# Pingala

# PINGALA

**PINGALA, c. fifth century b.c., Indian mathematician.** Binary numbers were known at the time of Pingala's

*Chhandah-shāstra*to classify Vedic meters. According to an old Indian tradition, Pingala was the younger brother of Pānini. This is so stated by Shadgurushishya in his

*Vedārtha Dīpikā*. If this tradition is correct, he should be assigned to the fifth century b.c., the most likely period of Pānini's life. The fact that the

*Chhandah-shāstra*is in the early unversified

*sūtra*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 (*pāda*), 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 *gāyatrī* 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, *vasantatilakā* (the ornament of spring) is 14 x 4, *mālinī* (the girl wearing a garland) is 15 x 4, *prithivī* (the earth) is 17 x 4,*mandākrāntā* (the slow stepper) and *harinī* (the doe) are 17 x 4, *shārdūla-vikrīdita* (the tiger's sport) is 19 x 4, and the sragdharā (the girl with a garland) is 21 x 4. The meters *gāyatrī* (24), *ushnik* (28), *atishakkarī* (30 for half), *anushtup* (32), *brihatī* (36), *pankti* (40), *trishtup* (44), and *jagatī* (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 *laghu*s and *guru*s 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, *prastāra*. 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 *laghu*s and *guru*s. This suggests that an understanding of the basis of the representation of numbers existed. The Pingala octal assignment has much similarity with the Katapayādi notation for representing numerals as words.

*Subhash Kak*

*See also***Pānini** ; **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.

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