Mathematics in Ancient and Medieval India c2.2

c2.2

Expression of very large numbers by means of indices to ten like that of Indian workers, and their use of fractions of various types were not accessible in the work of any contemporary nations. The concept of nine numerals and the decimal place-value with the introduction of zero is a significant Indian contribution to the development of Arithemetic. A systematic and straight-forward technique of fundamental mathematical operations and formulation of rule of three methods of calculation have been found to be of Indian origin. Reference has also been made to the comparatively more accurate results given by the Indian workers for the surd numbers. The geometrical knowledge of the ancient Indians have been derived from the instruments, units and the shape of the bricks used in altar constructions. Technical terms as found in the *Śulba-sūtras* and in later works are coined which interpret basic concepts of Indian nomenclature. A clear general statement of Pythagoras’ Theorem, calculation of the value of √2 by geometrical method and the classification of geometrical figures on the basis of angles and sides have been pointed out to be notable contribution to geometry by the Indians in the age of *Śulba-sūtras* ( 600 B. C. ). Brahmagupta’s theorem relating to the determination of diagonals of cyclic quadrilateral and his methods for the construction of a quadrilateral may also be regarded as remarkable achievements of Indian geometry. The growth of algebra in India has been discussed with particular reference to symbols of operation, equation, rule of false position, quadratic equation, indeterminate equations of first and second degree, progressive series, permutation and combination and Pascal’s triangle with binomial theorem. A comprehensive interpretation of *meru-prastāra* scheme ( 10th century A. D. ) due to Halāyudha ( commentator of Piṅgala's *Chandaḥ Sūtra*—200 B. C. ) as forming the basis of binomial expansion for any positive integral exponent in algebra has been given. The development of trigonometrical knowledge in ancient and medieval India has been discussed with particular reference to trigonometrical formulae, sine table, value of π and trigonometrical series. It has been shown that the Indians made a notable addition to the knowledge of trigonometry after Ptolemy relating to trigonometrical formulae which was systematised into a special branch of knowledge named *jyotpatti* ( trigonometry ). The work on trigonometrical series—π, sine, cosine and tan was first carried out by the Indians more than a century before Newton ( 1664 A. D. ) and Leibniz (1676 A.D.). The Indian concepts of infinitesimal calculus both differential and integral appeared in the work of Bhāskara II ( 1150 A. D. ). These are similar to those of Archimedes ( c. 225 B. C. ), and shows in some respects an improvement upon Archimedes' as to the infinitesimal character of the units chosen. Further development of this concept of infinitesimal ( integral ) calculus occurred in *Yukti-bhāṣā* in connection with the summation of infinite series ( π—series ) more than a century before Newton and Leibniz. The trend of Indo-Persian/Arabic literatures in the Medieval period in India has been discussed in a nut shell. The work closes with a general bibliography which serves as an important source in addition to original manuscript sources.