# Mathematics

**Mathematics** science of spatial and numerical relationships. The main divisions of pure mathematics include geometry, arithmetic, algebra, calculus, and trigonometry. Applied mathematics include statistics, numerical analysis, computing, mathematical theories of astronomy, electricity, optics, thermodynamics, and atomic studies.

Prehistoric human beings probably learned to count at least up to ten on their fingers. The Chinese, Hindus, Babylonians and Egyptians all devised methods of counting and measuring that were of practical importance in their everyday lives. The first theoretical mathematician is believed to be Thales of Miletus (580 BC) who is believed to have proposed the first theorems in plane geometry. His disciple Pythagoras established geometry as a recognised science among the Greek. The later School of Alexander Geometers (4th and 3rd centuries BC) included Euclid and Archimedes. The present decimal numbers are based on a Hindu-Arabic system that reached Europe about AD 100 from Arab mathematicians of the Middle East such as Khwarizmi. The basic development of mathematics in India (including Bengal) took place between 500 BC and 500 AD, marked as Buddhist and Jaina period.

* Mathematics in Buddhist and Jaina period* (500 BC-500 AD) The role of mathematics in the Buddhist and Jaina religions occupied a prominent place. Both these religious sects attached much importance to the study of mathematics. The Jainas considered the study of mathematics as an integral part of their religion. But no mathematical works of them since 500 BC to 500 AD have survived the ravages of time, although several mathematical concepts and materials are profusely scattered in their religious literature. Consequently, any discussion on mathematical materials should take into account of those that are found in the Suryaprajnapti (500 BC), Chandraprajnapti (500 BC), Jambudvipaprajnapti (500 BC), Sthananga-sutra (300 BC), Samavayanga-sutra (300 BC), Bhagavati-sutra (300 BC), Anuyogadvara-sutra (100 BC), Tattarthadhigama-sutra and its commentary (100 AD) by Umasvati, Sathkhandagama by Puspadanata and Bhutavali (200 AD), Trilokaprajnapti of Vativrsava (500 AD), Lalitavistara (100 BC) and Kaccayana's pali grammar of the same period.

The topics of mathematics, according to the Sthananga-sutra (sutra 747) are ten in numbers: parikarma (four fundamental operations), vyavahara (subjects of treatment), rajju (geometry), rashi (mensuration of solid bodies), kalasavarna (fractions), yavat-tavat (simple equation), varga (quadratic equation), ghana (cubic equation), varga-varga (biquadratic equation) and vikalpa (permutation and combination). However, the historians of mathematics differ in explaining some of the terms from the commentator, Abhayadeva Suri (1050 AD).

The Lalitavistara (100 BC) of the Buddhists speaks of a series of number-names based on centesimal scale right from koti, ayuta, niyuta ... to tallaksana=1053; Kaccayana's Pali grammar also furnishes us with the series of number-names, the last being asangkhyeya=10140. The Jainas, however, went further to the extent of number-names up to shirsaprahelika=(84000000)28, the second in the same timescale being purva=761011 years.

The above number-names and other positive evidences show that the decimal system of numeration with place-value and zero was known in Indian sub-continent long before the dawn of Christian era. Thus, the Anuyogadvara-sutra and Vyavahara-sutra refer to places of numeration as sthana and gananasthana respectively.

This fascination of naming large numbers, however, led the Jainas to an important conception of infinity, not crude, in fact, in the mathematical sense. Numbers are classified by the Jainas as Sangkhyeya (enumerable), Asangkhyeya (unenumerable) and Ananta (infinity), interestingly, like the Greeks, one (1) is not considered a number by the Jainas. Although the early Jainas admit that it is too difficult to reach the highest number amongst numerables, they were able to come to the concept of Alef-zero. Infinity, according to the Sthananga-sutra, is of five varieties: ekoto'nanta (infinite in one direction), dvidhananta (infinite in two directions), deshavistarananta (infinite in area), sarvavistarananta (infinite in entire space) and shashvatananta (infinite in eternity).

The Anuyogadvara-sutra (100 BC) and Sathkhanda-gama (200 AD) discuss the laws of indices. Bhanga or vikalpa (permutation and combination) was a favourite pursuit of the early Jainas, This branch of mathematics arises out of their various thoughts, viz in philosophy and religion. While commenting on the Sathkhanda-gama (200 AD) Virasena (816 AD) defined some terms as ardhaccheda, trakaccheda and caturthaccheda in which implied the concept of logarithm. In modern notation,

Ardhaccheda of x = logx, where the logarithm to the base 2; Trkaccheda of x = log3x, where the logarithm to the base 3; Caturthaccheda of x = log4x where the logarithm to the base 4.

That the Jainas were quite acquaint with some rules of logarithm is quite apparent as

Log (m/n) = 10gm - 10 gn

Log (m/n) = 10gm + 10gn

Log (Xx)x = Xx 10 gx

Considerable search has been made by the Jainas in mensuration as we come to the field of circle measurement. Tattarthadhigama-sutra of Umasvati (100 AD) describes the formulas for measuring the area, circumference and chord of circle and also the area of a segment of a circle less than a semi-circle.

The above information proves beyond doubt that during the period under consideration the study and research on mathematics in India were carried on mainly by the Jainas and in some areas of mathematics they were quite ahead of other culture areas. [Nandalal Maiti]

* Mathematical Concept in Bengal* The conceptions of mathematics in India are contained in vast literature she has produced since second millennium BC is well known. The Vedas (1500-800 BC), Vedanga-Jyotisa (800 BC), Shulba-sutras (800-500 BC), Suryaprajnapti (500 BC), Candraprajnapti (500 BC), Jambudvipaprajnapti (500 BC), Tattarthadhigamabhasya (100 AD), Anujogadvara sutra (100 BC), Bhagavati-sutra (300 BC), and so on, belonging to the pre-Christian era, contain mathematical conceptions and ideas, although in scattered form. That the decimal system of numeration with place-value and zero was discovered in India in or around 1st century is almost certain, although the symbol zero (0) for the first time appeared in 428 AD in Mankuwar stone inscription. India's contribution to the growth and development of mathematics in general is' immense, which can be found in the Bakshali Manuscript' (300 AD), Aryabhatiya (499 AD), Brahmasphutasiddhanta (628 AD), Patiganita and Trishatika (750 AD), Ganita-sara-sanggraha (850 AD), Ganitatilaka (1056 AD), Lilavati and Bijaganita (1150 AD), Tantrasanggraha (1500 AD), Karanapaddhati (1500 AD), and so on. The great mathematicians, Aryabhata (born in 476 AD), Brahmagupta (born in 598 AD), Sridhara (flourished in 750 AD), Mahavira (flourished in 850 AD), Bhaskara II (1150 AD) etc, had contributed much to the development of arithmetic, algebra, geometry, trigonometry and calculus. Their treatments of several types of equations, of indeterminate equations of first and second degree, of the theory of interpolations, of the theorem of cyclic quadrilateral etc, are highly valued in the worlds; even their anticipation of the concept of calculus and of Taylor series, Gregory-Leibnitz series etc be regarded as surprise to the world of mathematics.

In comparison with the classical mathematics developed in Indian sub-continent since antiquity through Aryabhata, Brahmagupta, Bhaskara II down to Madhava (1340-1425 AD), Nilkantha (1500 AD) and Putumana (1500 AD), the contribution of Bengal to this field is very little, practically, nil. But that some sort of mathematics had been prevalent in ancient Bengal since pre-Christian era is apparent from her history. The history tells us that Bengal had been under the rule of several kings and feudal rulers since 300 BC up to Laksmanasena, the last ruler of the Sena dynasty in the 12th century. That the kings and feudal rulers themselves, their children's, the Brahmins and scribes were devoid of mathematical learning would, indeed, be a wrong conclusion. Moreover, in the fifty and old Buddhist monasteries founded in Samatata, Radha, Varendra and Pundavardhana, chemistry, medical sciences and astronomy were taught there along with religious literature. Who can imagine that Shilabhadra and Dipankar Shrijnana were ignorant of mathematical learning, rudimentary or otherwiseFoodgrain It is said that Bhavadeva Bhatta, the powerful Prime Minister of Harisenavarman wrote a book on astronomy, which is, however, lost. Mathematics that was cultivated by the Bangalies in ancient time has left its traces only in some measures of lengths, weights etc. Several archeological evidences found out during excavations at mahasthangarh (300 BC) in Bogra district, Bangladesh, and at Berachanpa in South 24 Parganas, West Bengal and lots of cooper plate grants unquestionably prove the statement.

In Mahasthangarh, gandaka and kakanika are found to have been mentioned kakanika is indeed a copper coin, according to the Arthashastra of Kautilya. Gandaka or ganda and pana are, in fact, Austric words; pana and ganda mean in Santhali eighty (80) and four (4) respectively. In the copper plate grants of Vallalasena and Laksmanasena (1200 AD) are used such units as unmana, adhaka, drona, kaka, ganda etc. The Charyapada (1000-1200 AD) also mentions kaudi (= kada) and bodi (=budi).

The Srikrishnakirtana (1400-1500 AD), a poetical work, also mentions kaudi/koudi, pana etc. However, that the Bengalis cultivated practical mathematics meant for mass education is almost certain as is evident in many aryas (the rules of mathematics) composed in verses by Shubhangkar (1600-1700 AD) and many others. Unfortunately, no manuscripts or books written by Subhamkar, Anup Bhatta, Ratneshwar Bhatta, Shangkar Bhatta, Kshmananda, Dhulidanti, Dvija Ramdulal, Narasinggha, Dayaram and so on, have come down to us excepting in some aryas their names being mentioned. The aryas gained a wide popularity not only in Bengal but also in Assam where it is called Kayathali arya. A few of the Assamese authors are Gurudas, Raghupati, Jahunandan Chanda, Bakul Kayastha, Hriday' [Hrday] Kayastha, Subhangkar and others. The aryas are written in old Bengali with Prakrt, Apabhrangsha and Abahattha words; but due to their oral transmission, they have come down to us in revised forms. Thus, elementary mathematical education of Bengal in Pathashalas was carried on with the help of the aryas up to the end of the 18th century.

In the first half of the 19th century the first book which was published by calcutta school-book society was Robert May's Ankapustakam (1817), followed by John Harley's Ganitanka (1819) and Haladhar Sen's Anka-Pustaka (1839). May and Harley incorporated many aryas in their books. Prasanna Kumar Sarbadhikari made a breakthrough in traditional mathematics in his Patiganita (1855) and Bijganita in two parts (1859 and 1860). Several books on arithmetic, algebra, geometry and trigonometry were published before the end of the 19th century. Panchanan Ghosh published the Shubhangkari (1893) which gained so much popularity that its 80th edition was published in 1934. Sir Gurudas Banerjee's Saral Patiganit comprising arithmetic, algebra and geometry was published in 1913 and 1914. At the end of the 19th century Sir Ashutosh Mukherjee made a valuable contribution to modern mathematics (1880-1890).

Upto the first half of the 20th century (1947), Bangali mathematicians have made valuable contributions to geometry, theory of numbers, theory of functions and infinite series, differential equation, algebra, relativity, statistics and so on. A few names that can be mentioned here are Shyamadas Mukherjee, RC Bose, HN Datta, NB Mitra, NN Ghosh, SK Bhar, AB Datta, SC Dhar, pc mahalanabish, NR Sen, satyendra nath bose, quazi motahar husain and so on. [Nandalal Maiti]

* Mathematics in Bangladesh* The mathematical sciences have developed and progressed well since the emergence of Bangladesh as an independent and sovereign country. Imbued with the spirit of the war of liberation, Bangladeshi mathematicians founded bangladesh mathematical society in 1972, with the aim of promoting mathematical research and education at all levels. Since 1974 it has organised twelve national mathematical conferences (with international participation), a number of regional meetings and quite a few research workshops. It regularly publishes Ganit Parikrama, an educative college-level journal in Bangla, and a research journal GANIT: Journal of Bangladesh Mathematical Society in English.

In recent times curricula from primary to higher secondary levels have been revised quite substantially and new textbooks have been written. The new curriculum portrays mathematics as a living and useful subject; visualization and active participation by the pupils in the learning process are encouraged and emphasized. Some features of the new curriculum are worth mentioning: consistent use of decimal (metric) units, rendering arithmetic to almost a child's play, with the subsequent abolition of the time-honoured 'method of practice', restriction of pure arithmetic to the first eight year's of schooling; early introduction of geometric and algebraic ideas. On the whole, the curriculum at the higher secondary level has undergone the most significant changes since the introduction of elementary Calculus four decades ago. An innovation is the introduction of 'mathematics practical' as an essential component of the elective mathematics course at the secondary school certificate level and the higher secondary level.

Mathematics curricula at the universities are subject to periodic review and revision. Often special papers, especially at the Master's level, are introduced reflecting recent developments (eg Fuzzy Mathematics) or specialisation of faculty members (eg Lattice theory, Geometry of Numbers). The most recent curricula change provides for four-year integrated Honours course at the universities. The level and content of the present curriculum for this course compare favourably with those in advanced countries.

In recent years subjects having direct relevance to real-life problems, like Linear Programming, Operations Research have been introduced. In order to further emphasise this aspect and equip the students with tools for solving concrete problems with real-life data, Mathematics Practical has been introduced in the Honours curriculum. It provides for the use of computer for problem solving.

With the introduction of Honours course even in district and upazila level colleges in recent years, the number of students studying mathematics increased significantly during post- liberation period. This encouraged the writing of university (mostly undergraduate) level textbook (mostly in English, some in mixture of Bangla and English) by eminent mathematicians of the country. The textbook division of Bangla Academy has brought out a number of good quality Bangla books on various branches of mathematics.

During the nineteen fifties and sixties not much research work was carried out even in the universities. With the emergence of independent and sovereign Bangladesh the situation began to improve gradually. In the mid-seventies the research degree of MPhil was introduced which involves both coursework and a dissertation. Up to now more than a dozen PhD degrees in mathematics have been awarded by Dhaka and Rajshahi universities.

An important event in the development of mathematical sciences in Bangladesh in recent years has been the establishment in 1986 of the Research Centre for Mathematical and Physical Sciences (RCMPS) at the University of Chittagong. The Centre regularly organises national and international conferences, symposia and workshops over a very wide-ranging spectrum of subjects in the mathematical sciences and adjoining areas, from foundations of mathematics to chemical physics or bio-statistics or mathematical economics. The establishment of the Centre owes a great deal to the support extended to it by late Abdus Salam who shared the 1979 Nobel Prize for Physics with S Weinberg and S Glashow. The main thrust of the Centre is to provide opportunities for research leading to MPhil and PhD degrees.

Research papers by Bangladeshi mathematicians are regularly published in national and international journals. Besides GANIT: Journal of Bangladesh Mathematical Society the house journals published by the universities provide adequate publication opportunities to local researchers and authors. The Journal of Bangladesh Academy of Sciences, the Bangladesh Journal of Scientific Research and Bangladesh Journal of Science and Technology are, like GANIT whose authorship is open to all (local or foreign authors) and which publish papers in all scientific disciplines.

Though many Bangladeshi mathematicians have excellent foreign contacts, Bangladesh as a country is yet to enter the international mathematical forum as embodied in the International Mathematical Union (which is an organ of the International Council of Scientific Unions). The primary requirement is the existence of an active body of research mathematicians, having publications in international journals. Despite the fact that Bangladeshi mathematicians can boast of well over one hundred international publications over the past few years, Bangladesh has not yet applied for membership of International Mathematical Union. The International Mathematical Union celebrated the year 2000 as World Mathematical Year, with the principal objective of fostering the image of mathematics as a living and growing subject of ever greater significance.

Bangladeshi mathematicians are contributing their efforts for different national issues like economic planning, population planning, flood protection planning, computerisation and automation of management systems, etc. In the years to come, in addition to carrying out research work with increasing depth and significance, Bangladeshi mathematicians should strive to fulfil their obligation to the society as well. [Munibur Rahman Chowdhury]

**Bibliography** BB Datta, 'The Jaina School of Mathematics', Bulletin of Calcutta Mathematical Society, 21, 1929; PN Gosh, Shubhangkari, Ptrick Press, Calcutta, 1934; BB Datta and AN Singh, History of Hindu Mathematics, Parts-I and II, Lahore, 1935 and 1936; AN Singh, 'History of Mathematics from Jaina Sources', Jaina Antiquary, 15 (1949) and 16 (1950).