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Combinatorics and Chandas-śāstra‑1

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

1. A syl­la­ble (akṣara) is a vow­el or a vow­el with one or more con­so­nants pre­ced­ing it.

2. A syl­la­ble is laghu (light) if it has a short vow­el. The short vow­els in San­skrit are: अ , इ , उ , ऋ , लृ ।

3. Oth­er­wise the syl­la­ble is guru (heavy).

4. Even a short syl­la­ble will be a guru if what fol­lows is a con­junct con­so­nant, an anusvāra or a vis­ar­ga.

5. 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.