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Madhava's sine table is the table of trigonometric sines of various angles constructed by the fourteenth century Kerala mathematicianastronomer Madhava of Sangamagrama. The table lists the trigonometric sines of the twentyfour angles 3.75°, 7.50°, 11.25°, ... , and 90.00° (angles that are integral multiples of 3.75°, i.e. 1/24 of a right angle, beginning with 3.75 and ending with 90.00). The table is encoded in the letters of Devanagari using the Katapayadi system. This gives the entries in the table an appearance of the verses of a poem in Sanskrit.
Madhava's original work containing the sine table has not yet been traced. The table is seen reproduced in the Aryabhatiyabhashya of Nilakantha Somayaji (14441544) and also in the Yuktidipika/Laghuvivrti commentary of Tantrasamgraha by Sankara Variar (circa. 15001560). ^{[1]}
Contents 
The image below gives Madhava's sine table in Devanagari as reproduced in Cultural foundations of mathematics by C.K. Raju.^{[1]} 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".
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 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 arcseconds of the angle whose radian measure is the modern value of sin 22.50°. The modern numerical value of sin 22.50° is 0.382683432363 and,
and
In the Katapayadi system the digits are written in the reverse order. Thus in Madhava's table, the entry corresponding to 22.50° is 70435131.
For an angle whose measure is A, let
Then
Each of the lines in the table specifies eight digits. Let the digits corresponding to angle A (read from left to right) be
Then according to the rules of the Katapayadi system of Kerala mathematicians we have
To complete the numerical computations one must have a knowledge of the value of π. It is appropriate that we use the value of π computed by Madhava himself. Nilakantha Somayaji has given this value of π in his AryabhatiyaBhashya as follows^{[2]}:
A loose transliteration of the last two lines:
vibudhanētragajaahihutāśana
triguṇavēdabhavāraṇabāhavaḥ
navanikharvamitē vr̥tivistarē
paridhimānamidaṁ jagadur̥ budhāḥ
The various words indicate certain numbers encoded in a scheme known as the bhutasamkhya system. The meaning of the words and the numbers encoded by them are detailed in the following translation of the verse: "Gods (vibudha : 33), eyes (nētra : 2), elephants (gaja : 8), snakes (ahi : 8), fires (hutāśana : 3), three (tri : 3), qualities (guṇa : 3), vedas (vēda : 4), naksatras (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 navanikharva (900,000,000,000)." This yields the value π = 3.1415926535922 as the value of π used by Madhava. This value of π is accurate to 11 decimal places.
Madhava's table lists the following digits corresponding to the angle 45.00°:
This yields the angle with measure
The value of the trigonometric sine of 45.00° as given in Madhava's table is
Substituting the value of π computed by Madhava in the above expression, one gets sin 45° as 0.70710681.
This value may be compared with the modern exact value of sin 45.00°, namely, 0.70710678.
In table below the first column contains the list of the twentyfour angles beginning with 3.75 and ending with 90.00. The second column contains the values tabulated by Madhava in Devanagari in the form in which it was given by Madhava. (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.^{[1]}) The third column contains ISO 15919 transliterations of the lines given in the second column. The digits encoded by the lines in second column are given 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. These values are computed using the approximate value 3.1415926535922 for π obtained by Madhava. For comparison, the exact values of the trigonometric sines of the angles are given in the sixth column.
Angle A in degrees 
Madhava's numbers for specifying sin A  Value of sin A derived from Madhava's table 
Modern value of sin A 


in Devanagari script using Katapayadi system (as in Madhava's original table) 
in ISO 15919 transliteration scheme 
in Arabic numerals  







श्रेष्ठो नाम वरिष्ठानां  śreṣṭhō nāma variṣṭhānāṁ 

0.06540314  0.06540313 

हिमाद्रिर्वेदभावनः  himādrirvēdabhāvanaḥ 

0.13052623  0.13052619 

तपनो भानु सूक्तज्ञो  tapanō bhānu sūktajñō 

0.19509032  0.19509032 

मद्ध्यमं विद्धि दोहनं  maddhyamaṁ viddhi dōhanaṁ 

0.25881900  0.25881905 

धिगाज्यो नाशनं कष्टं  dhigājyō nāśanaṁ kaṣṭaṁ 

0.32143947  0.32143947 

छन्नभोगाशयांबिका  channabhōgāśayāṁbikā 

0.38268340  0.38268343 

मृगाहारो नरेशोयं  mr̥gāhārō narēśōyaṁ 

0.44228865  0.44228869 

वीरो रणजयोत्सुकः  vīrō raṇajayōtsukaḥ 

0.49999998  0.50000000 

मूलं विशुद्धं नाळस्य  mūlaṁ viṣuddhaṁ nāḷasya 

0.55557022  0.55557023 

गानेषु विरळा नराः  gāneṣu viraḷā narāḥ 

0.60876139  0.60876143 

अशुद्धिगुप्ता चोरश्रीः  aśuddhiguptā cōraśrīḥ 

0.65934580  0.65934582 

शम्कुकर्णो नगेश्वरः  śaṃkukarṇō nageśvaraḥ 

0.70710681  0.70710678 

तनूजो गर्भजो मित्रं  tanūjō garbhajō mitraṃ 

0.75183985  0.75183981 

श्रीमानत्र सुखी सखे  śrīmānatra sukhī sakhē 

0.79335331  0.79335334 

शशी रात्रौ हिमाहारौ  śaśī rātrou himāhārou 

0.83146960  0.83146961 

वेगज्ञः पथि सिन्धुरः  vēgajñaḥ pathi sindhuraḥ 

0.86602543  0.86602540 

छाया लयो गजो नीलो  chāya layō gajō nīlō 

0.89687275  0.89687274 

निर्मलो नास्ति सल्कुले  nirmalō nāsti salkulē 

0.92387954  0.92387953 

रात्रौ दर्पणमभ्रांगं  rātrou darpaṇamabhrāṁgaṁ 

0.94693016  0.94693013 

नागस्तुंग नखो बली  nāgastuṁga nakhō balī 

0.96592581  0.96592583 

धीरो युवा कथालोलः  dhīrō yuvā kathālōlaḥ 

0.98078527  0.98078528 

पूज्यो नारीजनैर्भगाः  pūjyō nārījanairbhagāḥ 

0.99144487  0.99144486 

कन्यागारे नागवल्ली  kanyāgārē nāgavallī 

0.99785895  0.99785892 

देवो विश्वस्थली भृगुः  devō viśvasthalī bhr̥ guḥ 

0.99999997  1.00000000 
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 like Nilakantha Somayaji (Tantrasangraha) and Jyeshtadeva (Yuktibhasa) 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.

