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Meru Prastara – Triangular Array

Meru Prastara is a triangular array of numbers. Pingala’s Chandah Sutra (C 200 BCE) deals with it for finding the combination of ‘n’ syllables taken 1, 2, 3, … at a time. This is used to calculate various possible permutations and combinations, which was one of the favorite mathematical pursuits of Vedic Hindus as well as the Jainas.

In Vedic literature one finds various Vedic meters, gayatri, anustubh, brihati, tristubh, jagati, to mention a few, with different numbers of syllables. The varnasangita, the music of sound variations of the Vedic and post-Vedic composers, depended only on the variation of two sounds – guru (long) and laghu (short). So they were concerned with finding different possible types of meters from those of varying syllables by changing the long and short sounds within each syllable group.

Halayudha (10th century CE) in his commentary on Chandah Sutra of Pingala, elaborated upon the meru-prastara (pyramidal scheme) technique for finding the number of combinations of syllables taken 1, 2, 3 … n at a time. He says: “After drawing a square on the top, two squares are drawn below so that half of each is extended on either side. Below it three squares, and below it four squares are drawn and the process is repeated till the desired pyramid is attained. In the very first square, the symbol for one is written. Then in each of the two squares of the second line, the symbol for one is marked. Then in the third line figure one is marked on each of the two extreme squares. In the middle square, the sum of the figures in the two squares immediately above is written. In the fourth line one is marked in each of the two extreme squares. In each of the two middle squares, the sum of the figures in the two squares immediately above, i.e., three is marked. Thus the second line gives the expansion of combinations of (short and long sounds forming) one syllable; the third line the same for two syllables, the fourth line for three syllables and so on.”

Emerging out of the intricacies of the Vedic meters, these rules found interesting practical applications, for example, for calculating the possible number of six tastes, taking one, two, three, etc., at a time; or the possible number of philosophical categories combining fundamental categories; or the total number of perfumes that can be produced from 16 different substances; and so on. In all these problems, the correct results have been obtained. Apart from giving a quick method clearly envisaged the binomial theorem.

In the Jaina texts, Jambudvipaprajanpati, Bhagavati Sutra and Anuyogadvara Sutra, the possible number of combinations out of ‘n’ fundamental categories taken one, two, three at a time, the possible selections out of a number of males, females and eunuchs and various groups to be constructed out of different senses have been given correctly.

Silanka, a Jaina commentator (9th century CE), reproduces, from some ancient mathematical works, rules for permutations and combinations with his own examples, which match well with the modern concept of the factorial. The Meru Prastara appeared in China in 1303 CE and in Europe in 1527 CE and was described later by some European mathematicians and in a work by Pascal in 1665 CE. This is now known as Pascal’s triangle.

Source – 

  • Mathematics in Ancient and Medieval India (1979) A K Bag – Chaukhambha Orientalia Varanasi
  • The Mathematical Achievements Of The Jaina – In Studies in the History of Science in India (1929) B B Datta – Debiprasad Chattopadhyaya (1982) New Delhi Editorial Enterprises.
  • ‘Mathematcis’. In A Concise History Of Science in India (1971) S N Sen – Indian National Science Academy : New Delhi
  • Encyclopedia of Hinduism Volume VII page 148 – 149 – Rupa IHRF