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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 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 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]]<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.

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.

[[File:Madhavasine.jpeg.jpg|thumb]]

[[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]].

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,

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,

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|&pi;]] 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>

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|&pi;]] 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>

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A transliteration of the last two lines:

A transliteration of the last two lines:

== Comparison of Madhava's and modern sine values ==

== 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.

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 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>

<center>

{| align="center" border="1" cellpadding="5" cellspacing="1"

{| align="center" border="1" cellpadding="5" cellspacing="1"

|-

|-

! rowspan="2"   |  Angle A <br/>in degrees  

! rowspan="2" |  Angle A <br />in degrees  

! colspan="3"   |  Madhava's numbers for specifying sin A  

! colspan="3" |  Madhava's numbers for specifying sin A  

! rowspan="2"   |  Value of sin A <br> derived from <br> Madhava's table

! rowspan="2" |  Value of sin A <br> derived from <br> Madhava's table

! rowspan="2"   |  Modern value <br/> of sin A  

! 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 [[Devanagari|Devanagari script]] <br /> using [[Katapayadi system]] <br />(as in Madhava's <br /> original table) '''

| '''in [[ISO 15919|ISO 15919 transliteration]]''' <br/> scheme

| '''in ISO 15919 transliteration''' <br /> scheme

| '''Decoded Values in'''<br>'''minutes seconds thirds'''

| '''Decoded Values in'''<br>'''minutes seconds thirds'''

|-

|-

[[Category:Indian mathematics]]

[[Category:Indian mathematics]]

[[Category:Kerala school of astronomy and mathematics]]

[[Category:Kerala school of astronomy and mathematics]]