|Born||possibly c. (6th century BCE)|
|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|
Little is known about Piṅgala himself. In later Indian literary tradition, he is variously identified either as the younger brother of Pāṇini (4th century BCE), or as Patañjali, the author of the Mahābhāṣya (2nd century BCE).
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 BCE or the early centuries CE, 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.
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[according to whom?] 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.
- A. Weber, Indische Studien 8, Leipzig, 1863.
- R. Hall, Mathematics of Poetry
- Maurice Winternitz, History of Indian Literature, Vol. III
- R. Hall, Mathematics of Poetry, has "c. 200 BC"
- Mylius (1983:68) considers the Chandas-shāstra as "very late" within the Vedānga corpus.
- Van Nooten (1993)
- Susantha Goonatilake (1998). Toward a Global Science. Indiana University Press. p. 126. ISBN 978-0-253-33388-9.
- 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. Retrieved 2010-05-06.