Madhava's sine table

The image below gives Madhava's sine table in Devanagari as reproduced in Cultural foundations of mathematics by C.K. Raju.[2]^: 120 The first twelve lines constitute the entries in the table. The last word in the thirteenth line indicates that these are "as told by Madhava".

Madhava's sine table in Devanagari

Diagram explaining the meaning of the values in Madhava's table

To understand the meaning of the values tabulated by Madhava, consider some angle whose measure is A. Consider a circle of unit radius and center O. Let the arc PQ of the circle subtend an angle A at the center O. 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. Let PS be an arc of the circle whose length is equal to the length of the segment RQ. For various angles A, Madhava's table gives the measures of the corresponding angles ${\displaystyle \angle }$POS in arcminutes, arcseconds and sixtieths of an arcsecond.

As an example, let A be an angle whose measure is 22.50°. In Madhava's table, the entry corresponding to 22.50° is the measure in arcminutes, arcseconds and sixtieths of an arcsecond of the angle whose radian measure is the value of sin 22.50°, which is 0.3826834;

multiply 0.3826834 radians by 180/π to convert to 21.92614 degrees, which is

1315 arcminutes 34 arcseconds 07 sixtieths of an arcsecond, abbreviated 13153407.

In the Katapayadi system the digits are written in the reverse order, so that the entry corresponding to 22.50° is 70435131.

For an angle whose measure is A, let

${\displaystyle \angle POS=m{\text{ arcminutes, }}s{\text{ arcseconds, }}t{\text{ sixtieths of an arcsecond}}}$

Then

${\displaystyle {\begin{aligned}\sin(A)&=RQ\\&={\text{length of arc }}PS\\&=\angle POS{\text{ in radians}}\\&={\frac {\pi }{180\times 60}}\left(m+{\frac {s}{60}}+{\frac {t}{60\times 60}}\right).\end{aligned}}}$

Each of the lines in the table specifies eight digits. Let the digits corresponding to angle A (read from left to right) be

${\displaystyle d_{1}\quad d_{2}\quad d_{3}\quad d_{4}\quad d_{5}\quad d_{6}\quad d_{7}\quad d_{8}}$

Then according to the rules of the Katapayadi system of Kerala mathematicians we have

${\displaystyle {\begin{aligned}m&=d_{8}\times 1000+d_{7}\times 100+d_{6}\times 10+d_{5}\\s&=d_{4}\times 10+d_{3}\\t&=d_{2}\times 10+d_{1}\end{aligned}}}$

To complete the numerical computations the value of pi (π) must be known. It is appropriate to use the value of π computed by Madhava himself (which is accurate to 11 decimal places). Nilakantha Somayaji has given this value of π in his Āryabhaṭīya-Bhashya as follows:[2]^: 119

Madhava's value of pi

A transliteration of the last two lines:

The various words indicate certain numbers encoded in a scheme known as the bhūtasaṃkhyā system. The meaning of the words and the numbers encoded by them (beginning with the units place) are detailed in the following translation of the verse: "Gods (vibudha : 33), eyes (netra : 2), elephants (gaja : 8), snakes (ahi : 8), fires (hutāśana : 3), three (tri : 3), qualities (guṇa : 3), vedas (veda : 4), nakṣatras (bha : 27), elephants (vāraṇa : 8), and arms (bāhavaḥ : 2) - the wise say that this is the measure of the circumference when the diameter of a circle is nava-nikharva (900,000,000,000)."

So, the translation of the poem using the bhūtasaṃkhyā system will simply read "2827433388233 is, as the wise say, the circumference of a circle whose diameter is nava-nikharva (900,000,000,000)". That is, divide 2827433388233 (the number from the first two lines of the poem in reverse order) by nava-nikharva (900,000,000,000) to get the value of π. This calculation yields the value π = 3.1415926535922. This is the value of π used by Madhava in his further calculations and is accurate to 11 decimal places.

Madhava's table lists the following digits corresponding to the angle 45.00°:

${\displaystyle 5\quad 1\quad 1\quad 5\quad 0\quad 3\quad 4\quad 2}$

This yields the angle with measure

${\displaystyle {\begin{aligned}m&=2\times 1000+4\times 100+3\times 10+0{\text{ arcminutes}}\\&=2430{\text{ arcminutes}}\\s&=5\times 10+1{\text{ arcseconds}}\\&=51{\text{ arcseconds}}\\t&=1\times 10+5{\text{ sixtieths of an arcsecond}}\\&=15{\text{ sixtieths of an arcsecond}}\end{aligned}}}$

The value of the trigonometric sine of 45.00° as given in Madhava's table is

${\displaystyle \sin 45^{\circ }={\frac {\pi }{180\times 60}}\left(2430+{\frac {51}{60}}+{\frac {15}{60\times 60}}\right)}$

Substituting the value of π computed by Madhava in the above expression, sin 45° is found to be 0.70710681. This is accurate to 6 decimal places.

The table below has entries for the twenty-four angles from 3.75° to 90° in steps of 3.75° (first column). The second column contains the values tabulated by Madhava in Devanagari in the form he used. (These are taken from Malayalam Commentary of Karanapaddhati by P.K. Koru[3] and are slightly different from the table given in Cultural foundations of mathematics by C.K. Raju.[2]^{: 114–123}) The third column contains ISO 15919 transliterations of the lines in the second column. The numbers in the second column are decoded into arcminutes, arcseconds, and sixtieths of an arcsecond in Arabic numerals in the fourth column. The values of the trigonometric sines derived from the numbers specified in Madhava's table are listed in the fifth column, using Madhava's value of 3.1415926535922 for π. The actual values of the sines to 8 places are listed in the sixth column.

No work of Madhava detailing the methods used by him for the computation of the sine table has survived. 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:

${\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }$

• Madhava series
• Madhava's correction term
• Āryabhaṭa's sine table
• Ptolemy's table of chords

• Bag, A.K. (1976). "Madhava's sine and cosine series" (PDF). Indian Journal of History of Science. Indian National Academy of Science. 11 (1): 54–57. Archived from the original (PDF) on 5 July 2015. Retrieved 21 August 2016.
• For an account of Madhava's computation of the sine table see : Van Brummelen, Glen (2009). The mathematics of the heavens and the earth : the early history of trigonometry. Princeton: Princeton University Press. pp. 113–120. ISBN 978-0-691-12973-0.
• For a thorough discussion of the computation of Madhava's sine table with historical references : C.K. Raju (2007). Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE. History of Philosophy, Science and Culture in Indian Civilization. Vol. X Part 4. Delhi: Centre for Studies in Civilizations. pp. 114–123.