Acharya Pingala[2] (Sanskrit: ??????, romanized: Pi?gala; c. 3rd–2nd century BCE)[1] was an ancient Indian poet and mathematician,[3] and the author of the Chanda??āstra (Sanskrit: ????????????, lit.?'A Treatise on Prosody'), also called the Pingala-sutras (Sanskrit: ?????????????, romanized: Pi?galasūtrā?, 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 "Chanda??āstra" (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody. Creator of Pingala's formula. |
| Notable ideas | mātrāmeru, binary numeral system. |
The Chanda??āstra 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 Chanda??āstra. According to some historians Maharshi Pingala was the brother of Pā?ini, 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
editThe Chanda??āstra 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
- Replicate list x as lists a and b
| 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 ?ūnya 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
editNotes
edit- ^ a b Plofker, Kim (2009). Mathematics in India. Princeton University Press. pp. 55–56. ISBN 978-0-691-12067-6.
- ^ 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 2025-08-06. Retrieved 2025-08-06.
- ^ "Pingala – Timeline of Mathematics". Mathigon. Retrieved 2025-08-06.
- ^ 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.
- ^ Fran?ois & Ponsonnet (2013: 184).
- ^ Van Nooten (1993)
- ^ 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.
- ^ Shah, Jayant. "A History of Pingala's Combinatorics" (PDF).
- ^ Plofker (2009), pp. 54–56: "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [...] Pingala's use of a zero symbol [?ūnya] 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."
- ^ Stakhov, Alexey; Olsen, Scott Anthony (2009). The mathematics of harmony: from Euclid to contemporary mathematics and computer science. World Scientific. ISBN 978-981-277-582-5.
- ^ B. van Nooten, "Binary Numbers in Indian Antiquity", Journal of Indian Studies, Volume 21, 1993, pp. 31–50
- ^ Susantha Goonatilake (1998). Toward a Global Science. Indiana University Press. p. 126. ISBN 978-0-253-33388-9.
Virahanka Fibonacci.
- ^ Chhanda Sutra – Pingala.
- ^ Pingalacharya (1938). Chand Shastra.
See also
editReferences
edit- 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. (2025-08-06). "Binary numbers in Indian antiquity". Journal of Indian Philosophy. 21 (1): 31–50. doi:10.1007/BF01092744. S2CID 171039636.
External links
edit- Math for Poets and Drummers, Rachel W. Hall, Saint Joseph's University, 2005.
- Mathematics of Poetry, Rachel W. Hall
Internet Archive, The Prosody of Pingala