Pingala (Devanagari: पिङ्गल piṅgala) (c. 3rd/2nd century BC), is the influential ancient scholar and the author of the Chandaḥśāstra (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody.
|Era||Maurya or post-Maurya|
|Main interests||Indian mathematics, Sanskrit grammar|
|Notable works||Author of the Chandaḥśāstra (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody|
|Notable ideas||mātrāmeru, 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. The 10th century mathematician Halayudha wrote a commentary on the Chandaḥśāstra and expanded it.
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. The discussion of the combinatorics of meter corresponds to the binomial theorem. Halāyudha's commentary includes a presentation of the Pascal's triangle (called meruprastāra). Pingala's work also contains the Fibonacci numbers, called mātrāmeru.
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. Pingala used the Sanskrit word śūnya explicitly to refer to zero.
- A. Weber, Indische Studien 8, Leipzig, 1863.
- Kim Plofker (2009). Mathematics in India. Princeton University Press. pp. 55–56. ISBN 0-691-12067-6.
- 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.
- R. Hall, Mathematics of Poetry, has "c. 200 BC"
- Mylius (1983:68) considers the Chandas-shāstra as "very late" within the Vedānga corpus.
- Van Nooten (1993)
- Susantha Goonatilake (1998). Toward a Global Science. Indiana University Press. p. 126. ISBN 978-0-253-33388-9.
- "Math for Poets and Drummers" (pdf). people.sju.edu.
- 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.
- Amulya Kumar Bag, 'Binomial theorem in ancient India', Indian J. Hist. Sci. 1 (1966), 68–74.
- George Gheverghese Joseph (2000). The Crest of the Peacock, p. 254, 355. Princeton University Press.
- Klaus Mylius, Geschichte der altindischen Literatur, Wiesbaden (1983).
- Van Nooten, B. (1993-03-01). "Binary numbers in Indian antiquity". Journal of Indian Philosophy. 21 (1): 31–50. doi:10.1007/BF01092744. Retrieved 2010-05-06.