Numbers: Their Tales, Types, and Treasures.

Chapter 5: Counting for Poets


In the first chapter we discussed various aspects of numbers and counting. As long as one just thinks of counting marbles, this actually was “counting for children.” But the abstraction principle, which states that you can count just about anything, even immaterial ideas, soon leads to counting tasks that are a whole lot more difficult. Consider, for example, the following problem: If at a New Year's Eve party everybody clinks glasses with everyone else, how many clinks will you hear? Problems like these can only be solved with elaborate and systematic thinking, and we give the answer to this and similar problems toward the end of this chapter. The need to learn more about these counting methods arose early in the history of humankind. Indeed, a very interesting solution can be traced back to the time of Vedic literature in old India, more than two thousand years ago.


The problem that we are going to consider in some detail shows how mathematical questions arise in quite unexpected ways. The problem has to do with the classification of verse meters in poetry. In order to understand why this problem (and its solution) arose first in old India, we have to understand what distinguishes Sanskrit poetry from modern English poetry.

The rhythm of speech is determined by the succession of stressed and unstressed syllables. A characteristic rhythmical structure is an important feature of most literature written in verses, which distinguishes poetry from prose. Poems are typically divided into lines that repeat a certain rhythmic pattern (with occasional variations, of course). The rhythmic structure of a line of verse is called its meter. In English (and many other languages, like, for example, German), most poetry uses highly regular meters, where stressed syllables occur in periodic intervals. Very often, each line of verse has a fixed number of syllables and stresses (“accentual-syllabic verse form”). This creates a rhythmic drive that in part is responsible for the fascination created by poems. As an example, observe the regular pattern of stressed and unstressed syllables in these lines from Shakespeare's Macbeth:


Double, double, toil and trouble;
Fire burn and caldron bubble.


In many cases, the rhythm can be described as the repetition of a short and simple basic pattern, which is called verse foot. In this example, the foot of the verse consists of two syllables, the first is stressed, the second unstressed. This is called a trochee, and here it is repeated four times in each line, which gives a meter called trochaic tetrameter. Other common feet are the iamb (unstressed–stressed), the dactyl (stressed–unstressed–unstressed), and the anapest (unstressed–unstressed–stressed).

The English way of emphasizing syllables is called accentuation. This means that stressed syllables are typically spoken louder than unstressed syllables. Thus the verse meter is given by a characteristic succession of loud- and soft-spoken syllables. Of course, the loudness is not the only parameter that could vary. One could, for example, also change the duration (quantity) of a syllable or modulate the voice pitch (intonation), and indeed, all these methods will occur simultaneously. But it is characteristic of a language that one of these methods dominates and is most important for creating the rhythm of speech. In accentuating languages like English, loudness is mainly used for that purpose. There are other languages, like Japanese, where the duration of a syllable is most important and others, like Chinese, who use intonation even for distinguishing words.

The type of a language has an influence on which verse meters are common. Accentuating languages tend to have meters that repeat simple feet in a regular pattern, as explained above. Even the structure of verse feet has certain restrictions, making the appearance of two stressed (or three unstressed) syllables in succession rather unusual. As a rule, accentuating languages are therefore rather limited concerning the diversity of possible verse forms. Other languages show a much greater variety of common verse meters and are thus able to convey subtleties that cannot be expressed in English.

The antique Indo-European languages Greek, Latin, and Sanskrit are all quantitative, which means that they distinguish syllables by their quantity or duration. Thus a verse meter would be defined as a characteristic succession of long and short syllables in a line of verse. These languages tend to have a much greater variety of common meters than modern accentuating languages. For example, a website of the University of Heidelberg on Sanskrit language resources presently lists 1,352 different meters. It appears to be easier to arrange long and short syllables in an arbitrary succession without disturbing the flow of speech, than to arrange stressed and unstressed syllables. As speakers of an accentuating language, we cannot really appreciate the wealth of antique meters because we tend to change the antique pronunciation following an implicit rule according to which a long syllable becomes stressed and a short syllable unstressed. But this is rather difficult and wouldn't give the right impression for the original verse.

The enormous number of verse meters made possible by the quantitative character of the Sanskrit language led old Indian scholars to raise questions about the theoretically possible number of verse meters and their classification.