Origins of Combinatorics in Chandas-śāstra
The Chandas-śāstra has some very interesting and intricate connection with mathematics. The word chandas means of prosody, the science of metres. It has been estimated by scholars that this Chandas-śāstra was composed by Piṅgala-nāga around 3rd century BCE, though there could be some uncertainty in his period. In his Chandas-śāstra, Piṅgala introduces some combinatorial tools called pratyayas which can be employed to study the various possible metres in Sanskrit prosody. The algorithms presented by him form the earliest examples of use of recursion in Indian mathematics.As Indic scholars have pointed out, Piṅgalācārya deserves to be credited with the following breakthroughs:The origin of binary arithmeticThe combinatorial techniquesThe Meru-prastāra (the so called Pascal’s triangle)The binomial coefficients and even the so called Fibonacci sequence
It is important to note that, all this have been arrived at by Piṅgala in the context of systematically studying the mathematics of poetry. The text Chandas-śāstra consists of 308 sūtras spread across eight chapters [I—15, II—16, III—66, IV—53, V — 44, VI—43, VII—36, VIII—35 ]. But for the first and the last chapters, the rest of the text presents an exhaustive account of different meters in Sanskrit. The enlisted meters essentially capture the metrical patterns that shall be followed in a given chandas.
Notion of a syllable in Sanskrit Prosody
A syllable (akṣara) is a vowel or a vowel with one or more consonants preceding it.
A syllable is laghu (light) if it has a short vowel. The short vowels in Sanskrit are: अ , इ , उ , ऋ , लृ ।
Otherwise the syllable is guru (heavy).
Even a short syllable will be a guru if what follows is a conjunct consonant, an anusvāra or a visarga.
Generally, the last syllable of a quarter irrespective of whether it is a short or a long vowel is considered to be guru.
The following example captures the metrical pattern called Sragdharā employed in the invocatory verse of Śataślokī composed by Śri Ādisankara Bhagavadpāda दृष्टान्तो नैव दृष्टः त्रिभुवनजठरे सद् गुरोर्ज्ञानदातुः
स्पर्शश्चेत् तत्र कल्प्यः स नयति यदहो स्वर् णतामश् मसारम् ।
GGG GLG GLL LLL LGG LGG LGG
In the Sanskrit texts, the syllable guru is denoted by ऽ and laghu is denoted by । .The pattern of a metre is usually characterised in term of eight gaṇas:Letter denoting the gaṇaPattern of the denoted gaṇasBinary FormMirror ImageDecimal
ValueमGGG / ऽऽऽ0000000यLGG / ।ऽऽ 1000011रGLG / ऽ।ऽ0100102सLLG / ।।ऽ1100113तGGL / ऽऽ।0011004जLGL / ।ऽ।1011015भGLL / ऽ।।0111106नLLL / ।।।1111117
It is clear from the above table that the mirror image of the binary representation of the gaṇas given by Piṅgala, directly gives the decimal number in a sequence.
It is indeed enthralling to know in depth about the rich scientific heritage of the Indian civilization. In the forthcoming articles we shall delve more into greater understanding of the various algorithms or pratyayas enunciated by Piṅgalācārya.