# Matra vrttas and Fibonacci series

The sequence 1,1,2.3.5.8,13……… in which nth term is given by U=U+ U is known as Fibonacci series .Leonardo of Pisa (1170-1250) was known as Fibonacci. But the sequence was well known in India before Leonardo’s time.
The Indian experts on metrical sciences used this sequence in work on metric.The basic unit in Sanskrit prosody are a letter having a single *matra* (*mora* or a syllabic instant) called*laghu*(light) and that having two morae called *guru*(heavy).And their role in metric is the same as that of 1 and 2 in combinatorics.
*Matra-vrttas* are meters in which numbers of *morae* remains constant and the number of letters is arbitrary.
The numbers of variations of meters having 1,2,3,4,5,6…………….morae are 1,2,3,5,8,13……………….respectively and these are the Fibonacci numbers.
**Acarya Pingala** is the first authority on the metrical sciences in India. **Pingala’s** rule * misrau ca* (ie; and the two mixed) is also meant for the expansion of

*matra vrttas*and matra vrttas should be expressed by combining the expansions of two earlier meters with a

*guru*and

*laghu*respectively.

**Acarya Virahanka**(between AD 600-800) given direction for knowing the number (of variations) of the next matra(vrtta).

**Gopala**(prior to AD 1135) is the first author on metric who specifically mentions the numbers of variations of

*(known as Fibonacci numbers) while commenting on this rule of Acharya Virahanka. Acarya Hemachandra(AD 1150) ,one of the greatest Jain writers is also an authority on metric,who specifically mentions the number of variations of*

**matra vrttas***matra vrttas*in his

*. Acarya Hemachandra’s rule may be translated as follows: “ Sum of last and the last but one numbers(of variations) is that of the*

**Chandonusasana***matra vrtta*coming next.” Hemachandra concludes his statement “1,2,3,5,8,13,21,34 and in this way, afterwards”.

*(GK) written by Narayana Pandita in AD 1356 was the first mathematical work in which the idea of the so called Fibonacci number were developed further Chapter 13 of GK defines*

**Ganitha kaumudi***(additive sequence) .The rule may be translated as “First Keeping unity twice ,write their sum ahead, write ahead of that ,the sum of numbers from the reverse order(and in) place equal to the greatest digit, write ahead the sum of those (in available places).Numbers at places (equal to) one more than ‘the sum of digits’ happen to be*

**Samasika Pankti***The relation between the so called Fibonacci numbers and the binomial coefficients as contained in the*

**Samasika Pankti***prakrta pingala*was established with the help of the

*matra meru*(figure of numbers).

**Narayana Pandita**established a relation between his

*samasika pankit*and the multinomial coefficients