Culture is that complex whole which includes knowledge , belief, art, morals, law, custom, and any other capabilities and habits acquired by man as a member of the society. Here , we are going to present one such branch of culture of the people living in this valley of the river Brahmaputra.
The literal meaning of mathematics is ‘learning through meditation’. So, mathematics is nothing but some sort of philosophical exercise.
That a glorious past of such a school of philosophy (or philosophical exercise ) flourished on the bank of the mighty Brahmaputra was still in the dark until late Dandiram Dutta of Belsor(Nalbari-Assam) first drew attention of the Assamese people through his book’Kaitheli Anka Aru Koutuk(Kaitheli mathematics and fun) in some sense a research based survey.
Before the British came, the Kayasthas (the highest class of Sudras whose occupation is penmanship-(the writers’ class ) gave education to the people in their local language. These Kayastha teachers, during their teaching, taught (mainly) Arithmetic for setting accounts. In the regional language of Kamrupa, the Kayasthas are known as ‘Kaith’. The school, in ancient Assam, run by these ‘Kaith’ teachers was known as school of (old) ‘Kaitheli Education’. As mainly the people belonging to this caste viz., ‘Kaith’ took this occupation and were engaged in the study and culture of this ancient branch of
Mathematics, such problems or techniques wee henceforth known as Kaitheli mathematics. With great care and hard labor, Dandiram Dutta , gathered the Sanchipatia-puthis (the book written on the bark of Sanchi tree) where this mathematical; work could be found.
In this context it reminds us of ‘Lilavati’,one chapter of Siddhanta Siromoni by Bhaskara, an Indian mathematician of the 12th century. ‘Llavati’ was in memory of his daughter Lilavati. Her husband died a few days after her marriage at the age of only six years. It was heard among the people that of one knows ‘Lilavati’ , then he could tell accurately the number of leaves in a tree. Interestingly , in Kaitheki mathematics too, such types if
problems are in vogue. There are problems regarding survey or measurements of land. These have been mentioned as ‘the survey of the east-land’(Purbadeshia) The style found in Lilavati or Subhankari tallies with that in Kaitheli mathematics. It is known that in these schools of education various types of lessons like Ka-phala, Swa-phala, Sddhi-phala, Aakar-phala, Naam –khari-bhanga,etc. were taught. Moreover in Arithmetical calculation aath(eight) katha, dashanke(ten) vidyarthi, piyal(survey) paanchak were taught. Among these , ‘Lilavati’ and ‘Subhankari’ were main books on mathematics at that time.
Jyotish Churamoni, Kachi Churamoni, Rasida Thakur,Asangar, Bakul Kayastha, Kartik-Mayur –Kai were some of the well-known scholars.
Subhankar Kayastha created many problems. The famous Subhankari oral calculations were also mainly his works. It is known that he was an inhabitant of Chamata village of Dharampur mouza of old Kamrup district of Assam. There were two scholars with same name Churamoni, one being the descendant of Chandibor Bhuyan. Durgabor or Durgabori Kayastha was well-known for his ballads. There are some literal symbols used in these Kaitheli mathematics, viz.,
1.Sashi, Chandra, Nisha-pati,Mahi,2. Netra, B huj,Kar,Pakhaar,3 Ram, Bahini,4.Veda,Yuga,5. Ban,6. Rasha, Rihi,Bita,7. Samudra,Muni,8.Basu, Naga,9.Graha,10.Disha,Dikpala,11.Rudra,12.Aditya,13. Bhuban,14. Gagan, Akash.
The language of these ‘Arithmetic-poems’ is undoubtedly Assamese, of course, not that as followed by the poets of the Vaishnava era. This indicates the independence of Assamese language even in the colloquial form though some Arabic words are found here and there. Problems are presented on the background of Indian culture and tradition. Hence, that the cultural identity of this land was distinctly linked with the rest or was a part of the vast Aryavarta is noteworthy to mention. Indication of the lifestyle of the people, the wordings like flowers (pushpa), elephants(hasti) ,crops (sashay) , cows(gai) land(mati) ,nawab, pasha, various astrological terms also may help to explore various types of cultural synthesis that took place during that time. Rigorous study along this line may give some hint regarding the cause for this type of synthetic of culture .That arithmetic or mathematics is beyond any culture , language or religion may be considered as a sufficient citation so far as this assorted or cosmopolitan background of presentation of these poems is concerned.
Among these writings, astrological calculations are also available. One may expect to find some relation with the well-known activities of ancient Pragjyotishpur, which may be called as a place for Jyotish Charcha (astrological study).
Here are a few of such arithmetic verses for the benefit of the readers. Some problems among these verses remind us of problems of large numbers.Interested readers may consult for this purpose some famous books of modern days viz., One,Two,Three,…,Infinity, Lore of large numbers, to mention a few.
- Ek roti sindoore no janik aate/ no mon sindoore koto janik baate/Kohio kayastha sobo kitapoto chaai/Lkoteko je teeri bhoilaa/ diyoko boojai/
Taharo bhangoni kothaa soono mohashoy/chollish sere mon jaaniba nischoy/Sehi chollishoko no diya poori/Seroko boojiba taak koho nisto kori/Botrish odhik chaari
shoto/janioka/Sianobboi poori take poonoh roti boojioko/ no diyaa poori likhi konyaka laok/Ito kothaa ehimaane xonghoria thaon/Soono paase yeno bhoilaa taro kothaa kaon.
- Gobinde sebibe mone konyaagono goilaa/Poromo aanonde goiya nikoto chaapila/Stooti Stotra kori sobe Krishnoko bhojilaa/Stootito Sontosha hooiyaa
bostro eko dilaa/Sei bostro konyagone chaoupase d horilaa/ choya angool bostro sobeo lobhilaa/griho goiya konyaa gone baatiya loilonto/prosthe dooi dighe aatho hosto porilonto/Suniyo kayastha sobo sthiro kori mon/Koto konyaa koto bostro kohiyo yotone/
Explanation: Angoole haat janibaa nischoy /36 konyaa jaana hatoto hoyoa/Sorosho haatoko sei 36 ere poori/Kayastho boloya konyaa/boojo sighro kori/Panchosoto soysotwori konyaa boojiyoko/Konyaro promaan sei bostroto laoka.
Answer: 576 konyaa.
The third chapter of Dutta’s book is with the method of multiplication. The methods are written in verse. Nonuse of mathematical symbols and numerals is an important characteristic of these methods.These verses are related with usual business calculation, land measurements together with fun with numbers, some techniques of multiplication (amisra-pooran)
Such types of problems in verse form reveal the tradition of mathematics education in ancient Kamrupa. Of these problems, of-course, almost all are well-known. Among these, some are on the simple art of multiplication, characteristic of decimal system of numeration, some algebraic formulae, etc. these may be categorized as mathematical recreations.
Last but not the least, an important point we would like to mention regarding Kaitheli mathematics is its rich potentiality for research. The scope for development of some of these techniques and problems leading to higher mathematical research should not be over-looked.It is our great pleasure that Prof. Dilip Kumar Dutta (retrd. Professor Rhode-island university,USA)is doing at least some work to his credit.We now discuss in short what Prof.Dutta has presented regarding this aspect of Kaitheli mathematics.
We first note the following two problems:
- Ek giri –tini bhai/Gohalit naota gaai/Ek ek xer barhi yaay/gaai gakhir xomaane paaye.
- Meaning: one man has three brothers. They have nine cows. If one cow gives one xer (old measure of approximately one litre) then the second cow gives two litres, 3rd , 4th, ..9th give three, four,…,9 liters of milk respectively . Now the problem is to divide these nine cows among the three brothers so that each of them gets three cows and equal measure of milk.
- One person has 25 bags of rice. One bag is of 1 kg. one is of 2kg,one of 3kg,thus up-to 25kg. To divide these 25 bags of rice equally among five persons so that each one gets equal number of bags and also same measure of rice.
Such type of problems deal with any squared number. For example: with 1,2,3-3×3=9, every number with-1,2,3,4-4×4=16 Similarly-to 5×5=25 is taken
so that each integral numbers comes once. But Prof. Dutta observes that this need not be. He sees with square matrix that it works with any number less than any squared number too.
An n –square matrix means a matrix with n-number of rows and with n-number of columns. From each row of an n-square matrix if we take one number in such a way that we never take two numbers from the same column., then we get a diagonal of the n-square matrix.
Among these, of the n-square matrix, the elements with same number of rows and columns are known-main diagonal. Similarly, beginning from the first element in the lower most row to the last element of the top most row the elements form the opposite diagonal. Similarly we get the other diagonals as shown.
As Professor Dutta’s Arithmetic matrix is of the following characteristics:
In ant two rows the difference between elements in the same column is same.
For example in the following matrix it is an arithmetic matrix, for in this matrix the difference between and is 3 and the difference between and is 3.
According to his one result we get that, in an arithmetic matrix-the sum of all the elements in each diagonal is same. And conversely, if this happens to be true in a square matrix, then the resulting matrix must be an arithmetic matrix-as he defined. In the following matrix the sum of the elements in each diagonal is 15.Moreover,if the rows and the columns are interchanged in such cases then also it remains till arithmetic.
Now we come to our problems in discussion, as he explains.
In the above two problems, the main trick would be to make an arithmetic matrix.for the first one, we consider an 3×3 square matrix and for the second we take an 5×5 matrix. For the first case we consider the numbers from 1 to 9 and for the second one we take numbers from 1 to 25 , such as-
Since the sum of the numbers in a diagonal of an arithmetic matrix is same, it would be sufficient to choose accordingly some diagonals. And therefore for the first problem answer would be :
And the answer to the second problem is as follows:
Now the question is are these the only answers to the questions?
For the purpose we note the following: two diagonals are parallel if they contain no common element or they do not intersect each other. If the numbers in the arithmetic matrix are possible to classify into some parallel diagonals, then the parallel diagonals are called a parallel collection. It is needless to say that in above examples each parallel collection is an answer.
For example in the first example there are two parallel collections.so the other answers are:
On the other hand for the other problem we 24 parallel collections. And so answer is at least 24.
Some other example:
In a govt. farm there are 16 cows for sale.
The account of the cows as they give milk is as follows:
Number of cows: ek seri 1, dui seri-2, tini seri-3, chari seri-3, paanch seri-2, soy seri-2, xaat seri-1, aath seri-1 and n-seri-1.
Four customers came to buy 4 cows. Those four customers wants cows in such manner that-each one gets same amount of milk from the cows that are in possession. How to divide the cows? In how many ways the answer would be?
We need not take squared numbers like 32,52,..etc. Here, we avoid rigorous mathematical or logical reasoning. We just try to give in the following diagram how to get the answer. Mathematical or our innate symmetrical sense will lead us to what reasoning is working behind the solution.
Answer (to the problem-a): (1,5,9)., (2,6,7) , (3,4,8)
Answer (to the problem b): (1,7,13,19,25), (2,8,14,20,21,).(3,9,15,16,22) , (4,10,11,17,23), (5,6,12,18,24)
Prof. Dutta generalized these types of problems in an elegant way which is capable of dealing not only with numbers of the type 32, 42,52,.. etc. but with any number less than any squared numbers .He introduced a structure named arithmetic matrix, and used it for solving such types of problems in a more general setting.
Reversed subtraction process of Arithmetic was another interesting and popular subject for the ancient Indian mathematicians. And these types of problems have also been dealt in Kaitheli mathematics. Interestingly, this aspect of Arithmetic is very rare to find in western mathematics. This aspect of mathematical problems is also noteworthy to mention for whom it may be possible to explore many interesting phenomenon of decimal system of numeration and may be considered stimulating so far as mathematical taste is concerned.