Acharya In Indian religions and society, an ''acharya'' (Sanskrit: आचार्य, IAST: ; Pali: ''ācariya'') is a religious teacher in Hinduism and Buddhism and a spiritual guide to Hindus and Buddhists. The designation has different meanings i ...

Pingala (; c. 3rd2nd century BCE) was an ancient Indian poet and

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

, and the author of the ' (), also called the ''Pingala-sutras'' (), the earliest known treatise on

Sanskrit prosody Sanskrit prosody or Chandas refers to one of the six Vedangas, or limbs of Vedic studies.James Lochtefeld (2002), "Chandas" in The Illustrated Encyclopedia of Hinduism, Vol. 1: A-M, Rosen Publishing, , page 140 It is the study of poetic met ...

. The ' is a work of eight chapters in the late

Sūtra ''Sutra'' ()Monier Williams, ''Sanskrit English Dictionary'', Oxford University Press, Entry fo''sutra'' page 1241 in Indian literary traditions refers to an aphorism or a collection of aphorisms in the form of a manual or, more broadly, a ...

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 '. According to some historians

Maharshi Maharishi (, ) is a Sanskrit word used for members of the highest order of ancient Indian sages, popularly known in India as "seers", i.e., those who engage in research to understand and experience nature, divinity, and the divine context of exis ...

Pingala was the brother of

Pāṇini (; , ) was a Sanskrit grammarian, logician, philologist, and revered scholar in ancient India during the mid-1st millennium BCE, dated variously by most scholars between the 6th–5th and 4th century BCE. The historical facts of his life ar ...

, the famous

Sanskrit grammarian Sanskrit (; stem form ; nominal singular , ,) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in northwest South Asia after its predecessor languages had Trans-cultural ...

, considered the first descriptive linguist''. François & Ponsonnet (2013: 184).'' Another think tank identifies him as

Patanjali Patanjali (, , ; also called Gonardiya or Gonikaputra) was the name of one or more author(s), mystic(s) and philosopher(s) in ancient India. His name is recorded as an author and compiler of a number of Sanskrit works. The greatest of these a ...

, the 2nd century CE scholar who authored Mahabhashya.

Combinatorics

The ' presents a formula to generate systematic enumerations of

metres The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...

, of all possible combinations of light (''laghu'') and heavy (''guru'') syllables, for a word of ''n'' syllables, using a recursive formula, that results in a partially ordered

binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two values (0 and 1) for each digit * Binary function, a function that takes two arguments * Binary operation, a mathematical op ...

representation. Pingala is credited with being the first to express the

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

Sanskrit metre Sanskrit prosody or Chandas refers to one of the six Vedangas, or limbs of Vedic studies.James Lochtefeld (2002), "Chandas" in The Illustrated Encyclopedia of Hinduism, Vol. 1: A-M, Rosen Publishing, , page 140 It is the study of poetic met ...

, eg. * Create a syllable list ''x'' comprising one light (''L'') and heavy (''G'') syllable: * Repeat till list ''x'' contains only words of the desired length ''n'' ** Replicate list ''x'' as lists ''a'' and ''b'' *** Append syllable ''L'' to each element of list ''a'' *** Append syllable ''G'' to each element of list ''b'' ** Append lists ''b'' to list ''a'' and rename as list ''x'' Because of this, Pingala is sometimes also credited with the first use of

zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...

, as he used the

Sanskrit Sanskrit (; stem form ; nominal singular , ,) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in northwest South Asia after its predecessor languages had Trans-cultural ...

word '' śūnya'' to explicitly refer to the number. Pingala's binary representation increases towards the right, and not to the left as modern

binary numbers A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A ''binary number'' may also ...

usually do. In Pingala's system, the numbers start from number one, and not zero. Four short syllables "0000" is the first pattern and corresponds to the value one. The numerical value is obtained by adding one to the sum of

place value Place may refer to: Geography * Place (United States Census Bureau), defined as any concentration of population ** Census-designated place, a populated area lacking its own municipal government * "Place", a type of street or road name ** O ...

s. Pingala's work also includes material related to the

Fibonacci numbers In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the s ...

, called '.

Editions

* A. Weber, ''Indische Studien'' 8, Leipzig, 1863. * Janakinath Kabyatittha & brothers, ''ChhandaSutra-Pingala'', Calcutta, 1931. * Nirnayasagar Press, Chand Shastra, Bombay, 1938

Notes

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 Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial ...

. * Klaus Mylius, ''Geschichte der altindischen Literatur'', Wiesbaden (1983). *

External links

*
Math for Poets and Drummers
', Rachel W. Hall,

Saint Joseph's University Saint Joseph's University (SJU or St. Joe's) is a Private university, private Jesuits, Jesuit university in Philadelphia, Lower Merion Township, Pennsylvania, and Lancaster, Pennsylvania. The university was founded by the Jesuits, Society of J ...

, 2005. *
Mathematics of Poetry
', Rachel W. Hall *
Internet Archive
', The Prosody of Pingala {{Authority control Fibonacci numbers Ancient Indian mathematicians Ancient Sanskrit grammarians Indian Sanskrit scholars 2nd-century BC mathematicians

Combinatorics

Editions

Notes

See also

References

External links