Āryabhaṭa's sine table
Āryabhaṭa's sine table is a set of twentyfour numbers given in the astronomical treatise Āryabhaṭīya composed by the fifth century Indian mathematician and astronomer Āryabhaṭa (476–550 CE), for the computation of the halfchords 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.^{[1]}^{[2]}
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 sinedifferences.^{[3]}^{[4]}
Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.^{[5]} The now lost tables of Hipparchus (c.190 BC – c.120 BC) and Menelaus (c.70–140 CE) and those of Ptolemy (c.AD 90 – c.168) were all tables of chords and not of halfchords.^{[5]} 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.
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.^{[6]} 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^{[7]}^{[8]}^{[9]} 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."^{[10]}
Contents
The tableEdit
The original tableEdit
The stanza in Āryabhaṭiya describing the sine table is reproduced below:
मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व 
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कलाअर्धज्यास् 
In modern notationsEdit
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 wordnumerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the HinduArabic 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 jyas 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 halfchord. 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 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 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.^{[11]}
Sl. No  Angle ( A ) (in degrees, arcminutes) 
Value in Āryabhaṭa's numerical notation (in Devanagari) 
Value in Āryabhaṭa's numerical notation (in ISO 15919 transliteration) 
Value in [HinduArabic numerals]] 
Āryabhaṭa's value of jya (A) 
Modern value of jya (A) (3438 × sin (A)) 

1  03° 45′

मखि

makhi

225

225′

224.8560

2  07° 30′

भखि

bhakhi

224

449′

448.7490

3  11° 15′

फखि

phakhi

222

671′

670.7205

4  15° 00′

धखि

dhakhi

219

890′

889.8199

5  18° 45′

णखि

ṇakhi

215

1105′

1105.1089

6  22° 30′

ञखि

ñakhi

210

1315′

1315.6656

7  26° 15′

ङखि

ṅakhi

205

1520′

1520.5885

8  30° 00′

हस्झ

hasjha

199

1719′

1719.0000

9  33° 45′

स्ककि

skaki

191

1910′

1910.0505

10  37° 30′

किष्ग

kiṣga

183

2093′

2092.9218

11  41° 15′

श्घकि

śghaki

174

2267′

2266.8309

12  45° 00′

किघ्व

kighva

164

2431′

2431.0331

13  48° 45′

घ्लकि

ghlaki

154

2585′

2584.8253

14  52° 30′

किग्र

kigra

143

2728′

2727.5488

15  56° 15′

हक्य

hakya

131

2859′

2858.5925

16  60° 00′

धकि

dhaki

119

2978′

2977.3953

17  63° 45′

किच

kica

106

3084′

3083.4485

18  67° 30′

स्ग

sga

93

3177′

3176.2978

19  71° 15′

झश

jhaśa

79

3256′

3255.5458

20  75° 00′

ङ्व

ṅva

65

3321′

3320.8530

21  78° 45′

क्ल

kla

51

3372′

3371.9398

22  82° 30′

प्त

pta

37

3409′

3408.5874

23  86° 15′

फ

pha

22

3431′

3430.6390

24  90° 00′

छ

cha

7

3438′

3438.0000

Āryabhaṭa's computational methodEdit
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.^{[11]}
 "When the second half[chord] partitioned is less than the first halfchord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sinedifferences] are less [than the previous ones] each by that amount of that divided by the first halfchord."
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 alsoEdit
ReferencesEdit
 ^ Selin, Helaine, ed. (2008). Encyclopaedia of the History of Science, Technology, and Medicine in NonWestern Cultures (2 ed.). Springer. pp. 986–988. ISBN 9781402044250.
 ^ Eugene Clark (1930). The Āryabhaṭiya of Āryabhaṭa: An ancient Indian work on mathematics and astronomy. Chicago: The University of Chicago Press.
 ^ Takao Hayashi, T (November 1997). "Āryabhaṭa's rule and table for sinedifferences". Historia Mathematica. 24 (4): 396–406. doi:10.1006/hmat.1997.2160.
 ^ B. L. van der Waerden, B. L. (March 1988). "Reconstruction of a Greek table of chords". Archive for History of Exact Sciences. 38 (1): 23–38. doi:10.1007/BF00329978.
 ^ ^{a} ^{b} J J O'Connor and E F Robertson (June 1996). "The trigonometric functions". Retrieved 4 March 2010.
 ^ "Hipparchus and Trigonometry". Retrieved 6 March 2010.
 ^ G. J. Toomer, G. J. (July 2007). "The Chord Table of Hipparchus and the Early History of Greek Trigonometry". Centaurus. 18 (1): 6–28. doi:10.1111/j.16000498.1974.tb00205.x.
 ^ B.N. Narahari Achar (2002). "Āryabhata and the table of Rsines" (PDF). Indian Journal of History of Science. 37 (2): 95–99. Retrieved 6 March 2010.
 ^ Glen Van Brummelen (March 2000). "[HM] Radian Measure". Historia Mathematica mailing List Archive. Retrieved 6 March 2010.
 ^ Glen Van Brummelen (2009). The mathematics of the heavens and the earth: the early history of trigonometry. Princeton University Press. ISBN 9780691129730.
 ^ ^{a} ^{b} Victor J Katz (Editor) (2007). The mathematics of Egypt, Mesopotamia, China, India, and Islam: a sourcebook. Princeton: Princeton University Press. pp. 405–408. ISBN 9780691114859.CS1 maint: Extra text: authors list (link)