 0& 61 42& 2\\ \newpage
  13&    5    2[2]    30    39&    62    14&    2\\
  14&    6    7    5    27&    62    51&    2\\
  15&    6    21    48    7&    63    18&    2\\
  16&    7    6    37    34&    63    44&    2\\
  17&    7    21    33    11&    64    7&    2 \\
  18&    8    6    33    34&    64    31&    0 $|$ \dha \\
  19&    8    21    35    47&    64    33&    0 $|$ \dha \\
  20&    9    6    39    40&    64    41&    0 $|$ \r \\
  21&    9    21    43    21&    64    30&    1\\
  22&    10    6    43    21&    64    16&    1\\
  23&    10    21    46    16&    6$\langle$4$\rangle$[3]    59&    1\\
  24&    11    6    39    35&    63    21&    1 $|$ \\
  25&    11    21    16    25&    62    56&    2 \\
  26&    0    5    53    20&    62    26&    2 \\
  27&    0    20    22    9&    61    56&    1 \r \\
  % \hline
  % \end{tabular}
  % \caption{The changes in elongation for the 14-day \avadhis\ during
  % the year}
  % \label{kkselongation}
\end{longtable}
\endgroup


Unlike the previous two tables, the \avadhi\ table contains a
non-linear sequence of values: that is, the differences between
successive entries are not constant, but vary sinusoidally.  It also
differs from them markedly in structure, with three rows of data below
each of its 27 argument values.  (See
figure~\ref{KKStablesf4relongation} for an example of its format in
the manuscript, and table~\ref{kkselongation} for a complete
reproduction of its content. In this reproduction, which displays the
original rows vertically rather than horizontally for convenience in
layout, an unmistakable scribal error is marked off by angle brackets
and missing correct values are supplied within square brackets.)

The first row contains the increment in elongation while the second,
labeled \textit{gataya\*h} `(daily) motions', gives the true velocity
of the elongation between sun and lunar node, presumably for the
beginning of each \avadhi.  The minimum value of this velocity is
60';5 at \avadhi\ 7 and the maximum $64';41$ at \avadhi\ 20.  These
extrema evidently fall respectively at the solar apogee, where the
sun's motion is slowest, and the perigee where it moves fastest; see
verse~1.3 below.  The third row, which is unlabeled, contains entries
of 0, 1 and 2 in a distribution whose rationale is unclear,
occasionally accompanied by symbols for the \nagari\ characters \dha\
and \r, standing for \dhana\ and \rna\ or `positive' and `negative'
respectively.  The positive and negative symbols appear to be applied
consistently with the transitions from increasing to decreasing values
of the velocity in the second row.  The entries in the third row may
thus form part of some second-difference interpolation rule that
\Bhaskara's text does not explain.

It may be that the reason for the non-constant differences between the
entries in the first row is that the variation in solar velocity has
been taken into account.  That is, the elongation per \avadhi\ could
have been computed via the following relation:
%
\begin{equation*}
  \hbox{{\rm Elongation per} \avadhi} = 14v\sb{\hbox{\rm sun}}^{(\circ/\hbox{\rm day})}
  + 14\overline{v}\sb{\hbox{\rm node}}^{(\circ/\hbox{\rm day})}
\end{equation*}
%
where $v\sb{\hbox{\rm sun}}$ is the true velocity of the sun and
$\overline{v}\sb{\hbox{\rm node}}$ is the mean velocity of the node.
The average difference between entries is about $14';30$, which more
or less agrees with the fact that the mean sun is moving approximately
$0\deg;59$ per day and the lunar node roughly $0\deg;44,20$ in the
opposite direction over 14 days:

\medskip

\begin{tabular}{l l}
  `Average' value of increments & $=0;59 \times 14 + 0;44,20$\\
  &$= 13;46 + 0;44,20 $\\
  &$= 14;30,20$\\
\end{tabular}
%

\medskip
\noindent
This reconstruction approximately describes but does not exactly
reproduce the entries in the table.

\begin{mula}{1.3 Lunar latitude and sizes of disks}%
%
  mitai\*h ko\*s\*thakair a\.ngul\=adi\*h \'sara\*h sy\=at \\
  sap\=atasya bhuktyupari helibimbam $||$\\
  tither m\=anagha\*tyopar\={\i} candrabh\=ubhe \\
  tul\=aj\=adi\*sa\*dbhonayuk spa\*s\*takubh\=a $||$ 3 $||$\\

  \sflight\medskip\noindent The latitude in digits and so on should be
  [determined] by the table entries commensurate with the degrees of
  the \textit{b\=ahu} [arc of elongation; see the end of the previous
  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 diameter of] the moon and the shadow of the earth
  [have as their] argument the measure of \ghatis\ in a \tithi.
  [That, i.e., the diameter of the shadow of the earth] is decreased
  or increased by [the appropriate tabular entry according as the sun
  is in] the six signs beginning with Libra or with Aries
  [respectively].  [This is] the corrected shadow of the earth.
\end{mula}

\subsection*{Verse Analysis}

Metre: \bhuj.%
%

The metre and/or the grammar in this verse are defective in several
instances, and we have tried to compromise between respecting their
requirements and accepting the manuscript readings.  In the second
\pada, the last syllable in \upari\ should be heavy for the sake of
the metre, but we leave it light as the correct ending for a neuter
nominative singular \textit{i}-stem \bahuvrihi\ compound agreeing with
the neuter noun \textit{bimba} that it modifies.  Likewise, the third
\pada\ contains a \dvandva\ compound ending in \textit{bh\=a} (f.\
`shadow') in the manuscript; we have corrected it to the feminine
nominative dual ending \textit{-bh\=e} and made the final vowel of
\textit{upari} long in the preceding \bahuvrihi\ compound to agree
with it.  In the fourth \pada, the syllable \textit{ku} (f.\ `earth')
should be long for metrical purposes.

The above compound forms with \upari, an indeclinable adverb meaning
`above, upon, on', are peculiar to this type of technical terminology.
Like other adverbs, \upari\ can be used as a modifying element in a
compound, as in \textit{upariku\*t\={\i}} meaning `upper room'. But in
this verse and in many of the table titles, it is used instead as the
\emph{primary} member of a \bahuvrihi\ compound: e.g.,
\textit{bhuktyupari} `having the velocity [as its] superscribed
[argument]'.%
%
\footnoteA{The use of \upari\ as a technical term meaning the argument
  of a table is attested in other table texts as well, including
  N\=agadatta's \textit{Brahmatulyas\=ara\*n\={\i}}; e.g., in MS.\
  Poleman 4735 (Smith Indic 45) f.~7v.}
%
The numbers indicating the values of that argument are written above
or at the tops of the respective columns.

Interestingly, the first part of the compound word \textit{heli-bimba}
meaning `disk of the sun' is a loan-word derived from the Greek
\textit{helios}.

\subsection*{Technical Analysis}


The apparent sizes of the disks of the eclipsed and eclipsing bodies
and the distance in latitude between their centers are necessary for
determining the amount of obscuration.  Qualitatively, the latitude of
the moon on its orbit is proportional to its elongation or distance
from its node, with the maximum latitude (conventionally $4\deg;30$ or
$270'$ \citep[16]{pingree81}, equivalent to 90 \angulas\ or digits of
linear measure) occurring when the moon is $90\deg$ from the node.
And the apparent size of the disk of sun or moon depends on its
distance from the earth, which is proportional to its apparent angular
velocity. The size of the earth's shadow also depends on the distances
of the moon and sun from the earth (as the moon gets closer it passes
through a wider circular cross-section of the shadow cone, while the
sun getting closer makes the shadow cone shorter; see
figure~\ref{ShadowConeDiagram} in section~\ref{EclipseTheoryGeom}).
The mean apparent size of the disk of the moon or the sun is
realistically assumed in Indian astronomy to be about 32 arcminutes or
10--11 digits, while that of the shadow disk is about $1;21\deg$ or 27
digits \citep[223]{montelle11}.  This verse explains how to adjust
these values to find their true sizes at a given time, which will
determine the appearance of the eclipse.

The first part of the verse refers to the table of lunar latitude on
f.~4r, which has three rows.  The first is labeled
\textit{bhuj\=a\*m\'sa} (`degrees of the \textit{bhuja}' or \bahu,
both meaning `arc' or `angular argument'), ranging from 0 to 16.  A
note in the margin asserts that this argument refers to the moon's
elongation from its node (\textit{atha
  sap\=atacandrabhuj\=a\*m\'sopari}\elp).  Evidently, any nodal
elongation greater than $16\deg$ is considered to negate the
possibility of an eclipse and thus is not tabulated.

The second row, labeled \textit{\'sar\=a\*mgul\=a\*h} `digits of
latitude', runs from $0;10$ (evidently a scribal error for $0;0$,
since the accompanying value of the difference between the first and
second entries is equal to the second entry itself) to $24;45$ digits.
These latitude values appear to have been computed more or less in
accordance with a rule from the \KK\ of \Bhaskara~II (verse 4.5,
\citealt[S67]{raouma08}) equivalent to setting the moon's latitude in
digits equal to 90 times the (modern) sine of its nodal elongation in
degrees. For instance, by this formula the maximum elongation of
$16\deg$ corresponds to a maximum latitude of $24;48$ digits, nearly
identical to the \kks's $24;45$.

The third row gives the differences between successive entries in
row~2, and it is noted in the row header that this difference is
always positive (\textit{'\*mtara sad\=a dhana\*m}).
%
% \footnoteA{\textit{tasya bhuj\=a\*m\'sopari \'sar\=a\*mgul\=adi 
% a\*mtara\*m sad\=a dhana\*m \'saragolopari dig j\~n\=ana\*m
% me\*s\=ady\=avuttaro golastulaadau dak\*si\*na\*h sm\*rta}  (f.~4r).}  
%
The differences for entries 0 to 8 are $1;34$ and from 9 to 16,
$1;30$.

The second \pada\ of the verse refers to the final table on f.~4r,
which tabulates the diameter of the disk of the sun
(\textit{ravibimba}) using as argument the daily elongation of
(presumably) the sun from the lunar node (\textit{sap\=atasya
  gati\*h}), which ranges from $0\deg;59,56$ to $0\deg;64,42$ over
twenty entries.  This diameter of the sun (also in digits, evidently)
ranges from $10;19$ to $11;11$.  Qualitatively, this relation makes
sense because the variation in the nodal-solar elongation depends on
the change in the sun's apparent angular velocity (that of the node,
which has no anomaly of its own, is constant).  That is, when the sun
is moving faster, the daily nodal-solar elongation is bigger and also
the sun is closer to the earth, meaning that the solar disk appears
larger.  (However, this adjustment may not have been indispensable in
a lunar eclipse computation: the compiler of the lunar-eclipse example
in \tsc{MS.~J}, for instance, neglects it.)

The third \pada\ refers to a table on f.~4v, which has three rows. The first
quantifies the length of the Indian time-unit called the \tithi, running from
52 to 67 \ghatis. (As remarked above in section~\ref{EclipseTheoryGeom}, a
\tithi\ on average is one-thirtieth of a synodic month or a little less than
one civil day of 60 \ghatis. The so-called `true' \tithi, on the other hand,
i.e., the time required for a $12\deg$ increment in the elongation between the
true longitudes of sun and moon,  is of variable length because it depends on
their true velocities: the faster they move apart in longitude, the shorter the
current \tithi\ will be.)

The table's second row enumerates values of the diameter of the moon's
disk from $11;57$ to $9;30$, and the third row the diameter of the
earth's shadow disk, ranging from $30;45$ to $23;11$. Both these
diameters, like that of the sun, are specified in digits.  Their
values seem to be based on the dependence of the apparent diameters of
moon and shadow upon the geocentric distance of the moon, which is
inversely proportional to its velocity.  As previously noted, fewer
\ghatis\ in a \tithi\ implies higher lunar velocity, and consequently
a smaller lunar distance and a larger apparent disk for both the moon
and the shadow.

The last \pada\ of this verse proposes a positive or negative
incremental correction to the size of the disk of the shadow,
depending on what zodiacal sign the sun is in (positive when the sun
is in the six signs beginning with Aries, negative otherwise; see
figure~\ref{kks-tables-f4v-correctiontable} and
table~\ref{shadowcorrectiontable}).  The correction amount ranges from
$0;0$ digits in signs ending at an equinox to $0;20$ digits in signs
ending at a solstice.


\begin{table}[t]
  \centering
  \footnotesize
  \begin{tabular}{|llllll|l|}\hline
    Me[\*sa]  &V\*r[\*sabha]     & Mi[thuna]   &Ka[rka\*ta] &Si[\*mha]
    &K$\langle$\=a$\rangle$[any\=a]     
    & Sign-entries \\
    0;11    &0;16   &0;20   &0;16   &0;1$\langle$6$\rangle$[1]   &0;0    &Positive
    earth-shadow\\ 
    Tu[l\=a]  &V\*r[\'scika]   &Dha[nus] &Ma[kara] &Ku\*m[bha]
    &M$\langle$i$\rangle$[\={\i}na]   
    &  Sign-entries \\
    0;11    &0;16   &0;20   &0;16   &0;11   &0;0    &Negative
    earth-shadow\\ 
    \hline
  \end{tabular}
  \caption{Transcription of the table in
    figure~~\ref{kks-tables-f4v-correctiontable}} 
  \label{shadowcorrectiontable}
\end{table}

%\begin{figure}[h]
\begin{wrapfigure}{l}{2.25in}
  \centering \vspace{-4pt}
  \includegraphics{KKe-tables-f4v-correctiontable}
  \caption{The table of correction to the size of the disk of the
    shadow, f.~4v}
  \label{kks-tables-f4v-correctiontable}
\end{wrapfigure}
How these values were determined is not entirely clear. But we can
reconstruct at least a qualitative justification for them by appealing
to \Bhaskara~II's treatment in the \KK\ of the position and effect of
the solar apogee and the sun's anomaly (that is, its mean position
minus the solar apogee position, which in \KK\ 2.1 he puts at about
$78\deg$ of sidereal longitude \citep[S15]{raouma08}).  We recall that
when the sun is closer to the earth and consequently more distant in
anomaly from the apogee, it is perceived as moving faster and shows a
larger apparent disk, as well as shortening the earth's shadow cone.
The formula for correcting the diameter of the shadow in \KK\ 4.7--8
\citep[S69--70]{raouma08} takes this effect into account by equating
the correction to $3/67$ of the true lunar daily motion minus $1/7$ of
the solar daily motion, implying that a larger true solar daily
velocity means a smaller shadow diameter.

The \kks\ likewise appears to locate the solar apogee somewhere
shortly before the beginning of Cancer, as the \avadhi\ elongation
table discussed in verse 1.2 suggests: that is, it takes the sun 6
\avadhis\ or about 84 days from the start of the year at or near the
beginning of Aries to attain its slowest velocity.  Consequently, a
solar anomaly between $90\deg$ and $270\deg$ would mean more or less
that the sun is somewhere between the beginning of Libra and the
beginning of Aries, i.e., it seems faster and larger.

We are told in the present verse of the \kks\ to decrease the diameter
of the shadow by the tabulated correction when the sun is between
Libra and Aries, and to increase it otherwise, which qualitatively
agrees with the rationale above.  Moreover, the table entries are also
qualitatively consistent with it, since the largest corrections (both
positive and negative) occur near a solstice (when the sun is roughly
at perigee or apogee, respectively), and the zero corrections near the
equinoxes where the true and mean solar velocity are about
equal. However, it remains unclear whether or to what extent this
rationale accounts quantitatively for the \kks's actual correction
values.

%%% VERSE 1.4 %%%%%
%%%%
\begin{mula}{1.4 Eclipse magnitude}%
%
  ch\=adyach\=adakama\*n\*dal\=anvitadala\*m k\=a\*n\*donachanna\*m puna\*h \\
  channa\*m gr\=ahyavivarjita\*m tu nikhile gr\=ase 'pi khachannakam $||$\\
  chann\=a\.nkair mitako\*s\*thake sthitir bhaven mardasya khachannata\*h \\
  \'sr\={\i}bh\=anor uday\=at tath\=astasamay\=at
  parv\=antamadhyagraha\*h $||$ 4 $||$

  \sflight\medskip\noindent 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.  Again, the obscuration is
  decreased by [the disk of] the eclipsed; and in a total eclipse
  there is obscuration of the sky.  In the table commensurate with the
  numbers [of the digits] of obscuration is the half-duration.  [There
  is also a table] of the half-duration of totality [resulting] from
  sky-obscuration [i.e., in the case of a total eclipse].  [A solar
  eclipse is visible] from the rising of the Lord Sun just as [a lunar
  eclipse] from [its] setting.  Mid-eclipse [is] the end of
  the \parvan.
\end{mula}

\subsection*{Verse Analysis}

Metre: \sard. The second syllable ought to be heavy, as should the
sixteenth syllable in each of \padas\ 2 and 3, suggesting that the
author thought of the following consonant \textit{cha} in each case as
doubled.

This verse illustrates \Bhaskara's tendency to prescribe the steps of
an algorithm before identifying the quantity it produces.
%
It also exemplifies his penchant for technically ungrammatical ways of
forming and modifying compounds.  For instance, he expresses
\textit{$\circ$dala\*m k\=a\*n\*dona\*m channa\*m} `the \elp\ half-sum
diminished by the latitude [is] the obscured amount' as
\textit{$\circ$dala\*m k\=a\*n\*donachanna\*m}, literally `the \elp\
half-sum [is] the obscured amount diminished by the latitude'.%
%
\footnoteA{This theoretically incorrect mixed construction is by no
  means unique to \Bhaskara, however; see, e.g.,
  \citet[189]{tubbboose}.}
%

\subsection*{Technical Analysis}

The amount of the \textit{channa} or obscuration ($C$) is stated to be
%
\begin{equation*}
C = {\displaystyle{{1}\over{2}}(R + r) - \beta},
\end{equation*}
%
where $R$ and $r$ are the radii of the obscurer and the obscured
respectively and $\beta$ is the lunar latitude, all of which the user
will have found from the procedures mentioned in the previous verse.
As illustrated in figure~\ref{ImmersionDiagram} in
section~\ref{EclipseTheoryComp}, $C$ represents not the extent of the
eclipsed body's own diameter which is obscured, but rather the depth
to which the leading edge of the eclipsed body penetrates within the
disc of the eclipser.  The second \pada\ defines `obscuration of the
sky', i.e., excess of the obscuration $C$ over the diameter $2r$ of
the obscured disk, which of course implies a period of totality in the
eclipse. (The author of the example in \tsc{MS.~J}, for reasons unknown,
instead (f.~1v, line 8) multiplies $C$ by 20 and divides it by the
lunar disk's diameter.) In principle, the greater the obscuration, the
longer the eclipse will last, and the greater the amount of
`sky-obscuration', the longer the period of totality will be.

It follows that the second half of the verse invokes two tables, on
f.~4v, that tabulate the half-duration of the eclipse from the
beginning of obscuration to mid-eclipse, and the half-duration of
totality (if any) from the beginning of total immersion to
mid-eclipse.  It also adds the fairly trivial observation that solar
eclipses are visible when the sun is above the horizon and lunar ones
when it is not.  The `end of the \parvan', as noted in
section~\ref{EclipseTheoryGeom}, refers to the syzygy moment at the
end of a \tithi\ when the luni-solar elongation is either exactly
$180\deg$ or exactly $0\deg$.

The argument of the half-duration table is the digits of `obscuration
of the moon' (\textit{ca\*mdrachinn\=a\-\*mgula}, for
\textit{ca\*mdrachann\=a\*mgula}?).  Its maximum value of 21 digits
evidently means that the maximum possible `obscuration', with the
center of the moon coinciding with the center of the shadow, is taken
to be equal to the maximum value of the moon's radius plus that of the
shadow's radius or approximately $6 + 15 = 21$ digits.  Its
corresponding table entries for the mean half-duration
(\textit{ca\*mdrasya madhyasthitigha\*ti}) are in \ghatis\ and run
from $57;30$ to $55;17$.  In fact, as the worked lunar eclipse example
in \tsc{MS.~J} (f.~1v, lines 9--10) makes clear, these tabulated values must
be subtracted from 60 to give the actual extent of the mean
half-duration, anywhere from $2;30$ to $4;43$ \ghatis.  A separate
table at 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
which seems to be only an extension of the third row of a different
table immediately to the left of it, and contains only a (meaningless)
zero in each cell.  The first row contains the table argument, the
digits of `sky-obscuration' (\textit{khachinn\=a\*mgulopari}), running
from 1 to 9.  Again, the maximum value suggests that if the moon's and
shadow's centers coincide there will be $15-6 = 9$ digits of shadow
beyond the edge of the moon.  The second row contains the
corresponding \ghatis\ running from $59;0$ to $57;54$, apparently
indicating a range of $1;0$ to $2;6$ \ghatis\ for the actual
half-duration of totality (\textit{mardagha\*tik\=a}).

The last table on f.~4v, which is not alluded to in the text, lists
the mean solar longitude (with corresponding solar velocities and
their interpolation differences) for each \avadhi\ from 1 to 27.  The
compiler of the \tsc{MS.~J} example uses this table to get the mean solar
position for his lunar eclipse date (f.~1v, line 10), which he then
adjusts to the true position (f.~2r, line 2).

%%%%%% For both these tables, it is extremely puzzling that the time
%%%%%% intervals \emph{decrease} as their respective obscuration
%%%%%% amounts increase.  As noted above, we would expect the trend of
%%%%%% the entries to be directly correlated with that of the
%%%%%% argument: the further the moon penetrates into the shadow, the
%%%%%% longer the half-duration of the eclipse. Likewise in the case
%%%%%% of totality, a larger sky-obscuration should mean a more
%%%%%% oversized shadow that the lunar disk remains in for a longer
%%%%%% time, and consequently a longer half-duration of totality.  We
%%%%%% have not succeeded in satisfactorily explaining either the
%%%%%% `backwards' trend of these tables or the values in their
%%%%%% ranges: the half-duration table, for example, seems to imply
%%%%%% that an eclipse is always more than 100 \ghatis\ or nearly two
%%%%%% days long, which is clearly absurd.  Evidently determining
%%%%%% eclipse durations from these table entries requires an
%%%%%% additional computational procedure that has so far eluded us.

%%% VERSE 1.5 %%%%%
%%%%
\begin{mula}{1.5 Beginning and ending of eclipses; zenith distance and
    \aksavalana}%
%
  sthity\=a mardena h\={\i}na\*h spar\'sasanm\={\i}lane syu\*h \\
  mardasthity\=anvite 'sminn unmila\*m mok\*sasa\*mj\~nam $||$\\
  kh\=a\.nkair nighna\*m 90 nan\=apta\*m ghasram\=ane nat\=a\*m\'s\=as \\
  t\=anai\*h 45 \'sodhy\=a\*m\'sako\*s\*the pr\=akpare saumyay\=amye
  $||$ 5 $||$\\ 

  \sflight\medskip\noindent [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 total immersion
  [respectively].  When that [i.e., 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 `time-arc',
  time from noon or hour-angle] by 90 [and] dividing by its own in
  [the sense of] the day-measure, [the result is] the degrees of
  [half] the zenith distance (\nata).
%% When 
  In the tabular entry for diminished degrees by 45 [one finds the
  \aksavalana].  [When the body is] East or West [of the meridian,
  respectively, the (\aksa)-deflection is laid off] in the North or
  South [respectively].
\end{mula}

\subsection*{Verse Analysis}

Metre: \marata.%
\footnoteA{This is a form of \sakvari, a 14-syllable met\-re. Each
  \pada\ in \marata\ is split into two groups of seven.  See
  \citet[vol.\,3, App.~A, p.\,19]{apte70}.}
%

Lexically speaking, the Sanskrit word \textit{unm\={\i}la} in \pada~2
(more usually \unmilana) should have a long `i'.  However, it occurs
in the verse with a short `i' which is required for the metre.

The word \textit{t\=ana} in the last \pada\ is associated in the
Sanskrit \bhut\ or `word-numeral' system with the number 49.%
\footnoteA{See, e.g., \citet[64]{sarma03}.}
%
But in this context, the number must mean 45 (and indeed in \tsc{MS}.~\ROne\
the numeral `45' is written out). \Bhaskara's nonstandard approach to
compounding is again apparent in the use of the instrumental
\textit{t\=anai\*h} `by 45' to modify only the individual element
\textit{\'sodhya} `diminished' in the following compound.

\subsection*{Technical Analysis} 

This verse deals with the determination of the five significant
moments in an eclipse (see section~\ref{EclipseTheoryComp}). %
%
% \footnoteA{%
% In fact, the information available to the user of the tables at this
% point, including the disk sizes, lunar latitude, half-duration, and
% extent of obscuration, is still insufficient for computing an
% eclipse; for instance, one also needs to know when the end of
% the \parvan\ occurs.}
%
The first part of the verse shows how to find the beginning of the
eclipse or first contact and the beginning of totality, if any:
namely, subtract the length of the half-duration or the half-duration
of totality, respectively, from the time of mid-eclipse or end of
the \parvan\ (which presumably the user is expected to know from a
calendar or some other source).  Conversely, when these same intervals
are added to the mid-eclipse moment, the results specify the instants
of release (end of the eclipse) and beginning of emersion,
respectively.

The second half of the verse concerns the \aksavalana\ or `deflection
due to latitude'.  The `deflection' or \valana, as discussed in
section~\ref{EclipseTheoryComp}, is the angle between the directional
axes determined by the more-or-less easterly movement of the eclipsed
body and the true local cardinal directions aligned with the prime
vertical circle and the prime meridian shown in
figure~\ref{ValanaEast}: it is traditionally considered to have
ominous significance.%
%
\footnoteA{For further details, see \citet[229--230]{montelle11}.
%
  The diagram in figure~\ref{ValanaEast} is based on a similar diagram
  in \citet[125]{chatterjee81}.}  The \valana\ is conventionally given
in the form of two components, one due to the tilting of the celestial
equator off the prime vertical by the amount of terrestrial latitude
(\aksavalana) and the other to the skew between equatorial east and
ecliptic east at the body's current tropical longitude (\ayanavalana).

Instead of identifying or defining the quantity \aksavalana, this
half-verse gives a rule to find the zenith distance (\nata\
`depression', $\zeta$)---or rather, half of it---which is manipulated
to become the argument in the table of \aksavalana. The zenith
distance depends on the hour-angle, or amount of time in \ghatis\
before or after noon (which, somewhat confusingly, is also
conventionally termed \nata), and the length of the body's `own day',
i.e., the amount of time it is above the horizon:
%
\begin{equation*}
  \hbox{\rm (half)}\, \zeta = {\displaystyle{\hbox{\rm hour-angle}
      \times 90}\over{\hbox{\rm own day measure}}} 
\end{equation*}
%
This rule is similar to a standard approximation appearing in, e.g.,
\KK\ 4.14 \citep[S80]{raouma08}, which more or less converts hour-angle
(time) into zenith distance (arc) by the simple proportion $\zeta :
90\deg$ :: hour-angle : half-day.%
%
\footnoteA{The relevant part of the \KK\ verse \citep[56]{mishra91} has
  clearly influenced the phrasing of the \kks's rule:
  \\
  \textit{kh\=a\.nk\=ahata\*m svadyudalena bhakta\*m spar\'se vimuktau
    ca nata\*m lav\=a\*h syu\*h}
%
} By using the whole `own day' in place of the half-day in the divisor
of this proportion, the \kks\ actually computes \emph{half} the arc of
zenith distance.  (Note also that this procedure does not include the
information needed to determine the length of the `own day', which
appears eventually in verse~1.9).

\begin{figure}[h]
  \includegraphics[height=2.5in]{KKeAksaAyanaValanaE}
  \centering
  \caption{The \aksavalana\ $\protect\angle NMP$ and \ayanavalana\
    $\protect\angle P'MP$ of a body at $M$}
  \label{ValanaEast}
\end{figure}

Theoretically, the relation of the zenith distance $\zeta$ to the
\aksavalana\ is based on the following geometric rationale.  In
figure~\ref{ValanaEast}, showing the celestial hemisphere above the
local horizon with zenith at $Z$, $M$ is the deflected body and $P$
and $P'$ are the poles of the equator and ecliptic respectively (the
latter is lifted off the rear surface of the hemisphere for ease of
viewing).  Qualitatively, as noted above, the local latitude or
elevation of the pole $P$ above the northern horizon is responsible
for the \aksavalana\ component of deflection, while the splay of the
ecliptic away from the equator produces the \ayanavalana\ component.
The total \valana\ is thus equal to the angle $\angle NMP'$.  To
compute its \aksavalana\ part $\angle NMP$, we note that angle $\angle
MNP$ or arc $ZQ$ is the projection of the body's zenith distance $ZM$
onto the prime vertical, while $PM$ is the complement of its
declination $\delta$ and $NP$, the elevation of the north celestial
pole above the horizon, is the local latitude $\phi$.  By the law of
sines for spherical triangles, $\angle NMP$ is trigonometrically
determined by
%
\begin{equation*}
  \displaystyle{{\sin\angle NMP}} =
  \displaystyle{{\sin\angle MNP\cdot{\sin NP}}\over{\sin PM}}
\end{equation*}
%
Assuming that the hour-angle or \nata\ converted to degrees of zenith
distance $\zeta$ as discussed above is approximately the same as
$\angle MNP$, this equation is equivalent to a typical form of the
rule for \aksavalana\ in Indian astronomy:%
%
\footnoteA{Variant forms of this rule also existed, using slightly
  different functions or rougher approximations. The differing
  formulas for \valana\ and the disagreements about it between rival
  authors are discussed in, e.g., \citet[246--247]{montelle11}.}
%
\begin{equation*}
  \displaystyle{{\sin\hbox{\rm \aksavalana}}} =
  \displaystyle{{\sin\zeta\cdot{\sin \phi}}\over{\cos\delta}}
\end{equation*}
%
Note that if the body is at the zenith and/or the latitude $\phi$ is
zero, there is no \aksavalana\ angle $\angle NMP$: the \aksavalana\
increases with both zenith distance and latitude.  The maximum
\aksavalana\ for a given $\phi$, when $\delta=0$ and $\zeta=90\deg$,
is just $\phi$ itself.%

%\begin{figure}[h]
The \kks's \aksavalana\ table, on the other hand, requires a rather
peculiar manipulation of the half-$\zeta$ computed in this verse: the
user is to subtract it from $45\deg$ 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 \textit{nat\=a\*m\'sa} or
`degrees of \nata', runs from 0 to 45.  The second row, labeled
\valana, is given in degrees, minutes and seconds and runs from
$22\deg;35,39$ to 0.  The third column, labeled \textit{antara}
`difference', gives the differences between successive entries in
row~2.

\begin{wrapfigure}[17]{l}{2.8in}
  \centering
  \includegraphics[width=2.75in]{KKeAksaValanaPlots}
  \centering
  \caption{Plot of tabulated values of \aksavalana\ (dots), and the
    graph of $\arcsin(\sin(90\deg - \zeta)\cdot \sin\phi)$ versus
    $\zeta/2 = $ 0 to $45\deg$ for latitude $\phi = 22\deg;35,39$
    (line)}
  \label{ValanaPlot}
\end{wrapfigure}


These \aksavalana\ values progress backwards with respect to the order
one would expect from the \aksavalana\ equation shown above: for a
given $\phi$ and $\delta$, the \aksavalana\ ought to \emph{increase}
as $\zeta$ increases.  As we can see in figure~\ref{ValanaPlot}, the
values in the table actually conform to the graph of a `reversed'
\aksavalana\ function, $\arcsin(\sin(90\deg -
\zeta)\cdot\sin(22\deg;35,39)$, where it is assumed that $\phi =
22\deg;35,39$ and $\delta = 0$.  This and the instruction
\textit{tanai\*h \'sodhya}$\circ$ `diminished by 45' inform us that
the actual table argument running from 0 to 45 is $45\deg - \zeta/2$:
in other words, the table entries are given for every \emph{two}
degrees of zenith distance starting at 90 (when the \aksavalana\
attains its maximum value $\phi$ if $\delta$ is 0) and ending at 0.

\Bhaskara's final remark about directions can be justified as follows:
When the eclipsed body is east of the meridian, the equator-east
direction skews north of the easterly direction parallel to the prime
vertical, so the \aksavalana\ is north.  When the body is west of the
meridian, on the other hand, the \aksavalana\ is south.

The awkwardness of the \kks's presentation as a guide to actually
using the tables is indicated by the way the author of the worked
example in \tsc{MS.~J} handles these calculations.  First he corrects for
precession (f.~2r, line 4) the true solar longitude that he found at
the end of the procedure in the previous verse. This
precession-corrected longitude is entered into a table described in
verse~1.9 to get the length of the half-day (f.~2r, line 5).  Then he
follows the directions in verse~1.5 to find the moment of first
contact (f.~2r, lines 10--11; there is no period of totality since the
example concerns a partial eclipse), and determines the \nata\ and its
direction in accordance with the rule in verse~1.8 (f.~2v, lines
3--4).  He then proceeds to compute the \valana\ as specified (f.~2v,
line 11--f.~3v, line 4).

%%% VERSE 1.6 %%%%%
%%%%
\begin{mula}{1.6 \textit{Ayanavalana}}%
%
y\=amyottare di\'si grahasya yut\=ayan\=a\*m\'s\=a\*h \\
ko\*ty\=a\*m\'sake\*su valan\=antaragu\*nyayuktam $||$\\
yog\=antar\=a\*m\'sapramite valana\*m sphu\*ta\*m sy\=at\\
s\=uryasya gr\=asavidhumok\*savilomadeyam $||$ 6 $||$

\sflight\medskip\noindent
In the south [or] north direction, the precession-increased degrees of [the longitude of] the planet
[determine the \ayanavalana\ in the same direction respectively].
[The difference between the precession-increased degrees and the nearest table argument is]
multiplied by the [tabulated] \valana-difference and applied to the degrees of the complement
[of the precession-increased longitude].
The \valana\ [determined] in the amount of the degrees of the sum or the difference
[of the two components] should [now] be accurate.
[In the case of an eclipse] of the sun, [the \valana] is given in the reverse order
of release [and first contact] from [the case of] the eclipsed moon.
\end{mula}

\subsection*{Verse Analysis}

Metre: \vas.

In \Bhaskara's prosody, a consonant conjoined with a following `r'
appears to be counted as a single consonant that does not make the
preceding syllable heavy.  This can be seen in the present verse in, e.g., 
\textit{graha} in the first \pada\ and \textit{pramite} in the third
\pada, each of which is preceded by a syllable that must still be
considered as `light' to fit the \vas\ metre.

\subsection*{Technical Analysis}

Like the last verse, this verse does not explicitly specify the
quantity it is computing, in this case the \ayanavalana\ or
`deflection due to tropical [longitude]'.  The \ayanavalana\ (angle
$\angle P'MP$ in figure~\ref{ValanaEast}) is the amount of deflection
of the easterly direction along the ecliptic from the easterly
direction along the equator due to the position of the body's tropical
longitude $\lambda^T$ in its quadrant of the ecliptic.  Qualitatively,
when the body is at a solstice there is no deflection from this
effect, since east is in the same direction along both great circles.
At an equinox, on the other hand, the deflection is equal to the
obliquity of the ecliptic (conventionally $24\deg$), since the local
directions of the two circles diverge by that amount.

Quantitatively, the value of the \ayanavalana\ $\angle P'MP$ between
those two extremes, again by the law of sines for spherical triangles,
is given by
%
\begin{equation*}
  \displaystyle{{\sin\angle P'MP}} =
  \displaystyle{{\sin\angle PP'M\cdot{\sin PP'}}\over{\sin PM}}.
\end{equation*}
%
The angle $\angle PP'M$ is the complement of the (tropical) longitude
of $M$, while $PM$ again is the complement of its declination $\delta$
and the arc $PP'$ between the two poles is the ecliptic obliquity.
The verse's algorithm for using the table to find the \ayanavalana\
$\angle P'MP$ simply prescribes adding the degrees of precession to
the degrees of the known (sidereal) longitude of the body to convert
it to tropical longitude, and using the \koti\ or complement of this
result to interpolate linearly in the table.  When the tropical
longitude is between the winter and summer solstices (called the
northern \ayana), the \ayanavalana\ is north because ecliptic-east
points 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 and solar eclipse.  In a lunar eclipse, the eastern part of the
moon's disk is eclipsed first; so at first contact it is the western
part of the shadow (the `eclipser') that is being affected, and the
eastern part for release.  This is reversed in the case of a solar
eclipse where the moon is the `eclipser' and the sun the `eclipsed'.

\enlargethispage{-2\baselineskip}

There are three tables related to this verse, on ff.~6r and 6v.  The
first is a table giving the \ayanavalana\ at `the time of contact [or]
the time of release' (which are the chief moments for which the
deflection is calculated).  Of its three rows, the first contains the
argument, running from 0 to 90 degrees, which is evidently the
complement of the precession-corrected longitude.  The second row
contains the corresponding values of the \ayana-deflection, from
$0;0,3$ when the argument is 0 (i.e., when the body is at a solstice)
to $24;0,1$ ($90\deg$ of longitude away, at an equinox). The third row
gives the differences between successive entries in the second row;
they change only at every tenth entry, indicating that \ayanavalana\
values were computed first for every tenth degree of argument and the
remaining entries found by linear interpolation.%
%
\footnoteA{In fact, the non-interpolated table entries seem to conform
  very closely to \Bhaskara~II's simple expression for \ayanavalana\
  in \KK\ 4.14--15 \citep[S81--82]{raouma08}, which is equivalent to
  $\sin(90\deg - \lambda^T)\cdot 24$.}
 

The other two tables on f.~6v give the `corrected' \valana\
(\textit{spa\*s\*tavalana}), which is not mentioned in the verse, and
which despite its name does not actually introduce any new elements to
give a more accurate value of the deflection. Rather, it simply
converts the combined deflection into a linear measure scaled
appropriately for the disks of the eclipsed and eclipsing
bodies. Consequently, there is one table for solar and one for lunar
eclipses.  The arguments for both tables run from 0 to 47, which
evidently represents the degrees of total \valana\ as the algebraic
sum of \aksavalana\ (whose maximum for the \kks's latitude is between
22 and 23 degrees) and \ayanavalana\ (which, as noted above, can be as
much as $24\deg$), giving a possible combined maximum of $47\deg$.%
%
\footnoteA{See the rule for `corrected' \valana\ in, e.g., \KK\ 4.16
  \citep[S81--82]{raouma08}.}
%
The table entries run from $0;7$ to $8;2$ digits for the sun and $0;4$
to $13;51$ digits for the moon.

\tsc{MS.~J}'s procedure confirms this interpretation; he adds his
\aksavalana\ and \ayanavalana\ components to get the argument he uses
for the lunar corrected-\valana\ table, in the case of both contact
(f.~3r, lines 1--2) and release (f.~3v, lines 1--3).

%%% VERSE 1.7 %%%%%
%%%%
\begin{mula}{1.7 Reduction of arcs to the first quadrant}%
%
  try\=una\*m bhuja\*h sy\=at tryadhika\*m ca \*sa\*dbh\=at \\
  vi\'sodhya bh\=ardh\=ad adhika\*m vibh\=ardham $||$\\
  nav\=adhika\*m tad ravip\=atita\*m ca \\
  ko\*tir bhavet sy\=at trigraha\*m bhujonam $||$ 7 $||$\\

  \sflight\medskip\noindent The \bhuja\ should be: [an arc] less than three
  [signs]; moreover, subtracting from six signs one [i.e., an arc]
  that is greater than three signs; [or, an arc] greater than a
  half-circle is diminished by a half-circle; [or, an arc] greater
  than nine [signs is] that taken away from 12 [signs].  The \koti\
  should be three signs diminished by the \bhuja.
\end{mula}

\subsection*{Verse Analysis}

Metre: \upajati.

The grammar in this verse is particularly inconsistent in its parallel
constructions, as frequently happens when \Bhaskara\ tries to revamp a
borrowed \KK\ verse with some original phrasing (see below).  However,
the intended meaning seems clear.

\subsection*{Technical Analysis}

This verse gives a rule for producing a \koti\ or complement from a
\bhuja/\bahu\ or given arc.%
%
\footnoteA{Compare \KK\ 2.4 \citep[20]{mishra91}: \\\bgroup\itshape
  try\=una\*m bhuja\*h sy\=at tryadhikena h\={\i}na\*m
  bh\=ardha\*m ca bh\=ardh\=ad adhika\*m vibh\=ardham $|$ \\
  nav\=adhikenonitam arkabha\*m ca bhavec ca ko\*tis trigraha\*m
  bhujonam $\|$ \egroup }
%  
Most commonly in geometry the \bhuja\ and \koti\ are understood as
perpendicular sides of a right triangle, but generally in astronomical
texts the \bhuja\ is an arc, e.g., of longitude.  This arc is
conventionally reduced to a trigonometrically equivalent amount less
than 90$^{\circ}$ or three zodiacal signs, and the complement arc of
this standard \bhuja\ is the \koti.

This verse is not directed specifically toward the use of any
particular table. Rather, it is a standard general procedure for
reducing arcs to their canonical forms in the first quadrant, for which alone
function values are tabulated. Note that \Bhaskara\ in the immediately
preceding verses has prescribed procedures assuming the user's ability
to reduce arcs, and now belatedly gives explicit directions for it.

%%% VERSE 1.8 %%%%%
%%%%
\begin{mula}{1.8 Position of \nata\ with respect to the meridian}%
%
  s\=uryagrahe divasam eva dina\*m svak\={\i}ya\*m\\
  r\=atris tathaiva \'sa\'siparva\*ni vaipari sy\=at $||$\\
  ahno 'rdhato ni\'sidalendunata\*m prat\={\i}cy\=a\*m\\
  r\=atridal\=ad dyudalato 'rkanatasya p\=urve $||$ 8 $||$\\

  \sflight\medskip\noindent In an eclipse of the sun, the day is just its own
  day. The night is likewise.  In a lunar eclipse, it should be
  reversed.  The [post-]midnight \nata\ of the moon after the middle
  of [its own] day [i.e., with `night' and `day' reversed in meaning
  as specified] is in the west.  From midnight to midday, [the place]
  of the sun's \nata\ is in the east.
\end{mula}

\subsection*{Verse Analysis}

Metre: \vas.

The last \pada\ is defective: the second syllable ought to be heavy
for the metre's sake.

\subsection*{Technical Analysis}

This verse conveys rather simple astronomical facts about the \nata\
or hour-angle/zenith distance of the moon and the sun, with respect to
the meridian (here indicated by midday or midnight). `Its own day'
means the period of time when the body in question is above the
horizon, which is of course daytime in its usual sense for the sun but
nighttime for the full moon. And indeed for both luminaries, from
`midnight' to `midday' of its own day the body is to the east of the
meridian, and from `midday' to `midnight' it is west of the meridian,
because of the westward daily rotation. Again, it might have been more
intuitive from the user's point of view to supply this information
earlier: namely, before prescribing the calculation of the
\aksavalana, whose direction depends precisely on this issue of
whether the body's \nata\ is in the east or the west.

%%% VERSE 1.9 %%%%%
%%%%
\begin{mula}{1.9 Precession; length of the half-day}%
%
prabh\=abdhi 445 \'sakon\=aptakh\=a\.ngo 60 'yan\=a\*m\'s\=a\*h\\
yuto 'rkasya r\=a\'sya\*m\'sam\=ane\*su ko\*s\*the $||$\\
din\=ardhavi\'sodhya\*m khar\=amair ni\'sardha\*m\\
din\=ardhe\*s\*tan\=a\*dyantara\*m ta\*m nata\*m sy\=at $||$ 9 $||$

\sflight\medskip\noindent
445 subtracted from the [current] \'Saka year [and] divided by 60 are the degrees
of precession. 
%
[That amount is] added to the
amounts in signs and degrees of the sun['s longitude]. 
The subtracted [complement] of the half-day in the table [entry] with
[respect to] 30  [\ghatis] is the half-night.  [Whatever is] the difference
between the desired \nadi\ [i.e., \ghati] and the half-day,
that should be the `hour-angle' (\nata).
\end{mula}

\subsection*{Verse Analysis}

Metre: \bhuj\ (see verse 1.2) with minor deviations: the third
syllable of the first and third \padas\ is light instead of heavy.

This unfamiliar \bhut\ numeration takes \textit{prabh\=a} `light' or
`shadow' to mean 45.  The related compound \textit{prabh\=avaka} is
elsewhere attested as meaning 8 \citep[66]{sarma03}.

A comprehensible translation of this verse requires taking substantial
liberties with its phrasing: its literal sense is more like `[The
amount] which has 60 as dividing the difference between the [current]
\'Saka year and 445 are the degrees of precession.'  The mathematical
meaning, however, is unmistakable.

\subsection*{Technical Analysis}

This verse explains how to correct the solar longitude for precession
or the difference between sidereal and tropical longitude (which it
would have been helpful to know before calculating the \ayanavalana\
requiring that correction).  The year \'Saka 444 or 522 \tsc{ce} is
frequently attested as the assumed date of zero precession, and the
canonical rate of precession is taken as 1 degree in 60 years.%
%
\footnoteA{See, for instance, \citet[30--31]{pingree72}.}
%
The difference between 445 and the current \'Saka year gives the
integer number of years elapsed between the end of 444 and the start
of the current year, each of which produces one additional arc-minute
of precession.

The resulting precession-corrected solar longitude in signs and
degrees is the argument of the table on f.~7r (see 2.2 below) that
tabulates the sun's `half-day' (\textit{dyu-dala}).  This means the
length in \ghatis\ of half the seasonally varying period of daylight
(the solar form of the `own day' previously invoked in 1.5).

%%% VERSE 1.10--11 %%%%%
%%%%
\begin{mula}{1.10--11 Graphical projection of eclipse appearance}%
%
  gr\=ahy\=ardhas\=utre\*na vidh\=aya v\*rtta\*m \\
  m\=anaikyakha\*n\*dena ca s\=adhit\=a\'sam $||$\\
  b\=ahye 'tra v\*rtte valana\*m yath\=a\'sa\*m\\
  pr\=ak spar\'sika\*m pa\'scimata\'s ca mok\*sam $||$ 10 $||$

  \smallskip\noindent
  deya\*m rave\*h pa\'scimap\=urvata\'s ca\\
  jy\=avac ca b\=a\*nau valan\=agrak\=abhy\=am $||$\\
  utp\=adya matsya\*m valan\=agrak\=abhy\=a\*m\\
  madhya\*h \'saras tanmukhapucchas\=utre $||$ 11 $||$

  \noindent\sflight
  When one has produced a circle by means of a [radius] string half
  [the size] of the eclipsed body, and [another circle by a radius
  string equal in length to] the [half] part of the sum of their
  measures [i.e, diameters of eclipsed and eclipser: this is the
  circle of first contact (\sparsa) and release (\moksa)] with
  cardinal directions 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 lunar eclipse];

  \medskip\noindent [and] in the case of the sun, west and east
  [respectively]. And having extended two latitude [values] like a
  chord at the two extremities of the \valana, a fish \mbox{[-figure} is
  drawn with arcs centered] at the two extremities of the \valana.
  The mid-eclipse latitude is [laid off from the center] on its
  head-tail-line [i.e., the axis of the fish-figure].
\end{mula}


\subsection*{Verse Analysis}

Metre:  \indravajra.

The first of these two verses is identical to \Bhaskara~II's \KK\
4.18, and the second nearly the same as \KK\ 4.19, except for the
syllable \textit{ca} instead of \textit{te} at the end of its first
\pada.%
%
\footnoteA{\KK\ 4.18--19 \citep[60]{mishra91}: \\\bgroup\itshape
  gr\=ahy\=ardhas\=utre\*na vidh\=aya v\*rtta\*m\\
  m\=anaikyakha\*n\*dena ca s\=adhit\=a\'sam $|$\\
  b\=ahye 'tra v\*rtte valana\*m yath\=a\'sa\*m\\
  pr\=ak spar\'sika\*m pa\'scimata\'s ca mok\*sam $||$ 18 $||$\\
  deya\*m rave\*h pa\'scimap\=urvatas te\\
  jy\=avac ca b\=a\*nau valan\=agrak\=abhy\=am $|$\\
  utp\=adya matsya\*m valan\=agrak\=abhy\=a\*m\\
  madhya\*h \'saras tanmukhapucchas\=utre $||$ 19 $||$ \egroup }
%
The word \textit{\=a\'s\=a} can mean `quarters of the compass'; this
adverbial form \textit{\=a\'sam} evidently means `relating to cardinal
directions'.

\begin{figure}%[h]
  % \begin{wrapfigure}[17]{l}{2.8in}

\includegraphics[height=3.5in]{KKeParilekha1}
\centering
\caption{Initial circles and lines for the graphical projection of an
  eclipse}
\label{KKeParilekha1}
\end{figure}


\subsection*{Technical Analysis}

These verses and the two following give directions for producing the
graphical projection of the eclipse.  The procedure does not refer
directly to any of the data given in the tables, as it is a practical
description of how to represent the eclipse configuration using the
quantities previously computed.  The following interpretation of these
two verses is illustrated in figure~\ref{KKeParilekha1}.%
%
\footnoteA{As noted in the Verse Analyses for this and the following
  group of verses, this procedure is directly copied from that of the
  \KK, which in turn is a simplified version of its counterpart in the
  same author's \SidSir\ 5.26--32 \citep[121--124]{sastriSidSir}.  }
The author of the lunar-eclipse example in \tsc{MS.~J} briefly echoes the
text's instructions for this diagram (f.~3v, lines 4--8), but alas,
the manuscript preserves no drawing in illustration of them.

The central circle represents the disk of the eclipsed body. At the
moment of \sparsa\ or \moksa, the center of the eclipsing body is on
the circumference of a second, larger concentric circle with radius
equal to the sum of the radii of eclipsed and eclipser, so the two
bodies are tangent.  The trajectory between \sparsa\ and \moksa\ 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 figures~\ref{KKeParilekha1} and \ref{KKeParilekha2}.  Note
that this direction grid represents a left-right-reversed image of the
eclipsed body as it would appear in the southern part of the sky,
where the southward-facing observer would see east on the left hand
and west on the right.%
%
\footnoteA{See, e.g., \SurSid\ 6.12, \textit{viparyayo di\'s\=a\*m
    k\=arya\*h p\=urv\=aparakap\=alayo\*h} `a reversal of the
  directions of the east and west hemispheres is to be made'
  \citep[69]{pandeya91}.  The graphical projection shows how the
  eclipse would look in a mirror image, e.g., in a tray of water or
  other reflector set on the ground next to the diagram.}

The amount of the appropriately scaled linear combined \valana\ (see
1.6 above) for \sparsa\ or \moksa\ is marked off north or south of the
appropriate extremity on the east-west line.  The radii between those
two points will not be exactly collinear unless the \sparsa\ and
\moksa\ values of \valana\ are exactly equal, but they will indicate
approximately the direction of the ecliptic with respect to local
east-west.

Then the corresponding values of the latitude are marked off from the
\valana\ points `like a chord' so that they fall on the circumference
of the outer circle, to indicate the positions of the center of the
eclipser at \sparsa\ and \moksa.  (Presumably, since the possible
values of latitude are small, the fact that the `chord-like' line
segments representing them are not quite perpendicular to the
approximate line of ecliptic direction determined by the \valana\
points is considered negligible.)  Then a `fish-figure' or
perpendicular bisector to the ecliptic direction is constructed
through the center of the circle, and a line segment representing
mid-eclipse latitude is laid off along it.

\begin{figure}%[h]
  \includegraphics[height=3in]{KKeParilekha2}
  \centering
  \caption{Depicting the three moments of contact, mid-eclipse, and
    release in the projection}
  \label{KKeParilekha2}
\end{figure}

%%% VERSE 1.12--13 %%%%%
%%%%
\begin{mula}{1.12--13 Graphical projection, continued}%
%
  kendr\=ad yath\=a\'so 'tha \'sar\=agrakebhyo\\
  v\*rttai\*h k\*rtair gr\=ahakakha\*n\*dakena\\
  syu\*h spar\'samadhyagrahamok\*sasa\*msth\=a\\
  ath\=a\.nkayen madhya\'sar\=agracihn\=at $||$ 12 $||$\\

  \noindent
  m\=an\=antar\=ardhena vidh\=aya v\*rtta\*m\\
  kendre 'tha tanm\=argayutidvaye 'pi $||$\\
  tamo 'rdhas\=utre\*na vilikhya v\*rtta\*m\\
  sanm\={\i}lanonm\={\i}lanake ca vedye $||$ 13 $||$

  \sflight\medskip\noindent By means of circles constructed from a centre in
  the [appropriate] direction at the tips 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
  contact, mid-eclipse and release should be [shown].

  \medskip\noindent Now, producing a circle with [radius equal to]
  half the difference of the amounts [of the diameters of the two
  bodies], one should draw [it] from the point of the tip of the
  mid-eclipse latitude.  Now [there are] two centres at the two
  intersections of that [circle] and the path [of the eclipser].
  [Upon] drawing a circle [at each centre] with radius [equal to that
  of] the eclipser, the beginning of totality and end of totality too
  are to be known.
\end{mula}

\begin{figure}%[h]
  \includegraphics[height=3in]{KKeParilekha3}
  \centering
  \caption{Depicting the beginning and end of totality}
  \label{KKeParilekha3}
\end{figure}

\bigskip %DW

\subsection*{Verse Analysis}

Metre: \upajati.

These verses, like the preceding ones, closely track their
counterparts in the \KK.%
\footnoteA{\KK\ 4.20--23\\ \citep[61]{mishra91}:\\
  \bgroup\itshape
  kendr\=ad yath\=a\'sa\*m sva\'sar\=agrakebhyo \\
  v\*rttai\*h k\*rtair gr\=ahakakha\*n\*dakena $|$ \\
  syu\*h spar\'samadhyagrahamok\*sasa\*msth\=a \\
  ath\=a\.nkayen madhya\'sar\=agracihn\=at $||$ 20 $||$ \\
  \=adyantyab\=a\*n\=agragate ca rekhe \\
  j\~ney\=av imau pragrahamuktim\=argau $|$ \\
  m\=an\=antar\=ardhena vilikhya v\*rtta\*m \\
  kendre 'tha tanm\=argayutadvaye 'pi $\|$ 21 $\|$ \\
  bh\=ubh\=ardhas\=utre\*na vidh\=aya v\*rtte \\
  samm\={\i}lanonm\={\i}lanake ca vedye $|$  \\
  m\=argapram\=a\*ne viga\*nayya p\=urva\*m \\
  m\=arg\=a\.ngulaghna\*m sthitibhaktam i\*s\*tam $\|$ 22 $\|$ \\
  i\*s\*t\=a\.ngul\=ani syur atha svam\=argo \\
  dady\=adam\=uni\*s\*tava\'s\=at tadagre $|$  \\
  v\*rtte k\*rte gr\=ahakakha\*n\*dakena \\
  sy\=ad i\*s\*tak\=ale graha\*nasya sa\*msth\=a $\|$ 23 $\|$ \egroup
}
%% 
The \kks\ version has introduced a couple of unusual or awkward
grammatical constructions, such as the nominative\slash accusative form of
\textit{tamas} `darkness' that we take to refer to the eclipser in a
genitive sense, and what appears to be an instance of the particle
\textit{atha} `now' more or less prepended to \textit{\'sara}
`latitude', instead of \textit{sva} `its own', to imply `the
now-latitude' or `latitude at that time.'

\subsection*{Technical Analysis}

Figure~\ref{KKeParilekha2} shows an arc drawn through the
aforementioned three latitude points to mark the `path' of the
eclipse, and the disk of the eclipsing body centred at each of them.
To find where the eclipsing body sits at the remaining two principal
moments of the eclipse, i.e., the beginning and end of totality, the
user is instructed to draw a circle (shown shaded in
figure~\ref{KKeParilekha3}) whose radius equals the difference between
the radii of eclipsed and eclipser.  The two points where this circle
intersects the `path' mark the positions of the centre of the eclipser
at those two moments.

\bigskip
\begin{tika}%
%
  iti candraparv\=adhik\=ara\*h $||$\\

  \medskip\noindent Thus, the chapter on lunar eclipses.
\end{tika}

% SOLAR ECLIPSE CHAPTER

%%% VERSE 2.1 %%%%%
%%%%
\begin{mula}{2.1 Didactic aphorism}%
%
  \'s\=astr\=a\*ny anek\=ani mah\=arthas\=utr\=a-\\
  \*ny anantavidy\=alpamatir jano 'yam $||$\\
  kalau na d\={\i}rgh\=ayur ato hi [gr\=a]hya\*m\\
  tattva\*m yath\=a k\*s\={\i}ravidhau ca ha\*msa\*h $||$ 1 $||$\\

  \sflight\medskip\noindent [There are] many treatises (\textit{\'s\=astras}),
  rules (\textit{s\=utras}) with great purpose, [and] endless
  knowledge; this person [i.e., 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] in the act of [separating]
  milk [from water].
\end{mula}

\subsection*{Verse Analysis}

Metre: \upajati?

We have taken considerable liberties with the arrangement of the
\padas\ and especially with the reconstruction and translation of the
third \pada, in which the penultimate syllable seems to be missing.
The final syllable looks in the manuscript more like \textit{hva\*m}
than \textit{hya\*m}, so the final word in the original might have
been something quite different. But the reference is clearly to the
well-known analogy between the soul discerning valid knowledge from a
mixture of truth and untruth and the proverbial ability of a goose or
swan to separate out the components of a mixture of milk and water.%
%
\footnoteA{\label{SubhasitaRefs} This simile is attested in Sanskrit
  literature as far back as, e.g.,
  \MBharat\ 1.69.10 \cite{smithMBharat}:\\
  \bgroup\itshape
  \mbox\null pr\=aj\~nas tu jalpat\=a\*m pu\*ms\=a\*m \'srutv\=a\\
  \mbox\null\qquad v\=aca\*h \'subh\=a\'subh\=a\*h $|$ \\
  \mbox\null gu\*navad v\=akyam \=adatte ha\*msa\*h k\*s\={\i}ram \\
  \mbox\null\qquad iv\=ambhasa\*h $\|$
  \egroup\\
  But the direct inspiration for the \kks's verse was apparently
  closer to the form found
  in, e.g., \GP\ 16.84 \citep[164]{woodGP}:\\
  \bgroup\itshape
  anek\=ani ca \'s\=astr\=a\*ni svalp\=ayur vighrako\*taya\*h  $|$ \\
  tasm\=at s\=ara\*m vij\~n\=an\={\i}y\=at k\*s\={\i}ra\*m ha\*msa
  iv\=ambhasi $\|$ \egroup }
%
The Kaliyuga or Indian `Iron Age' is traditionally considered a time
of degeneration and decay, in which, for example, the standard human
lifespan is much shorter than it is said to have been in earlier ages.

%%% VERSE 2.2 %%%%%
%%%%
\begin{mula}{2.2 Half-day and oblique ascension}%
%
  s\=ayan\=arkabhalav\=adiko\*s\*thak\=a\*h \\
  yukte ce\*s\*thagha\*tiyuktatatsam\=an $||$\\
  ko\*s\*thak\=a\.nkamiti\*m r\=a\'sip\=urvaka\*m \\
  labdhalagnam ayan\=a\*m\'sakonnatam $||$ 2 $||$\\

  \sflight\medskip\noindent The table entries [for half-day and oblique
  ascension have as argument] the precession-increased [longitude of
  the] sun in zodiacal signs and degrees and so on.
%
  And when [the longitude in signs and degrees] is increased [by
  minutes, use the digits] corresponding to those [minutes] increased
  by the \textit{gha\*tik\=as} [corresponding to] the desired
  [argument: these digits are] commensurate with the number in the
  table cell headed by [its] zodiacal sign.
%
  The obtained oblique ascension [has] the precession-increased [solar
  longitude] beginning with zodiacal signs [as its] argument
  (\textit{unnata}).
\end{mula}

\subsection*{Verse Analysis}

Metre: \rath.

As usual with the verses that appear to be original to the \kks, there
are a few metrical irregularities: namely, in \pada~2 the second
syllable should be light, as should the sixth syllable in \pada~3.

\enlargethispage{-12pt}

\subsection*{Technical Analysis}

This verse apparently applies to two tables, although it explicitly
mentions only the second of them.  The first, on f.~7r, tabulates half
the seasonally varying length of daylight in \ghatis\ (see 1.9 above),
and is titled \textit{atha s\=ayana\-ravi\-r\=a\'syopari\-dyu\-dala\*m} `Now
[the length of] the half-day with argument the signs [and degrees] of
the precession-corrected [longitude of the] sun'.  Its values for each
degree of the sun's (tropical) longitude are arranged in double entry
format, zodiacal signs against degrees: along the vertical axis are
the signs numbered 1--11 followed by 0, and along the horizontal,
degrees numbered 0--29.  The maximum entry $16;48$ \ghatis\ occurs at
Cancer $0\deg$ and the minimum is $13;13$ \ghatis\ in Capricorn
$0\deg$; the equinoctial half-day is of course 15 \ghatis, one-quarter
of a full nychthemeron or 60 \ghatis.%
% 
\footnoteA{We assume that zodiacal sign 1 in the first row is Taurus
  and zodiacal sign 0 (the last row) is Aries.  The half-day extreme
  values imply that the so-called `maximum half-equation of daylight'
  $\omega_{\hbox{\rm max}}$, i.e., the difference between half the
  equinoctial day and half the solstitial day, is $1;47$ \ghatis.  We
  used this quantity to confirm the value of the terrestrial latitude
  $\phi = 22;35,39\deg$ that we inferred from the \aksavalana\ table
  used in verse 1.5: plugging it into the formula for the terrestrial
  latitude $\phi = \hbox{\rm arctan}(\sin\omega_{\hbox{\rm
      max}}/\tan\varepsilon)$, where $\varepsilon$ is the canonical
  ecliptic obliquity of $24\deg$, yields a value for $\phi$ of
  approximately $22;38\deg$.}
%

As noted above, this table seems more relevant to computations
involving the \nata\ in the lunar eclipse chapter.  Its placement here
may have been intended to allow the user to check whether the sun will
be visible above the horizon at the time of the predicted solar
eclipse.

The remaining instructions in the verse deal with computing the
\lagna, usually meaning the `horoscopic point' or ascendant where the
ecliptic intersects the eastern horizon, but in this case evidently
intended to signify the oblique ascension in \ghatis\ of the ecliptic
arc between the sun and the vernal equinox.  The computation employs a
two-part set of tables: the first part, on ff.~7v--8r, is another
double entry table for each degree of (tropical) solar longitude, with
zodiacal signs 0--11 on the vertical axis and degrees 0--29 on the
horizontal.  The tabulated values begin with 0 at $0\deg$ of argument
and increase to 60 \ghatis\ at $360\deg$, meaning that these oblique
ascensions are cumulative.  The differences between tabular entries
for successive degrees within a given zodiacal sign are constant and
their values are symmetrical about the equinoxes, as shown in
table~\ref{LagnaDiffs}.

\begin{table}[h]
  \centering
  \begin{tabular}{l l}
    Zodiacal sign & Constant difference (\ghatis) \\
    \noalign{\smallskip}
    Aries/Pisces &0;7,36\\
    Taurus/Aquarius &0;8,38\\
    Gemini/Capricorn &0;10,12\\
    Cancer/Sagittarius &0;11,20\\
    Leo/Scorpio &0;11,18\\
    Virgo/Libra &0;10,56\\
  \end{tabular}
  \caption{The constant differences between oblique ascension values
    for successive degrees} 
  \label{LagnaDiffs}
\end{table}

The other table in this set, on ff.\,8v--9v, allows the user to make the
oblique ascension calculation precise for arcminutes of longitude: it
is somewhat misleadingly titled \textit{atha lagnasya
  kal\=ako\*s\*tak\=a} `Now, the tabular entries for the arcminutes of
the oblique ascension'.  Actually, in each of its six sub-tables (one
for each pair of zodiacal signs from Aries/Pisces to Virgo/Libra), it
is the \emph{argument} that is measured from 1 to 60 \kalas\ or
arcminutes within a degree, not the tabulated oblique ascension
amounts.

The table entries, in sixtieths of a \ghati, are the corresponding
increments added by those extra arcminutes to the oblique ascension
already computed for the degree of tropical solar longitude specified
by the previous table's argument.  Again, the differences between
entries in each sub-table are constant: the entries for successive
arcminutes differ by just one-sixtieth of the previous table's
constant difference between entries for successive degrees of the
appropriate zodiacal sign.  (Thus, for example, every additional
arcminute in a solar longitude falling in the sign Aries adds
$0;0,7,36$ \ghatis\ to the corresponding oblique ascension.)  By the
combination of the table for degrees and the one for arcminutes, the
accumulated oblique ascension can be determined for each of the
possible 21,600 arcminutes of tropical solar longitude.

The version of this procedure in the solar eclipse example in \tsc{MS.~J}
starts by finding the usual \lagna\ or longitude of the ascendant and
computing the longitude difference between that and the tropical solar
longitude to give the argument for the algorithm in the following
verse.

%%% VERSE 2.3 %%%%%
%%%%
\begin{mula}{2.3 Longitudinal parallax}%
%
  dar\'s\=antak\=ale 'rkatanovi\'se\*sa\*m \\
  k\=arya\*m tada\*m\'samitispa\*s\*thako\*s\*the $||$\\
  yallambana\*m sve gu\*nakena gu\*nya\*m \\
  khaveda 40 bhakta\*m sphu\*talambana\*m sy\=at $||$ 3 $||$

    \sflight\medskip\noindent At the time of the end of the conjunction
  [\tithi], the difference [of the longitudes] of the sun and the
  ascendant (? \textit{tanas}) is to be computed.  Whatever
  longitudinal parallax (\lambana) [corresponds to that difference],
  multiplied by the multiplier in its own true-table entry
  commensurate with the degree of that [ascendant] and divided by 40,
  should be the corrected longitudinal parallax.
\end{mula}

\subsection*{Verse Analysis}

Metre: \upajati.

In \pada~2, the second syllable of \textit{a\*m\'sa} ought to be heavy
and the second syllable of \textit{mitispa\*s\*ta} ought to be light
for the sake of the metre.  We know of no other use of the word
\textit{tanas} as a synonym for \lagna\ or ascendant, but that is the
meaning required by the procedure.

\subsection*{Technical Analysis}


\begin{figure}[h]
%\begin{wrapfigure}[17]{l}{2.8in}
\includegraphics[width=3in]{KKeLambanaMean}
\centering\normalfont
\caption{\normalfont Plots of `mean' \lambana\ in \ghatis\ 
against elongation $\lambda_A - \lambda_S$ in degrees, in three forms:
the trigonometric expression $4\sin(\lambda_A - \lambda_S)$ (line),
the short list of values in the \KK\ (large dots), and the tabulated
values in the \kks\ (small dots)}
\label{LambanaMean}
\end{figure}

The first of the two tables referenced in this verse lists the `mean'
longitudinal parallax or \lambana\ in \ghatis\ for values of
elongation between the sun and the ascendant ranging from 0 to
$91\deg$.  The table entries start with $3;40$ \textit{gha\*tik\=as}
at $0\deg$, maximising to $4;0$ at $24\deg$, and decreasing to $0;1$
at $91\deg$.  The next table, which gives the
\textit{la\*mbanaspa\*s\*tagu\*nak\=a\*h} `multipliers for correcting
the longitudinal parallax', has as its argument the longitude of the
ascendant (\lagna) adjusted for precession.  This is a double-entry
table with 0--11 zodiacal signs along the vertical and 0--29 degrees
along the horizontal.  The multiplier's maximum is $35,59$ (for Virgo
9--$12\deg$) and its minimum is $27,21$ (for Pisces
$29\deg$). According to the instructions in the verse, the multiplier
is divided by 40 when applied to the `mean' longitudinal parallax to
correct it.

\begin{figure}[h]
\includegraphics[height=2.35in]{KKeLambanaHsph}
\centering
\caption{Celestial hemisphere with the autumnal equinox 
at the ascendant and the nonagesimal on the meridian near the zenith}
\label{LambanaHemisphere}
\end{figure}

\normalfont This application of table data roughly parallels the
algorithmic approach to parallax calculation in \KK\ 5.2--3
\citep[S92--S94]{raouma08}, which also begins with finding a so-called
`mean' \lambana.  This `mean' or approximate \lambana, as in the
corresponding \kks\ table, depends only on the elongation of the sun's
position $\lambda_S$ (which in a solar eclipse more or less coincides
with that of the moon) from a given point on the ecliptic.  In the
\KK's formula the reference point for elongation is the so-called
nonagesimal with longitude exactly $90\deg$ less than the longitude of
the ascendant $\lambda_A$; a body at the nonagesimal has no
longitudinal parallax.  The `mean' \lambana\ is just the sine of that
sun-nonagesimal elongation linearly scaled from its minimum of 0 when
the sun and nonagesimal coincide to a maximum absolute value of 4
\ghatis\ when the sun is on the horizon. However, the \KK\ also
provides a brief versified table of nine `mean' \lambana\ values that
do not exactly correspond to this relation, but which apparently
inspired the \kks's tabulated values on f.\,9v.
Figure~\ref{LambanaMean} shows the two sets of values along with the
curve representing the scaled sine of the elongation.

\begin{figure}[h]
  \includegraphics[height=2.35in]{KKeLambanaAsc0} \quad
  \includegraphics[height=2.35in]{KKeLambanaAsc90}
  \centering
  \caption{Plots of `mean' and corrected \lambana\ in \ghatis\ against
    elongation $\lambda_A - \lambda_S$ in degrees with
    $\phi=22;35,39\deg$, for $\lambda_A = 0\deg$ (left) and $\lambda_A
    = 90\deg$ (right): the corrected quantity in both trigonometric
    (dashed line) and tabulated (dots) form is smaller than its
    uncorrected counterpart}
  \label{LambanaTrue}
\end{figure}

The `mean' \lambana\ is subsequently corrected to its `true'
counterpart by taking into account the contribution from the
depression of the nonagesimal from the zenith.
Figure~\ref{LambanaHemisphere} illustrates a case where the
nonagesimal $V$ is at the summer solstice and consequently high in the
sky near the zenith $Z$: the small nonagesimal zenith distance $ZV$
does not greatly affect the parallax due to the elongation between $V$
and the sun.  \Bhaskara~II's correction represents this by multiplying
the `mean' \lambana\ by the cosine of $ZV$, which will make the
\lambana\ zero if the nonagesimal $V$ is on the horizon but leave the
`mean' \lambana\ unchanged if it is at the zenith. The zenith distance
arc $ZV$ is taken as approximately equal to the local latitude $\phi$
or $ZQ$ diminished by the nonagesimal's ecliptic declination or $VQ$.
(Note that this combination usually only approximates the exact zenith
distance, because the declination and latitude are measured along
different arcs unless the nonagesimal, the zenith and the celestial
pole all fall on the same great circle as shown in
figure~\ref{LambanaHemisphere}.)  So this true \lambana\ or
longitudinal parallax correction to the time of mid-eclipse is given
by the equation
%
\begin{equation*}
  \hbox{\textrm{(true)} \lambana} = 4 \cdot \sin | \lambda_S -
  (\lambda_A - 90\deg)|  
  \cdot \cos(\phi - \arcsin(\sin 24\deg \sin(\lambda_A - 90\deg)) ).
\end{equation*}
%

The individual `multipliers' in the second \kks\ table on f.~10r,
which must be divided by 40 and then multiplied by the `mean'
\lambana, have been computed to imitate this correction.  The graphs
in figure~\ref{LambanaTrue} reproduce the `mean' \lambana\ values from
figure~\ref{LambanaMean} and also show their corrected versions for
two sample values of $\lambda_A$ at the \kks's terrestrial latitude
$\phi=22;35,39\deg$.  The above trigonometric expression for true
\lambana\ is graphed as a dashed line, while the \kks's tabulated
\lambana\ values corrected by their appropriate tabulated multipliers
are shown as small dots.  As $\lambda_A$ increases from $0\deg$ and
$V$ gets closer to the zenith, each corrected version of \lambana\
approaches its original `mean' value, so that by about $\lambda_A =
150\deg$ the `mean' and true versions become indistinguishable on the
graph.

%%% VERSE 2.4 %%%%%
%%%%
\newpage %DW
\begin{mula}{2.4 Longitudinal and latitudinal parallax}%
%
  vi\'sle\*sas tribhyo 'bhyadhikonaka\'s cet\\
  tithy\=antad\=a svar\*nam ida\*m kram\=at sy\=at $||$\\
  lagnasya r\=a\'sya\*m\'samite\*su ko\*s\*the\\
  natis tath\=a lambanako gu\*no 'sti $||$4$||$\\

      \sflight\medskip\noindent If the difference [between the longitudes of the
  sun and the ascendant] is greater or less than three [signs] at the
  end of the tithi, this [longitudinal parallax] should be [applied]
  positively or negatively respectively.  The latitudinal parallax
  (\textit{nati}) is in the table [entry] among those [entries]
  determined by the signs and degrees of the ascendant.  Now, the
  lambana is [to be] multiplied: [see next verse.]
\end{mula}

\subsection*{Verse Analysis}

Metre: \upajati.

The third syllable in \pada~1 ought to be light.  The rather odd
construction \textit{tithy\=antad\=a} in the second \pada\ appears to
be intended as an adverb of time, along the lines of \textit{ta\-d\=a}
`then'.

\subsection*{Technical Analysis}

\Bhaskara's discussion of longitudinal parallax continues with a rule
for applying it to the time of conjunction.  If the luminaries are
westward of the nonagesimal (roughly, post meridian, or more
precisely, farther than one quadrant away from the ascendant), it
makes the conjunction later than initially calculated, since the
parallax depresses the moon toward the (western) horizon. Likewise,
parallax applied to a position east of the nonagesimal makes the
conjunction earlier.  So the longitudinal parallax in \ghatis\
calculated in the previous verse is applied to the mid-eclipse time
positively or negatively, respectively.

The other parallax component, latitudinal parallax or \nati, accounts
for the fact that the nonagesimal or midpoint of the visible
semicircle of the ecliptic usually does not coincide with the local
zenith. The parallactic depression of the nonagesimal toward the
horizon is treated as a correction to the moon's latitude north or
south of the ecliptic.

\begin{figure}[h]
  \includegraphics[height=2.5in]{KKeNati}
  \centering
  \caption{Plot of \nati\ in digits (northward positive, southward
    negative) against $\lambda_A$ in degrees, showing the \kks's
    tabulated values of \nati\ at the initial degree of each sign
    (dots), and the \KK's \nati\ function with $\phi=22;35,39\deg$
    (line)}
  \label{NatiPlot}
\end{figure}

From the graph in figure~\ref{NatiPlot} it seems almost certain that
the \kks's \nati\ values in the table on f.~10v, whose argument is the
sign and degree of the longitude $\lambda_A$ of the ascendant, were
derived from a formula in \KK\ 5.3 \citep[S92--93]{raouma08}.  In this
rule the ecliptic declination of the nonagesimal is again
arithmetically combined with the local latitude $\phi$ to produce the
nonagesimal's approximate zenith distance, whose sine is then linearly
scaled to give the \nati\ in digits:
%
\begin{equation*}
  \hbox{\nati} = \frac{1}{8}\cdot\frac{13}{12}\cdot 120
  \sin(\arcsin(\sin 24\deg \sin(\lambda_A + 90\deg)) - \phi) 
\end{equation*}
%
Qualitatively, we can understand the behavior of this function by
recollecting that the nonagesimal is the point where longitudinal
parallax is zero, so the latitudinal component of parallax or \nati\
depends only on how much (and in which direction) the nonagesimal is
depressed from the zenith.

Note that in the above expression, subtracting $\phi$ from the
declination means that the zenith distance $ZV$ will be negative if
the nonagesimal $V$ falls south of the zenith, and positive if it is
north (see figure~\ref{LambanaHemisphere}).  Consequently, the
nonagesimal will approximately occupy the zenith point and the
latitudinal parallax will vanish when the northern declination $VQ$ is
equal to the local terrestrial latitude $\phi = ZQ$, here about
$22;35,39\deg$. This value is not far from the maximum declination of
$24\deg$ that the nonagesimal reaches in the southern or northern
direction when it falls at the winter or summer solstice respectively,
while the ascendant occupies the following equinox.  In other words,
when the ascendant falls a little before or a little after the
autumnal equinox at tropical Libra $0\deg$, we should expect the
\nati\ for this latitude to be minimised.

Indeed, the values in the \kks's \nati\ table reach their minimum of
$0;1$ digits when the argument $\lambda_A$ is Virgo $12\deg$ or Libra
$18\deg$; between these two positions, the \nati\ is to be applied in
the northern direction (because then the nonagesimal is slightly north
of the zenith), although it is a southward correction everywhere
else. Conversely, the \nati's maximum (absolute) value of $11;46$
digits occurs when the ascendant is around the vernal equinox and the
nonagesimal around the winter solstice, producing the largest possible
depression of the nonagesimal south of the zenith.%
%
\footnoteA{If $\phi$ were zero, the \nati\ would fluctuate
  symmetrically between zero and equal north and south extremes of
  about $6.6$ digits, corresponding to the maximum nonagesimal
  depression of $24\deg$ north or south (see
  section~\ref{EclipseTheoryComp} for the relation between arcs and
  digits of parallax).}
%
Most of the other \nati\ table entries appear to have been linearly
interpolated between trigonometrically computed values at $15$-degree
intervals.

The final reference to the longitudinal parallax goes with the
subsequent verse.

%%% VERSE 2.5 %%%%%
%%%%
\begin{mula}{2.5 Adjusting eclipse data for parallax}%
%
  vi\'sva 13 ghnalambanakal\=as tithivad yutona-\\
  p\=at\=acca k\=a\*n\*damata\*sa\*dgu\*nalamban\=a\*m\'sai\*h $||$\\
  yugya\*m vilagnatanati\'s ca \'sarau vidady\=at\\
  spa\*s\*to bhavec ca vi\'say\=at sthitichannas\=adhyam $||$ 5 $||$\\

     \sflight \medskip\noindent [There are] arcminutes [produced by] the 
     \lambana\
  multiplied by 13, and [by them the elongation of the sun] from the
  node [is] increased or decreased, [according] as [the time to the
  end of] the \tithi\ [is increased or decreased by the \lambana\ to
  give the moment of apparent conjunction].  By the degrees [produced
  by] the \lambana\ considered as a [separate] quantity [and]
  multiplied by 6, [the longitude of the nonagesimal] is to be
  increased [or decreased, respectively].  One should apply the
  nonagesimal-[derived] \nati\ to the [lunar] latitude, and [it]
  should be correct.  The determination of the half-duration and the
  obscuration is [derived] from [the corrected time of] the middle [of
  the eclipse].
\end{mula}

\subsection*{Verse Analysis}

Metre:  \vas. 

%% The tenth syllable \textit{ti} in the last \pada\ ought to be light
%% for metrical consistency, but \sandhi\ requires \textit{cch} rather
%% than \textit{ch} as the following consonant, which makes it heavy.
Our translation understands \nati\ in the accusative in the third
\pada, which is grammatically incorrect but represents our best guess
at the sense.  The unusual term \textit{vilagnata}, if we have not
misread or misinterpreted it, seems to be intended as a sort of
portmanteau word implying the correction of the nonagesimal
(\textit{vilagna}) by the \lambana\ as well as the recalculation of
its \nata\ or zenith distance and its resulting new \nati\ value.

\subsection*{Technical Analysis}

This and the following verse rather sketchily describe a procedure
evidently modeled on that of the \GL\ for correcting the eclipse based
on the computed values of the parallax components.%
%
\footnoteA{The key verses are \GL\ 6.3--5 \citep[S394]{raouma06}:\\
%
  \bgroup\itshape
  trikunighnavilambana\*m kal\=as tatsahitonas \\
  \mbox\null\quad tithivad vyagu\*h \'saro 'ta\*h  $|$ \\
  atha \*sa\*dgu\*nalambana\*m lav\=as tair\\
  \mbox\null\quad  yugayugvitribhata\*h punar nat\=a\*m\'s\=a\*h  $||$ 3 $||$\\
  da\'sah\*rtanatabh\=agon\=ahat\=a\*s\*tendavastad \\
  \mbox\null\quad rahitasadh\*rtiliptai\*h \*sa\*dbhir \=apt\=asta eva $|$ \\
  svadigiti natir etatsa\*msk\*rta\*h so '\.ngul\=adi\*h sphu\*ta \\
  \mbox\null\quad  i\*suramuto 'tra sy\=at sthiticchannap\=urvam $||$ 4 $||$\\
  sthitirasahatiram\'s\=a vitribha\*m tai\*h p\*rthakstha\*m \\
  \mbox\null\  rahitasahitam\=abhy\=a\*m lambane ye tu t\=abhy\=am $|$ \\
  sthitivirahitayukta\*h samsk\*rto madhyadar\'sa\*h\\ 
  \mbox\null\  krama\'sa iti bhavet\=a\*m spar\'samuktyos tu k\=alau $||$5$||$
  \egroup\\
  Compare the \KK\ counterpart quoted in the discussion of the
  following verse, and the worked example in
  \citet[S99--S108]{raouma08}.  As noted in \citet[68]{raovenu08}, the
  solar eclipse methods in the \KK\ and \GL\ give quite similar
  results.}
%
The flowchart in figure~\ref{FlowChart} illustrates the main sequence
of steps in such a procedure, starting with the time of true
conjunction and the positions of the bodies at that time. These
quantities and the corresponding longitudinal parallax are computed in
turn until they become fixed at the time of apparent conjunction.  The
resulting latitudinal parallax corrects the mid-eclipse latitude and
magnitude and consequently the half-duration of the eclipse. A similar
cycle of correction is then applied to each half to find the positions
and times of the apparent contact and release.

The present verse starts the \kks's version of this process by
computing a quantity more or less equivalent to the change in
arcminutes in the moon's longitude between true and apparent
conjunction.  Specifically, the \lambana\ in \ghatis\ multiplied by 13
is taken as arcminutes applied to the elongation of the sun from the
node, positively if the \lambana\ in \ghatis\ is positive (i.e.,
increasing the time to conjunction) and negatively if it is negative.
This scale factor is qualitatively justified by recollecting that the
sun's position at the time of a solar eclipse is nearly coincident
with the moon's, and if the moon travels approximately 13 degrees per
day then it moves approximately 13 arcminutes per \ghati.  Of course,
strictly speaking the \lambana\ corresponds to an arc of time-degrees
or right ascension along the equator rather than to elongation along
the ecliptic, but this distinction is disregarded for the sake of
convenience.

Meanwhile, the original \lambana\ in \ghatis\ is also converted to
degrees by mutiplying it by 6.  Again, this quantity is applied as
though it were an arc of longitude to correct an ecliptic position, in
this case the longitude of the nonagesimal.  From the new position of
the nonagesimal a new value of \nati\ is to be computed, which then
requires recomputing the times of contact and release and the extent
of the obscuration.

\begin{figure}[h]
  \includegraphics[height=4in]{KKeIteration}
  \centering
  \caption{The cycles of parallax correction to get from the time of
    true conjunction to the moments and positions of apparent
    conjunction, contact, and release}
  \label{FlowChart}
\end{figure}

\newpage %DW
%%% VERSE 2.6 %%%%%
%%%%
\begin{mula}{2.6 Adjusting eclipse data for parallax, continued}%
%
  sthiti\'s\=ulas\=uta 6 ghnalavonayukt\=a\*h \\
  p\*rthaksthatanvo\*h k\*rtalambana\*m svam $||$\\
  \*r\*na\*m sthitih\={\i}nayuter vidar\'se \\
  tata\*h sphu\*tasa\*msp\*r\'si[mo]k\*sik\=alau $||$ 6 $||$\\

      \sflight\medskip\noindent [The degrees of the nonagesimal's longitude] are
  [separately] diminished and increased by the degrees [produced by]
  multiplying the half-duration by 6.  For [each of] the two separate
  results [? \textit{tanu}], the \lambana\ [is] made [and applied]
  additively [or] subtractively [to the half-duration, to get the
  interval between mid-eclipse and release and the interval between
  contact and mid-eclipse, respectively].  When the mid-eclipse [time]
  is decreased or increased by the [corresponding] half-duration, then
  the two times of true contact and release [result].
\end{mula}

\subsection*{Verse Analysis}

Metre: \upajati? 

%\begin{figure}[h]
%  \includegraphics[height=0.5in]{KKeCh2Vs6Pada1}
%  \centering
%  \caption{The first \pada\ of verse~2.5 as it appears in the
%    manuscript}
%  \label{Ch2Vs6Pada1}
%\end{figure}

The metre and sense of this verse as transcribed here are both far
from satisfactory, but it seems to borrow some of its phrasing not
only from the abovementioned \GL\ verses but also from the \KK.%
%
\footnoteA{\KK\ 5.6--8\\ \citep[68--69]{mishra91}:\\
  \bgroup\itshape
  spa\*s\*to 'tra b\=a\*no natisa\*msk\*rta\*h sy\=ac channa\*m \\
  \mbox\null\qquad tata\*h pr\=agvad ata\*h sthiti\'s ca $|$ \newline
  sthityonayukt\=ad ga\*nit\=agat\=ac ca tithyantato \\
  \mbox\null\qquad lambanaka\*m p\*rthakstham $\|$ 6 $\|$ \newline
  svar\*na\*m ca tasmin pravidh\=aya s\=adhyas t\=atk\=alika\*h \\
  \mbox\null\qquad spa\*s\*ta\'sara\*h sthiti\'s ca $|$ \newline
  tayonayukte ga\*nit\=agate tat svar\*na\*m p\*rthakstha\*m \\
  \mbox\null\qquad muhur evam etau $\|$ 7 $\|$ \newline
  sy\=at\=a\*m sphu\*tau pragrahamuktik\=alau sak\*rt k\*rte\\
  \mbox\null\qquad lambanake sak\*rtsna\*h $|$ \newline
  tanmadhyak\=al\=antarage sthit\={\i} sphu\*te \'se\*sa\*m \\
  \mbox\null\qquad \'sa\'s\=a\.nkagraha\*noktam atra hi $\|$ 8 $\|$
  \egroup }

As reconstructed in \upajati\ metre, the first syllable makes the
first \pada\ hypermetric and the fourth syllable in each of the last
two \padas\ ought to be heavy instead of light, but neither the
manuscript nor the sense gives any pretext for enforcing corrections.
We are not sure how to interpret \textit{\'s\=ula\-s\=uta} to supply
the required content of a word-numeral meaning six.  Perhaps this form
is a corruption of the reading \textit{\'s\=ulisuta 6} `son of the
spear-bearer' quoted twice (f.~5r, line 11 and f.~5v, line 4) in the
worked example of \tsc{MS.~J:} we assume this alludes to the six-headed
deity Skanda but we know of no other instance of his name invoked as a
word-numeral.  The word \textit{tanu} in, presumably, the genitive
dual must mean the adjusted nonagesimal longitude, although we are not
familiar with this meaning (compare \textit{tanas} for `ascendant' in
verse~2.3 above).

\subsection*{Technical Analysis}

Although the verse's grammar is very hard to follow, there seems no
doubt that it is attempting to continue describing the above
\GL-inspired procedure.  Namely, the corrected half-duration in
\ghatis\ is converted to degrees by multiplying it by 6, and then, as
before, that time-degree arc is treated as an arc of the ecliptic,
applied negatively or positively to the longitude of the nonagesimal
to give approximate positions of the nonagesimal at the moments of
contact and release, respectively.  A new \lambana\ is computed for
each of those nonagesimal positions, and the time of half-duration is
adjusted separately by each of those \lambana\ values to give the
actual duration of each `half' of the eclipse.  These two distinct
`half' durations, no longer equal halves of the total eclipse
duration, give the moments of contact and release when applied with
the appropriate sign to the moment of mid-eclipse.

This completes the \kks's determination of the times of the beginning,
middle and end of a solar eclipse.  Indefinite cycles of iterative
recalculation are discarded in favour of a few specified corrections
and re-corrections.  Nor are there specific directions stated for
computing the times of beginning and end of totality, if any.

The example in \tsc{MS.~J} appears to follow these directions faithfully, finishing with 
the sum of the disks' radii (presumably for the purpose of an eclipse diagram) and 
identifying the eclipse `lord' (\textit{parve\'sa}) as Brahm\=a. 

%%% VERSE 2.7 %%%%%
%%%%
\begin{mula}{2.7 Conclusion}%
%
  asti vai\*s\*navadh\=amni sajjanavati saud\=amik\=ahve pure\\
  \'srautasm\=arttavic\=aras\=aracaturo mo\*dho hi r\=am\=ahvaya\*h $||$\\
  jyotirvittilakopamanyava iti khy\=ata\*h k\*sitau svair gu\*nai\*h\\
  tats\=unu\*h kara\*n\=akhyake\'sarim ima\*m cakre kavir
  bh\=askara\*h $||$ 7 $||$

     \sflight \medskip\noindent In the city called Saud\=amik\=a, filled with good
  people, in the Vai\*s\*nava clan, there is one called R\=ama, a
  Mo\*dha, learned in the essence of investigations of \sruti\ and
  \smrti, a [member of the] Aupamanyava [\gotra, who is] an ornament
  of the \textit{jyo\*ti\*h}-knowers, renowned on the earth for his
  own good qualities.  His son, \Bhaskara, the poet, wrote this
  [work], the \kks.
\end{mula}

\subsection*{Verse Analysis}

Metre: \sard\ with minor discrepancies: the second and twelfth
syllables of the first \pada\ should be heavy rather than light.

\Bhaskara's idiosyncratic method of compounding appears to be at work
in the last compound where the name of the text is unpacked as
`\textit{Ke\'sari} called \textit{Kara\*na}' instead of the other way
around.  The term \textit{mo\*dha} seems to refer to a Brahmana group
primarily associated with the Gujarat region (which is consistent with
the terrestrial latitude used in the \kks), but we have no other
information on \Bhaskara's location or any identification of the
placename Saud\=amik\=a with a modern locality.

\begin{tika}%
%
  iti \'sr\={\i}daivaj\~nar\=am\=atmajabh\=askara\-vi\-racite
  kara\*na\-ke\'sariye s\=urya\-parv\=adhi\-k\=ara\*h $||$ sa\*m\-p\=u\-r\*no
  'ya\*m grantha\*h $||$ \'su\-bha\*m lekhaka\-p\=a\*thaka\-yo\*h $||$
  \'subha\*m bhavet kaly\=a\*nam astu $||$ sa\*m. 1819 var\*se
  \'s\=ake 1684 mit\={\i} a\'svina\-\'sudi 14 \'sanau dine
  lip\={\i}\-k\*rta\*m $||$


     \sflight \medskip\noindent Thus the chapter on solar eclipses in the \kks\
  composed by \Bhaskara, the son of R\=ama \Daivajna.  This book is
  complete. Good fortune to the writer and the reader; may there be
  good fortune, may there be prosperity. [The manuscript was] written
  in the year Sa\*mvat 1819, \'Saka 1684, on the 14th [\tithi] of the
  bright [fortnight] of \=A\'svina, on Saturday [i.e., Saturday 2
  October 1762 \xtsc{ce}].
\end{tika}

\section*{Acknowledgements}

\lettrine[lines=2,loversize=.1]{W}{e} are indebted to the authors and maintainers of Davide~P.\ Cervone's
Object\TeX{} and Crit\TeX{} macro packages,
the Ledmac macro package, and Frans Velthuis's
\texttt{devnag} preprocessor at \texttt{sarovar.org}, in conjunction
with the \TeX Live distribution, for enabling us to produce the
critical edition and accompanying commentary.  Our thanks are also due
to Michio Yano for his Pancanga calendar conversion program and to the
developers of the Inkscape and Sage open source software tools
supporting graphics and computation,
as well as to the typesetters for their elegant and meticulous work. 
We are especially grateful to
Anuj Misra, our doctoral student at the University of Canterbury, for
his efforts in procuring copies of manuscript materials.

The libraries of Brown University and Columbia University kindly
supplied images of manuscript copies (from the John Hay Library's
David~E.\ Pingree Collection and the Smith Indic Collection
respectively), as did the Rajasthan Oriental Research Society (Jodhpur
and Jaipur).  We wish to thank the Royal Society of New Zealand for
their generous support, Agathe Keller for the initial inspiration,
and the anonymous reviewers of the paper for their helpful corrections 
and comments.

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\bigskip

\appendix

\section{Glossary of technical terms}
\label{Glossary}

\begin{description}
\item[\aksavalana] The component of \valana\ or directional
  deflection/inclination that depends on the \aksa\ or local
  terrestrial latitude.

\item[\angula] Literally `digit' (in the sense of finger-breadth), a
  linear measure applied to quantities like disk diameter and lunar
  latitude; taken as equivalent to three arc-minutes.

\item[\avadhi] A period of fourteen mean solar days.

\item[\ayanavalana] The component of \valana\ that depends on the
  tropical longitude of the bodies.

\item[\bahu, \bhuja] Arc or angle, usually referring to some table
  argument measured in units of arc.

\item[\ghati] One-sixtieth of a day, equal to twenty-four minutes.

\item[\karana] Usually a brief astronomical handbook in verse (may
  also mean one-half of a \tithi).

\item[\kala] One-sixtieth of a degree, or arc-minute.

\item[\koti] The complement of a given arc or angle (\bahu\ or
  \bhuja).

\item[\kostha, \kosthaka] A numerical table or an entry in such a
  table.

\item[\lagna] Ascendant: the point of intersection of the ecliptic and
  the local horizon in the east.

\item[\lambana] Longitudinal component of parallax, usually measured
  in time-units.

\item[\madhya] The moment of mid-eclipse.

\item[\moksa] The moment of release or end of eclipse, when the
  eclipsed body is no longer obscured.

\item[\nadi] A synonym for \ghati.

\item[\nagari] The most common North Indic script, in which these
  manuscripts of the \kks\ are written.

\item[\nata] Literally `depression' or angular distance from an apex;
  used both for zenith distance and for hour-angle separation from the
  meridian.

\item[\nati] Latitudinal component of parallax, usually measured in
  digits.

\item[\nimilana] The moment of immersion or beginning of totality.

\item[\parvan] Either an eclipse itself or the instant of syzygy when
  mid-eclipse occurs.

\item[\pada] One line of a (usually) four-line Sanskrit verse.

\item[\sayana] Precession-corrected, referring to tropical longitude.

\item[\sanmilana] A synonym for \nimilana.

\item[\sparsa] The moment of first contact between eclipsed and
  eclipsing bodies.

\item[\tana] Used idiosyncratically in the \kks\ to mean ascendant.

\item[\tithi] One-thirtieth of a synodic month.

\item[\unmilana] The moment of emersion or end of totality.

\item[\upari] Literally `above', but in this case referring to the
  argument at the top of a table.

\item[\valana] `Deflection' or `inclination', i.e., angle between the
  local east-west direction and the eastward movement of the moon in
  an eclipse.

\item[\vilagna] Nonagesimal: the point of the ecliptic $90\deg$ west
  of the ascendant.

\end{description}

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