Pingala

For the subtle energy channel described in yoga, see Nadi (yoga).

Pingala
Born	possibly c. 500 BC (unknown)
Died	(unknown)
Residence	Indian subcontinent
Academic work
Era	Maurya or post-Maurya
Main interests	Indian mathematics, Sanskrit grammar
Notable works	Author of the Chandaḥśāstra (also Chandaḥsūtra), the earliest known Sanskrit treatise on prosody
Notable ideas	mātrāmeru, binary numeral system, arithmetical triangle

Pingala (Devanagari: पिङ्गल piṅgala) is the traditional name of the author of the Chandaḥśāstra (also Chandaḥsūtra), the earliest known Sanskrit treatise on prosody.

Little is known about Piṅgala himself. In later Indian literary tradition, he is variously identified either as the younger brother of Pāṇini (4th century BC), or as Patañjali, the author of the Mahabhashya (2nd century BC).

The Chandaḥśāstra is a work of eight chapters in the late Sūtra style, not fully comprehensible without a commentary. It has been dated to either the final centuries BC or the early centuries AD, at the transition between Vedic meter and the classical meter of the Sanskrit epics. This would place it close to the beginning of the Common Era, likely post-dating Mauryan times. The 10th century mathematician Halayudha wrote a commentary on the Chandaḥśāstra and expanded it.

Combinatorics

The Chandaḥśāstra presents the first known description of a binary numeral system in connection with the systematic enumeration of meters with fixed patterns of short and long syllables. The discussion of the combinatorics of meter corresponds to the binomial theorem. Halāyudha's commentary includes a presentation of the Pascal's triangle (called meruprastāra). Pingala's work also contains the Fibonacci numbers, called mātrāmeru.

Use of zero is sometimes mistakenly ascribed to Pingala due to his discussion of binary numbers, usually represented using 0 and 1 in modern discussion, while Pingala used light (laghu) and heavy (guru) syllables. As Pingala's system ranks binary patterns starting at one (four short syllables—binary "0000"—is the first pattern), the nth pattern corresponds to the binary representation of n-1 (with increasing positional values). Positional use of zero dates from c. the 7th century (Brahmagupta) and would have been known to Halāyudha but not to Pingala.

Editions

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

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