Panchanga- Tantra - Department of Mathematics - National
30
Panchanga- Tantra: The Magic of the Indian Calendar System / 1 Foreword to the Second Edition The fable of Apara Ganita and the Mystical Garden of Enchanted Numbers is obviously fictional. The inspiration is Leelavati Ganitam, a chapter in the ancient mathematical treatise, the Siddhanta Siromani, written by Bhaskaracharya in 1150CE. The Leelavati Ganitam is fascinating not only for its treatment of indeterminate analysis and a method to solve Pell’s Equation, but also, as a Canadian university’s website on mathematical history puts it, for its poetic conversation between the narrator and a narratee named Leelavati 1 . The similarity between this poetic construct and the conversation between Apara Ganita and the dwara palika is probably noticeable. Frame stories are not common for scientific research papers, but they certainly have a historical precedent. 1 “Bhaskaracharya”, History of Mathematics, Simon Fraser University, <http://www.math.sfu.ca/histmath/India/12thCenturyAD/Bhaskara.html > (21st September, 2002.)
Transcript of Panchanga- Tantra - Department of Mathematics - National
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.
![Astro-Vision LifeSign Horoscope Panchanga Predictions Name : … · 2020. 4. 20. · Astro-Vision LifeSign Horoscope Panchanga Predictions Name : Sample [Male] Om Sri During Uttarayana](https://static.fdocuments.net/doc/165x107/603a690f6a18fa2e8f265b54/astro-vision-lifesign-horoscope-panchanga-predictions-name-2020-4-20-astro-vision.jpg)
