Pingala
Template:Short description Script error: No such module "For". Template:Infobox scholar Acharya Pingala[1] (Template:Langx; c. 3rdTemplate:En dash2nd century BCE)[2] was an ancient Indian poet and mathematician,[3] and the author of the Template:Transliteration (Template:Langx), also called the Pingala-sutras (Template:Langx), the earliest known treatise on Sanskrit prosody.[4]
The Template:Transliteration 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 Template:Transliteration. According to some historians Maharshi Pingala was the brother of Pāṇini, 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 Template:Transliteration 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 śūnya 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 Template:Transliteration.[14]
Editions
- A. Weber, Indische Studien 8, Leipzig, 1863.
- Janakinath Kabyatittha & brothers, ChhandaSutra-Pingala, Calcutta, 1931.[15]
- Nirnayasagar Press, Chand Shastra, Bombay, 1938[16]
Notes
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).
- Script error: No such module "Citation/CS1".
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
- ↑ Script error: No such module "Citation/CS1".
- ↑ Script error: No such module "citation/CS1".
- ↑ Script error: No such module "citation/CS1".
- ↑ Script error: No such module "citation/CS1".
- ↑ 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)
- ↑ Script error: No such module "Citation/CS1".
- ↑ Script error: No such module "citation/CS1".
- ↑ 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."
- ↑ Script error: No such module "citation/CS1".
- ↑ B. van Nooten, "Binary Numbers in Indian Antiquity", Journal of Indian Studies, Volume 21, 1993, pp. 31–50
- ↑ Script error: No such module "citation/CS1".
- ↑ Script error: No such module "citation/CS1".
- ↑ Script error: No such module "citation/CS1".