# Pingala

Pingala
Bornunclear, 3rd or 2nd century BCE[1]
EraMaurya or post-Maurya
Main interestsSanskrit prosody, Indian mathematics, Sanskrit grammar
Notable worksAuthor of the "Chandaḥśāstra" (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody. Creator of Pingala's formula.
Notable ideasmātrāmeru, binary numeral system.

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

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]

## Notes

1. ^ a b Plofker, Kim (2009). Mathematics in India. Princeton University Press. pp. 55–56. ISBN 978-0-691-12067-6.
2. ^ Singh, Parmanand (1985). "The So-called Fibonacci Numbers in Ancient and Medieval India" (PDF). Historia Mathematica. 12 (3). Academic Press: 232. doi:10.1016/0315-0860(85)90021-7. Archived from the original (PDF) on 2019-07-24. Retrieved 2018-11-29.
3. ^ "Pingala – Timeline of Mathematics". Mathigon. Retrieved 2021-08-21.
4. ^ Vaman Shivaram Apte (1970). Sanskrit Prosody and Important Literary and Geographical Names in the Ancient History of India. Motilal Banarsidass. pp. 648–649. ISBN 978-81-208-0045-8.
5. ^ R. Hall, Mathematics of Poetry, has "c. 200 BC"
6. ^ Mylius (1983:68) considers the Chandas-shāstra as "very late" within the Vedānga corpus.
7. ^
8. ^ Van Nooten (1993)
9. ^ Hall, Rachel Wells (February 2008). "Math for Poets and Drummers". Math Horizons. 15 (3). Taylor & Francis: 10–12. doi:10.1080/10724117.2008.11974752. JSTOR 25678735. S2CID 3637061. Retrieved 27 May 2022 – via JSTOR.
10. ^ Shah, Jayant. "A HISTORY OF PIṄGALA'S COMBINATORICS" (PDF).
11. ^ Plofker (2009), pages 54–56: "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [...] Pingala's use of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero. ... In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value “n”. [...] The answer is (2)7 = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where “n” is large. Pingala’s use of a zero symbol as a marker seems to be the first known explicit reference to zero."
12. ^ Stakhov, Alexey; Olsen, Scott Anthony (2009). The mathematics of harmony: from Euclid to contemporary mathematics and computer science. ISBN 978-981-277-582-5.
13. ^ B. van Nooten, "Binary Numbers in Indian Antiquity", Journal of Indian Studies, Volume 21, 1993, pp. 31–50
14. ^ Susantha Goonatilake (1998). Toward a Global Science. Indiana University Press. p. 126. ISBN 978-0-253-33388-9. Virahanka Fibonacci.
15. ^
16. ^ Pingalacharya (1938). Chand Shastra.