The Positive Sciences of the Ancient Hindus c4.s2.49

c4.s2.49

This makes 1 Kshana of the Nyáya-Vaiseshika equal to 2/45 of a second. The Nyáya assumes that the unit of physical change (or the time occupied by any single antecedent step in a causal series before the succeeding step is ushered in) is equal to a Kshana (or 2/45 of a second). The astronomers were familiar with far smaller measures of time. The astronomical Truti of time measures about the thirty-four-thousandth part of a second. This is of special value in determining the exact character of Bháskara's claim to be regarded as the precursor of Newton in the discovery of the principle of the Differential Calculus, as well as in its application to astronomical problems and computations. This claim, as I proceed to show, is absolutely established; it is indeed far stronger than Archimedes' to the conception of a rudimentary process of Integration. Bháskara, in computing the "instantaneous motion" (तात्कालिकी गति) of a planet, compares its *successive* positions, and regards its motion as constant during the interval (which of course cannot be greater than a Truti of time, though it may be indefinitely less). This Tátkálika motion is no other than the differential of the planet's longitude, and Bápudeva Śástrí claims that both the conception of the instantaneous motion and *the method of determining it* plainly show that Bháskara was acquainted with the principle of the Differential Calculus. On the data before him, Mr. Spottiswoode remarks that Bápudeva Śástrí "overstates the case." Bháskara "makes no allusion to one of the most essential features of the Differential Calculus, viz. the infinitesimal magnitude of the intervals of time and space therein employed. Nor indeed is anything specifically said about the fact that the method is an approximate one." "With all these reservations," Mr. Spottiswoode continues, "it must be admitted that the formula he establishes and the method of establishing it bear a strong analogy to the corresponding process in modern mathematical astronomy" (viz., the determination of the differential of the planet's longitude—by no means the first step in transcendental analysis or in its application to astronomy). And Mr. Spottiswoode concludes by stating that mathematicians in Europe will be surprised to learn of the existence of such a process in the age of Bháskara (*circa* 1150 A.D.—born 1114 A.D.). Mr. Spottiswoode's second objection, that Bháskara does not specifically state that the method of the Calculus is only approximative, cannot be taken seriously. The conception of limit and the computation of errors came late in the history of the Calculi of Fluxions and Infinitesimals. For the rest, Bháskara introduces his computation expressly as a "correction" of Brahmaand one Pala as much as four *Suvarnas* or Tolás. A Pala of gold therefore weighs 320 *Krishnalas* (Manu, Chap. VIII. ; Vishnu, Chap. IV. ; and Yájñavalkya, Chap. I.).