The ``error'' in the Indian ``Taylor series approximation'' to the sine (Q1599820)

In 1974, R. C. Gupta published a Sanskrit approximation for the sine function, whose second-order term coincides with that of a Taylor expansion, while the third-order term has 4 in the denominator where the Taylor expansion has 6. This has given rise, on one hand to the claim that the Keralese school founded mathematical analysis in the 15th century, on the other to the question about the origin of the ``error'' in the Indian formula. Basing herself on related commentary writings Kim Plofker explains in the present lucid article how the approximation is likely to have been obtained by ``creative manipulation of the geometry peculiar to sines'' and cosines involving ``a judicious dexterity in handling negligible quantities''; the fact that these negligible quantities are small but not analyzed as infinitesimals (alias ``ghosts of departed quantities'', that is, supposed in the argument to vanish) explains the discrepancies between the Indian rules. The Taylor polynomials are thus shown to ``represent not so much `errors' as an entirely different approach to the problem''. Understanding the Indian formulae through the lens of later European mathematics, useful though it may be for certain purposes, makes us lose insight into ``an extremely right legacy of insight about and experimentation with other approaches to approximation'' -- as emphasized in the conclusion.