Chandaḥśāstra

Acharya Pingala ('; c. 3rd2nd century BCE) was an ancient Indian poet and mathematician, and the author of the ' (also called the ''Pingala-sutras''), the earliest known treatise on Sanskrit prosody.

The ' 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. In the 10th century CE, Halayudha wrote a commentary elaborating on the '. Pingala Maharshi was also said to be the brother of Pāṇini, the famous Sanskrit grammarian, considered the first descriptive linguist''. François & Ponsonnet (2013: 184).''

Combinatorics

The ' 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. Pingala's discussion of the combinatorics of metre corresponds to the binomial theorem. Halāyudha's 10th-century commentary on the ' includes a presentation of this theorem in what is now called Pascal's triangle (called ' in the commentary), named after French mathematician Blaise Pascal despite its discovery by Halayudha and others centuries before. Pingala's work also includes material related to the Fibonacci numbers, called '.

Pingala is credited with the first use of binary numbers, using light (''laghu'') and heavy (''guru'') syllables to describe combinatorics of Sanskrit metre. 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., pages 54–56: "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, ..Pingala's use of a zero symbol �ūnyaas 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." 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

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).
*

External links

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

{{Authority control

Fibonacci numbers
Ancient Indian mathematicians
Ancient Sanskrit grammarians
Indian Sanskrit scholars
2nd-century BC mathematicians