PINGALA

PINGALA, c. fifth century b.c., Indian mathematician. Binary numbers were known at the time of Pingala's Chhandah-sh?stra to classify Vedic meters. According to an old Indian tradition, Pingala was the younger brother of P?nini. This is so stated by Shadgurushishya in his Ved?rtha D?pik?. If this tradition is correct, he should be assigned to the fifth century b.c., the most likely period of P?nini's life. The fact that the Chhandah-sh?stra is in the early unversified s?tra style makes this conclusion quite plausible. In Europe, a rediscovery of the binary notation, in a slightly different form, was made by Gottfried Leibniz (1646–1716) at the end of the seventeenth century.

Most Vedic hymns are in stanzas of four quarters (p?da), though there are some with three or five divisions. The most popular meters have quarters that have 8, 11, or 12 syllables. The usual way to classify meters is by counting the number of syllables in each line. Thus the g?yatr? consists of 8 syllables in 3 lines (8 x 3), anushtubh is 8 x 4, trishtubh and indravajra (Indra's thunderbolt) is 11 x 4, indravamsha (Indra's family) is 12 x 4, vasantatilak? (the ornament of spring) is 14 x 4, m?lin? (the girl wearing a garland) is 15 x 4, prithiv? (the earth) is 17 x 4,mand?kr?nt? (the slow stepper) and harin? (the doe) are 17 x 4, sh?rd?la-vikr?dita (the tiger's sport) is 19 x 4, and the sragdhar? (the girl with a garland) is 21 x 4. The meters g?yatr? (24), ushnik (28), atishakkar? (30 for half), anushtup (32), brihat? (36), pankti (40), trishtup (44), and jagat? (48) were used most frequently in the Vedic texts.

The syllables are prosodically either short (laghu) or long (guru). A laghu syllable is a short vowel followed by at most one consonant; any other syllable is a guru. Within each quarter verse, a sequence of laghus and gurus defines the meter; this is much like the representation of a number by a succession of 0s and 1s used in the binary arithmetic of computers.

Pingala presented a method where all the binary laghu-guru sequences were shown as a matrix, prast?ra. Given a specific sequence, he showed how it could be converted into an equivalent decimal number; he also showed how a given decimal number could be expanded into the sequence of laghus and gurus. This suggests that an understanding of the basis of the representation of numbers existed. The Pingala octal assignment has much similarity with the Katapay?di notation for representing numerals as words.

Subhash Kak

See alsoP?nini ; Vedic Aryan India

BIBLIOGRAPHY

Kak, Subhash. "Indian Binary Numbers and the Katapayadi Notation."Annals of the Bhandarkar Oriental Research Institute 81 (2000): 269–272.

Nooten, Barend van. "Binary Numbers in Indian Antiquity." In Computing Science in Ancient India, edited by T. R. Rao and S. Kak. New Delhi: Munshiram Manoharlal, 2000.