Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 1
Fore
wor
d to
the
Seco
nd E
ditio
n Th
e fa
ble
of A
para
Gan
ita a
nd t
he M
ystic
al G
arde
n of
Enc
hant
ed N
umbe
rs i
s
obvi
ousl
y fic
tiona
l. Th
e in
spira
tion
is L
eela
vati
Gan
itam
, a
chap
ter
in t
he a
ncie
nt
mat
hem
atic
al t
reat
ise,
the
Sidd
hant
a Si
rom
ani,
writ
ten
by B
hask
arac
hary
a in
115
0CE.
The
Leel
avat
i Gan
itam
is fa
scin
atin
g no
t onl
y fo
r its
trea
tmen
t of i
ndet
erm
inat
e an
alys
is
and
a m
etho
d to
sol
ve P
ell’s
Equ
atio
n, b
ut a
lso, a
s a
Can
adia
n un
iver
sity’
s w
ebsit
e on
mat
hem
atic
al h
istor
y pu
ts i
t, fo
r its
poe
tic c
onve
rsat
ion
betw
een
the
narr
ator
and
a
narr
atee
na
med
Le
elav
ati1 .
The
sim
ilarit
y be
twee
n th
is po
etic
co
nstru
ct
and
the
conv
ersa
tion
betw
een
Apa
ra G
anita
and
the
dwar
a pa
lika
is pr
obab
ly n
otic
eabl
e.
Fr
ame
stor
ies
are
not
com
mon
for
sci
entif
ic r
esea
rch
pape
rs,
but
they
cer
tain
ly
have
a h
istor
ical
pre
cede
nt.
1 “B
hask
arac
hary
a”,
His
tory
of
M
athe
mat
ics,
Sim
on
Fras
er
Uni
vers
ity,
<http
://w
ww
.mat
h.sf
u.ca
/hist
mat
h/In
dia/
12th
Cen
tury
AD
/Bha
skar
a.ht
ml>
(21s
t Sep
tem
ber,
2002
.)
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 2
Prol
ogue
–
The
Mys
tical
Gar
den
of E
ncha
nted
Num
bers
O
nce
upon
a t
ime,
in
the
mag
ical
mys
tical
city
of
Suva
rnap
uri2 ,
ther
e liv
ed a
stud
ent c
alle
d Ap
ara
Gan
ita3 . A
para
Gan
ita w
as v
irtu
ous
and
devo
ted
to h
is s
cien
ces.
Hav
ing
spen
t con
side
rabl
e am
ount
of t
ime
lear
ning
the
shas
tras
from
his
gur
u, h
e wa
s
surp
rise
d wh
en o
ne d
ay h
is g
uru
calle
d hi
m u
p.
“You
hav
e pe
rfor
med
wel
l, O
sis
hya4 m
ine”
, the
gur
u sa
id, “
but t
he ti
me
has
now
com
e fo
r you
to ta
ke le
ave”
.
Apar
a G
anita
was
at
once
sad
, fo
r he
had
lea
rned
a l
ot u
nder
him
. Bu
t he
rem
aine
d qu
iet a
nd c
ontin
ued
liste
ning
to h
is g
uru.
“Lis
ten,
Apa
ra G
anita
, I sh
all n
ow te
ll yo
u so
met
hing
that
my
guru
told
me
when
I
finis
hed
my
stud
ies.
For,
a st
udy
in G
anita
Sas
tra
(mat
hem
atic
s) is
not
com
plet
e, u
nles
s
one
visi
ts th
e M
ystic
al G
arde
n of
Enc
hant
ed N
umbe
rs”
“You
mus
t go
and
find
this
pla
ce fo
r you
r edu
catio
n to
be
trul
y co
mpl
ete”
.
And
so A
para
Gan
ita w
ent
abou
t se
arch
ing
for
this
pla
ce.
Inde
ed,
afte
r m
uch
trav
ellin
g an
d se
arch
ing,
he
was
final
ly s
hown
the
way
to
the
Mys
tical
Gar
den
of
Ench
ante
d N
umbe
rs.
And
lo, w
hat
a be
autif
ul s
ight
it w
as!
For
it wa
s si
tuat
ed i
n th
e m
idst
of
a lu
sh
gree
n va
lley,
sad
dled
by
mou
ntai
ns o
n ei
ther
sid
e. D
own
ther
e, A
para
Gan
ita c
ould
see
fam
ous
mat
hem
atic
ians
exp
ositi
ng t
heir
the
orie
s an
d sk
ills,
like
hawk
ers
on a
baz
aar
stre
et. T
here
was
Euc
lid s
tand
ing
on a
rec
tang
le, e
xpla
inin
g th
e be
auty
of
the
Gol
den
Ratio
in c
lass
ic G
reco
Cal
dean
arc
hite
ctur
e. P
ytha
gora
s wa
s st
andi
ng n
ext t
o hi
m a
s a
2 Suv
rnap
uri =
City
of G
old
3 A
para
Gan
ita =
som
eone
with
a lo
t of m
athe
mat
ical
tale
nt.
4 Sish
ya =
stud
ent
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 3
part
of
the
Gre
ek e
xhib
it, e
xpla
inin
g th
e vi
rtue
s of
a r
ight
-ang
led
tria
ngle
to a
cur
ious
crow
d. F
rom
the
far
end
of th
e O
rato
r’s
Cor
ner,
Zha
o Ju
n Q
ing
look
ed a
t Pyt
hago
ras
and
smile
d. H
e wa
s hi
mse
lf ho
ldin
g a
righ
t-ang
led
tria
ngle
and
was
exp
lain
ing
his
proo
f
for
the
Pyth
agor
as’
Theo
rem
. M
ande
lbro
t wa
s de
cora
ting
the
Gar
den
with
flo
wers
of
frac
talla
te b
eaut
y. J
ohn
Nas
h wa
s cl
ose
by;
he w
as p
oint
ing
at a
gro
up o
f wo
men
,
prob
ably
exp
lain
ing
gam
e th
eory
to
onlo
oker
s ar
ound
him
. In
ano
ther
cor
ner
of t
he
gard
en,
(Sec
tor
1729
), Sr
iniv
asa
Ram
anuj
an w
as v
ocife
rous
ly a
rgui
ng a
poi
nt w
ith
Thom
as H
ardy
.
It wa
s suc
h an
env
iron
men
t tha
t Apa
ra G
anita
wan
ted
to e
nter
.
How
ever
, as
he w
as a
bout
to e
nter
thro
ugh
the
grea
t doo
rs g
uard
ing
the
gard
en,
he h
eard
a so
noro
us v
oice
cal
ling
out h
is n
ame.
He
stop
ped
and
turn
ed a
roun
d to
see
who
was
cal
ling
him
onl
y to
saw
a y
oung
wom
an c
omin
g to
ward
s hi
m. W
ith e
yes
burn
ing
with
cur
iosi
ty a
nd a
voi
ce s
weet
er th
an
a ni
ghtin
gale
, she
said
: -
O S
tude
nt E
rudi
te,
Wha
t is
it th
at y
ou st
udy
toni
ght?
Just
wha
t I
need
ed,
a m
ystic
al d
war
a pa
lika
(fem
ale
door
kee
per)
, he
sai
d to
him
self.
Sha
king
his
hea
d in
wry
am
usem
ent,
he lo
oks
at th
e bo
oks
in h
is h
and
and
take
s
a de
ep b
reat
h to
beg
in h
is d
isse
rtat
ion.
...
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 4
Con
tent
s
Fore
word
1
Prol
ogue
– T
he M
ystic
al Ga
rden
of E
ncha
nted
Num
bers
2
Cont
ents
4
Stha
ana P
raka
rana
– H
ow th
e cale
ndar
is d
iffer
ent i
n di
ffere
nt re
gion
s Err
or! B
ookm
ark
not
defin
ed.
The S
outhe
rn A
maan
ta Ca
lenda
r E
rror
! Boo
kmar
k no
t def
ined
. W
ester
n Ama
anta
Calen
dar
Err
or! B
ookm
ark
not d
efin
ed.
Purn
imaa
nta C
alend
ar
Err
or! B
ookm
ark
not d
efin
ed.
The M
alaya
li Cale
ndar
E
rror
! Boo
kmar
k no
t def
ined
. Ta
mil C
alend
ar
Err
or! B
ookm
ark
not d
efin
ed.
Beng
ali C
alend
ar
Err
or! B
ookm
ark
not d
efin
ed.
Oriya
Cale
ndar
E
rror
! Boo
kmar
k no
t def
ined
. Th
e Nan
aksh
ahi C
alend
ar
Err
or! B
ookm
ark
not d
efin
ed.
Natio
nal C
alend
ar of
1957
E
rror
! Boo
kmar
k no
t def
ined
.
Maas
a Naa
mak
aran
a - H
ow th
e Mon
ths g
ot th
eir N
ames
. Er
ror!
Boo
kmar
k no
t def
ined
. 1)
Mo
nths n
amed
after
Nak
shat
ras
Err
or! B
ookm
ark
not d
efin
ed.
2)
Month
s nam
ed af
ter ra
asis
Err
or! B
ookm
ark
not d
efin
ed.
Parv
a Din
a Nirn
aya –
How
the d
ays o
f fes
tivals
are d
ecid
ed.
Erro
r! B
ookm
ark
not d
efin
ed.
Sam
vad
Sand
esha
– Ho
w Er
as co
me i
nto
play
Er
ror!
Boo
kmar
k no
t def
ined
.
Ksha
ya S
utra
– H
ow ce
rtain
mon
ths a
re d
ropp
ed.
Erro
r! B
ookm
ark
not d
efin
ed.
Epilo
gue –
The
Beg
inni
ng
25
Bibl
iogr
aphy
26
Ackn
owled
gem
ents
27
Appe
ndix-
The
Stru
ctur
e of t
he In
dian
Cale
ndar
Sys
tem
28
Appe
ndix
– Ksh
aya U
ntan
gled
. 29
Appe
ndix
– Why
Ksh
aya d
idn’
t occ
ur b
etwe
en 18
41 an
d 19
64CE
. 30
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 5
Sth
aana
Pra
kara
na –
H
ow th
e ca
lend
ar is
diff
eren
t in
diffe
rent
regi
ons
In a
sono
rous
voi
ce, t
he d
wara
pal
ika
said
, “I
n or
der t
o as
certa
in y
our d
isse
rtatio
n’s v
erac
ity,
can
I hea
r you
talk
abo
ut th
e ca
lend
ar’s
regi
onal
com
plex
ity?”
To
whi
ch A
para
Gan
ita li
stene
d to
the
mul
titud
es o
f voi
ces i
n th
e G
arde
n, a
nd re
plie
d th
us:-
Prob
ably
the
easi
est w
ay to
cla
ssify
Ind
ian
cale
ndar
s is
by th
e re
gion
of u
sage
. It
mus
t be
reite
rate
d th
ough
, tha
t suc
h an
exe
rcise
mig
ht b
e m
isle
adin
g. T
he c
lass
ifica
tion
is
inde
ed n
ot w
ater
tight
; al
l ca
lend
ars
are
intr
insi
cally
int
er-li
nked
with
one
ano
ther
. A
flow
char
t of
the
var
ious
Ind
ian
cale
ndar
s an
d th
e lin
ks b
etw
een
them
is
give
n in
the
App
endi
x.
W
ith t
his
cave
at,
we’
ll no
w t
rave
rse
Indi
a on
a c
alen
dric
al v
ehic
le o
f so
rts. I
n
parti
cula
r, w
e try
to
asce
rtain
the
fol
low
ing
elem
ents
in
each
reg
ion’
s ca
lend
rical
prac
tices
: -
B
asis
of th
e C
alen
dar
Lo
cal V
aria
tion.
W
hen
does
the
year
beg
in?
Er
a Fo
llow
ed
We’
ll fin
d th
e fo
llow
ing
cale
ndar
s def
ined
with
thes
e m
etric
s: -
The
Sout
hern
Am
aant
a C
alen
dar
Th
e So
uthe
rn A
maa
nta
Luni
sola
r C
alen
dar
is p
redo
min
antly
fol
low
ed in
the
Sout
h an
d So
uth-
Wes
t Ind
ian
stat
es o
f And
hra
Prad
esh,
Kar
nata
ka a
nd M
ahar
asht
ra.
It is
esse
ntia
lly a
lun
isola
r on
e; i
.e.,
its d
ays
and
mon
ths
are
calc
ulat
ed b
ased
on
the
mot
ions
of t
he m
oon.
Lik
e th
e C
hine
se c
alen
dar,
the
mon
th is
cal
cula
ted
from
new
moo
n
to n
ew m
oon.
It
how
ever
, di
ffers
fro
m t
he C
hine
se c
alen
dar
in t
hat
the
luna
r da
y
(“th
ithi”
) of t
he n
ew m
oon
is c
onsi
dere
d th
e la
st d
ay o
f the
pre
viou
s m
onth
. Aga
in, a
s in
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 1
1
(Cha
itrad
i) la
st
full
moo
n be
fore
M
esha
Sa
nkra
nti
Tam
il N
adu
Sola
r K
ali,
Jovi
an
cycl
e (S
outh
ern
Scho
ol)
Mes
ha S
ankr
anti
Tri
pura
So
lar
Kal
i, B
enga
li Sa
n So
lar
Day
af
ter
Mes
ha S
ankr
anti
Ben
gali
rule
s fo
r be
ginn
ing
of
mon
th
Utt
aran
chal
Pu
rnim
aant
a V
ikra
ma
Era
(Cha
itrad
i) O
ne d
ay a
fter
the
last
fu
ll m
oon
befo
re
Mes
ha
Sank
rant
i
Utt
ar P
rade
sh
Purn
imaa
nta
Vik
ram
a Er
a (C
haitr
adi)
One
day
afte
r th
e la
st
full
moo
n be
fore
M
esha
Sa
nkra
nti
Wes
t Ben
gal
Sola
r K
ali,
Ben
gali
San
Sola
r D
ay
afte
r M
esha
San
kran
ti B
enga
li ru
les
for
begi
nnin
g of
m
onth
Ta
ble
1: -
Cal
endr
ical
pra
ctic
es in
diff
eren
t Ind
ian
stat
es
Not
e:
1)
The
tabl
e is
exh
aust
ive
neith
er i
n te
rms
of c
alen
dars
nor
in
term
s of
sta
tes.
Aru
nach
al P
rade
sh, M
anip
ur, M
egha
laya
, Miz
oram
, Nag
alan
d an
d Si
kkim
wer
e
left
out.
2)
Cha
tterje
e m
entio
ns th
at th
e O
rissa
Sch
ool f
or d
ecid
ing
the
begi
nnin
g of
the
sola
r
mon
th i
s al
so u
sed
in P
unja
b an
d H
arya
na “
whe
re t
he s
olar
cal
enda
r is
also
used
”.19
19 C
hatte
rjee,
SK
p. 1
4
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 1
2
Maa
sa N
aam
akar
ana
- H
ow th
e M
onth
s go
t the
ir N
ames
. Li
sten
ing
to th
is, s
he sa
id,
“Sin
ce w
e ar
e de
ep in
this
gam
e,
Mig
ht I
ask
how
eac
h m
onth
got
its n
ame?
” To
whi
ch A
para
Gan
ita st
ared
at a
gul
moh
ar fl
ower
with
twen
ty-s
even
bud
s and
repl
ied
thus
:-
Th
e co
mpl
exity
of
the
Indi
an c
alen
dar
syst
em i
s no
t ju
st i
n th
e pl
etho
ra o
f
cale
ndar
s av
aila
ble,
but
also
in
the
man
ner
in w
hich
the
y lin
k up
with
one
ano
ther
. A
prin
cipa
l poi
nt o
f lin
kage
of m
ost I
ndia
n ca
lend
ars
is in
thei
r nam
es o
f the
mon
ths;
as
we
shal
l se
e, t
he s
imila
r se
ts o
f m
onth
nam
es a
re u
sed
in m
ore
than
one
cal
enda
r. In
thi
s
sect
ion,
we
aim
to fo
rmul
ate
rule
s det
erm
inin
g th
e na
min
g of
the
mon
ths.
Our
mot
ivat
ion
is no
t jus
t tax
onom
ic; m
onth
nam
es, w
e sh
all s
ee, a
re c
ritic
al to
und
erst
andi
ng th
e In
dian
cale
ndar
sys
tem
.
W
e pr
opos
e th
at th
ere
are
two
type
s of m
onth
nam
es: -
1) M
onth
s na
med
afte
r Nak
shat
ras
Th
e se
t of
mon
th n
ames
nam
ed a
fter
naks
hatr
as i
s us
ed b
y bo
th s
olar
and
luni
sola
r cal
enda
rs, a
ddin
g to
see
min
g co
mpl
exity
of t
he In
dian
cal
enda
r sys
tem
. Ind
eed,
as w
e sh
all
see,
thi
s ty
pe s
houl
d ac
tual
ly c
alle
d as
‘M
onth
s in
itial
ly n
amed
afte
r
Nak
shat
ras’
; th
ere
has
been
an
in
fusio
n of
so
lar
rule
s in
to
an
esse
ntia
lly
luna
r
conv
entio
n.
Le
t us
then
, firs
t con
side
r the
orig
inal
rule
. Sah
a an
d La
hiri
men
tion
that
pak
shas
or fo
rtnig
hts
wer
e di
ffer
entia
ted
base
d on
the
naks
hatr
a “w
here
the
moo
n is
full”
.20 T
hat
is to
say
, if
a pa
rticu
lar
full
moo
n oc
curs
nea
r, sa
y, th
e lu
nar
aste
rism
, Vis
akha
, the
full
moo
n w
ould
be
calle
d as
Vai
sakh
a Pu
rnim
aasi
, and
the
mon
th w
ould
be
Vais
akha
. The
earli
est
luni
sola
r m
onth
s, th
en,
wer
e pu
rnim
aant
a, t
hat
is, t
he n
ame
of t
he f
ull
moo
n
20 S
aha
et a
l. p.
221
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 1
3
corr
espo
nded
to th
e na
me
of th
e m
onth
. Of c
ours
e, th
e fu
ll m
oon
occu
rs a
t all
naks
hatr
as.
Fifte
en w
ere
take
n in
to a
ccou
nt fo
r nam
ing
of th
e m
onth
, spa
ced
mor
e or
less
equ
ally
.
W
e th
us h
ave
the
follo
win
g se
t of n
ames
alo
ng w
ith th
eir r
espe
ctiv
e na
ksha
tras
21:
-
Nak
shat
ra o
n Pu
rnim
a M
onth
Nam
e C
hitra
C
haitr
a V
isakh
a V
aisa
kha
Jyes
tha
Jyai
stha
(P
urva
& U
ttara
) Aas
haad
ha
Aas
haad
ha
Srav
ana
Sraa
vana
(U
ttara
& P
urva
) Bha
adra
pada
B
haad
rapa
da
Asv
ini
Asv
ayuj
a (A
asvi
na)
Krit
tika
Kaa
rthik
a M
ruga
sira
Maa
rgha
sira
Pu
shya
mi
Paus
a (P
ushy
am)
Mag
haa
Maa
gha
(Utta
ra a
nd P
urva
) Pha
lgun
i Ph
algu
na
It m
ay b
e no
ted
that
the
mon
ths
of A
asha
adha
, Bha
drap
ada
and
Phal
guna
are
linke
d to
tw
o na
ksha
tras
res
pect
ivel
y. C
hatte
rjee
and
Cha
krav
arth
y gi
ve t
he f
ollo
win
g
crite
ria fo
r cho
osin
g na
ksha
tras
for m
onth
nam
es22
: -
1)
The
yoga
taar
as o
r th
e id
entif
ying
sta
rs o
f th
e na
ksha
tras
are
pro
min
ent o
r ha
ve
tradi
tiona
l sig
nific
ance
.
2)
The
y ar
e sp
aced
mor
e or
less
equ
idist
ant f
rom
one
ano
ther
.
It m
ust b
e m
entio
ned
that
this
rule
is n
ow a
n ap
prox
imat
ion
larg
ely
due
to E
arth
’s
prec
essio
n; fo
r in
stan
ce, t
his
year
’s C
hitr
a Pu
rnim
aasi
had
Swa
ti as
its
naks
hatr
a. A
lso,
poss
ibly
for
hist
oric
al r
easo
ns, a
nd a
llow
ing
for
regi
onal
var
iatio
n in
pro
nunc
iatio
n, th
e
Oriy
a, B
enga
li, A
ssam
ese,
Pun
jabi
and
Tam
il so
lar
cale
ndar
s al
so u
se t
he s
ame
set
of
mon
th n
ames
. To
reco
ncile
all
this,
we
mig
ht fr
ame
a ne
w r
ule;
that
, the
am
aant
a lu
nar
21Sa
ha e
t al.
p. 2
21
22 C
hakr
avar
thy
et a
l, p.
281
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 1
4
mon
th ta
kes
its n
umbe
r fro
m th
e so
lar
mon
th th
at s
tarts
in it
, but
its
nam
e fro
m th
e so
lar
mon
th in
whi
ch it
sta
rts, w
hile
follo
win
g th
e pu
rnim
aant
a m
onth
s in
chr
onol
ogic
al o
rder
.
That
is to
say
, sin
ce C
hitr
a oc
curr
ed d
urin
g th
e pu
rnim
a of
this
yea
r’s
first
pur
nim
aant
a
mon
th,
we
call
this
mon
th a
s ‘C
haitr
a’.
Con
sequ
ently
, th
e fir
st a
maa
nta
mon
th w
ould
also
be
‘Cha
itra’
, w
hich
also
wou
ld b
e th
e na
me
of t
he s
olar
mon
th d
urin
g w
hich
the
amaa
nta
‘Cha
itra’
sta
rted.
How
ever
, the
‘nu
mbe
r’ o
f the
sol
ar m
onth
(‘1
’ in
the
case
of
amaa
nta
and
purn
imaa
nta
Cha
itra)
is n
ot q
uite
the
sam
e; th
e so
lar C
haitr
a is
the
last
(i.e
.,
12th)
mon
th o
f the
yea
r. Th
e lu
niso
lar
Cha
itra’
s nu
mbe
r is
take
n by
the
sola
r m
onth
that
begi
ns in
it, n
amel
y th
e so
lar
Vai
sakh
a. A
ll th
is ca
n be
see
n in
the
gra
phic
in t
he n
ext
page
.
The
rela
tions
hips
for a
ll th
e m
onth
s m
ay b
e m
appe
d ac
cord
ing
to th
e fo
llow
ing
tabl
e23: -
Raa
si
App
roxi
mat
e na
ksha
tra
on
Purn
ima
Lun
ar
Mon
th
Nam
e
Sola
r M
onth
N
ame
Ass
ames
e V
ersi
on
Tam
il V
ersi
on
Punj
abi
Ver
sion
24
Mes
ha
Chi
tra
Cha
itra
Vai
sakh
a B
ahag
C
hitta
rai
Vai
sakh
V
rsha
va
Visa
kha
Vai
sakh
a Jy
aist
ha
Jeth
V
aika
si
Jeth
M
ithun
a Jy
esth
a Ja
isht
a A
asha
adha
A
har
Aan
i H
arh
Kar
kata
(P
urva
&
Utta
ra)
Aas
haad
ha
Aas
haad
ha
Sraa
vana
Sa
on
Aad
i Sa
wan
Sim
ha
Srav
ana
Sraa
vana
B
haad
rapa
daB
had
Aav
ani
Bha
don
Kan
ya
(Pur
va &
U
ttara
) B
haad
rapa
da
Bha
adra
pada
Asv
ayuj
a (A
asvi
na)
Ahi
n Pu
ratta
asi
Asu
Tula
A
svin
i A
svay
uja
(Aas
vina
) K
aarth
ika
Kat
i A
rppi
si
Kat
ik
Vris
chik
a K
rittik
a K
aarth
ika
Maa
rgha
sira
A
ghon
K
arth
igai
M
agha
r D
hanu
s M
ruga
sira
Maa
rgha
sira
Pa
usa
(Pus
hyam
) Pu
ha
Maa
rgal
i Po
h
Mak
ara
Push
yam
i Pa
usa
(Pus
hyam
) M
aagh
a M
agh
Thaa
i M
agh
Kum
bha
Maa
gha
Maa
gha
Phal
guna
Ph
agun
M
aasi
Ph
agun
M
ina
(Utta
ra a
nd
Purv
a)
Phal
guna
C
haitr
a C
hait
Pang
uni
Che
t
23 C
hakr
avar
thy,
et a
l. p.
280
24
Pal
Sin
gh P
urew
al, N
anak
shah
i Sam
at. <
http
://w
ww
.sikh
.net
/sik
hism
/Nan
aksh
ahi.h
tm >
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 1
5
Phal
guni
The
Ass
ames
e, P
unja
bi a
nd T
amil
vers
ions
hav
e be
en p
rovi
ded
to g
ive
an id
ea o
f
the
lingu
istic
var
iatio
n. I
t is
also
int
eres
ting
to o
bser
ve t
hat
the
Nat
iona
l C
alen
dar
sugg
este
d by
Sah
a an
d La
hiri
also
use
s th
e sa
me
set
of m
onth
nam
es,
incr
easi
ng t
he
pote
ntia
l con
fusio
n. A
s is
pro
babl
y ob
viou
s by
now
, the
rul
e do
es n
ot c
orre
spon
d to
the
Tam
il, N
atio
nal a
nd N
anak
shah
i cal
enda
rs.
2) M
onth
s na
med
afte
r raa
sis
Onl
y so
lar
mon
ths
shar
e th
eir
nam
es w
ith r
aasi
s. SK
Cha
tterje
e an
d A
purb
a K
umar
Cha
krav
arth
y gi
ve th
e fo
llow
ing
nam
es a
long
with
the
asso
ciat
ed ra
asis
25.
Raa
si
Sans
kriti
sed
Ver
sion
M
alay
alam
Ver
sion
M
esha
M
esha
M
edam
V
rsha
va
Vrs
hava
Ed
avam
M
ithun
a M
ithun
a M
idhu
nam
K
arka
ta
Kar
kata
K
arita
ka
Sim
ha
Sim
ha
Chi
ngam
K
anya
K
anya
K
anni
Tu
la
Tula
Th
ulam
V
risch
ika
Vris
chik
a V
risch
ikam
D
hanu
s D
hanu
s D
hanu
M
akar
a M
akar
a M
akar
am
Kum
bha
Kum
bha
Kum
bham
M
ina
Min
a M
inam
Th
at is
to s
ay, t
he m
onth
sha
res
its n
ame
with
that
of i
ts c
orre
spon
ding
San
kran
ti.
For
inst
ance
, if
Mes
ha S
ankr
anti
occu
rs o
n a
certa
in d
ay, t
hen
the
perio
d un
til th
e ne
xt
Sank
rant
i wou
ld b
e M
esha
maa
sa (M
edha
m m
aasa
m).
This
nam
ing
rule
is f
ollo
wed
prim
arily
in t
he M
alay
alam
cal
enda
r. In
cide
ntal
ly,
Abh
ayan
kar s
ays t
hat t
he O
riya
cale
ndar
also
follo
ws t
his r
ule.
26
25 C
hakr
avar
ty, e
t al.
p. 2
80
26 A
bhay
anka
r, p.
55
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 1
6
Parv
a D
ina
Nirn
aya
–
How
the
days
of f
estiv
als
are
deci
ded.
H
eari
ng h
im sp
eak,
she
aske
d,
“The
cul
tura
l com
plex
ity is
inte
rest
ing,
bu
t per
haps
you
hav
e a
fest
ival
s lis
ting?
To
whi
ch A
para
Gan
ita lo
oked
at b
irds
chi
rpin
g an
d re
plie
d th
us:-
We
prov
ide
a lis
t of
Ind
ian
fest
ival
s, al
ong
with
the
ir (I
ndic
) da
tes
and
the
cale
ndar
use
d to
rec
kon
the
parti
cula
r fe
stiv
al.
The
list
of f
estiv
als
is by
no
mea
ns
exha
ustiv
e; th
e en
tries
are
mos
tly p
ublic
hol
iday
s in
Indi
a.
Fest
ival
27
Indi
c D
ate
Add
ition
al R
ules
C
alen
dar
Use
d
Mak
ara
Sank
rant
i/
Pong
al
Mak
ara
Sank
rant
i N
one
Sola
r
Mah
a Si
va R
aatr
i M
agha
K 1
4 M
ust c
over
a n
isita
Lu
niso
lar
Hol
i Ph
algu
na P
urni
ma
Hol
ika
Dah
ana
is
obse
rved
on
the
nigh
t
of t
he P
urni
ma;
Hol
i
is ob
serv
ed
on
the
sola
r da
y af
ter
Hol
ika
Dah
ana
Luni
sola
r
Uga
di /
Gud
i Pad
wa
Cha
itra
S 1
Non
e Lu
niso
lar
Ram
a N
avam
i C
haitr
a S
9 M
ust c
over
Mad
yahn
a
Tam
il N
ew
Yea
r,
Vish
u,
Ben
gali
New
Yea
r
Mes
ha S
ankr
anti
R
espe
ctiv
e Sa
nkra
nti
rule
s
Sola
r
Gan
esh
Cha
turti
B
hadr
apad
a S
4 M
ust c
over
Mad
yahn
a Lu
niso
lar
Bud
dha
Purn
ima
Vai
sakh
i Pur
nim
a
Luni
sola
r
Rak
sha
Ban
dan
Sr
avan
a Pu
rnim
a
Luni
sola
r
Janm
asht
ami
Srav
ana
K 8
Luni
sola
r
27 C
hatte
rjee,
p. 6
0-68
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 1
7
Ona
m
Moo
n is
in S
rava
na
naks
hatra
in
So
lar
Bha
drap
ada
Lu
niso
lar
and
Sola
r
Mah
anav
ami
Asv
ayuj
a S
9 (M
ahan
avam
i is
reck
oned
be
fore
th
e
othe
r 8
days
of
Dus
sehr
a28)
Luni
sola
r
Vija
yada
sam
i (T
he
thith
i af
ter
Mah
anav
ami)
Mus
t cov
er a
Nis
ita
Luni
sola
r
Dee
pava
li A
svay
uja
Am
avas
ya
Mus
t cov
er p
rado
sha
Luni
sola
r
A
bit
of e
xpla
natio
n is
nece
ssar
y. F
irst,
the
term
s. “N
isita
” is
defin
ed to
be
a tim
e-
perio
d m
easu
red
by t
wo
ghat
ikas
(1/
60th o
f a
sola
r da
y; a
ppro
xim
atel
y 20
min
utes
)
stre
tchi
ng o
n ei
ther
sid
e of
mid
nigh
t. “P
rado
sha”
is
the
time-
perio
d st
retc
hing
for
tw
o
muh
urta
s (1
/15th
of
the
time
betw
een
sunr
ise a
nd s
unse
t; ap
prox
imat
ely
1 ho
ur 3
6
min
utes
) af
ter
suns
et. “
Mad
hyah
na”
is on
e-th
ird o
f the
tim
e-pe
riod
betw
een
sunr
ise a
nd
suns
et. T
his
fract
ion
cove
rs m
id-d
ay.
Seco
nd,
thes
e da
tes
are
valid
onl
y on
non
-inte
rcal
ary
thith
is f
or a
ll lu
niso
lar
fest
ival
s. B
oth
leap
day
s an
d no
n-le
ap d
ays
in le
ap m
onth
s ar
e de
emed
unf
it fo
r fes
tival
s.
(Ksh
aya
maa
sas
are
not
an i
ssue
her
e be
caus
e a)
jug
ma
mon
ths
are
deem
ed f
it fo
r
relig
ious
obs
erva
nce
and
b) in
the
East
ern
and
Nor
thw
este
rn sc
hool
s, th
e ex
tra in
terc
alar
y
mon
th is
dee
med
to b
e no
rmal
).
And
fin
ally
, if
the
give
n th
ithi d
oesn
’t co
ver
the
give
n tim
e, o
r co
vers
the
give
n
time
on tw
o so
lar d
ays,
then
the
seco
nd so
lar d
ay is
reck
oned
to b
e th
e re
spec
tive
fest
ival
.
28 S
ivas
ri Sa
rma,
Mad
ugul
a. In
terv
iew
by
auth
or. H
yder
abad
, Ind
ia. 4
th Ja
nuar
y, 2
002.
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 1
8
Sam
vad
Sand
esha
–
How
Era
s co
me
into
pla
y Pe
rcei
ving
the
resp
onse
, she
que
stio
ned,
“I
don
’t kn
ow if
this
is a
n im
porta
nt p
art,
but f
rom
whe
n do
all
cale
ndar
s sta
rt?”
To w
hich
Apa
ra G
anita
look
ed a
t a fo
unda
tion
stone
and
repl
ied
thus
:-
Th
e In
dian
cal
enda
r sy
stem
fol
low
s a
wid
e ra
nge
of e
ras,
som
e of
hist
oric
al
inte
rest
. A
lso,
we
do n
ot a
ttem
pt t
o lin
k in
divi
dual
cal
enda
rs t
o er
as,
for
the
sam
e
cale
ndar
may
be
reck
oned
with
two
diffe
rent
era
s in
two
diffe
rent
pla
ces.
Her
e’s t
he li
stin
g29: -
Era
Z
ero
Yea
r B
egin
ning
of
Era
with
res
pect
to
indi
vidu
al y
ear
Saka
78
CE
Mes
ha S
ankr
anti,
Cha
itra
S 1
Vik
ram
a 57
CE
Mes
ha
Sank
rant
i, C
haitr
a S
1,
Kar
tika
S 1,
Ash
adha
S 1
Kal
i 31
01 B
CE
Mes
ha S
ankr
anti,
Cha
itra
S 1
Kol
lam
82
4 C
E K
anya
San
kran
ti, S
imha
San
kran
ti
Ben
gali
San
963
+ so
lar y
ears
sin
ce 1
556
CE
Mes
ha S
ankr
anti
In a
dditi
on, s
ome
regi
ons
also
nam
e th
eir
year
s ac
cord
ing
to th
e na
mes
of t
he J
ovia
n
year
s. Sa
ha a
nd L
ahiri
poi
nt o
ut th
at th
ere
are
two
scho
ols
for
this
; the
Sou
ther
n sc
hool
nam
es i
ts y
ears
in
cont
inuo
us s
ucce
ssio
n, w
hile
the
Nor
ther
n sc
hool
nam
es i
ts y
ears
corr
espo
ndin
g to
the
pres
ent J
ovia
n ye
ar30
.
29 S
aha
et a
l. p.
252
– 2
58.
30 Ib
id. p
272
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 1
9
Ksh
aya
Sutr
a –
H
ow c
erta
in m
onth
s ar
e dr
oppe
d.
Obs
ervi
ng th
e re
actio
n, sh
e en
quir
ed,
To c
alen
dars
you
seem
to b
e an
act
ive
saak
shya
31,
But h
ave
you
stud
ied
the
ephe
mer
ally
con
foun
ding
ksh
aya?
To
whi
ch, A
para
Gan
ita lo
oked
at s
ome
falle
n le
aves
and
repl
ied
thus
:-
O
ne o
f the
mos
t int
eres
ting
aspe
cts
of th
e In
dian
luni
sola
r ca
lend
ar is
its
ksha
ya
maa
sas,
liter
ally
“de
caye
d m
onth
s”. O
ccas
iona
lly, c
erta
in m
onth
s ar
e dr
oppe
d fro
m th
e
luni
sola
r cal
enda
r. W
e no
w tr
y to
und
erst
and
the
mod
aliti
es b
ehin
d th
is o
mis
sion;
we
try
to a
nsw
er h
ow, w
hy, w
hen
and
whe
re it
hap
pens
.
Fi
rst,
let’s
try
to
defin
e a
ksha
ya m
onth
. C
hatte
rjee,
in
his
wor
k on
Ind
ian
cale
ndar
s, sa
ys th
at a
cer
tain
luna
r mon
th “
may
com
plet
ely
over
lap
any
of th
e sh
ort t
hree
nira
yana
sola
r mon
ths o
f Mar
gasi
ra, P
aush
a an
d M
agha
”, w
ith th
e re
sult
that
ther
e w
ill b
e
no n
ew m
oon
in th
e re
spec
tive
sola
r m
onth
. Con
sequ
ently
, the
re w
ill b
e no
luna
r m
onth
nam
ed “
afte
r …
this
sola
r m
onth
”.32
A g
raph
ic d
escr
ibin
g th
is in
tera
ctio
n is
give
n in
App
endi
x C
.
We
lear
n th
e fo
llow
ing
from
thi
s st
atem
ent:
- a)
tha
t th
e so
lar
mon
ths
of
Mar
gasir
a, P
ausa
and
Mag
ha a
re s
mal
l, b)
that
at a
cer
tain
tim
e, th
ere
mig
ht b
e no
new
moo
n in
the
se m
onth
s, an
d c)
the
cor
resp
ondi
ng l
unar
mon
th i
s dr
oppe
d fro
m t
he
cale
ndar
. Not
e th
at C
hatte
rjee
is si
lent
on
whe
ther
the
drop
ped
luna
r mon
th is
am
aant
a or
purn
imaa
nta;
a n
aïve
ass
umpt
ion
wou
ld b
e th
at s
ince
he
talk
s ab
out
new
moo
ns,
the
mon
th w
ould
be
amaa
nta.
But
, a
stud
y of
the
(C
haitr
adi)
amaa
nta
and
purn
imaa
nta
cale
ndar
s fo
r th
e pr
esen
t ye
ar r
evea
ls th
at th
e di
ffer
ence
bet
wee
n th
ese
two
cale
ndar
s is
still
two
wee
ks. T
here
fore
, it’s
saf
e to
con
clud
e th
at k
shay
a m
onth
s w
ere
drop
ped
from
the
purn
imaa
nta
cale
ndar
as w
ell.
31 sa
aksh
ya =
witn
ess (
in S
ansk
rit)
32 C
hatte
rjee,
p. 3
4
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 2
0
Mor
eove
r, th
e st
atem
ent
abou
t “c
orre
spon
ding
lun
ar m
onth
” is
unc
lear
; ar
e w
e
talk
ing
abou
t th
e lu
nar
mon
th w
ith t
he s
ame
num
ber
as t
he n
ew-m
oon-
lack
ing
sola
r
mon
th?
Or a
re w
e ta
lkin
g ab
out t
he lu
nar m
onth
with
the
sam
e na
me
of th
e so
lar m
onth
?
Run
ning
the
cal
endr
ica
code
pro
vide
d by
Der
show
itz a
nd R
eing
old
with
the
ir bo
ok
Cal
endr
ical
Cal
cula
tions
– T
he M
illen
ium
Edi
tion
(see
tabl
e fo
r val
ues)
, we
see
that
it’s
the
luna
r mon
th w
ith th
e sa
me
nam
e th
at g
ets d
ropp
ed.
To
acc
ount
for a
pur
nim
aant
a ks
haya
, and
to fu
rther
cla
rify
whi
ch m
onth
to d
rop,
we
re-p
hras
e th
e de
finiti
on o
f a
ksha
ya m
onth
to b
e th
us: -
“in
any
giv
en lu
nar
year
, if
two
cons
ecut
ive
Sank
rant
is oc
cur
betw
een
two
cons
ecut
ive
new
moo
ns, t
hen
the
luna
r
mon
th, w
heth
er a
maa
nta
or p
urni
maa
nta,
with
the
sam
e na
me
as th
e so
lar m
onth
in w
hich
this
occu
rs, i
s dr
oppe
d.”
As
we
shal
l see
, suc
h a
re-p
hras
ing
is us
eful
for
com
puta
tiona
l
purp
oses
.
In
deed
, as
w
e m
entio
ned
earli
er,
we
ran
the
Der
show
itz
and
Rei
ngol
d’s
cale
ndri
ca p
acka
ge to
get
val
ues
for
the
occu
rren
ce o
f a k
shay
a m
onth
. Sin
ce s
earc
hing
for
a ks
haya
mon
th is
com
puta
tiona
lly v
ery
heav
y33, w
e us
ed a
tab
le p
repa
red
by S
aha
and
Lahi
ri (ta
ble
22 in
the
book
)34 a
s a
star
ting
poin
t. W
e al
so ta
bula
ted
resu
lts fo
r no
n-
ksha
ya m
onth
s, sp
ecifi
cally
yea
rs w
ith g
aps
of 1
9, 4
6, 6
5, 7
6, 1
22 a
nd 1
41 y
ears
resp
ectiv
ely.
The
resu
lts a
nd th
e gr
aphs
from
thes
e re
sults
are
tabu
late
d in
the
appe
ndix
.
It
mus
t be
note
d th
at a
ll ca
ses t
abul
ated
pre
viou
sly
have
bee
n ca
lcul
ated
acc
ordi
ng
to S
urya
Sid
dhan
tic r
ules
and
tha
t w
e m
ay g
et a
diff
eren
t se
t of
res
ults
if
calc
ulat
ed
acco
rdin
g to
eph
emer
is ca
lcul
atio
ns. I
ndee
d, a
s C
hatte
rjee
has
poin
ted
out,
ther
e w
as a
diffe
renc
e in
196
4 C
E; e
phem
eris
calc
ulat
ions
sho
wed
Mar
gasir
a to
be
ksha
ya (
and
33 D
ersh
owitz
, Nac
hum
and
Rei
ngol
d. C
alen
dar T
abul
atio
ns –
190
0 to
220
0. (2
002:
Cam
brid
ge) C
ambr
idge
U
nive
rsity
Pre
ss. p
. 24
34 S
aha
et a
l. p.
250
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 2
1
Kar
thik
a, C
haitr
a to
be
adhi
ka),
whi
le a
s w
e’ve
see
n, S
urya
Sid
dhan
tic c
ompu
tatio
n
show
ed P
ausa
to b
e ks
haya
(and
Asv
ina
and
Cha
itra
to b
e ad
hika
).35 C
hatte
rjee,
how
ever
,
seem
s to
be
in a
gree
men
t w
ith D
ersh
owitz
and
Rei
ngol
d in
say
ing
that
the
re w
as a
ksha
ya in
Mag
ha in
198
3 C
E36, d
espi
te h
is us
e of
eph
emer
is ca
lcul
atio
ns.
Wha
t do
we
get f
rom
all
this
? W
e se
e th
at a
ksh
aya
mon
th c
an o
ccur
eve
ry 1
9, 4
6,
65,
76,
122
or 1
41 y
ears
. In
deed
, Sa
ha a
nd L
ahiri
’s t
abul
atio
n pr
ovid
e us
with
the
follo
win
g fre
quen
cies
of o
ccur
renc
es fo
r gap
s bet
wee
n ks
haya
mon
ths:
-
Inte
rval
Num
ber o
f tim
es o
ccur
ing
1911
463
651
761
122
114
16
Ta
ble
– N
umbe
r of t
imes
a p
artic
ular
inte
rval
gap
occ
urre
d
We
ther
efor
e se
e th
at b
etw
een
525
CE
and
1985
CE,
ksh
aya
occu
rred
11
times
with
a g
ap o
f 19
year
s, th
rice
with
a g
ap o
f 46
year
s, si
x tim
es w
ith a
gap
of 1
41 y
ears
,
and
once
eac
h w
ith g
aps o
f 65,
76
and
122
year
s. Th
e ob
viou
s qu
estio
n on
e w
ould
like
to
ask
wou
ld b
e w
hy. W
hy d
oes k
shay
a oc
cur o
nly
in th
ese
gaps
?
To a
nsw
er t
his
bette
r, w
e re
-iter
ate
wha
t ca
uses
ksh
aya
in t
he f
irst
plac
e. W
e
alre
ady
said
that
a k
shay
a w
ould
occ
ur w
hen
two
cons
ecut
ive
Sank
rant
is oc
cur b
etwe
en
two
Am
avas
yas.
That
is
to s
ay,
whe
n a
sola
r m
onth
is
shor
ter
in l
engt
h th
an,
and
is
com
plet
ely
encl
osed
by,
a (
an A
maa
nta)
luna
r m
onth
. Sah
a an
d La
hiri
go o
n to
say
that
the
“max
imum
dur
atio
n of
a lu
nar m
onth
exc
eeds
the
leng
ths
of th
e so
lar m
onth
s on
ly in
35 C
hatte
rjee,
SK
. p. 3
8 36
Der
show
itz,
Nac
hum
and
Edw
ard
M.
Rein
gold
. Ca
lend
rical
Cal
cula
tions
– T
he M
illen
nium
Edi
tion.
(2
001:
Cam
brid
ge) C
ambr
idge
Uni
vers
ity P
ress
. p. 2
69
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 2
2
the
case
of M
arga
sira,
Pau
sa a
nd M
agha
”37 a
nd th
at, t
here
fore
, ksh
aya
is p
ossi
ble
only
in
thes
e m
onth
s.
This
wou
ld e
xpla
in t
he s
olar
mon
th p
art,
but
wha
t of
luna
r? H
ow c
an t
he lu
nar
mon
th b
e bi
gger
than
the
sola
r m
onth
? A
la’a
Juw
ad h
as s
ome
answ
ers;
in h
is ar
ticle
, he
sugg
ests
tha
t th
e ca
noni
cal
syno
dic
mon
th,
a lu
nar
mon
th b
etw
een
two
cons
ecut
ive
phas
es o
f the
moo
n, is
not
con
stan
t in
leng
th. I
ndee
d, h
e go
es o
n to
say
that
bet
wee
n 16
00
and
2400
CE,
the
syno
dic
mon
th e
xten
ds in
leng
th fr
om 2
9 da
ys 6
hou
rs a
nd 3
1 m
inut
es
to 2
9 da
ys 1
9 ho
urs
and
59 m
inut
es.38
Mor
eove
r, he
say
s th
at th
e “l
onge
st lu
nar
mon
ths
… o
ccur
whe
n th
e da
te o
f the
new
Moo
n co
inci
des
with
apo
gee”
.39 A
bru
te-fo
rce
sear
ch
for t
he lo
nges
t syn
odic
mon
th d
efin
itely
won
’t gi
ve u
s a
ksha
ya; f
or k
shay
a to
occ
ur, t
he
luna
r m
onth
nee
ds t
o be
onl
y bi
gger
tha
n its
sol
ar c
ount
erpa
rt an
d m
ore
impo
rtant
ly,
com
plet
ely
enco
mpa
ss it
. Ind
eed,
Jaw
ad s
ays
that
the
long
est s
ynod
ic m
onth
occ
urre
d in
1610
CE,
a y
ear w
hich
occ
urs w
ithin
the
141
year
long
ksh
aya
hiat
us b
etw
een
1540
-154
1
CE
and
1680
– 8
1 CE
.
We
ther
efor
e se
arch
for o
ther
clu
es to
uns
cram
ble
ksha
ya. O
n a
pure
ly a
rithm
etic
pers
pect
ive,
we
obse
rve
the
follo
win
g: -
19
= 19
* 1
46
=
19 *
2 +
8
65
= 19
* 3
+ 8
76
=
19 *
4
122
= 19
* 6
+ 8
14
1 =
19 *
7 +
8
That
is to
say,
the
year
-gap
s are
in th
e fo
rm 0
, 8 m
od 1
9.
37 S
aha
et a
l. p.
250
38
Jaw
ad, A
la’a
. “H
ow L
ong
Is a
Lun
ar M
onth
?” in
Sky
& T
eles
cope
, Nov
embe
r 199
3. p
. 76
39 Ib
id.
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 2
3
Is it
pos
sibl
e th
en, t
hat t
he k
shay
a m
onth
has
som
ethi
ng to
do
with
the
Met
onic
cycl
e? T
he M
eton
ic C
ycle
is a
fairl
y w
ell d
ocum
ente
d ph
enom
enon
; firs
t obs
erve
d by
the
Gre
ek a
stro
nom
er M
etos
, eve
ry 1
9 ye
ars,
the
luna
r da
tes
over
lap
with
the
tropi
cal o
nes.
The
unde
rlyin
g m
athe
mat
ical
reas
on is
sim
ple:
- 19
sid
erea
l yea
rs c
onta
in 1
9*36
5.24
2189
= 69
39.6
sola
r day
s, w
hile
235
syn
odic
mon
ths (
with
a m
ean
of 2
9.53
sola
r day
s) c
onta
in
235*
29.5
3058
8853
= 6
939.
68 s
olar
day
s. Th
e le
ngth
s ov
erla
p. B
ut t
his
obvi
ousl
y is
neith
er n
eces
sary
nor
suf
ficie
nt; i
t mig
ht b
e us
eful
for t
he d
ates
to re
peat
, but
it d
efin
itely
does
n’t f
ulfil
the
requ
irem
ent f
or k
shay
a.
One
sug
gest
ion
ther
efor
e, m
ight
be
that
the
ksh
aya
occu
rs w
hen
the
num
ber
of
sola
r day
s of
a s
ider
eal y
ear i
s eq
ual t
o th
at o
f a s
ynod
ic m
onth
, whi
ch in
turn
is e
qual
to
that
from
an
anom
alist
ic m
onth
. An
anom
alist
ic m
onth
is d
efin
ed to
be
the
time
– pe
riod
betw
een
two
cons
ecut
ive
perig
ee p
assa
ges
and
has
a m
ean
valu
e of
27.
5545
5 da
ys.
Taki
ng t
hese
ave
rage
val
ues,
we
calc
ulat
e th
e av
erag
e va
lues
of
sola
r da
ys i
n w
hole
num
bers
of
sy
nodi
c an
d an
omal
istic
m
onth
s (c
anon
ical
ks
haya
ye
ars
shad
ed
for
refe
renc
e): -
Inte
rval
Occ
urre
nce
Mod
ulo
Sola
r Yea
rSy
nodi
c M
onth
sAn
omal
istic
Mon
ths
1911
1*19
6939
.601
591
6939
.688
3869
43.7
466
270
1*19
+898
61.5
3910
398
63.2
1667
798
64.5
289
380
2*19
1387
9.20
318
1387
9.37
676
1388
7.49
3246
32*
19+8
1680
1.14
069
1680
2.90
506
1680
8.27
5557
03*
1920
818.
8047
720
819.
0651
420
831.
2398
651
3*19
+823
740.
7422
923
742.
5934
423
752.
0221
761
4*19
2775
8.40
636
2775
8.75
352
2777
4.98
6484
04*
19+8
3068
0.34
388
3068
2.28
182
3069
5.76
8795
05*
1934
698.
0079
634
698.
4419
3471
8.73
310
30
5*19
+837
619.
9454
737
621.
9702
3763
9.51
5311
40
6*19
4163
7.60
955
4163
8.13
028
4163
4.92
505
122
16*
19+8
4455
9.54
706
4456
1.65
858
4455
5.70
735
133
07*
1948
577.
2111
448
577.
8186
648
578.
6716
514
16
7*19
+851
499.
1486
551
501.
3469
651
499.
4539
5
B
road
ly s
peak
ing,
we
mig
ht s
umm
ariz
e th
e ab
ove
tabl
e as
thu
s: -
for
the
mos
t
part,
the
num
ber
of s
olar
day
s in
sol
ar y
ears
, syn
odic
and
ano
mal
istic
mon
ths
over
lap
in
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 2
4
ksha
ya y
ears
. H
owev
er,
this
over
lap
does
n’t
occu
r on
ly i
n ks
haya
yea
rs;
as t
he t
able
show
s, th
ere’
s an
ove
rlap
for
133
year
s as
wel
l. D
oes
this,
the
n, e
xpla
in t
he k
shay
a
phen
omen
on?
We
mig
ht s
umm
ariz
e it
as b
eing
stro
ngly
sug
gest
ive,
but
def
inite
ly n
ot
conc
lusi
ve.
Tre
atm
ent o
f Ksh
aya
Mon
ths40
W
e m
ay c
ompl
ete
our
disc
ussio
n of
ksh
aya
mon
ths
by d
escr
ibin
g th
e th
ree
Ksh
aya
Scho
ols o
f tho
ught
.
The
Nor
th W
este
rn S
choo
l is
fol
low
ed in
the
nor
th-w
este
rn p
art
of t
he c
ount
ry,
pres
umab
ly
in
Guj
arat
an
d/
or
Raj
asth
an,
whe
re
the
luni
sola
r ca
lend
ar
is us
ed.
Esse
ntia
lly, t
he N
orth
Wes
tern
Sch
ool t
reat
s the
adh
ika
mon
th b
efor
e ks
haya
as
a no
rmal
mon
th a
nd t
he o
ne a
fter
the
ksha
ya m
onth
to
be i
nter
cala
ry.
This
con
trast
s w
ith t
he
East
ern
Scho
ol w
here
the
rev
erse
is
follo
wed
; th
e ad
hika
mon
th b
efor
e th
e ks
haya
is
deem
ed i
nter
cala
ry,
whi
le t
he o
ne a
fter
it is
deem
ed n
orm
al.
The
East
ern
Scho
ol i
s
follo
wed
in th
e ea
ster
n pa
rts o
f the
cou
ntry
, whe
re th
e lu
niso
lar c
alen
dar i
s fo
llow
ed. T
he
final
of
the
trio,
the
Sou
ther
n Sc
hool
, tre
ats
both
adh
ika
maa
sas
as i
nter
cala
ry,
inst
ead
reck
onin
g th
e ks
haya
mon
th a
s a
“jug
ma”
, i.e
., th
e fir
st h
alf o
f the
thith
i of t
his
mon
th is
deem
ed t
o be
tha
t of
the
firs
t m
onth
, and
the
sec
ond
half
as t
hat
of t
he s
econ
d m
onth
.
This
is p
resu
mab
ly f
ollo
wed
in
the
Sout
hern
par
ts o
f th
e co
untr
y w
here
the
lun
isola
r
cale
ndar
is fo
llow
ed.
40 C
hatte
rjee,
SK
. p 3
7- 4
0
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 2
5
Epi
logu
e –
Th
e B
egin
ning
By th
is ti
me,
onl
ooke
rs a
ll si
des
gath
ered
aro
und
the
two.
The
y we
re a
ttent
ivel
y
liste
ning
to th
e co
nver
satio
n be
twee
n th
em. A
long
with
Apa
ra G
anita
, the
y we
re w
aitin
g
for
the
dwar
a pa
lika
to a
sk o
nce
agai
n. B
ut s
he d
idn’
t. Sh
e st
ood
and
smile
d. H
er fa
ce
was
radi
ant,
glow
ing
like
the
moo
n on
a P
urni
ma
and
the
hars
h su
mm
er su
n en
teri
ng th
e
Mith
una
raas
i.
Sh
e st
ill s
aid
noth
ing.
She
got
up
and
walk
ed a
way
from
the
crow
d. S
till s
mili
ng.
Still
gra
cefu
l.
Th
e sp
arks
cam
e sl
owly
, bu
t su
dden
ly.
All
arou
nd t
hem
, th
e la
ndsc
ape
was
chan
ging
. The
gat
e wa
s mel
ting
into
the
walls
, the
wal
ls in
to th
e gr
ound
. The
gro
und
was
chan
ging
into
gra
ss, t
he g
rass
cov
erin
g th
e en
tire
grou
nd.
Exce
pt t
he g
roun
d un
dern
eath
Apa
ra G
anita
. H
e fo
und
him
self
stan
ding
on
an
elev
ated
pod
ium
, fac
ing
liste
ners
all
arou
nd h
im, a
ll wa
iting
to h
ear h
im sp
eak.
For o
nce,
he
didn
’t kn
ow w
hat t
o sa
y.
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 2
6
Bib
liogr
aphy
1.
S.K
. Cha
tterje
e. In
dian
Cal
endr
ic S
yste
m, P
ublic
atio
ns D
ivis
ion,
Min
istry
of
Info
rmat
ion
and
Bro
adca
stin
g, G
over
nmen
t of I
ndia
, 198
8.
2.
C
hakr
avar
ty,
Apu
rba
Kum
ar a
nd S
K C
hatte
rjee.
“In
dian
Cal
enda
r fro
m P
ost-
Ved
ic P
erio
d to
AD
190
0” in
Hist
ory
of A
stro
nom
y in
Ind
ia, e
d. S
.N. S
en a
nd
K.S
. Shu
kla,
. Ind
ian
Nat
iona
l Sci
ence
Aca
dem
y, N
ew D
elhi
, 198
5.
3.
M
.H. S
aha
and
N.C
. Lah
iri, R
epor
t of t
he C
alen
dar R
efor
m C
omm
ittee
, Cou
ncil
of S
cien
tific
and
Indu
stria
l Res
earc
h, N
ew D
elhi
, 199
2.
4.
Jaw
ad, A
la’a
H. “
How
Lon
g is
a Lu
nar M
onth
?” in
Sky
and
Tel
esco
pe, N
ovem
ber
1993
. 5.
A
bhay
anka
r, K
D. “
Our
Deb
ts t
o ou
r A
nces
tors
” in
Tre
asur
es o
f A
ncie
nt I
ndia
n A
stro
nom
y. e
d. K
D A
bhay
anka
r and
Dr.
BG
Sid
harth
. Aja
nta
Publ
icat
ions
, Del
hi.
1993
. 6.
Des
how
itz,
Nac
hum
and
Edw
ard
M.
Rei
ngol
d. C
alen
dric
al C
ompu
tatio
ns:
The
Mill
enni
um E
ditio
n. C
ambr
idge
Uni
veris
ty P
ress
, Cam
brid
ge. 2
001.
7.
Ven
kata
Ram
ana
Saas
tri,
Chi
vuku
la.
Kal
yana
Gan
itham
. Sr
inge
ri V
irupa
aksh
a Pe
etha
m, S
ringe
ri. 1
942.
(Thi
s is a
Tel
ugu
lang
uage
reso
urce
)
Panc
hang
a- T
antra
: The
Mag
ic of
the I
ndian
Cale
ndar
Sys
tem / 2
7
Ack
now
ledg
emen
ts
Th
is re
port
is 51
50 w
ords
long
, mak
ing
it th
e bi
gges
t rep
ort I
’ve
ever
writ
ten.
It’s
been
in
the
mak
ing
for
the
last
one
-yea
r in
tw
o co
untri
es.
Obv
ious
ly,
ther
e ar
e m
any
peop
le w
ho’v
e he
lped
me,
and
I’d
like
to th
ank
ever
yone
of t
hem
.
Fi
rst,
I’d
like
to th
ank
Dr.
Hel
mer
Asl
akse
n, m
y re
sear
ch s
uper
viso
r; sir
, it’s
bee
n
a pl
easu
re w
orki
ng w
ith y
ou. I
t was
gre
at s
yner
gy a
ll th
e w
ay. A
khil
Deo
gar a
nd A
ksha
y
Pras
ad, t
he o
ther
two
stud
ents
who
wor
ked
on th
is, a
lso n
eed
a gr
atef
ul th
ank-
you
here
.
Way
to g
o gu
ys, w
e m
ade
it. T
o D
r. D
esho
witz
, for
hel
ping
me
out w
ith th
e ca
lend
rica
code
, jus
t w
hen
I w
as s
tuck
. To
Dr.
Subr
aman
yam
and
all
the
won
derf
ul p
eopl
e at
the
Dep
artm
ent
of A
stro
logy
, Po
tti S
riram
ulu
Telu
gu U
nive
rsity
, H
yder
abad
, it’
s be
en a
plea
sure
mee
ting
you
all a
nd I
’m g
rate
ful y
ou s
at th
roug
h m
y pr
esen
tatio
n on
that
col
d
Dec
embe
r ev
enin
g. T
o D
r. B
G S
iddh
arth
, Dire
ctor
Gen
eral
, BM
Birl
a Sc
ienc
e C
ente
r,
your
com
men
ts w
ere
inva
luab
le.
To D
r. V
alla
bh,
Prof
esso
r an
d H
ead,
Ast
rono
my
Dep
artm
ent,
Osm
ania
Uni
vers
ity, s
ir, I
tha
nk y
ou f
or s
pend
ing
som
e tim
e w
ith m
e. T
o
Dr.
Mad
ugul
a Si
vasr
i Sar
ma,
it’s
bee
n a
plea
sure
mee
ting
you
and
I’m
gra
tefu
l for
you
r
com
men
ts o
n th
e ru
les
for
fest
ival
s. To
Dr.
CV
L N
aras
imha
m,
for
thos
e w
onde
rful
even
ings
on
the
bank
s of
the
Riv
er M
usi d
iscus
sing
pan
chan
gam
trad
ition
s. A
nd la
st b
ut
not l
east
, I’d
like
to th
ank
my
pare
nts a
nd m
y lit
tle b
roth
er fo
r kee
ping
up
with
me
durin
g
all t
hose
late
-nig
ht se
ssio
ns I
spen
t pou
ring
over
cal
enda
rs.
I o
nce
agai
n th
ank
ever
yone
who
’s h
elpe
d m
e. O
f cou
rse,
it b
ears
no
need
to s
ay
that
all
erro
rs a
re m
ine.
