# Pingala

Acharya Pingala[2] (piṅgala; c. 3rd–2nd century BCE)[1] was an ancient Indian poet and mathematician,[3] and the author of the Chandaḥśāstra (also called the Pingala-sutras), the earliest known treatise on Sanskrit prosody.[4]

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

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. Pingala Maharshi was also said to be the brother of Pāṇini,[citation needed] the famous Sanskrit grammarian, considered the first descriptive linguist.[7]

## Combinatorics

The Chandaḥśāstra presents the first known description of a binary numeral system in connection with the systematic enumeration of metres with fixed patterns of short and long syllables.[8] Pingala's discussion of the combinatorics of metre corresponds to the binomial theorem. Halāyudha's 10th-century commentary on the Chandaḥśāstra includes a presentation of this theorem in what is now called Pascal's triangle (called meruprastāra in the commentary), named after French mathematician Blaise Pascal despite its discovery by Halayudha and others centuries before. Pingala's work also includes material related to the Fibonacci numbers, called mātrāmeru.[9]

Pingala is credited with the first use of binary numbers, using light (laghu) and heavy (guru) syllables to describe combinatorics of Sanskrit metre.[10] 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 system of metre starts with four light laghu syllables as the first pattern ("0000" in binary), three light laghu and one heavy guru as the second pattern ("0001" in binary), and so on, so that in general the ${\displaystyle n}$ -th syllable pattern corresponds to the binary representation of ${\displaystyle n-1}$  (with increasing positional values).

## Editions

• A. Weber, Indische Studien 8, Leipzig, 1863.

## 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. Academic Press. 12 (3): 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. ^ Susantha Goonatilake (1998). Toward a Global Science. Indiana University Press. p. 126. ISBN 978-0-253-33388-9. Virahanka Fibonacci.
10. ^ Hall, Rachel Wells (February 2008). "Math for Poets and Drummers". Math Horizons. Taylor & Francis. 15 (3): 10–12. doi:10.1080/10724117.2008.11974752. JSTOR 25678735. S2CID 3637061. Retrieved 27 May 2022 – via JSTOR.
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."