Pingala
Encyclopedia
Pingala is the traditional name of the author of the (also ), the earliest known Sanskrit
treatise on prosody
.
Nothing is known about Piṅgala himself. In Indian literary tradition, he is variously identified either as the younger brother of Pāṇini (4th century BCE), or as Patañjali
, the author of the Mahābhāṣhya (2nd century BCE).
The 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 and expanded it.
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 ). Pingala's work also contains the basic ideas of Fibonacci number
, called , 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.
Sanskrit
Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
treatise on prosody
Sanskrit prosody
Versification in Classical Sanskrit poetry is of three kinds.# Syllabic verse : meters depend on the number of syllables in a verse, with relative freedom in the distribution of light and heavy syllables...
.
Nothing is known about Piṅgala himself. In Indian literary tradition, he is variously identified either as the younger brother of Pāṇini (4th century BCE), or as Patañjali
Patañjali
Patañjali is the compiler of the Yoga Sūtras, an important collection of aphorisms on Yoga practice. According to tradition, the same Patañjali was also the author of the Mahābhāṣya, a commentary on Kātyāyana's vārttikas on Pāṇini's Aṣṭādhyāyī as well as an unspecified work of medicine .In...
, the author of the Mahābhāṣhya (2nd century BCE).
The is a work of eight chapters in the late Sūtra
Sutra
Sūtra is an aphorism or a collection of such aphorisms in the form of a manual. Literally it means a thread or line that holds things together and is derived from the verbal root siv-, meaning to sew , as does the medical term...
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
Vedic meter
The verses of the Vedas have a variety of different meters. They are divided by number of padas in a verse, and by the number of syllables in a pada. Chandas , the study of Vedic meter, is one of the six Vedanga disciplines, or "organs of the vedas".*: 3 padas of 8 syllables-Principles:The main...
and the classical meter
Sanskrit prosody
Versification in Classical Sanskrit poetry is of three kinds.# Syllabic verse : meters depend on the number of syllables in a verse, with relative freedom in the distribution of light and heavy syllables...
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
Halayudha
Halayudha was a 10th century Indian mathematician who wrote the ', a commentary on Pingala's Chandah-shastra, containing a clear description of Pascal's triangle ....
wrote a commentary on the and expanded it.
Combinatorics
The presents the first known description of a binary numeral systemBinary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
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
Binomial theorem
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...
. Halāyudha's commentary includes a presentation of the Pascal's triangle
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
(called ). Pingala's work also contains the basic ideas of Fibonacci number
Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
, called , and now known as the Gopala–Hemachandra number.
Use of zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
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
Halayudha
Halayudha was a 10th century Indian mathematician who wrote the ', a commentary on Pingala's Chandah-shastra, containing a clear description of Pascal's triangle ....
but not to Pingala.
Editions
- A. WeberAlbrecht WeberAlbrecht Friedrich Weber was a German Indologist and historian.He was born in Breslau, where his father was a Professor of Political Economy. He studied in that town, Bonn, and in Berlin, 1842-1845, busying himself especially with literature and Sanskrit archaeology. He received a doctor's degree...
, Indische Studien 8, Leipzig, 1863. - Bibliotheca Indica, Calcutta 1871-1874, reprint 1987.
External links
- Math for Poets and Drummers, Rachel W. Hall, Saint Joseph's UniversitySaint Joseph's UniversitySaint Joseph's University is a private, coeducational Roman Catholic Jesuit university located partially in the Wynnefield section of Philadelphia and partially in Lower Merion Township and located in the Pennsylvania Main Line, Pennsylvania, United States.The school was founded in 1851 as Saint...
, 2005. - Mathematics of Poetry, Rachel W. Hall