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Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in [[Surya Siddhanta]] and another text is the astronomical treatise [[Āryabhaṭīya]] which was composed during the fifth century by the  [[Indian mathematician]] and astronomer  [[Aryabhata|Āryabhaṭa]] (476–550 CE),  for the computation of the [[Jya|half-chords]] of certain set of arcs of a circle. The table found in [[Surya Siddhanta]] is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the [[Finite difference|first differences]] of the values of [[Trigonometric functions|trigonometric sines]] expressed in [[arcminutes]], and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.<ref>{{cite journal|doi=10.1006/hmat.1997.2160|last=Takao Hayashi|first1=T|date=November 1997|title=Āryabhaṭa's rule and table for sine-differences|journal=Historia Mathematica |volume=24|issue=4|pages=396–406 }}</ref><ref>{{cite journal|doi=10.1007/BF00329978|last=B. L. van der Waerden|date=March 1988|first1=B. L.|title=Reconstruction of a Greek table of chords|journal=Archive for History of Exact Sciences|volume=38|issue=1|pages=23–38|title-link=table of chords}}</ref>

Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the  [[Indian mathematician]] and astronomer  [[Aryabhata|Āryabhaṭa]] (476–550 CE),  for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.<ref>{{cite journal|doi=10.1006/hmat.1997.2160|last=Takao Hayashi|first1=T|date=November 1997|title=Āryabhaṭa's rule and table for sine-differences|journal=Historia Mathematica |volume=24|issue=4|pages=396–406 }}</ref><ref>{{cite journal|doi=10.1007/BF00329978|last=B. L. van der Waerden|date=March 1988|first1=B. L.|title=Reconstruction of a Greek table of chords|journal=Archive for History of Exact Sciences|volume=38|issue=1|pages=23–38|title-link=table of chords}}</ref>

Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the [[history of mathematics]].<ref name="mcs">{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |accessdate=4 March 2010}}</ref> Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the [[Madhava series|power series expansions]] of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the [[Kerala school of astronomy and mathematics]], and the tabulation of a [[Madhava's sine table|sine table by Madhava]] with values accurate to seven or eight decimal places.

Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.<ref name="mcs">{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |accessdate=4 March 2010}}</ref> Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.

== Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==

== Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==

'''Jyā''', '''koti-jyā''' and  '''utkrama-jyā''' are three [[trigonometric functions]] introduced  by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is [[Surya Siddhanta]].<ref name="Datta">{{cite journal|last=B.B. Datta and A.N. Singh|date=1983|title=Hindu Trigonometry|journal=Indian Journal of History of Science|volume=18|issue=1|pages=39&ndash;108|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdf|accessdate=1 March 2010}}</ref> These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern [[trigonometric functions]] of [[sine]] and [[cosine]]. In fact, the origins of the modern terms of "sine" and "cosine" have been  traced back to the [[Sanskrit]] words jyā and kotijyā.<ref name="Datta" />

'''Jyā''', '''koti-jyā''' and  '''utkrama-jyā''' are three trigonometric functions introduced  by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta.<ref name="Datta">{{cite journal|last=B.B. Datta and A.N. Singh|date=1983|title=Hindu Trigonometry|journal=Indian Journal of History of Science|volume=18|issue=1|pages=39&ndash;108|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdf|accessdate=1 March 2010}}</ref> These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of "sine" and "cosine" have been  traced back to the [[Sanskrit]] words jyā and kotijyā.<ref name="Datta" />

[[File:Jya Figure.jpg|thumb|419x419px]]

[[File:Jya Figure.jpg|thumb|419x419px]]

===Definition===

===Definition===

Let 'arc AB'  denote an [[Arc (geometry)|arc]] whose two extremities are A and B of a circle  with center O. If a perpendicular BM be dropped from B to OA, then:

Let 'arc AB'  denote an arc whose two extremities are A and B of a circle  with center O. If a perpendicular BM be dropped from B to OA, then:

* ''Vyāsardhā'' = Radius (R)

* ''Vyāsardhā'' = Radius (R)

===From jyā to sine===

===From jyā to sine===

The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',<ref>{{cite web|url=http://mathforum.org/library/drmath/view/54053.html|title=How the Trig Functions Got their Names|work=Ask Dr. Math|publisher=[[Drexel University]]|accessdate=2 March 2010}}</ref><ref>{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last= J J O'Connor and E F Robertson|date=June 1996 |accessdate=2 March 2010}}</ref>   

The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',<ref>{{cite web|url=http://mathforum.org/library/drmath/view/54053.html|title=How the Trig Functions Got their Names|work=Ask Dr. Math|publisher=[[Drexel University]]|accessdate=2 March 2010}}</ref><ref>{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last= J J O'Connor and E F Robertson|date=June 1996 |accessdate=2 March 2010}}</ref>  or more specifically to its synonym  ''jīva''. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means "bosom". The text's 12th-century  Latin translator used the Latin equivalent for "bosom", ''[[wikt:sinus|sinus]]''.<ref>Various sources credit the first use of ''sinus'' to either:  

or more specifically to its synonym  ''jīva''.

This term was [[Indian influence on Islamic science|adopted in medieval Islamic mathematics]], transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means "bosom". The text's 12th-century  [[Medieval Latin|Latin]] translator used the Latin equivalent for "bosom", ''[[wikt:sinus|sinus]]''.<ref>Various sources credit the first use of ''sinus'' to either:  

* [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]

* [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]

* [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]

* [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]

a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.<ref name="Katz" />

a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.<ref name="Katz" />

* "When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord."

* "When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord."

This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function.

This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table.  The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes  = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle.  

Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table.  The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in [[Degree (angle)|degrees]], [[arcminutes|minutes]], [[arcseconds|seconds]], etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes  = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle.  

The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the [[Aryabhata numeration|numerical scheme]] explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in [[ISO 15919]] transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of  ''[[jya]]'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations,  the modern values of ''jya''s are given in the last column of the table.

The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of  ''jya'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations,  the modern values of ''jya''s are given in the last column of the table.

===Comparing different sine tables===

===Comparing different sine tables===

{| class="wikitable" style="margin: 1em auto 1em auto;"

{| class="wikitable" style="margin: 1em auto 1em auto;"

!Sl. No

!Sl. No

!Angle<br />(in [[Degree (angle)|degrees]],<br />[[arcminutes]])

!Angle<br />(in degrees,<br />arcminutes)

!Āryabhaṭa's <br />value of Jyā (R.sine)

!Āryabhaṭa's <br />value of Jyā (R.sine)

![[Surya Siddhanta]] value of Jyā (R.sine)

![[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] value of Jyā (R.sine)

!Surya Siddhanta versed sines

!Surya Siddhanta versed sines

!Modern value <br />of ''[[Jyā]]'' R.sine

!Modern value <br />of ''Jyā'' R.sine

!Madhava's sine values

!Madhava's sine values

!Madhava's derived sine values

!Madhava's derived sine values