>PrimeBOT: /* The original table */Task 24 - replacement of a template following a TFD

2020-06-23T19:56:35Z

The original table: Task 24 - replacement of a template following a TFD

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'''''{{Use dmy dates|date=October 2014}}'''''
The astronomical treatise [[Āryabhaṭīya]] was composed during the fifth century by the [[Indian mathematics|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. It is not a table in the modern sense of a mathematical table; that is, it is not a set of numbers arranged into rows and columns.<ref>{{cite book|editor-first=Helaine|editor-last=Selin|editor-link=Helaine Selin|title= Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |url=https://archive.org/details/encyclopaediahis00seli|url-access=limited|publisher=Springer|year=2008|edition=2|pages=[https://archive.org/details/encyclopaediahis00seli/page/n1002 986]–988|isbn=978-1-4020-4425-0}}</ref><ref>{{cite book|last=Eugene Clark|title=Theastronomy|publisher=The University of Chicago Press|location=Chicago|year=1930}}</ref>
[[File:Arc and chord.svg|thumb|right|250px|Arc and chord of a circle]]
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>

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> The now lost tables of [[Hipparchus]] (c.190 BC – c.120 BC) and [[Menelaus of Alexandria|Menelaus]] (c.70–140 CE) and [[Ptolemy's table of chords|those of]] [[Ptolemy]] (c.AD 90 – c.168) were all tables of [[chord (geometry)|chord]]s and not of half-chords.<ref name="mcs"/>
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 historians of mathematics have argued that the sine table given in Āryabhaṭiya was an adaptation of earlier such tables constructed by mathematicians and astronomers of ancient Greece.<ref>{{cite web|url=http://131.111.104.4/starry/hippmaths.html|title=Hipparchus and Trigonometry |accessdate=6 March 2010}}</ref> [[David Pingree]], one of America's foremost historians of the exact sciences in antiquity, was an exponent of such a view. Assuming this hypothesis, [[G. J. Toomer]]<ref>{{cite journal|doi=10.1111/j.1600-0498.1974.tb00205.x|last=G. J. Toomer|first1=G. J.|authorlink=G. J. Toomer|date=July 2007 |title=The Chord Table of Hipparchus and the Early History of Greek Trigonometry|journal=Centaurus|volume=18|issue=1|pages=6–28}}</ref><ref>{{cite journal|last=B.N. Narahari Achar|year=2002|title=Āryabhata and the table of Rsines|journal=Indian Journal of History of Science|volume=37|issue=2|pages=95–99|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol37_2_1_BNNAchar.pdf|accessdate=6 March 2010}}</ref><ref>{{cite web|url=http://sunsite.utk.edu/math_archives/.http/hypermail/historia/mar00/0038.html|title=[HM] Radian Measure|last=Glen Van Brummelen |date=March 2000|publisher=Historia Mathematica mailing List Archive|accessdate=6 March 2010}}</ref> writes, "Hardly any documentation exists for the earliest arrival of Greek astronomical models in India, or for that matter what those models would have looked like. So it is very difficult to ascertain the extent to which what has come down to us represents transmitted knowledge, and what is original with Indian scientists. ... The truth is probably a tangled mixture of both."<ref>{{cite book|last=Glen Van Brummelen|title=The mathematics of the heavens and the earth: the early 0|url=http://press.princeton.edu/titles/8956.html}}</ref>

==The table==

===The original table===
{{Contains special characters|Indic}}
The stanza in Āryabhaṭiya describing the sine table is reproduced below:
<poem>
मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||
</poem>

===In modern notations===

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.

In the Indian mathematical tradition, the sine ( or ''jya'') of an angle is not a ratio of numbers. It is the length of a certain line segment, a certain half-chord. The radius of the base circle is basic parameter for the construction of such tables. Historically, several tables have been constructed using different values for this parameter. 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. It has not yet been established who is the first ever to use this value for the base radius. But [[Aryabhatiya]] is the earliest surviving text containing a reference to this basic constant.<ref name="Katz">{{cite book|last=Victor J Katz (Editor)|title=The mathematics of Egypt, Mesopotamia, China, India, and Islam: a sourcebook|publisher=Princeton University Press|location=Princeton|year=2007|pages=405–408|isbn=978-0-691-11485-9}}</ref>

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
! Sl. No
! Angle ( A ) (in [[Degree (angle)|degrees]], [[arcminutes]])
! Value in [[Aryabhata's numerical notation|Āryabhaṭa's numerical notation]] (in [[Devanagari]])
! Value in [[Aryabhata's numerical notation|Āryabhaṭa's numerical notation]] (in [[ISO 15919]] transliteration)
! Value in [[Hindu-Arabic numerals]]
! Āryabhaṭa's value of ''[[jya]]'' (A)
! Modern value of ''[[jya]]'' (A) (3438 × sin (A))
|-
|   1
| {{center|03°   45′}}
| {{center|मखि}}
| {{center|makhi}}
| {{center|225}}
| {{center|225′}}
| {{center|224.8560}}
|-
|   2
| {{center|07°   30′}}
|{{center|भखि}}
| {{center|bhakhi}}
| {{center|224}}
| {{center|449′}}
| {{center|448.7490}}
|-
|   3
| {{center|11°   15′}}
|{{center|फखि}}
| {{center|phakhi}}
| {{center|222}}
| {{center|671′}}
| {{center|670.7205}}
|-
|   4
| {{center|15°   00′}}
|{{center|धखि}}
| {{center|dhakhi}}
| {{center|219}}
| {{center|890′}}
| {{center|889.8199}}
|-
|   5
| {{center|18°   45′}}
|{{center|णखि}}
| {{center|ṇakhi}}
| {{center|215}}
| {{center|1105′}}
| {{center|1105.1089}}
|-
|   6
| {{center|22°   30′}}
|{{center|ञखि}}
|{{center|ñakhi}}
| {{center|210}}
| {{center|1315′}}
| {{center|1315.6656}}
|-
|   7
| {{center|26°   15′}}
|{{center|ङखि}}
| {{center|ṅakhi}}
| {{center|205}}
| {{center|1520′}}
| {{center|1520.5885}}
|-
|   8
| {{center|30°   00′}}
|{{center|हस्झ}}
| {{center|hasjha}}
| {{center|199 }}
| {{center|1719′}}
| {{center|1719.0000}}
|-
|   9
| {{center|33°   45′}}
|{{center|स्ककि}}
| {{center|skaki}}
| {{center|191 }}
| {{center|1910′}}
| {{center|1910.0505}}
|-
|   10
| {{center|37°   30′}}
|{{center|किष्ग}}
| {{center|kiṣga}}
| {{center|183 }}
| {{center|2093′}}
| {{center|2092.9218}}
|-
|   11
| {{center|41°   15′}}
|{{center|श्घकि}}
| {{center|śghaki}}
| {{center|174 }}
| {{center|2267′}}
| {{center|2266.8309}}
|-
|   12
| {{center|45°   00′}}
|{{center|किघ्व}}
| {{center|kighva}}
| {{center|164 }}
| {{center|2431′}}
| {{center|2431.0331}}
|-
|   13
| {{center|48°   45′}}
|{{center|घ्लकि}}
| {{center|ghlaki}}
| {{center|154 }}
| {{center|2585′}}
| {{center|2584.8253}}
|-
|   14
| {{center|52°   30′}}
|{{center|किग्र}}
| {{center|kigra}}
| {{center|143 }}
| {{center|2728′}}
| {{center|2727.5488}}
|-
|   15
| {{center|56°   15′}}
|{{center|हक्य}}
| {{center|hakya}}
|{{center|131 }}
|{{center| 2859′}}
| {{center|2858.5925}}
|-
|   16
| {{center|60°   00′}}
|{{center|धकि}}
| {{center|dhaki}}
| {{center|119}}
| {{center|2978′}}
| {{center|2977.3953}}
|-
|   17
| {{center|63°   45′}}
|{{center|किच}}
| {{center|kica}}
| {{center|106}}
| {{center|3084′}}
| {{center|3083.4485}}
|-
|   18
| {{center|67°   30′}}
|{{center|स्ग}}
| {{center|sga}}
| {{center|93 }}
| {{center|3177′}}
| {{center|3176.2978}}
|-
|   19
| {{center|71°   15′}}
|{{center|झश}}
| {{center|jhaśa}}
| {{center|79}}
| {{center|3256′}}
| {{center|3255.5458}}
|-
|   20
| {{center|75°   00′}}
|{{center|ङ्व}}
| {{center|ṅva}}
| {{center|65}}
| {{center|3321′}}
| {{center|3320.8530}}
|-
|   21
| {{center|78°   45′}}
|{{center|क्ल}}
| {{center|kla}}
| {{center|51 }}
| {{center|3372′}}
| {{center|3371.9398}}
|-
|   22
|{{center|82°   30′}}
|{{center|प्त}}
| {{center|pta}}
| {{center|37}}
| {{center|3409′}}
| {{center|3408.5874}}
|-
|   23
| {{center|86°   15′}}
|{{center|फ}}
| {{center|pha}}
| {{center|22}}
| {{center|3431′}}
| {{center|3430.6390}}
|-
|   24
| {{center|90°   00′}}
|{{center|छ}}
| {{center|cha}}
| {{center|7}}
| {{center|3438′}}
| {{center|3438.0000}}
|-
|}

==Āryabhaṭa's computational method==

The second section of Āryabhaṭiya titled Ganitapādd
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."

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

==See also==
* [[Madhava's sine table]]
* [[Bhaskara I's sine approximation formula]]

==References==

{{reflist}}

{{DEFAULTSORT:Aryabhata's Sine Table}}
[[Category:Trigonometry]]
[[Category:Indian mathematics]]

Āryabhaṭa's sine table - Revision history

>PrimeBOT: /* The original table */Task 24 - replacement of a template following a TFD