# Pingala

With thanks to Phil Krause

Pingala (पिङ्गल piṅgala) is the traditional name of the author of the Chandaḥśāstra (also Chandaḥsūtra), the earliest known Sanskrit treatise on prosody.

Nothing is known about Piṅgala himself. In Indian literary tradition, he is variously identified either as the younger brother of Pāṇini (4th century BCE), or as Patañjali, the author of the Mahābhāṣhya (2nd century BCE).

The chandaḥśāstra is a work of eight chapters in the late Sūtra style, not fully comprehensible without a commentary. It has been dated to either the final centuries BCE or the early centuries CE, at the transition between Vedic meter and the classical meter of the Sanskrit epics. This would place it close to the beginning of the Common Era, likely post-dating Mauryan times. The 10th century mathematician Halayudha wrote a commentary on the chandaḥśāstra and expanded it.

The chandaḥśāstra presents the first known description of a binary numeral system in connection with the systematic enumeration of meters with fixed patterns of short and long syllables. The discussion of the combinatorics of meter corresponds to the binomial theorem. Halāyudha’s commentary includes a presentation of the Pascal’s triangle (called meruprastāra). Pingala’s work also contains the basic ideas of Fibonacci number, called mātrāmeru, and now known as the Gopala–Hemachandra number.

Use of zero is sometimes mistakenly ascribed to Pingala due to his discussion of binary numbers, usually represented using 0 and 1 in modern discussion, while Pingala used short and long syllables. As Pingala’s system ranks binary patterns starting at one (four short syllables—binary “0000”—is the first pattern), the nth pattern corresponds to the binary representation of n-1, written backwards. Positional use of zero dates from later centuries and would have been known to Halāyudha but not to Pingala

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### 2 Responses to Pingala

1. alfy says:

Do you remember the Ishango bone, one of deskarati’s earlier posts? This suggests that people in the Mesolithic had got hold of the idea of geometrical series, among other mathematical ideas. None of this should surprise us as people of towering intellect are likely to emerge at any stage in history.

However, re-naming stuff is not a good idea. Who decided that the Fibonacci series is “now known as the GH number.”? Was there any discussion of the matter? I am going to decide that geometric series are now known as ” Ishango bone boy’s magic trick”. So go rewrite the maths textbooks.

One of the problems with a lot of this early stuff is that the distinctions between subject areas recognised in later millennia did not exist then. From a cursory look at the Pingala work it seems that he thought he was writing a book on metre in poetry. Did he actually realise that he was producing a mathematical theory with a universal applicability? I rather doubt it. The reason for saying this is that not much came out of it.
By contrast, Fibonacci was aware that he was analysing a mathematical series with quite wide applications and his ideas were taken up by others and expanded and continue to be a source of research in the present day.

Let’s look at a different but parallel issue. A chap called Hero of Alexander invented a steam engine around 50 BC or thereabouts. (For accurate details see wikipedia). It was a toy and enormous fun at the time, flying round and round at speed and spraying steam and hot water over the silly sods standing by it. Now I am going to insist that all steam engines by Trevithick, Newcomen, Watt et al are “now known as Hero engines”.

Why is this a daft idea? Very simply because although Hero invented the first steam engine as far as we know, absolutely nothing came of it. The Greeks saw it as a toy. Who needed a new source of power when there were lots of slaves available?

The men who helped create the industrial revolution in Britain saw steam as a source of power in itself. They continually tried to improve the efficiency of their machines and continually saw new uses for them. Having begun with mine pumping they realised steam could be used to drive mills of various kinds. Then they saw steam could be used to haul coal and iron ore. Drawing passengers was another revelation, and the ultimate was steam powered shipping.

What has Hero to do with this? Absolutely bugger all. He was not linked technically or intellectually to the industrial revolution of the 18 and early 19 centuries. He was just a historical curiosity. Similarly, I am afraid is Pingala. Just a curiosity. Nothing links him to the broad sweep of mathematics, as does Fibonacci who rediscovered the series much later.

2. Deskarati says:

We like the sound of Hero of Alexandria. Just because the need wasn’t there at the time, surely doesn’t diminish the discovery of what turned out to be a turning point in our industrial progression some many centuries later. We think they should all be called ‘Hero Engines’, it’s got a nice ring to it.