PINGALA

Contents

• Introduction
• His works
• Elements of Prosody
• Combinatory
• ‘Zero’, not the Brainchild of Pingala
• Halayudha’s commentary on Chanda-shastra

Introduction

Pingala was a 4^th century BCE Indian mathematician and Sanskrit grammarian. While nothing specific is known about the birth and death of Pingala, records prove that he contributed greatly in the fields of science and literature during the Vedic period.

Pingala’s Chanda-shastra has been recorded as the most ancient Sanskrit treatise on prosody. Pingala’s contribution in the field of mathematics is tremendous and his vital inventions in this field include Binary numeral system, Pascal’s triangle, Binomial theorem, Fibonacci number (Matrameru).

Little is known about the life of Pingala; many Indian scholars identify him as the younger brother of the 4^th century BCE grammarian, Panini. On the contrary, another think tank identifies him as Patanjali, the 2^nd century CE scholar who authored Mahabhashya.

His works

Pingala is mainly known for his famous work Chanda-shastra, the earliest recorded treatise on Sanskrit prosody. Chanda-shastra has been dated to either the later part of BCE, or the rising CE; this was the time when an evident transition took place from the Vedic meter to the Classical meter in Sanskrit literature. This period is specific to the Mauryan era when the Arthashastra and the Edicts of Ashoka were written and appreciated by many.

The Sutra style content of Pingala’s Chanda-shastra is incomprehensible without a commentary and the famous of all commentaries written on the text is the one composed by the 10^th century mathematician, Halayudha.

Elements of Prosody

Gana – The method of Ganas was first propounded by Pingala for describing meters and is found in treatises on Sanskrit prosody. Ganas describe the pattern of light and heavy syllables in an order of three.

A mnemonic – In fact, a mnemonic for Pingala’s ganas was coined by medieval commentators as yamātārājabhānasalagāḥ or yamātārājabhānasalagam.

Combinatory

The term ‘Chanda’ literally means the study of Vedic meter and is one of the six Vedangas (limbs of Vedas). Unfortunately, not a single treatise dealing exclusively with Chanda or Vedic meter has been survived. Thus, it can be said that Chanda-shastra, is the first recorded text describing the Vedic meter and its transition into Classical Sanskrit metrical form. The famous epic Agni Purana is also based on Chanda-shastra. In this text, Pingala introduces the first known description on binary numeral system. The listing of Vedic meters (with short and long syllables) is accurately linked to the binary numeral system. Again, Pingala’s explanation on combinatorics of meter also precisely matches with the binomial theorem. His works also presents a brief sketch on Fibonacci number (Matrameru).

In his article ‘Binary numbers in Indian Antiquity’ published in Journal of Indian Studies 1993, B. Van Nooten, a Vedic scholar, defines the works of Pingala and his art of exploring the binary numbers, “Instead of giving names to the meters he constructs a prastāra, a ‘bed’, or matrix, in which the laghus and gurus are listed horizontally… He (Pingala) knew how to convert that binary notation to a decimal notation and vice versa. We know of no sources from which he could have drawn his inspiration, so he may well have been the originator of the system…this knowledge was available to and preserved by Sanskrit students of metrics. Unlike the case of the great linguistic discoveries of the Indians which directly influenced and inspired Western linguistics…”

 ‘Zero’, not the Brainchild of Pingala

Pingala used short and long syllable to explain the binary numbers, while the modern system uses 0 and 1 for the same. Hence, the invention of Zero is sometimes wrongly attributed to Pingala. However, as per Pingala’s system, the usage of four short syllables (described as ‘0000’ in binary) stands for number ‘One’ and not ‘Zero’. It was in the later centuries that the positional usage of Zero became prevalent, possibly in the times when Halayudha composed his commentary on Pingala’s work.

Halayudha’s commentary on Chanda-shastra

Halayudha is specifically known for his lucid commentary on Pingala’s Pascal triangle (Meru-prastaara). Again, the term ‘Meru-prastaara’ literally stands for ‘the staircase to Mount Meru’. Halayudha has logically referred to the Pascal triangle in following phrase -

"Draw a square. Beginning at half the square, draw two other similar squares below it; below these two, three other squares, and so on. The marking should be started by putting 1 in the first square. Put 1 in each of the two squares of the second line. In the third line put 1 in the two squares at the ends and, in the middle square, the sum of the digits in the two squares lying above it. In the fourth line put 1 in the two squares at the ends. In the middle ones put the sum of the digits in the two squares above each. Proceed in this way. Of these lines, the second gives the combinations with one syllable, the third the combinations with two syllables ...”