Pingala: Difference between revisions

Pingala
Pingala
Born	unclear, 3rd or 2nd century BCE
Academic work
Era	Maurya or post-Maurya
Main interests	Sanskrit prosody, Indian mathematics, Sanskrit grammar
Notable works	Author of the "Chhandaḥśāstra" (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody. Creator of Pingala's formula.
Notable ideas	mātrāmeru, binary numeral system, arithmetical triangle

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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]

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,^{[citation needed]} 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 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 work also includes material related to the Fibonacci numbers, called mātrāmeru.^[9]

Pingala is credited with the use of the system similar to 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 $n$ -th syllable pattern corresponds to the binary representation of $n-1$ (with increasing positional values).

Editions

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

Notes

^ ^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 2019-07-24. Retrieved 2018-11-29.
^ "Pingala – Timeline of Mathematics". Mathigon. Retrieved 2021-08-21.
^ 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.
^ François & Ponsonnet (2013: 184).
^ Van Nooten (1993)
^ Susantha Goonatilake (1998). Toward a Global Science. Indiana University Press. p. 126. ISBN 978-0-253-33388-9. Virahanka Fibonacci.
^ 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.
^ 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."

References

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. S2CID 171039636.

External links

Math for Poets and Drummers, Rachel W. Hall, Saint Joseph's University, 2005.
Mathematics of Poetry, Rachel W. Hall

[plofker55-1] 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.

[FPencyclo-7] François & Ponsonnet (2013: 184).

[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. 15 (3). Taylor & Francis: 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."

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 The ''{{IAST|Chandaḥśāstra}}'' presents the first known description of a [[binary numeral system]] in connection with the systematic enumeration of [[Metre (poetry)|metres]] with fixed patterns of short and long syllables.<ref>Van Nooten (1993)</ref> Pingala's work also includes material related to the [[Fibonacci numbers]], called ''{{IAST|mātrāmeru}}''.<ref>{{cite book |title = Toward a Global Science | author = Susantha Goonatilake |publisher = Indiana University Press |year = 1998 |page = [https://archive.org/details/towardglobalscie0000goon/page/126 126] |isbn = 978-0-253-33388-9 |url = https://archive.org/details/towardglobalscie0000goon |url-access = registration |quote = Virahanka Fibonacci. }}</ref>
-Pingala is credited with the first use of [[Binary number|binary numbers]], using [[Sanskrit prosody#Light and heavy syllables|light (''laghu'') and heavy (''guru'') syllables]] to describe [[combinatorics]] of [[Sanskrit prosody|Sanskrit metre]].<ref>{{Cite journal |last=Hall |first=Rachel Wells |date=February 2008 |title=Math for Poets and Drummers |url=https://www.jstor.org/stable/25678735 |journal=Math Horizons |publisher=[[Taylor & Francis]] |volume=15 |issue=3 |pages=10{{en dash}}12 |doi=10.1080/10724117.2008.11974752 |jstor=25678735 |s2cid=3637061 |access-date=27 May 2022 |via=JSTOR}}</ref> Because of this, Pingala is sometimes also credited with the first use of [[0|zero]], as he used the [[Sanskrit]] word ''[[Śūnyatā|śūnya]]'' to explicitly refer to the number.<ref>{{harvtxt|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)<sup>7</sup> = 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."</ref> 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 <math>n</math>-th syllable pattern corresponds to the binary representation of <math>n-1</math> (with increasing positional values).
+Pingala is credited with the use of the system similar to [[Binary number|binary numbers]], using [[Sanskrit prosody#Light and heavy syllables|light (''laghu'') and heavy (''guru'') syllables]] to describe [[combinatorics]] of [[Sanskrit prosody|Sanskrit metre]].<ref>{{Cite journal |last=Hall |first=Rachel Wells |date=February 2008 |title=Math for Poets and Drummers |url=https://www.jstor.org/stable/25678735 |journal=Math Horizons |publisher=[[Taylor & Francis]] |volume=15 |issue=3 |pages=10{{en dash}}12 |doi=10.1080/10724117.2008.11974752 |jstor=25678735 |s2cid=3637061 |access-date=27 May 2022 |via=JSTOR}}</ref> Because of this, Pingala is sometimes also credited with the first use of [[0|zero]], as he used the [[Sanskrit]] word ''[[Śūnyatā|śūnya]]'' to explicitly refer to the number.<ref>{{harvtxt|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)<sup>7</sup> = 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."</ref> 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 <math>n</math>-th syllable pattern corresponds to the binary representation of <math>n-1</math> (with increasing positional values).
 ==Editions==

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