NPTEL C OURSE ON

M ATHEMATICS IN I NDIA :
F ROM V EDIC PERIOD TO M ODERN TIMES
Lecture 5
Pingalas

Chandah
s
astra
.

M. D. Srinivas
Centre for Policy Studies, Chennai

Outline
I

Development of Prosody or Chandah.s

astra

Long (guru) and short (laghu) syllables

Scanning of varn.avr.tta (syllabic metres) and the eight

gan.as

Pratyayas in Pingalas

Chandah.s
astra
I
I
I

Prast
ara or enumeration in the form of an array
Sankhy

a: The total number of metrical forms of n syllables

Nas..ta and Uddis..ta: The association between a metrical
form and the row-number in the prast
ara through binary
expansion
Lagakriy
a: Number of metrical forms in the prast
ara with a
given number of laghus
Varn.ameru and the Pascal Triangle

Development of Chandah
s
astra
.
In his Chandah.s
astra (c.300 BCE), Pingala

introduces some
combinatorial tools called pratyayas which can be employed to
study the various possible metres in Sanskrit prosody.
Following are some of the important texts which include a
discussion of various pratyayas:
I

Pingala

(c.300 BCE): Chandah.s

astra

Bharata (c.100 BCE): N

a.tyas
astra

Brahmagupta (c.628 CE): Br

ahmasphut.asiddh
anta

Virahanka
(c.650): Vr.ttaj
atisamuccaya
Mahavra (c.850): Gan.itas
arasangraha

I
I

Halayudha (c.950): Mr.tasa

njvan Commentary on
Pingalas

Chandah.s
ast.ra

Development of Chandah
s
astra
.

Kedarabhat.t.a (c.1000): Vr.ttaratn

akara
Yadavaprakasa (c.1000): Commentary on Pingalas

Chandah.s
ast.ra
Hemacandra (c.1200): Chandonus
asana

Pr
akr.ta-Paingala

(c.1300)

Narayan.a Pan.d.ita (c.1350): Gan.itakaumud

Damodara (c.1500): V
an.bh
u.san.a

I
I

I
I

Narayan.abhat.t.a (c.1550): N
ar
ayan. Commentary on
Vr.ttaratn
akara

Varn
. a-Vr
. tta
I

A syllable (aks.ara) is a vowel or a vowel with one or more

consonants preceding it.

A syllable is laghu (light) if it has a short vowel.

Even a short syllable will be a guru if what follows is a conjunct

consonant, an anusv
ara or a visarga.

Otherwise the syllable is guru (heavy).

The last syllable of a foot of a metre is taken to be guru

optionally.

The first verse of K

alid
asas Abhij
n
anas
akuntalam:
ya.a .sxa:a.H

.~:a:
u .=:a:d;a.a

va:h: a.ta ;a.va: a.Da:hu:tMa ya.a h:a.va:ya.Ra ..ca h.ea.a.a

yea :d
e k+a:lM ;a.va:Da.aH
ya.a:ma.a:huH

(rua: a.ta:a.va:Sa:ya:gua:Na.a ya.a ;a.~Ta:ta.a

////
v.ya.a:pya ;a.va.(;a:m,a

x +. a.ta:a=; a.ta ya:ya.a :pra.a:a.Na:naH :pra.a:Na:va:ntaH

.sa:vRa:ba.a.ja:pra:k

:pra:tya:[a.a:a.BaH :pra:sa:a:~ta:nua:a.Ba.=;va:tua va:~ta.a:a.Ba.=;.a:a.Ba.=:a:ZaH

GGG GLG GLL LLL LGG LGG LGG

The Eight Gan

. as
A.a:a.d:ma:Dya.a:va:sa.a:nea:Sua ya.=;ta.a ya.a:a.nta
// l;a:Ga:va:m,a
Ba.ja:sa.a ga.Ea.=;vMa ya.a:a.nta
// ma:na.Ea tua gua.+.l;a:Ga:va:m,a

Ya: LGG Ra: GLG Ta: GGL

Bha: GLL Ja: LGL Sa: LLG
Ma: GGG Na: LLL
The pattern of a metre is usually characterised in term of these gan.as.
For instance the verse of K
alid
asa cited earlier is in Sragdhar
a metre:
.aE:ya.Ra:na.Ma a:yea:Na ;aa:mua:
a.na:ya: a.ta:yua:ta.a .~:a:gDa.=:a

ma:B
k
 +a: a.tRa:tea:ya:m,a

Thus Sragdhar
a is characterised by the pattern: MaRaBhaNaYaYaYa,
with a break (yati) after seven syllables each.
GGGGLGG LLLLLLG GLGGLGG

A Mnemonic for the Gan

. as
There is the mnemonic attributed to P
an.ini

ya:ma.a:ta.a.=:a.ja:Ba.a:na:sa:l+ga:m,a
LGGGLGLLLG
If we replace G by 0 and L by 1, we obtain a binary sequence of
length 10
1000101110
The above linear binary sequence generates all the 8 binary
sequences of length 3. We can remove the last pair 1, 0 and
view the rest as a cyclic binary sequence of length eight.
In modern mathematics such sequences are referred to as
De Bruijn cycles.

A Mnemonic for the Gan

. as

Figure: De Bruijn cycles for Patterns of three and four letters.

stand for L, G or 1,0]

[ and Y

Pratyayas in Pingalas

Chandah
s
astra
.
In chapter eight of Chandah.s
astra, Pingala

introduces the
following six pratyayas:
Prast
ara: A procedure by which all the possible metrical
patterns with a given number of syllables are laid out
sequentially as an array.
Sankhy

a: The process of finding total number of metrical

patterns (or rows) in the prast
ara.
Nas..ta: The process of finding for any row, with a given number,
the corresponding metrical pattern in the prast
ara.
Uddis..ta: The process for finding, for any given metrical pattern,
the corresponding row number in the prast
ara.
Lagakriy
a: The process of finding the number of metrical forms
with a given number of laghus (or gurus).
Adhvayoga: The process of finding the space occupied by the
prast
ara.

Prast
ara
;a.d
/

:k+Ea gl;Ea ;a.ma:(ra.Ea ..ca :pxa:Ta:gl;a:a.ma:(ra.aH va:sa:va:a.~:a:k+aH

C+ndH Za.a:~:a:m,a

8 20 23

Form a G, L pair. Write them one below the other.

Insert on the right Gs and Ls.

[Repeating the process] we have eight (vasavah.) metric forms in

the 3-syllable-prast
ara.

Single syllable prast

ara
1
2

G
L

Two syllable prast

ara
1
2
3
4

G
L
G
L

G
G
L
L

Prast
ara

Three syllable prast

ara
1
2
3
4
5
6
7
8

G
L
G
L
G
L
G
L

G
G
L
L
G
G
L
L

G
G
G
G
L
L
L
L

Another Rule for Prast

ara
:pa.a:de .sa:vRa:gua.=:a:va.a:d;a.a:+.GM

ua nya:~ya gua.=:ea.=;DaH
ya:Ta.ea:pa:a= ta:Ta.a Zea:SMa BUa:yaH

k
u +.ya.Ra:d:mMua ;a.va: a.Da:m,a

+.nea d:d;a.a:
:+nea:va ya.a:va:tsa:vRa:l+Gua:BRa:vea:t,a
u

vxa.a.=;a.a:k+=;m,a 6 2 3

Start with a row of Gs. Scan from the left to identify

the first G. Place an L below that. The elements to the
right are brought down as they are. All the places to
the left are filled up by Gs. Go on till a row of only Ls is
reached.
Example: The following are five successive rows in 4-syllable
prast
ara
G
L
G
L
G

G
G
L
L
G

G
G
G
G
L

L
L
L
L
L

Four-Syllable Prast
ara
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

G
L
G
L
G
L
G
L
G
L
G
L
G
L
G
L

G
G
L
L
G
G
L
L
G
G
L
L
G
G
L
L

G
G
G
G
L
L
L
L
G
G
G
G
L
L
L
L

G
G
G
G
G
G
G
G
L
L
L
L
L
L
L
L

If we set G=0 and L=1, then we see that each metric pattern is the mirror
reflection of the binary representation of the associated row-number-1.

Sankhy

a
;a.d.=;DeRa .+pea ZUa:nya:m,a ;a.dHZUa:nyea ta.a:va:d:DeRa ta:u;a.Na:ta:m,a

(C+ndHZa.a:~:a:m,a 8.28-31)
The number of metres of n-syllables is Sn = 2n .
Pingala

gives an optimal algorithm for finding 2n by means of

multiplication and squaring operations that are much less than
n in number.
I

Halve the number and mark 2

If the number cannot be halved deduct one and mark 0

[Proceed till you reach zero. Start with 1 and scan the
seqence of marks from the end]

If 0, multiply by 2

If 2, square

Sankhy

a
Example: Six-syllable metres (n = 6)
I 6
2
I 3
I 2
2
I 1

= 3 and mark 2
cannot be halved. 3-1=2 and mark 0
= 1 and mark 2
1 = 0 and mark 0

Sequence 2, 0, 2, 0 yields
1x2, (1x2)2 , (1x2)2 x2, ((1x2)2 x2)2 = 26
Pingalas

algorithm became the standard method for computing

powers in Indian mathematcis.

Sankhy

a
Next s
utra of Pingala

gives the sum of all the sankhy

as Sr for
r = 1, 2, . . . n.
;a.d
:d;aUR :nMa ta:d:nta.a:na.a:m,a

(C+ndHZa.a:~:a:m,a

8 32

S1 + S2 + S3 + . . . + Sn = 2Sn 1
Then comes the s
utra:
:pa:=e :pUa:NRa:m,a

(C+ndHZa.a:~:a:m,a

8 33

Sn+1 = 2Sn
Together, the two s
utras imply
Sn = 2n
and
1 + 2 + 22 + . . . + 2n = 2n+1 1
This clearly is the formula for the sum of a geometric series.

Sankhy

The sankhy

a 2n discussed above is for the case of syllabic

metres of n-syllables which are sama-vr.ttas metres which
have the same pattern in all the four p
adas or quarters.

Ardhasama-vr.ttas are those metres which are not sama,

but whose halves are the same.

Vis.ama-vr.ttas are those which are neither sama nor

vis.ama.

In the fifth Chapter of Chandah.-s

astra, Pingala

has dealt
with the sankhy

a ofArdhasama and Vis.ama-vr.ttas.

Sankhy

a
.sa:mMa ta.a:va:tkx+.tvaH kx+.ta:ma:DRa:sa:ma:m,a ;a.va:Sa:mMa . ca .=:a:ZyUa:na:m,a
(C+ndHZa.a:~:a:m,a 5.3-5)
The number of Ardhasama-vr.ttas with n-syllables in each
p
ada is
(2n )2 2n
In the same way, the number of Vis.ama-vr.ttas with n-syllables
in each p
ada is
h

i
(22n )2 (2n )2 2n + 2n = (22n )2 22n

Nas..ta

l+DeRa .sEa:ke g,a (C+ndHZa.a:~:a:m,a 8.24-25)

I

To find the metric pattern in a row of the prast

ara, start with
the row number

Halve it (if possible) and write an L

If it cannot be halved, add one and halve and write a G

Proceed till all the syllables of the metre are found

Nas..ta
Example: Find the 7th metrical form in a 4-syllable prast
ara
(7+1)
= 4 Hence G
2
I 4 = 2 Hence GL
2
I 2 = 1 Hence GLL
2
(1+1)
I
= 1 Hence GLLG
2
I

If we set G = 0 and L = 1, we can see that Pingalas

nas..ta
process leads to the desired metric form via the binary
expansion
7 = 0 + 1.2 + 1.22 + 0.23

Uddis..ta
:pra: a.ta:l;ea:ma:ga:NMa ;a.d:l;Ra:d;a:m,a ta:ta.ea:gyea:kM .ja:h:a.a:t,a
(

C+ndH Za.a:~:a:m,a

8 26 27

To find the row number of a given metric pattern:

I
I

I
I

Start with number 1

Scan the pattern from the right beginning with the first L
from the right
Double it when an L is encountered
Double and reduce by 1 when a G is encountered

Example: To find the row-number of the pattern GLLG in a

4-syllable prast
ara:
I
I
I
I

Start with 1.
Skip the G and go to L. So we get 1x2 = 2
Then we find L. So we get 2x2 = 4
Finally we have G. We get 4x2 1 = 7

Uddis..ta
Another Method

o+a.;M ;a.d:gua:Na.a:na.a:d;a.a:du:pa:yRa:*:;a:n,
a .sa:ma.a:
a.l+Kea:t,a

l+Gua:~Ta.a yea tua ta.a.a:*:;a:~tE

aH .sEa:kE+.a.mRa:a.(ra:tEa:BRa:vea:t,a
(

vxa.a.=;a.a:k+=;m,a

6 5

Place 1 on top of the left-most syllable of the given metrical

pattern

Double it at each step while moving right.

Sum the numbers above L and add 1 to get the

row-number

Uddis..ta

Example: To find the row-number of the pattern GLLG

1
G

2
L

22
L

23
G

Row-Number = 0.1 + 1.2 + 1.22 + 0.23 + 1 = 7

Both the nas..ta and uddis..ta processes of Pingala

are essentially
based on the fact that every natural number has a unique
binary representation: It can be uniquely represented as a sum
of the different sankhy

a Sn or the powers 2n .

Lagakriy
a
:pa:=e :pUa:NRa:a.ma: a.ta

(C+ndHZa.a:~:a:m,a

8 34

Pingalas

s
utra on lagakriy
a process is too brief. Hal
ayudha, the tenth
century commentator explains it as giving the basic rule for the
construction of a table of numbers which he refers to as the
Meru-prast
ara.

o+pa:a=;.a:de:k
M ..ca:tua.=;~:a:k+e
a:M ;
a.l+ a.Ka:tva.a ta:~ya.a:Da:~ta.a:d
u :Ba:ya:ta.eaY:DRa:
a.na:Sk
+a:ntMa k+ea:;d
:yMa

;
a.l+Kea:t,a ta:~ya.a:pya:Da:~ta.a. +;a:yMa ta:~ya.a:pya:Da:~ta.a:a:tua::yMa ya.a:va:d:a.Ba:ma:tMa .~Ta.a:na:a.ma: a.ta

+;a.M
mea.+.pra:~ta.a.=H ta:~ya :pra:Ta:mea k+ea:e O;:k+.sa:*

a v.ya:va:~Ta.a:pya l+[a:Na:a.ma:dM :pra:va:tRa

yea:t,a ta.a :pa:=e k+ea:e ya:d

x .a:sMa:K.ya.a.ja.a:tMa ta:t,a :pUa:vRa:k+ea:;ya.eaH
ta.a.ea:Ba:ya.eaH

:pUa:Na ;
a.na:vea:Za:yea:t,a

k+ea:;k+.ya.ea;=e;k
E +.k+.ma:*:

a ma:Dyea k+ea:e tua :pa.=;k+ea:;d

a:k
 +a
:ya.a:*:+me
M d:d;a.a:t,

x +.tya :pUa:Na ;
a.na:vea:Za:yea:a.d: a.ta :pUa:NRa:Za:b.d.a:TRaH ..ca:tua:Tya.a :pa:*:;a:va:

k
a.pa :pa:yRa:nta:k+ea:;ya.ea

x +.tya :pUa:Na
:=e;k
E +.k+.mea:va .~Ta.a:pa:yea:t,a ma:Dya:ma:k+ea:;ya.ea:~tua :pa.=;k+ea:;d
a:k
 +a:k
:ya.a:*:+me

+;a.a.+pM
;aa:sa:*
a .~Ta.a:pa:yea:t,a

...

Varn

. a-Meru of Pingala

Clearly the number of metrical forms with r gurus (or laghus) in

the prast
ara of metres of n-syllables is the binomial coefficient
nC
r
The above passage of Hal
ayudha shows that the basic rule for
the construction of the above table, is the recurrence relation
n

Cr =n1 Cr 1 +n1 Cr

Pascal Triangle

The above Varn.a-Meru is actually a rotated version of the so

called Pascal Triangle (c.1655) shown below:
1
1
1
1
1
1
1

1
2
3
4
5
6

1
3
6
10
15

1
4
10
20

1
5
15

1
6

References

stra of Pingala

with Comm. Mr.tasa

njvan of
1. Chandah.sa
rd
Halayudha Bhat.t.a, Ed. Kedaranatha, 3 ed. Bombay 1938.
2. Vr.ttaratn
akara of Ked
ara with Comms. Nar
ayan. and Setu,
Ed. Madhusudana Sastri, Chaukhambha, Varanasi 1994.
3. B. van Nooten, Binary Numbers in Indian Antiquity, Jour.
Ind. Phil. 21, 1993, pp.31-50.
4. R. Sridharan, Sanskrit Prosody, Pingala

S
utras and Binary
Arithmetic, in G. G. Emch et al Eds., Contributions to the
History of Indian Mathematics, Hindustan Book Agency,
Delhi 2005, pp. 33-62.

Thanks!

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