Madhava's sine table

The table is encoded in the letters of the Sanskrit alphabet using the Katapayadi system, giving entries the appearance of the verses of a poem.

The table is reproduced in the Aryabhatiyabhashya of Nilakantha Somayaji[1](1444–1544) and also in the Yuktidipika/Laghuvivrti commentary of Tantrasamgraha by Sankara Variar (circa.

The last verse means: “These are the great R-sines as said by Madhava, comprising arcminutes, seconds and thirds.

The only difference is that in the earlier days they had not standardized on decimal values (or powers of ten as denominator) for fractions.

However, since these tables were mostly used for astronomy, and distance on the celestial sphere is expressed in angle measures, these values are also given likewise.

However, this also leads to the usage of sexagesimal subdivisions in Madhava's refining the earlier table of Aryabhata.

Instead of choosing a larger R, he gave the extra precision determined by him on top of the earlier given minutes by using seconds and thirds.

Drop the perpendicular QR from Q to OP; then the length of the line segment RQ is the value of the trigonometric sine of the angle A.

However from the writings of later Kerala mathematicians including Nilakantha Somayaji (Tantrasangraha) and Jyeshtadeva (Yuktibhāṣā) that give ample references to Madhava's accomplishments, it is conjectured that Madhava computed his sine table using the power series expansion of sin x: