Pingala

For the subtle energy channel described in yoga, see Nadi (yoga).

Acharya Pingala^[2] (Sanskrit: पिङ्गल, romanized: Piṅgala; c. 3rd–2nd century BCE)^[1] was an ancient Indian poet and mathematician,^[3] and the author of the Chandaḥśāstra (Sanskrit: छन्दःशास्त्र, lit. 'A Treatise on Prosody'), also called the Pingala-sutras (Sanskrit: पिङ्गलसूत्राः, romanized: Piṅgalasūtrāḥ, lit. 'Pingala's Threads of Knowledge'), the earliest known treatise on Sanskrit prosody.^[4]

Pingala
Born	unclear, 3rd or 2nd century BCE[1]
Academic work
Era	Maurya or post-Maurya
Main interests	Sanskrit prosody, Indian mathematics, Sanskrit grammar
Notable works	Author of the "Chandaḥśāstra" (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody. Creator of Pingala's formula.
Notable ideas	mātrāmeru, binary numeral system.

The Chandaḥśāstra is a work of eight chapters in the late Sūtra style, not fully comprehensible without a commentary. It has been dated to the last few centuries BCE.^[5][6] In the 10th century CE, Halayudha wrote a commentary elaborating on the Chandaḥśāstra. According to some historians Maharshi Pingala was the brother of Pāṇini, the famous Sanskrit grammarian, considered the first descriptive linguist.^[7] Another think tank identifies him as Patanjali, the 2nd century CE scholar who authored Mahabhashya.

Combinatorics

The Chandaḥśāstra presents a formula to generate systematic enumerations of metres, of all possible combinations of light (laghu) and heavy (guru) syllables, for a word of n syllables, using a recursive formula, that results in a partially ordered binary representation.^[8] Pingala is credited with being the first to express the combinatorics of Sanskrit metre, eg.^[9]

• Create a syllable list x comprising one light (L) and heavy (G) syllable:
• Repeat till list x contains only words of the desired length n
  • Replicate list x as lists a and b
    • Append syllable L to each element of list a
    • Append syllable G to each element of list b
  • Append lists b to list a and rename as list x

Possible combinations of Guru and Laghu syllables in a word of length n^[10]

Word length (n characters)	Possible combinations
1	G L
2	GG LG GL LL
3	GGG LGG GLG LLG GGL LGL GLL LLL

Because of this, Pingala is sometimes also credited with the first use of zero, as he used the Sanskrit word śūnya to explicitly refer to the number.^[11] Pingala's binary representation increases towards the right, and not to the left as modern binary numbers usually do.^[12] In Pingala's system, the numbers start from number one, and not zero. Four short syllables "0000" is the first pattern and corresponds to the value one. The numerical value is obtained by adding one to the sum of place values.^[13] Pingala's work also includes material related to the Fibonacci numbers, called mātrāmeru.^[14]

Editions

• A. Weber, Indische Studien 8, Leipzig, 1863.
• Janakinath Kabyatittha & brothers, ChhandaSutra-Pingala, Calcutta, 1931.^[15]
• Nirnayasagar Press, Chand Shastra, Bombay, 1938^[16]