Pingala
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Acharya In Indian religions and society, an ''acharya'' (Sanskrit: आचार्य, IAST: ; Pali: ''ācariya'') is a preceptor and expert instructor in matters such as religion, or any other subject. An acharya is a highly learned person with a ...
Pingala ('; c. 3rd2nd century
BCE Common Era (CE) and Before the Common Era (BCE) are year notations for the Gregorian calendar (and its predecessor, the Julian calendar), the world's most widely used calendar era. Common Era and Before the Common Era are alternatives to the or ...
) 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, structure, space, models, and change. History On ...
, 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 metr ...
. The ' is a work of eight chapters in the late
Sūtra ''Sutra'' ( sa, सूत्र, translit=sūtra, translit-std=IAST, translation=string, thread)Monier Williams, ''Sanskrit English Dictionary'', Oxford University Press, Entry fo''sutra'' page 1241 in Indian literary traditions refers to an ap ...
style, not fully comprehensible without a commentary. It has been dated to the last few centuries BCE. In the 10th century CE,
Halayudha Halayudha (Sanskrit: हलायुध) was a 10th-century Indian mathematician who wrote the ',Maurice Winternitz, ''History of Indian Literature'', Vol. III a commentary on Pingala's ''Chandaḥśāstra''. The latter contains a clear descri ...
wrote a commentary elaborating on the '. Pingala
Maharshi Maharishi is a Sanskrit word, written as "महर्षि" in Devanagari (formed from the prefix mahā- meaning "great" and r̥ṣi - sage, poet or a singer of sacred hymns), indicating members of the highest order of ancient Indian sages, po ...
was also said to be the brother of
Pāṇini , era = ;;6th–5th century BCE , region = Indian philosophy , main_interests = Grammar, linguistics , notable_works = ' (Sanskrit#Classical Sanskrit, Classical Sanskrit) , influenced= , notable_ideas=Descript ...
, the famous
Sanskrit grammarian Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late ...
, considered the first descriptive linguist''. François & Ponsonnet (2013: 184).''


Combinatorics

The ' presents the first known description of a
binary numeral system A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one). The base-2 numeral system is a positional notatio ...
in connection with the systematic enumeration of
metres The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its prefi ...
with fixed patterns of short and long syllables. Pingala's discussion of the
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
of metre corresponds to the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
. Halāyudha's 10th-century commentary on the ' includes a presentation of this theorem in what is now called
Pascal's triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
(called ' in the commentary), named after French mathematician
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...
despite its discovery by Halayudha and others centuries before. Pingala's work also includes material related to the
Fibonacci numbers In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
, called '. Pingala is credited with the first use of binary numbers, using light (''laghu'') and heavy (''guru'') syllables to describe
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
of
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 metr ...
. Because of this, Pingala is sometimes also credited with the first use of
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
, as he used the
Sanskrit Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late ...
word '' śūnya'' to explicitly refer to the number., pages 54–56: "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, ..Pingala's use of a zero symbol ūnyaas a marker seems to be the first known explicit reference to zero. ... In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value “n”. ..The answer is (2)7 = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where “n” is large. Pingala’s use of a zero symbol as a marker seems to be the first known explicit reference to zero." Pingala's binary system of metre starts with four light ''laghu'' syllables as the first pattern ("0000" in binary), three light ''laghu'' and one heavy ''guru'' as the second pattern ("0001" in binary), and so on, so that in general the n-th syllable pattern corresponds to the binary representation of n-1 (with increasing positional values).


Editions

* A. Weber, ''Indische Studien'' 8, Leipzig, 1863.


Notes


See also

*
Chandas 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 metr ...
*
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 metr ...
*
Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...
* Indian mathematicians * History of the binomial theorem *
List of Indian mathematicians chronology of Indian mathematicians spans from the Indus Valley civilisation and the Vedas to Modern India. Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and 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 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 su ...
. *
Klaus Mylius Klaus Albert Robert Curt Mylius (born 24 August 1930 in Berlin) is a German indologist. He was Professor of Sanskrit Studies and Indian Archaeology at the University of Leipzig until 1990, after which he was a lecturer at the universities of Bay ...
, ''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 {{Authority control Fibonacci numbers Ancient Indian mathematicians Ancient Sanskrit grammarians Indian Sanskrit scholars 2nd-century BC mathematicians