________________
(iv) Aryabhata (476 AD) gave the algorithm for finding cube roots, sum of squares and cubes of natural numbers, approximation of л by 3.1416, mesuration formulae and tables for sin 0.
(v) Brahmagupta (598 AD) solved the indeterminate equation Nx2 + 1 = y2 in integers, gave the sum of geometrical progression, determined the right angled triangles and cycle quadrilaterals with rational sides, gave interpolation formulae and the formulae for the area of a cyclic quadrilateral.
(vi) Mahāvīrcārya wrote his famous work, Gaṇitā-Sāra Samgraha, 850 AD. It contains problems on arithemetic, on quadratic equations, on the gereral formulae for cube root, on unit fractions, on construction of cyclic quadrilaterals of any given area and circumference for the ellipse.
(vii) Bhaskarācārya (Born 1114 AD) wrote siddhanta Śiromani at the age of thirty six, which was in the systematized mathematics known at that time. It gave formulae for surface area of a sphere and its volume, volume of frustums of a pyramid, problems on permutations and combinations, problems on quadratic equations and indeterminate equations. Bhaskara also gave some results which show that he could almost have discovered the calculus.
(viii) After Bhaskara, new mathematics was not searched except in Kerala, where a great deal of mathematical work was done. Ganita Kaumudi by Nārāyaṇa Pandita of the fourteenth century, and the work of Madhava, are pioneers. These were followed by Tantra Samgraha and Aryabhatiyam of Sankara Vāriyār, Karana Paddhati, Drgganita and Kriyākramakari, In these works, series for sin x, cosx, tan1x, etc., are given at least two hundreds years before they were found in Europe.
(ix) The result,
lim
1
74
1+2+...+ (n-1)", 1
P+1
DP+1
was also given by these authors. Karaṇapaddhati gives the value of correct to 10 places of decimals, and Sadaratnamālā gives it correct to 17 places. Nilakantha also clearly stated, in about 1500 AD, that the ratio of the circumference to the diameter can never be expresses as the ratio of two integers.
Hindu Mathematicians in ancient India were highly influenced by the Greeks at an early stake and by the Chinese at a later stage, but they also contributed a lot to the development and research of mathematics.
The nature of mathematics is two-fold, On the one side, it deals with quantitative relationships between sciences, etc. On the other, It postulates and seeks to carry these theorems to their logical conclusions, without any practical
अर्हत् वचन, 23 (3), 2011
