Pingala

Template:Short description Template:For Template:Infobox scholar Acharya Pingala<ref>Template:Cite journal</ref> (Template:Langx; c. 3rdTemplate:En dash2nd century BCE)<ref name=plofker55>Template:Cite book</ref> was an ancient Indian poet and mathematician,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and the author of the Template:IAST (Template:Langx), also called the Pingala-sutras (Template:Langx), the earliest known treatise on Sanskrit prosody.<ref>Template:Cite book</ref>

The Template:IAST 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.<ref>R. Hall, Mathematics of Poetry, has "c. 200 BC"</ref><ref>Mylius (1983:68) considers the Chandas-shāstra as "very late" within the Vedānga corpus.</ref> In the 10th century CE, Halayudha wrote a commentary elaborating on the Template:IAST. According to some historians Maharshi Pingala was the brother of Pāṇini, the famous Sanskrit grammarian, considered the first descriptive linguist.<ref name="FPencyclo">François & Ponsonnet (2013: 184).</ref> Another think tank identifies him as Patanjali, the 2nd century CE scholar who authored Mahabhashya.

Combinatorics

The Template:IAST 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.<ref>Van Nooten (1993)</ref> Pingala is credited with being the first to express the combinatorics of Sanskrit metre, eg.<ref>Template:Cite journal</ref>

• 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

citation

CitationClass=web }}</ref>
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 śūnya to explicitly refer to the number.<ref>Template:Harvtxt, pp. 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."</ref> Pingala's binary representation increases towards the right, and not to the left as modern binary numbers usually do.<ref>Template:Cite book</ref> 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.<ref>B. van Nooten, "Binary Numbers in Indian Antiquity", Journal of Indian Studies, Volume 21, 1993, pp. 31–50</ref> Pingala's work also includes material related to the Fibonacci numbers, called Template:IAST.<ref>Template:Cite book</ref>

Editions

• A. Weber, Indische Studien 8, Leipzig, 1863.
• Janakinath Kabyatittha & brothers, ChhandaSutra-Pingala, Calcutta, 1931.<ref>Template:Cite book</ref>
• Nirnayasagar Press, Chand Shastra, Bombay, 1938<ref>Template:Cite book</ref>

See also

• Chandas
• Sanskrit prosody
• Indian mathematics
• Indian mathematicians
• History of the binomial theorem
• List of Indian mathematicians

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).
• Template:Cite journal

External links

• 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