**Use of Mathematics in Business in the ancient and mediaeval Bangladesh**

**Dr. Ramit Azad**

** Excavations at Mainamoti, Mohasthangar, Paharpur, Naogaon and other sites of Bengal civilization have uncovered evidence of the use of practical mathematics in Bangladesh. The people of Bangladesh of that period manufactured bricks whose dimensions were in a correct proportion. In these constructions they used regular geometrical shapes included hexahedra, cones, cylinders, etc. thereby demonstrating knowledge of basic geometry. The “Somapuri Vihara” at Paharpur site is said to have been the biggest Buddhist monastery in south of the Himalayas. The site consists of a large quadrangle of some 170 monastic cells set in a high wall/earthwork and looking inward to a 3 level Stupa now largely ruined but originally containing, presumably, Buddha statues in shrines set deep into each face of the structure. There are also various other structures within the quadrangle. A visit is likely to consist of a walk around the outer wall and then round each of the 3 levels of the stupa whilst taking in some of the other structures. Around the Stupa itself we can see examples of the terracotta “tiles” which decorated the walls of each level. These types of decorations are also found at “Salban Vihara” at Mainamoti. **

** From ancient to medieval period important contributions to mathematics were made by Bangladeshi mathematicians. Unfortunately original works of many of our mathematicians are lost. It is because of the almost 200 years’ repression of the British rule. However history keeps some of the names immortal. One of them is Sridhara **

** Sridhara is now believed to have lived in Bangladesh in the ninth and tenth centuries. However, there has been much dispute over his date and in different works the dates of the life of Sridhara have been placed from the seventh century to the eleventh century. The best present estimate is that he wrote around 900 AD, a date which is deduced from seeing which other pieces of mathematics he was familiar with and also seeing which later mathematicians were familiar with his work.**

** Sridhara is known as the author of two mathematical treatises, namely the Trisatika (sometimes called the Patiganitasara ) and the Patiganita. However at least three other works have been attributed to him, namely the Bijaganita, Navasati, and Brhatpati. Information about these books was given the works of Bhaskara II (writing around 1150), Makkibhatta (writing in 1377), and Raghavabhatta (writing in 1493). We give details below of Sridhara’s rule for solving quadratic equations as given by Bhaskara II.**

** There is another mathematical treatise Ganitapancavimsi which some historians believe was written by Sridhara. Hayashi in, however, argues that Sridhara is unlikely to have been the author of this work in its present form. **

** The Patiganita is written in verse form. The book begins by giving tables of monetary and metrological units. Following this algorithms are given for carrying out the elementary arithmetical operations, squaring, cubing, and square and cube root extraction, carried out with natural numbers. Through the whole book Sridhara gives methods to solve problems in terse rules in verse form which was the typical style of Indian sub-continental texts at this time. All the algorithms to carry out arithmetical operations are presented in this way and no proofs are given. Indeed there is no suggestion that Sridhara realised that proofs are in any way necessary. Often after stating a rule Sridhara gives one or more numerical examples, but he does not give solutions to these example nor does he even give answers in this work. In his book Pati Ganita Shridhara gave a good rule for finding the volume of a sphere. He dealt with various operations like elementary operations, extracting square and cube roots, fractions, eight rules given for operations involving zero, methods of summation of different arithmetic and geometric series.**

** After giving the rules for computing with natural numbers, Sridhara gives rules for operating with rational fractions. He gives a wide variety of applications including problems involving ratios, barter, simple interest, mixtures, purchase and sale, rates of travel, wages, and filling of cisterns. Some of the examples are decidedly non-trivial and one has to consider this as a really advanced work. Other topics covered by the author include the rule for calculating the number of combinations of n things taken m at a time. There are sections of the book devoted to arithmetic and geometric progressions, including progressions with a fractional numbers of terms, and formulae for the sum of certain finite series are given. **

** The book ends by giving rules, some of which are only approximate, for the areas of a some plane polygons. In fact the text breaks off at this point but it certainly was not the end of the book which is missing in the only copy of the work which has survived. We do know something of the missing part, however, for the Patiganitasara is a summary of the Patiganita including the missing portion. **

** In Shukla examines Sridhara’s method for finding rational solutions of Nx2 ± 1 = y2, 1 – Nx2 = y2, Nx2 ± C = y2, and C – Nx2 = y2 which Sridhara gives in the Patiganita. Shukla states that the rules given there are different from those given by other Hindu mathematicians. **

** Sridhara wrote on practical applications on algebra.**

** Babylonian mathematicians, as early as 2000 BC could solve a pair of simultaneous equations of the form **

** x + y = p, xy = q **

** Which are equivalent to the equation**

** X2 + q = px**

** Sridhara was one of the first mathematicians to give a rule to solve a quadratic equation. Unfortunately, as we indicated above, the original is lost and we have to rely on a quotation of Sridhara’s rule from Bhaskara II:- **

** Multiply both sides of the equation by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the coefficient of the unknown; then take the square root. **

** To see what this means take **

** ax2 + bx = c. **

** Multiply both sides by 4a to get **

** 4a2x2 + 4abx = 4ac **

** then add b2 to both sides to get **

** 4a2x2 + 4abx + b2= 4ac + b2 **

** and, taking the square root **

** 2ax + b = √(4ac + b2). **

** There is no suggestion that Sridhara took two values when he took the square root.**

** Usually it is told that there is no original work on mathematics in Indian sub-continent in the 2nd millennium. But basically in that period the development of mathematics in this region occurred to some other directions namely computational mathematics and mathematical logic. Some Bengali mathematicians worked on mathematical logic.**

** Bengali mathematician Raghunatha Siromoni (1477-1547) was a philosopher and logician He was born at Navadvipa in present day Nadia district of West Bengal. He was the grandson of Śulapāṇi (c. 14th century CE), a noted writer on Smṛti from his mother’s side. He brought the new school of Nyaya, Navya Nyāya, representing the final development of Indian formal logic, to its zenith of analytic power.**

** Raghunatha’s analysis of relations revealed the true nature of number, inseparable from the abstraction of natural phenomena, and his studies of metaphysics dealt with the negation or nonexistence of a complex reality. His most famous work in logic was the Tattvacintāmaṇidīdhiti, a commentary on the Tattvacintāmaṇi of Gangeśa Upādhyāya, founder of the Navya Nyāya school.**

** Mathuranath Tarkavagish (c 1550) scholar of navya nyaya (new logic), was born at Navadwip, son of Sriram Tarkalankar, also a famous logician. Mathuranath studied logic under Rambhadra Sarvabhauma. Jagadish Tarkalankar, a renowned logician of the time, was his classmate and Harihar Tarkalankar, another renowned logician, was his pupil.**

** Mathuranath’s commentaries on new logic expanded the study of logic in Bengal. By virtue of his extraordinary wisdom and writing skill he won a distinguished place in learned society. Mathuri, his commentary on Chintamani, was very famous and regarded as an essential text for the study of logic. It is held in high esteem all over India. He also wrote commentaries and explanatory notes on a number of books by Raghunath Shiromoni, Udayan and Bardhamanopadhyay, among them Anumanadidhitimathuri, Gunadidhitimathuri, Dravyakiranavalitika, Dravyaprakashatika, and Gunaprakashavivrti. One of his original books was named Siddhantarahasya. **

** Jagadish Tarkalankar Nyaya philosopher and Sanskrit scholar from Navadwip in the 16th century. Jagadish Tarkalankar’s ancestors were originally from Sylhet, Bangladesh. His father, Jadavchandra Vidyavagish, was a nyaya scholar at Navadwip and his great-grandfather, Sanatan Mishra, was the father-in-law of Sri Chaitanya. Jagadish was taught nyaya scriptures at Bhavananda’s chatuspathi (religious school), where he became well-versed in nyaya philosophy and was awarded the title of ‘Tarklankar’.**

** Jagadish Tarkalankar was a college teacher. Mayukh, Jagadish’s annotation of Raghunath Shiromani’s Tattvachintamanididhiti is a four volume discourse: Pratyaksamayukh, Anumanmayukh, Upamanmukh and Shabdamayukh. He also wrote Anumandidhititika, Pratyaksadidhititika and Lilavatididhititika, annotations on Shiromani’s didhiti.**

** Jagadish’s Shabdashaktiprakashika was once taught as a textbook at all chatuspathi in Bengal. Two of his other books are Tarkamrta and Nyayadarshan. He was awarded the title of ‘Jagadguru’ for his scholarship.**

** Few words on business and mathematics:**

** Though the mathematization of economics began in earnest in the 19th century. Most of the economic analysis of the time was what would later be called classical economics. Subjects were discussed and dispensed with through algebraic means, but calculus was not used. One of the earliest use of mathematics was in trading. **

** Algebra is widely used in business. Since Sridhara was the first mathematician who made practical application of algebra, so it can be guessed that the use of algebra in business started then.**

** Muslin was bought by the pharaohs of Egypt. Weaving muslin cloth needed a good knowledge of measurement, where use of mathematics is vital.**

** First century B.C. to first century A.D. Kushan and Gupta era the trade became more developed. Use of currency spread over. The money-goods-money relation became a practice. The use of mathematics became important in this case. **

** The merchant fleet floated in the Bay of Bengal. They had their business in Sindh, Koromandal and Bengal. Trade items were silk, precious stones, spices, metal goods, clothes etc. The Bengalese had their sea trade with Ceylon, Malabar, Koromandal, Pegu, etc. during Mughal period. On the land way they had the business from Persia to Volga river. During the reign of Emperor Akbar there was a professional group called kusid who used to give money on interest. Mathematics played a role in calculation of this type of business. The Emperor Akbar also introduced ‘common system of measurement’ and ‘currency unit’.**

** Conclusion**

** So we see that the use of mathematics in business and trade was wide in ancient and medieval Bengal. The Bengali mathematicians made enormous contribution in world mathematics as well.**

** References **

** 1. D Pingree, Biography in Dictionary of Scientific Biography (New York 1970-1990). **

** Books:**

** 2. G G Joseph, The crest of the peacock (London, 1991). **

** Articles:**

** 3. K Shankar Shukla, The Patiganita of Sridharacarya (Lucknow, 1959). **

** 4. B Datta, On the relation of Mahavira to Sridhara, Isis 17 (1932), 25-33. **

** 5. Ganitanand, On the date of Sridhara, Ganita Bharati 9 (1-4) (1987), 54-56. **

** 6. T Hayashi, Sridhara’s authorship of the mathematical treatise Ganitapancavimsi, Historia Sci. (2) 4 (3) (1995), 233-250. **

** 7. K Shankar Shukla, On Sridhara’s rational solution of Nx^2+1=y^2, Ganita 1 (1950), 1-12. **

** 8. A I Volodarskii, Mathematical treatise Patiganita by Sridhara (Russian), in 1966 Phys. Math. Sci. in the East (Russian) ‘Nauka’ (Moscow, 1966), 141-159 **

** 9. A I Volodarskii, Notes on the treatise Patiganita by Sridhara (Russian), in 1966 Phys. Math. Sci. in the East (Russian) ‘Nauka’ (Moscow, 1966), 182-246. **

** 10. A I Volodarskii, Outlines of the history of medieval Indian Mathematics (Russian), ‘Nauka’ (Moscow, 2009). **

** 11. Bongard Levin, Antonova, Kotovski, History of India, Progress Publisher, Moscow,1982**