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'''Madhava's sine table''' is the table of sines of various angles constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama. The table lists the trigonometric sines of the twenty-four angles 3.75°, 7.50°, 11.25°, ..., and 90.00° (angles that are [[integer|integral]] [[Multiple (mathematics)|multiples]] of 3.75°, i.e. 1/24 of a right angle, beginning with 3.75 and ending with 90.00). The table is [[character encoding|encoded]] in the [[Letter (alphabet)|letters]] of [[Devanagari]] using the [[Katapayadi system]]. This gives the entries in the table an appearance of the [[Verse (poetry)|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]]<ref name=":0">''The Aryabhatiam of Aryabhattacharya with the Bhashya of Nilakantha Somasutvan, Part1-Gaṇitapāda,'' Edited by K. Sambasiva Sastri, Trivandrum Sanskrit Series No.101. p. 55.
https://ia601902.us.archive.org/28/items/Trivandrum_Sanskrit_Series_TSS/TSS-101_Aryabhatiya_With_the_Commentary_of_Nilakanta_Somasutvan_Part_1_-_KS_Sastri_1930.pdf
http://www.sanskritebooks.org/2013/02/trivandrum-sanskrit-series-anantasayana-samskrita-granthavali/</ref>(1444–1544) and also in the ''Yuktidipika/Laghuvivrti'' commentary of [[Tantrasamgraha]] by [[Sankara Variar]] (circa. 1500-1560).<ref name="Raju">{{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=114–123}}</ref>
== The table ==
The image below gives Madhava's sine table in [[Devanagari]] as reproduced in ''Cultural foundations of mathematics'' by C.K. Raju.<ref>{{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=120}}</ref> 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".
[[File:Madhava sine tabe in Devanagari.jpg|thumb|center|350px]]
==Values in Madhava's table==
[[File:Madhavasine.jpeg.jpg|thumb]]
To understand the meaning of the values tabulated by [[Madhava of Sangamagrama|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 <math>\angle</math>POS in [[Minute of arc|arcminutes]], [[arcsecond]]s 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,
:0.382683432363 radians = 180 / π × 0.382683432363 degrees = 21.926145564094 degrees.
and
:21.926145564094 degrees = 1315 arcminutes 34 arcseconds 07 sixtieths of arcsecond.
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.
==Derivation of trigonometric sines from Madhava's table==
For an angle whose measure is ''A'', let
:<math>\angle POS = m \text{ arcminutes, } s \text{ arcseconds, } t \text{ sixtieths of an arcsecond}</math>
Then
:<math>
\begin{align}
\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{align}
</math>
Each of the lines in the table specifies eight digits. Let the digits corresponding to angle A (read from left to right) be
:<math> d_1\quad d_2\quad d_3\quad d_4\quad d_5\quad d_6\quad d_7\quad d_8 </math>
Then according to the rules of the [[Katapayadi system]] of Kerala mathematicians we have
:<math>
\begin{align}
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{align}
</math>
== Madhava's value of pi ==
To complete the numerical computations one must have a knowledge of the value of [[pi]] (π). It is appropriate that we use the value of [[pi|π]] computed by Madhava himself. [[Nilakantha Somayaji]] has given this value of π in his [[Āryabhaṭīya]]-Bhashya as follows:<ref>{{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=119}}</ref>
<br>
[[File:Madhava value of pi.jpg|thumb|left|350px]]<br>
<br>
<br>
<br>
<br>
<br>
A transliteration of the last two lines:
vibudha-netra-gaja-ahi-hutāśana<br>
tri-guṇa-veda-bha-vāraṇa-bāhavaḥ<br>
nava-nikharva-mite vr̥tivistare<br>
paridhi-mānam idaṁ jagadur budhāḥ
<br>
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 pi (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.
==Example==
Madhava's table lists the following digits corresponding to the angle 45.00°:
:<math>5\quad 1\quad 1\quad 5\quad 0\quad 3\quad 4\quad 2</math>
This yields the angle with measure
:<math>
\begin{align}
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{align}
</math>
The value of the trigonometric sine of 45.00° as given in Madhava's table is
:<math>
\sin 45^\circ = \frac{\pi}{180\times 60}\left( 2430 + \frac{51}{60} + \frac{15}{60\times 60}\right)
</math>
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.
== Comparison of Madhava's and modern sine values ==
In table below the first column contains the list of the twenty-four 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<ref>{{cite book|last=[[Puthumana Somayaji]]|title=[[Karanapaddhati]] (with a commentary in [[Malayalam]] by P.K. Koru)|publisher=Astro Printing and Publishing Company|location= [[Cherpu]], [[Kerala]], [[India]]}} (Published in 1953)</ref> and are slightly different from the table given in ''Cultural foundations of mathematics'' by C.K. Raju.<ref name="Raju"/>) The third column contains [[ISO 15919|ISO 15919 transliterations]] of the lines given in the second column. The digits encoded by the lines in second column are given in [[Arabic numeral]]s 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.
<center>
{| align="center" border="1" cellpadding="5" cellspacing="1"
|-
! rowspan="2" | Angle A <br/>in degrees
! colspan="3" | Madhava's numbers for specifying sin A
! rowspan="2" | Value of sin A <br> derived from <br> Madhava's table
! rowspan="2" | Modern value <br/> of sin A
|-
| '''in [[Devanagari|Devanagari script]] <br/> using [[Katapayadi system]] <br/>(as in Madhava's <br/> original table) '''
| '''in [[ISO 15919|ISO 15919 transliteration]]''' <br/> scheme
| '''Decoded Values in'''<br>'''minutes seconds thirds'''
|-
| <center>'''(1)'''</center>
| <center>'''(2)'''</center>
| <center>'''(3)'''</center>
| <center>'''(4)'''</center>
| <center>'''(5)'''</center>
| <center>'''(6)'''</center>
|-
| <center>03.75</center>
| श्रेष्ठं नाम वरिष्ठानां
| śreṣṭhaṁ nāma variṣṭhānāṁ
| <center>224ʹ 50ʹʹ 22ʹʹʹ</center>
| 0.06540314
| 0.06540313
|-
| <center>07.50</center>
| हिमाद्रिर्वेदभावनः
| himādrirvēdabhāvanaḥ
| <center>448ʹ 42ʹʹ 58ʹʹʹ</center>
| 0.13052623
| 0.13052619
|-
| <center>11.25</center>
| तपनो भानु सूक्तज्ञो
| tapanō bhānu sūktajñō
| <center>670ʹ 40ʹʹ 16ʹʹʹ</center>
| 0.19509032
| 0.19509032
|-
| <center>15.00</center>
| मध्यमं विद्धि दोहनं
| maddhyamaṁ viddhi dōhanaṁ
| <center>889ʹ 45ʹʹ 15ʹʹʹ</center>
| 0.25881900
| 0.25881905
|-
| <center>18.75</center>
| धिगाज्यो नाशनं कष्टं
| dhigājyō nāśanaṁ kaṣṭaṁ
| <center>1105ʹ 01ʹʹ 39ʹʹʹ</center>
| 0.32143947
| 0.32143947
|-
| <center>22.50</center>
| छन्नभोगाशयाम्बिका
| channabhōgāśayāmbikā
| <center>1315ʹ 34ʹʹ 07ʹʹʹ</center>
| 0.38268340
| 0.38268343
|-
| <center>26.25</center>
| मृगाहारो नरेशोयं
| mr̥gāhārō narēśōyaṁ
| <center>1520ʹ 28ʹʹ 35ʹʹʹ</center>
| 0.44228865
| 0.44228869
|-
| <center>30.00</center>
| वीरो रणजयोत्सुकः
| vīrō raṇajayōtsukaḥ
| <center>1718ʹ 52ʹʹ 24ʹʹʹ</center>
| 0.49999998
| 0.50000000
|-
| <center>33.75</center>
| मूलं विशुद्धं नाळस्य
| mūlaṁ viṣuddhaṁ nāḷasya
| <center>1718ʹ 52ʹʹ 24ʹʹʹ</center>
| 0.55557022
| 0.55557023
|-
| <center>37.50</center>
| गानेषु विरळा नराः
| gāneṣu viraḷā narāḥ
| <center>2092ʹ 46ʹʹ 03ʹʹʹ</center>
| 0.60876139
| 0.60876143
|-
| <center>41.25</center>
| अशुद्धिगुप्ता चोरश्रीः
| aśuddhiguptā cōraśrīḥ
| <center>2266ʹ 39ʹʹ 50ʹʹʹ</center>
| 0.65934580
| 0.65934582
|-
| <center>45.00</center>
| शङ्कुकर्णो नगेश्वरः
| śaṅkukarṇō nageśvaraḥ
| <center>2430ʹ 51ʹʹ 15ʹʹʹ</center>
| 0.70710681
| 0.70710678
|-
| <center>48.75</center>
| तनुजो गर्भजो मित्रं
| tanujō garbhajō mitraṃ
|<center> 2584ʹ 38ʹʹ 06ʹʹʹ</center>
| 0.75183985
| 0.75183981
|-
| <center>52.50</center>
| श्रीमानत्र सुखी सखे
| śrīmānatra sukhī sakhē
| <center>2727ʹ 20ʹʹ 52ʹʹʹ</center>
| 0.79335331
| 0.79335334
|-
| <center>56.25</center>
| शशी रात्रौ हिमाहारौ
| śaśī rātrou himāhārou
| <center>2858ʹ 22ʹʹ 55ʹʹʹ</center>
| 0.83146960
| 0.83146961
|-
| <center>60.00</center>
| वेगज्ञः पथि सिन्धुरः
| vēgajñaḥ pathi sindhuraḥ
| <center>2977ʹ 10ʹʹ 34ʹʹʹ</center>
| 0.86602543
| 0.86602540
|-
| <center>63.25</center>
| छाया लयो गजो नीलो
| chāya layō gajō nīlō
| <center>3083ʹ 13ʹʹ 17ʹʹʹ</center>
| 0.89687275
| 0.89687274
|-
| <center>67.50</center>
| निर्मलो नास्ति सत्कुले
| nirmalō nāsti satkulē
| <center>3176ʹ 03ʹʹ 50ʹʹʹ</center>
| 0.92387954
| 0.92387953
|-
| <center>71.25</center>
| रात्रौ दर्पणमभ्राङ्गं
| rātrou darpaṇamabhrāṅgaṁ
| <center>3255ʹ 18ʹʹ 22ʹʹʹ</center>
| 0.94693016
| 0.94693013
|-
| <center>75.00</center>
| नागस्तुङ्ग नखो बली
| nāgastuṅga nakhō balī
| <center>3320ʹ 36ʹʹ 30ʹʹʹ</center>
| 0.96592581
| 0.96592583
|-
| <center>78.75</center>
| धीरो युवा कथालोलः
| dhīrō yuvā kathālōlaḥ
| <center>3371ʹ 41ʹʹ 29ʹʹʹ</center>
| 0.98078527
| 0.98078528
|-
| <center>82.50</center>
| पूज्यो नारीजनैर्भगः
| pūjyō nārījanairbhagaḥ
| <center>3408ʹ 20ʹʹ 11ʹʹʹ</center>
| 0.99144487
| 0.99144486
|-
| <center>86.25</center>
| कन्यागारे नागवल्ली
| kanyāgārē nāgavallī
| <center>3430ʹ 23ʹʹ 11ʹʹʹ</center>
| 0.99785895
| 0.99785892
|-
| <center>90.00</center>
| देवो विश्वस्थली भृगुः
| devō viśvasthalī bhr̥ guḥ
| <center>3437ʹ 44ʹʹ 48ʹʹʹ</center>
| 0.99999997
| 1.00000000
|}
</center>
==Madhava's method of computation==
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]] ([[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''.
:<math>
\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots
</math>
==See also==
*[[Indian sine tables]]
*[[Surya Siddhanta]]
*[[Indian standard circle]]
==References==
{{reflist}}
==Further references==
*{{cite journal|last=Bag|first=A.K.|year=1976|title=Madhava's sine and cosine series|journal=Indian Journal of History of Science|publisher=Indian National Academy of Science|volume=11|issue=1|pages=54–57|url=http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20005af4_54.pdf|accessdate=21 August 2016|archive-url=https://web.archive.org/web/20150705200732/http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20005af4_54.pdf|archive-date=5 July 2015|url-status=dead}}
*For an account of Madhava's computation of the sine table see : {{cite book|last=Van Brummelen|first=Glen |title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|location=Princeton|year=2009|pages=113–120|isbn=978-0-691-12973-0|url=http://press.princeton.edu/titles/8956.html}}
*For a thorough discussion of the computation of Madhava's sine table with historical references : {{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=114–123}}
{{Indian mathematics}}
{{DEFAULTSORT:Madhava's Sine Table}}
[[Category:Trigonometry]]
[[Category:Indian mathematics]]
[[Category:Kerala school of astronomy and mathematics]]
Madhava's original work containing the sine table has not yet been traced. The table is seen reproduced in the ''Aryabhatiyabhashya'' of [[Nilakantha Somayaji]]<ref name=":0">''The Aryabhatiam of Aryabhattacharya with the Bhashya of Nilakantha Somasutvan, Part1-Gaṇitapāda,'' Edited by K. Sambasiva Sastri, Trivandrum Sanskrit Series No.101. p. 55.
https://ia601902.us.archive.org/28/items/Trivandrum_Sanskrit_Series_TSS/TSS-101_Aryabhatiya_With_the_Commentary_of_Nilakanta_Somasutvan_Part_1_-_KS_Sastri_1930.pdf
http://www.sanskritebooks.org/2013/02/trivandrum-sanskrit-series-anantasayana-samskrita-granthavali/</ref>(1444–1544) and also in the ''Yuktidipika/Laghuvivrti'' commentary of [[Tantrasamgraha]] by [[Sankara Variar]] (circa. 1500-1560).<ref name="Raju">{{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=114–123}}</ref>
== The table ==
The image below gives Madhava's sine table in [[Devanagari]] as reproduced in ''Cultural foundations of mathematics'' by C.K. Raju.<ref>{{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=120}}</ref> 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".
[[File:Madhava sine tabe in Devanagari.jpg|thumb|center|350px]]
==Values in Madhava's table==
[[File:Madhavasine.jpeg.jpg|thumb]]
To understand the meaning of the values tabulated by [[Madhava of Sangamagrama|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 <math>\angle</math>POS in [[Minute of arc|arcminutes]], [[arcsecond]]s 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,
:0.382683432363 radians = 180 / π × 0.382683432363 degrees = 21.926145564094 degrees.
and
:21.926145564094 degrees = 1315 arcminutes 34 arcseconds 07 sixtieths of arcsecond.
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.
==Derivation of trigonometric sines from Madhava's table==
For an angle whose measure is ''A'', let
:<math>\angle POS = m \text{ arcminutes, } s \text{ arcseconds, } t \text{ sixtieths of an arcsecond}</math>
Then
:<math>
\begin{align}
\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{align}
</math>
Each of the lines in the table specifies eight digits. Let the digits corresponding to angle A (read from left to right) be
:<math> d_1\quad d_2\quad d_3\quad d_4\quad d_5\quad d_6\quad d_7\quad d_8 </math>
Then according to the rules of the [[Katapayadi system]] of Kerala mathematicians we have
:<math>
\begin{align}
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{align}
</math>
== Madhava's value of pi ==
To complete the numerical computations one must have a knowledge of the value of [[pi]] (π). It is appropriate that we use the value of [[pi|π]] computed by Madhava himself. [[Nilakantha Somayaji]] has given this value of π in his [[Āryabhaṭīya]]-Bhashya as follows:<ref>{{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=119}}</ref>
<br>
[[File:Madhava value of pi.jpg|thumb|left|350px]]<br>
<br>
<br>
<br>
<br>
<br>
A transliteration of the last two lines:
vibudha-netra-gaja-ahi-hutāśana<br>
tri-guṇa-veda-bha-vāraṇa-bāhavaḥ<br>
nava-nikharva-mite vr̥tivistare<br>
paridhi-mānam idaṁ jagadur budhāḥ
<br>
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 pi (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.
==Example==
Madhava's table lists the following digits corresponding to the angle 45.00°:
:<math>5\quad 1\quad 1\quad 5\quad 0\quad 3\quad 4\quad 2</math>
This yields the angle with measure
:<math>
\begin{align}
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{align}
</math>
The value of the trigonometric sine of 45.00° as given in Madhava's table is
:<math>
\sin 45^\circ = \frac{\pi}{180\times 60}\left( 2430 + \frac{51}{60} + \frac{15}{60\times 60}\right)
</math>
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.
== Comparison of Madhava's and modern sine values ==
In table below the first column contains the list of the twenty-four 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<ref>{{cite book|last=[[Puthumana Somayaji]]|title=[[Karanapaddhati]] (with a commentary in [[Malayalam]] by P.K. Koru)|publisher=Astro Printing and Publishing Company|location= [[Cherpu]], [[Kerala]], [[India]]}} (Published in 1953)</ref> and are slightly different from the table given in ''Cultural foundations of mathematics'' by C.K. Raju.<ref name="Raju"/>) The third column contains [[ISO 15919|ISO 15919 transliterations]] of the lines given in the second column. The digits encoded by the lines in second column are given in [[Arabic numeral]]s 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.
<center>
{| align="center" border="1" cellpadding="5" cellspacing="1"
|-
! rowspan="2" | Angle A <br/>in degrees
! colspan="3" | Madhava's numbers for specifying sin A
! rowspan="2" | Value of sin A <br> derived from <br> Madhava's table
! rowspan="2" | Modern value <br/> of sin A
|-
| '''in [[Devanagari|Devanagari script]] <br/> using [[Katapayadi system]] <br/>(as in Madhava's <br/> original table) '''
| '''in [[ISO 15919|ISO 15919 transliteration]]''' <br/> scheme
| '''Decoded Values in'''<br>'''minutes seconds thirds'''
|-
| <center>'''(1)'''</center>
| <center>'''(2)'''</center>
| <center>'''(3)'''</center>
| <center>'''(4)'''</center>
| <center>'''(5)'''</center>
| <center>'''(6)'''</center>
|-
| <center>03.75</center>
| श्रेष्ठं नाम वरिष्ठानां
| śreṣṭhaṁ nāma variṣṭhānāṁ
| <center>224ʹ 50ʹʹ 22ʹʹʹ</center>
| 0.06540314
| 0.06540313
|-
| <center>07.50</center>
| हिमाद्रिर्वेदभावनः
| himādrirvēdabhāvanaḥ
| <center>448ʹ 42ʹʹ 58ʹʹʹ</center>
| 0.13052623
| 0.13052619
|-
| <center>11.25</center>
| तपनो भानु सूक्तज्ञो
| tapanō bhānu sūktajñō
| <center>670ʹ 40ʹʹ 16ʹʹʹ</center>
| 0.19509032
| 0.19509032
|-
| <center>15.00</center>
| मध्यमं विद्धि दोहनं
| maddhyamaṁ viddhi dōhanaṁ
| <center>889ʹ 45ʹʹ 15ʹʹʹ</center>
| 0.25881900
| 0.25881905
|-
| <center>18.75</center>
| धिगाज्यो नाशनं कष्टं
| dhigājyō nāśanaṁ kaṣṭaṁ
| <center>1105ʹ 01ʹʹ 39ʹʹʹ</center>
| 0.32143947
| 0.32143947
|-
| <center>22.50</center>
| छन्नभोगाशयाम्बिका
| channabhōgāśayāmbikā
| <center>1315ʹ 34ʹʹ 07ʹʹʹ</center>
| 0.38268340
| 0.38268343
|-
| <center>26.25</center>
| मृगाहारो नरेशोयं
| mr̥gāhārō narēśōyaṁ
| <center>1520ʹ 28ʹʹ 35ʹʹʹ</center>
| 0.44228865
| 0.44228869
|-
| <center>30.00</center>
| वीरो रणजयोत्सुकः
| vīrō raṇajayōtsukaḥ
| <center>1718ʹ 52ʹʹ 24ʹʹʹ</center>
| 0.49999998
| 0.50000000
|-
| <center>33.75</center>
| मूलं विशुद्धं नाळस्य
| mūlaṁ viṣuddhaṁ nāḷasya
| <center>1718ʹ 52ʹʹ 24ʹʹʹ</center>
| 0.55557022
| 0.55557023
|-
| <center>37.50</center>
| गानेषु विरळा नराः
| gāneṣu viraḷā narāḥ
| <center>2092ʹ 46ʹʹ 03ʹʹʹ</center>
| 0.60876139
| 0.60876143
|-
| <center>41.25</center>
| अशुद्धिगुप्ता चोरश्रीः
| aśuddhiguptā cōraśrīḥ
| <center>2266ʹ 39ʹʹ 50ʹʹʹ</center>
| 0.65934580
| 0.65934582
|-
| <center>45.00</center>
| शङ्कुकर्णो नगेश्वरः
| śaṅkukarṇō nageśvaraḥ
| <center>2430ʹ 51ʹʹ 15ʹʹʹ</center>
| 0.70710681
| 0.70710678
|-
| <center>48.75</center>
| तनुजो गर्भजो मित्रं
| tanujō garbhajō mitraṃ
|<center> 2584ʹ 38ʹʹ 06ʹʹʹ</center>
| 0.75183985
| 0.75183981
|-
| <center>52.50</center>
| श्रीमानत्र सुखी सखे
| śrīmānatra sukhī sakhē
| <center>2727ʹ 20ʹʹ 52ʹʹʹ</center>
| 0.79335331
| 0.79335334
|-
| <center>56.25</center>
| शशी रात्रौ हिमाहारौ
| śaśī rātrou himāhārou
| <center>2858ʹ 22ʹʹ 55ʹʹʹ</center>
| 0.83146960
| 0.83146961
|-
| <center>60.00</center>
| वेगज्ञः पथि सिन्धुरः
| vēgajñaḥ pathi sindhuraḥ
| <center>2977ʹ 10ʹʹ 34ʹʹʹ</center>
| 0.86602543
| 0.86602540
|-
| <center>63.25</center>
| छाया लयो गजो नीलो
| chāya layō gajō nīlō
| <center>3083ʹ 13ʹʹ 17ʹʹʹ</center>
| 0.89687275
| 0.89687274
|-
| <center>67.50</center>
| निर्मलो नास्ति सत्कुले
| nirmalō nāsti satkulē
| <center>3176ʹ 03ʹʹ 50ʹʹʹ</center>
| 0.92387954
| 0.92387953
|-
| <center>71.25</center>
| रात्रौ दर्पणमभ्राङ्गं
| rātrou darpaṇamabhrāṅgaṁ
| <center>3255ʹ 18ʹʹ 22ʹʹʹ</center>
| 0.94693016
| 0.94693013
|-
| <center>75.00</center>
| नागस्तुङ्ग नखो बली
| nāgastuṅga nakhō balī
| <center>3320ʹ 36ʹʹ 30ʹʹʹ</center>
| 0.96592581
| 0.96592583
|-
| <center>78.75</center>
| धीरो युवा कथालोलः
| dhīrō yuvā kathālōlaḥ
| <center>3371ʹ 41ʹʹ 29ʹʹʹ</center>
| 0.98078527
| 0.98078528
|-
| <center>82.50</center>
| पूज्यो नारीजनैर्भगः
| pūjyō nārījanairbhagaḥ
| <center>3408ʹ 20ʹʹ 11ʹʹʹ</center>
| 0.99144487
| 0.99144486
|-
| <center>86.25</center>
| कन्यागारे नागवल्ली
| kanyāgārē nāgavallī
| <center>3430ʹ 23ʹʹ 11ʹʹʹ</center>
| 0.99785895
| 0.99785892
|-
| <center>90.00</center>
| देवो विश्वस्थली भृगुः
| devō viśvasthalī bhr̥ guḥ
| <center>3437ʹ 44ʹʹ 48ʹʹʹ</center>
| 0.99999997
| 1.00000000
|}
</center>
==Madhava's method of computation==
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]] ([[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''.
:<math>
\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots
</math>
==See also==
*[[Indian sine tables]]
*[[Surya Siddhanta]]
*[[Indian standard circle]]
==References==
{{reflist}}
==Further references==
*{{cite journal|last=Bag|first=A.K.|year=1976|title=Madhava's sine and cosine series|journal=Indian Journal of History of Science|publisher=Indian National Academy of Science|volume=11|issue=1|pages=54–57|url=http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20005af4_54.pdf|accessdate=21 August 2016|archive-url=https://web.archive.org/web/20150705200732/http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20005af4_54.pdf|archive-date=5 July 2015|url-status=dead}}
*For an account of Madhava's computation of the sine table see : {{cite book|last=Van Brummelen|first=Glen |title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|location=Princeton|year=2009|pages=113–120|isbn=978-0-691-12973-0|url=http://press.princeton.edu/titles/8956.html}}
*For a thorough discussion of the computation of Madhava's sine table with historical references : {{cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|pages=114–123}}
{{Indian mathematics}}
{{DEFAULTSORT:Madhava's Sine Table}}
[[Category:Trigonometry]]
[[Category:Indian mathematics]]
[[Category:Kerala school of astronomy and mathematics]]