A History of Hindu Chemistry from the Earliest Times to the Middle of the Sixteenth Century A.D. v2.c34.257

Language: English

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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's to the conception of a rudimentary process of Integration. Bhāskara in computing the "instantaneous motion" ( तात्कालिको 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 differentia of the planet's longitude, and Bápudeva S'ástrí claims that both the conception of the instantaneous motion and the method of determining it plainly show that Bhāskara was acquainted with the principle of the Differential Calculus. On the data before him, Mr.

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Spottiswoode remarks that Bápudeva S'ástrí "overstates the case." Bhāskara "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 Bhāskara (circa 1150 A.D.—born 1114 A.D.) Mr. Spottiswoode's second objection that Bhāskara 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 Calculuses of Fluxions and Infinitesimals. For the rest, Bhāskara introduces his computation expressly as a "correction" of Brahmagupta's rough simplification.

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The first objection (viz., that Bhāskara makes no allusion to the infinitesimal magnitude of the intervals of space and time employed) would be more to the point, if it were well founded. But it is not, and Mr. Spottiswoode's error was due to the insufficiency of the data supplied to him. As a matter of fact, even Bhāskara's unit, the Truti of time (or Paramānu), is exceedingly small as the very name implies, being about one thirty-four-thousandth of a second of time. And in the passage in which Bhāskara describes the process, he distinguishes between Sthúla-gati and Súkshma-gati (velocity roughly measured, and measured accurately i.e., by reference to indefinitely small quantities, for Súkshma, as we have seen, has always a reference to the Anu, the indefinitely small). Indeed he expressly mentions that the Sthúla-gati takes only Sthúla-kála (finite time) into consideration, and that the determination of the Tátkálikí Gati (Súkshma-gati) must have reference to the moment (प्रतिक्षणम्), which is an indefinitely small quantity of time being of course smaller than his unit, the Truti. (Cf. इयं किल स्थूला गतिः अथ सूक्ष्मा तात्कालिकी कथ्यते.......यदा आसन्नस्थितस्तदा तात्कालिक्या गत्या तिथिसाधनं कर्तुं पुज्यते। तया समीपचालनञ्च। यदा तु दूरतरस्थितस्तः दूरचालनं वा चन्द्रस्य तदा आदाय स्थूलया कर्तुं युज्यते स्थूलकालत्वात्।

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यतश्चन्द्रगतिः महत्त्वान् प्रतिक्षणं समा न भवति अतस्तदर्थं अयं विशेषोऽभिहितः;— nothing can be clearer than this conception of ‘momentary’ motion.—Bháskara, Siddhánta-S’iromani, Ganitádhyáya, Gatisphutiprakarana: cf. also प्रतिक्षणं सा न समा महत्त्वतः, ibid., cf. also Goládhyáya, Tátkálikí-karana-vásaná-prakarana, where Bháskara points out that the mode of computing adopted by the A’cháryya (Brahmagupta) is a rough simplification. The computation of relative motion and the idea of resolved components of motion were of course familiar to the astronomers.—(Cf. एवं सत्याचार्येण लाघवाथं इष्टघटीसम्बन्धिन्यो गतिकला अर्क प्रक्षिप्ताः। ibid.) I may add *en passant* that Bhāskara's formula for the computation of a table of Sines also implies his use of the principle of the Differential Calculus.