Āryabhaṭa's sine table

Āryabhata's sine table is a set of twenty-four numbers given in the astronomical treatise Āryabhatiya composed by the fifth century Indian mathematician and astronomer Āryabhata (476–550 CE), for the computation of the half-chords of a certain set of arcs of a circle. The set of numbers appears in verse 12 in Chapter 1 Dasagitika of Aryabhatiya. It is not a table in the modern sense of a mathematical table; that is, it is not a set of numbers arranged into rows and columns. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as Āryabhaṭa's table of sine-differences. Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics. The now lost tables of Hipparchus (c. 190 BC – c. 120 BC) and Menelaus (c. 70–140 CE) and those of Ptolemy (c. AD 90 – c. 168) were all tables of chords and not of half-chords. Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by Madhava of Sangamagrama (c. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places. Some historians of mathematics have argued that the sine table given in Āryabhaṭiya was an adaptation of earlier such tables constructed by mathematicians and astronomers of ancient Greece. David Pingree, one of America's foremost historians of the exact sciences in antiquity, was an exponent of such a view. Assuming this hypothesis, G. J. Toomer writes, "Hardly any documentation exists for the earliest arrival of Greek astronomical models in India, or for that matter what those models would have looked like.