Hindu Achievements in Exact Science: A Study in the History of Scientific Development c12: VI. Differential Calculus
Hindu Achievements in Exact Science: A Study in the History of Scientific Development c12
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VI. Differential Calculus
c12.2
BHASKARACHARYA anticipated Newton (1642–1727) by over five hundred years (1) in the discovery of the principles of differential calculus and (2) in its application to astronomical problems and computations.
c12.3
According to Spottiswoode, the formula established by Bhaskara and "the method of establishing it bear a strong analogy to the corresponding process in modern mathematical astronomy,"⁵⁹ viz., the determination of the differential of the planet's magnitude.
c12.4
According to Bapudeva Shastri,⁵⁹ Bhaskara's conception of instantaneous motion and the method of determining it indicate that he was acquainted with the principle of differential calculus.
c12.5
According to Seal, Bhaskara's claim is indeed far stronger than Archimedes' to the conception of a rudimentary process of integration.
c12.6
"Bhaskara, in computing the instantaneous motion 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, i.e., 1/3375th part of a second, though it may be infinitely less).”
c12.7
This process is not only “analogous to, but virtually identical with, that of the differential calculus.” As Spottiswoode remarks, mathematicians in Europe will be surprised to hear of the existence of such a process in the age of Bhaskara (twelfth century).
c12.8
The claim for Bhaskara is, however, limited to the historically imperfect form of the calculus. Bhaskara does not specifically state that the method of the calculus is only approximative. But it must be remembered that the conception of limit and the computation of errors came late in the history of the calculi of fluxions and infinitesimals. For the rest, Bhaskara introduces his computation expressly as a “correction” of Brahmagupta’s rough simplification.⁵⁹
c12.9
Further, Bhaskara’s formula for the computation of a table of sines also implies his use of the principle of differential calculus.