# Pingala

(Redirected from Chandah-shastra)
Pingala
Bornunclear, 3rd / 2nd century BCE[1]
ResidenceIndian subcontinent
EraMaurya or post-Maurya
Main interestsIndian mathematics, Sanskrit grammar
Notable worksAuthor of the Chandaḥśāstra (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody
Notable ideasmātrāmeru, binary numeral system, arithmetical triangle

Acharya Pingala[2] (Devanagari: पिङ्गल piṅgala) (c. 3rd/2nd century BCE)[1] was an ancient Indian mathematician who authored the Chandaḥśāstra (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody.[3]

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.[4][5] The 10th century mathematician Halayudha wrote a commentary on the Chandaḥśāstra and expanded it.

## Combinatorics

The Chandaḥśāstra presents the first known description of a binary numeral system in connection with the systematic enumeration of meters with fixed patterns of short and long syllables.[6] The discussion of the combinatorics of meter corresponds to the binomial theorem. Halāyudha's commentary includes a presentation of Pascal's triangle (called meruprastāra). Pingala's work also includes material related to the Fibonacci numbers, called mātrāmeru.[7]

Use of zero is sometimes ascribed to Pingala due to his discussion of binary numbers, usually represented using 0 and 1 in modern discussion, but Pingala used light (laghu) and heavy (guru) rather than 0 and 1 to describe syllables. As Pingala's system ranks binary patterns starting at one (four short syllables—binary "0000"—is the first pattern), the nth pattern corresponds to the binary representation of n-1 (with increasing positional values).

Pingala is credited with using binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), a notation similar to Morse code.[8] Pingala used the Sanskrit word śūnya explicitly to refer to zero.[9]

## Editions

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

## Notes

1. ^ a b Kim Plofker (2009). Mathematics in India. Princeton University Press. pp. 55–56. ISBN 0-691-12067-6.
2. ^ Singh, Parmanand (1985). "The So-called Fibonacci Numbers in Ancient and Medieval India" (PDF). Historia Mathematica. Academic Press. 12: 232.
3. ^ 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.
4. ^ R. Hall, Mathematics of Poetry, has "c. 200 BC"
5. ^ Mylius (1983:68) considers the Chandas-shāstra as "very late" within the Vedānga corpus.
6. ^ Van Nooten (1993)
7. ^ Susantha Goonatilake (1998). Toward a Global Science. Indiana University Press. p. 126. ISBN 978-0-253-33388-9.
8. ^ "Math for Poets and Drummers" (pdf). people.sju.edu.
9. ^ Kim Plofker (2009), Mathematics in India, Princeton University Press, ISBN 978-0691120676, page 54–56. Quote – "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." Kim Plofker (2009), Mathematics in India, Princeton University Press, ISBN 978-0691120676, 55–56. "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.