|Born||possibly c. 200 BC
|Main interests||Indian mathematics, Sanskrit grammar|
|Notable ideas||mātrāmeru, binary numeral system, arithmetical triangle|
|Major works||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 BCE), or as Patañjali, the author of the Mahabhashya (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 number, called mātrāmeru, and now known as the Gopala–Hemachandra number.
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 short and long 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, written backwards. Positional use of zero dates from later centuries and would have been known to Halāyudha but not to Pingala.
- A. Weber, Indische Studien 8, Leipzig, 1863.
- Bibliotheca Indica, Calcutta 1871-1874, reprint 1987.
- 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.
- Math for Poets and Drummers, Rachel W. Hall, Saint Joseph's University, 2005.
- Mathematics of Poetry, Rachel W. Hall