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Mathematics comprising geometry (jyamiti), algebra (bija-ganita) and arithmetic (ganita or ankasastra) are direct descendants from Astronomy. The origion of geometry is traced to the architectural measurement of altars, which were required for the great Soma sacrifice. They were made in ten different shapes as enumerted in the Sulba-satras (200 B.C.) or supplementary portions of Kalpa sutra of Baudhayana and Apastamba to the Taittiriya Samhita. These altars refer to the construction of squares and triangles; the relation of the diagonal to the sides; the equivalence of rectangles and squares; and the construction of equivalent squares and circles. Aryabhatta was the first to insert a definitely mathematical section (ganita) in his astronomy. 'He deals in it with evolution and involution, area and volumes, progression, algebraic identities, and indeterminate For instance Panahara, Apoklima, Hibuka, trikona jimitra, Meshurana, signs of zodiac including kriya, tavuri, jituma, leya, pathona juka, kampya, tankshika, anokero, Hridroga and Itthya, etc.

equations of the first degree (ax + by = c). It defines that 'the product of three equal numbers is a cube and it also has twelve edges. His notation is expressed in consonants, viz. K and M for 1 to 25, Y to H for 30 to 100, vowels denoting multiplication by powers of 100, A being 100 and B 1000. Brahmagupta's work covers 'the ordinary arithmetical operations, square and cube rules, rule of three, interest, progressions, geometry, including treatment of the rational right-angled triangle, and the elements of the circle, elementary mensuration of solids, shadow, problems, negative and positive quantities, cipher, surds, simple algebraic identities, indeterminate equations of the first and second degrees (in considerable detail), and simple equations of the first and second degrees, and cyclic quadrilaterals being specially treated.' The Ganita-sara-sangraha (9 th century) of Mahaviracharya gives many examples of solutions of indeterminates but not the cyclic method of Brahmagupta ; introduces geometrical progressions, and alone deals with ellipses, but has no formal algebra. The Trisati of Sudhana (born 991) deals, in addition, with quadratic equations. The Bija-ganita of Bhaskaracharya, which agrees with Brahmagupta, contains the fullest and most systematic account of algebra, and his Lilavati includes combinations. The Bakhshali Ms. of the 3 rd or 4 th century also refers to Hindu mathematics. Professor Keith does not believe in the Greek influence on Indian Mathematics. "The facts are that, as regards indeterminates equations, the Greeks by the 4 th century had achieved rational solutions, not necessarily integral, of the the equations of the first and second degree and of some cases of the third degree. The Indian records go distinctly beyond this. Brahmagupta shows a complete grasp of the integral solution ax byc, and indicates the method of composition of the solution of on " = q. Bhaskaracharya adds the cyclic method. The combination of these two methods gives integral solutions, the finest thing achieved in the theory of these numbers. "To find an ultimate Greek origine for these discoveriese," concludes Professor Keith, "seems due rather to a parti pris than to justice." - 2 -

¹ PRACTICAL SCIENCES 265 In regard to Geometry both the Indian and Grecian mathematics shows from 300 A.D. 'an absence of definitions, and does not deal with angles, nor mentions parallels, nor gives a theory of proportion, while the traditional inaccuracies are common'. The independ- nce and originality of Indian mathematics have been defended on the score that the love of dealing with large numbers and making calculations is recorded early for India'. The abacus inverted in India and the nun bers of the west were borrowed from India, words for numbers are used in the unique system of Aryabhatta. The figures of the Brahm or Kharoshthi notation in Asoka Inscriptions have not place value which is actually found in Inscriptions from the ninth century onwards. But the Indian figures were figures were known in Syria in 662 A. D. "The probability still remains that India did render a great service in this regard and in any case excelled Greece" concludes Professor Keith. India has also inspired the Arabic mathematics. The Algebra of Md.ibl Musa (782) bears the Indian influence. Arabian science from 771 "borrowed freely from Indian astronomy, translating and adapting both Aryabhatta and Brahmagupta". Coincidences with the Chinese mathematics are numerous and interesting. The so-colled Chinese invention of the system of Nakshatras found in early Indian Astronomy is undeniable. Indian influence on China is "proved sufficiently by the history of Chinese Buddhism and the discoveries in Central Asia". The original contributions of the Hindus in the practice of arithmetic, algebra and geometry may be briefly illustrated. They discovered the cardinal numbers 1 to 9, and also the zero (bindu). They knew the eight-fold system of aldition (yoga), substraction (viyoga), multiplication (purana), square (varga), cube ghana),.square-root (vargamula), ard cube-root (ghana-mula). They discovered the modern method of division and the rule of three. They knew the fraction and its addition and substraction by the method 1 While the western system of counting does not go beyona some six figures (billion), the Indian system counts up to eighteen or nineteen figures ending sigara. 34

of L. C. C., called Niruddha in the Ganita-sara-samgraha of Mahavira (9 th century). Pingala (second century B. C.) used in his Chhanda-sutra the method of permutation and combination (chhandaganita). Arya Bhatta refers to this and also to arithmetical and geometrical progression. In his Lilavati Bhaskarachirya has demonstrated that when a figure is divided by zero the result is infinite number. Bija ganita is the title of the two chapters of the Siddhanta-siromani of Bhaskaracharya (1150). In English it is called Algebra because it was borrowed from the Aljeb-oyal-mokabela of Md. Musa-al-Khoya-rejmi (825). But the Arabs had learnt it from the Hindus. The Hindus called this called this science both Bija-ganita and Avyakta-ganita. They discovered the positive (dhana) and negative (rina) numbers. Brahmagupta (628) discovered equation (sam karana). Its four varieties were in use: they are known as simple (ekavar- na), simultaneous (aneka-varna), quadratic (madhyamaharana) and Bhavita or equation involving products of two unknown quantities. Kuttama first solved the indeterminate equation of the first degree (eka-varnasamikarana). Aryabhatta, Brahmagupta, Sridhara, Padmanibha, and Bhaskaracharya solved such equations of algebra as could be done in Europe as late as the 17 th r 18 th centuries. In Jyamiti or geometry Baudhayana (second century B. C. ) actually solved the theorem long before it was associated with the name of Grecian Pythagorus, viz, the square on the hypotenuse of a right angled triangle is equal to the sum of squares on the other two sides. He also proved the theorem that the square on the diagonal of a rectangle is twice the area of the rectangle. The Sulva sutras also explain how to draw a square equal to the area of a triangle, and a circle equal in area of a square. The Surya-siddhanta (5 th century) found out the area of a triangle from its sides, which in Europe was discovered in the 16 th century by Clouvius. Brahmagupta and Bhaskaracharya worked out the area of a quadrangle from its sides. Baudhayana and Apastamba worked out the proportion between the diagonal and sides of a square (1:1.42156) which corresponds to the fifth decimal of modern finding (1/2 = 1.41423...).

Hindus In Trikonamiti or trigonometry, the discovered jya (sine), koti-jya (co-sine), utkrama-jya (versed sine), of which there are tables as in Suryasiddhanta (5 th century) which was discovered in Europe in the 16 th century by Briggs. Bhaskaracharya in the Lilavati explained the method to find out the length of the sides of equilateral and equiangular triangle, quadrangle, pentagon, hexagon, septagon, octagon and nine-sided figure drawn within a circle in comparison with its diameter. These exactly correspond to the modern formula.