The Bakhshall Mamuscript

In 1881, a farmer of lndia found- an old-manuscript inside a tree trunk in the village called "Bakhshali" within fifty miles of the city of Pesahar. The farmer handed this unknown manuscript to the local government and it was exported to the Bodelien library in oxford, England for safe keeping. Since then a lot of research work has been done on this manuscript which was written on birch-bark and only thirtyfive leaves out of seventy, were in legible condition. Written in early Sharada script {a modified version of Sanskrit} this manuscript has been claimed as a valuable mathematical treatise and its author is yet to be known. The mathematical rules in this ancient manuscript were written in the form of verses-in "Anushtubh" meter where the sentence consists of thirtytwo syllables.

Recently, a Japanese scholar named Takao Hayashi from Brown University, USA has published his doctoral thesis entitled: "The Bakhshali Manuscript" {Groningen Oriental Studies-1995}. In this manuscript, the ~"Rule of Three" [traisika] was applied to solve the problems for the business community, travelers and goldsmiths. This "Rule of Three" was known to the European in the thirteen century as the famous "Golden Mean" from the book written by the Italian Mathematician Leonardo of Pisa, known as Fibonacci [filius bonacci]. Fibonacci obtained the necessary informations from the Arabic book on Hindu Mathematics. In the Bakhshali Manuscript the "Rule of Three" was applied to prove a theorem of transforming a rectangle into a series of squares [continued fraction]. The "Golden Mean" is the ratio of the larger to the smaller side of the rectangle and it is always close to a constant value of 1.618 ,provided the sides of the rectangle are proportional to two consecutive numbers such as 2, 3, 5, 8,13, 21 so on, these numbers are called Fibonacci numbers and they are generated by adding the two preceding numbers.

Most of the Bakhshali Manuscript contains problems and their final solutions. A simplified version of a problem can be stated as follows: A certain King asked his goldsmith to distribute eleven gold coins among his three daughters in the ratio of a half,a quarter and one sixth. The goldsmith added one more gold coin from his own pocket and out of total twelve gold coins,he took the ratio of one half which was six and gave it to the first daughter. The second daughter got a quarter of twelve which was three gold coins. Finally, the third daughter received one sixth of twelve which amounts to two gold coins. The goldsmith took back his own gold coin but satisfied everyone. The application ofthe"Rule of Three" can be traced in the 20th century "Quantum Physics"where any basic model can be built by using THREE operators, named as The Creation Operator[Brahma], The Number Operator [Vishnu] and The Annihilation Operator [Shiva] {JJ Sakurai, Advanced Quantum Mechanics, Addison- Wesley, 1967}. Similarly, the basic constituents of the particles such as proton, neutron, mesons etc., consists of the THREE fundamental particles "QUARK" {this name quark was taken from the name ofa character of the Irish writer James Joyces's "Finnegans Wake"}.

The ''Rule of Three" has been extended to other prime numbers such as 5,7,etc., and was applied not only in science but also in various other sectors of the society. In fact, the number 3 was considered to be a divine number in most of the ancient civilizations. After the excavation in India where the two cities Mohenjo-Daro and Harappa of the Indus-Sarasvati Civilization {~5000 B.C.} were discovered, the "TREFOIL" motif has been found on the robe of the statue ofa priest {J.M. Kenoyer, Ancient cities of the Indus Valley Civilization, Oxford, 1998}. The next prime number 5 is mentioned in the "Taittiriya" and "Brihadaranyaka" Upanishads as the most important number because the entire universe is FIVE-FOLD. In fact, in the Hindu ritualist worship, one needs minimum five items such as: flowers, sandelpaste, incense stick, light and food [naivedya] which represent the Five basic elements of the constituents of the universe: Space, Earth, Air, Fire and Water respectively.

In Physics" a quasi-periodic tiling with a crystallographically impossible Five-fold quasi-symmetry" has been observed {Roger Penrose, The Emperor's New Mind, 1989, Oxford}. In the Rig-Veda, the SEVEN sages are considered as the founder of the Vedic Aryans linages and they are represented as the SEVEN bright stars associated with northern celestial pole, known as " The Great Bears" {Ursa Major}. Most of the "Hindu Wedding" follows the Rig-Vedic framework of marriage where "The Seven Steps" {Saptapadi} is one of the most important part of this ritual. Thus, at the dawn of the civilization,the prime numbers were very auspicious even at present age it is very significant to understand the Universe yet, it remains as mysterious as before.

------ Source: Dr. Ramananda Chatterjee, Emeritus Professor of Physics, The University of Calgary \