Vedic Mathrix 2024
Pingala’s Chandaùçästra
Vinay Nair
Outline
• Development of Prosody or Chandaùçästra
• Short (laghu) & Long (guru) syllables
• Varëa Våtta
• Eight Gaëas
• Pratyayas in Pingaÿa’s Chandas
– Prastära
– Sankhyä
– Nañöa
– Uddiñöa
– Lagakriya
– Adhvayoga
• Varëa meru (so-called Pascal’s triangle)
Development of Chandas
• Development of Chandaùçästra
– Pingaÿa (300 BCE) Chandas
– Bharata’s (100 BCE) Naöyaçästra
– Mathematical works like Braùmasphuöasiddhänta of
Braùmagupta (628 CE)
– Virahanka’s Prakrit work Våttajätisamuccaya (650 CE)
– Mahävéra’s Gaëitasärasangraha (850 CE)
– Haläyudha (10th Cent CE) wrote a commentary on
Chandas called ‘Måtasanjévani’
– Kedärabhaööa’s Våttaratnäkara (14th Cent CE) – the
most commonly studied book by students of Sanskrit
– ‘Yädavaprakäça’ (1000 CE), commentary on
Chandas
– Hemachandra’s (1200 CE) Chandonuçäsana on
Prakrit metres
– Präkrita-Paingaÿa (1300 CE)
– Närayaëa Paëdita’s Gaëitakaumudi (1350 CE)
– Dämodara’s Vänibhüñaëa (1500 CE)
– Närayaëabhaööa’s ‘Näräyaëé’ (1550 CE) a
commentary on Våttaratnäkara
Varëa Våtta
• Basic building blocks of studying Sanskrit metres
• Syllable is a vowel by itself or a vowel with one or more
consonants preceding it
• A syllable is Laghu if it has a short vowel
• Even a short syllable will be a Guru if what follows is a
conjunct consonant, an anusvära or visarga
• Otherwise the syllable is Guru (heavy)
• The last syllable of a foot of a metre is taken to be Guru
(optionally)
Kälidäsa’s Abhijïänaçäkuntalam
The first verse of Kälidäsa’s Abhijïänaçäkuntalam
Eight Gaëas
Ya: LGG Ra: GLG Ta: GGL
Bha: GLL Ja: LGL Sa: LLG
Ma: GGG Na: LLL
• Gaëas are units of three syllables each with a structure of L & G.
• Thus Sragdhara is characterised by the pattern:
MaRaBhaNaYaYaYa with a break (yati) after seven (muni)
syllables each
• GGGGLGG LLLLLLG GLGGLGG
• There is the mnemonic attributed to Päëini
• 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.
Pratyayas (Chapter 8)
• Prastära: a rule by which you can write down all the possible metres of a
possible length.
• Sankhyä: The process of finding total number of metrical patterns (or rows)
in the prastära (Formula: 2^n where ‘n’ is the number of syllables in a
prastära).
• Nañöa: The process of finding any row, with a given number, the
corresponding metrical pattern in the prastära. If you have written down
the prastära on the ground and the wind has blown, prastära has gone
away and I want to know the 20th row of a 5 syllable prastära. Without
doing the prastära once again, what is the sequence.
• Uddiñöa: The process of finding, for any given metrical pattern, the
corresponding row number in the prastära (Ranking an unranking process
in modern terminology).
• Lagakriya: The process of finding the number of metrical forms with a
given number of L or G.
• Adhvayoga: The process of finding the space occupied by the prastära.
Prastära
• Single syllable prastära G
L
• Two syllable prastära
G G
L G
G L
L L
Prastära
• Three syllable prastära G G G
L G G
• Pingaÿa’s rule: G L G
– Form a G, L pair. L L G
– Write them one below the other. G G L
– Insert on the right Gs and Ls. L G L
– Repeating the process, G L L
we have eight (vasavaù) metric forms L L L
in the 3-syllable-prastära.
Another rule for Prastära
• 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 empty places to the left are filled up by Gs.
• Go on till a row of only Ls is reached.
• GGG
• LGG
• GLG
• LLG
• GGL
• LGL
• GLL
• LLL
Link
Sankhyä
• The number of metres in an n-syllable prastära is 2^n
• Pingaÿa’s optimum algorithm for finding Sn = 2^n
– Half the number ‘n’ and mark 2. If ‘n’ cannot be halved, deduct 1
from it and mark 0.
– Continue this process till you get the result zero.
– For every “0”, multiply by 2.
– For every “2”, square.
– Start with 1 and scan from the right.
Sankhyä
• E.g. 2^8
• Here, n=8
• Since 8 can be halved, write 2.
• 8/2 = 4 2
• 4/2 = 2 22
• 2/2 = 1 222
• 1-1 = 0 2220
Sankhyä
• Sum of all sankhyäs Sr for r = 1, 2, 3, …n
• 21 + 22 + 23 + 24 +…+ 2n-1 = 2n – 2
• S0 + S1 + S2 + S3 + … + Sn = 2Sn – 1
• 1 + 21 + 22 + 23 + 24 +…+ 2n-1 = (2n – 1)/(2-1)
• Thus, we get the formula for sum of geometric series.
• Sn+1 = 2Sn
Nañöa
• To find the metric pattern in a row of the prastära, 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.
Nañöa
• Example: Find the 9th metrical form in a 4-syllable
prastära
• (9+1) = 5. Hence G
2
(5+1) = 3. Hence G
2
(3+1) = 2. Hence G
2
2 = 1. Hence L
2
GGGL
Uddiñöa
• GGGL
• If we replace all G by 0 and all L by 1 and expand using
binary expansion,
• 9 = 0*20 + 0*21 + 0*22 + 1*23 + 1
• E.g.14th metrical row in a 4-syllable prastära
• LGLL
1011
1*20 + 0*21 + 1*22 + 1*23 + 1 = 14
Uddiñöa
• To find the row number of a given metric pattern, scan
the pattern from the right.
• Start with number 1.
• Double it when an L is encountered.
• Double and reduce by 1 when a G is encountered.
Uddiñöa
• Example: GLGL
• Start with 1. First is L. So, double it. 1*2 = 2
• Next is G. So double it and reduce 1. 2*2-1 = 3
• Next is L. So, double it. 3*2 = 6
• Last is G. So double it and reduce 1. 6*2-1 = 11.
• So, GLGL is the 11th metrical form.
Uddiñöa
• Another method from Våittaratnäkara:
• 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
Uddiñöa
• Example: GLGL
1,2,4,8
Numbers below L are 2 and 8. Add both the numbers
and add 1 to it. Thus, we get it as 11th row of the metrical
form.
Observation
• Both the nañöa and uddiñöa processes of Pingaÿa 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 sankhya Sn or the
powers 2n.
Lagakriyä
Varëa Meru of Pingaÿa
nC = n-1Cr-1 + n-1Cr
r
So-called Pascal’s Triangle
• Varëa Meru is just the rotated version of the Pascal’s
Triangle (1655 CE)
Comments
• Pingaÿa was the great originator of all the binary
mathematics. This was rediscovered in 1990s.
• Sankhya algorithm – one of the most efficient algorithm
for calculating the nth power of a number. This algorithm
of Pingaÿa became the standard algorithm for calculating
the nth power of a number.
References
• NPTEL course on Mathematics in India: from Vedic period to
modern times, Lec Notes by Prof. M.D.Srinivas.
• Chandaùçästra of Pingaÿa with Comm. Mrtasanjévani of
Haläyudha Bhaööa, Ed. Kedarnath, 3rd ed. Bombay 1938.
• Vrttaratnäkara of Kedara with Comms. Näräyaëé and Setu,
Ed. Madhusudana Sastri, Chaukhambha, Varanasi 1994.
• B. van Nooten, Binary Numbers in Indian Antiquity, Jour.
Ind. Phil. 21, 1993, pp.31-50.
• R. Sridharan, Sanskrit Prosody, Pingaÿa Sütras 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.
