There are many myths surrounding the Fibonacci series. For one, the name of the person who introduced this series to the Western world is taken to be Leonardo Fibonacci, except that was not his name – he was actually Leonardo of Pisa. He called himself the son of Bonacci (filius Bonacci) – it may even be a clan name. However, because a historian, Guillaume Libri called him that in 1838, it was almost after 500 years that he came to be called Fibonacci. So much for the historically conscious West.
In those days, Pisa had rich trade with North African ports. There was a Pisan colony in Bugia in North Africa. And it is said that Leonardo’s father was a customs official there and hence Leonardo could see the trade transactions and the way the Arab and African traders kept their account books. They employed essentially a better and more efficient system. They used the Hindu numerals.
So, by 1202 CE Leonardo (who was later known by the name Fibonacci) wrote a book ‘Liber abbaci’ and it can said that it was this was book that ushered West into the modern era.
The famous Fibonacci series comes in this book in the form of a recreational mathematics question: Suppose we have a mixed pair (one male and the other female) of newborn rabbits. Each pair takes a month to become mature. Starting at the beginning of the following month, each adult pair produces a mixed pair every month. Assuming that the rabbits are immortal, find the number of pairs of rabbits we will have at the end of the year.
The answer, of course, as we all know, produces the famous Fibonacci series.
The series, however, was well known in India. It discovered by Virahanka between 600 and 800 CE; by Gopala before 1135 CE and was employed by Jain polymath Hemachandra around 1150 CE and also appears in a special case of a formula discovered by Narayana Pandit (1340-1400 CE). (Thomas Koshy, Pell and Pell–Lucas, Numbers with Applications, p.18)
What is interesting in all these is the way mathematical knowledge was taught– combined with concrete natural examples.
Kanakathikaram is a 15th century text of mathematics in Tamil. The book was written by one Kari Nayanar who was the son of a person called Buddhan and he lived in a village ‘Korukaiyur’ in the region called ‘Ponni Nadu’. The book says that it has given in Tamil verses the collective wisdom of pan-Indian mathematical pedagogy.
The specialty of traditional popular mathematical works in India is the way everyday phenomena are associated with mathematical problems so that students can relate the math to his or her own lived experience. Consider the following problem presented in ‘Kanakathikaram’:
A palm tree is of height 32 ‘muzham’. A chameleon on each day climbs 1 ‘chaan’ and slips down 4 ‘virarkadai’. Thus in how many days will it climb the tree? (1 muzham = 2 chaan = 12 virarkadai = 13/8 foot)
How can one calculate the number of fruitlets inside a Jack fruit without cutting it? Kanathikaram provides a mathematical solution: Count the number of small thorn like projection around the stalk of the fruit. Multiply it by 6 and divide the product by 5 and you will get the number of fruitlets inside. Whether the statement is empirically true or not, the point is students learned that their mathematical knowledge is something which manifests in nature around them.
Then there is the problem of a dog catching a porcupine:
A porcupine runs 30 kädams a day. A dog starts chasing it. Dog runs 1 kädam the 1st day, 2 kädams 2nd day, 3 kädams the 3rd day and so on. If every day the dog runs 1 kädam more than the previous day, then in how many days the dog would catch the porcupine? (kädam: an ancient measure of distance).
In working out the student naturally gets the idea of arithmetic progression. Now one can clearly visualize how such Indian mathematical riddles – one of which involving the Hemachandra-series should have reached the Pisan colony of Bugia in North Africa and then the rest became, as the cliché says, history.
Clearly Indian mathematical education was highly localized and yet well networked. The network of small community schools in which the education was more decentralized was also more egalitarian. So most probably each village school should have had mathematical pedagogy texts which while relevant to the local problems, were also related to a pan-Indian treatise. In a sustained way local problems and innovations should have contributed to the body of pan-Indic mathematical knowledge. By 16th century, mathematician Krishna Daivajna had come up with the idea of representing the positive and negative numbers along a straight line. Today mathematical text books call it the ‘number line’.
At some point we lost it all. When we today look at the classic ‘Mathematics can be fun’ by Yakov Perelman or ‘The Scientific American Book of Mathematical Puzzles & Diversions’ by Martin Gardner, one cannot but think of them as the updated versions of Kanakathikaram. And one cannot stop wondering with a bit of pain, why it does not happen anymore in India.
References on next page
Dr.K.Ramakalyani, Tamil Mathematical Works, (Paper-Presented at the Seminar on Science in Ancient India’, organized by the Physics Department of DGV College and Kuppuswami Sastri Research Institute, Chennai. Courtesy: Prof. Uthara Durairajan, HOD, Physics, DGV College
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